Lie Groups: New Research

MATHEMATICS RESEARCH DEVELOPMENTS SERIES

LIE GROUPS: NEW RESEARCH No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

MATHEMATICS RESEARCH DEVELOPMENTS SERIES Boundary Properties and Applications of the Differentiated Poisson Integral for Different Domains Sergo Topuria 2009. ISBN 978-1-60692-704-5 Quasi-Invariant and Pseudo-Differentiable Measures in Banach Spaces Sergey Ludkovsky 2009. ISBN 978-1-60692-734-2 Operator Splittings and their Applications Istvan Farago and Agnes Havasiy 2009. ISBN 978-1-60741-776-7

Geometric Properties and Problems of Thick Knots Yuanan Diao and Claus Ernst 2009. ISBN: 978-1-60741-070-6 Measure of Non-Compactness for Integral Operators in Weighted Lebesgue Spaces Alexander Meskhi 2009. ISBN: 978-1-60692-886-8 Mathematics and Mathematical Logic: New Research Peter Milosav and Irene Ercegovaca (Editors) 2009. ISBN: 978-1-60692-862-2 Lie Groups: New Research Altos B. Canterra 2009. ISBN: 978-1-60692-389-4

MATHEMATICS RESEARCH DEVELOPMENTS SERIES

LIE GROUPS: NEW RESEARCH

ALTOS B. CANTERRA EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Canterra, Altos B. Lie groups : new research / Altos B. Canterra. p. cm. Includes index. ISBN 978-1-61668-164-7 (E-Book) 1. Lie groups. I. Title. QA387.C35 2009 512'.482--dc22 2009015095

Published by Nova Science Publishers, Inc. Ô New York

CONTENTS Preface

vii

Chapter 1

Lie Group Guide to the Universe Bernd Schmeikal

Chapter 2

Rotation Manifold SO(3) and Its Tangential Vectors Jari Mäkinen

61

Chapter 3

Asymptotic Homology of the Quotient of PSL2(R) by a Modular Group Jacques Franchi

89

Chapter 4

Group Analysis of Solutions of 2-Dimensional Differential Equations Sergey I. Senashov and Alexander Yakhno

123

Chapter 5

The Module Structure of the Infinite-Dimensional Lie Algebra Attached to a Vector Field Guan Keying

139

Chapter 6

Lie Group Methods for Modulus Conserving Differential Equations Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

169

Chapter 7

Singularities and Stability of a Work Function Jean Lerbet

187

Chapter 8

The Conformal-Affine Structure of open Quantum Relativity, Its Physical Realization and Implications G. Basini and S. Capozziello

199

Chapter 9

Twisted Balanced Metrics Julien Keller

267

Chapter 10

Reduction, Hydrodynamics and Control for Geodesics of Left- or Right Invariant Metrics on Lie Groups Mikhail V. Deryabin

283

1

vi

Contents

Chapter 11

Some Approximation Theorems for Quasimetric, Induced by C1-smooth Non-commutative Vector Fields A.V. Greshnov

307

Chapter 12

Lie Theory in Physics Gabriela P. Ovando

325

Chapter 13

Lévy Processes in Lie Groups and Homogeneous Spaces Ming Liao

351

Chapter 14

Symmetry Classification of Differential Equations and Reduction Techniques Giampaolo Cicogna

385

Chapter 15

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras up to Dimension Eight R. Campoamor-Stursberg and J. Guerón

401

Chapter 16

The Automorphism Groups of Some Geometric Structures on Orbifolds A.V. Bagaev and N.I. Zhukova

447

Chapter 17

Wrap Groups of Connected Fiber Bundles, Their Structure and Cohomologies S.V. Ludkovsky

485

Chapter 18

Groups of Diffeomorphisms and Wraps of Manifolds over Non-archimedean Fields S.V. Ludkovsky

563

Index

601

PREFACE This new book is dedicated to recent and important research on Lie groups. A Lie Group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. They are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. As discussed in Chapter 1, reconstructing physics in Clifford algebra brought to light that unity of physics that we were after, at least to some considerable extent. In 1990 David Hestenes published “Clifford Algebra and the Interpretation of Quantum Mechanics” [19]. Therein he represented the spin of an electron by the exterior product −½ γ1∧γ2 or bivector −½ γ12. He did that over and over again between 1980 and 2000. But he too did not mention Lipkin. Quite obviously, he had not read »Lie Groups for Pedestrians«. In my 2004 publication [20] on “Transpositions in Clifford Algebra” I pointed out Harry Lipkin had found that out already in 1965. Lipkin [21] understood the ½γij were angular 4-momenta, and he even identified the γ-matrices as linear combinations of baryon creation- and annihilation operators. I have carefully surveyed the Clifford algebra literature. No one went so far as Harry J. Lipkin with his famous »Lie Groups for Pedestrians«. And later, no one of us rediscovered the simplicity and beauty of angular momentum algebra in quadratic Clifford algebras. We may say that many of us discovered Clifford algebra, but only few of us who were good enough in geometric algebra understood the deep meaning of Lie algebra in Clifford algebra. It is therefore that I decided to lift Lipkins Lie Groups for Pedestrians up to Clifford algebra. I will tell us a story. The story has the title »Lie Group Guide To The Universe«. It is a booklet about Clifford-Lie-Algebra, quantum geometry, and the standard model of matter. Suppose the preceding section would have had to represent Lipkins introduction. Then next there should follow a review of angular momentum algebra. To lift that topic to geometric algebra let us first consider the Pauli algebra which is a representation

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of the Clifford algebra Cl3,0 in Mat(2, ℂ). This will not prevent us from constructing some more general concept of quantum geometry. In Chapter 2, we prove that incremental material rotation vectors belong to different tangent spaces of the rotation manifold SO(3) at a different instant. Moreover, we show that the material tangent space as the tangent space at unity is not a possible definition yielding geometrically inconsistent results, although this kind of definition is widely adopted in applied mechanics community. In addition, we show that the standard Newmark integration scheme for incremental rotations neglects first order terms of rotation vector, not third order terms. Finally, we show that the rotation interpolation of extracted nodal values on the rotation manifold is not an objective interpolation under the observer transformation. This clarifies controversy about the frame-indifference of geometrically exact beam formulations in their finite element implementations. Consider G := PSL2( ) ≡ T1Η2, a modular group Γ , and the homogeneous space

Γ \G ≡ T1( Γ \Η2 ). Endow G, and then Γ \G, with a canonical left-invariant metric, thereby equipping it with a quasi hyperbolic geometry. Windings around handles and cusps of Γ \G are calculated by integrals of closed 1-forms of Γ \G. The main results express, in both Brownian and geodesic cases, the joint convergence of the law of these integrals, with a stress on the asymptotic independence between slow and fast windings. The non-hyperbolicity of Γ \G is responsible for a difference between the Brownian and geodesic asymptotic behaviours, difference which does not exist at the level of the Riemann surface Γ \Η2 (and generally in hyperbolic cases). Identification of the cohomology classes of closed 1-forms with harmonic 1-forms, and equidistribution of large geodesic spheres, are also addressed in Chapter 3. It is well known [4, 9] that if a system of differential equations admits the Lie group of point transformations (point symmetry), then any solution of the system is transformed to a solution of this system. This property permits the construction of new solutions without integrating the given system of partial differential equations (PDEs), by means of group transformations alone under known solutions. This is an effective method if we have a sufficiently rich group of point transformations. By applying point transformations to exact solution, a family of so-called Ssolutions can be constructed, i.e., obtained by means of symmetries. This family of S-solutions is dependent on the group parameter. If this parameter is equal to zero, then we have an initial solution. This procedure is called the production [9] or reproduction of solutions [4]. Moreover, it is easy to show that under a group transformation characteristic curves of the system of PDEs of the hyperbolic type are transformed to the characteristics curves. The evolution of characteristic curves permits to find out the boundary conditions for new Ssolutions. In Chapter 4 authors will show some applications of this procedure for the system of the theory of ideal plane plasticity, developing results obtained in [12]. In particular, we shall use an infinite subgroup of the group of symmetries for deformation of characteristics curves of the considered hyperbolic system of PDEs to construct a new analytical solutions. From the system of PDEs an automorphic system will be deduced, which permits find out some relations between different solutions by means of group transformations. In Chapter 5, based on a generalized definition on the admittance of a Lie group by a vector field, it is proved that, attached to any given smooth vector field X on a n-dimensional

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ix

manifold M, there is an infinite-dimensional Lie algebra L(X) formed by infinitesimal generators of all one-parameter Lie groups admitted by X. As a compound module, through its any given basis (X,V1,V2, ...,Vn−1), L(X) can be treated as a direct sum of two modules L(X) = L<X> L where L<X> is generated by X and is a module of rank 1 over the coefficient ring formed by smooth functions, and L is spanned by (V1,V2, ...,Vn−1), and is a module of rank (n−1) over the coefficient ring formed by all first integrals of the autonomous system determined by X. This module structure is useful in the study of integrating the autonomous system. Based on this structure, examples in seeking exact travelling wave solutions for three famous nonlinear wave equations are given. Lie group methods are new geometric numerical methods, which were proposed to solve the Lie group differential equations on manifolds. The famous Lie group methods are the RKMK method and the Magnus method. The Lie group methods can preserve the numerical solutions of the differential equations on the same manifolds. The preservation of the modulus square conserving property is very important for the modulus conserving differential equations, which has good stability. In Chapter 6, we applied the Lie group methods, such as the RKMK method and the Magnus method, to the modulus conserving differential equations, such as the ferromagnet equation, the Euler equation of the rigid body problem, the nonlinear Schrodinger equation and the vorticity equation. Numerical results showed that Lie group methods can preserve the modulus square conserving property of the modulus conserving differential equations and have the same accuracy as the classical explicit RungeKutta methods. Lie group methods are ideal methods for constructing the explicit square conserving schemes of the modulus conserving differential equations. Chapter 7 is the beginning of a systematical analysis of singularities and stability conditions of a product of exponential mappings. More precisely, let f be defined as f : θ = (θ1 ,..., θ n ) a f (θ ) = exp(θ1 X 1 )...exp(θ n X n ) defined from the n-space

S n = K1 × ... × K n of parameters to a n-dimensional Lie group G where (X1,...,Xn) is a basis of the Lie algebra Γ of G and Kk = S1 or Ik is the 1- torus or a compact interval of R (according to the nature of the corresponding joint in applications). We are looking for conditions (about (X1,...,Xn)) for which f is a stable mapping according to the theory of singularities. This means that the orbit of f under the action of diffeomorphisms in the source and in the target is an open set in the set of differential mappings from (S1)n to G. First, we prove that the set Σ ( f ) of singularities is a (n-1)-dimensional submanifold of (S)n. 1

Secondly, we analyse the conditions so that f is a submersion with fold. Using the fact that f is inf-stable if and only if g = f|Σ1 ( f ) is an immersion with normal crossings, we analyse this property and we highlight some consequences. Applications to robotics are suggested. Beside the post-relativistic theories, Open Quantum Relativity is a gauge theory of interactions based on a nonlinear realization (NLR) of the local Conformal-Affine (CA) group of symmetry transformations. Such a theory, thanks to a covariantsymplectic formulation, succeeds in treating General Relativity and Quantum Mechanics under the same standard. In Chapter 8, we obtain the coframe fields and the gauge connections of the theory while the tetrads and Lorentz group metric are used to induce the spacetime metric. The inhomogenously transforming (under the Lorentz group) connection coefficients serve as

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gravitational gauge potentials used to define covariant derivatives accommodating the couplings of matter and gauge fields. On the other hand, the tensor valued connection forms serve as auxiliary dynamical fields associated with the dilation and as special conformal and deformation (shear) degrees of freedom inherent in the bundle manifold. As a consequence, the bundle curvature of the theory is determined and the boundary topological invariants are then constructed. They serve as a prototype (source free) gravitational Lagrangian to derive the following dynamics. Finally, the Bianchi identities, covariant field equations and gauge currents are obtained. These mathematical tools give rise to a compact, self-contained approach to physical interactions (in particular gravitation), based on the local gauge invariance. Starting from this general invariance principle, we discuss the global and the local Poincar´e invariance, developing the spinor, vector and tetrad formalisms. This covariantsymplectic approach allows to construct the curvature, torsion and metric tensors starting from the covariant derivative. The resulting theory describes a spacetime endowed with nonvanishing curvature and torsion, while the gravitational field equations are Yang-Mills-like equations of motion, with the torsion tensor playing the role of the Yang-Mills field strength. Besides other physical consequences and the reliable reproduction of several physical experiments and astrophysical observations described elsewhere [1], such field equations provide, in principle, the theoretical device to achieve Close Time Curves and, consequently, the conceivability of time travels. In Chapter 9 we introduce the notion of twisted balanced metrics. These metrics are induced from specific projective embeddings and can be understood as zeros of a certain moment map. We prove that on a polarized manifold, twisted constant scalar curvature metrics are limits of twisted balanced metrics, extending a result of S.K. Donaldson and T. Mabuchi. In contrast to the Euler-Poincaré reduction of geodesic flows of left- or rightinvariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid with a constant pressure. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. In Chapter 10, we give explicit general expressions for the reduced vector field and its symmetry fields, provide examples of such reduction and discuss two applications of this approach. As the first application, we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group, and assume that the mass geometry of the system may change under the action of internal control forces. Such system can also be reduced to the Lie group. With no controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and thus its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions, under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. The hydrodynamic interpretation of the system both provides a convenient ”language” and sharpens the controllability results: we show that by changing the mass geometry, one can bring one vortex manifold to any other vortex manifold. As an example we consider the n-dimensional Euler top. The other application is the reduction for the Euler equations of an ideal fluid, that describe the geodesics of a right-invariant metric on a Lie group SDiff(M) of the volume-

Preface

xi

preserving diffeomorphisms of a Riemannian manifold M, to the group SDiff(M). For a typical coadjoint orbit we find all symmetry fields of a reduced flow, and, as a corollary, we get a simple proof for nonexistence of new invariants of coadjoint orbits, which are the integrals of local densities over the flow domain. In Chapter 11, on some domain O ∈

N

we consider some collection of C1-smooth non-

commutative vector fields X = { X i }i =1,..., N such that rank(X1,…,XN)(g) = N, for every

g ∈ O , equipped with the graduation. Let us denote by θ g the canonical (exponential) mapping induced by X in some neighborhood O of g, acting on some neighbourhood of origin to O. We suppose that the vector fields {(θ g−1 )* X i }i =1,..., N are satisfying some special conditions of homogeneity. Using the properties of the mapping

θ g and the conditions of X

homogeneity we define some anisotropic metric (quasimetric) d cc which agrees with our Xˆ

graduation and consider the metric spaces (quasispases) ( O, d cc ), ( Og , d cc g ), where X

Xˆ

( O, d cc g ) is the local homogeneous approximation of ( O, d cc ) with respect to the action of X

the homogeneous operator of dilatation which agrees with our graduation in some Xˆ

neighborhood of g (( O, d cc g ) is some analogue of so-called nilpotent tangen cone). For the Xˆ

quasispaces ( O, d cc g ), ( O, d cc ) we develop some technique, which help us to get the local X

approximation theorem for quasimetric. As the consequence, we get some results for quasispaces induced by the collection of C1 basis canonical non-commutative vector fields, by the collection of C2 basis non-commutative vector fields. The purpose of Chapter 12 is to review the Adler Kostant Symes scheme as a theory which can be developed successfully in different contexts. It was useful to describe some mechanical systems, the so called generalized Toda, and now it was proved to be a tool for the study of the linear approach to the motion of n uncoupled harmonic oscillators. The complete integrability of these systems has an algebraic description. In the original theory this is related to ad-invariant functions, but new examples show that new conditions should be investigated. A Lévy process in a Lie group is a process that possesses independent and stationary multiplicative increments. The theory of such processes is not merely an extension of Lévy processes in Euclidean spaces. Because of the unique structures possessed by noncommutative Lie groups, these processes exhibit certain interesting properties which are not present for their counterparts in Euclidean spaces. The concept of Lévy processes may be extended to include Markov processes in a homogeneous space that are invariant under the group action. More generally, we will also study processes in Lie groups and homogeneous spaces that possess independent, but not necessarily stationary, increments, called nonhomogeneous Lévy processes. These processes appear naturally when studying a decomposition of a general Markov process in a manifold invariant under a group action. In Chapter 13, we will provide an introduction to Lévy processes in Lie groups and homogeneous spaces, and present some selected results in this area. The reader is referred to

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the literature for the most of proofs, but some explanation will be given to the results not yet published. The symmetry classification of differential equations containing arbitrary functions can be a source of several interesting results. In Chapter 14 we study two particular but significant examples: a nonlinear ODE and a linear PDE (the 1-dimensional Schr¨odinger equation). We provide first of all a necessary, but very restrictive, simple condition involving the arbitrary functions in order that the given equation does admit Lie symmetries. In the first example, we show that some symmetry appears only if a precise numerical relation between the involved parameters is satisfied. In the case of Schrödinger equation, we see that only for a very limited class of potential functions some symmetry is admitted, and that the Lie generators of these symmetries are precisely recursion operators and are related to the Dirac step up - step down operators, well known in Quantum Mechanics. In connection with all these symmetries, we also discuss the important problem of the reduction of the differential equations, in both the different contexts of ODE’s and of PDE’s. As presented in Chapter 15, contractions and deformations of Lie algebras and their relations have played an important role in many fields since their introduction in the 1950’s, and many progress has been done in understanding their structural and geometrical properties. Although being a rather active research field, there remain various important problems concerning contractions and deformations that have still not be satisfactorily solved. The notion of contraction appeared first in physical context by Segal [1], and was soon recognized to have important consequences, like the possibility of switching off interactions, or analyzing the precise effect of some physical quantities when others are disregarded. The formal introduction of contractions, done by Inönü and Wigner [2], was soon defined more generally by Saletan and Kupczyński [3], in order to cover other limiting processes observed in symmetry groups used in Physics, like the transition from relativistic to non-relativistic physics. Other, more or less specifical, types of contractions have been introduced in the literature since, and their structural properties analyzed [4–7,9,10]. In addition, the contractions among Lie algebras of fixed dimension have been studied in detail [11–16], as well as important classes of algebras, like those of kinematical groups [17,18]. However, the lack of complete classifications for Lie algebras from dimension six onwards is an important obstruction that motivated different approaches to the problem. The relation of deformation theory, a formalism born in Differential Geometry, with contractions of Lie algebras, was first observed in [5], and has offered a kind of “inverse” procedure to study contractions. This point of view also suggested a geometrical interpretation of contractions in terms of orbits in a manifold, the points of which correspond to Lie algebras [19]. An advantage of this approach is a definition of contractions that includes all special types used in the literature, and that allow to establish different sufficiency criteria for the existence of contractions. Moreover, this motivated the application of specific techniques like cohomology of Lie algebras, which have proven to be an essential tool in many problems [20–23]. In Chapter 16, the Ehresmann’s theorem about a Lie structure in the hole automorphism group of a finite type G-structure on manifold is generalized to orbifolds.

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Estimates for dimension of such Lie group are established, depending on stratifications of orbifolds. Particular attention is devoted to affine connected, pseudo- Riemannian and Riemannian orbifolds. The content is illustrated by examples. Chapter 17 is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real R, complex C numbers, the quaternion skew field H and the octonion algebra O. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition ( f , g ) a f

−1

g is continuous or differentiable depending on a class of

smoothness of groups. Moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. Nevertheless, it is proved that these groups does not necessarily satisfy the Campbell-Hausdorff formula even locally. Iterated wrap groups are studied as well. Their smashed products are constructed. Cohomologies of wrap groups and their structure are investigated. Sheaves of wrap groups are constructed and studied. Moreover, twisted cohomologies and sheaves over quaternions and octonions are investigated as well. CW-groups associated with wrap groups are studied. Chapter 18 is devoted to the investigation of groups of diffeomorphisms and wraps of manifolds over non-archimedean fields of zero and positive characteristics. Different types of topologies are considered on groups of wraps and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies pairwise incomparable are found as well. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. There are proved theorems about projective limit decompositions of these groups and their compactifications for compact manifolds. Moreover, an existence of one-parameter local subgroups of diffeomorphism groups is investigated.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 1-59

ISBN: 978-1-60692-389-4 © 2009 Nova Science Publishers, Inc.

Chapter 1

LIE GROUP GUIDE TO THE UNIVERSE Bernd Schmeikal Am Platzl, Garsten, Austria

Introduction When we were freshmen in physics some of us worked as scanners in the High Energy group led by Walter Thirring. We set in a small room called Schrödingerzimmer in the Institute for Theoretical Physics, Boltzmanngasse 5, Vienna. We did not yet have the social competence to realize what that meant. We calculated Clebsch-Gordon coefficients which to me appeared as some browbeating luxury that kept me away from understanding physics. We were searching after K-vertices and -vortices, the Omega minus in the morning 6 to 9 AM and late during night shift between 6 PM and shy of midnight. We were young and swamped by some incoming papers on the unitary symmetry SU(3) of strong interaction. Some written by George Zweig [1] from CERN, others authored by Murry Gell-Mann [2] from Pasadena, some by military attaché Yuval Ne’eman [3], then Imperial College, still others by Wolfgang Kummer, one of our teachers, later Erwin Schrödinger Laureate1 [4]. Our head Walter Thirring cared for us, like his father Hans had done so for Albert Einstein. He supported us in every concern. Then, there was a catholic seminary just around the corner. We didn’t know, Thirring would some day convert that seminary into a Schrödinger Institute. That meant some more discipline, indeed. There were sociologists around me some miles away in the Institute for Advanced Studies, Vienna, peace loving people like Robert Jungk, Oskar Morgenstern, Robert Reichardt, Anatol Rapoport, James S. Coleman. They taught me sociology and mathematics. I found out there were many more people like us calculating Clebsch-Gordon coefficients all over the world and in many different kinds of situations. One of my later friends, Zbigniew Oziewicz, from the College of Mathematics and Physics at the University of Wroclaw in Poland, did so under the most pressing conditions of political imprisonment. There was something important one could learn from sociology, namely that the whole unitary knowledge of mankind can only be found in all the heads. One had to master boarder 1

Sure you will soon understand why, in this writing, I prefer to mention Wolfgang Kummer in connection with the so(2, 1) invariant gravitational fields to be found in Vienna, Preprint ESI 1110 (2001)

2

Bernd Schmeikal

disputes. There had emerged many different divisions of theory. They could not be brought together without trans-national access to research. Their unification needed some radical mind opening effort: radical constructivism, creativity and cooperation. I would not have known that those efforts would lead me back to Lie groups. I came back to the work of Sophus Lie [5] and the beautiful theory that follows due to Elie Cartan [6] and Hermann Weyl [7]. But this time I came back on the unusual road built by the efforts of many colleagues who completed geometry [8] as was begun with by William Kingdon Clifford in London in about 1870. He discovered what nowadays is called Clifford algebra, only a few years before he died in Madeira 1879. That discovery – which is better called a construction – underwent a lexical fixation in the year 1878. But then, in 1965, we felt for the first time the Dirac equation had a deep geometric meaning. Only thirty years later those of us advanced enough in geometric algebra understood why. In 2005 in ‘Letter 02’2 to the author Professor Osziewicz [9] could be sure enough to write “see, the Dirac equation is nothing but Clifford algebra, rediscovered by Dirac in 1928. However Dirac himself in so many re-editions of his Quantum Mechanics book never mentioned Clifford.” Whether Dirac wanted to mystify the insights of Clifford, or if he just did not see the connection remains an open question of phenomenology. Dirac published quantum theory of the electron in 1928 [10] where he first presented his unusual equation of motion. Dirac found this equation by playing around with the relativistic Klein-Gordonequation. He needed a differential operator of first order which squared would recover the d’Alembert operator. This could be achieved only by using spinors and gamma-matrices. By that equation the anomalous Zeeman-effect and fine structure of atomic absorption spectra could be explained. Note all that was based on the assumption spin was a relativistic phenomenon. Today we know that it is not.3 But then Dirac could conclude with the prognosis: there must exist a positron, an anti-electron with positive charge. When Anderson in 1932 established the identity of the positron in cosmic radiation he supported, at the same time, the belief system that began to form around spin and relativity and the myth of the Dirac-sea with a huge negative energy. The Principles of Quantum Mechanics was released first in 1930 [11]. In the same year (1930/2) Juvet published Opérateurs de Dirac et équations de Maxwell in the second book of Commentarii Mathematici Helvetici [13]. Today we would say he based the rigor on a matrix representation of the Dirac- and d’Alembert operators in the Clifford algebra Cl4,0. He explained why the Γ-matrices he introduced in the stile of Dirac were hypercomplex numbers also already known as Clifford numbers (p. 227). He realized the wave equations and their spinorial solutions lived in a 16-dimensional space of Clifford numbers which he for that special purpose preferred to denote as Lorentz numbers. Juvet even wrote down the general element of the Clifford algebra Cl4,0 as is presented today by the Clifford software of MAPLE [14]. For the first time Juvet and Sauter [15] made use of algebraic spinors, that is, they replaced column spinors by matrix spinors where only the first column was non-zero. After 2

‘Letter 02’ had four contents dwelt by their author: 1. Forbidden to talk: Forwarded message, 2. Your letter arrived in Mexico on 16th of July 2005, 3. Hestenes popularized Clifford, 4. Relativistic addition of relative velocities. 3 In the Clifford algebra Cl3,0 generated by the Euclidean 3-space the construction of a universal covering group of the rotation group SO(3) goes back to Lipschitz and is denoted as Spin(3). In the matrix formulation provided by the Pauli spin matrices, the spin group has an isomorphic image which is the special unitary group SU(2). This is a two-fold covering group of the rotation group or double-cover. This statement can be generalized to higher dimensions and is the mathematical cause for the appearance of spin. It is not bound to relativity. (Lounesto 2003, p. 59, 220) [12]

Lie Group Guide to the Universe

3

all it makes me contemplative why Dirac in so many editions of his Principles of Quantum Mechanics not even mentioned Clifford numbers. Did he want sole reign? We have no strong enough empirical indicators for a definite vote. But as a matter of fact the resulting time-lag determined upon progress and unity of science. It took us a long time, until the cognitive gaps could be filled, and our work is still patchwork. In 1965 the young could hardly forebode a gamma-matrix would someday represent a base unit in Clifford algebra. In the 60s, we could see by the eyes, Lie group multiplets of SU(3) were pure geometry. Still, we did not dare to correlate that geometry with outer spacetime symmetries. We would not even have guessed such a correlation could turn out one-to-one. We just did not know where that geometry came from. But there were many more such unbridgeable proximities. To mention the next one: the state ψ of a physical system and some observable, say angular momentum L or Hamiltonian H, had to live in different spaces, ψ lived in a function space and L in some matrix algebra. Beginning in the 1980s Hiley from Birkbeck College [16], advised by Bohm, Frescura and some others pressed ahead with the algebraization and by 2000 ended up with the Gelfand-Naimark-Segal construction (GNS) [17] in what they denoted as ‘generalised Clifford algebra’ or ‘discrete Weyl algebra’, some useful types of *-algebras. In those pilot projects and test results there was made use of construction plans which later turned out superfluous since the existence of (anti)involutions and closedness – the C in C* - were natural properties of Clifford algebra. Clearly, all the advantages gained from the instruments of Clifford algebra would automatically transpose onto Lie algebra, once we decided to construct Lie algebra and groups and manifolds within the geometric algebra. In the beginning the efforts made to arrive at some algebraic quantum mechanics were rather arduous. Using some types of Clifford algebras several types of mistakes were made. Sometimes nilpotents were displayed as idempotents. Hiley suddenly said that for any element A of a *-algebra A with A*A = 0 allowing for the GNS-construction there should have followed A = 0 which could not make sense as that would have resulted in the diminishing of the Gel’fand ideal and forbid orthogonality of pure states. Because, consider the Clifford algebra of the Minkowski spacetime Cl3,1. This is a C*-algebra where two pure states f1 and f4 can be represented by the primitive idempotents f1= ½(1+e1)½(1+e24) and f4= ½(1−e1)½(1−e24), with unit bivector e24. It is quickly verified that f4 is the main involuted of f1 and at the same time those two are orthogonal primitive idempotents, that is, the Clifford product f1 f4 = 0 vanishes, but neither do we have f1= 0 nor is f1* = 0. As we found out, those primitive idempotents represent fermion quark states [18]. Indeed, reconstructing physics in Clifford algebra brought to light that unity of physics that we were after, at least to some considerable extent. In 1990 David Hestenes published “Clifford Algebra and the Interpretation of Quantum Mechanics” [19]. Therein he represented the spin of an electron by the exterior product −½ γ1∧γ2 or bivector −½ γ12. He did that over and over again between 1980 and 2000. But he too did not mention Lipkin. Quite obviously, he had not read »Lie Groups for Pedestrians«. In my 2004 publication [20] on “Transpositions in Clifford Algebra” I pointed out Harry Lipkin had found that out already in 1965. Lipkin [21] understood the ½γij were angular 4-momenta, and he even identified the γmatrices as linear combinations of baryon creation- and annihilation operators. I have carefully surveyed the Clifford algebra literature. No one went so far as Harry J. Lipkin with his famous »Lie Groups for Pedestrians«. And later, no one of us rediscovered the simplicity and beauty of angular momentum algebra in quadratic Clifford algebras. We may say that

4

Bernd Schmeikal

many of us discovered Clifford algebra, but only few of us who were good enough in geometric algebra understood the deep meaning of Lie algebra in Clifford algebra. It is therefore that I decided to lift Lipkins Lie Groups for Pedestrians up to Clifford algebra. I will tell us a story. The story has the title »Lie Group Guide To The Universe«. It is a booklet about Clifford-Lie-Algebra, quantum geometry, and the standard model of matter. Suppose the preceding section would have had to represent Lipkins introduction. Then next there should follow a review of angular momentum algebra. To lift that topic to geometric algebra let us first consider the Pauli algebra which is a representation of the Clifford algebra Cl3,0 in Mat(2, ℂ). This will not prevent us from constructing some more general concept of quantum geometry.

The Clifford Algebra Cl3,0 of Euclidean 3-Space The Clifford algebra Cl3,0 is generated by the 3-dimensional Euclidean space having unit vectors e1, e2, e3 with positive signature (1) and satisfying anti-commutation relations (2)

e12 = 1 , e22 = 1 , e32 = 1

(1)

{e1,e2}= {e1,e3} ={e2,e2} = 0

(2)

We use to establish a 1-1 correspondence with the Pauli algebra of unitary 2 x 2 matrices with complex entries.

Cl 3, 0

Mat(2, ℂ)

Id

I2

e1

e2

e3

σ1

σ2

e12

e13

e23

σ1 σ 2

σ1σ 3

e123

σ1 σ 2 σ 3

σ3

scalar vector

σ 2σ3

bivector director

As you see, any exterior product ej ∧ ek can indeed be represented by a matrix product of Pauli spin matrices σj σk. The unary e1 ∧ e2 ∧ e3 represents something like an imaginary unit. It is a unit matrix with non-vanishing diagonal entries i, the pseudo-scalar. What about history of spin and quantized rotation? In 1924 Wolfgang Pauli suggested to introduce a new dichotomous degree of freedom for the electron. You remember how Ralph Kronig and Alfred Landés led that back to a quantized rotation of the electron, an idea which Pauli did not like. Because of Pauli’s critics Kronigs suggestion remained unpublished. To explain the fine structure of spectra and the anomalous Zeeman effect George Uhlenbeck and Samuel Goudsmit postulated the existence of spin in 1925. In 1927 Pauli constructed the quantum theory of the electron spin using the SU(2) matrices and 2-component spinors. Since then we have been using σ 3 as a symbol for either the spin or the angular momentum of the spinning electron. However, the situation

Lie Group Guide to the Universe

5

became somewhat unclear as soon as we were able to lift the theory into geometric algebra. Rather generally and for quite a while, it seemed all rotation was generated by bivectors such as e12. On the other hand, the well tried and proven old candidate e3 = σ3 repeatedly reentered the rigor. To give you a few examples, in [19a] Hestenes found out (p. 156), in the gauge group of the Dirac current, “iσ3 = i γ3γ0 = γ2γ1 is the generator of rotations in a spacelike plane related to physical currents.” In many writings he said that the quantity S = (½)ħe2e1 relates the bivector in the Dirac equation to the electron spin. In “Spin and uncertainty in the interpretation of quantum mechanics”, Hestenes found “that the average ‘internal angular momentum’ has the constant value ħ σ3”. A similar ambivalence can be found in the writings of William Baylis [22]. But in the end, he seems to give a definite vote on ħ σ3 at least where the Stern-Gerlach experiment is concerned. He confirmes “evidently sz = ħe3/2 is the spin operator for ψ ” (Baylis 2004, p. 389f.) Surprisingly, hardly any author has carried out the quantization of angular momentum and calculated spectra. Therefore my crafty question: which one should we take, e12 or e3 ? It is exactly that question with which we begin to lift Lie groups for pedestrians up to a Clifford level of geometry. By the way, the answer is: both!

New Review of Angular Momentum Algebra Therefore, consider the three angular momentum operators J1 = ½ e1, J2 = ½ e2 and J3 = ½ e3 given by the base unit vectors that generate the Clifford algebra Cl3,0 of Euclidean 3-space. We recall the well known commutation relations with the imaginary unit i. We substitute the i by the unit director or pseudo scalar e123 = e1 ∧ e2 ∧ e3 ∈ Cl3,0. Then the shift operators (Jx ± i Jy) are transposed onto

J+ =

1 2

(e1 − e13 ) and J − =

1 2

(e1 + e13 ) .

(3)

Though they are graded, their commutators with J3 are preserved

[J 3 , J + ] = J + and [J 3 , J − ] = − J −

(4)

From there we can go on. Understanding physical motion as graded motion, the story unfolds until to the standard model. Depending on the identity Id of the Clifford algebra Cl3,0, we obtain for the sum of squared components

J 2 = J 12 + J 22 + J 32 = 34 Id

(5)

Have we reached our arrival point? Are the J+, J− the ultimate solution? Are they definite? It is interesting that J± mix grades 1 and 2. Are there, may be, even more general shift operators associated with J3 ? Is it true that step operators J± shift eigenfunctions J, M〉 of J² and J3 to J ± J , M =

J ( J + 1) − M ( M ± 1) J , M ± 1 , from J, M〉 toJ, M ± 1〉?

With the J± , are we climbing up and down a ladder?

6

Bernd Schmeikal

Clifford Manifold of Step Operators and Spin space We confirm the significance of the angular momentum ½ ħ σ3, abstractly: quantity ½ e3 ∈ Cl3,0. First we realize that the normalized Clifford product of J+ with J− is equal to that primitive idempotent 1 2

J + J − = 12 ( Id + e3 ) = f 3 with f 3 f 3 = f 3

(6)

which brings forth a well known minimal left ideal: the algebraic Pauli spinor space corresponding with the angular momentum J3. def  ψ S = Cl 3,0 f 3 =  1  ψ 2

0   with ψ1 ,ψ 2 ∈ ℂ 0 

(7)

We shall call ½ J+ J− an eigenform of the angular momentum algebra associated with the spinor space S. Consider the general multivector element of the Clifford algebra Cl3,0 with the eight coordinates xj:

X = x1 Id + x 2 e1 + x3 e2 + x 4 e3 + x5 e12 + x6 e13 + x7 e23 + x8 e123

(8)

We are searching for general elements J±(X) of angular algebra satisfying commutation relations (4). Separate solutions can be calculated for the { J+(X) } and { J−(Y) } independent of each other. We obtain J+ = x 2 (e1 − e13 ) + x3 (e2 − e23 ) , J− = y 2 (e1 + e13 ) + y 3 (e2 + e23 )

(9)

Using a Clifford algebra calculator, you can easily confirm these elements indeed satisfy commutation relations (4). Their normalized product is equal to ½ J+(X) J−(Y ) = ( x 2 y 2 + x 3 y 3 )( Id + e3 ) + ( x 2 y 3 − x3 y 2 )(e12 + e123 )

(10)

Please, recall what Kauffman [23] said about eigenforms, and verify that the quantity f = ½ J+(X) J−(Y ), a Clifford product of J+(X) and J−(Y), is an eigenform in the Clifford algebra Cl3,0. Namely we first define recursively by the aid of the Clifford product the bilinear term

f n +1 = f n f n having form f n = a ( Id + e3 ) + b(e12 + e123 ) Clearly that form is recursively preserved, that is, we obtain after the next step

f n +1 = A( Id + e3 ) + B (e12 + e123 ) with A = 2(a 2 − b 2 ), B = 4ab

(11)

Lie Group Guide to the Universe

7

Therefore, the fn moves stepwise on a plane within the same real vector space, spanned by {Id, e3, e12, e123}.

E ( f ) = span R {Id , e3 , e12 , e123 } the space of spinor eigenforms

(12)

Observe that E(f) is itself an eigenform within Cl3,0. It reproduces itself by Clifford multiplication. It is a closed 4-dimensional real associative subalgebra of Cl3,0. We have E(f)E(f) = E(f) which signifies the closedness and it adopts the * (either reversion or conjugation) from the Clifford algebra. The space of spinor-eigenforms E(f) is therefore a C*algebra too. Those four elements represent the essential components of an angular momentum algebra associated with J3. If you remember, the coefficient a was equal to x2 y2 + x3 y3 and the b was x2 y3 − x3 y2. So the fn performs a nonlinear motion in the 4-dimensional subspace generated algebraically by the core spin space J=

{ e3 , e12 } ⊂ Cl3,0

(13)

Thus we can be convinced: for an angular momentum algebra of spin in Euclidean 3space we need both, bivector e12 and spinvector e3. Generally, such a form may diverge or collapse towards zero or converge to a fixed point of the eigenform. We obtain the fixed point for fn at first under the condition J−(Y) ≡ J+(X)* = J+(X)† (J+ reverted) J+ = x 2 (e1 − e13 ) + x3 (e2 − e23 ) , J− = x 2 (e1 + e13 ) + x3 (e2 + e23 )

(14)

f = ( x 2 + x3 ) Id + ( x 2 + x 3 ) e3

(15)

2

2

2

2

We know that at a given basis {e1, e2, e3} the Cl3,0 contains six primitive idempotents ½ (Id ± ej). One of those is ½ (Id + e3). Clearly, the f = f3 represents the fixed point to form (12). That fixed point occurs on the Thales’ circle where

x 22 + x32 = and the diameter is equal to

1 2

1 2

(16)

. From this we can learn something important that we seem to

have overlooked, namely the significance of the unitary group U(1) or SO(2) within the angular momentum algebra of su(2). Now, we have said the Clifford algebra is a C*-algebra in two ways. First we have f3* = ½ (Id − e3) where * is Clifford conjugation. But the star can be interpreted in two ways: 1.) as Clifford conjugation and 2.) as reversion († the dagger).

8

Bernd Schmeikal

Both * and † are anti-automorphisms (not so the grade- or main involution) and therefore those two provide us with a *-operation. If we take reversion, we get J−(Y) = J+(X)† as a payoff and at the fixed point we have now f = ½ J+J+†. The algebraic spinor space turns over into

S = Cl 3,0 12 J + J +* provided that * means †

(17)

Since * is an anti-involution, we obtain from (17) that S * = S. The spinor-space is preserved under Clifford reversion. The above J+(X) and its Clifford reverse give us the most general Lie manifold L = {J3, J+(X), J+(X)*}of shift operators which together with the preferred J3 generate the su(2) as a form su(2, X) ⊂ Cl3,0 and at the same time preserve the idempotent – or pure state – and its minimal left ideal. Note that it does so under the condition that the coefficients are real, that is, we work with a real Pauli algebra. Although the entries in the matrices are complex, the coefficients are real valued. In case that we worked with a complexified algebra, we obtained a third solution apart from the null solution and the real one. That is, we had to have

x 2 y 2 + x3 y 3 =

1 4

(18)

x 2 y 3 − x3 y 2 = ± 14 i

We differ between the space of eigenforms (12), the generating core spin space as is given by (13) and the algebraic spinor space (7, 17). The spinor space is a minimal left ideal of the algebra while the space of eigenforms is a linear subspace E(f) ⊂ Cl3,0. This houses a manifold of primitive idempotents either real as given by (15, 16) or complex according to (18) if the span in (12) is taken over the complex number field. Is there any relation to Hilbert space?

GNS Construction of Hilbert Space for L in Cl3,0 We can carry out an interesting GNS construction with the E(f). Let us first simply proceed with the rigor and go into the theory of GNS later. Consider a linear functional φ over the Clifford algebra Cl3,0

φ : Cl 3, 0 → ℝ real with φ(αA + βB ) = αφ( A) + βφ( B )

(20)

α, β real and A, B ∈ Cl3,0. We should have φ(Id) = 1 and φ(A *B) ≥ 0. Consider the general element A∈ Cl3,0 as we have written down in equation (8). Suppose the *-operation is represented by reversion †. In that case we obtain A†A = (a1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + a8 ) Id + 2

2

2

2

2

2

2

+ 2( a1 a 2 − a3 a5 − a 4 a 6 + a 7 a8 )e1 +

2

9

Lie Group Guide to the Universe + 2( a1 a3 + a 2 a5 − a 4 a 7 − a 6 a8 )e2 + + 2( a1 a 4 + a 2 a 6 + a3 a 7 + a 5 a8 )e3

(21)

In order to debar φ(A *A) < 0 we define the functional φ(X) for the general element X ∈ Cl3,1 as

φ( X ) = x1 + x 4 real

φ:

(22)

For the element A∈ Cl3,0 this imposes conditions on the coordinates aj. But we are interested in the Lie manifold of Lie algebras L = {J3, J+(X), J−(X)} and in the space E(f) of spinor eigenforms. That is we restrict the equation (21) to representations of L by J3 = ½ e3, and J+(X), J−(X) : J+ = a 2 (e1 − e13 ) + a3 (e2 − e23 ) , J− = b2 (e1 + e13 ) + b3 (e 2 + e 23 ) We obtain

φ( J+† J+) = 4( a 2 + a 3 ) 2

(23)

φ( J−† J−) = 4(b2 + b3 )

2

2

φ(J3† J3) = φ(¼ Id) = ¼

2

(24)

Let the general element in the space of eigenforms be

f = c1 Id + c 4 e3 + c5 e12 + c8 e123 this gives us

(25)

f † f = (c1 + c 4 + c5 + c8 ) Id + 2(c1c 4 + c 5 c8 )e3

(26)

2

2

2

2

Restricting f to the orbit f = ½ J+ J− we obtain the measure φ(f † f) = 4(a 2 + a 3 )(b2 + b3 ) 2

2

2

2

Therefore, all Hermitian scalar products are positive definite: 〈J3J3〉 ≥ 0

〈 J+ J+〉 ≥ 0

〈 J− J−〉 ≥ 0

〈ff〉≥0

(28)

and provide the same norm. What have we learned from this? We have learned that the GNSconstruction by functional (22) endows the Lie algebra L = {J3, J+(X), J−(X)} and its group SU(2, X) with a pre Hilbert space having quadratic norm (24). This can be completed to the Hilbert space of square integrable functions Hφ = L2(X). Classical theory started off with the concept of Hilbert space. But today the existence of such spaces can be derived by the aid of the GNS construction. Even an explicit relegation to Hφ is no longer necessary, since all the mathematics is done within the Clifford algebras.

10

Bernd Schmeikal

For completeness we represent the Clifford algebra Cl3,0 and the Lie algebra L = {J3, J+(X), J−(X)} by the Pauli algebra as was proposed in chapter 2. We have

0 e1 =  1

1  0 

1 0   J 3 = 12   0 − 1

 0 i  e2 =  − i 0  

1 0   basis e3 =  0 − 1  

(29)

0 J + =  0

 0 J − =   2

0  angular 0 

(30)

0 . i 

(31)

2  0 

1 f 3 =  0

0  and the space of spinor eigenforms spanned by 0 

1 Id =  0

0  , e3 , 1 

i 0   e12 =  0 − i

i e123 =  0

Quantum Spacetime Important purpose of the Lie group guide to the universe is a deeper understanding of the non-commutative geometry of the fundamental forces of nature.4 Quantum spacetime is a special object of quantum geometry. It unifies classical geometric methods with noncommutative C*-algebras and their associated functional analysis. By establishing quantum geometric methods we diminish some classic differences such as the one between operator and quantum state, observer field and observed field. We start from the assumption that inner symmetries of matter and outer symmetries of spacetime are essentially the same. But we give up the limitation to points and trajectories made of points. We shall rather show how the confinement of strange fermions to point locations – as delta-functions – can be derived ‘wlog’ from the field properties. Proceeding in this way we arrive at new concepts such as of space and quantum groups – Lie groups of spacetime-matter – by enriching mathematics from the side of quantum theory. To derive exact statements about phenomena and structure of space and time from our experience with force fields is one of the main concerns of quantum geometry. Briefly put, we try to derive space from the fields rather than reverse. Micho Ðurñevich has put a terrific paper [24] into the net, a “Brief Introduction to Quantum Geometry”. This provides a good basis for understanding the mathematics of this undertaking, namely to get space from the fields. In traditional differential geometry a compact topological space X can be reconstructed entirely in terms of the associated *-algebra A of continuous complex-valued functions on X. Every point x ∈ X gives rise to a linear functional χ which is a character of A. In this 4

In 2007 I gave out a limited edition of ten booklets with title »Spacetime Matter« and subtitle » non-commutative geometry of the fundamental forces of nature « wherein I compiled four selected works appeared in Applied Clifford Algebras from 2005 to 2007. Therein I introduced the concept of the surabale and force categories in the Clifford algebra of Minkowski spacetime in the Lorentz metric.

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Lie Group Guide to the Universe

way we obtain bijections between points and characters. In our case the A shall be given by a Clifford algebra. In such Clifford quantum geometry a single location may involve vectors, areas and spacetime volumes. This has consequences for both compactness and commutation. There exist partitions of Clifford quantum geometry into commuting and non-commuting subspaces. Those are by no means artificial, but follow from the observed properties of the fields. Classically, following the theorem by Gelfand and Naimark, the category of compact topological spaces repeats in the category of commutative C*-algebras. Continuous maps between spaces X → Y are transposed onto unary *-homomorphisms between their algebras B and A. Symmetries of spaces being homomorphisms or monomorphisms of X become automorphism of A. Now, suppose the standard model Lie algebra of the forces of nature turns out to be nothing else than the *-algebra associated with spacetime, the main properties of space and time should be derived from that algebra. That is, the symmetries of strong and electroweak interactions as described by the standard model Lie groups should be led back to the automorphisms of geometric Clifford algebra. Classically, by the Riesz representation theorem we can construct a correspondence between probability measures on X and the positive linear functionals. A similar statement should hold true in quantum spaces.

Primitive Idempotents in Quantum Geometry We begin with some associative unary (or unital) linear *-algebra A representing space. Probability measures on that space are given by positive normalized linear functionals φ:

A → ℂ

satisfying

φ(A*A) ≥ 0

for all A ∈ A

(32)

Such functionals are called states. Normalization is achieved by sending the unit Id ∈ A to φ(Id) = 1. The set of all linear functionals over A build up the dual space A*. The positive linear functionals form the positive cone of the dual space A*+. A functional φ ∈ A* is called extremal if it cannot be decomposed into a linear combination of two others. Those extremal functionals are called pure states. In the Clifford algebras we are using there exist full sets of mutually annihilating primitive idempotents fi , that is,

f i f j = δ ij f j N

∑f

i

=1

symmetric orthogonality

(33)

complete set of orthogonal idempotents

(34)

i =1

where in those algebras, determination of the N usually follows the calculation of the RadonHurwitz number as outlined by Lounesto in [12, p. 226]. Such procedure has important consequences. Namely, in general, states can be decomposed according to

12

Bernd Schmeikal N

N

φ( A) = ∑ λ i φ i ( A)

with

i =1

∑ i =1

λi = 1

(35)

From that there follows a decomposition of the identity in A as N N  N  N φ( Id ) = ∑ λ i φ i ( Id ) = ∑ λ i φ i  ∑ f j  = ∑ λ i φ i ( f j ) i =1 i =1  j =1  i , j =1

(36)

Thus we have established a 1-1 correspondence between extremal states and primitive idempotents in A.

*- Irreducible Representations with Pure States *-algebra representations are connected with states. Let D:

A → B(H) be a Hilbert space representation of A

(37)

Every unit vector ψ ∈ H brings forth a state φ according to (32) via the scalar bracket

φ( A) = ψ D ( A)ψ

(38)

Even for cyclic vectors ψ the representation D is fully determined by the associate state φ. Thus we obtain a bijection between states in A and equivalence classes of triples (H, D, ψ). It is due to this so called GNS construction that the irreducible representations of a Clifford algebra A are those that are associated with the orthogonal primitive idempotents.

The CNC Duality in Quantum Geometry This is the shortest chapter but not the least important. As far as I can figure it out, the algebra of spacetime can be understood in terms of the matter algebra only then, if we consider the duality between commutative and non commutative subspaces in Clifford algebras (CNC duality). We may observe graded motion as a trajectory which involves changing grade while gaining repayments in terms of commutativity and the associated quantization rules. Nevertheless, the whole quantum geometry remains non-commutative. Historically, this duality is reflected in the battles and rebuttals around the Sakata Model and Nambu qcd. In an extremely interesting and almost personal article Harry Lipkin has recently reported how right experiments disproved wrong models and wrong experiments could lead theorists astray [25], - how the Sakata model which incorporates a fundamental SU(3) triplet was used, misused, killed and reemployed in hypernuclear physics. This is not by fortune.

Lie Group Guide to the Universe

13

Considering wisely the back-step into commutative subspaces, we have to have for the algebra: A = C(X). The GNS-representation D of a state φ is now operating in a Hilbert space H = L²(X, µ φ). The cyclic vector is represented by the unit function. The operators D are given by left multiplication. The irreducible representations are 1-dimensional. The associated characters of A are again the points of X . Pure states are given by characters. Probability measures are Dirac δ-functions located at the ‘points’ of X. Most surprisingly, those δfunctions will be accompanied, - in the whole non-commutative Clifford algebra which accommodates the commutative subspace, - by idempotents of fermion pure states having baryon number ⅓.

Lie Groups in Clifford Quantum Geometry Considering developments in high energy and mathematical physics during the last half century has convinced me of the necessity of a definite decision on the form of geometric quantum algebra A. This has been proposed by several authors: Ðurñevich [24], Woronowicz [26], Majid [27], Owczarek [28] and Oziewicz [29]. In my own work I have favoured quadratic Clifford algebra. That is, I choose A = Clp,q having p spatial base units in the standard basis and q time-like ones. Practically, in what follows we shall use Cl3,1, the ℂ⊗Cl3,1, ‘de Sitter’-Clifford algebras Cl4,1 ∼ Mat(4, ℂ) and Cl4,2 ∼ Mat(8, ℝ).

The Functor Cl gL Knowing that the Clifford algebra is itself a vector space it would be the most simple to denote any representation of Lie algebra in some Clifford algebra Cl by symbols such as γλ(Cl), or γλn(Cl) indicating the dimension n of the ground space. But this would not explain the construction. Therefore, in this section we have to go slowly and show how a Lie group in Clifford algebra is a composition of two functors, namely Cl and gL. We shall find out that Cl is an injective functor from the category of quadratic spaces Quad into the category of associative unary algebras AlgF. For F a field, let LieF be the category of small Lie algebras over F. Then gL is a functor which, by the Lie bracket, assigns to each associative algebra A ∈ AlgF the Lie algebra γλ(A) on the same vector space, but not necessarily associative. Thus we obtain the following composition of functors

Cl

gL

Quad → AlgF →  LieF

(39)

Historically, algebras were defined in terms of generators and relations. Grassmann used unit vectors {e1, e2, …, en} to define his “Hauptgebiet“ (primary domain), the exterior algebra ∧V (also V∧) by the exterior or ‘wedge’ product. The V∧ has basis

14

Bernd Schmeikal

(40) ∧V ≡ V

∧

def

= ⊕ in=0 ∧i V … Grassmanns exterior algebra

(41)

There are various ways to transform some Grassmann exterior algebra into a Clifford algebra: by stepwise definition, by factorization, by deformation and Cliffordization. We do it by the aid of the Clifford map [30], [12]. Consider a bilinear form B: V × V → F and the left contraction ┘. A Clifford map γx: ∧V→∧V is an endomorphism parameterized by a 1vector x ∈ V in the form γx = x ┘ + x ∧ with calculation rules (x, y ∈ V; u, v, w ∈∧V)

(42)

(43) where û is the main involuted û = (-1)∂(u) u and ∂(u) grade of u. We decompose B into a symmetric part g and an antisymmetric a. Then the Clifford maps of the generators {ei} of V generate the Clifford algebra Cl(V, B). With Id the identity morphism, we obtain in a basis free notation

γ x γ y + γ y γ x = 2 g ( x, y ) Id

(42)

In the anticommutator only the symmetric component of B occurs. But in the commutation relations there appears the antisymmetric a.

γ x γ y − γ y γ x = 2 x ∧ y + 2 a ( x, y ) Id

(43)

A very brief definition has been given by Lounesto (p. 190). It “is suitable for nondegenerate quadratic forms, especially the real quadratic spaces ℝp,q ”. An associative algebra

15

Lie Group Guide to the Universe

over F with unity Id is the Clifford algebra Cl(Q) of a non-degenerate Q on V if it contains V and F = F.Id as distinct subspaces so that

(44)

....

The algebra product is a Clifford product
. It can be decomposed into a symmetric inner part and an antisymmetric exterior product u
v = u . v + u ∧ v . The dot product represents the symmetric part u v = = 12 (u
v + v
u ) and the wedge product the antisymmetric

u ∧ v = = 12 (u
v − v
u ) . Once Cliffordization has provided the algebraic Clifford product, we can define the Lie commutator

[u, v] = u
v − v
u def

the Lie bracket

(45)

fulfilling the following conditions for all a, b ∈ F and u, v, w∈ Cl

(46) from those there follows antisymmetry [u, v] = −[v, u]. Adapting to tradition, we may denote this Lie algebra as γλ(Cl, ○). Then, historically, the construction led us from the Grassmann algebra ∧V to the category of general linear Lie algebras over Clifford algebras, that is we have a functor Cl gL that takes us from the general associative Grassmann algebras to the category of general non-associative linear Lie algebras over linear spaces of Clifford algebras

Cl gL

∧V   → γλ(Cl, ○)

(47)

with this in mind we can go further. In what follows let V be a quadratic space of dimension n over ℝ.

V ≅ x12 + x 22 + ... + x 2p − x 2p +1 − ... − x 2p + q

having detailed signature

with measure s = p − q.

with p + q = n

16

Bernd Schmeikal

Maximal Cartan Subalgebra in Lp,q From any quadratic Clifford algebra Clp,q we construct the Lie algebra γλ(Clp,q) calculating by the standard basis all possible normalized commutators in accordance with (46). We denote this Lie algebra as Lp,q. Take any positive non-scalar e from the standard basis of Clp,q, that is, e² = 1. Consider f = ½ (1 + e) and g = ½ (1 – e). These are orthogonal idempotents which sum up to unity, f g = g f = 0, g + f = 1. Thus the Clp,q decomposes into a sum of two left ideals

Cl p ,q = Cl p ,q f ⊕ Cl p ,q g

(48)

Note that while Clp,q has dimension 2n the ideals Clp,q f and Clp,q g both have dimension 2n-1. If we further have a maximal set of k positive non-scalar, commuting base unit monomials e(1), . . . , e(k), we can construct 2k mutually annihilating (symmetrically orthogonal) primitive idempotents which sum up to unity. In this way one decomposes the algebra into a sum of minimal left ideals which cannot be further decomposed. We obtain a minimal left ideal by forming a maximal product of non-annihilating commuting idempotents. The number k is obtained as

k = q − rq − p with the recursion r j +8 = r j + 4

(49)

for negative j, that is, p > q, we have

r−1 = −1 and r− j = 1 − j + r j − 2

with integer j

Table 1. Radon-Hurwitz number r j rj

(50)

j

0 1 2 3 4 5 6 7 8 9 10 11 12 0122333345 6 6 7

All the elements e(j), e(k) commute and generate a finite group of order 2k. The products of the corresponding idempotents

f ( α ) = 12 (1 ± e(1) ) 12 (1 ± e( 2) ) . . . 12 (1 ± e( k ) )

(51)

are primitive in Clp,q. Therefore each S(α) = Clp,q f(α) is a minimal left ideal in Clp,q. We have decomposed the algebra with respect to its pure states. Note that α = 1, . . . , k. Clearly, in the polynomial (51) there appear exactly 2k − 1 commuting Grassmann monomials which squared give unity. Those form the standard Cartan algebra of Lp,q. Theorem 1: The Lie algebra Lp,q ≡ γλ(Clp,q) contains a maximal Cartan subalgebra η = {e(1), . . . , e(c)} of Grassmann monomials e(η) with |η|= c = 2

( q − rp − q )

−1

(52)

and rp-q the Radon-Hurwitz number to Clp,q. Each such standard Cartan algebra gives rise to one family of pure states. All the families in Lp,q are mutually isomorphic. Thus each of the 2k

17

Lie Group Guide to the Universe

pure states is defined in a vector space with commuting geometry and gives rise to a minimal left ideal of the Clifford algebra. Note: this theorem substantiates the existence of qcd-fermion families. Proof: is given by the above rigor of section 7 and Cartans theorem number one, saying all maximal Abelian subalgebras of a semi-simple Lie algebra are mutually isomorphic. Theorem 1 has a series of important consequences into which we shall be going in the next two sections. First, it is interesting to notice that the Clifford algebras Clp,q with p + q = n for even n = 2m accommodate maximally the special unitary group SU(2m). This is resulting from the order of Cl(n) which equals 2n whereas that of SU(n) is n²-1. Especially the Cl(4) accommodates as a maximal special unitary subgroup the SU(4) since m = n/2 = 2 and 2² = 4, and the order of SU(4) equals 4² − 1 = 15. Thus we obtain a series of maximal inclusion relations as shown in Table 2. of maximal special unitary subgroups

Cl ( 2)

Cl ( 4)

Cl ( 6)

Cl (8)

Cl (10)

...

Cl ( 2 m )

SU (2)

SU (4)

SU (8)

SU (16)

SU (32)

...

SU (2 m )

The Sakata Model In chapter 3 of the Lie Groups for Pedestrians Harry Lipkin gave a representation of the Lie algebra of the special unitary group SU(3) in terms of the Sakata model of high energy physics. In this old model the isospin transformations had been extended to include the lambda hyperon as well as the proton and neutron. The Lie algebra was constructed by forming all possible bilinear products of creation- and annihilation operators that do not change the number of particles. Let us begin with the creation- and annihilation operators

a †p , a p , a †n , a n , a †Λ , a Λ of the proton, neutron and lambda hyperon. Consider bilinear forms

(53)

18

Bernd Schmeikal Those are exactly the ones given in Lipkin 1965, formulas (3.1).

The Spacetime Algebra Cl3,1 and its Lie group L 3,1 From the spacetime algebra Cl3,1 there is constructed the Lie algebra L3,1. Following theorem 1, (52) we calculate k = 1 – r–2 = 1 – (r6 – 4) = 2. We have |η| = 2k –1 = 3 commuting Grassmann monomials which form the maximal Cartan subalgebra η. The Lie algebra L3,1 has dimension 15 and rank 3 and is a Clifform slCL(4) of sl(4, ℝ). η = {e1 , e24 , e124 }

(54)

Now if ĥ1 denotes the complement of ĥ1 in L3,1 we have to have dim(ĥ1) = 15 – 3 = 12. This ratifies that space L3,1 has 12 root vectors. The six isomorphic Cartan algebras are essentially given by equations (55). Roots of A3 are sketched in All maximal Abelian subalgebras of a semi-simple Lie algebra are mutually isomorphic. In the Cl3,1 we have six color spaces

ch1 = {1, e1 , e24 , e124 }

ch4 = {1, e2 , e14 , e124 }

ch2 = {1, e1 , e34 , e134 }

ch5 = {1, e3 , e14 , e134 }

ch3 = {1, e2 , e34 , e234 }

ch6 = {1, e3 , e 24 , e 234 }

(55)

Each of these spaces can be generated by two units of grades 1 and 2. Taking out the scalar they represent the 6 isomorphic Cartan algebras of Cl3,1 here denoted by “ch roma”. From the arrangement of roots in A3 there follows the existence of six Lie subalgebras L(2) ∼ sl(3) with rank 2. Each has a root space A2 as shown in figure 2.

Figure 1. The root system A3 .

Lie Group Guide to the Universe

19

Those 12 roots of A3 belong to the algebras so(6, ℂ) ≅ sl(4, ℂ) and to the real forms of sl(4, ℂ) which are the algebras su(4, ℂ), sl(4, ℝ), su(p, q; ℂ) with p+q = 4. Imaging Euclidean unit vectors, we can see in A3 a sketch of one special rootspace A2 e.g. for the su(3, ℂ). The A3-cube of figure 1 with its 12 roots gives rise to six hexagons. Those represent root systems A2 which characterize the SU(3) and its various forms. Take a look at

Figure 2. The root system A2 in ℝ3.

Space of Isospinoreigenforms and Isospin Space The results found in chapter 3, the New Review of Angular Momentum Algebra concerning spin space and spinor eigenforms can be generalized. We shall work this out for the L3,1⊂ Cl3,1. In [20, p. 362] and [30, p. 276f.] it has been demonstrated that the graded quantity t3 = ¼ (e24 – e124) represents the general isospin element – (the operator τ0 in Lipkins notation (53 iv)) – in L3,1 and respectively its Clifford algebra. We first seek after general solutions of equations

[t3 , τ + ] = τ + these are:

and

[t3 , τ − ] = −τ −

(56)

20

Bernd Schmeikal

τ + = a (e23 + e34 − e123 − e134 )

with real a

τ − = b(e 23 − e34 − e123 + e134 )

with real b

Observe, the product τ + τ − = −4 a b( Id − e1 + e24 − e124 ) and thus

−

τ+τ− 1 = ( Id − e1 + e24 − e124 ) = 14 ( Id − e1 )( Id + e24 ) 16ab 4

(57)

This equation is analogous to (6), (10) in angular momentum algebra. The product of isospin shift operators (Lipkins 53 ii, iii) is equal to the primitive idempotent f13 in [20] and [30] where it represents the u-quark. This a a pure state in chromatic space ch1. It is obvious that, - once we have chosen to fixate the τ0 according to the standard basis, - we find two manifolds T+ = {τ+} and T− = {τ−} which obey the isospin commutation relations, and not just two operators.

U- and V-Spin Surprisingly it has never become quite clear how the isospin components of the SU(3) follow naturally from both the seemingly obvious spacetime geometry and quantum mechanics. We are looking for a general multivector w fulfilling commutation relations

[ τ 3 , w+ ] = 12 w+

[τ 3 , w− ] = − 12 w−

(58)

This is analogous to the v-spin in the SU(3). Solutions are

w+ = a e2 − b e3 − a e 4 + c e12 − d e13 − c e14 − d e 234 − b e1234

(59)

w− = a e2 − b e3 + a e4 + c e12 − d e13 + c e14 + d e234 + b e1234

(60)

Similar as in section 3.1 we find an eigenform

f = 12 w+ w−

(61)

Again, recall what Kauffman [23] said about eigenforms, and verify quantity f = ½ w+w−, the Clifford product of w+ and w−, is an eigenform in the Clifford algebra Cl3,1. We define recursively the bilinear term

f n+1 =

1 2

fn fn

with f n = A( Id + e24 ) + B ( Id − e124 ) + C (e 23 + e123 ) + D (e34 + e134 )

(62)

Lie Group Guide to the Universe

21

and A = a − c , B = b − d , C = ad − bc , D = cd − ab and find that this form is recursively preserved, just as it was in the angular momentum algebra. Therefore, the fn moves within the space of isospinoreigenforms 2

2

2

2

E ( f ) = span R {Id , e1 , e23 , e24 , e34 , e123 , e124 , e134 }

(63)

Observe E( f ) is itself an eigenform within Cl3,1. It reproduces itself by Clifford multiplication. A form fn, as was said, may diverge or collapse towards zero or converge towards a fixed point. With the input of (61), (62), we obtain the general solution form

f n+1 = α Id + β e23 + γ e123 + δ e24 + ε e34 + φ e134

(64)

The regained fn+1 is a polynomial with nonlinear coefficients having a special structure, e. g. for the first and last coefficients we have

α = a 4 + b 4 + c 4 + d 4 + 2( a 2 b 2 + c 2 d 2 − a 2 c 2 − a 2 d 2 − b 2 c 2 − b 2 d 2 ) , φ = abc 2 + abd 2 + a 2 cd + b 2 cd − a 3 b − ab 3 − c 3 d − cd 3 Anyway, as long as the w+, w− have the symmetric structure (59), (60), the isospinor eigenform moves in 6-dimensional space

E ( f ) = span R {Id , e23 , e24 , e34 , e123 , e134 }

(65)

which is a bit smaller than in (63). That larger space comes in as soon as the symmetry of equations (59), (60) is broken and coefficients in (59) for w+ deviate from those for w−. Only the space of (63) is closed for Clifford multiplication. Therefore the fn are eigenforms. That linear space which generates the space of isospinor eigenforms by Clifford multiplication is called the core isospin space. J=

{ e1 , e123 , e124 } ⊂ Cl3,1

core isospin space of L3,1

(66)

Actually there is a stable manifold of fixed points fn in E(f). The equation (61) denotes only one of those, though apparently a rather prominent one. This situation must be investigated more closely. In the 2004 work on transposition involutions there have been used two isospin multivectors that were denoted as u and v. Those are special solutions within the manifolds of (59) and (60). Since these manifolds can be split such that the SU(3) commutation relations are preserved. Split u + = b e3 − d e13 + d e 234 − b e1234

22

Bernd Schmeikal

u − = b e3 + d e13 + d e234 + b e1234 with [τ 3 , ± u ± ] = ∓ 12 u ±

(67)

(68) and take all constants a = b = c = d = ¼ . Then we obtain the special forms of u+, u− , v+, v− in the standard representation of L3,1.

u + = 14 (e3 + e234 ) − 14 (e13 + e1234 ) = λ 4 + λ 5

(69)

u − = 14 (e3 + e234 ) + 14 (e13 + e1234 ) = λ 4 − λ 5

(70)

v+ = 14 (e2 + e14 ) − 14 (e4 + e12 ) = λ 6 + λ 7

(71)

v− = 14 (e2 + e14 ) + 14 (e4 + e12 ) = λ 6 − λ 7

(72)

Please, observe quantities λ4, λ5, λ6, λ7 are already generators of the Lie algebra L(2) ∼ a real Clifform slCl(3) of the Lie algebra sl(3, ℝ). For completeness, consider the isoform fn = ½ v+ v− as derived from general v-spin (68). We have

f n = 12 v + v − = (a 2 + c 2 )( Id + e24 ) − 2ac(e1 + e124 ) which approaches a fixed point in the proximity of a = c =

1 2 2

(73)

.

Only in that case is the quantity ½ v+ v− = const.(Id – e1)(Id + e24) a primitive idempotent. In the current work this has been denoted as f13 ∈ ch1. It is fit to represent the pure state of a u-quark. It is therefore that we say that A pure state gives rise to an isospin split in the Lie manifold. The isospin manifolds form pure state equivalence classes.

We can carry out the same rigor for the general u± spin given by formulas (67). We obtain

f n = (b 2 + d 2 )( Id − e124 ) + 2bd (e1 − e 24 ) which converges towards a pure state, iff b = d =

1 2 2

(74)

.

In that case quantity ½ u+ u− = ½(Id + e1)½(Id − e24) is the Clifford conjugate of f13, namely f12, a fermion with strangeness.

Lie Group Guide to the Universe

23

The Spacetime Lie Algebra L(2) With equations (56) and (69) to (72) we have collected all but one generator of the Lie algebra L(2). As we know that the real spacetime algebra Cl3,1 is isomorphic with the Majorana algebra of 4×4-matrices with real entries Mat(4, ℝ), it is natural to identify the Lie algebra L(2) with sl(3). This appears as a 3×3 subalgebra of sl(4, ℝ). Yet, we shall prefer to denote L(2) by the word slCl(3) since the Cl3,1 may be defined over complex, hypercomplex and other number fields. In accordance by the same custom we shall denote L3,1 by slCl(4). The slCl(3) can be spanned by elements of spacetime algebra Cl3,1

(75) What we need to show is that this is indeed a real normal form of the Lie algebra to the special unitary group SU(3). We can do this without going into the whole classification machinery [31], [32]. Consider the real representation

(76) The Gell-Mann matrices are now identical with the following set

{−2 λ1 , 2 i λ 2 ,−2 λ 3 ,−2 λ 4 , 2 i λ 5 , 2 λ 6 ,−2 i λ 7 , 2 λ 8 } ⊂ ℂ⊗ Cl3,1

(77)

Observe the analogy between Lipkin 1965 and Schmeikal 2008

τ 0 = 12 (a †p a p − a †n a n ) , λ 3 = 14 (e24 − e124 ) N = 13 ( −2a †Λ a Λ + a †p a p + a †n a n ) λ 8 = with baryon numbers

1 2 3

(−2e1 + e 24 + e124 )

(78)

24

Bernd Schmeikal

B = a †p a p + a †n a n + aΛ† a Λ b = 121 (3Id − e1 − e24 − e124 ) and b= y−s,

y = 16 (−2e1 + e24 + e124 ) =

1 3

λ 8 , hypercharge

s = − f12 = − 14 (1 + e1 )(1 − e24 ) strangeness f12 primitive idempotent ∈ ch1 †

Watch the Cartan subalgebra {τ0, λ8} to derive the interpretation of e24 ∼ a p a p as proton- and e124 ∼ a n a n as neutron-number-operators. Compare Lipkins N with λ 8 to see †

field quantization on the preferred direction e1 brings forth the Λ -Hyperon. Verify correct matrix representations of quantum numbers in accordance with representation (76), as for example 0

0  B=   

1 3 1 3

 0    1   and λ = 1   8   1 3   1 − 2   3

(79)

Pure Space and Constitutive Group In unitary geometry, traditionally, we used to solve a special problem. Namely, given a complex vector fj of dimension 3, we had to find those transformations

fj  → U k j f j

that preserved the norm

∑ f j* f j

(80)

j

Such matrices satisfy the relation U† = U−1 and form a group, in accordance with the defining equation for the inverse:

(UV )† = V †U † = V −1U −1 = (UV ) −1

(81)

This group of matrices ⊂ Mat(3, ℂ) is the unitary group U(3). Unitarity imposes nine constraints on the 18 real degrees of freedom in a 3×3 matrix with complex entries. So the U(3) has dim = 9. The norm is invariant under a transposition by a phase eiα . Therefore U(3)

Lie Group Guide to the Universe

25

can be decomposed into a direct product U(1) × SU(3), where SU(3) has unit determinant and one less degree of freedom, that is, dim = 8. This result could be generalized. The unitary group U(n) incorporates the special unitary group via the equation U (n) = U(1) × SU(n). In terms of some predicate logic calculus that whole problem could be reformulated as follows: Consider an associative unary algebra. Find those transformations that preserve the predicate “unary”. As we know that from every unit element e there can be derived two idempotents f = ½ (1 ± e) we can just as well demand the existence of a group which preserves the quality of »idempotent«. This shall be exactly the way we shall pose this problem: Find the group which unfolds the primitive idempotent manifold! We begin with pure states. Pure states in Clifford algebras Clp,q are given by the 2k primitive idempotents as formulated in (51)

f ( α ) = 12 (1 ± e(1) ) 12 (1 ± e( 2) ) . . . 12 (1 ± e( k ) ) with k = q − rq − p These pure states can be unfurled to manifolds, − with mutual orthogonality relations preserved. The tools to unfold the idempotent manifolds in that way, are subalgebras L in the Lie algebras Lp,q ≡ γλ(Clp,q). Consider any Clifford number λ∈ Lp,q and the quantity obtained by the exponential map

g = e λ ∈ L p ,q group element

(80)

A stranded braid of primitive idempotents is given by equivalence classes def

∆ = {∆ (α ) } = {g f (α ) g −1 / g ∈ L ⊆ L p ,q }

(81)

brought forth by conjugation. The invariance of orthogonality relations, therefore, means that

f j f k = 0 ⇒ ( g f j g −1 )( g f k g −1 ) = 0

(82)

which is evident. There is a maximal subset of primitive idempotents in a maximal subspace Pp,q with positive definite signature [33, p. 125 ff.]. The stranded braid of primitive idempotents with positive definite signature shall be called the »pure space« of the quantum algebra. It is constituted by pure states and their specific orthogonality relations. The L ⊂ Lp,q is called the »constitutive group« of the pure space. This riddle of finding the pure space and constitutive group of a quadratic Clifford algebra has been fully solved by the author for the case of the spacetime algebra Cl3,1 . For the general case it has been solved only by (small) parts. The problem is very challenging because of the grading. The Lipschitz group has to be replaced by a graded group which unfolds each general primitive idempotent into the whole positive definite subspace of Clp,q. For the spacetime algebra the solution can be articulated in the following way.

26

Bernd Schmeikal

Pure space of Clifford Algebra Cl3,1 In the standard representation of the Cl3,1 the positive definite subspace is P3,1 = span {Id, e1, e2, e3, e14, e24, e34, e124, e134, e234}. The constitutive group of the 6 color spaces chχ with χ= 1, . . . , 6 and their pure states is the Lie group L(2) = SLCl(3) with generating elements (75) from algebra L3,1 and elements (77) for the complexified algebra ℂ⊗Cl3,1. Let us investigate this in-depth. Any idempotent f ∈ chχ has the property f f = f. Two distinct mutually annihilating primitive idempotents fulfil f1 f2 = 0. Solving four multilinear equations, we find there prevails a pattern of only four orthogonal primitive idempotents in each space chχ. In ch1 those are

f11 = 12 (1 + e1 ) 12 (1 + e24 )

f12 = 12 (1 + e1 ) 12 (1 − e 24 )

f13 = 12 (1 − e1 ) 12 (1 + e24 )

f14 = 12 (1 − e1 ) 12 (1 − e24 )

(83)

The first is taken to be a representative of the electron neutrino νe. The others are interpreted as fermion states. (Schmeikal 2004). The neutrino is fixed by its stabilizer algebra, namely L(2). Therefore the f11 annihilates the Lie algebra [32, p. 81f.]

f11 L(2) = {0} or equivalently f11 expL(2) = { f11 }that is, f11 absorbs group SLCl(3)

(84) (85)

The other primitive idempotents are not fixed, but transformed by carrying out the conjugation. Algebra L(2) contains an important element, namely the universal generator l of both the trigonal color- and flavor rotations. This is the multivector

l=(

2 3

arccos(− 1 2 ))(λ 2 + λ 5 + λ 7 ) and exponential T = e l

(86)

The universal T is brought forth by two reflections which on their part are given by standard primitive idempotents in the pure space

g = 14 (1 + e3 )(1 + e24 )

h = 14 (1 + e1 )(1 − e34 )

(87)

together with transpositions τ(g) = 1– 2g and τ(h) = 1 – 2h. Those are Coxeter reflections in a large finite automorphism group [20, p. 358] of transpositions. We calculate

τ ( g ) 2 = τ ( h) 2 = 1

Coxeter reflections and rotator

T = τ ( g )τ ( h) with l = ln (T)

(88) (89)

Lie Group Guide to the Universe

27

T is trigonal, that is T 3 = 1 and T 2 = T −1 (figure 3)

Figure 3. Trigonal rotation of fermion pure states.

Some SU(3) History Full comprehension of the meaning of the SU(3) symmetry in HEPhy was and still is not easy. But the Clifford algebra view presented here will help us to see the whole more clearly. Namely, those symmetries of the standard model are indeed one with the geometry of spacetime. Models using the SU(3) symmetry of interacting fields go back to Sakata (1956) [34] and Lipkin (1959) [35], [36]. Their relevance was substantiated by the works of Goldberg and Ne’eman [37], [38], [39] and well established by Nambu’s construction of the color gauge Lagrangian. There was in the beginning some considerable competition between “The eightfold Way” and “The Tenfold Way”. Seen from the nowadays viewpoint those represented the alternatives between SU(3) multiplets, that is, between the Baryon octet and the decuplet. At that time the difference between those two was a bit unclear. That quasi Buddhist label5 supported the perennial mystification of the meaning of the SU(3) within the prevailing model, – thereby favouring Gell-Manns model. But the eightfold path and some wrong experiment regrettably blocked up timely understanding some important heavy hadrons. Lipkin [25] recently reported that sometime in the academic year 1961-62 Hayim Goldberg and Yuval Ne’eman showed how the Baryon octet could be constructed from SU(3) triplets with baryon number 1/3. They placed the ∆(1238)- and Σ(1385)-resonances in the tendimensional representation of SU(3) and used the Gell-Mann-Okubo mass formula to predict the Ξ and Ω− with masses close to 1500 MeV. A wrong experiment indicated that the decay Σ → Σ π was forbidden and disposed theorists to foolishly classify and experimenters to search for Ξ and Ω− in the SU(3) 27-plet. A second crucial experience was the discovery of the φ vector meson and the unexpected suppression of the φ → ρ π decay. That decay should have occurred instead of a strong interaction of the form

5

„Gell-Mann nannte dieses Schema Eightfold Way, eine Bezeichnung die die Oktette des Modells mit dem Achtfachen Pfad des Buddhismus verbindet. Er prägte auch den Namen Quark, den er aus dem Satz ‚Three quarks for Muster Mark’ aus James Joyce’s Roman Finnegans Wake entnahm. Da einzelne Quarks in Experimenten nie beobachtet wurden, bezeichnete Gell-Mann selbst sie als mathematische Fiktion.“ (Wikipedia 2007/08: „Quark+Physik“)

28

Bernd Schmeikal

K − + p+  → Λ + φ  → K − + K + or equivalently u s + uu d → u d s + s s → u d s + u s + su In a 1999 private communication to Harald Fritzsch [40], [41] George Zweig wrote: “ In the April 15, 1963 Physical Review Letters there is a paper [42] titled ‘Existence and Properties of the φ - Meson’ . I remember being very surprised by Figure 1, which showed a Dalitz plot for the reaction K− + p → Λ + K− + K+. There was an enormous peak at about (1020 MeV)² in the M²-plot for K K , right at the edge of the phase space. The fact that the φ decayed predominantly into K K and not ρ π was totally unintelligibly despite the authors’ assurance that this suppression ‘need not be disconcerting’. [ …] Here was a reaction that was allowed but did not proceed!” 6 Those experiments convinced Zweig that baryons had constituents and were not just “hypothetical objects carrying the symmetries of the theory, but real objects that moved in spacetime from hadron to hadron.” One just had to assign the correct constituents to pseudoscalar and vector mesons in order to, among other things, explain the ρ π-suppression. Zweig then reported how Richard Feynman at Nino’s 1964 Erice summer school did not believe his arguments. But, “later that fall, when I gave Gell-Mann my explanation of φ-decay and drew my diagram for φ → ρ π (which involved polygon blocklike icons for the constituents), I can still hear him saying “Oh, the concrete quark model!” Despite the mystification of the authority of the person Gell-Mann it must be realized that the classification of hadrons in the tenfold way which predicted the existence of the isoscalar baryon with strangeness −1 (the Ω −) was not only understood by Goldberg and Ne’eman, but it was also noted by Gell-Mann and others. Lipkin points out, the right classification goes back to Glashow and Sakurai. They predicted the existence of the Ω−. When I was young I was proud to hear that, obviously without knowing, the scanners of our HEPhy group in Vienna were among those three groups who had correctly scanned the omega-minus. Some of us studied the tenfold way and various mathematical models that should explain the decays. Then, everything was very fascinating and unclear at the same time. As a scholar of Walter Thirring I became busy with a thought which did not let me go until today. Those symmetries were properties of spacetime. Then, the most important problem was that the Ω − = s s s, ∆− = d d d and ∆++ = u u u violated the Fermi statistics because they contained three identical spin ½ fermions in a space symmetric state coupled symmetrically to spin 3/2. This was first solved by Greenbergs parastatistics [43] and later led to the introduction of a new degree of freedom, namely color. The new theory was supported and further developed by Yōichirō Nambu [44], [45]. His qcd answered the fundamental questions of high energy physics. With this now in mind we can solve the qcd-problems in geometric algebra.

6

quoted after Harald Frizsch 2002 [40]

Lie Group Guide to the Universe

29

The Spacetime Oscillator Color spaces are 4-dimensional commutative subspaces of the Clifford algebras consisting of a scalar and 3 units of different grades. In a recent writing I have denoted those by letters a, b, c in order to indicate their algebraic equivalence. The unit vectors ej, ek4, ejk4 with j≠k≠4 form commuting triples in corresponding color spaces. They satisfy the multiplication table of the smallest non-cyclic group V. Table 3. Klein 4 group table Id a b c

a Id c b

b c Id a

c b a Id

This is the multiplication table of the Klein-4 group denoted as K4 or equivalently V (ierergruppe). We can insert for a, b, c any of the triples of the fundamental isomorphic maximal Cartan algebras η of a rank-3 algebra L(3) such as for instance {e1, e24, e124}. Consider any geometric element ξ∈ch1 having form

ξ = x0 + x1 a + x 2 b + x3 c with x j ∈ ℝ and a, b, c ∈ K4

(90)

This can be represented in the space of symmetrically orthogonal, primitive idempotents in the form

ξ = A0 ψ 0 + A1ψ 1 + A2 ψ 2 + A3 ψ 3

(91)

with coefficients

A0 = x0 + x1 + x 2 + x3 A1 = x0 + x1 − x 2 − x3 A2 = x0 − x1 + x 2 − x3 A3 = x 0 − x1 − x 2 + x3

(92)

and the bilateral7 orthogonal, primitive idempotents ψ 0 = 12 (1 + a) 12 (1 + b) ψ 2 = 12 (1 − a ) 12 (1 + b) 7

ψ1 = 12 (1 + a) 12 (1 − b) ψ 3 = 12 (1 − a) 12 (1 − b)

(93)

It is important to sometimes mention symmetric or bilateral orthogonality because there also exist other types of asymmetric unilateral orthogonality

30

Bernd Schmeikal

Equations (90) to (92) make clear that, in a unary algebra, we have two ways to represent the multivector ξ. First, we can represent it in the ground space of base units as span{Id, a, b, c} and second, we can represent it in the idempotent space span{ψ0, ψ1, ψ2, ψ3}. I have variously called those units “extensions”8 because they really reflect what Spinoza and Descartes meant by extension in contrast to cognition. What is so special about these quantities? Each of the three extensions a, b, c have different grade. The grade (or length) is increasing from 1 to 3. Time appears in grades 2 and 3 as wrapped up and disguised in a space-like unit which squared gives one. Surprisingly, the grade doesn’t seem to make any difference. Each quantity appears as equally important within its color space. Moreover, the quadratic form can be extended over the whole positive definite subspace (10-dimensional in Cl3,1). The quadratic form has a continuation over different grades. Last not least, the universal trigonal SU(3) operation applies equally to both the unit vectors and the pure states. That is, the T acts on the states and on the extensions in the same definite way:

Figure 4. Trigonal flavor rotation in Cartan algebra η.

This means while we color-rotate the fermions wave functions – supposed as antisymmetric according to some type of Pauli exclusion principle – we SU(3)-rotate the basis at the same time. Clearly, in such a 3-dimensional commutative space like ch1 it is most natural to quantize three coupled boson-oscillators alongside the a, b and c or respectively e1, e24 and e124. This is not a paradox, because the pure states constituted by the boson fields behave like fermions. We quantize fields with a color-space representation (90) to (93). We do so by first quantization of the three dimensional oscillator in space η spanned by the Cartan subalgebra {e1, e24, e124}. We follow Lipkins quasispin classification (chapter 4) and SU(3) representation by first finding the preferred direction in the fermion u d s-space, namely the sdirection 9. We have to locate the quantum oscillators and use the basic inner correspondence

8

being aware of other such denotations as for example “algebraic extension” in Mathematics Subject Classification 2000: 12F05 9 This is the Λ-direction in Lipkins Sakaton model.

Lie Group Guide to the Universe

31

Next we derive the cylindrical annihilation- and creation operators in three directions and bilinear forms which satisfy the SU(3) relations

H = ℏω (a †d a d + au† au + a †s a s + 32 ) Hamiltonian

τ + = a d† au

τ − = au† a d

τ 0 = 12 ( a d† a d − au† au ) with the total spin τ

τ 2 = 12 ( τ + τ − + τ − τ + ) + τ 02 and u-,v-spin forms

B+ = a †d a s

B− = au† a s

C + = a †s au

C − = a †s a d

Y = 13 ( a †d a d + a u† au − 2a †s a s ) The harmonic oscillators are located at the arrowheads represented below in

Figure 5. Locating the SU(3) oscillators in η.

(94)

32

Bernd Schmeikal

This model is essentially the same as the Sakata model. It is related to Elliott’s Nuclear Shell Modell [46]. Hadrons appear as degenerate eigenstates of the oscillator Hamiltonian. The bilinear forms of τ±, B± and C± shift quanta between the primitive idempotents ψ1 = f12 or selected direction e1, the plane e24 or respectively primitive idempotent ψ2 = f13 and spacetime volume e124 or primitive idempotent ψ3 = f14. Quantum number Y is one-third the difference between the number of quanta in the isodublet space {u, d} and twice the number of quanta in the distinguished “strange direction” s. We have Y = 0 when the number of quanta in all three directions are equal. Y > 0 when the average number of quanta in the u-, d- directions is larger than in the s-direction, and Y < 0 when strange fermion energy dominates. Thus Y measures the departure from spherical symmetry, the deformation of the field. This has a special importance because the three directions have different grades within the Clifford algebra. Though their grade is unimportant as long as we restrict our rigor to one distinguished color space ch1. The special importance of the grading is the following. Recalling chapter 5, it is clear we have a commutative space within a C*-algebra. The GNS-representation of the fermion states is definitely operating in a Hilbert space H = L²(X). The irreducible representations are 1-dimensional. The associated characters of the algebra are points. Probability measures are Dirac δ-functions located at the ‘points’ of X. But we have not only point locations, but also definite areas and sharp spacetime volumes. Between those there occur graded transpositions which can best be compared with maps between discontinua such as (Peano) fractals having dimensions 1, 2 and 3. Dislocations as annihilations of point locations in favour of sharp “δ-functioned” spacetime areas and spacetime-volumes are possible. I denote this process as annihilation and creation of extensions. Suppose we count three quanta on the spacetime oscillator. That would allow for ten fermion combinations which form the well known baryon decuplet. hadron

S

B

Y

τ0

sss

Ω−

−3

1

−2

0

ssu

Ξ0

−2

1

−1

1 2

ssd

Ξ−

−2

1

−1

− 12

suu

−1

1

0

1

−1

1

0

−1

sud

Σ+ Σ− Σ0

−1

1

0

0

ddu

∆0

0

1

1

− 12

uud

∆+

0

1

1

1 2

uuu

∆+ +

0

1

1

3 2

ddd

∆−

0

1

1

− 32

quarks

sdd

Lie Group Guide to the Universe

33

Figure 6. The SU(3) Decuplet.

The Spacetime Color Degree of Freedom In case of three equal quanta sss = Ω−, ddd = ∆− and uuu = ∆++ the spacetime oscillator shows maximal departure from spherical symmetry and violates the Pauli principle. Therefore it needs a further degree of freedom which is color. The state function has the possibility to evade into another space. Which space is it where into the fermion state function from ch1 quibbles? It’s spaces ch3 and ch5. Consider the idempotent f34 from color space ch3 and a second one f52 from ch5 and, recalling forms (55), form the product

TC = (1 − 2 f 34 )(1 − 2 f 52 ) where f 34 =

1 2

(95)

(1 − e2 ) 12 (1 − e34 ) and f 52 = 12 (1 + e3 ) 12 (1 − e14 ) −1

Verify TC f12TC =

1 2

(1 + e2 ) 12 (1 − e34 ) = f 32 . That means TC carries the s-quark

pure state from color space ch1 to color space ch3. There exist universal trigonal rotations which transport quarks of each family in 3-cycles from color to color. Each color space provides a definite color and contains three quarks such as u, d, s. Considering u, d, s as one ternary fermion family, spaces ch1, ch3, ch5 provide three different colors for that family. However, consider the transition from ch1 to ch4. It represents a special involutive automorphism of the Clifford algebra, namely a transposition or exchange; in the present case a transposition of base unit of type e1 onto e2, so that the triple {e1, e24, e124} goes over into {e2, e14, e124}. The spacetime unit-volume e124 is preserved. We may assume that Transpositions of form

ch1 = {1, e1 , e24 , e124 } ⇒ ch4 = {1, e2 , e14 , e124 }

34

Bernd Schmeikal

ch3 = {1, e2 , e34 , e234 } ⇒ ch6 = {1, e3 , e 24 , e 234 }

(96)

ch5 = {1, e3 , e14 , e134 } ⇒ ch2 = {1, e1 , e34 , e134 } which wrap and unwrap time on a distinguished unit that indicates strangeness represents an energy consuming symmetry breakage. This could be a reason why we observe the up, down and strange quarks in spaces 1, 3, 5 and the high energy fermions charm, bottom and top in representation spaces ch2, ch4 and ch6. So we have six different types of strong interacting fields which constitute the spacetime algebra.

CPT and Spacetime Area Oscillation Spacetime areas may change direction. That affects the CPT of pure states. Clifford algebra representations (83), (93) of the fermion states u, d, s as shown in figures 4 and 5 are not only the most convenient ones, but they are also the only ones compatible with the demands of quantum geometry.

Figure 7. Oscillating oriented spacetime area.

Once recognized, this has enormous theoretical and empirical consequences. To see this, consider the directed spacetime area e24 in a dynamic oscillation with real scalar x such as x e24 = (sin ωt) e24. Clearly, this quantity changes sign in accordance with the sine function. Taking into account the anticommutation relation

e24 + e42 = 0 plus ψ = x e24

(97)

the sign change of ψ can be interpreted in two ways, namely first as a turnaround of x and second as a reversal of the spacetime area. The latter includes time reversal and all the rest of it. Let us go into this further because it concerns CP-violation and the Cabibbo-KobayashiMaskawa−matrix, CKM−matrix, a 3×3 unitary matrix describing mass-mixtures of fermion eigenstates in strong transitions with W+ Boson emission . The Cl3,1 pure states have detailed forms

Lie Group Guide to the Universe

35

From this we learn that orientation reversal of bivector e24 can be transposed onto a reversal of the spacetime volume e124 which compensates the necessity of a time-reversal, since the turning back can occur on the bivector part of the exterior product e124 = e12 ∧ e4. Therefore, provided such geometric interpretation is valid, within the isodublet {u, d} causality violation can always be compensated for by an exchange of u and d-quarks. A similar argument, however, does not apply for the strange quark; pairs {s, u}, {s, d}. Once the oscillator on its spacetime volume is in phase on the area e12 a simultaneous excitation of the oscillator on e24 causes a split between an advanced and a retarded wave. The strange fermion excitation got to be accompanied by a cloud of virtual processes. Note, for the singlet {s} parity on the distinguished direction e1 is opposite in the dublet {u, d}. The same statement holds for the spacetime areas e14 and e34 and the other isodublets {c, s} and {t, b}. We should expect transition rates u↔d, c↔s, t↔b significantly larger than the others. Our theory confirms the 2006 matrix of the Particle data group [47].

(V ) ij

Vud = Vcd Vtd

Vus Vub   d  Vcs Vcb   s Vts Vtb   b

  d′      =  s′    b ′ 

(97)

with

(V ) = 00,97383 ,2271 ij

0,00396  0,97296 0,04221   0,00814 0,04161 0,999100 0,2272

Going further into the cosmos, we can model antimatter either by space inversion or by complexifying the Clifford algebra. Some detailed bookkeeping makes clear that we have to proceed with the complex matrix algebra Mat(4, ℂ) which is isomorphic with the Cl4,1. These coherences have been shown in [48, p. 73] and [12, p. 217]. The implementation of Higgs bosons and dark matter demands Cl4,2.

Out of the Light . . . The standard model is based on chiral decompositions of the form SU(3)⊗SU(2)⊗U(1) and therefore does not bring on any unification. Warren Siegel [49, p. 245] in his magnum opus »Fields« points out how grand unified theories force those three gauge groups to be subgroups of another group which is broken to SU(3)⊗SU(2)⊗U(1) by the Higgs mechanism and then further down to SU(3)⊗U(1). This would introduce yet unobserved spin-1 particles

36

Bernd Schmeikal

with very large masses. The simplest such model, he argues, could use the special unitary group SU(5). Then he shows how the pentaplets decompose in accordance with the standard model. Lipkin in his 1965 work does not consider the SU(5), but concentrates on the multiplets of the SU(4), SU(6) and SU(12). The SU(4) fits perfectly into the limits of the Clifford algebra Cl3,1 of the Minkowski spacetime in the Lorentz metric. Lipkin shows us, how we can go on from SU(4) to find the SU(5) multiplets. Its just the same as going from SU(3) on to SU(4), just one step further. All these considerations become very interesting, as soon as we can relate fields to geometric algebra of spacetime. Its actually the way Einstein and Mach dealt with theory: it’s the fields that bring forth the spacetime. The matter makes the observed properties of space and time. We can read out the features of space and time by understanding the experiments we make with matter. This was Mach’s idea. [50] It is clear that the quadratic Clifford algebra Cl4,1 having dimension 32 cannot embed any appropriate unifying symmetry group because its maximal Cartan subalgebra also provides maximal rank 3 to the Lie groups, just like the spacetime algebra Cl3,1. From such rank 3 algebra we must construct the rank 2 Lie groups as Clifforms of SU(3) which are graded and therefore allow for a norm- and othogonality preserving transformation of pure states within the whole ten-dimensional positive definite subspace. This procedure is not altered by moving from Cl3,1 to Cl4,1 because of the form of the basic primitive idempotent. Just the number of isomorphic Cartan subalgebras is doubled. This can indeed be used to construct some sort of additional Baryon like dark matter fermions which contribute to Massive Compact Halo Objects (MACHOs). But such approach does not meet the main riches of the present conception, nor does it cope with the astrophysics findings. Dark matter and energy, hot and cold, are not baryonic. The most preferred candidates of cold dark matter are weakly interacting massive particles (WIMPs). Taking into account the simple ΛCDM10 model which describes cosmic evolution by the cosmological constant Λ and six more parameters, cosmologists found the contingent of baryonic matter on the critical density (9,7 · 10−26 kg m-3) equal to only 4,44% and that of matter, CDM inclusive 26,6%. Hot dark matter provided by the known three neutrinos would lead to a false scenario. The best explanation is given by the WIMPs. This is in perfect correspondence with a model based on a Lie algebra γλ(Clp,q) of a rank higher than 3. We shall have to investigate the Clifford algebra generated by a space similar to a de Sitter space: »adS6« and respectively γλ(Cl4,2) having two time coordinates. Because of its peculiar primitive idempotent structure Clifford algebra Cl4,2 allocates some invariant massive neutrino pure states. Those are relatives of the known neutrinos, but have higher energy. Clearly, we are then operating in a theoretical domain related to the Randall-Sundrum model [51], [52]. The pure states of wimps are consistently incorporated into the Lie algebra as fixed points in the manifold, and we can establish the unified field, supersymmetry and the Higgs mechanism [53]. We shall slowly go into this. But first I have to introduce us into a laudable approach – small paths that have been trodden by some of us − before we left the light and correctly stepped out into the dark.

10

CDM = Cold Dark Matter

Lie Group Guide to the Universe

37

A Scheme of Symmetry Breakage Sometime in 2007 I reviewed an interesting paper by Robert Gordon Wallace [54] from Australia. It seemed somehow extraterrestrial like my own, - realized Clifford algebra in some strange matrix modules carrying out basic rigor by a spreadsheet. So I first had to translate everything into the languages I used. Those were Maple Clifford by Rafal Ablamowicz and Bertfried Fauser and my old “handy” Clical by our deceased friend Pertti Lounesto. Both programs use indices for concrete base units, whereas Wallace used mere capital letters for a bunch of matrices, - why not? Verbatim his abstract stated “A scheme of symmetry breakage can be imposed on orthogonal directed lineelements for the algebra sl(4, ℂ) which, for Cl+3,1, Cl+4,0 and Cl+2,2 subalgebras results in a pattern corresponding to the standard model, together with elements corresponding to fundamental particles of dark matter.” Wallace had correctly picked up some algebraic investigation of my own 2001 work [48] and tried to lift the Clifford algebraic version of the Weinberg Salam theory of electroweak interaction worked out by David Hestenes into those Cl4,1 spaces that incorporated the Clifform of SU(3) developed by me. But he was not fully aware of the consequences of the fact that the Clifform was not a Lipschitz group, but was based on some graded algebra, the grading of which became invisible, indeed, by using matrices from sl(4, ℂ). It behaved different than the familiar spinor models. So he did his best to structurally lift Hestenes’ approach into the appropriate Clifford algebra. He practically found all the essential electroweak modules, but wisely did not fix the standard model. Nevertheless he visualized a possible origin of hot dark matter and the asymmetry between light photons and heavy Higgs bosons. The mathematical origin of the symmetry breakage is so original – almost witty – that I want to give us a sketch. It seems Wallace used a basis as follows

Cl 4,1 is generated by W = {e1 , e2 , e3 , ie1235 , e5 }

(98)

which has a real component

V = span{e1 , e2 , e3 , e5 } and Cl3,1 = ⊕ k ∧kV ∼ Mat(4, ℝ)

(99)

and an imaginary part which is simply equal to iCl3,1. Thus the basis vector e4 ∈ Cl4,1 is the complexified Graßmann product ie1235 and the Cl4,1 gets a real and an imaginary part, that is, it is decomposed as

Cl 4,1 ∼ Cl3,1 ⊕ iCl3,1

(100)

which is in perfect correspondence with the isomorphism ℂ⊗ Cl3,1 ∼ Mat(4, ℂ) ∼ Cl 4,1 If we consider any multi-index η and its complement η we have

(101)

38

Bernd Schmeikal

eη = ± i e η examples: e45 = − i e123 or e134 = i e25 The 10-dimensional subspace P3,1 with positive definite signature turns into a 10dimensional subspace with negative definite signature. Both taken together give a module isomorphic with the Cl4,1. Using a basis

0 0 e1 =  − 1  0 − 1 0 e3 =  0  0

0 − 1 0 0  0 −1 0   0 0 1 −1 0 0 0  e2 =  0 0 0 0 0 0 − 1    1 0 0 0 −1 0  0

0 0 0 i 0 0    − 1 1 0 0 0 0 0 − i  e4 = e = − i 0 0 0  5  0 0 1 0     0 0 − 1 0 i 0 0 0 0 0

(102)

0 0 0 0 0 0 1  0 − 1 0

1

0

It can easily be verified how a general element of some special electroweak group SU(2)×U(1) can be constructed for instance as

(aId − be23 + ce13 + de12 ) × ( Id cos φ − e123 sin φ)

(103)

Wallace gives matrix representations of the electroweak special unitary symmetry for some fermions like the red u-quark. Using his spreadsheet he detected six fermionic subalgebras of the type s(u(2)×u(1)) with real pseudoscalar base units, corresponding to quark families of ordinary matter and nine such algebras with imaginary pseudoscalars for dark matter fermions. To give an example, a red up quark can be represented in space

(104) which can be provided by definitions a = cosh θ cos 1 2ψ , b = cosh θ cos 1 2ψ , c = sinh θ cos 1 2ψ , d = sinh θ sin 1 2ψ j = cos φ , k = sin φ

(105)

Lie Group Guide to the Universe

39

The model also contains bosonic subalgebras su(2)⊕su(2) for Higgs bosons W+, W−, Z0 and photons. It has space for supersymmetry by two times three special (pseudo)unitary bosonic subalgebras su(1,1)⊕su(1,1). Wallace argues that the four anti-commuting triples {e21, e13, e23}, {e234, e134, e12}, {e234, e124, e13}, {e134, e124, e23} represent building blocks for Goldstone- and Higgs bosons. The symmetry breaking distinguishes the first triple from the other. It is indeed possible to test their behaviour in strong interacting processes by applying t, some universal trigonal operator as in (86), (87). Take

t = u1u2 with t −1 = t 2 , u12 = u22 = Id and

(106)

u1 = ½ (Id + e3+e25 + e235) u2 = ½ (Id+e1–e35+e135) It turns out that up to direction only the first triple is preserved under a t-rotation which says nothing else than that the triple {e21, e13, e23}can be denoted as an electromagnetic triple or photon, since it does not interact strongly with matter. So far the strength of the Wallace model, - it actually can describe a structure of hot dark matter,- but let us also see the weakness of it in case we want to describe the wimps, the cold and weak dark matter. The model is still too strong to describe the frail. We got to understand the meaning of primitive idempotents and the special role of heavy neutrinos.

Into the Dark Why a Cl4,1-model cannot work has been said at the beginning of the previous chapter by the voice of Warren Siegel. Also we got to understand the special structure of a graded standard model. Consider the rotation group SO(3,1). It has a double cover Spin(3, 1), that is, we have SO(3, 1) ∼ Spin(3, 1)/{±1}. In Cl3,1 there exist elements s of the graded orthogonal group L(2) that rotate, for example

s −1 f12 s = f13 but the left (or right) –sided equation

(107)

ρ f12 = f13

(108)

has no solution for ρ. This is different to the elder Lipschitz approach. What is the difference? We have a spin group, namely L(2), but no rotation group of which it is a double cover. Yet, this does not impair the action of the Lie group which is based on the Clifford commutator. It also imposes no limitation on formulating the standard model. But it needs a new interpretation of motion. Strictly, we have an inner Lagrange function in the commutative subspaces and an outer Lagrange in the whole space. We can formulate two norms and two equations of motion for inner and outer spacetime. We understand why, because of the role of pure states and the neutrino fix point Ansatz, we have to have a heavy neutrino in the Clifford algebra Cl4,2 which satisfies the equation

40

Bernd Schmeikal

f11 L(6) = {0}

or equivalently

(109)

f11 expL(6) = { f11 }

that is, f11 absorbs group exp L(6)

(110)

We have solved this problem entirely for the Cl3,1. To comprehend its logic for the de Sitter space: »adS6« in L(6) we first go back to the “Lie groups for Pedestrians” investigation. Harry Lipkin began with the bottom up construction with the diagonal isospin su(2) operators

B = a †p a p + a †n a n

baryon number

τ 0 = 12 (a †p a p − a †n a n )

isospin

(111)

and from there proceeded to su(3)

B = a †p a p + a †n a n + a Λ† a Λ

baryon number

τ 0 = 12 (a †p a p − a †n a n )

isospin

N = 13 (−2a †Λ a Λ + a †p a p + a n†a n )

the later hypercharge

(112)

and from there to su(4)

B = a †p a p + a n†a n + a †Λ a Λ + a †Χ a Χ

baryon number

Z = 14 (−3a†Χ aΧ + a†p a p + a†n an + aΛ† aΛ ) =

1 4

B+C

(113)

with a new quantum number C then called ‘charm’ while Z was simply “a new operator Z”. We might call it Z-charge, but we are aware of the terminological changes that have occurred since 1965. Today we pack both charm and strangeness into the su(3) algebra and the indices in the Sakata SU(3) have turned from p, n, Λ to d, u, s. Lipkin’s bottom up construction can be continued, and we wish to go further to the SU(7) in Cl4,2. Before doing so we take a look at the su(3) in (112) and its relation to the L(2) in Cl3,1. Recall the analogy in equations (78) between Lipkin’s and the pure state Ansatz of Cl3,1. After having found the graded ‘quasi Lipschitz’ group for the Clifform of su(3) generated by the λ-Clifford numbers (75) we realized that the pure state f11 representing the e-neutrino annihilated each λ i and therefore the whole algebra, that is, f11 L(2) = {0}. Now, that we have a clear image of the algebra L(2) we can make a test and find out by a computer program how those manifolds X look like that satisfy an equation of the form f11 X = {0} in Cl3,1. Maple Clifford gives us a return like the following: starting off with a manifold

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Lie Group Guide to the Universe X =

{g1 Id + g 2 e1 + g 3 e2 + g 4 e3 + g 5 e4 + g 6 e12 + g 7 e13 + g 8 e14 + + g 9 e23 + g10 e24 + g11e34 + g12 e123 + g13 e124 + g14 e134 + g15 e234 + g16 j}

with spacetime-volume j, we obtain the general solution for X. Coordinates g2, g5, g6, g7, g8, g 10, g 11, g 12, g 13, g 14, g 15, g 16 can be chosen freely. But g1, g3, g4 and g9 have to satisfy the four equations: Table 4. Manifold partition for Lie algebra L(2)

g1 = − g 2 − g10 − g13

base units occupied Id , e1 , e24 , e124

λ’s used

λ3 , λ8

Y, η0

g 3 = g5 − g 6 + g8

e2 , e4 , e12 , e14

λ6 ,λ7

v-spin

g 4 = − g 7 + g15 + g16

e3 , e13 , e234 , j

λ 4 , λ5

u-spin

g 9 = − g11 − g12 − g14

e23 , e34 , e123 , e134

λ1 ,λ 2

t-spin

Manifold partition

showing how this manifold which is absorbed by the pure neutrino state decomposes into exactly 4 parts. The first is given by the Cartan algebra, that is, 3rd component isospin plus hypercharge, the second by v-spin, the third by u-spin and the last by isospin t. Note, we have a partition of the manifold into blocks of 4 times 4 coordinates of X. It is tempting to ask what kind of partition we would obtain for the group L(6) in Cl4,2? We are aware, according to Table 2, the largest su(n) within Cl4,2 is of course the su(8) having dim = 65 and rank 7 (8 minus 1). The group SU(8) ⊂ L4,2 is virtually clung to the Cl4,2. To rotate the pure states we need to construct the group L(6) from L4,2 which has a rank by 1 smaller than that of L4,2 . A standard neutrino fix point is the Clifford number

f11 = 18 ( Id + e1 + e25 + e36 + e125 + e136 − e2356 − e12356 )

(114)

We have to test the equation

f11 X = 0

(115)

Since Cl4,2 has dimension 64 the general multivector element has 64 coordinates x1, x2, …, x64 to base units Id, e1, …, j = e123456. Solving the equation (115) brings forth 8 manifolds as shown below. Each having 7 degrees of freedom while a single coordinate is determined by the seven chosen ones.

− x1 − x 2 − x15 + x19 − x 25 − x 29 + x55 + x60 = 0 − x3 + x6 + x8 + x11 − x35 − x 41 − x 45 − x51 = 0 + x 4 − x7 + x9 + x12 − x34 − x38 − x 44 − x48 = 0

42

Bernd Schmeikal

− x5 + x10 + x36 + x 40 + x 46 + x50 + x63 + x64 = 0 + x13 + x16 + x18 + x 22 + x 23 − x 26 + x 28 + x32 = 0 − x14 + x 20 − x 24 − x30 + x54 − x57 + x59 − x62 = 0 + x17 − x 21 − x 27 − x31 − x53 + x56 − x58 + x61 = 0 − x33 − x37 + x39 − x 42 − x 43 + x 47 + x 49 − x52 = 0

(116)

Consider the first equation. It is in correspondence with the first row in table 4. This characterized the Cartan algebra of L(2). Now it stands for the Cartan algebra in L(6). Consider Table 5. Cartan manifold for f11 X = 0 Manifold

− x1 − x 2 − x15 + x19 − x 25 − x 29 + x55 + x60 = 0

Base units occupied

{Id } , η0 = { e1 , e25 , e36 , e125 , e136 , e2356 , e12356 }

Those unit vectors are exactly the commuting base units which constitute the invariant neutrino pure state (114) of SU(7). Following the bottom up construction employed by Lipkin, we obtain a W-charge for L(6) which corresponds perfectly with a field hypercharge number operator W constructed by bilinear products of creation and annihilation operators, namely

w=

1 (e 14 1

+ e25 + e36 + e125 + e136 + e2356 − 6e12356 ) constituted by

W = 17 (a †d a d + au† au + a †s a s + a c† a c + ab† ab + a t† a t − 6a †w a w )

(117)

We have added to flavor a new quantum number w . The W-charge measures one seventh the difference between the number of subnuclear particles with SU(3) baryon number ⅓ and six times the number of w-fermions. We have W = 0 when the average number of quanta constituting the six bottom multidirections equals the contribution to the top spacetime volume e12356, the “top multivector unit” in the Cartan algebra. There is W > 0 when the average number of quanta on the u-, d, s, c, b, t flavor-directions is larger than on the wvolume, and Y < 0 when w-fermion energy dominates. Thus as in the case of SU(3), quantity W measures departure from spherical symmetry and respectively deformation of the field. Following the SU(n) construction plan, we have added a new, say, very strange quantum number ‘beyond’ YO = − a w a w such that the w-fermion has ‘beyondness’ −1 while the other †

fermions have YO = 0. So the W-charge satisfies

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Lie Group Guide to the Universe

W=

1 B + YO 7

(118)

The algebra contains 7 creation- and 7 annihilation operators for spacetime oscillators and respectively subnuclear fields. Those bring on 49 bilinear products. The Lie algebra L(6) is of rank 6 since we can now find six Clifford numbers which commute with one another. There is a total number operator B together with the following 6 commuting Clifford-charges. See the bottom up construction: Table 6. Bottom up construction of L(6) and commuting SU(n) charges

τ 0 = 12 (a †d a d − a u† au )

SU(2) ⊂ L(6)

t 3 = (e25 − e125 ) 1 4

Y = 13 (−2a †s a s + a †d a d + au† au )

y = (−2e1 + e25 + e125 ) = 1 6

1 3

SU(3) ⊂ L(6)

λ8 ,

Z = 14 (a †d a d + au† au + a †s a s − 3a c† a c ) z = 18 ( + e1

SU(4) ⊂ L(6)

+ e25 + e36 − 3e125 )

U = 15 (a †d a d + au† au + a †s a s + ac† ac − 4ab† ab ) 1 u = 10 (e1

SU(5) ⊂ L(6)

+ e25 + e36 + e125 − 4e136 )

V = 16 ( a†d ad + au† au + a†s a s + ac† ac + ab†ab − 5at† at ) 1 v = 12 (e1

SU(6) ⊂ L(6)

+ e25 + e36 + e125 + e136 − 5e2356 )

W = 17 (a †d a d + au† au + a †s a s + a c† a c + ab† ab + a t† a t − 6a †w a w ) w

1 = 14

SU(7) ⊂ L(6)

(e1 + e25 + e36 + e125 + e136 + e2356 − 6e12356 )

Clearly, we could just as well have used any other linear combination of these quantities. The reason for this particular choice follows the argument given by Lipkin in [21 a, b]: namely to allow the subgroups SU(2), SU(3) until to SU(6) to be used in the classification of the SU(7) multiplets, one after the other. Subtracting the six plus one (for B) there remains the †

†

†

†

algebra of 42 bilinear products having forms like a d au , au a d , …, a d a w , a w a d and so forth. All those operators which annihilate a baryon and create a beyond state or annihilate a beyond and create a baryon, change the eigenvalue of W, and respectively w, by ±1. There are 12 of them; the other 30 commute with W. From the expression of W in table 6 we conclude that it takes eigenvalues W . . . n, n ± 17 , n ± 72 , n ±

3 7

with integer n

(119)

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Bernd Schmeikal

With those eigenvalues there correspond seven different types of SU(7)-multiplets. In analogy with the SU(3) decuplet in figure 6 we can also classify baryons according to their beyondness YO and W. Then we combine 7 baryons and vary YO between 0 and 7 and obtain W = 1, 0, −1, . . ., −5. The 7-baryon combination forms a 1716-plet. It is important to realize †

that the operators such as a d a w which change the eigenvalues of W by ±1 not only change YO, but also change the SU(3) quantum number Y by one third. Any application of a bilinear generator changing YO entails a top down amendment of quantum numbers from W down to V, U, Z and Y. For a group L(6) having rank 6 the drawing of multiplet diagrams in 2 dimensions is not feasible. But one first has to specify a specific question, and then look for a means of representation. Modelling spacetime and matter in Clifford algebra has an enormous advantage. First of all, we can understand how the standard model symmetries come upon, secondly we can literally see how the spacetime oscillators bring about both subnuclear matter and spacetime with one stroke, thirdly, all the states we use are pure states, that is, primitive idempotents in the Clifford algebra. Further, the whole dynamics of HEPhy is bound to the existence of a heavy neutrino which is constituted by the basis multivector spacetime oscillators. We have def

w0 = 12 (1 + e1 ) 12 (1 + e25 ) 12 (1 + e36 ) wimp

(120)

identical with the primitive idempotent f11 in equation (114). The wimp annihilates its Lie group and becomes invisible to interaction. It partakes only in electroweak interaction. By balancing out the contributions of the constitutive oscillators it segregates from strong interaction. Thereby it switches off for itself the generalized Pauli principle which holds for fermions. As untouchable as the wimp appears to be in the Clifford algebra, as cool and invisible it may appear to us in outer space. But the reason why this model works so well is less in its objectivity, - since there are no objects beyond our experience, - but in the fact that it can be thought so well. The central equation is indeed »wimp×L(6) = 0 «. If we start from this equation we can construct a concrete form of the Lie algebra and its group just as we have done in the case of L(2) for the SU(3). Clearly, the form (119) is bound to the standard representation. The Lie group L(6) unfolds from this form a manifold having the dimension of the positive definite subspace dim(P4,2) = 36 [33, p. 128, table 4]. So we have a manifold of wimps just as we must have a manifold of fermions, baryons and bosons. But there is not only one such manifold, but there are 24 equivalence classes which arise from the permutation of the 6 indices in the standard representation of Cl4,2. In the Minkowski spacetime algebra Cl3,1 we had only six leptons, in the Cl4,2 we have 24 of them. This can easily be verified by realizing the matrix representations of Cl3,1 ∼ Mat(4, ℝ) and Cl4,2 ∼ Mat(8 ℝ). Matrices of Cl4,2 are 8× 8 and have 4 times as many elements as Cl3,1 matrices.

Lie Group Guide to the Universe

45

Towards a New Concept of Motion: Appendix A Most of us are aware that groups may work pretty independent of the equations of motion. But the discovery of the universal graded color and flavor rotation T is so comprehensive that it advises a revision of our concepts. Understanding the meaning of Clifford algebra is not a trivial thing. To understand that the gamma matrices are indeed unital base elements in geometric Clifford algebra was a first step. But it took quite a while until the concept of bivector was fully comprehended. The use of bilinear exterior forms such as e12 or γ12 has been probed in depth by David Hestenes. For instance in Quantum Mechanics from Self-Interaction [55] Hestenes showed how the Dirac equation yields two distinct plane wave solutions for a particle with a definite proper momentum p and spin polarization e3. Those are the electron- and positron plane waves which satisfy p² = m²c²

Ψ ∓ = const. e

± γ 12 p . x / ℏ

(120)

The bivector e12 = e1∧e2 is the directed unit area which squared gives −1 and therefore often merely replaces the imaginary unit. The situation becomes more complicated as we have to interpret multivectors with a positive signature such as e24 in Cl3,1 or e25, e125 in Cl4,2. Those are space-like but contain time as a factor. They have their own algebra as depicted in chapter 14 and give rise to some peculiar logic of quantumchromodynamic projectors [56]. From a thorough investigation of the algebras of spacetime it has become evident to me that the phenomena of qcd have a definite origin in the organization of space and time. This origin has been scanned in all my recent works until to this one. Let me first point out how the Pauli principle can be realized by the following motion picture

Figure 8. Time wrap by universal flavor rotation T in Cl4,2.

46

Bernd Schmeikal

The generalized Pauli principle is realized by carrying a space-like unit vector e1 to a space like directed spacetime area e25. This area cannot be interpreted as a (hyper)complex number, but squared it gives +1. It behaves almost like a space unit vector. Almost, because there is a difference, namely, it commutes with the other space unit vectors unequal e2. Therefore we need a new understanding of quantized oscillation on the area e25. The same argument holds indeed for the smaller algebra Cl3,1. What is the phenomenological difference between a unit time vector and a unit spacetime area? The answer is in both mathematics and phenomenology. The unit e5 (or respectively e4 in Cl3,1) stands for real time, time in motion, or what we call in anthropology the diachronic time. Wrapped time such as e25, e126 (or e24 in Cl3,1) represent synchronous time. Both diachronic and synchronous time are bound to material structures. Both are in motion. But diachronic time implies linear order whereas synchronous time is motion beyond linear order. The synchronous time can best be understood as a “pattern in the here and now”. Anthropologists have invented the idea. A concert unfolds in time, the man playing the concert grand needs time. His fingers move in time. But the score is a synchronous arrangement. A chromosome is a synchronous structure. It is not by fortune that the nucleotide triplets can be represented by an SU(3)-64-plet. A synchronous structure helps energy to unfold a temporal process. The genes represent synchronous templates for the diachronic process of morphogenesis. Counting is a life process, diachronic, but natural numbers are synchronous structures which regulate the counting. The synchronous pattern triggers the diachronic time-evolution of events. That’s the fundamental difference we need to start with. Let us try to understand this with the most simple real plane waves. Consider only two directions, namely the space unit e1 with coordinate x and the time, say, e5 measured by some real ct. Consider a classical real plane wave

Ψ ( x, t ) = Ψ0 sin(ωt − kx + ϕ)

(121)

with circular frequency ω, angular wave number k and phase ϕ.

Elements of Motion When the phase is zero and kx = ωt the wave reaches maximum amplitude Ψ0. For light waves we have c = ω/k. We use to interpret this image by fixing either the locus x or the point in time t. At any fixed x such as x = 0 the wave function Ψ performs an oscillation Ψ0 sin (ωt + ϕ).

The Synchronous On the other hand, if we fix time, we obtain a periodic spatial pattern

Ψ ( x) = Ψ0 sin(− kx + ϕ) at fixed time t = 0

(122)

This pattern represents the time process incarnated in a synchronous spatial structure.

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Lie Group Guide to the Universe

The Universal Trigonal Rotation Next consider the universal flavor rotation acting on a vector in the Minkowski space ℝ3,1 and respectively Cl3,1 T: x e1 + t e4 → x e24 − t e123 → x e124 + t e1234 → x e1 + t e4

(123)

Therefore the periodic pattern alongside the unit vector e1 is turned into a periodic pattern alongside the spacetime area e24 (recall figure 6). However, while x increases linearly, the area measure increases with its quadratic form. The equidistant nodal points turn into closed nodal lines with decreasing distance. Yet, all three quantities xe1, xe24, xe124 behave the same way despite the grading. They commute and allow for 1-norm dynamics (see table 3 in chapter 14, and [56]). As long as we consider motion within one algebraically closed color space such as ch1, those quantities are absolutely equivalent.

Pattern Convolution and Deconvolution Consider spaces such as ch1 or plane spanned by {e1, e4} There are various movements we want to consider, first the movements of elements in those spaces, second the movement of locations, such as the oscillation of spacetime areas, third the time evolution of scalar functions such as (121). In case of Clifford algebra, the movement of elements is the most general and comprises all three. A pattern in ch1 projected onto e1 may look like

Figure 9. Standing wave pattern alongside

e1 .

By a process (123) two things are ensured. First during strong force action the Pauli principle is preserved. Second the wave pattern on e1 is carried onto a space-like extension quantity e24 and further to e124. To see what happens thereby we must not only understand the meaning of an oriented space area, - a bivector which squared gives −1, - but also that of a space-like bivector which squared gives +1. This bivector which is the exterior product of a space-like vector e1 and a time-like vector e4, does not behave like a bivector e12 in the euclidean Pauli algebra. See the difference in figure 10! The pattern can be deconvoluted further onto the spacetime volume e124. By T it can also be convoluted back into e1. You see, the generalized Pauli principle offers an impressive degree of freedom for qcd. The situation becomes even more interesting once we realize the deep origin of those degrees of freedom which transpose base units of the Clifford algebras

48

Bernd Schmeikal

independent of grade. Those we have denoted as involutive transposition automorphisms [20]. The whole standard model has its deep origin in these finite automorphism groups which, in a sense, stem from the abeyance of nature where the identity of base unit vectors is concerned. Wallace [54] justly cited my conviction that at its deepest level, nature does not distinguish between multivector components. Thereby I mean that there is a natural uncertainty as to whether the observer encounters a space line element or an extension area or a spacetime extension volume.

Figure 10. Convolution of spatial pattern in a spacetime extension.

Despite the reflections 1−2f (with idempotent f) some of the most important transpositions in the Clifford algebras are the Weyl reflections [20, p. 356]. These transpose the Euclidean unit vectors onto each other. Weyl reflections contribute to the elements of motion in qcd. Namely, suppose there is convoluted into one of the euclidean base units e1, e2, e3 some synchronous pattern originating from areas or respectively volumes e14, e24, e34, e124, e134, e234. Such a pattern can be distributed by the Weyl automorphisms among the three manifolds equivalent with base units e1, e2, e3. Once a pattern has become imprinted into the volume e123 by the generalized Pauli principle it is transposed onto real synchronic time by strong interaction. In order to see this process more clearly, we have to investigate what I called the Zeit Dreibein or orthogonal chromatic time-like space or time 3-space.

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49

The well known Cartan structure η0 is not repeated, yet reflected in the temporal triangle e4, e123, e1234. Commutators of these quantities do not vanish, but bring on a closed Lie subalgebra, while their anticommutator vanishes. These base units form an orthogonal Zeit Dreibein in Cl3,1, just like the unit vectors of Euclidean 3-space. Observe the following relations carefully: Table 7. Spinoff of an orthogonal chromatic time-like space

If this structure has any physical meaning, - and I feel it has a very deep meaning, - a local temporal oscillation can be fully transposed onto a periodic pattern in volume e123 and respectively spacetime volume e1234. It would further mean that in strong interaction a field may be excited and thereafter folded into and deconvoluted from periodic spacetime structures. This could be the explanation for the appearance of various unexpected effects like a light velocity spectrum, violation of RT [57] and principles of RT, or quantum vacuum forces [58] which alter the metric and orientation of time. When I first became aware of these matters, I suddenly saw the empirical meaning of the decomposition theorem by Banach and Tarski [59], [60]. If the actualities of the climatic situation permit some more research in HEPhy and if we regain our scientific virtue and become a bit more careful, then one of our next discoveries might be some interesting non-decidability: as a matter of principle where extension and age of the universe are concerned. Creation is not compatible with the concept of nature in a stable isometric spacetime.

50

Bernd Schmeikal

Relativity Revisited: Appendix B While I laid down the main streams of thought for this work, Professor Zbigniew Oziewics has gone further with his investigation of the foundations of physics [61]. This concerns the isometry of Lorentz group-invariance and the Relativity principle. Let us go into this and find out about some relation between our works. If such Lie algebras like L(2) and respectively L(6) elucidate the emergence of the qcd Pauli principle and transformations of the standard model of HEPhy, they will also explain convolution and deconvolution of scale invariant or conformal phenomena of motion. This would connect the model of spacetime with the decomposition theorem by Banach and Tarski as well as conformal transformations. It would confirm the view of categorical relativity as is elaborated by Oziewicz [62]. Categorical relativity theory describes transformations of massive observer fields, - or massive reference systems, - by a groupoid. The groupoid approach to relative motion does not automatically foreclose the quadratic Clifford algebraand Lie group- Ansatz by two main reasons. The Clifford algebra as a mathematical instrument is a unital graded algebra, mostly applied in its associative versions, but never commutative and providing extremely rich spaces of representation. It is, moreover, not exactly the same as the symmetry group of the metric tensor. Second, the groups investigated here represent auxiliaries for classification and counting. They are cognitional with Lipkin’s Lie groups for pedestrians, yet on a geometric algebra level of conception. The groupoid theory of relativity has more degrees of freedom and also incorporates many more representation categories than the Lie group approach. That is, we shall have to specify the relevant conditions which legitimate the geometric algebra classification within the groupoid and differences as well as deviations that become important. In cases where reciprocity v∧v−1 = 0 is violated, v denoting velocity between two massive observer fields, we should explore the magnitude of the difference. At present we are aware that in the categorical relativity the non-isometric transformation of electric fields is slightly different from that of the magnetic fields. Oziewicz has reviewed a series of works so to say pre-historic to Relativity, beginning with Voigt (1887) [63] and Heaviside (1888) [64] which implicitly or explicitly disclose that the Relativity principle and observer-independence of the speed of light are not the same as the isometries of the metric tensor field in four dimensions. But the principles of Special Relativity are independent of the isometric Lorentz invariance. Oziewicz points out that the metric tensor field was not explicit in the Einstein 1905 paper [65]. But it was Minkowski who observed the isometric invariance in the transformation equations deduced by Einstein in 1908. It is true that observer independence as is postulated by the RP and Lorentz-invariance of equations of motion are essentially different concepts. Oziewicz correctly verifies that the Lorentz- and Poincare groups are isometry groups for a metric tensor in an empty spacetime. They do not connect massive reference systems. Consequently he insists on calling the traditional belief a dogmatic trend [61] something which can be comprehended by all means if one studies the papers he quotes. Oziewicz often repeats their calculations by nowadays instruments in order to decide upon their correctness and legitimacy. Special relativity is very essentially based on a concept of ternary velocity, a fact which we have not been fully aware of once we had the tool-box provided by the representations of the Lorentz-groups. Repeating the rigor by Voigt 1887, Oziewicz confirms, Voigt had proven that the d’Alembert-Laplace wave equation for a scalar field in two dimensions is

Lie Group Guide to the Universe

51

conformally invariant. Next, he states, Cunningham (1909/10) and Bateman (1910) already had verified, the Maxwell equations in four dimensions are also conformally invariant. In Voigt’s proof the d’Alembert operator acquires a scalar factor.11 Voigt considered coordinate transformations with real constants a, c, v, and f = a²(1-(v/c)²) ∈ℝ

x ′ = a ( x − vt ) dt ′ ∧ dx ′ = fdt ∧ dx

t ′ = a (t − vx / c 2 )

(124)

See the appearance of exterior products of spacetime differentials involving relative coordinate times of local clocks

dt ′ ∧ dt ≠ 0

(125)

Hence, Oziewicz accentuates, the inverse transformation for f ≠ 1 is not reciprocal,

x = af

−1

( x ′ + vt ′)

t = af −1 (t ′ + vx ′ / c 2 ) in other terms

{x, t , x ′, t ′, v} ≠ {x ′, t ′, x, t ,−v}

The d’Alembert-Laplace operator acquires a scalar factor f. The metric tensor on vector fields is

g −1 = −c 2 dt ⊗ dt + dx ⊗ dx = f

−1

{− c dt ′ ⊗ dt ′ + dx ′ ⊗ dx ′} and 2

2 2   ∂  2  ∂   ∂  ∂  − + f =  d’Alembert-Laplacian     + −    ∂ct ′   ∂x ′   ∂ct   ∂x 

So the wave equation is invariant relative to Voigt’s transformation and thus preserved by conformal transformations in two dimensions. Equation (125) was one of the reasons why I introduced a second time coordinate and chose Cl4,2 instead of Cl4,1. A second important cause is in my belief in the existence of a constant symmetry breaking giving rise to the emergence of a proper time arrow contrasting an almost vanishing reverse arrow. The third reason that motivates me to favour that algebra is in the process of deconvolution of antagonistic advanced and reverse waves. This arose already in electroweak interaction. It is of even greater importance in strong force events. Sure my readers have realized the astonishing jump from the special unitary groups SU(4) and SU(3) relevant in the Clifford algebras Cl3,1 and Cl4,1 to SU(8) and SU(7) in the 11

This rigor is quoted after Oziewicz [61] section 1.1

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Bernd Schmeikal

Cl4,2 involving a de Sitter space adS6 ⊂ γλ(Cl4,2) for orthogonal transformations of vectors and a Lie group L(6) for orthogonal transformations of the pure state manifolds. It is fascinating to observe how far we can backtrack the meaning of bilinear exterior forms such as f dt ∧ dx, dt ∧ dt’, x ∧ t and unitals e1 ∧ e4. The Lorentz scalar factor γv = 1/(1−v²/c²)½ was introduced by Oliver Heaviside in order to relate electric fields in mutual relative motion. The same factor was introduced one year later to Heaviside in a letter by George Francis FitzGerald [61, section 2.1] and independently by Hendrik Antoon Lorentz in 1893. In 1887 he used it to explain the results of the Michelson-Moreley experiment disproving the existence of the aether drift [66]. Oziewicz [61] evaluated a re-derivation of the Heaviside transformation by Hajra and Gosh (2005) [67]. “The re-derivation starts from the following formulas for the constant relative velocity u [67, formulas (11)-(12)]

dx = udt ⇒

x − ut = constant ⇒

∂ ∂ =u ∂t ∂x

∂x 0 if x and t are independent = ∂t u if x and t are dependent Therefore the coordinates x and t here are not independent variables, contrary to the assumption made on page 64. The derivation of the Heaviside transformation […] seems to need some more justification in differential geometry”. On 11th November 1903, the Proceedings of the London Mathematical Society received a paper written by E. T. Whittaker [68] in which he derived the electromagnetic field equations by means of two scalar potential functions instead of and collateral to the scalar and vector potential function. He could not refer to any principle of relativity and did not use the Lorentz transformation. Still he obtained two typically Lorentz invariant massless Klein Gordon equations for - we would say today - longitudinal and time-like scalar field components of the four potential, the second quantization of which straightforwardly leads us to longitudinal massless bosons, quanta of pure scalar energy [69], [70]. Then it was known that the equations of motion of the dielectric displacement and the magnetic field strength had to be connected with two orthogonal circular motions, the curls of the electric and magnetic fields. These curls were needed in addition to time derivatives and linear translations of charge density. Starting off with the vector potential A and Stratton potential S Whittaker calculated two scalar fields F and G capable to provide the ED equations of motion which, then, took the form

d1 =

∂ 2 F 1 ∂ 2G ∂ 2 F 1 ∂ 2G ∂2F 1 ∂2F + − − , d2 = , d3 = ∂x∂z c ∂y∂t ∂y∂z c ∂x∂t ∂z 2 c 2 ∂t 2

∂ 2G ∂ 2G 1 ∂ 2 F ∂ 2G 1 ∂ 2 F ∂ 2G , h2 = − , h3 = + h1 = − − c ∂y∂t ∂x∂z c ∂x∂t ∂y∂z ∂x 2 ∂y 2 which are equivalent to vector equations for d and h in traditional mathematical form:

(126)

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Lie Group Guide to the Universe

d = curl curl f + curl

1 1 gɺ , h = curl fɺ − curl curl g c c

(127)

Here he introduced vectors f and g parallel to the axis z with magnitudes F and respectively G. The quantities d and h are the electric displacement vector and magnetic field. Whittaker calculated explicit forms for the scalar functions of F and G. He ascertained that those would have singularities at the points occupied by the electrons and found by differentiation the Klein Gordon equations for massless boson fields away from singularities.

□F = 0

□G = 0

with

□= ∆+

1 ∂2 c 2 ∂t 2

(128)

In the context of Whittaker’s gauge theory which is no other than the Maxwell-Heaviside theory, the field equations for the scalar and vector potentials are recovered from the equations (126). We should be aware that Whittaker’s calculation is electrodynamic, yet bare of the method of isometric Lorentz transformation. From a geometric algebra perspective, going into details, we observe rotation as curls alongside the plane unit areas e13 and e23 and a surprising asymmetry of the field equations in the third vector components alongside e3 which cannot be removed. Contrasting nowadays conviction, this approach leads to three types of field quanta, namely 1.) transverse photons as well as 2.) longitudinal and 3.) timelike bosons. The latter are physically real, but show little interaction with matter. However, interference of scalar bosons leads to particle creation. Whittaker started classically but, apparently, ended up with relativistic equations of motion. However, the equations are conformally invariant and therefore not necessarily bound to reciprocity. This again indicates the importance of scale invariance. A last question concerns the use of base units in Clifford algebras. It is true that the physics in its essence is base-free and coordinate free. This is what Oziewicz emphasizes over and over again. In the beginning of section 2 in [61] he states: “An invertible endomorphism of a vector space, a transformation of a vector space, is said to be the ‘passive’ transformation, if the domain is the manifold of all basis of the vector space. ‘Passive’ means the active action on the coordinate-free basis, or an action on the coordinate basis, but not on individual vectors.” I wish to make this point clear at the end of this work. We are using the unital base elements in order to construct the equivalence classes of idempotent manifolds, Lie groups and similar objects. These mathematical elements are important for both mathematics and physics. They form manifolds, but not any distinguished basis or generators. For instance the 24 primitive idempotents which form lattices of idempotents within Cl4,2 give rise to 24 equivalence classes of lattice-manifolds strictly separate from each other. Each lattice contains 8 primitive idempotents or pure states which are cleanly disaggregated throughout their manifolds. So we distinguish 192 primitive idempotents in Cl4,2. At present I have no other means at hand to find those equivalence classes than to use the standard basis. But may be some day we shall have better tools to solve such problems of fundamental structure. To see the beauty of those structures which give rise to the Lie groups similar to SU(3), take a look at the 48 primitive idempotents in Cl4,1. The figure may be a late tribute to Harry Lipkin. May be some of us find themselves suddenly motivated to study the SU(4) multiplets

54

Bernd Schmeikal

on the cover page of this wonderful unabridged Dover edition . . . a piece of fine art. I add to it this one:

Figure 11. Bilateral orthogonality lattice with definite maximal element pure states in the Clifford algebra

f1

and tetragonal structure of

Cl 4,1 .

The 48-primitive idempotent structure of the Cl4,1 is extremely rich. It has first been used in a minimal spin gauge theory of quantumchromodynamics [48] to work out graded analoga to the su(3). Probably my readers have spied out in the meantime that I rarely use explicit equations of motion. In [33] I designated the Dirac-Hestenes Equation in order to visualize the equivalence classes of the surabale. In [56] I used the Dirac equation to demonstrate the meaning of ternary logic projector fields. I wrote this in memory of Carl Friedrich von Weizsäcker. In the abstract to this lecture I said that we are familar with the ℤ2 -grading of the Clifford algebra and the double cover of orthogonal groups. With this we associate the projector equation and decomposition of unity 1 = P1 + P0 according to spin decomposition or chirality. But we have not yet comprehended the importance of the quaternary decomposition of unity 1 = P0 + P1 + P2 + P3. A binary decomposition is characteristic for qed. It has first been used by John von Neumann and interpreted by von Weizsäcker as »logic alternative«. Weyl has for sometime pondered over the meaning of the Klein-4 group and the rays as compared with vectors. Then he could not yet realize the importance of a quaternary decomposition of unity and the K4grading. The equation 1 = P0 + P1 + P2 + P3 characterizes qcd dynamics in quite general algebras. What a binary decomposition is for qed, the quaternary is for qcd. In this paper the algebraic foundations are given by what is denoted here as maximal ternary Cartan decomposition in noncommutative algebras. The natural norm in a Cartan extension is not derived from the Minkowski metric, but from the fact that the Clifford product becomes the inner product while the exterior product vanishes. The special Clifform L3,1 = sl(4, ℝ) together with the decomposition of L(2) = slCl(2, ℝ)×soCl(3, ℝ) with a quasi relativistic factor suggests to consider the Iwasawa decomposition

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55

as in [32] and the Iwasawa-Whittaker-Liouville wave functional for the groups SL(n) together with the asymptotic Harish-Chadra function. But then, again we are led back to conformal invariance but not Lorentz-invariance. For my understanding this is not irrelevant, but rather a matter of fact and empirical probe. The Liouville type of wave equations in the SL(3) and SL(n) framework has been thoroughly investigated by a group of russian scientists [71]. Their paper provides all the means necessary to solve equations of motion and scattering problems. They followed the lines of Liouville field theory, developed further the Whittaker model which is partly responsible for the quantization of the Liouville field theory. They investigated the wave functions of the open Toda model in group theoretical terms and explained the simple relation to the Whittaker function for the quantum Lorentz group [72]. Briefly spoken, the solutions to our dynamic problems are still burrowed in the Liouville quantum mechanics. They wait for their chance to deconvolute into cognitive spaces.

References [1]

[2] [3]

[4]

[5] [6] [7]

[8]

[9] [10] [11] [12] [13]

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[14] Ablamowicz, R. CLIFFORD - A Maple xx Package for Clifford Algebra Computations. http://math.tntech.edu/rafal/cliff xx/index.html. [15] Sauter, F. Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren. Z. Phys. 63, 1930, 803-814. [16] (a) Frescura, F. A. M., Hiley, B. J. The Implicate Order, Algebras, and the Spinor. Found Phys 10, 1980, 7-31. (b) Frescura, F. A. M., Hiley, B. J. Algebraization of Quantum Mechanics and the Implicate Order, Found. Phys 10, 1980, 705-722. (c) Frescura, F. A. M., Hiley, B. J. Algebras, Quantum Theory and Pre-space. Rev Brasil Fis, Volume Especial, 1984, Os 70 anos de Mario Schönberg, 49-86. (d) Hiley, B. J. Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space. http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic%20Quantum%20Mechanic%205.pdf (10 May 2003). [17] Bordemann, M., Waldmann, S. Formal GNS Construction and WKB Expansion in Deformation Quantization. In: Sternheimer, D., Rawnsley, J., Gutt, S. (eds.): Deformation Theory and Symplectic Geometry. Mathematical Physics Studies no. 20, 315-319. Dordrecht-Boston-London 1997. [18] (a) Schmeikal, B. The generative process of spacetime and strong interaction - quantum numbers of orientation. In: R. Ablamowicz, P. Lounesto, J.M. Parra (eds.): Clifford Algebras with Numeric and Symbolic Computations. Birkhäuser, Boston, 1996, 83100. (b) Schmeikal, B. Clifford Algebra of Quantum Logic. In: R. Ablamowicz, B. Fauser (eds.): Clifford Algebras and their Application in Mathematical Physics. Birkhäuser, Boston 2000, 219-241. [19] (a) Hestenes, D. Space-Time Structure of Weak and Electromagnetic Interactions. Found Phys 12, 1982, 153-168. (b) Hestenes, D. Clifford Algebra and the Interpretation of Quantum Mechanics. In: J.S.R. Chisholm, A. K. Commons (eds.): Clifford Algebras and their Applications in Mathematical Physics. Dordrecht-Boston, 1986, 321-346. (c) Hestenes, D. Universal Geometric Algebra. Simon Stevin (ed.) A Quarterly Journal of Pure and Applied Mathematics, Volume 62, 1988, No. 3-4, 1-15. (d) Hestenes, D., The Zitterbewegung Interpretation of Quantum Mechanics. Found Phys 10, 1990, 12131232. (e) Hestenes; D. A Homogeneous Framework for Computational Geometry and Mechanics. Department of Physics and Astronomy, Arizona State University, Tempe, Arizona, USA, http:://modelingnts.la.asu.edu, 2001 (12 December 2001) (f) Hestenes, D. Spacetime Calculus for Gravitation Theory. 1996 PACS, http://modelingnts.la.asu.edu/pdf/NEW_GRAVITY.pdf (20 March 2008). [20] Schmeikal, B. Transposition in Clifford Algebra. In: R. Ablamowicz (ed.): Clifford Algebras – Applications to Mathematics, Physics and Engineering. Birkhäuser, BostonBasel-Berlin 2004, 351-372. [21] (a) Lipkin, H.J. Anwendung von Lieschen Gruppen in der Physik, Mannheim 1967. (b) Lipkin, H.J. Lie Groups for Pedestrians. Dover-New York 2002. [22] Baylis, W. E. The Quantum/Classical Interface: Insights from Clifford’s (Geometric) Algebra. In: R. Ablamowicz (ed.): Clifford Algebras – Applications to Mathematics, Physics and Engineering. Birkhäuser, Boston-Basel-Berlin 2004, 375-391. [23] Kauffman, L.H. Eigenform, Kybernetes. Vol. 34, No. 1/2, 2005, 129-150. [24] Ðurñewich, M. Quantum Geometry and New Concept of Space. http://www.matem.unam.mx/~micho/qgeom.html (10 May 2003).

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[25] Lipkin, H. J. From Sakata Model to Goldberg-Ne´eman Quarks and Nambu QCD – Phenomenology and “Right” and “Wrong” experiments. [hep-ph/0701032v2], (31 January 2007). [26] (a) Woronowicz, S. L., C*-algebras Generated by Unbounded Elements, Rev Math Phys 7 (3), 1995, 481-521. (b) Woronowicz, S. L. Twisted SU(2) group: An example of a non-commutative differential calculus. Publ RIMS, Kyoto Univ, 23, 1987, 117-181. [27] Majid, S. Noncommutative Physics on Lie Algebras. In: R. Ablamowicz (ed.): Clifford Algebras – Applications to Mathematics, Physics and Engineering. Birkhäuser, BostonBasel-Berlin, 2004, 491-518. [28] Owczarek, R. M. Dirac Operator on Quantum Homogeneous Spaces and Noncommutative Geometry. In: R. Ablamowicz (ed.): Clifford Algebras – Applications to Mathematics, Physics and Engineering. Birkhäuser, Boston-Basel-Berlin, 2004, 519530. [29] Cruz Guzmán, J. de J., Oziewicz, Z. Föhlicher-Nijenhuis Algebra and four Maxwell’s Equations for Non-Inertial Observer. Bulletin de la Société de Sciences et des Lettres de Lódź, LIII, Série : Recherches sur les Déformations, XXXIX, 2003, 107-140. [30] Schmeikal, B. Algebra of matter. Advances in Applied Clifford Algebras 15, No 2, 2005, 271-290. [31] Magnea, U. An Introduction to Symmetric Spaces. Department of Mathematics, University of Torino, Italy, [cond-mat/0205288 v1] 14 May 2002 (19 August 2003). [32] Schmeikal, B. Symmetric spaces of Matter and Real Fermion Manifolds. Advances in Applied Clifford Algebras 16, No 1, 2006, 69-83. [33] Schmeikal, B. The Surabale of Spacetime and its Algebra Signature Split. Advances in Applied Clifford Algebras 17, No 1, 2007, 107-135. [34] Sakata, S. On a Composite Model for the New Particles. Prog. Theor. Phys. 16, 1956, 686-688. [35] Goshen, S. Lipkin, H. J. A Simple Independent-Particle System having Collective Properties. Ann. Phys. (NewYork) 6, 1959, 301-309. [36] Lipkin, H. J., Pairing and quadrupole forces in a two-dimensional soluble model. Nucl. Phys. 26, 1961, 147-160. [37] Ne’eman, Y., 1961, see [3a] [38] Goldberg, H., Ne’eman, Y. Nuovo. Cim. 27, 1963, 1. [39] Ne’eman, Y. The spectrum generating group program and the string. Found Phys. 18, 1988, 245-275. [40] Fritzsch, H. Mesons, Quarks and Leptons. [hep-ph/0207279 v1] 23 Jul 2002, *To be published in a commemorative volume at Bologna University in honor of N. Zichichi (4 September 2003). [41] Zweig, G. private communication to Harald Fritzsch: quoted in [38]. [42] Connolly, P. L., et al. Existence and Properties of the φ - Meson. Phys. Rev. Lett. 10, 1963, 371. [43] Greenberg, O. W. Spin and Unitary-Spin Independence. Phys. Rev. Lett. 13, 1964, 598602. [44] Nambu, Y., Jona-Lasinio, G. A dynamical model of elementary particles based on an analogy with superconductiviy. part 1, Phys. Rev. 122, 1961, 345, part 2, ibid., 124, 1961, 246.

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[45] Nambu, Y. A three triplet model with double SU(3) symmetry. Phys. Rev. 130, 1965, B 1006. [46] Elliott, J. P. The Nuclear Shell Model and its Relation with Other Models. in: F. Janouch, ed., Selected Topics in Nuclear Theory, IAEA, Vienna 1963. [47] particle data group, website: http://pdg.lbl.gov/2006/reviews/kmmixrpp. [48] Schmeikal, B. Minimal Spin Gauge Theory. Clifford Algebra and Quantumchromodynamics. Advances in Applied Clifford Algebras 11, No 1, 2001, 6380. [49] Siegel, W. Fields. December 1999 [hep-th/9912205] (19 August 2003). [50] Cohen, R. S.; Ed.; Ernst Mach – Physicist and Philosopher. Dordrecht 1975. [51] (a) Randall, L. Warped Passages. Unraveling the Mysteries of the Universe's Hidden Dimensions. New York 2005. (b) Randall, L., Sundrum, R. Large Mass Hierarchy from a Small Extra Dimension. R. Phys Rev Lett 83, 1999, 3370-3373 [hep-ph/9905221]. (c) Randall, L., Sundrum, R. An Alternative to Compactification. R. Phys Rev Lett 83, 1999, 4690-4693 [hep-ph/9906064]. [52] Dvergsnes, E. The Randall-Sundrum Radion: Production Through Gluon Fusion, and Two Photon Decay. Thesis, Department of Physics, University of Bergen, Norway 2000. http://hdl.handle.net/1956/1801 (4 September 2003). [53] Higgs, P. W. Broken Symmetries and the Masses of Gauge Bosons. Phys Rev Lett 13, 1964, 508-509. [54] Wallace, G. W. The Pattern of Reality. Advances in Applied Clifford Algebras 18, No 1, 2007, 115-133. [55] Hestenes, D. Quantum Mechanics from Self-Interaction. Found Phys 15, No. 1,1983, 63-87. [56] Schmeikal, B. Pregeometry of Extensions and Eigenfields. Prepared for the Conference on Clifford Algebras and their Applications in Mathematical Physics in Brazil, May 2008, In memory of Carl Friedrich von Weizsäcker. [57] Unnikrishnan, C. S. Precision measurement of the one-way speed of light and implications to the theory of motion and relativity. Gravitation Group/FI-Lab, Tata Institute of Fundamental Research, Mumbai, India [2006]; personal copy, available: [email protected]. [58] Maclay, J., Hammer, J. George, M., Ilic, R. Leonard, Q., Clark, R. Measurement of Repulsive Quantum Vacuum Forces. Published as: AIAA/ASME/SAE/ASEE 37th Joint Propulsion Conference, Salt Lake City, July 2001, 1-9. AIAA-2001-3359. [59] Banach, S., Tarski, A. Sur la décomposition des ensembles de points en parties respectivement congruents. Fundamenta Mathematica 6, 1924, 244-277. [60] Stewart, I. The ultimate jigsaw puzzle. in: New Scientist 13, April 1991, 30-33. [61] Oziewicz, Z. Electric field and magnetic field in moving reference system. International Symposium on Recent Advances in Mathematics and its Applications, Calcutta, India, December 2006. [62] Oziewicz, Z. What is categorical relativity? International Journal of Geometric Methods in Modern Physics 4, (1), 2007, math. CT / 0608770 [63] Voigt, W. Über das Doppler’sche Prinzip. Nachrichten Ges Wiss Göttingen 41, 1887. [64] (a)Heaviside, O., The electro-magnetic effects of a moving charge. The Electrician 22, 1888, 147-148. (b) Heaviside, O. Electrical Papers. Providence, R.I., AMS Chelsea Publishing 2003.

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[65] Einstein, A. Zur Elektrodynamik bewegter Körper. Annalen der Physik (Leipzig) 17, 1905, 891-921. [66] (a) Lorentz, H. A. Zichtbare en onzichtbare bewegingen. Leiden 1901. (b) Lorentz, H. A. Lectures on Theoretical Physics. (vol. I-III), Macmillan and Co, London 1931 (c) Lorentz, H. A. The intensity of radiation and the motion of earth. Proceedings of the Royal Netherlands Academy of Arts and Sciences 4, 1901-1902, 678-681. [67] Hajra, S., Ghosh, A. Collapse of SRT 1: derivation of electrodynamics equations from the Maxwell field equation. Galilean Electrodynamics 16, (4), 2005, 63-70. [68] Whittaker, E. T. On an Expression of the Electromagnetic Field due to Electrons by means of two scalar Potential Functions. Proceedings of the London Mathematical Society 1, 1904, 367-372. [69] AIAS, Authors. Representation of the Vacuum Electromagnetic Field in Terms of Longitudinal and Time-like Potentials: Canonical Quantization. Journal of New Energy 4, No 2, 2000, 82-91. [70] AIAS, Authors. An Experimental Test of the Existence of Whittaker’s g and f Fluxes in the Vacuum. Journal of New Energy 4, No 2, 2000, 92-96. [71] Gerasimov, A., Kharchev, S., Mershakov, A., Mironov, A., Morozov, A. Olshanetsky, M. Liouville Type Models in Group Theory Framework I. Finite-Dimensional Algebras. ITEP M4/TH-7/95, FIAN/TD-18/95, [hep-th/9601161 v1] (24 January 2003). [72] Olshanetsky, M., Rogov, V. Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group. Journ. Phys. Math. Gen. 27, 1994, 4669-4683.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 61-88

ISBN: 978-1-60692-389-4 © 2009 Nova Science Publishers, Inc.

Chapter 2

ROTATION MANIFOLD SO(3) AND ITS TANGENTIAL VECTORS Jari Mäkinen∗ Tampere University of Technology, Department of Mechanics and Design, FIN-33101 Tampere, FINLAND

Abstract In this paper, we prove that incremental material rotation vectors belong to different tangent spaces of the rotation manifold SO(3) at a different instant. Moreover, we show that the material tangent space as the tangent space at unity is not a possible definition yielding geometrically inconsistent results, although this kind of definition is widely adopted in applied mechanics community. In addition, we show that the standard Newmark integration scheme for incremental rotations neglects first order terms of rotation vector, not third order terms. Finally, we show that the rotation interpolation of extracted nodal values on the rotation manifold is not an objective interpolation under the observer transformation. This clarifies controversy about the frame-indifference of geometrically exact beam formulations in their finite element implementations.

Keywords: finite rotation, rotation manifold, rotation interpolation, objectivity, Newmark scheme.

1. Introduction A finite rotation (See [2] [3]), is a vector quantity, or more precisely, the finite rotation belongs to a tangent space on a manifold. This manifold is a Lie group of the special orthogonal tensors SO (3) , also called the manifold of finite rotations or shortly the rotation manifold. In general, Lie-groups are noncommutative groups, which are also differentiable manifolds such that their differentiable structures are compatible with their group structure for the definitions of general Lie-groups, (See details in [13]). As it is shown in [14], material ∗

E-mail address: [email protected]; Fax: +358 3 3115 2107; Phone: +358 3 318 3851

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Jari Mäkinen

incremental rotation vectors, material angular velocity vectors, and material angular acceleration vectors belong to the different tangent spaces of the rotation manifold SO (3) at a different instant. Hence, the direct application of the material incremental rotation vector, with standard time integration methods, yields serious problems: adding quantities which belong to the different tangent spaces. This approach, the standard Newmark integration scheme for incremental rotations, is widely adopted and comes from [19] and is also the point of departure for [9] and is recently used in [18]. In the following, we name this Newmark scheme for incremental rotations as the simplified Newmark scheme. There are two main approaches for the time-integration of finite rotation with a threeparametric presentation. One approach, which is called Eulerian formulation [4], directly applies an incremental rotation vector, angular velocity vector and angular acceleration vector in the formulation. This approach suffers from problems cited above, if standard timestepping schemes are used. The modified Newmark time-stepping for the Eulerian formulation scheme, which overcomes this difficulty, is given in [14]. Another approach, which is referred to as the updated Lagrangian formulation [4], applies a current reference placement, which is updated incrementally. In this approach, updated rotation vectors and their time derivatives belong to the same tangent space of rotation and do not suffer problems, as found in the Eulerian formulation. It is shown in [14] that the modified Newmark time-stepping scheme and the updated Lagrangian formulation with the standard Newmark scheme are equivalent time stepping methods. Recently, in the paper [18] it is shown that the simplified Newmark integration scheme neglects third order terms of rotation vector. However, the proof suffers from the fact that the material incremental rotation vector and its time derivatives cannot be integrated directly by the Newmark scheme. We show that the simplified Newmark integration scheme for finite rotations neglects first order terms of rotation vector, as opposed to third order terms [18]. Moreover, we could present the third approach, called total Lagrangian formulation [4] and [16], where a reference placement is permanently the initial placement and a total rotation vector and its time derivatives are used as unknown variables. Well-known singularity problems at full-angle and its multiples can be bypassed by introducing a complement rotation vector [16]. A rotation vector and its complement rotation vector are the parameterization charts of the rotation manifold SO (3) . We could represent the rotation manifold globally with these two parametrization charts. When a rotation angle exceeds straight-angle, we accomplish the change of parametrization, giving a new rotation angle smaller than a straight-angle. Thus, we get out of the singularity problems at full-angle. Objectivity, or frame-invariance, is another issue that we are considering. We show that the interpolation of the total rotation vector is an objective interpolation under the observer transformation. Regardless, contrary results are also given in [6] where, with the aid of counterexample, it is shown that the finite element discretization violates objectivity. This counterexample suffers serious problems: extracted nodal vectors are interpolated, which way is not generally allowed and is never used in the finite element discretization of the total and updated Lagrangian formulations. In connection with this, we show that the rotation interpolation of extracted nodal values on the rotation manifold SO (3) is not an objective interpolation under the observer transformation. This paper is organized as follows: We give some necessary definitions and results about manifolds and their tangent spaces in Chapter 2. In Chapter 3, we give generalized definitions

Rotation Manifold SO(3) and Its Tangential Vectors

63

for the rotation manifold SO (3) ; and we prove that the material incremental rotation vector belongs to different tangent spaces of the rotation manifold SO (3) at a different instant. In Chapter 4, definitions for angular velocity and acceleration vectors are given. In Chapter 5, Euler equations for rotation motion, which are differential equations on the tangent bundle, are given in the material and spatial representations. In Chapter 6, we show that the simplified Newmark integration scheme for incremental rotations neglects first order terms of rotation vector. Finally, we show that the rotation interpolation on the rotation manifold is an objective interpolation under the observer transformation.

2. Manifolds and Their Tangents In this Chapter, we introduce mathematical preliminaries that are necessary to understand the rotation, angular velocity, and angular acceleration vectors, which are vectors on the rotation manifold SO (3) . Some elementary knowledge of differential geometry [22] is necessary to understand the rotation vector that is a vector of a tangent space of a manifold. The rotation manifold SO (3) is a Lie-group of special orthogonal tensors (See textbooks on manifolds and Lie groups [13], [12], [1], and [5].) In Figure 1, a fundamental notion is introduced for a differentiable manifold. A differentiable manifold can be mapped from a chart in a parameter space into a chart of manifold in an embedding space. The change of parameterization is differentiable for differentiable manifolds. Definition 2.1. (manifold) A set M ⊂ E is a manifold with dimension d if there exists n

a bijection ϕ i : Ui → E from an open domain Ui ⊂ E n

d

in a d -dimensional Euclidean

parameter space onto some open set in the manifold, ϕ i : Ui → ϕ i ( Ui ) ⊂ M , such that every point of the manifold is an image under a mapping, (See [Figure 1]). A pair (Ui , ϕ i ) is called a chart or a parameterization chart. Definition 2.2. (differentiable manifold) A manifold M is a differentiable manifold if for every point x ∈ M there exist images ϕ1 (U1 ) and ϕ 2 (U2 ) where the point x ∈ M belongs to such that the composite map

ϕ 2−1
ϕ1 is a diffeomorphism from

ϕ1−1 (ϕ1 (U1 ) ∩ ϕ 2 (U2 )) onto ϕ 2−1 (ϕ1 (U1 ) ∩ ϕ 2 (U2 )) . The composite map is called the change of parametrization (See Figure 1). A mapping is a bijection if it is injective and surjective, i.e. one-to-one and onto mapping, and a diffeomorphism is a bijection with a continuously differentiable mapping and its inverse mapping. We note that generally a chart map is defined by an inverse map from an open set of a manifold into a parameter space.

64

Jari Mäkinen

Ì E

m a n ifo ld M

n

j

j 1

U

1

bU g Ç

c h a n g e o f p a ra m e triz a tio n j 2- 1 o j 1 1

p a ra m e te r s p a c e E

1

j 2

bU g U

2

j 2

2

d

Figure 1. Geometric interpretation for parametrization of the manifold when n = 3 and d = 2.

Definition 2.3. (tangent vector) Let ϕ (η ) be a parametrized vector-valued curve in the manifold M through the base point x ∈ M such that ϕ (η = 0) = x . The tangent of the curve (or the equivalent class of curves) ϕ (η ) at η = 0 to the manifold M is defined as

t = lim

ϕ (η ) − ϕ (0)

η →0

η

, where ϕ (0) = x, ϕ (η ) ∈ M .

The tangent vector t belongs to a tangent space of the manifold, namely t ∈ Tx M , (See Figure 2). The tangent (vector) space Tx M is the set of the tangent vectors at the base point

x ∈M . Definition 2.4. (tangent bundle) A tangent bundle T M is defined as a union of the tangent spaces on the manifold M at its every point

T M :=

∪ ( x, TxM ) .

x∈M

The dimension of the tangent bundle is twice the dimension of the manifold M . The pair of state vectors, the placement x ∈ M and its velocity vector v ∈ Tx M , belongs to the tangent bundle, ( x, v ) ∈ T M .

65

Rotation Manifold SO(3) and Its Tangential Vectors

T x M

x t

j (t)

M

Figure 2. Tangent vector t and its tangent space Tx M on the manifold M at the point x.

We note that usually a chart mapping is defined by an inverse mapping from an open set of a manifold into a parameter space. We have defined a chart mapping differently since we could use this terminology when constraint equations are parametrized. A vector space, where n

a manifold is embedded, is called an embedding space; the Euclidean space E in Figure 1.

3. Rotation Manifold and Its Tangents In this Chapter, we give conventional notation, which aids the comprehension of the geometric structure of the rotation manifold SO (3) , (See textbooks [13], [5], and [7]). A rotation motion can be represented by rotation operators R that form a group. This special noncommutative Lie-group of the proper orthogonal linear transformations is defined as

{

}

SO (3) := R : E3 → E3 R T R = I, det R = 1

(1)

where E3 indicates three-dimensional Euclidean vector space. It can be demonstrated that SO (3) is indeed a group and satisfies all the group properties with internal operation (product). Rotation tensor can be represented minimally by three parameters, which parametrize the rotation tensor only locally. It is well known that there exists no single threeparametric global presentation of rotation tensor because the rotation group is a compact group [21]. The rotation group is also a three-dimensional manifold (i.e. Lie-group) with differentiable structure. Euler angles are the most extensive three-parametric presentations in the literature of analytical dynamics. However, simpler and more useful parametrization can be obtained if the parameters are canonical, i.e. the rotation vector parametrization. We are attempting to find an expression for the rotated vector p1 in the terms of the

original vector p 0 , the unit rotation axis n , and the non-negative rotation angle ψ about the rotation axis, (See Figure 3).

66

Jari Mäkinen

r 1

r y

E 3

p

p

0

1

0

n r r r

= - n 0 1

= r

0

bn

c o s y

p

0

n

g

n 1

y p 0

r

s in y

p 0

s in y

0

Figure 3. A rotation about n -axis where p0 is the original vector and p1 is the rotated vector.

The original projection vector r0 and the rotated projection vector r1 in the rotation plane are given in Figure 3, where × denotes the cross product on E . Now, the rotated vector p1 3

can be expressed by

p1 = p0 − r0 + r1

= p0 + (1 − cosψ ) n × ( n × p 0 ) + n × p0 sinψ

(2)

p1 , p0 , n, r0 , r1 ∈ E , ψ ∈ R +

= Rp0

3

Now, the rotation operator R can be written in terms of the rotation vector that is defined by

Ψ := ψ n,

n ∈ E3 ,ψ ∈ R + .

(3)

Euler’s theorem states that any rotation tensor R is a rotation through an angle ψ about an axis n (unit vector). Thus, we have the rotation vector Ψ := ϕ n . This yields the expression of the rotation operator

R := I +

sinψ ɶ 1 − cosψ ɶ 2 Ψ+ Ψ , ψ = Ψ 2

ψ

ψ

ɶ , called the rotation tensor, is defined by the formula where the skew-symmetric tensor Ψ

(4)

Rotation Manifold SO(3) and Its Tangential Vectors

ɶ h = Ψ × h, Ψ

∀h ∈ E 3 ,

67 (5)

ɶ := Ψ × . or more formally Ψ It is well known that the rotation operator and the rotation vector are related by the exponential mapping, (See [7]; p. 70]).

ɶ ) := I + Ψ ɶ +1Ψ ɶ2+1Ψ ɶ3+ 1Ψ ɶ4… R = exp ( Ψ 2! 3! 4!

(6)

ɶ is a skew-symmetric tensor and its axial vector is the rotation vector Ψ . where Ψ ɶ ) , (See Def. 2.3.), with Differentiating the parametrized expression ϕ (η ) := exp(ηΨ respect to the parameter η at η = 0 gives the tangent of the rotation operator at the identity, yielding ɶ) d exp (ηΨ dη

ɶ =Ψ

(7)

η=0

ɶ belongs to the tangent space of the rotation Thus, the skew-symmetric tensor Ψ ɶ ∈ T SO (3) , where the identity I ∈ SO (3) represents the base manifold, the notation Ψ I point of the rotation manifold. It is clear that the base point is the identity I ∈ SO (3) since

ɶ ) with η = 0 is equal to the identity I , (See Def. 2.3.). exp(ηΨ In general, the skew-symmetric tensors form Lie-algebra with Lie-brackets defined as

ɶ , Bɶ  := AB ɶ ɶ − BA ɶ ɶ,  A 

ɶ , Bɶ ∈ so(3) ∀A

(8)

where the set of the skew-symmetric tensors so(3) are defined as

{

ɶ : E3 → E3 linear A ɶ T = −A ɶ so(3) := A

}

(9)

Note that the elements of Lie-algebra so(3) do not need to be infinitesimal quantities. Instead, they may form a vector space at the identity of the rotation group. It can be verified that the set so(3) 9 meets all the Lie-algebra properties, (See [5] or [13]).

3.1. Compound Rotation A rotation operator is an element of a Lie group that is a differentiable manifold as well as a non-commutative group. A compound of successive rotations is also a rotation itself and induces a Lie group structure with underlying Lie algebra. Compound rotation can be defined

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Jari Mäkinen

by two different, though equivalent ways: by the material description, and by the spatial description. Definition 3.1. We define the material description of a compound rotation by the left translation map Lef tR : SO (3) → SO (3) as mat mat ɶ) := RR inc Lef tR R inc = R exp ( Θ

mat , R ∈ SO (3) , R inc

mat is an incremental material rotation operator, and Θ is an incremental material where R inc

rotation vector with respect to the base point R ∈ SO (3) . This description is called material since the incremental rotation operator acts on a material vector space. Definition 3.2. We define the spatial description of a compound rotation by the right translation map RightR : SO (3) → SO (3) as

()

spat ɶ RightR R spat inc := R inc R = exp θ R

R spat inc , R ∈ SO (3) ,

where R inc is an incremental spatial rotation operator, and θ is an incremental spatial spat

rotation vector with respect to the base point R ∈ SO (3) . This description is called spatial since the incremental rotation operator acts on a spatial vector space. We use majuscules for material vectors and minuscules for spatial vectors. Both the material and spatial rotation incremental tensors, and the rotation vectors are related by mat R inc = R T R spat inc R ,

ɶ R T and θ = RΘ , θɶ = RΘ

(10)

where the first relation is called inner automorphism that is an isomorphism onto itself. The

ɶ = RΘ ɶ R , and the last second relation is a Lie algebra so(3) adjoint transformation Ad R Θ T

relation is another Lie algebra adjoint transformation in the Euclidean space with the vector 3 cross product as the Lie algebra (E , ⋅ × ⋅ ) , (See e.g. [13]).

3.2. Tangent Spaces of Rotation Manifold According to Def. 3.1, differentiating the material expression of a compound rotation

ɶ ) with respect to the parameter η gives at η = 0 the tangent of the ϕ(η ) := R exp(η Θ rotation manifold SO (3) at the base point. It is clear that the base point is the rotation operator R ∈ SO (3) since ϕ (η = 0) is equal to the rotation operator R , (See Def. 2.3). We may write the material tangent tensor of the rotation manifold by

69

Rotation Manifold SO(3) and Its Tangential Vectors

ɶ ), ϕ (η = 0) = R , ϕ(η ) := R exp(η Θ d ϕ(η ) ɶ . = RΘ dη η = 0

(11)

ɶ and t := RΘ ɶ at the base point R , we Now, if we have two tangent tensors t1 := RΘ 1 2 2 may add them together, giving a linear combination

ɶ + β RΘ ɶ = R (αΘ ɶ + βΘ ɶ ), α, β ∈R , α t1 + β t 2 = α RΘ 2 1 2 1

(12)

ɶ and Θ ɶ can be added up when we as expected. Note that the incremental rotation tensors Θ 1 2 are at the same base point of the rotation manifold. This yields a definition for material tangent space of rotation. Definition 3.3. (material tangent space) Differentiating the material expression of the

ɶ ) with respect to the parameter η and setting η = 0 yields compound rotation R exp(ηΘ the material tangent space at the base point R ∈ SO (3) . This material tangent space on the rotation manifold SO (3) , at any base point R , is defined as

{

}

ɶ := (R , Θ ɶ ) with RΘ ɶ ; R∈ SO (3), Θ ɶ ∈ so(3) , T SO (3):= Θ R

mat R

ɶ ∈ T SO (3) is a skew-symmetric where an element of the material tangent space Θ R mat R ɶ ∈ so(3) . tensor, i.e. Θ R ɶ ) , the pair of the rotation operator R , and the skew-symmetric The notation ( R , Θ ɶ , represents the material skew-symmetric tensor at the base point R ∈ SO (3) , (See tensor Θ ɶ is a skew-symmetric tensor, or a tangent tensor, Figure 4). Hence, we may express that Θ R at the base point R on the rotation manifold SO (3) . For simplicity, we could omit the base

ɶ ∈ T SO (3) if there is no danger of confusion. Especially, the point R by denoting Θ mat R ɶ and Θ ɶ in Eqn. (12), which belong to the same material incremental rotation tensors Θ 1 2 material tangent space

T SO (3) , are additive quantities.

mat R

This definition is rather different than the definition found in [20] or in [19] that reads in the form

ɶ := RΘ ɶ for any Θ ɶ ∈ so(3)} . T SO (3) := {Θ R

mat R

(13)

Basically, Def. 3.3 and (13) are similar in that (13) the rotation tensor R could be

ɶ ∈ so(3) . However, we regarded as the base point of the material skew-symmetric tensor Θ

70

Jari Mäkinen

ɶ , in Def. 3.3, is a (material) skew-symmetric tensor while the product RΘ ɶ is note that Θ R not. This can be comprehended by noticing that the left-translation in Lie groups is defined by a product and not by an addition as in linear spaces. In linear spaces, the left-translation to the vector Y reads Lef t X Y := X + Y where the vector X is a base point. We consider (13) rather confusing and prefer Def. 3.3. If we have composite rotation of three successive material rotations Θ1 , Θ 2 and Θ 3

R := exp( Θ1 ) exp( Θ 2 ) exp(Θ 3 ) ,

(14)

then according to Def. 3.3 we have

ɶ ∈ T SO (3), Θ ɶ ∈ T SO (3), Θ ɶ ∈ T SO (3), Θ 1 mat I 2 mat R1 3 mat R12

(15)

ɶ ), R := exp( Θ ɶ ) exp( Θ ɶ ). R1 := exp( Θ 1 12 1 2

ɶ ,Θ ɶ and Θ ɶ belong to the different tangent spaces and This means that the tensors Θ 1 2 3 thus must not be added up. Correspondingly, we may write for the spatial tangent tensor of the rotation manifold

ϕ(η ):= exp(η θɶ )R , d ϕ(η ) = θɶ R . 0 = η dη

(16)

ɶ R and t := θɶ R at the base Likewise above, if we have two tangent tensors t1 := θ 1 2 2 point R , we may add them together, giving a linear combination

α t1 + β t 2 = αθɶ 2 R + β θɶ 1R = (αθɶ 2 + βθɶ 1 ) R , α , β ∈ R .

(17)

ɶ and θɶ can be added up when we are at the Note that the incremental rotation tensors θ 1 2 same base point of the rotation manifold. This yields definition for the spatial tangent space of rotation. Definition 3.4. The spatial tangent space on the rotation manifold SO (3) at any base point R is defined

{

}

T SO (3) := θɶ R := ( R , θɶ ) with θɶ R; R ∈ SO (3), θɶ ∈ so(3) ,

spat R

where an element of the material tangent space θɶ R ∈ spatTR SO (3) is a skew-symmetric tensor, i.e. θɶ R ∈ so(3) .

71

Rotation Manifold SO(3) and Its Tangential Vectors T IS O (3 )

sp a t

d~ i

e x p Y

m a t

T R S O (3 )

Y

T IS O (3 )

m a t

~

I

I

R Q

~

q

~

e x p y~

b g

y~

R R

sp a t

T R S O (3 )

S O (3 ) R

S O (3 )

Figure 4. Geometric representation of the material tangent space (on the left) and the spatial tangent space (on the right) on the rotation manifold SO (3) .

ɶ ) , the pair of the rotation operator R and the skew-symmetric tensor The notation (R , θ

θɶ , represents a spatial skew-symmetric tensor at the base point R, (See Figure 4). Again, we could omit the base point R , i.e. θɶ ∈ T SO (3) if there is no danger of confusion. spat R

Rotation operators, the elements of the Lie group SO (3) , are defined as linear operators

R ∈ SO (3) . Equations (10b,c) give another interpretation for a rotation operator, it is an adjoint transformation between the material and spatial tangent spaces. In addition, a rotation motion induces the rotation operator, since the rotation operator maps the material place vector X ∈ B0 into the spatial place vector x ∈ B by the equation x (t ) = R (t )X , i.e.

R ∈ L (B0 , B ) . More generally, a rotation operator transforms material vectors into spatial vectors, that is R ∈ L (TX B0 , TxB ) , (See Figure 5 and [22]).

R

d R

R

q T

T

m R

T I

I

y I

T

a tT R

R

R T

T

sp a t

T

R

T m a t

T I

I

sp a t

T

T

I

Figure 5. Commutative diagram of variations of material and spatial rotation vectors on the rotation manifold (on the left), and corresponding vector spaces (on the right).

If we have composite rotation of three successive spatial material rotations θ1 , θ2 and θ3

R := exp( θɶ 3 ) exp( θɶ 2 ) exp( θɶ 1 ) ,

(18)

72

Jari Mäkinen

then, according to Def. 3.3, we have

θɶ 1 ∈ spatTI SO (3), θɶ 2 ∈ spatTR1 SO (3), θɶ 3 ∈ spatTR 21 SO (3), R1 := exp( θɶ 1 ), R 21 := exp( θɶ 2 ) exp( θɶ 1 ) .

(19)

ɶ , θɶ and θɶ belong to different tangent spaces and thus This means that the tensors θ 1 2 3 must not be added up. Let us consider the material form of compound rotation, given in Def. 3.1, with the aid of η -parametrized exponential mappings

ɶ +η δΨ ɶ ) = exp ( Ψ ɶ ) exp (η δ Θ ɶ ), exp ( Ψ R

(20)

ɶ , such that it belongs to the same tangent where we are finding the virtual rotation tensor δ Ψ ɶ , i.e. such that δ Ψ ɶ ,Ψ ɶ ∈ T SO (3) with the identity as the space as the rotation tensor Ψ mat I ɶ ) , and δ Θ ɶ ∈ T SO (3) . The base point omitted for simplicity. Note that R = exp (Ψ R mat R ɶ is called the total material associated rotation vector for the skew-symmetric tensor Ψ rotation vector whose base point is the identity. Taking the derivatives of (20) with respect to the parameter η at η = 0 gives, (See e.g. [10])

δ ΘR = T ⋅ δ Ψ , sinψ 1 − cosψ ɶ ψ − sinψ Ψ+ Ψ ⊗ Ψ, T := I− ψ ψ2 ψ3 ψ := Ψ ,

ɶ ), R = exp( Ψ

(21)

lim T( Ψ ) = I ,

Ψ →0

where the material tangential transformation T = T( Ψ ) is a linear mapping between the virtual material tangent spaces

T SO (3) →

mat I

T SO (3) . Now, we could make another

mat R

verification that the virtual rotation vector δ Θ R and the virtual total rotation vector δ Ψ belong to different vector spaces on the rotation manifold. This is because the tangential transformation T is equal to the identity only at Ψ = 0 . Note that the transformation T has an effect on the base points, changing the base point I into R . Definition 3.5. (material vector space) For convenience, we define a material vector space on the rotation manifold at any base point R as

{

}

ɶ ∈ T SO (3) , T := Θ R ∈ E3 Θ R mat R

mat R

Rotation Manifold SO(3) and Its Tangential Vectors

73

where an element of the material vector space is Θ R ∈ matTR . The space is an affine space with the rotation vector Ψ as a base point and the incremental rotation vector Θ as a tangent vector, then T : matTI →

T , (See Figure 5). Def. 3.5 gives a practical notation for

mat R

sorting rotation vectors in different tangent spaces. Correspondingly, we could determine the spatial tangential transformation, yielding

δ θR = T T ⋅ δ ψ , T = T( ψ ),

ψ := ψ where T : spatTI → T

(22)

( = Ψ ),

T is the same linear operator as in the material form (21), (See

spat R

Figure 5). Definition 3.6. (spatial vector space) We define a spatial vector space on the rotation manifold at any point R as

{

}

T := θR ∈ E3 θɶ R ∈ spatTR SO (3) .

spat R

(23)

An element of the spatial vector space is θR ∈ spatTR .

3.3. Where Does the Material Incremental Rotation Vector Belong? Traditionally it has been assumed that the material incremental rotation vector belongs to the same tangent space at any point in time, e.g. [19], [9] and [8]. This assumption has not been proven anywhere and it cannot be proven because it is false. We give three separate proofs against this assumption. We note that this is not a matter of definition because incremental rotation vectors have a strict geometric meaning. Hence, the definitions of tangent spaces 3.3 and 3.4. have to be geometrically consistent with Def. 2.3. Proof I: The material incremental rotation vector Θ or more precisely its skewsymmetric counterpart, belongs to the tangent space of the rotation manifold SO (3) . If we

ɶ belongs to the same tangent space at any assume that the skew-symmetric rotation tensor Θ point, then we have a manifold whose tangent spaces are identical at any point. Manifolds whose tangent spaces are identical at any point are flat, but this is a contradiction since the rotation manifold SO (3) is a non-flat manifold, i.e.

α R1 + β R 2 ∉ SO (3), ∀R1 , R 2 ∈ SO (3), α , β ∈ R .

(24)

Proof II: This proof has given by the author in the paper [14]. The Newmark scheme reads for the total rotation vector

74

Jari Mäkinen

Ψn( i +1) = Ψn( i ) + ∆Ψn( i ) ∈matTI , ɺ ( i +1) = Ψ ɺ ( i ) + γ ∆Ψ ( i ) , Ψ n n n

(25)

hβ

ɺɺ ( i +1) = Ψ ɺɺ ( i ) + 1 ∆Ψ ( i ) . Ψ n n n 2 h β

Substituting (25b,c) for the material angular velocity and angular acceleration vectors, Eqns. (35a) and (36a), we have the modified time-stepping scheme:

Ψn( i +1) = Ψn( i ) + ∆Ψn( i ) Ωn(+i +11) = Ωn(+i )1 +

γ T( Ψn( i +1) ) ⋅ ∆Ψn( i ) , hβ

(26)

γ ɺ ɺ (i +1) ) ⋅ ∆Ψ (i ) . Α (ni++11) = Α n( i+)1 + 21 T( Ψn( i +1) ) ⋅ ∆Ψn( i ) + T ( Ψn(i +1) , Ψ n n hβ

h β

We note that the modified Newmark scheme (26) is not the original Newmark scheme because of the appearance of the tangential transformation T and its time derivative. Hence, we cannot use the original Newmark scheme for the material quantities, as it is the wellknown fact for the spatial case. Therefore, an assumption that we could use the original Newmark scheme for material quantities yields a contradiction, because an incremental rotation vector and its time derivatives belong to the same tangent space at any point in time. Proof III: Studying the material form of compound rotation

ɶ + ηδ Ψ ɶ ) = exp( Ψ ɶ ) exp(ηδΘ ɶ ), exp( Ψ we find out that Ψ , δ Ψ belong to the same vector space, namely

(27)

T . This is clear

mat I

because the total rotation vector Ψ is a parametrization of the rotation manifold SO (3) and the total rotation vector lives in that parametrization space, (See Figure 1 and Figure 6). Additionally, we have a relation between δ Ψ and δ Θ that reads δ Θ = TδΨ , where

T( Ψ ) is the tangential transformation Eqn. (21). Since the tangential transformation T( Ψ ) depends on the total rotation vector, the vector δ Θ also depends on it. Hence, δ Θ depends ɶ ) that is a base point of the rotation manifold SO (3) . on the rotation operator R = exp( Ψ However, δ Ψ always occupies in the same fixed vector space

T and thus δ Θ cannot belong to the same vector space, unless at a specific point when T( Ψ ) is equal to the identity I at Ψ = 0 . According to our notation, the vector δ Θ or more precisely δ Θ R , which is the same vector, belongs to the vector space of rotation

mat I

T , which depends on the rotation operator

mat R

Rotation Manifold SO(3) and Its Tangential Vectors

75

ɶ ) . On the other hand, the vector δ Ψ belongs to the vector space of rotation R = exp( Ψ T , which is a fixed vector space, (See Figure 5).

mat I

ro ta tio n m a n ifo ld S O (3 )

p a ra m e triz a tio n ~ m a p p in g e x p ( Y )

2 p

c h a n g e o f p a ra m e triz a tio n

p

p a ra m e triz a tio n ~ C m a p p in g e x p ( Y ) 2 p

p

p a ra m e triz a tio n c h a rt

c o m p le m e n t p a ra m e triz a tio n c h a rt

Figure 6. The change of parametrization in the parameter space E3 for the canonical representation of rotation manifold.

ɶ and δ Θ ɶ do not belong to the same tangent space of The skew-symmetric tensors Ψ R ɶ ) exp( Θ ɶ ) ≠ exp( Ψ ɶ +Θ ɶ ) , generally. rotation as it can be verified that exp( Ψ 3.4. Complement Rotation Vector Let a rotation vector Ψ with a rotation angle larger than zero and less than full-angle, i.e.

0 < ψ < 2 π , thus its complement rotation vector Ψ C is defined Ψ C := Ψ − 2π Ψ ,

ψ

ψ := Ψ .

(28)

After substituting the complement rotation vector into the rotation operator (4), we notice that the rotation vector and its complement represent the same rotation operator, i.e.

R ( Ψ C ) = R ( Ψ ) . Def. (28) is a change of parametrization in the parameter space E 3 , (See Figure 1 and Figure 6). This change of parametrization is a continuously differentiable mapping on the open domain 0 < ψ < 2 π , giving a smooth construction of the rotation manifold SO (3) at this domain. Note that the complement of a complement rotation vector

76

Jari Mäkinen

is a rotation vector itself, i.e. ( Ψ C )C = Ψ . Hence, there is no priority over these parametrization charts. We could represent the rotation manifold globally with these two parametrization charts. When a rotation angle exceeds straight-angle (ψ > π ), we accomplish the change of parametrization according to (28), giving a new rotation angle smaller than straight-angle. Thus, we never get into trouble with singularity at ψ = 2 π . As illustrated in Figure 6, the change of parametrization maps the rotation angle outside of straight-angle into inside of straight angle. Note that there exists no other canonical parametrization with rotation less than perigon such as those parametrizations given in (28). The null rotation vector is an isolated point, at the centre of the domain, for the parametrization change. Using a limit process, we find out that the rotation operator approaches to the identity element when the rotation angle is decreased. Hence, we could modify the domain of the parametrization where the rotation angle is less than perigon, i.e. ψ < 2π including the null rotation angle. This domain is still an open domain in the Euclidean 3

3

space E ; indeed, it is an open ball in E with 2π-radius. When the rotation vector Ψ ∈matTI is switched to the complement rotation vector

Ψ C ∈matTIC , in dynamic analysis we need its time derivatives that are ɺ C = BΨ ɺ ∈matTIC , Ψ ɺɺ C = BΨ ɺɺ + B ɺ ∈ T C, ɺΨ Ψ mat I

(29)

where the symmetric kinematic operator, defined by B := D Ψ Ψ , and its time derivative are C

B = (1 − 2π ) I + 2π n ⊗ n ,

ψ

ψ

ɺ )I + ( Ψ ɺ ⊗n+n⊗Ψ ɺ ) − 3( n ⋅ Ψ ɺ )n ⊗ n , ɺ = 2π ( n ⋅ Ψ B  ψ2 

(30)

and where the rotation axis is n = Ψ / ψ .

4. Angular Velocities, Accelerations Vectors In this Section, we give definitions for material as well as spatial angular velocities and accelerations. Definition 4.1. The material angular velocity (skew-symmetric) tensor is defined with the aid of rotation operator R ∈ SO (3) and its time derivative by

ɶ := R T R ɺ, Ω R

Rotation Manifold SO(3) and Its Tangential Vectors

77

where the dot denotes the time derivative. (See justification in [13; Ch. 8.6 and 15.2]). The rotation tensor can be viewed as a mapping, a push-forward of a material vector,

R : matTR → spatTR between the material and spatial vector spaces, (See [22] and [12]). Then ɶ : T → T . Thus, the material angular the material angular tensor is a mapping Ω R mat R

mat R

velocity tensor is indeed a true material tensor. The skew-symmetry can be obtained by taking a derivative for the equation R T R = I . If the rotation operator is expressed with the aid of exponential mapping by

ɶ (t ) + O(Θ ɶ 2 (t ))) , where the fixed rotation R is superimposed by an R new = R ( I + Θ R R ɶ (t ) plus higher order terms, then substituting this into Def. infinitesimal rotation I + Θ R

4.1. yields T T ɶ = RT R ɶ ɶ2 ɶɺ ɶ ɺ Ω new new = ( I − Θ R ( t ) + O( Θ R ( t )) )R R ( Θ R (t ) + O( Θ R (t ))) R ɺɶ (t ) + O( ɶ (t )), =Θ ΘR R

(31)

ɶ → 0ɶ after the limit process Θ R ɺɶ ɶ =Θ ɺ Ω R R ⇔ ΩR = Θ R .

(32)

This states that the angular velocity vector is the time derivative of the incremental

ɺ , Ω ∈ T , which is the rotation vector Θ R , moreover, (if the base point is omitted) Θ, Θ mat R material rotation vector space on the rotation manifold. The result in (32) is often given as a definition for the angular velocity vector in elementary text books. Similar expression and derivation can be accomplished for the spatial angular velocity tensor and vector, yielding

ɺ T, ɶ R := RR ω ɶ R = θɺɶ R ⇔ ωR = θɺ R , ω

(33)

ɺ and the where the spatial incremental rotation vector θR , its time derivative vector θ R spatial angular velocity vector ωR belong to the same spatial vector space on the manifold,

θ, θɺ , ω ∈ spatTR ; the base point R is omitted. The material and spatial quantities have connections:

ɶ RT , ɶ R = RΩ ω R

ωR = RΩR ,

such like for incremental rotation vectors (10), (See Figure 5).

(34)

78

Jari Mäkinen

Definition 4.2. The material and spatial angular acceleration tensor and the corresponding vector are defined as the time derivative of the angular velocity terms, giving

ɶ := Ω ɶɺ , Α R R ɺ , Α =Ω R

R

ɶɺ R , αɶ R := ω ɺ R, αR = ω

ɶ ∈ T SO (3), Α R mat R Α R ∈ matTR , αɶ R ∈ spatTR SO (3), α R ∈ spatTR ,

where Α R is the material angular velocity vector, and α R is the spatial angular velocity vector at the base point R . Note that the incremental material rotation vector Θ R , the material angular velocity vector ΩR and the material angular acceleration vector Α R (majuscule of alpha-letter) belong to the same material vector space on the rotation manifold, i.e. Θ R , ΩR , Α R ∈ matTR with the base point R = exp( ΨI ) . At separate time moments, these vectors, however, occupy different vector spaces because the rotation operator depends on time, namely R = R (t ) . The base point is moving as time travels. Vector quantities of this kind may be called spin vectors. Spin vectors are rather tricky in the numerical sense as they always occupy a distinct vector space on a manifold. Correspondingly, the spatial spin vectors are θR , ωR , α R ∈ spatTR . Angular velocity vectors and the time derivative of total rotation vectors are related by, (See [7,] p. 70])

ɺ , where Ω ∈ T , Ψ ɺ , Ψ ∈ T for material ΩR = T( ΨI ) ⋅ Ψ I R mat R I I mat I α R = TT (ψ I ) ⋅ ψɺ I , where ωR ∈ spatTR , ψɺ I , ψ I ∈ spatTI for spatial

(35)

where the tangential transformation depends on the total rotation vector, and the rotation

ɶ ) = exp(ψɶ ) . Similar expression for the angular acceleration vector operator is R = exp( Ψ I I can be obtained by differentiating the above formulas, giving

ɺɺ + Tɺ ⋅ Ψ ɺ , where Α ∈ T , Ψ , Ψ ɺ ,Ψ ɺɺ ∈ T for material ΑR = T ⋅ Ψ I I R mat R I I I mat I T T ɺ ɺɺ I +T ⋅ ψɺ I , where α R ∈ spatTR , ψ I , ψɺ I , ψ ɺɺ I ∈ spatTI for spatial αR = T ⋅ ψ

(36)

where the tangential transformation depends on the total rotation vector; and the rotation

ɶ ) = exp(ψ ) . Note that the tangential transformations R = exp( Ψ ɺ T ∈ L ( T , T ) operate with different base points. T, Tɺ ∈ L ( matTI , matTR ) and T T , T spat I spat R

operator

is

The time derivative of the tangential transformation can be written

Rotation Manifold SO(3) and Its Tangential Vectors ɺɶ + c Ψ ɺ , Ψ) = c (Ψ ⋅ Ψ ɺ ) I − c (Ψ ⋅Ψ ɺ )Ψ ɶ + c (Ψ ⋅Ψ ɺ ) Ψ⊗Ψ + c Ψ ɺ ɺ , Tɺ ( Ψ 1 2 3 4 5 ( ⊗Ψ + Ψ⊗Ψ )

79 (37)

where coefficients ci are given by

ψ cosψ − sinψ ψ sinψ + 2 cosψ − 2 , , c2 := 3 ψ ψ4 3sinψ − 2ψ − ψ cosψ cosψ − 1 ψ − sinψ , c4 := , . c3 := c5 := 5 2 ψ ψ ψ3 c1 :=

(38)

ɺ is The limit value of the tensor T

ɺ , Ψ) = − 1 Ψ ɶɺ . ɺ (Ψ lim T  →0

(39)

2

5. Euler’s Equations In this Chapter, Euler equations for rotation motion, which are differential equations on the tangent bundle, are given in the material and spatial representations. Lets consider virtual work for a body V that can be given in a form

∫ δ x ⋅ ( f − ρ ɺɺx ) dV = 0,

∀δ x ∈ Tx0 M ,

(40)

V

where f is the external body force field, ρ is the density, ɺxɺ is the acceleration vector field, and δ x is the virtual displacement field. The holonomic constraints g – such like constraints arisen from rigid body assumptions – form a constraint manifold M into the placement space. The constraint manifold can be defined by

{

}

M := x ∈ X g ( x ) = 0 ,

(41)

whose the tangent space at the point x 0 = x (t0 ) is

{

}

Tx0 M := δ x ∈ X D xg ( x = x 0 ) ⋅ δ x = 0 .

(42)

The virtual displacement field δ x therefore belongs to a tangent space of the constraint manifold M . It should be noted that the virtual displacement could be any size, infinitesimal or finite. A parametrization that parametrizes general spherical motion can be given in a form

80

Jari Mäkinen

x = RX ,

(43)

where R a rotation operator and X is initial placement vector field. Now the parametrization (43) satisfies the rigid body constraint equations g ( RX ) = 0 . If we substitute the parametrization (43), whose variation and time derivative are

ɶ X = − RX ɶδΘ , δ x = Rδ Θ R R ɶ X + RΩ ɶ 2 X = −RX ɶ X ɶ Α − RΩ ɶ ɺɺ x = RΑ R R R R ΩR ,

(44)

into the principle of virtual work (40), we get

ɶ JΩ ) = 0 δ Θ R ⋅ ( M R − JΑ R − Ω R R

(45)

where we have used notation for the inertial tensor J and for the external force vector M R

ɶ TX ɶ dV , J := ∫ ρ X

ɶ T f dV M R := ∫ XR

V

(46)

V

Equation (45) represents the material form of the virtual work principle for Euler’s differential equation under a rotational motion. We also notice that Euler’s equation is defined on the tangent space of the rotation manifold, namely

T

mat R ,

since the virtual incremental

rotation vector δ Θ R ∈matTR belongs to this tangent space. While the rotation operator R = R ( t ) – a base point of the tangent space

T –

mat R

depends on time, also the tangent space varies. Traditional time-stepping methods, like the Newmark scheme, have been designed for ordinary differential equations in linear spaces. Therefore, these integration schemes are not suited for Euler’s equation (45), which is a differential equation on the non-flat manifold SO (3) . However, the Newmark time-stepping scheme can be consistently modified for the rotation manifold as given in [14]. This modified Newmark scheme is equivalent with the different approach developed by [4] where the updated total rotation vector Ψinc ∈matTR ref is used for an unknown vector. In this formulation, the base point R ref is constant during the time integration procedure and the base point is updated when a new solution is obtained. This formulation is called an updated Lagrangian formulation. Euler’s equation in the spatial representation can be obtained by substituting parametrization (43) for the virtual work principle (40). The virtual displacement and time derivative in the spatial representation are

δ x = δ θɶ R RX = −xɶ δ θR , ɶ R2 RX = − xɶ α R − ω ɶ R xɶ ωR . ɺɺ x = αɶ R RX + ω

(47)

81

Rotation Manifold SO(3) and Its Tangential Vectors Thus, we have

ɶ R jω R ) = 0 , δ θ R ⋅ ( m R − jα R − ω

(48)

where we have denoted for the spatial inertial tensor j and for the spatial external moment

mR j:= ∫ ρ xɶ T xɶ dV ,

ɶ dV mR := ∫ xf

V

(49)

V

Material and spatial quantities have relations (10) and (34). Thus, the principle (48) can be rewritten

ɶ JΩ ) = 0 δ θ R ⋅ R ( M R − JΑ R − Ω R R

(50)

where the terms in the brackets correspond to Euler’s equation in the material representation. On the other hand, when we linearize the spatial quantities, we have to use Lie derivatives such as

R D ( R T
) ,

(51)

where
denotes the place for a spatial vector. Applying this Lie derivative to the principle of virtual work (48) and especially to the formula (50), we find out that derivative is accomplished to material quantities. This is because the derivative of an objective material quantity is always an objective quantity. From this reality and identifying ψ = Ψ it follows that spatial tangent tensors receive the same form as the corresponding material quantities. Fully consistent tangential tensors are found in [15].

6. Simplified Newmark Integration Scheme Here we name the standard Newmark scheme with incremental rotations as the simplified Newmark scheme. In the paper [14] it has been shown that the standard Newmark scheme is indeed a simplified version of the correct one. Recently, in paper [18], it is shown that the simplified Newmark integration scheme neglects third order terms of rotation vector. However, the proof suffers from the fact that material incremental rotation vector and its time derivatives cannot be integrated directly by the Newmark scheme. We show that the simplified Newmark integration scheme for incremental rotations neglects first order terms of rotation vector, not third order terms on the contrary [18]. The updated total rotation vector and its time derivatives can be correctly integrated by the standard Newmark scheme since the updated total rotation vector Ψinc ∈matTR ref for any (i )

82

Jari Mäkinen

iteration step (i). The Newmark iteration scheme for the updated total rotation vector reads, (See [14]) ( i +1) (i ) (i ) Ψinc ∈matTR ref n n = Ψinc n + ∆Ψ

ɺ ( i +1) = Ψ ɺ ( i ) + γ ∆Ψ ( i ) Ψ inc n inc n

(52)

hβ

ɺɺ ( i +1) = Ψ ɺɺ ( i ) + 1 ∆Ψ ( i ) Ψ inc n inc n 2 h β

with the zero-acceleration predictor (0) h 1 − 2β Α Ψinc ( ) n n = hΩn + 2

2

ɺ (0) = Ω + h (1 − γ ) Α Ψ inc n n n (0) ɺɺ Ψ inc n = 0

(53)

where the angular velocity and acceleration vectors Ωn , Α n ∈ matTn are obtained from the previously converged solution (or they are initial values). At the next iteration step, we have, after elimination of ∆Ψ Ψ (0) , for the velocity equation

ɺ (1) = Ω + h (1 − γ ) Α + γ hΨ ɺɺ (1) Ψ inc n n n inc n

(54)

Next we assume that the iteration converges at the first iteration step. This has to be assumed since the modified Newmark scheme (26) is nonlinear because of the tangential ( i +1)

tensor and its time derivative depend on the iterated solution Ψinc n . Now, multiplying (54) by the tangential transformation Tn = T( Ψinc n ) and using the formulas (35a) and (36a), we (1)

(1)

have

ɺ (1) . ɺ (1) Ψ Ωn +1 = Tn(1)Ωn + h (1 − γ ) Tn(1) Α n + γ h Α n+1 − γ hT n inc n (1)

The tangential transformation Tn

(55)

and its time derivative can be expanded by a serial

development

ɶ + O( Ψ 2 ) T = I − 12 Ψ ɶɺ + 1 ΨΨ ɶɺ ɶ + ΨΨ ɶ ɶɺ + O( Ψ 2 ) ɺ =−1Ψ T 2 6

(

)

(56)

where O( Ψ ) indicates the second and higher order terms. With this serial development, 2

Eqn. (55) reads

Rotation Manifold SO(3) and Its Tangential Vectors

ɶ (1) ( Ω + h (1 − γ ) Α ) + Ωn +1 = Ωn + h (1 − γ ) Α n + γ hΑ n +1 − 12 Ψ inc n n n ɺ (1)

ɶ Ψ ɶ ɺ − γ h 16 Ψ inc n inc n Ψinc n + O( Ψinc n ) (1)

(1)

83

(57)

(1) 2

Note that the simplified Newmark scheme reads for angular velocity Ωn+1

Ωn +1 = Ωn + h (1 − γ ) Α n + γ h Α n +1

(58)

Hence comparing Eqns. (57) and (58), we find out that the simplified Newmark scheme for finite rotations neglects terms of the first order of rotation vector. It is interesting to notice that at the plane rotation (2D rotation) the first and higher order terms of rotation vector vanish, as expected. Similar results can be obtained for acceleration equation, yielding

Α n+1 =

1

βh

(

1 h

(1) (1) 1− 2 β Ψinc n − Ωn ) − 2 β Α n + O( Ψinc n ) .

(59)

Hence, the simplified Newmark scheme for accelerations also neglects the first order terms of rotation vector. Generally, when a convergent solution is achieved at the iteration step ( m ) , the rotation tensor is computed by

ɶ (m) ) . R n +1 = R n exp( Ψ inc n

(60)

The updated total rotation vector Ψinc n is small for small time steps h but not negligible. ( m)

The initial value of Ψinc n is computed by (53a). (0)

7. Interpolation of the Rotation Field and Its Objectivity In this Chapter, we show what sort of rotation interpolation on the rotation manifold is an objective interpolation under the observer transformation. This objectivity may be called observer frame-indifference, (See [17, Ch. 2] or [12]). The interpolation can be consistently accomplished if the rotation vectors of each node belong to the same tangent space. Let the observer transformation to the rotation operator R and to the placement x c be

R + = QR ,

x c+ = Q ( x c + c ) ,

(61)

where the orthogonal operator Q ∈ SO (3) corresponds to the rigid body rotation and the vector c ∈ E corresponds to the rigid body translation, respectively. Note that the rotation 3

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Jari Mäkinen

operator R ∈ Tx B

⊗ TX B 0

is a two-point tensor and it acts in the observer transformation

like the deformation gradient F . Let Ψ1 ,Ψ2 ∈matTI be nodal vectors of total material rotation for a beam element, which has a linear interpolation, and let ΨQ be a total rotation vector for an objective

ɶ ) , i.e. for a rigid body rotation. These rotation vectors transformation operator Q = exp( Ψ Q have the following component values [6] with respect to the global frame {O , e1 , e 2 , e 3}

[Ψ 1i ]

 1   -0.4   0.2      = -0.5 , [Ψ 2i ] = 0.7 , Ψ Qi  =  1.2  ,        0.25  0.1   -0.5

(62)

hence e.g. Ψ1 = Ψ 1i e i with conventional summation. Linear interpolation functions for the nodal rotation vectors Ψ1 and Ψ2 read

N1 ( s ) = 1 −

s , L

N 2 ( s) =

s , L

s ∈ [0, L] .

(63)

The interpolated rotation field is therefore Ψ = N iΨ i ∈ C ( matTI ) . This interpolation is clearly acceptable because the nodal rotation vectors belong to the same tangent space of rotation. The observer transformation to the rotation operator R is given in (61a), this yields

ɶ extr ) = Q exp( Ψ ɶ ), exp( Ψ 1 1 The nodal rotation vector Ψi

extr

ɶ extr ) = Q exp( Ψ ɶ ). exp( Ψ 2 2

(64)

∈ matTI can be obtained by extracting it from the rotation

operator via Spurrier’s algorithm. We note that the original rotation vectors Ψ1 and Ψ2 in the observer transformation (64) occupy the tangential vector space extract the transformed nodal rotation vectors Ψi

extr

T Although we may

mat Q .

such that they satisfy the observer

transformation relations (64), the linear interpolation is not preserved, (See Figure 7)

ɶ extr ) ≠ Q exp( N ( s ) Ψ ɶ ), exp( N i ( s ) Ψ i i i

∀s ∈ [0, L ], Q ∈ SO (3) .

(65)

This arises from the fact that the rotation manifold SO (3) has a curved character. Indeed, a linear vector valued function in the tangential vector space different tangential vector space

T is not linear in the

mat Q

T . Result (65) indicates that the linear interpolation does

mat I

not preserve under the observer transformation as shown in Figure 7, settings after (62).

85

Rotation Manifold SO(3) and Its Tangential Vectors

Components of rotation vector

2

1.5

1

0.5

0

−0.5

−1 0

0.2

0.4

0.6

0.8

1

0.8

1

Length parameter s/L 2

Norm of rotation vector

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 0

0.2

0.4

0.6

Length parameter s/L Figure 7. The components of the rotation vectors Ψ+(s) and Ψextr(s) interpolated through the length of beam (left), and the norm ||Ψ+(s)|| and ||Ψextr(s)|| (right). The solid line indicates the rotation field Ψ+(s) under the observer transformation and the broken line the rotation interpolation of the extracted rotation values Ψextr(s) . Solid lines correspond to the proper values.

Dispute Eqn. (65) states that the interpolation is not preserved in the observer transformation. This does not mean that the rotation interpolation is non-objective. Indeed, this property is never required for being an objective formulation. It is sufficient that the

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Jari Mäkinen

rotation interpolation satisfies the condition (61a). Generally speaking, a global interpolation on a non-flat manifold never preserves under an observer transformation. This is because of the nonlinear character of the manifold; a parametrization mapping is a nonlinear mapping for a non-flat manifold. In the paper [6], the authors utilize a corotational interpolation, where the interpolation is carried out with respect to an element-attached frame. Hence, this interpolation naturally preserves in a rigid body motion. Therefore, we may pronounce that extracting the nodal rotation vectors from the corresponding rotation operations may not be interpolated. There are infinite possibilities for the interpolation since the transformation operator Q ∈ SO (3) is arbitrary. The rotation interpolation after an observer transformation reads formally

ɶ + ( s ) = log ( Q exp( N ( s ) Ψ ɶ )) Ψ i i

(66)

where log -operator is the inverse operator of exp -operator. Now, the transformed rotation + + field Ψ is not a (vector-valued) linear function. If we substitute Eqn. (66) for R , we find out that

ɶ + ( s )) = Q exp( N ( s ) Ψ ɶ ) = QR ( s ) . R + ( s ) = exp( Ψ i i

(67)

Hence we have Eqn. (61a) as expected. Note that the rotation interpolation of extracted nodal values Ψ extr ( s ) , the left side of Eqn. (65), is not objective and does not realize the condition (61a), (See Figure 7). We have assumed above that the observer transformed rotation interpolation

Ψ + ∈ C ( matTI ) keeps the base point I fixed. However, it also makes sense and can be assumed that the base point transforms under the observer transformation ( I → Q ) giving

Ψ + ∈ C ( matTQ ) . Then we notice that Ψ + ( s ) = Ψ ( s ) and an interpolation is preserved under an observer transformation. This is an important issue and clarifies the frameindifference of geometrically exact beam formulations.

Conclusions In this paper, we have shown that incremental material rotation vectors belong to different tangent spaces of the rotation manifold SO (3) at a different instant. Moreover, we have proven that the material tangent space as the tangent space at unity is not a possible definition yielding geometrically inconsistent results, although this kind of definition is widely adopted in applied mechanics community. We have shown that simplified Newmark integration scheme for finite rotations neglects first order terms of rotation vector. Hence, the direct application of the material incremental rotation vector with standard time integration methods neglects first order terms of incremental rotation vector, likewise in the spatial case. In addition, we have shown that the rotation interpolation of extracted nodal values is not an objective interpolation under the observer transformation.

Rotation Manifold SO(3) and Its Tangential Vectors

87

Acknowledgements Financial support for this research was provided by the Academy of Finland under the project number 206020. This support is gratefully acknowledged.

References [1] [2] [3] [4] [5] [6]

[7] [8] [9] [10]

[11]

[12] [13] [14] [15]

[16]

[17]

Abraham R, Marsden J, Ratiu T (1983) Manifolds, Tensor Analysis and Applications. Addison-Wesley Reading. Argyris J (1982) An Excursion into Large Rotations. Comp. Methods Appl. Mech. Engng 32: 85-155. Argyris J, Poterasu VF (1993) Large Rotation Angles Revisited Application of Lie Algebra. Comput. Methods Appl. Mech. Engng 103: 11-42. Cardona A, Géradin M (1988) A Beam Finite Element Non-Linear Theory with Finite Rotations. Int. J. Num. Meth in Engng 26: 2403-2438. Choquet-Bruhat Y, DeWitt-Demorette C, Dillard-Bleick M, (1989) Analysis, Manifolds and Physics, Part I: Basics, North-Holland Amsterdam. Crisfield MA, Jelenic G (1999) Objectivity of Strain Measures in the Geometrically Exact Three-Dimensional Beam Theory and Its Finite-Element Implementation. Proc. Royal Society of London A 455: 1125-1147. Géradin M, Cardona A (2001) Flexible Multibody Dynamics: A Finite Element Approach, John Wiley and Sons Chichester. Ibrahimbegović A (1997) On the choice of finite rotation parameters. Comput. Methods Appl. Mech. Engrg 149: 49-71. Ibrahimbegović A, Al Mikdad M (1998) Finite rotations in dynamics of beams and implicit time-stepping schemes. Int. J. Num. Meth Engrg 41: 781-814. Ibrahimbegović A, Frey F, Kozar I (1995) Computational Aspects of Vector-Like Parametrization of Three-Dimensional Finite Rotations. Int. J. Num. Meth. Engng 38: 3653-3673. Jelenić G, Crisfield MA (1999) Geometrically Exact 3D Beam Theory: Implementation of a Strain-Invariant Finite Element for Statics and Dynamics. Comp. Meth. Appl. Mech. Engng 171 (1999) 141-171. Marsden JE, Hughes TJR (1994) Mathematical Foundation of Elasticity Dover, New York. Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer-Verlag New York. Mäkinen J (2001) Critical Study of Newmark-Scheme on Manifold of Finite Rotations. Comp. Meth. Appl. Mech. Engng 191: 817-828. Mäkinen J, Marjamäki H (2005) Total Lagrangian Parametrization of Rotation Manifold, In: Proc. The Fifth EUROMECH Nonlinear Dynamics Conference, ENOC2005, Eindhoven, 2005, 522-530. http://www.tut.fi/~jmamakin/ENOC2005.pdf. Mäkinen J (2007) Total Lagrangian Reissner’s Geometrically Exact Beam Element without Singularities. International Journal for Numerical Methods in Engineering 70(9) 1009-1048. Ogden RW (1984) Non-Linear Elastic Deformations, Ellis Horwood Chichester.

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[18] Rubin MB (2007) A simplified implicit Newmark integration scheme for finite rotations, Computers and Mathematics with Applications 53(2): 219-231. [19] Simo JC, Vu-Quoc L (1988) On the Dynamics in Space of Rods Undergoing Large Motion - A Geometrically Exact Approach, Comp. Meth. Appl. Mech. Engng 66: 125161. [20] Simo JC, Marsden JE, Krishnaprasad PS (1988) The Hamiltonian Structure of Nonlinear Elasticity: The Material and Convective Representation of Solids, Rods, and Plates, Arch. Rat. Mech. Anal. 104: 125-183. [21] Stuelpnagel J (1964) On the Parametrization of the Three-Dimensional Rotation Group”, SIAM Review 6: 422-430. [22] Stumpf H, Hoppe U (1997) The Application of Tensor Algebra on Manifolds to Nonlinear Continuum Mechanics – Invited Survey Article, ZAMM Applied Mathematics and Mechanics 77: 327-339.

In : Lie Groups : New Research Editor : Altos B. Canterra, pp. 89-122

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 3

A SYMPTOTIC H OMOLOGY OF THE Q UOTIENT OF P SL2 (R) BY A M ODULAR G ROUP Jacques Franchi∗ I.R.M.A., Universit´e Louis Pasteur et CNRS, 7 rue Ren´e Descartes, 67084 Strasbourg cedex. France

Abstract 2

Consider G := P SL2 (R) ≡ T 1 H , a modular group Γ, and the homogeneous 2 space Γ\G ≡ T 1 (Γ\ H ). Endow G , and then Γ\G , with a canonical left-invariant metric, thereby equipping it with a quasi hyperbolic geometry. Windings around handles and cusps of Γ \ G are calculated by integrals of closed 1-forms of Γ \ G . The main results express, in both Brownian and geodesic cases, the joint convergence of the law of these integrals, with a stress on the asymptotic independence between slow and fast windings. The non-hyperbolicity of Γ\G is responsible for a difference between the Brownian and geodesic asymptotic behaviours, difference which does not 2 exist at the level of the Riemann surface Γ\ H (and generally in hyperbolic cases). Identification of the cohomology classes of closed 1-forms with harmonic 1-forms, and equidistribution of large geodesic spheres, are also addressed.

Keywords : Brownian motion, Geodesics, Geodesic flow, Ergodic measures, Asymptotic laws, Modular group, Quasi-hyperbolic manifold, Closed 1-forms. Mathematics Subject Classification 2000 : primary 58J65 ; secondary 60J65, 37D40, 37D30, 37A50, 20H05, 53C22.

1.

Introduction

Consider G := P SL2(R) ≡ T 1H2 , a modular group Γ, and the homogeneous space Γ\G ≡ T 1 (Γ\H2 ). Endow G , and then Γ\G , with a canonical left-invariant metric, thereby equipping it with a quasi-hyperbolic geometry, which pertains to the 6th 3-dimensional ∗

E-mail address : [email protected]

90

Jacques Franchi

geometrical structure of the eight described by Thurston [T]. The non-hyperbolic manifold Γ\G has finite volume, finite genus, and a finite number of cusps. It is natural in this setting to study the asymptotic behaviour of the Brownian motion and of the geodesic flow (under some Liouville-like measure), by means of their asymptotic homology, calculated by the integrals of the harmonic 1-forms of Γ\G along their paths. The main results here express, in both Brownian and geodesic case, the joint convergence in law of these integrals, with asymptotic independence of slow and fast windings. This same results yield in fact also the asymptotic law of the normalised integrals, along Brownian and geodesic paths, of any C 2 closed 1-form. Indeed, it holds true on Γ\G that the cohomology classes of closed 1-forms can be identified with harmonic 1-forms. The non-hyperbolicity of Γ \ G is responsible for a difference between the Brownian and geodesic asymptotic behaviours, difference which does not exist at the level of the Riemann surface Γ\H2. Lifting to the unit tangent bundle has also the advantage to free the harmonic forms of the constraint to have a null sum of their residues at the cusps of Γ\ H2 . Counter to the hyperbolic setting, the geodesic flow on Γ \ G is not ergodic with respect to the normalised Liouville measure of Γ \ G, so that the Liouville-like law governing the geodesic, for which ergodicity holds, has to be supported by some leaf. Moreover asymptotic equidistribution of large geodesic spheres holds for such measure. This article, in which the modular group Γ is arbitrary, is mainly a generalisation of [F3], which deals with the particular case of the modular group Γ being the commutator subgroup of P SL2(Z) ; in which case the quotient manifold (which is interestingly linked to the trefoil knot) has a unique cusp and a unique handle ; for example, we have now to take into account the pullbacks of singular harmonic forms on Γ\ H2 , which did not exist in [F3]. However, the identification of the cohomology classes with harmonic spaces H 1 and the equidistribution of large geodesic spheres, two questions which are addressed here, were not discussed in [F3]. Brownian and geodesic asymptotic behaviours were already studied in a similar way, but in an hyperbolic setting, in [E-F-LJ1], [E-F-LJ2], [E-LJ], [F2], [G-LJ], [LJ1], [LJ2], [W]. As in [F3], hyperbolicity (which is replaced by quasi-hyperbolicity) does not hold in the present setting, nor ergodicity of the Liouville measure (which has to be replaced by Liouville-like measures, supported by leaves), and the asymptotic Brownian and geodesic windings are no longer the sames, though comparable (the spiral windings of the geodesics of G about their projections on H2 is mainly responsible for this feature). Moreover fast and slow windings are addressed jointly. As in [LJ1], [G-LJ], [F3], the aim is here to study asymptotic homology, meaning that only closed forms are considered. Consequently no foliated diffusion is needed. Whereas [LJ2], [E-LJ], [F2], [E-F-LJ1], [E-F-LJ2] dealt with non necessarily everywhere closed 1-forms, so that the showing up of a spectral gap at the level of the stable foliation was needed.

1.1.

Outline of the Article

The framework of this article is along the following sections, as follows. 2.) Iwasawa coordinates and metrics on G = P SL2 (R)

Asymptotic Homology of a Modular Quotient

91

Taking advantage of the global Iwasawa coordinates on G , a canonical one-parameter family of left-invariant Riemannian metrics on G is exhibited, endowing it with a nonhyperbolic, but quasi-hyperbolic geometry (of 3-dimensional tangent bundle). 3.) Geometry of a modular homogeneous space Γ\G A basis of harmonic 1-forms on Γ\G is described in Theorem 3.1, together with their asymptotics in the cusps. A particular role is played by a form ω0 , which is not the pullback of a form on Γ\ H2. Comparing with the two dimensional case of Γ\ H2 , lifting to the unit tangent bundle Γ\G has then also the advantage to free the harmonic forms of the constraint to have a null sum of their residues at the cusps. 4.) Closed forms and harmonic forms ˜ are The geometries associated with Γ and with a free normal subgroup of finite index Γ compared. The identification of closed 1-forms with harmonic 1-forms modulo exact forms is deduced. This has the important consequence that the asymptotic study of (integrals of) closed 1-forms will reduce to the asymptotic study of harmonic 1-forms. 5.) Left Brownian motion on G = P SL2(R) The natural left Brownian motion is seen to decompose into a planar hyperbolic Brownian motion and a correlated angular Brownian motion. 6.) Asymptotic Brownian windings in Γ\G The harmonic forms of the basis (ωj , ω ˜ l )j,l exhibited by Theorem 3.1 are integrated along the Brownian paths, run during a same time t going to infinity. This yields on one ˜ l ), accounting for the Brownian windings around the handles, hand slow martingales (M t and on the other hand fast martingales (Mtj ), accounting for the Brownian windings around the cusps. Theorem 6.1 gives the joint asymptotic law of all these normalised martingales. Its statement is mainly as follows :  ν∞  (Γ)  . .√  X j l ` l ˜ t converges in law towards rj Q` , N , Theorem 6.1 Mt t , Mt j,l

`=1

j,l

,Nl

are independent, each Q` is Cauchy with parameter where all variables Q` h` l , N is centred Gaussian with variance h˜ ωl , ω ˜ l i, and the rj` are residues in the 2 V (Γ\H2 ) cusps. 7.) Geodesics of G = P SL2 (R) and ergodic measures The geodesics of G are described. They project on H2 as quasi-geodesics having constant speed. Thus T 1(Γ\G) appears as naturally foliated, with on each leaf an ergodic measure, image of the Liouville measure of Γ\G ≡ T 1 (Γ\ H2 ), introduced in Definition 7.1. On the contrary, in this non-hyperbolic structure, the geodesic flow is not ergodic with respect to the Liouville measure on T 1 (Γ\G). 8.) Asymptotic geodesic windings The martingales analysed in Theorem 6.1 are in this section replaced by the integrals of ˜ l)j,l , but along geodesic segments (of length t) instead of Brownian the same forms (ωj , ω paths. The geodesics are chosen according to the natural ergodic measures introduced in Section 7.. Theorem 8.1 describes the asymptotic law of the normalised geodesic windings produced in this way. Its statement is mainly as follows : Z  Z  −1 −1/2 Theorem 8.1 t ωj , t ω ˜l conγ[0,t]

γ[0,t]

j,l

92

Jacques Franchi

verges in law, under the ergodic measure µkε (dγ), to   ν (Γ) ∞  1 − k2 1/2 X  4(1 − k2 ) 1/4 (1 + a2 )k ` √ 1{j=0} + 2 r Q , N l , where ` j 1 + a2 k2 1 + a2k2 1 + a2 k 2 `=1 j,l

the limit random variables Q` , N l are as in Theorem 6.1, and a is the parameter of the metric. An interesting feature is the difference between the Brownian and geodesic behaviours, in noteworthy contrast with the hyperbolic case : counter to the Brownian case, the dθ-part of the form ω0 is responsible for a non-negligible asymptotic contribution, and the metric parameter a now appears in the limit law. 9.) Equirepartition in Γ\G of large geodesic spheres Corollary 9.1 asserts that the ergodic measures µkε of Section 7. and Theorem 8.1, are weak limit of the uniform law on large geodesic quasi-spheres. This is easily deduced from the following. Theorem 9.1 The normalized Liouville measure µΓ on T 1 (Γ\H2 ) ≡ Γ\G is the weak limit as R → ∞ of the uniform law on the geodesic sphere Γg P SO(2)ΘR of Γ \ G having radius R and fixed center Γg ∈ Γ\G : for any compactly supported continuous function f on Γ\G , denoting by d% the uniform law on P SO(2), we have Z Z f (Γg % ΘR ) d% . f dµΓ = lim R→∞ P SO(2)

The equidistribution theorem 9.1 and its proof (based on the mixing theorem) were already given by Eskin and McMullen in [E-MM]. 10.) Synthetic proof of Theorem 6.1 This proof is an adaptation of an analogous proof in ([F3], Section 10). Thus some details are here somewhat eluded, for which we refer to [F3]. However all ingredients are given, with a stress on differences with the particular case addressed in [F3], and the most involved arguments are detailed to a certain extent. The main difficulty of the whole proof, widely responsible for its length, is to establish the asymptotic independence between slow windings (about the handles) and singular windings (about the cusps). This demands in particular to get good approximation of the contribution of both type, and then to analyse carefully the successive excursions of Brownian motion in the core and in the cusps of the quotient hyperbolic surface. 11.) Proof of Theorem 8.1 The strategy for this proof is mainly to replace the geodesic paths by the Brownian paths, as in hyperbolic case ([LJ1], [LJ2], [E-LJ], [F2], [E-F-LJ1], [E-F-LJ2]), and as in [F3], in order to reduce Theorem 8.1 to Theorem 6.1. Here again, the analogous proof in ([F3], Section 14) is adapted.

2.

Iwasawa Coordinates and Metrics on G = P SL2 (R)

This section is mainly taken from [F3]. Consider the group√ G := P SL2 (R), which is classically parametrized by the Iwasawa coordinates (z = x + −1 y , θ) ∈ H2 × (R/2π Z) (H2 denotes as usual the hyperbolic

Asymptotic Homology of a Modular Quotient

93

plane, identified with the Poincar´e half-plane), in the following way : each g ∈ G writes uniquely g = g(z, θ) := ± n(x)a(y)k(θ) , where n(x) , a(y) , k(θ) are the one-parameter subgroups defined by : n(x) :=



1 x 0 1



, a(y) :=

√

y 0

0 √ 1/ y



, k(θ) :=



cos(θ/2) sin(θ/2) − sin(θ/2) cos(θ/2)



, (1)

and generated respectively by the following elements of the Lie algebra s`2 (R) : ν :=



0 1 0 0



, α :=



1/2 0 0 −1/2



, κ :=



0 1/2 −1/2 0



.

i h √ √ √ g = g(z, θ) ⇐⇒ g( −1 ) = z and g 0( −1 ) = y e −1 θ .   0 1/2 Set also λ := ν − κ = , which is natural, since α, λ are symmetrical 1/2 0 while κ is skew-symmetrical, the basis (α, λ, κ) of s`2 (R) the Killing form  and since in −2 0 0 is diagonal : it has matrix  0 −2 0  . 0 0 2

Note that

For this reason, we take on s`2 (R) the inner product such that the basis (α, λ, aκ) is orthonormal, for some arbitrary parameter a ∈ R∗. And since we want to work on an homogeneous space Γ\G , the Riemannian metric to be considered on G must be a least Γleft-invariant, and then a natural choice for the Riemannian metric on G is the left-invariant a metric, say ((gij )) , generated by the above inner product on s`2 (R) . The simple lemma below shows that this choice of metric(s) is geometrically canonical (up to a trivial multiplicative constant), G being seen as T 1H2 . This equips G ≡ T 1H2 , and its homogeneous spaces as well, with the 6th of the eight 3-dimensional geometries described by Thurston ([T]), and actually with a quasi-hyperbolic but not hyperbolic structure. Let us denote by Lν , Lα , Lκ , Lλ the left-invariant vector fields on G generated respectively by ν , α , κ , λ . A standard computation shows that Lλ = y sin θ

∂ ∂ ∂ ∂ ∂ ∂ ∂ +y cos θ −cos θ , Lα = y cos θ −y sin θ +sin θ , Lκ = . ∂y ∂x ∂θ ∂y ∂x ∂θ ∂θ

a )) defined above are, up to a multiplicative Lemma 2.1 The Riemannian metrics ((gij constant, the only ones on G which are left-invariant and also invariant with respect to the action of the (Cartan compact subgroup) circle exp(Rκ) = {k(θ)} . They are given in Iwasawa coordinates (y, x, θ) by :



y −2 a ((gij )) :=  0 0

0 (1 + a−2 )y −2 a−2 y −1

 0 a−2 y −1  . a−2

(2)

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Jacques Franchi

Proof

The left-invariant metrics on G are those which are given by a constant matrix ∂ ∂ ∂  ((aij )) in the basis L := (Lα , Lλ, Lκ ) . Set I := , , . We have I = LA , with ∂y ∂x ∂θ   −1 y cos θ −y −1 sin θ 0 A :=  y −1 sin θ y −1 cos θ 0 , so that the left-invariant metrics are given in the ba0 y −1 1 ∂ t t A((aij ))A = 0 . sis I by A((aij ))A . Among them, the ones we want have to satisfy ∂θ   1 0 0 0 , and A direct computation shows that this is equivalent to ((aij )) = c2  0 1 0 0 a−2 a then to ((gij )) being as in the statement.  Note that with these metrics any holomorphic form f (z)dz is coclosed, and then harmonic. The left Laplacian on G corresponding to the basis (α, λ, aκ) is the Beltrami Laplacian a )), and is given by associated with the metric ((gij ∆a := L2λ + L2α + a2L2κ = y 2

 ∂2 2 ∂2 ∂2  2 ∂ − 2y + (1 + a + ) . ∂y 2 ∂x2 ∂θ∂x ∂θ2

(3)

Note that Lλ and Lα generate the canonical horizontal left-invariant vector fields lifted from H2 to G, H2 being endowed with its Levi-Civita connexion, so that ∆0 is the Bochner ∂2 horizontal left Laplacian, and ∆a = ∆0 + a2 ∂θ 2 . dx dy dθ is bi-invariant, hence this is both the Haar measure The measure µ(dg) := 4π 2 y 2 of G and the Liouville measure of T 1 H2 . Recall that an isometry γ ∈ G is called respectively elliptic, parabolic, or loxodromic, according as it fixes a point in H2 , no point in H2 and a unique point in ∂ H2 = R ∪ {∞}, or no point in H2 and two points in ∂ H2 = R ∪ {∞}, respectively. Any isometry of H2 is either elliptic, or parabolic, or loxodromic. We shall use the following easy lemma. Lemma 2.2 Any parabolic or loxodromic isometry γ ∈ G can be written γ = ± exp(σ), for a unique σ ∈ sl2 (R). Proof Consider first a loxodromic γ ∈ G , and an isometry g ∈ G mapping two fixed boundary points of γ to {0, ∞}, so that we have for some t ∈ R∗ :  t      0 e t 0 −1 −1 g = ± exp g g . γ = ±g 0 e−t 0 −t     1 0 u v+w 2 2 2 2 , and then And if σ = ∈ sl2(R), then σ = (u + v − w ) 0 1 v−w −u     sin % sh % 1 0 1 0 σ or exp(σ) = (cos %) + σ, exp(σ) = (ch %) + 0 1 0 1 % % according as (u2 + v 2 − w2 ) =: ±%2 is non-negative or negative.

Asymptotic Homology of a Modular Quotient 95   t e 0 = ± exp(σ) , which Hence (setting σ := gσ 0g −1) : γ = ± exp(σ 0) ⇔ 0 e−t in the first case implies at once v = w = 0 , whence u = t , and in the second case : sin % = 0 , whence et = e−t , an impossibility, establishing the unicity of σ 0 . Consider then a parabolic γ ∈ G , and an isometry g ∈ G mapping its fixed boundary point to ∞ , so that we have for some x ∈ R∗ :       −1 1 x −1 0 x g = ± exp g g . γ = ±g 0 1 0 0   1 x 0 And γ = ± exp(σ ) ⇔ = ± exp(σ) , which in the first case (for exp(σ)) implies 0 1 at once u = v − w = 0 , whence v = x/2 , and in the second case : sin % = 0 , whence an impossibility, establishing again the unicity of σ 0. 

3.

Geometry of a Modular Homogeneous Space Γ\G

Consider the group G := P SL2(R), its full modular subgroup Γ(1) := P SL2(Z), and another modular subgroup Γ, that is, a subgroup of Γ(1) having finite index [Γ(1) : Γ]. As usual, let us identify G with the unit tangent bundle T 1H2 ≡ H2 × S1 of the hyperbolic plane H2 , and also with the group of M¨obius isometries (homographies z 7→ 2 2 az+b cz+d with ad − bc = 1) of H , that is the group of direct isometries of H . The elements u := (z 7→ −1/z) , v := (z 7→ (z − 1)/z) generate the group Γ(1), which admits the presentation {u, v | u2 = v 3 = 1}. Γ(1) is of course also generated by {u, vu = (z 7→ z + 1)}. Note that [Γ, Γ] is a free ofDΓ(1) := [Γ(1), Γ(1)], which is the  group,as a subgroup  2 1 1 1 free group generated by ± and ± . Generally [Γ : [Γ, Γ]] has not to 1 1 1 2 be finite, as shows the counterexample DΓ(1), for which DΓ(1)/[DΓ(1), DΓ(1)] ≡ Z2 . But since DΓ(1) is a normal subgroup of Γ(1) such that Γ(1)/DΓ(1) ≡ Z/6Z , then ˜ := Γ ∩ DΓ(1) Γ . ˜ ≡ Γ · DΓ(1) DΓ(1) is a subgroup of is a free and normal subgroup of Γ such that Γ/Γ ˜ is isomorphic to a subgroup of Z/6Z . Γ(1)/DΓ(1), so that Γ/Γ We are interested in the modular homogeneous space Γ\G . The identification of G with the unit tangent bundle T 1 H2 allows to identify similarly this modular homogeneous space with the unit tangent bundle of the corresponding Riemann surface Γ\ H2 : Γ\G ≡ Γ\ T 1H2 ≡ T 1(Γ\ H2). dx dy dθ (which is also a right and 4π 2 y 2 left Haar measure on G) onto Γ\G is proportional to the volume measure V of Γ\G . By The projection of the Liouville measure µ(dg) =

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a the choice of the metric ((gij )), the volume of Γ\G is clearly V (Γ\G) = 2

2π |a| 2

× covol(Γ),

where covol(Γ) = V (Γ\ H ) denotes the finite hyperbolic volume of Γ\ H . 2π Let µΓ := covol(Γ) µ denote the normalized projection of µ on Γ\G , identified Γ\G with a law on left Γ-invariant functions on G.

3.1. From Γ\H2 to Γ\G The following lemma ensures that the lift to the unit tangent bundle increases the first Betti number of the Riemann surface Γ \ H2 by exactly one. I thank T. Delzant for having explained to me why, so that I owe to him this lemma. Lemma 3.1 The modular homogeneous space Γ\G ≡ T 1(Γ\ H2 ) is diffeomorphic to (Γ\ H2 ) × S1 . Consequently, we have the following simple relation between the first Betti numbers of Γ\G and of Γ\ H2 : dim [H 1(Γ\G)] = 1 + dim [H 1(Γ\ H2 )]. Proof As a cover of Γ(1)\H2, the Riemann surface Γ\H2 is orientable and non-compact. As is known for any orientable non-compact smooth manifold, it carries a smooth non  vanishing vector field, hence a smooth cross section x 7→ (x, ~vx) ∈ Tx1(Γ \ H2 ) of \ vx, ~v) in the oriented plane Tx1 (Γ\H2 ), we T 1 (Γ\H2 ). Denoting by α = αx (~v) the angle (~ get the diffeomorphism : (x, ~v) 7→ (x, α) from T 1(Γ\ H2) onto (Γ\ H2 ) × S1 .  Furthermore, under the canonical projection π : Γ\G → Γ\H2 ≡ Γ\G/ exp(Rκ), the harmonic space H 1(Γ\ H2 ) (that is, the space of real harmonic forms on Γ\ H2) is pulled back to the subspace π∗ [H 1(Γ\H2 )] of the harmonic space H 1(Γ\G), which is isomorphic to H 1(Γ\ H2 ). Thus, to describe the harmonic 1-forms of the modular homogeneous space Γ\G , once the harmonic 1-forms of the Riemann surface Γ\ H2 are known, by the above lemma 3.1 it / π∗ [H 1(Γ\H2)]. is sufficient to produce a harmonic 1-form ω0 ∈ H 1(Γ\G), such that ω0 ∈ Now, such harmonic 1-form ω0 was computed in ([F3], Section 6), as the restriction to Γ\G of a harmonic 1-form on Γ(1)\G :   ω0 := dθ + 4 Im(η 0(z)/η(z)) dx + 4 Re(η 0(z)/η(z)) dy = d θ + 4 arg(η(z)) , (4) where η denotes the Dedekind function, defined on H2 (seen as the Poincar´e half-plane) by : √ √ Y (1 − e −1 2π n z ) . η(z) := e −1 π z/12 × n∈N ∗

3.2. Geometry of the Riemann Surface Γ\H2 Let us describe now the harmonic space H 1(Γ\H2) of the Riemann surface Γ\H2 . Denote by g(Γ) the genus of Γ\ H2, and by ν∞ (Γ) := Card(Γ\ Q) the number of its cusps, that is, the number of Γ-inequivalent parabolic points of Γ. Note that clearly ν∞ (Γ) ≥ 1 . Let us denote these cusps by C1 , .., Cν∞(Γ) , choosing Cν∞ (Γ) = Γ∞ to be the one cusp associated to the particular parabolic point ∞ .

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97

The Riemann surface Γ \ H2 decomposes into the disjoint union of a compact core, which is a compact surface having genus Γ\ H2 and a boundary made of ν∞ (Γ) pairwise disjoint circles S1 , and of ν∞ (Γ) pairwise disjoint ends, each being diffeomorphic to S1 × R∗+ and associated with one of the cusps C` , which we call “solid cusp” and denote also by C` . Recall now that the harmonic space H 1(Γ\ H2 ) is the dual of the first real singular homology space H1 (Γ\H2 ) (see for example ([D], 24.33.2)), so that the above decomposition implies the formula : dim [H 1(Γ\ H2 )] = 2 g(Γ) + ν∞ (Γ) − 1 .

(5)

On the other hand, an automorphic form f of weight 2 with respect to Γ induces the holomorphic differential f (z) dz on Γ\H2. More precisely, let us denote as Miyake ([M]) by G2 (Γ) the complex vector space of those automorphic forms f (of weight 2) which are holomorphic on H2 and at the cusps of Γ, and by S2(Γ) the subspace of so-called “cusp forms”, that is of forms in G2 (Γ) which vanish at the cusps of Γ. The so-called Petersson inner product (see for example ([M], Section 2.1)) is defined for (f1 , f2) ∈ S2(Γ) × G2(Γ), by : Z 2 −1 hf1 (z)dz, f2(z)dzi = hf1 , f2i := V (Γ\ H ) f1(z) f2 (z) dz . (6) Γ\H 2

Note that y |f (z)| is the natural norm of f (z)dz ∈ Tz∗ (Γ \ H2 ), induced by the volume dxdy measure of Γ\H2 ; so that the differential |f (z)|2 dz = kf (z)dzk2 2 , integrated over y 2 ∗ Γ\ H , indeed computes precisely the global norm of f (z)dz ∈ T (Γ\ H2 ). As in [M] again, let N2 (Γ) denote the orthogonal complement of S2 (Γ) in G2 (Γ), with respect to the Petersson inner product. Then ([M], Theorem 2.5.2) states the following : dimC [S2(Γ)] = g(Γ)

and

dimC [N2 (Γ)] = ν∞ (Γ) − 1 .

(7)

Note that the −1 in the second formula is natural, since the sum of the residues of a harmonic differential form has to be zero. As a consequence, comparing (5) and (7), we see that the regular part of the harmonic space H 1(Γ\ H2 ), that is the part due to the handles, admits a basis made of 2 g(Γ) real harmonic forms Re[f (z) dz], with f ∈ S2(Γ) a cusp form ; and that the singular part of the harmonic space H 1(Γ \ H2 ), that is the part due to the cusps and having residues at the cusps which are not all null, admits a basis made of (ν∞ (Γ) − 1) real harmonic forms Re[f (z) dz] , with f ∈ N2 (Γ). Notice that, specifying that the singular harmonic forms must be real and orthogonal to the regular harmonic forms (and then in particular have non all vanishing residues), we get indeed (ν∞ (Γ) − 1) independent such differential forms, and not 2(ν∞ (Γ) − 1).

3.3. Geometry of the Modular Space Γ\G Note that y is naturally the height in the solid cusp Cν∞ (Γ), or in the corresponding end T 1 Cν∞ (Γ) of Γ\G as well. Similarly, for any solid cusp C` , by using a M¨obius isometry

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mapping it on Cν∞ (Γ) , we get a Γ-invariant function y˜` on G or on H2 , which yields a canonical height in the solid cusp C` (or in T 1 C` as well). Similarly yet, we have canonical coordinates (˜ y` , x ˜` , θ) on the end T 1 C` , with |d˜ x` | = |d˜ y` | = y˜` and |dθ| ≡ 1 . Let us sum up the above, ([F3], Theorem 1) and ([M], Corollary 2.1.6) in the following. Theorem 3.1 The real harmonic space H 1(Γ\G) of the  modular homogeneous space ˜1 , .., ω ˜ 2g(Γ) , where ω0 is given by (4), each Γ\G admits a basis ω0 , ω1 , .., ω(ν∞(Γ)−1) , ω ωj (for 1 ≤ j < ν∞ (Γ)) equals Re[fj (z) dz] for some automorphic form fj ∈ N2 (Γ), and the ω ˜ j are pairwise orthogonal and can be writen Re[f˜j (z) dz] for some cusp form ˜ fj . Moreover, the ω ˜ j are bounded, and we have the following behaviours near the cusps : π ω0 = dx + dθ + O(ye−2π y ) near the cusp Cν∞ (Γ) , id est for y → ∞ ; 3 x` + O(˜ y` e−π y˜` /h` ) ωj = rj` d˜

near the cusp C` , id est for y˜k → ∞ ;

rj` denoting the residue of the harmonic form ωj at the cusp C` (so that, in particular, we have for ω0 : r0` = π3 1{`=ν∞ (Γ)}), and h` > 0 denoting the width of the solid cusp C` , determined by : ± n(h` ) (recall Formula (1)) is conjugate in G to a generator of ν∞ (Γ) P rj` = 0 , for the parabolic subgroup of Γ associated with the cusp C` . We have `=1

1 ≤ j < ν∞ (Γ). Note in particular that, while the regular harmonic forms ω ˜l belong to L2 (Γ\G), on the contrary the singular harmonic forms ωj do not belong to L1 (Γ\G). This is not surprising, since the ω ˜ l calculate slow windings about the handles of the manifold Γ\G, whereas the ωj calculate fast windings about the cusps of the manifold Γ\G. Note also that lifting to the unit tangent bundle has also the advantage to free the harmonic forms of the constraint to have a null sum of their residues at the cusps of Γ \ H2 ν∞ (Γ) P (which are also the cusps of Γ\G) : the constraint rj` = 0 , which holds for harmonic `=1

(and closed, by Proposition 4.1 below) forms on Γ\H2 , does not hold any longer at the level of Γ\G, since it breaks down in particular for the harmonic form ω0 . Furthermore, the genus and the volume of Γ\H2 can be expressed in terms of two more parameters, the numbers ν2 (Γ) and ν3 (Γ) of Γ-inequivalent elliptic points of Γ, of order 2 and 3 respectively. The genus of Γ\ H2 is given by the formula ([M], Theorem 4.2.11) : g(Γ) = 1 +

1 12

[Γ(1) : Γ] −

1 4

ν2 (Γ) −

1 3

ν3 (Γ) − 12 ν∞ (Γ) .

(8)

The volume of Γ\ H2 is given by the formula ([M], Theorem 2.4.3) : V (Γ\ H2 ) = 2π × [2 g(Γ) − 2 + ν∞ (Γ) +

1 2

ν2 (Γ) + 23 ν3 (Γ)] .

(9)

In the particular case of principal congruence groups Γ(N ), very explicit formulae for [Γ(1) : Γ(N )] and for νj (Γ(N )) are known (see ([M], Section 4.2)), and there exists a precise description of the space N2 (Γ) in terms of analytic continuations of Eisenstein series (see ([M], Section 7.2)).

Asymptotic Homology of a Modular Quotient

4.

99

Closed Forms and Harmonic Forms

Recall that Γ was neither supposed to be a congruence subgroup, nor to be normal. But ˜ := Γ ∩ DΓ(1) is a free and normal subgroup recall from the beginning of Section 3. that Γ ˜ is isomorphic to a subgroup of Z/6Z. Beginning by a comparison of Γ, such that Γ/Γ ˜ , we shall in this section deduce that between the geometries associated with Γ and Γ their cohomology spaces (of 1-forms) identify with their harmonic spaces H 1, so that the asymptotic study of closed 1-forms will reduce to the asymptotic study of harmonic 1forms. By Formulas (5) and (8) we have : 2

d := dim [H 1 (Γ\ H )] = 2g(Γ) + ν∞ (Γ) − 1 and g(Γ) = 1 +

1 12

[Γ(1) : Γ] −

1 4

˜ H2 )] = 2g(Γ) ˜ + ν∞ (Γ) ˜ − 1, d˜ := dim [H 1 (Γ\

ν2 (Γ) −

1 3

ν3 (Γ) −

1 2

ν∞ (Γ) ;

and then ˜ = 1 + 1 [Γ(1) : Γ] ˜ − 1 ν2 (Γ) ˜ − 1 ν3 (Γ) ˜ − 1 ν∞ (Γ) ˜ g(Γ) 12 4 3 2   1 ˜ ≥ 1 + [Γ : Γ](g(Γ) ˜ [Γ(1) : Γ] − 12 ν∞ (Γ) × [Γ : Γ] − 1) , ≥ 1 + 12

˜ cannot have any elliptic point, and on the other hand, since on one hand the free group Γ recalling the definition of ν∞ (Γ) (in Section 3.2.), we must have : ˜ = Card(Γ\ ˜ Q) ∈ [ν∞ (Γ), [Γ : Γ] ˜ × ν∞ (Γ)] . ν∞ (Γ) From the above we deduce at once : d = dim [H 1(Γ\ H2 )] = 1 + [Γ(1) : Γ]/6 −

1 2

ν2 (Γ) −

2 3

ν3 (Γ) ≤ 1 + [Γ(1) : Γ]/6 ,

and ˜ H2 )] = 1 + [Γ(1) : Γ]/6 ˜ d˜ = dim [H 1(Γ\ ≥ 1 + [Γ(1) : Γ]/6 ≥ d . Note that we can have d < d˜, as shows the simple example Γ = Γ(1), for which ˜ = g(Γ) ˜ = 1 , d˜ = 2 . d = g(Γ) = 0 , ν∞ (Γ) = ν∞ (Γ) ˜ and if γ ∈ Γ represents a generator Furthermore, if F is a free set of generators of Γ, ˜ then Γ is generated by F t{γ}, with a relation γ n = φ(F ) (where n ∈ {1, 2, 3, 6} of Γ/Γ, ˜ Hence the Abelianized Ab(Γ) := Γ/[Γ, Γ] admits the same repreis the order of Γ/Γ). . ˜ := Γ ˜ [Γ, ˜ Γ] ˜ is sentation as a free Abelian group, meaning that the Abelianized Ab(Γ) isomorphic to a subgroup of Γ/[Γ, Γ], such that the quotient be cyclic (of order dividing b of additive (real) characters of Γ, we find that : n). Hence, considering the group Γ \ \ ≡ Ab( b ≡ Ab(Γ) ˜ ≡ Zd0 , Γ Γ)

. ˜ =Γ ˜ [Γ, ˜ Γ] ˜ ≡ Zd0 , or equivawhere d0 is the dimension of the free Abelian group Ab(Γ) ˜ lently, the number of generators of Γ. Now we have the following important fact, which generalizes Proposition 2.2 of [G-LJ], and, allowing the identification of the cohomology and of the harmonic space of Γ \ H2 , allows mainly to focus on harmonic forms the asymptotic study of integrals of closed forms.

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˜ equals the Proposition 4.1 (i) The number of generators of every free modular group Γ 2 0 1 ˜ ˜ dimension of the corresponding harmonic space : d = d := dim [H (Γ\ H )]. (ii) For any modular group Γ, every C 1 closed form on Γ\ H2 is cohomologous to a (unique) harmonic form, element of H 1(Γ\ H2 ). Proof ` from defines ω ˜ (`) :=

˜ \ H2 , b ∈ ˜b , and 1) As ([G-LJ], Prop 2.2), fix a base-point ˜b ∈ Γ 2 ˜b to ˜b , associate its lift `¯ in H started from b , and `(b) ¯ = `˜b 1 0 0 ˜ ˜ ˜ e ˜ ˜ on `Z ∈ Γ satisfying Z Zll = l l . Then for any C closed form ω ω ˜=

`

ω ˜=

π ˜ ◦`¯

`¯

to any loop ∈ ˜b . This ˜ \ H2, set Γ

π ˜∗ω ˜ , where π ˜ denotes the covering projection from H2 onto

˜ H2 , which induces a pullback π ˜∗ , mapping ω ˜ to a closed form on H2 . As H2 is simply Γ\ ¯ ˜ thereby defining an connected, ω ˜ (`) is a function of `(b), so that we can set ω ˜ (`) =: ω b (`), ˜ additive character ω b on Γ. Z ·

Now if ω b = 0 , meaning that ω ˜ is exact, then the primitive

ω ˜ is defined on the

˜ b

˜ H2 (since it is arcwise connected), and then constant (as any holomorphic modular whole Γ\ function on Γ\ H2 ; see for example ([L], VI.2.E)), proving that ω ˜ = 0 . Hence, the linear 1 ˜ \ H2 into Γ, b and, a map ω ˜ 7→ ω b is one-to-one from the space of C closed forms on Γ 2 1 0 ˜ ˜ b fortiori, from H (Γ\ H ) into Γ, proving that d ≤ d . 2) Reciprocally, notice that the map ` 7→ `˜ defined above induces a morphism ϕ from ¯ ˜ H2 ) into Γ, ˜ since any homotopy (`s ) from ` to `0 defines a path (`¯s (b)) from `(b) π1 (Γ\ 0 0 ˜ ˜ ˜ ˜ ¯ ¯ to ` (b) in the discrete Γ b , forcing ` = ` . And if ` = 1 , then `(b) = b in the simply connected H2 , so that `¯ is homotope to the constant loop b , ` is homotope to the constant ˜ H2 ) into Γ. ˜ loop ˜b , and thus we have a one-to-one morphism ϕ from π1(Γ\ ˜ Moreover, since Γ is discrete and free, it contains only parabolic and loxodromic isome˜ can be written γ = ± exp(σ), for a unique tries, and then by Lemma 2.2, any γ ∈ Γ σ ∈ sl2(R). Setting `(γ)(s) := ± exp(s σ)(˜b) for 0 ≤ s ≤ 1 and taking the homotopy ˆ ∈ π1 (Γ\ ˜ H2 ) of `(γ), we get a pre-image for γ, with respect to the above morclass `(γ) phism ϕ , which is thus onto. By duality ([D], 24.33.2), this yields a one-to-one morphism ˜ H2), hence d0 ≤ d˜. b into H 1(Γ\ from Γ 0 ˜ H2 ) onto Hence d˜ = d , and we have exhibited an isomorphism ω ˜ 7→ ω b from H 1(Γ\ b Γ. ˜ H2), and consider any C 1 closed form ω ˜ H2 , ˜ d0 ) of H 1(Γ\ ˜ on Γ\ 3) Fix a basis (˜ ω1, .., ω b By the above, we have reals α1 , .., αd0 , such which, as described above, defines ω b ∈ Γ. 0 d d0 X X αj ω bj , implying, as seen above, that ω ˜− αj ω ˜ j be exact. that ω b= j=1

j=1

So far, we have proved (i) of the statement, and (ii) for the case of a free modular group ˜ Γ. ˜ \ H2 onto Γ \ H2 , which induces a 4) Let p˜ denote the covering projection from Γ pullback p˜∗ , mapping closed smooth differential forms on Γ \ H2 to closed smooth dif˜ \ H2 , and harmonic forms on Γ \ H2 to harmonic forms on Γ ˜ \ H2 , ferential forms on Γ 2 2 1 1 ˜ that is, H (Γ \ H ) into H (Γ \ H ). Note that p˜∗ is necessarily one-to-one (proving ˜ : indeed, if a closed smooth form ω belongs to its kernel, then we have againZ that d ≤Zd) ˜ H2 . Now, since the order of the covering group ω for any loop `˜ on Γ\ 0 = p˜∗ω = `˜

p˜◦`˜

Asymptotic Homology of a Modular Quotient

101

˜ divides 6, for any ` ∈ H1(Γ\ H2 ) the lift of `6 to Γ\ ˜ H2 is also a loop, meaning that Γ/Γ Z Z ˜ H2). Hence 0 = `6 ∈ p˜ ◦ H1 (Γ\ ω = 6 ω for any loop ` on Γ\ H2, implying that `6

`

ω must be exact. Considering a primitive of ω and using that any holomorphic modular function on Γ\ H2 is constant (see for example ([L], VI.2.E)), we get ω = 0 . ˜ H2 ) such that ω1 , .., ω ˜ d0 ) of H 1 (Γ\ 5) The injectivity of p˜∗ allows to choose the basis (˜ 2 ˜ d) be the image under p˜∗ of a basis (ω1 , .., ωd) of H 1 (Γ\ H ) : ω ˜ j = p˜∗ ωj for (˜ ω1 , .., ω ˜ ˜ ˜ 1 ≤ j ≤ d . Let us fix a dual basis (`1, .., `d0 ), that is, a basis of H1(Γ \ H2) such that Z `˜k

ω ˜j = 1{j=k} for 1 ≤ j, k ≤ d0 .

In particular, for 1 ≤ j ≤ d we have : 1{j=k} =

Z `˜k

ω ˜j =

Z

ωj , implying on one

p˜◦`˜k ˜ 2

hand, for k > d , that p˜◦ `˜k ≡ 0 , and then that `˜k is a lift to H1(Γ\H ) of a loop homotope ˜ \ H2 ) of a basis to 0 in Γ \ H2 , and on the other hand, that (`˜1, .., `˜d) is the lift to H1 (Γ (`1, .., `d) of H1 (Γ\ H2 ). 6) Consider any C 1 closed form ω on Γ\ H2 . By 3) above, we can write 0

p˜∗ω =

d X

αj ω ˜j + dF ,

˜ H2 ). for some F ∈ C 2 (Γ\

j=1

IntegratingZthis relation along the loop `˜k , using 5) above, gives at once αk = 0 for k > d , and αk =

ω for 1 ≤ k ≤ d .

`k

Hence ω ¯ := ω −

d X

αj ωj is a C 1 closed form ω on Γ\H2 , vanishing on H1 (Γ\H2 ),

j=1

˜ H2 , γ ∈ Γ, and any arc c joining ˜b to γ ˜b in and such that p˜∗ ω ¯ = dF . For any Z˜b ∈ Γ\ Z Z 2 ˜ ˜ ˜ ¯= ω ¯ = 0 , since p˜ ◦ c is a loop in Γ\H , we have : F (γ b) − F (b) = dF = p˜∗ ω c

c

p˜◦c

Γ\H2 . This proves that F is Γ-invariant, hence that F = F˜ ◦ p˜ for some F˜ ∈ C 2 (Γ\H2 ). d X ω − dF˜ ) = 0 , whence ω = αj ωj + dF˜ , by 4) above.  Finally we have got : p˜∗(¯ j=1

Moreover, the preceding proposition 4.1 lifts to the modular homogeneous space Γ\G : we can also identifiy the cohomology and the harmonic space of Γ\G , and then focus on harmonic forms the asymptotic study of integrals of closed forms on Γ\G . Proposition 4.2 For any modular group Γ, every C 1 closed form on Γ\G is cohomologous to a harmonic form, element of H 1(Γ\G). Proof Recall from Section 3.1. the canonical projection π : Γ \ G → Γ \ H2 , whose pullback π∗ maps in a one-to-one way the closed forms on Γ\H2 to closed forms on Γ\G , and the harmonic space H 1(Γ \ H2 ) to the isomorphic subspace π∗ [H 1(Γ \ H2 )] of the harmonic space H 1(Γ\G). Fix b = Γ · (y, x, θ) ∈ Γ \ G, and denote by `0 a loop above π(b), generating H1 (π −1(b)). Fix also a basis (`1, .., `d) of H1 (Γ \ H2 ), dual to the basis (ω1 , .., ωd) of

102

Jacques Franchi

H 1(Γ\H2). Identify (`1, .., `d) with its lift to H1(Γ\G), prescribing merely to each `j the constant third Iwasawa coordinate θ . By Theorem 3.1 and by duality ([D], Z (see for example Z 24.33.2)), (`0, `1, .., `d) is a basis of H1 (Γ\G) : indeed, we have π∗ ω = ω=0 `Z π◦`0 0 Z ω0 = dθ = 2π . for any closed form ω on Γ\ H2, whereas by (4) we have Consider now any C 1 closed form ω on Γ\G , and set : 0

ω := ω −

0



1 2π

Z



ω ω0 − `0

1

ω is a C closed form on Γ\G such that

d Z X j=1

Z

`0

`0



ω π∗ ωj .

`j

ω = 0 for any ` ∈ H1(Γ\G), and then for `

any loop ` on Γ\G. Hence it is exact. 

5.

Left Brownian Motion on G = P SL2 (R)

This short section is taken from [F3]. Brownian motion gs = g(zs , θs ) = g(ys , xs, θs ) on G has infinitesimal generator 12 ∆a and is the left Brownian motion solving the Stratonovitch stochastic differential equation d gs = gs ◦ (λ dYs + α dXs + κ a dWs) , where (Ys , Xs , Ws) denotes a 3-dimensional standard Brownian motion. Since a direct calculation shows that g(z, θ)−1dg(z, θ) = (sin θ dy + cos θ dx)

α κ λ + (cos θ dy − sin θ dx) + (y dθ + dx) , y y y

we get the differential system dys = ys sin θs ◦ dYs + ys cos θs ◦ dXs = ys sin θs dYs + ys cos θs dXs , dxs = ys cos θs ◦ dYs − ys sin θs ◦ dXs = ys cos θs dYs − ys sin θs dXs , dθs = a dWs − cos θs ◦ dYs + sin θs ◦ dXs = a dWs − cos θs dYs + sin θs dXs . Setting dUs := sin θs dYs + cos θs dXs and dVs := cos θs dYs − sin θs dXs , we get a standard 3-dimensional Brownian motion (Us , Vs, Ws) such that dys = ys dUs , dxs = ys dVs , dθs = a dWs − dVs . Hence we see that the projection of our Brownian motion gs = (ys , xs, θs ) on the hyperbolic plane H2 , that is to say on the Iwasawa coordinates (y, x) , is simply the standard hyperbolic Brownian motion of H2, and that the angular component (θs ) is just a real Brownian motion with variance (1 + a2 ). Remark 5.1 The degenerate limit-case a = 0 is quite possible for the left Brownian motion (gs ) . It corresponds to the Carnot degenerate metric on G, and to the horizontal left Brownian motion on G, associated with the Levi-Civita connexion on H2.

Asymptotic Homology of a Modular Quotient

6.

103

Asymptotic Brownian Windings in Γ\G

Let us denote by Mtj

:=

Z

ωj

and

˜ l := M t

g[0,t]

Z

ω ˜l

(10)

g[0,t]

˜l along the paths of the martingales obtained by integrating the harmonic forms ωj and ω the left Brownian motion (gs ). Note that we may as well consider the Brownian motion (gs ) as living on G or on Γ\G . Section 5. and Theorem 3.1 show that Z t  0 0 0 4 Re ηη (xs , ys ) ys dUs + (4 Im ηη (xs , ys ) ys − 1) dVs , Mt = a Wt + 0

Mtj =

Z t

Refj (xs , ys ) ys dVs − Imfj (xs , ys ) ys dUs



for 1 ≤ j < ν∞ (Γ),



for 1 ≤ l ≤ 2g(Γ).

0

and ˜l = M t

Z t

Re f˜l (xs , ys ) ys dVs − Im f˜l (xs , ys ) ys dUs

0

˜ l )1≤l≤2g(Γ) converges towards the centred Gaussian Lemma 6.1 The law of t−1/2 (M t ˜ li. law with diagonal covariance matrix having (on its diagonal) the variances h˜ ωl , ω Proof

˜sl ) such that By the above, we have some real Brownian motion (B   Z t  l l l l 2 ˜ ˜ ˜ ˜ |˜ ωj (zs )| ds , Mt = B hM it = B 0

and then by scaling, we have the following identity in law (for each t > 0) : Z t     2 ˜ l hM ˜ l t−1 ˜ lit /t = B ˜l ≡ B |˜ ω (z )| ds , t−1/2 M l s t 0

which by ergodicity converges almost surely to Z     2 −1 l ˜ ˜ l h˜ B V (Γ\ H ) |˜ ωl |2 dV = B ˜li . ωl , ω Γ\H2

Moreover, by ergodicity and by orthogonality of the different ω ˜ l , for 1 ≤ l < ` ≤ 2g(Γ), we have : ˜ l, M ˜ l0 it −→ h˜ ωl , ω ˜ l0 i = 0 . t−1 hM Hence Knight’s Theorem (see ([R-Y], XIII, 2, Corollary (2.4))) implies the asymptotic ˜ l).  independence of the martingales (M t The following theorem describes the asymptotic Brownian windings in Γ\G .

104 

Theorem 6.1 As t → ∞, Mtj wards

 ν∞ (Γ) X

rj` Q`

,N

l

Jacques Franchi .√  ˜l t t, M t

.

0≤j<ν∞ (Γ), 1≤l≤2g(Γ)



converges in law to-

, where all variables Q` , N l are indepen-

0≤j<ν∞ (Γ), 1≤k≤2g(Γ)

`=1

h` dent, each Q` is Cauchy with parameter 2 V (Γ\ , and N l is centred Gaussian with H2 ) ˜ li. Here h` > 0 denotes the width of the solid cusp C` (already defined in variance h˜ ωl , ω Theorem 3.1).

Observe the irrelevance of the parameter a in this theorem, which is valid as well in the ∂ degenerate case a = 0 . The reason is that a was initially the inverse norm of Lκ = ∂θ , which comes in only in the differential form ω0 , and which contributes there only to a second order term. Remark 6.1 Theorem 6.1 wellfor all finite dimensional marginals : . is true as .√  j l ˜ t , for any given as t → ∞, M cn t t , M cn t N ∈  ν∞ (Γ) X `=1

N∗ and 0 rj` Q`cn , Ncln



0≤j<ν∞ (Γ),1≤l≤2g(Γ), 1≤n≤N

<

c1

<

..

<

cN ,

converges jointly towards

, where all processes Q` , N l are inde-

0≤j<ν∞ (Γ), 1≤k≤2g(Γ),1≤n≤N

pendent, each Q` is Cauchy with parameter variance h˜ ωl , ω ˜ l i, started from 0.

h` 2 V (Γ\H2 )

, and N l is real Brownian with

Note that such statement gives at once the asymptotic law of any finite family of stochastic integrals (along Brownian paths) of harmonic forms, merely by decomposing them in ˜ k ). the basis (ωj , ω By Proposition 4.2, this gives also the asymptotic behavior of any finite family of stochastic integrals (along Brownian paths) of smooth closed forms, since the normalized contribution of any exact form is clearly negligible in probability (at least for the stationary Brownian motion).

7.

Geodesics of G = P SL2 (R) and Ergodic Measures

This section is similar to ([F3], Sections 11,12).

7.1. Description of These Geodesics a )) (indexed by a ∈ R∗ ) of G , as exRecall from Section 2. and Lemma 2.1 the metric ((gij ˙ , given in Iwasawa coordinates (y, x, θ) pressed by the Lagrangian L = L(y, x, θ, y, ˙ x, ˙ θ) by :

2 L = y −2 y˙ 2 + (1 + a−2 ) y −2 x˙ 2 + 2a−2 y −1 x˙ θ˙ + a−2 θ˙2 .   ∂ ∂L ∂L = ∂z The equation of geodesics ∂s j reads here : ∂ z˙ j ˙ · = −y −3 y˙ 2 − (1 + a−2 ) y −3 x˙ 2 − a−2 y −2 x˙ θ˙ , (y −2 y)

(11)

(12)

Asymptotic Homology of a Modular Quotient

105

and (1 + a−2 ) y −2 x˙ + a−2 y −1 θ˙ = c0

y −1 x˙ + θ˙ = c ,

and

(13)

for two constants c0, c . Eliminating θ˙ , this gives : x˙ = c0y 2 − c a−2 y

and

Eliminating then c a−2 gives :

(y −2 y) ˙ · = −y −3 (y˙ 2 + x˙ 2 ) − c a−2 y −2 x˙ ,

(14)

· = −c0 x (y/y) ˙ ˙ , or equivalently, for some constant c00 :

y/y ˙ = c00 − c0 x . Now this implies

1 2

 x˙ 2 + y˙ 2 

d

y2

(15)

0 0 = (x/y)c ˙ dy − (y/y)c ˙ dx = 0 , whence for some non-

negative constant C : (c0 x − c00)2 + (c0y − c a−2)2 =

x˙ 2 + y˙ 2 = C2 , y2

θ˙ = c (1 + a−2 ) − c0 y . (16)

and

In the particular case c0 = 0 , we find c00x + c a−2 y = c0 constant and θ˙ = c(1 + a−2 ). Hence we see that any geodesic projects on a Euclidian circle or line of H2, and that its projection has constant speed C (or energy C 2 ). Precisely if c0 6= 0 : c0 x = c00 + C sin ϕ ,

c0 y = c a−2 + C cos ϕ , and

ϕ˙ = c0 y ,

(17)

where the last formula results at once from the two preceding ones and from Formula (15). If C 6= 0 , set (18) k := c a−2C −1 . Then if k 6= 1 , according as |k| > 1 or |k| < 1, we have : ϕs = 2 arctg

"

q

k+1 k−1



tg C

#

√ k 2 −1 2

(s − s0 )

or ϕs = 2 arctg

"

q

1+k 1−k



th C

#

√ 1−k 2 2

(s − s0 )

.

For |k| > 1, the geodesic projects on a (Euclidian and hyperbolic) circle totally included in H2, and for |k| < 1, the geodesic projects on a quasi-geodesic of H2 (which is a geodesic if and only if k = 0). In the limiting case |k| = 1, the geodesic projects on an horocycle of H2, and ϕs = 2 arctg [C(s − s0 )]. Note moreover that by Equations (11), (13), (16), and (18), we have : 2L =

˙ 2 x˙ 2 + y˙ 2 −2 x ˙ = C 2 + a−2 c2 = (1 + a2 k2 )C 2 , + θ + a y2 y

so that prescribing constant speed one to the geodesics implies : C = (1 + a2 k2 )−1/2 .

(19)

We have in particular established that the energy of any geodesic of Γ\G splits into the constant energy C 2 of its projection on Γ\ H2 and the constant energy 1 − C 2 = a2 k2 C 2 of its angular windings about its projection.

106

Jacques Franchi

Remark 7.1 The case of main interest for the following is |k| < 1, that is, when the quasi-geodesic γ ˜ of H2 we get intersects ∂ H2 in two end-points. It is sufficient to consider the case of these two end-points are on the real line, that is, when the quasi-geodesic γ˜ is 0 a circle, of radius R = C/|c0| and centre at √ height y0 = kC/c . Then the geodesic ψ(γ) 2 having the same end-points has radius R 1 − k , and the orthogonal projection ps on ψ(γ) of the point ms ∈ √ γ˜ having angular coordinate ϕs has angular coordinate αs , 1−k2 sin ϕs ˜ is C = |m ˙ s | = R|ϕ˙ s |/(y0 + determined by : sin αs = 1+k cos ϕs . As the speed of γ R cos ϕs ) = |ϕ˙ s |/(k + cos ϕs ), we find that q the geodesic ψ(γ) must be run at constant √ 1−k2 2 speed |p˙s | = |α˙ s / cos αs | = C 1 − k = 1+a 2 k 2 , in order that the distance from ms to ps remain constant ; necessarily equal to argch[(1 − k2 )−1/2], by standard computation. Let us consider the coordinate system (y, x, θ, u, v, w) on T 1G , where (y, x, θ) are the Iwasawa coordinates of the base point γ ∈ G , and (u, v, w) are the coordinates of the unit ∂ ∂ ∂ , y ∂x , ∂θ ) of Tγ1G . Thus we have u2 +v 2 +a−2 (v +w)2 ≡ tangent vector in the basis (y ∂y 1. Now, consider the geodesic (γs ) determined by the initial value (γ0, γ00 ) ∈ T 1 G having coordinates (y0 , x0, θ0, u0, v0, w0), and let k be the unique real number determined by the k (1+a2 ) k a2 , and ε := sign((1 + a2 )v0 + w0) = sign( √ − w0 ) = equation v0 + w0 = √1+a 2 k2 1+a2 k 2 0 sign(c ). Then by Equations (16), (13), (18), (19), the above implies that (γs, γs0 ) remains in the leaf L(k, ε) having equations : L(k, ε) :=



u2 + v 2 =

1 1 + a2 k 2

\

v+w = √

k a2 1 + a2 k 2

\  sign((1 + a2 )v + w) = ε . (20)

7.2. Ergodic Measures for the Geodesic Flow on Γ\G We know by the above section 7.1. that any ergodic invariant measure for the geodesic flow on Γ\G must be carried by a leaf L(k, ε), for some real k and ε = ±1 . Note that each geodesic corresponding to the case |k| > 1 projects on a periodic curve (circle) in H2 , so that there is too few to be said on the asymptotic geodesic behaviour in that case. Therefore we shall henceforth suppose that |k| < 1. Lemma 7.1 For each k fixed in ] − 1, 1[ and each  = ±1 , there is a natural one-toone map ψ = ψεk from the leaf L(k, ε) (seen as made of geodesics of Γ\G having initial value θ0 = 0 for their angular part θs ) onto the set of geodesics of Γ\ H2 . This map goes as follows : with any geodesic γ of Γ \ G , associate successively the projection γ˜ on H2 of its lift to G, and the projection ψ(γ) on Γ\H2 of the geodesic of H2 at bounded distance of γ ˜. This map makes sense as well at the level of line-elements, and thus defines a homeomorphism from L(k, ε) onto T 1(Γ\ H2 ) ≡ Γ\G . Proof The analysis made in Section 7.1. ensures that the map ψ = ψεk is well defined. Note indeed the necessary Γ-invariance : if two geodesics γ, γ 0 of G can be identified

Asymptotic Homology of a Modular Quotient

107

modulo some g ∈ Γ, then indeed the same g identifies also the geodesics of H2 at bounded distance of the projections γ˜, γ ˜ 0 of γ, γ 0 on H2 . In the reverse direction, to any (oriented) geodesic ψ(γ) of Γ \ H2 correspond two quasi-geodesics in Γ\ H2 at constant distance argch[(1 − k2 )−1/2], and, owing to the sign ε which by Formula (20) and Equation (13) prescribes the sign of c0 (and then the sign of the height of the centre y0 = kC/c0 by Remark 7.1), in fact a unique one. And to this unique quasi-geodesic in Γ \ H2 is associated by (13) or (16) (for any prescribed initial value θ0 of the angular part) a unique geodesic Γγ of Γ \ G , obviously included in the leaf L(k, ε). By using furthermore the orthogonal projection in H2 between our quasigeodesics and their associated geodesic, we get at once the analogous map at the level of line-elements, with a clear continuity in both directions.  Remark 7.2 Note that in fact each leaf L(k, ε) splits into a continuum of sub-leaves : S L(k, ε) = θ0 ∈R/2πZ L(k, ε, θ0), taking into account the initial value θ0 of the angular √ part (either at time 0, or above the orthogonal projection of the fixed point −1 on the quasi-geodesic γ ˜ ) of the geodesic γ. Thus this is indeed the set of its line-elements of each sub-leaf L(C 2 , ε, θ0), which is set in one-to-one correspondence with T 1 (Γ\ H2 ) ≡ Γ\G k . Note that L(k, ε, θ ) has indeed 3 dimensions, as G. However, by the map ψ = ψε,θ 0 0 this initial value θ0 will not matter anyway in the following, so that we drop it henceforth, going on with the shorter notation L(k, ε), ψεk . Remark 7.3

According to Section 7.1. and Lemma 7.1 , we have h  i k2 = th 2 dist π(L(k, ε)), ψεk(L(k, ε)) ,

π denoting as in Section 3.1. the canonical projection from Γ \ G onto Γ \ H2 . As ψεk (L(k, ε)) is the only geodesic of Γ \ H2 which is asymptotic to the quasi-geodesic π(L(k, ε)), we see that |k| is fully determined by the leaf L(k, ε), and furthermore, that |k| is necessarily preserved by any isometry applied to L(k, ε). In particular, if 0 ≤ k < k0 < 1 and ε, ε0 = ±1 , then L(k, ε) ∩ L(k0 , ε0) = ∅ . As a consequence, note that, counter to the hyperbolic setting, the geodesic flow on S Γ \ G is not ergodic (with respect to the normalised Liouville measure of Γ \ G) : 0
Note that, by Remark 7.3, for |k| < |k0 | < 1 the measures µkε , µkε0 have disjoint supports.

108

8.

Jacques Franchi

Asymptotic Geodesic Windings

We fix here a leaf L(k, ε) , and endow it with the ergodic invariant probability measure µkε of Definition 7.1. We want to obtain the asymptotic law under µkε of Z Z   −1 −1/2 ωj , t ω ˜l as t → ∞ , t γ[0,t]

γ[0,t]

0≤j<ν∞ (Γ), 1≤l≤2g(Γ)

where the geodesic γ of Γ\G is chosen (at time 0) according to µkε , and γ[0, t] denotes this geodesic γ run during the time-interval [0, t] . ˜ l , it makes no difference to think of Note that by the Γ-invariance of the forms ωj , ω the geodesics γ as started in a fundamental domain D and living on G, the forms being harmonic on G as well. The following theorem describes the asymptotic geodesic windings in Γ\G , under the ergodic measures of Section 7.2.. A minor mistake in [F3], concerning the contribution of dθ , is corrected here. Theorem 8.1 Let us consider a fixed leaf L(k, ε) (defined in Section 7.1.) of Γ\G , with |k| < 1 , endowed with the ergodic invariant probability measure µkε of Definition 7.1. Then the under µkε = µkε (dγ) of Z Zlaw   ωj , t−1/2 ω ˜l t−1 γ[0,t]

γ[0,t]

0≤j<ν∞ (Γ), 1≤l≤2g(Γ)

converges as t → ∞ to the law of 

 (Γ)  1 − k2 1/2 ν∞  4(1 − k2 ) 1/4 2 X )k (1 + a ` l √ 1{j=0} + 2 rj Q` , N  1 + a2 k2 1 + a2k 2 1 + a2 k 2 `=1

,

0≤j<ν∞(Γ), 1≤l≤2g(Γ)

where all variables Q` , N l are independent, each Q` is Cauchy with parameter h` , and N l is centred Gaussian with variance h˜ ωl , ω ˜ li. Here h` > 0 denotes 2 V (Γ\H2 ) the width of the solid cusp C` (defined in Theorem 3.1). Note a clear difference between the Brownian and geodesic behaviors : mainly, here (counter to the Brownian case) the dθ-part of the form ω0 is responsible for a nonnegligible asymptotic contribution. Moreover the parameter a now appears in the limit law. This makes a noteworthy contrast with the hyperbolic case (see [E-F-LJ1], [E-F-LJ2], [F2]). This difference appears in Lemma 8.1 below, whereas once the dθ-part has been moved away, the remaining asymptotic law is essentially the same as the Brownian one, given by Theorem 6.1. So the remaining task, following [F3], will be then (in Section 11. below) mainly to compare on H2 the geodesic paths to the Brownian paths, somewhat in the spirit of the methods already employed in [E-F-LJ1], [E-LJ], [F2], [LJ2], but in a simpler way, ˜ l , somewhat as in [LJ1]. taking advantage of the harmonicity of the forms ωj , ω The following lemma (in which the minor mistake in [F3] concerning the contribution of dθ is corrected) reduces the study along the geodesics of G to a study along the geodesics of H2 .

Asymptotic Homology of a Modular Quotient Z Z   Lemma 8.1 The asymptotic law of t−1 ωj , t−1/2 ω ˜l γ[0,t]

µkε (dγ)

under H2) of :

(1+a2 )k √ 1+a2 k 2

γ[0,t]

is the same as the asymptotic law under the Liouville

1{j=0} +



1−k2 1+a2 k2

1/2

t

−1

Z

ωj ,



γ[0,t]

1−k 2 1+a2 k 2

1/4

t

−1/2

109

0≤j<ν∞ (Γ), 1≤l≤2g(Γ) measure µΓ on T 1(Γ\

Z

ω ˜l

γ[0,t]

!

.

0≤j<ν∞(Γ), 1≤l≤2g(Γ)

Let us deal ˜ l = Zπ ∗ ω ˜ l . By Definition 7.1 Z first with ω Z and Lemma 7.1, we just have −1/2 −1/2 −1/2 ω ˜l = t ω ˜l with t ω ˜l . to compare t Proof

γ[0,t]

γ ˜[0,t]

ψ(γ)[0,t]

Now use that on H2, ω ˜l = dFl is exact, and recall from Remark 7.1 that q i h  1−k 2 γ (t)], Fj ψ(γ)( 1+a = (1 − k2)−1/2 , dist Fj [˜ 2 k 2 t) to get : Z   Z q 1−k 2 = Fj [˜   ω ψ(γ)( ω ˜ − ˜ γ (t)] − F t) − F [˜ γ (0)] + F [ψ(γ)(0)] l l j j j 2 2  1/2 1+a k 1−k2 ψ(γ) 0, t γ˜ [0,t] 2 2 1+a k

≤ 2 k˜ ωl k∞ 

Z

This shows that t−1/2 

ω ˜l − γ˜[0,t]

Z

.p

1 − k2 . 

  ψ(γ) 0,

1−k 2 1+a2 k 2

˜l  1/2  ω t

goes uniformly to 0, prov-

ing the result relating to the regular forms ω ˜l . Now we have to deal with the singular forms ωj , which from Theorem 3.1 equal π ∗ωj0 for 1 ≤ j < ν∞ (Γ), whereas ω0 = dθ + π ∗ ω00 . Thus we can handle the ωj0 as the ω ˜l above, using Theorem 3.1 again, to get : Z Z Z   0 0 ˜ (ψ ◦ γ, 0) + Y˜ (ψ ◦ γ, t) ,    ω0 + O Y ωj = ω0 =  j 1/2 2 γ[0,t]

where

γ ˜ [0,t]

Y˜ (g, t) :=

ψ(γ) 0,

max 1≤`≤ν∞ (Γ)

1−k 1+a2 k 2

t

n  hq i  o √ 1−k 2 2 . sup y˜` (z) dist g t , z ≤ 1/ 1 − k 1+a2 k 2

On the other hand we have by Equations (16), (18), and (19) : Z Z −1 −2 −1 dθ = c (1 + a ) − t c0ys ds −→ c (1 + a−2 ) − c0 lim y(γ(t)) t γ[0,t]

t

γ[0,t]

.p 1 + a2 k2 . Z −1 Therefore, by Definition 7.1, the asymptotic law of t ωj under µkε (dγ) is the = c (1 + a−2 ) = (1 + a2 )k

γ[0,t]

same as the asymptotic law under µ(dg) of : Z  . (1 + a2)k −1 ˜ (g, 0) + Y˜ (g, t) t .    ωj0 + O Y √ 1{j=0} + t  1/2 2 1 + a2 k 2 g 0, 1−k t 1+a2 k 2

110

Jacques Franchi

Observe further that under µΓ (dg) the process Y˜ (g, t) is stationary, so that the last term above asymptotically vanishes in probability as t → ∞ . Hence we have shown that the Z asymptotic law of t−1

ωj under µkε (dγ) is the same as the asymptotic law under

γ[0,t]

µ(dg) of (1 + a2)k √ 1{j=0} + t−1 1 + a2 k 2

Z

  g 0,

1−k 2 1+a2 k 2

0 1/2  ωj t

.

Finally the result is valid jointly for the terms with singular forms ωj and regular forms ω ˜l , since for each the negligible contributions vanish in probability. It remains only to replace −1/2  1−k 2 t.  t by 1+a 2 k2

9.

Equirepartition in Γ\G of Large Geodesic Spheres

dy dθ Recall that µ (defined in Section 3. by µ(dg) := dx ) denotes the Liouville measure 4π 2 y2 Γ on G , and that µ denotes its normalized projection on Γ\G , identified with a law on left Γ-invariant functions on G ; and covol(Γ) denotes the finite hyperbolic volume of Γ\ H2 .

Fix some g ∈ G , and a Poincar´e half-plane model P2 for H2 such that π(g) = e0 = (0, 1). For any R > r > 0 , let Br := B(e0 ; r) denote the geodesic ball of radius r in P2 , centred at e0 = (0, 1), and (denoting by Θ the geodesic flow) consider : TR(r) := (T 1Br ) ΘR = P SO(2) Θ[0,r] P SO(2) ΘR ⊂ T 1(BR+r \BR−r ) ⊂ T 1 P2 ≡ G . Set then : ϕR :=

X

1γ TR (r) = ϕ0 ◦ ΘR ,

so that

Z

ϕR dµΓ =

γ∈Γ

2π (ch r − 1) vol(Br ) = ∈ ]0, ∞[. covol(Γ) covol(Γ)

Indeed, by definition of µ, µΓ , considering some fundamental domain D of Γ, we have : Z Z X X Γ ϕR dµ = ϕ0 dµΓ = µΓ (T 1 D ∩ γ T 1Br ) = covol(Γ)−1 vol(D ∩ γ Br ) γ∈Γ

= covol(Γ)

−1

X γ∈Γ

γ∈Γ

vol(Br ) vol(Br ∩ γ D) = = covol(Γ)

R 2π R r 0

sh % d% dθ 2π (ch r − 1) = . covol(Γ) covol(Γ) 0

Similarly, for any f ∈ L2 (µΓ ) we have : covol(Γ) 2π |a|

×

Z

Γ

f × ϕR dµ =

Z

X T 1 D γ∈Γ

1γ TR (r) f dµ =

XZ γ∈Γ

1TR (r)∩γT 1 D f dµ =

Z

f dµ . TR (r)

Moreover, for any r > 0 , by the discontinuity of the action of Γ on H2 , ϕ0 ∈ L∞ (µΓ ). Applying now the mixing theorem, for any f ∈ L2 (µΓ ) and r > 0, we have as R → ∞: Z Z Z Z Γ Γ Γ ϕR × f dµ = ϕ0 ◦ ΘR × f dµ −→ ϕ0 dµ × f dµΓ ,

Asymptotic Homology of a Modular Quotient

111

or equivalently : 2π |a| vol(Br )

×

Z

f dµ = TR (r)

covol(Γ) vol(Br )

×

Z

Γ

f × ϕR dµ −→

Z

f dµΓ .

Therefore, for any r > 0, the probability law µΓ on Γ\G is the weak limit, as R → ∞, 2π |a| 1 µ of µ to the large of the projection on Γ\G of the normalized restrictions vol(B r ) TR (r) shells TR (r). Suppose then that the function f on Γ\G is continuous and compactly supported, and then uniformly continuous as a (left Γ-invariant) function on G = P SL2 (R). Then, it is easily seen (and verified by a standard computation) that the maximal angular deviation in a section of diameter r of a large thin shell TR(r), between geodesics arriving 2Rr , as r & 0 and R ≥ 2 . This implies that from T 1Br , is equivalent to R−1 ξ ∈ T 1Br , % ∈ P SO(2) , dist H2 [π(ξΘR ), π(%ΘR )] < r ⇒ dist P SL2 (R) (ξΘR , %ΘR ) = O(r) .

Hence, denoting by d% the normalized uniform measure on P SO(2), we have : Z Z 2π |a| f dµ − f (% ΘR) d% −→ 0 as r & 0 , vol(Br ) TR (r)

P SO(2)

uniformly with respect to R ≥ 2 . This means that the mean of f on the large thin shell TR(r) converges, as r & 0 and R ≥ 2 , to its mean on the geodesic sphere SR := (Te10 P2)ΘR ≡ P SO(2)ΘR ⊂ T 1 (S(e0; R)). Finally, we get that : Z Z Z Z Z Z 2π |a| 2π |a| Γ Γ f ≤ f dµ − vol(Br ) f dµ + vol(Br ) f dµ − f f dµ − SR

TR (r)

TR (r)

SR

is arbitrary small for r fixed so small that the first term on the right hand side be small enough, and then for large enough R . Forgetting the irrelevant choice of the Poincar´e model P2 for H2 and of its base point e0 , and projecting on Γ \ G, we can see SR as the sphere of radius R in Γ\G , centred at Γg ∈ Γ\G . This proves the following. Theorem 9.1 The normalized Liouville measure µΓ on T 1(Γ\ H2 ) ≡ Γ\G is the weak limit as R → ∞ of the uniform law on the geodesic sphere Γg P SO(2)ΘR of Γ\G having radius R and fixed center Γg ∈ Γ\G : for any compactly supported continuous function f on Γ\G , denoting by d% the uniform law on P SO(2), we have Z Z Γ f (Γg % ΘR ) d% . f dµ = lim R→∞ P SO(2)

Remark 9.1 This result and its proof are contained in [E-MM]. Such equidistribution result goes back to [R].

112

Jacques Franchi

For any compactly supported continuous function h on L(k, ε) (recall Definition 7.1), applying Theorem 9.1 to h ◦ (ψεk )−1 (this is licit according to Lemma 7.1), we get : Z

h dµkε

=

Z

L(k,ε)

h

◦ (ψεk )−1 dµΓ

Z

= lim

Γ\G

R→∞ P SO(2)

h[(ψεk )−1 (Γg % ΘR)] d% ,

hence the following equidistribution result, reminiscent of a multi-dimensional ergodic theorem. Corollary 9.1 For any (k, ε) fixed in ] − 1, 1[×{±1} and any g ∈ G, the probability measure µkε on the leaf L(k, ε) (recall Definition 7.1) is the weak limit as R → ∞ of the uniform law on the geodesic quasi-sphere (ψεk )−1 (Γg P SO(2)ΘR).

10. Synthetic Proof of Theorem 6.1 The proof of Theorem 2 in ([F3], Section 10) essentially applies here, with minor modifications. Thus this section presents a somewhat sketched proof of Theorem 6.1, containing all ingredients, but not all details, for which we refer to [F3]. ˜ l were already easily handled in Lemma 6.1. The slow windings (about the handles) M t Hence we must now deal with the singular windings (about the cusps) Mtj , and then establish the asymptotic independence of both types, which is the main difficulty of the whole proof and is widely responsible for its length. 1) To proceed, we first cut the solid cusps at some high level r > 0, considering (for 0 ≤ j < ν∞ (Γ) and 1 ≤ ` ≤ ν∞ (Γ)) the martingales j,`,r Mt

:=

1{r` 6=0} (rj` )−1 j

Z

t 0

1{˜y` (s)>r} dMsj , where y˜` (s) := 1{gs∈C`} y˜` (gs),

(21)

y˜` (defined in Section 3.3.) being the height in the cusp C` . ν∞ (Γ) X j Observe that the martingale Mt − rj` Mtj,`,r , locally constant out of the compact `=1 ν∞ (Γ)

\

{˜ y` ≤ r}, has bounded quadratic variation, so that

`=1



ν∞ (Γ)

Mtj

−

X

rj` Mtj,`,r

.√

t

`=1



converges in law and

ν∞ (Γ)

Mtj

−

X

rj` Mtj,`,r

.

t goes to 0 in L2 -norm, as t → ∞ .

`=1

Set Mt`,r :=

Z 0

t

1{˜y` (s)>r} d˜ x` (s) , where x ˜` (s) := 1{gs∈C`} x ˜` (gs ).

Owing to TheoremZ 3.1 and Section 5., we .  t O(1) dVs` goes also to 0 in L2 -norm. Mtj,`,r − Mt`,r t = t−1

(22) see

0

We have therefore only to study the martingales (Mt`,r ), instead of the (Mtj ).

that

Asymptotic Homology of a Modular Quotient

113

2) Consider then a discretization of the excursions of the Brownian motion (gt) in the √ y` ≥ r} for the nth time cusps : it enters the shortened solid cusp {˜ y` > r + r} and exits {˜ within the interval of time say [τn` , σn` ], during which it performs an elementary winding ϕ`n = ϕ`n (r) :=

Z

` σn

d˜ x` (s).

(23)

τn`

Depending only on planar hyperbolic Brownian motions (recall Section 5.), these elementary windings are independent, and independent from the points on the level {˜ y` = r} at which the excursions start, and are easily (and classically) seen to have a Cauchy law, of √ parameter r . A random number λ`t = λ`t (r) of these windings is performed till time t . By ergodicity, we have lim λ`t /t =: %`r almost surely. Otherwise the Markov property t→∞

implies the independence of the excursion durations {σn` − τn` | n ∈ N∗ }, so that by the law N X of large numbers N −1 (σn` − τn` ) goes almost surely to E(σ1` − τ1` ) as N → ∞. By an n=1 λ`t Z t Z t X ` ` (σn − τn ) and 1{˜y` (s)>r} ds , 1{˜y` (s)>r+√r} ds , obvious comparison between 0

n=1

0

and by the ergodic theorem, we deduce that √ V ({˜ y` > r + r}) V ({˜ y` > r}) ≤ %`r × E(σ1` − τ1` ) ≤ . 2 V (Γ \ H ) V (Γ \ H2) Now, on one hand it is easily computed that E(σ1` − τ1` ) = 2 log(1 + r−1/2), and on the Z other hand, we have : V ({˜ y` > r}) =

[0,h` ]×]r,∞[

y˜`−2 d˜ x` d˜ y` = h` /r , by definition of

the width h` (recall Theorem 3.1). Hence we find that lim

r→∞

√ √ h` λ` (r) r lim t . = lim r %`r = r→∞ t→∞ t 2 V (Γ \ H2 )

(24)

`,r

3) Let us now analyse further the behaviour of the martingales Mt of formula (22), by means of the above excursions. There are possibly two incomplete excursions, namely the very first one, the winding contribution (divided by the normalisation t) of which almost surely vanishes, and the very last one, which exists only when the Brownian motion at time t visits the solid cusp {˜ y` > r}, which is the case only with probability O(1/r), so that its winding contribution (letting r → ∞) eventually vanishes in probability. Hence the λ`t X `,r only non-negligible contribution of the martingale Mt comes from ϕ`n . Then, n=1 N X ϕ`n using again that lim λ`t /t = %`r , taking advantage of the above observation that t→∞ n=1 √ constitutes a discretized Cauchy process (of parameter r ), and using the scaling property λ`t X −1 ϕ`n , hence Mt`,r /t , has, and the right continuity of a Cauchy process, we see that t n=1

114

Jacques Franchi [%`r t]

in probability, the same asymptotic behaviour as t

−1

X

ϕ`n ; and as t → ∞, this last

n=1 √ process converges in law towards a Cauchy variable of parameter r %`r . 4) To establish the asymptotic independence of Theorem 6.1, we need to approach also ˜ tk (recall Formula 10), by martingales that are supported the slow windings martingales M √ in the complement Kr of all solid cusps {˜ y` > r + r }, in order to be able to take advantage of the Markov property, from which independence can then derive. Precisely, let us order all stopping times {τn` , σn` | 1 ≤ ` ≤ ν∞ (Γ), n ∈ N∗ } into a unique strictly increasing sequence (.. < τn < σn < ..) , fix any q˜ ∈ R2g(Γ) , and con2g(Γ) ν∞ (Γ) X Z τn+1 X q k ˜ sider Jn := q˜k dMt . Let λt := λ`t be the total number of excursions k=1

σn

`=1

performed till time t . The same argument as for Lemma 6.1 proves that, as t → ∞ , 2g(Γ) λt X X ˜ tk − t−1/2 q˜k t−1/2 M Jnq is asymptotically O(1/r), provided we can handle the n=1

k=1

last excursion in the compact core Kr , alive at time t ; now, considering the quadratic variation and using the integrability of (τn+1 − σn )2 and that λt/t is bounded in probability, it is easily seen that this last excursion in Kr has a contribution which vanishes in probability. 2g(Γ) λt X X q ˜k. Jn for the martingale q˜k M Hence we can asymptotically substitute t n=1

k=1

Consider now the Markov chain (Zσn , Zτn+1 ) induced, for any fixed r , by the Brownian motion (Zt) on H 2/Γ , which is known to be stationary and ergodic under the so-called Palm probability measure χ induced by the volume measure on the union of all bound√ y` = r + r } of the solid cusps. The transition operator of this induced aries {˜ y` = r}, {˜ Markov chain has a sprectral gap in L2 (χ), which implies that correlations between durations (τn+1 − σn ) decay exponentially fast. [%r t] λt X X q This implies in turn that (the quadratic variation of) Jn − Jnq goes to 0 in n=1

n=1 [%r t]

ν∞ (Γ)

probability, where %r :=

X

%`r is deterministic : we can substitute

Jnq for

n=1

`=1 ν∞ (Γ)

5) Consider any (q, q˜) ∈ R " 

√ −1 Aq,˜q := lim E exp t→∞

X

×R

2g(Γ)

 νX ∞ (Γ)

λt X

Jnq .

n=1

, and 2g(Γ) −1

q` t

Mt`,r

`=1

+

X

q˜k t

−1/2

˜k M t

# ,

(25)

k=1

which by Item 1) above is the quantity to calculate to get the asymptotic law of Theorem 6.1. Items 3) and 4) above show that we have : " #  ν∞  (Γ) [%`r t] [%r t] X X X  √ −1 q` t−1 ϕ`n + t−1/2 Jnq Aq,˜q = lim lim E exp . (26) r→∞ t→∞

`=1

n=1

n=1

Let us apply now the Markov property : conditionally on the σ-field F generated by the induced Markov chain (Zσn , Zτn+1 ), the random variables {ϕ`n , Jnq | 1 ≤ ` ≤

Asymptotic Homology of a Modular Quotient

115

ν∞ (Γ), n ∈ N∗ } are independent. Therefore, denoting by EF the conditional expectation with respect to F , we have : " ν∞ (Γ) [%` t] r Y Y

Aq,˜q = lim lim E r→∞ t→∞

h

√

EF e

−1

(q` /t)ϕ`n

i



[%r t]

×

`=1 n=1

Y

√

EF e

√

−1

q/ Jn

t

# .

(27)

n=1

get rid of the conditioning on F . To do this, we work on each h6)√ We must `finally i −1 (q /t)ϕ ` n E e , depending on a single excursion in a given solid cusp ; to analyse such quantity, we can drop for a while the irrelevant index ` , and suppose that the width h` of the cusp is 1, for the sake of notational simplicity. Now by Section 5., during Z s  each excursion near the cusp, we have x ˜s = x ˜0 + B y˜t2 dt , for some Brownian mo0 Z σ1 ys ). Set Y := yt2 dt . We have tion (Bs ) independent from the height component (˜ τ1 h √ i h i √ −1 q B(Y ) −q 2 Y /2 −|q| r E e . Then for any real q and any n ∈ N∗ we = E e = e have : i h √ i h √ i h √ ` ` EF e −1 q ϕn = E e −1 q ϕn Zσn , Zτn+1 = E e −1 q B(Y ) B(Y ) modulo 1 . F

We have thus to make sure that the knowledge of the value of B(Y ) modulo 1 will perturb the law of B(Y ) only in a negligible way. For this, let us fix u ∈ R and ε > 0, and write : h

E e

√ −1 q B(Y )

i B(Y ) ∈ ]u, u + ε[+Z − 1 =



√

E e

−1 q B(Y )



X

−1 ×

X

E

1{u
k∈Z

1{u




k∈Z



−1/2

E (2πY )



=

Z

u+ε

X

√

e

u

E (2πY

−1 q (x+k)

  −(x+k)2 /(2Y ) −1 e dx

k∈Z

)−1/2

Z

u+ε

u

X

2 e−(x+k) /(2Y )



.

dx

k∈Z

o √ − 1| × e Then observing that sup |e k ∈ R ≤ |q| Y , we can replace the Riemannian sum above by a Riemannian integral + an error term, in order to get :   √  √  X √ 2 2 2πY + O |q| Y . e −1 q (x+k) − 1 e−(x+k) /(2Y ) = e− q Y /2 − 1 n

√ −1 q k

−k 2 /(2Y )

k∈Z

Hence we have : i e −1 q B(Y ) B(Y ) ∈ ]u, u+ε[+Z −1 =

h

E

√

whence :

h

EF e

i √ −1 q ϕ`n

h

i √ − 1 + O(|q|) e−|q| r − 1 + O(|q|)  =  √ , 1 + O(1/ r ) 1 + O(E(Y −1/2))

E e− q

2

Y /2

 √  √ = 1 − 1 + On (1)/ r |q| r ,

(28)

116

Jacques Franchi

for some uniformly bounded function On (1) of Zσn . 7) To conclude the proof of Theorem 6.1, we note that by Birkhoff’s ergodic Theorem applied to the Markov chain (Zσn ) (via the sequence On (1)), Formula (28) implies : [%`r t]

Y

E

F

h

√

e



 √ [%`r t] √ |q` | r X = exp − (1 + On (1)/ r ) + o(1) t n=1  √ √  t→∞ −→ exp − |q` | r %`r (1 + O(1/ r )) .

−1 (q`/t)ϕ`n

n=1

i

Hence we get from Formula (27) : # " [%r t]  ν∞  (Γ) √  Y F √ X q/ t √ ` √ −1 Jn × exp − E e |q` | r %r (1 + O(1/ r )) Aq,˜q = lim lim E r→∞ t→∞

n=1

= lim E t→∞

`=1

" 2g(Γ) X

−1/2

q˜k t

˜k M t

#

× exp



ν∞ (Γ)

X

−

k=1

= exp



`=1 2g(Γ)

−

X

ν∞ (Γ) 1 ˜k 2q

h˜ ωk , ω ˜k i −

k=1

X

|q` | h` 2 V (Γ \ H2 )

|q` | h` 2 V (Γ \ H2 )

`=1





,

by Item 4), Formula (24), and Lemma 6.1. This achieves the proof of Theorem 6.1, owing to Formula (25) defining Aq,˜q and to Item 1). 

11. Proof of Theorem 8.1 The strategy for this proof is mainly to replace the geodesic paths by the Brownian paths, as in [E-LJ], [F2], [LJ1], in order to reduce Theorem 8.1 to Theorem 6.1. As in [F3], we shall ˜ l , somewhat as in [LJ1], to get a here take advantage of the closedness of the forms ωj , ω simple enough proof, without using a spectral gap, nor rising to the stable foliation ; simultaneously disintegrating the Liouville and the Wiener measures, we condition the Brownian motion (starting from a given point z ∈ H2 ) to exit the hyperbolic plane at the same point as a given geodesic (starting also from z). We essentially follow ([F3], Section 14). Because of Lemma 8.1, the asymptotic law we are looking for is given by the asymptotic behavior, as t → ∞ and for (λ0, λ) ∈ Rν∞ (Γ) × R2g(Γ) , of the following quantity :    Z ν∞ (Γ)−1 0 Z 2g(Γ) X λj X λl Z √ √ Jtλ := exp  −1  ωj0 + ω ˜l  µΓ (dg) (29) t t Γ\G g[0,t] g[0,t] j=0 l=1 = covol(Γ)

−1

Z Γ\H

Z 2

2π

Z h√  −1 exp −1 t

0

0

−1/2

ω +t

Z

g(y,x,θ)[0,t]

g(y,x,θ)[0,t]

ω ˜

i

dθ

dx dy , 2π y 2

where we use the notations of Section 2. and set : ν∞ (Γ)−1 0

ω :=

X j=0

2g(Γ)

λ0j

ωj0

,

ω ˜ :=

X l=1

λl ω ˜l .

(30)

Asymptotic Homology of a Modular Quotient

117

11.1. Conditioning by End-points √ For (z = x + −1 y , θ) ∈ H2 × (R/2π Z) , denote by (ztθ ) the geodesic of H2 defined by g(z, θ), and by Pθz the law of the Brownian motion (Ztθ ) of H2 , started from z and θ conditioned to exit H2 at the positive end z∞ of (ztθ ). Consider then the the hitting time, say ht , by the coordinate process (Zt), of the stable θ horocycle defined by (z∞ , ztθ ). It is defined precisely by z,θ t ht = ht := inf{s > 0 | Bz∞ θ (z, Zs ) = e } , where (z, z 0) 7→ Bu (z, z 0) = p(z 0, u)/p(z, u) denotes the Busemann function based at u ∈ ∂ H2 , p denoting the Poisson kernel. The following lemma ensures that the disintegration of the Liouville and Wiener meaθ . A reason is that sures is simultaneous, by conditioning with respect to the end-point z∞ 2 the harmonic measures at ∂ H are the same for both, namely p(z, u)du . Z 2π dθ Lemma 11.1 is the Wiener measure started from z, for any Pz := Pθz 2π 0Z z ∈ Γ\ H2, and

Pθz dµΓ (z, θ) is the stationary Wiener measure on Γ\ H2.

Pµ :=

Proof (Ztθ ) is by definition the h-process of the unconditioned Brownian motion, with θ ) , p(z, u) = y/|z − u|2 still denoting the Poisson kernel. h(z) = p(z, z∞ Hence we have for any (z, θ) , any t and any Ft-measurable positive functional Ft : Eθz [Ft ] = Ez [Bz∞ θ (z, Zt ) × Ft ] . The first identity of the lemma follows, since for any z, θ, Z Zwe have Z Z 2π

Bz∞ θ (z, Z)dθ = 2

0

Bu (z, Z)p(z, u)du = 2

p(Z, u)du = 2π .

R

R

Integrating this first identity with respect to the normalized Liouville measure µΓ gives immediately the second identity of the lemma. 

11.2. From Geodesics to Brownian Paths We perform here the substitution of the Brownian paths for the geodesics. Our first aim is to establish the following, to the proof of which this section is devoted. As t → ∞ , Jtλ (defined by Formula (29)) behaves as  Z Z 2π Z Zh Z Zh i h√ √ t t dx dy θ −1 −1 −1 0 √ := covol(Γ) Ez exp t ω + t ω ˜ dθ . 2 2π y 2 0 Γ\H z z

Proposition 11.1 0

Ktλ ,λ

˜ being closed, we have the following expression for Jtλ : The forms ω 0, ω λ

Jt =

Z

Z 2π

H2

Γ\

0





Eθz exp 

√ −1 t

 

Z Z h

t

z

0

ω +

Z zθ t Zh t



0 ω +

√ −1 √ t

 

Z Z h

t

z

ω ˜ +

Z zθ t Zh t

 

ω ˜  

dθ dx dy 2πcovol(Γ)y 2

.

Applying the isometry fz,θ of H2 which maps g(1, 0) to g(z, θ) , we see that the law Z zθ Z et t √ ∗ of ω ˜ under Pθz is the same as the law of fz,θ ω ˜ , where et := −1 et and Zh0t Zht

Zh0

t

118

Jacques Franchi

is the point at which the Brownian motion (Zt0 ) started from at ∞ hits the horizontal horocycle having equation y = et .

√ −1 and conditioned to exit

Now (Zt0 ) is the h-process of the unconditioned Brownian motion, with h(z) = p(z, ∞) ≡ y , so that its infinitesimal generator is 12 y −1 ∆ ◦ y = 12 ∆ + y∂y , Z t √ 2 0 wt +t/2 + ews +s/2 dWs , ∆ denoting the Laplacian of H . Thus we have Zt = −1 e 0

for two independent standard real Brownian motions (wt) and (Wt ). As a consequence, using the boundedness of ω ˜ , we have Z

et Zh0

t

Z  ∗ fz,θ ω ˜ = O e−t ×

inf{s | ws +s/2=t} 0

 ews +s/2 dWs .

The technical Brownian behavior we need now and after is given by the following. As t → ∞ , e−t

Lemma 11.2

Z

inf{s | ws +s/2=t}

ews +s/2 dWs converges in law, and

0

inf{s | ws + s/2 = t} = 2t + o(tq ) almost surely, for any q ∈]1/2, 1] . Fix c ∈ R , set yt0 := ewt +t/2 , and look for a C 2 function f on R+ such that Z t h i 2 Rt := exp −(c /2) (ys0 )2ds f (yt0) be a martingale. (yt0) having generator 12 y 2 ∂y2 + Proof

0

y∂y , we have by Itˆo’s formula : Rt = f (1)+mart+ 12

Z

t

2 /2)

e−(c

Rs 0

(yv0 )2 dv

h i ×(ys0 )2× f 00 (ys0)+2(ys0)−1 f 0 (ys0)−c2f (ys0 ) ds ,

0

Setting f1 (y) := whence the equation : f 00(y) + 2y −1 f 0(y) − c2f (y) = 0 . √ 00 −1 0 2 −2 yf (y) , this gives f1 (y) + y f1 (y) − (c + (2y) )f1(y) = 0 . Since f1 must be bounded near 0, we have, up to some multiplicative constant : X (cy)2k f (y) = (cy)−1/2I1/2(cy) = , where Ir denotes the usual modified 1 2k+ 2 k!Γ(2k + 32 ) k≥0 2 Bessel function. The optional sampling theorem then gives h

E exp

√

−1 c

Z

inf {s | ws +s/2=t} ws +s/2

e

i

dWs

0

=E

h

i  c2 Z inf {s | ys0 =et } f(1) (ys0 )2ds = . exp − 2 0 f(et )

Changing c into ce−t , we get as t → ∞ : h

√

−t

E exp( −1 c e

Z

inf {s | ws +s/2=t}

i X ews +s/2 dWs −→

0

which proves the first sentence of the lemma.

Γ(3/2) c2k −1 ∈ L2 (R, dc) , 2kk!Γ(2k + 3 ) 2 2 k≥0

Asymptotic Homology of a Modular Quotient

119

Finally, the second sentence of the lemma is straightforward from the following ob√ −1 ,0 = inf{s | ws + s/2 = t} = inf{s | ys0 = et } , we servation : setting again ht = ht have t = log yh0t = 12 ht + wht = 12 ht + o((ht)q ) .  As a consequence of this lemma and of the above, we see that t−1/2

Z

ztθ

ω ˜ goes to 0

Zht

in Pθz -probability. This proves half of Proposition 11.1. Z zθ t −1 We have now to deal with the law of t ω 0 under Pθz , or equivalently by the same Zht Z et −1 ∗ fz,θ ω 0 . This cannot be handled further reason as above for ω ˜ , with the law of t Zh0

ω0

t

is unbounded. But integrating along the horizontal horocycle y = et as above, since θ containing et , Zht , we have the following estimate :   Z e  Z Z  −t ht ws +s/2 −t ht ws +s/2 t ∗ 0 ∗ 0 fz,θ ω ≤ e e dWs × sup |fz,θ ω |(√ −1 +x)et ; |x| ≤ e e dWs ,  0 Zh  0 0 t

√

−1 ,0

= inf{s | ys0 = et } = inf{s | ws + s/2 = t}. Z ht ews +s/2 dWs , for large Fix any r > 0 . Lemma 11.2 shows that the laws of e−t 0 h Z ht i −t t , are tight, and then provides some R > 0 such that P e ews +s/2 dWs > R < r where again ht = ht

for any large enough positive t . We deduce from these last two estimates that 

Pθz  t−1

 0 ω > r  = Zh t

Z zθ t



P t−1

0

 ∗ 0   , fz,θ ω > r  ≤ r + 1     Z0 ht t−1 sup |f ∗ ω 0 | √ |x|≤R >r/R ( −1 +x)et   z,θ  

Z e t

and then by integrating against µ and using Lemma 11.1 : Z ztθ n i h o i h −1 Pµ t ω 0 > r ≤ r + µ t−1 sup |ω 0|Hx (ztθ ) |x| ≤ R > r/R Zht

o i h n = r + µ t−1 sup |ω 0 |Hx (z) |x| ≤ R > r/R ,

where (Hx , x ∈ R) denotes the positive horocycle flow. For the last equality, we used the invariance of the Liouville measure µ under the geodesic flow. o n 0 0 By continuity of |ω | , sup |ω |Hx (z) |x| ≤ R is finite for every z, and thus we just proved : Z ztθ h i −1 Pµ t ω 0 > r ≤ 2r for large enough t . Zht

Since in the last expression above for Jtλ (immediately after Z Proposition 11.1), we were

not only under the law Pθz , but indeed under the law Pµ = proved Proposition 11.1.

Pθz dµ(z, θ) , we have so far

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11.3. End of the Proof of Theorem 8.1 Section 5. allows to denote also by Pµ the stationary Wiener measure on Γ \ G , since the Brownian motion of G projects on the Brownian motion of H2 (and similarly for the volume measures). Recall also that the forms ωj0 , ω ˜ l come from Γ\ H2 : they are defined on Γ \ G and on Γ \ H2 as well, in other words are invariant under pull back π ∗ by the canonical projection. Hence the joint laws of their integrals along the Brownian paths are the same, no matter whether they are understood on Γ\G or on Γ\ H2 . Moreover we have seen in Section 5. also that the angular Brownian component θs is aZmere one-dimensional Brownian motion. As a consequence, it is immediate that t−1

dθ = (θt − θ0 )/t goes to 0 Pµ -almost surely.

Therefore we can replace in

g[0,t]

Theorem 6.1 the form ω0 by the form ω00 = ω0 − dθ . These remarks show that the following is merely an alternative version of Theorem 6.1 (with the notations of Formula (30) and Theorem 6.1). Corollary 11.1

We have for any (λ0, λ) ∈ Rν∞ (Γ) × R2g(Γ) : lim Eµ

t→∞

exp

h√

−1 t

Z

0

ω + Z[0,t]

√ −1 √ t

Z

ω ˜

i

!

Z[0,t]



   ν∞ (Γ)−1 2g(Γ) ν∞ (Γ) X X X √ λ0j rj` Q` + λl N l  . = Λ(λ0, λ) := E exp  −1  j=0

`=1

l=1

Now Lemma 11.2 asserts that the time-change ht = hz,θ t appearing in the expression λ0 ,λ θ uniformly with of Kt in Proposition 11.1, satisfies ht = 2t + o(t) Pz -almost surely, √ z,θ −1 ,0 θ in Lemma respect to (z, θ) . Indeed, the law under Pz of ht equals the law of ht 11.2. So that, with arbitrary large probability, we can write ht = 2t + o(t) with a uniform deterministic o(t) . This allows to replace t by ht in the formula of Corollary 11.1 above, getting then (using also the definition of Pµ in Lemma 11.1) : ! Z Z √ h√ i √ 0 λ /2,λ/ 2 −1 = lim Eµ exp 2t ω 0 + √−1 ω ˜ = Λ(λ0, λ). lim Kt 2t t→∞

t→∞

Z[0,ht ]

Z[0,ht ]

Therefore, using Proposition 11.1 we have proved that √ lim Jtλ = Λ(2λ0, 2 λ). t→∞

This concludes the proof, since by the definition of Λ in Corollary 11.1, by the very definition (29) of Jtλ , and by Lemma 8.1, this formula is equivalent to Theorem 8.1.

References ´ ements d’Analyse 9. Gauthier-Villars, Paris, 1982. [D] Dieudonn´e J. El´

Asymptotic Homology of a Modular Quotient

121

[E-F-LJ1] Enriquez N. , Franchi J. , Le Jan Y. Stable windings on hyperbolic surfaces. Prob. Th. Rel. Fields 119, 213-255, 2001. [E-F-LJ2] Enriquez N. , Franchi J. , Le Jan Y. Central limit theorem for the geodesic flow associated with a Kleinian group, case δ > d/2. J. Math. Pures Appl. 80, 2, 153-175, 2001. [E-LJ] Enriquez N. , Le Jan Y. Statistic of the winding of geodesics on a Riemann surface with finite volume and constant negative curvature. Revista Mat. Iberoam., vol. 13, no 2, 377-401, 1997. [E-MM] Eskin A. , McMullen C. Mixing, counting, and equidistribution in Lie groups. Duke Math. J., vol. 71, no 1, 181-209, 1993. [F1] Franchi J. Asymptotic singular windings of ergodic diffusions. Stoch. Proc. and their Appl., vol. 62, 277-298, 1996. [F2] Franchi J. Asymptotic singular homology of a complete hyperbolic 3-manifold of finite volume. Proc. London Math. Soc. (3) 79, 451-480, 1999. [F3] Franchi J. Asymptotic windings over the trefoil knot. Revista Mat. Iberoam., vol. 21, no 3, 729-770, 2005. [G-LJ] Guivarc’h Y. , Le Jan Y. Asymptotic windings of the geodesic flow on modular ´ Norm. Sup. 26, no 4, 23-50, 1993. surfaces with continuous fractions. Ann. Sci. Ec. [H] Hopf E. Ergodicity theory and the geodesic flow on a surface of constant negative curvature. Bull. Amer. Math. Soc. 77, 863-877, 1971. [I-W] Ikeda N. , Watanabe S. Stochastic differential equations and diffusion processes. North-Holland Kodansha, 1981. [L] Lehner J. Discontinuous groups and automorphic functions. Amer. Math. Soc, math. surveys no VIII, Providence, 1964. [LJ1] Le Jan Y. Sur l’enroulement g´eod´esique des surfaces de Riemann. C.R.A.S. Paris, vol 314, S´erie I, 763-765, 1992. [LJ2] Le Jan Y. The central limit theorem for the geodesic flow on non compact manifolds of constant negative curvature. Duke Math. J. (1) 74, 159-175, 1994. [M] Miyake T. Modular forms. Springer, Berlin 1989. [R] Randol B. The behavior under rojection of dilating sets in a covering space. Trans. Amer. Math. Soc. 285, 855-859, 1984. [R-Y] Revuz D. , Yor M. Continuous martingales and Brownian motion. Springer, 1999. [SL-M] de Sam Lazaro J. , Meyer P.A. Questions de th´eorie des flots. S´em. Probab. IX, Lect. Notes no 465, P.A. Meyer editor, Springer 1975.

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[Sh] Shimura G. Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan, Princeton University Press, 1971. [Sp] Spitzer F. Some theorems concerning two-dimensional Brownian motion. Trans. A. M. S. vol. 87, 187-197, 1958. [T] Thurston W.P. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6, 357-381, 1982. [W] Watanabe S. Asymptotic windings of Brownian motion paths on Riemannian surfaces. Acta Appl. Math. 63, no 1-3, 441-464, 2000.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 123-138

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 4

G ROUP A NALYSIS OF S OLUTIONS OF 2- DIMENSIONAL D IFFERENTIAL E QUATIONS Sergey I. Senashov and Alexander Yakhno Univ. de Guadalajara, Univ. Centre of Sciences and Ingeneira, Mexico

Abstract It is well known [4, 9] that if a system of differential equations admits the Lie group of point transformations (point symmetry), then any solution of the system is transformed to a solution of this system. This property permits the construction of new solutions without integrating the given system of partial differential equations (PDEs), by means of group transformations alone under known solutions. This is an effective method if we have a sufficiently rich group of point transformations. By applying point transformations to exact solution, a family of so-called Ssolutions can be constructed, i.e., obtained by means of symmetries. This family of S-solutions is dependent on the group parameter. If this parameter is equal to zero, then we have an initial solution. This procedure is called the production [9] or reproduction of solutions [4]. Moreover, it is easy to show that under a group transformation characteristic curves of the system of PDEs of the hyperbolic type are transformed to the characteristics curves. The evolution of characteristic curves permits to find out the boundary conditions for new S-solutions. In the present chapter authors will show some applications of this procedure for the system of the theory of ideal plane plasticity, developing results obtained in [12]. In particular, we shall use an infinite subgroup of the group of symmetries for deformation of characteristics curves of the considered hyperbolic system of PDEs to construct a new analytical solutions. From the system of PDEs an automorphic system will be deduced, which permits find out some relations between different solutions by means of group transformations.

1.

Introduction

Symmetry theory or the group analysis is of a fundamental importance in studying systems of partial differential equations. At present algebras of classical infinitesimal symmetry transformations are known for many equations of continuum mechanics, see, for example

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[1, 7, 9]. Methods for finding these algebras go back to S. Lie’s works written about 130 years ago. In particular, the knowledge of symmetry algebras makes it possible to construct effectively wide classes of exact solutions for equations under consideration. Unfortunately, the classical group analysis, being a semi-inverse method of the resolution of PDEs, does not give a direct algorithm to solve boundary value problems (BVPs). Nevertheless, there are some results showing an invariance (or partial invariance) of BVP with respect to admitted symmetries [3]. It is necessary to take into consideration that in the theory of Lie groups of transformations the character of transformations is local, while some boundary conditions are the global ones. In the case of the hyperbolic system of quasi-linear PDEs there are theorems about the invariance of a characteristic surface under the action of the point transformations [9]. These results are used in the present chapter to find out some new solutions of the system of 2-dimensional ideal plasticity and to give the appropriate boundary conditions. The chapter is structured as follows. In the Introduction, we start with some basic definitions and statements of the theory of Lie group of point transformations, which are necessary for the construction of new exact solutions. In Section 2., we provide a concept of the reproduced solution, obtained as the transformation of the known initial solution by a symmetry and the concept of transformed characteristic curves. Section 3. contains the information on the system of plane ideal plasticity, which will be analyzed as an example. As a result, in Section 4. we obtain a number of exact solutions of plasticity equations and we set for them suitable boundary conditions. Let us consider the homogeneous system of two quasilinear equations of two independent variables x, y and two dependent ones u1 , u2 : aij (u1 , u2 )

∂uj ∂uj + bij (u1 , u2 ) = 0, i, j = 1, 2. ∂x ∂y

(1)

The system of such a form are widely used in the mechanics of a continuum media [10], for example in the gas dynamics for describing isoentropic plane-symmetry flows; in the theory of plane plasticity for the stresses of a deformed region under the different yield criterion; for the motion of granular materials [14] and so on. Note, that system (1) can be linearized by a so-called hodograph transformation of the form x = x(u1 , u2 ), y = y(u1 , u2 ) when corresponding Jacobian ∆ = ∂(x, y)/∂(u1 , u2 ) is not zero. This transformation is just an interchange of roles of the unknown functions and the independent variables. Thus, the system (1) takes the linear form: ∂x ∂x ∂y ∂y − b11 − a12 + a11 = 0, ∂u1 ∂u2 ∂u1 ∂u2 ∂x ∂x ∂y ∂y b22 − b21 − a22 + a21 = 0. ∂u1 ∂u2 ∂u1 ∂u2

b12

(2)

We call any solution of the system (1) U = (u1 (x, y), u2 (x, y)) nonsingular one, if its transformation to the corresponding solution χ = (x(u1 , u2 ), y(u1 , u2 )) for the linearized system (2) is not degenerate. Let us define the space R of the variables {x, y, u1 , u2 }, then a point transformation

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S : R → R of the following form: x′ = f1 (x, y, u1 , u2 , ai ), y ′ = f2 (x, y, u1 , u2 , ai ), u′1 = g1 (x, y, u1 , u2 , ai ), u′2 = g2 (x, y, u1 , u2 , ai ),

(3)

f1 |ai =0 = x, f2 |ai =0 = y, g1 |ai =0 = u1 , g2 |ai =0 = u2 , acting on the space R is called the symmetry of the system of partial differential equations (1), if this system has the the same form with respect to the transformed variables. In such a case one says that the system (1) admits the above transformation. Here ai ∈ R is some parameter (i = 1, . . . , r). In other words, the change of variables (3) transforms the system (1) to itself. It means, if U = (u1 , u2 ) is a solution to (1), then function U ′ transformed by S : U → U ′ = (u′1 , u′2 ) is another solution to the system (1), whenever U ′ is well defined [8]. Acting admitted symmetries over the one known exact solution of the system of PDEs, we have an opportunity to construct a family of new solutions. If the parameters ai ∈ I ⊂ R are sufficiently small ones from an open interval I containing zero, and the functions f1,2 , g1,2 are sufficiently smooth, then the transformations (3) form the local one-parametric Lie group of the point transformations G1 with respect to the composition of transformations for every ai . In general, we have r-parametric Lie group Gr . The problem of constructing a complete set of the symmetries for the given system of PDEs was successfully solved for a lot of systems. The principal method consists in the determination of coefficients of infinitesimal generators Xi (i = 1, . . . , r): Xi = ξi1 (x, y, u1 , u2 )

∂ ∂ ∂ ∂ +ξi2 (x, y, u1 , u2 ) +ηi1 (x, y, u1 , u2 ) +ηi2 (x, y, u1 , u2 ) , ∂x ∂y ∂u1 ∂u2

which form the basis of Lie algebra Lr associated with group Gr . Any operator Xi generates one-parameter group G1 and is related to the functions of transformation (3) by means of the Lie equations: dfj dgj j j ξi = , η = , i = 1, . . . , r; j = 1, 2. (4) dai ai =0 i dai ai =0

The knowledge of the Lie algebra of symmetries, admitted by the system of PDEs, permits to construct some classes of exact solutions. To any solution U = (u1 , u2 ) of the system (1) one can associate its orbit UG [9] as a set of all solutions obtained from U by an application of all transformations (3) of the admitted group Gr . The solution U = (u1 , u2 ) is called H-invariant solution if u′1 = u1 , u′2 = u1 for some subgroup H of Gr , i.e. the orbit UH coincides with U . The classification of essentially different invariant solutions (that is those H-solutions which are not related by some transformation from Gr ) is based on the concept of the equivalence of subalgebras of Lr with respect to a well known adjoint representation of Lie algebra. The construction of invariant solutions is simpler due to a less number of independent variables, of course if there exist such H-invarian t solution. For example, for the system (1) all invariant solutions can be obtained from the system of ordinary differential equations.

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Sergey I. Senashov and Alexander Yakhno

Reproduction of Solutions

For some differential equations it is convenient to construct all H-invariant solutions firstly, and then try to transform them by applying the rest of symmetries from Gr . Sometimes it is the only way to obtain explicit formulas. This procedure is called reproduction or deformation of solutions [4]. For example, some interesting results on the deformation of invariant solutions for Kadomtsev – Pogutse equation were obtained in Ref. [5]. Formally, if we have the symmetry S ∈ / H and the initial H-invariant solution U 0 =
0 0 u1 = u1 (x, y), u2 = u2 (x, y) , then acting by (3) on U 0 we obtain an implicit formula for the unknown functions u1 and u2 : g1 (x, y, u1 , u2 , ai ) = u01 (f1 (x, y, u1 , u2 , ai ), f1 (x, y, u1 , u2 , ai )) , g2 (x, y, u1 , u2 , ai ) = u02 (f1 (x, y, u1 , u2 , ai ), f1 (x, y, u1 , u2 , ai )) .

(5)

Formula (5) gives the family of solutions that depends on the group parameter ai . Let us call this family S-solution, that is obtained from the initial solution U 0 by means of symmetry S. If parameter ai is equal to zero, then S-solution coincides with the initial one. As it was noted in Ref. [4] when reproducing a regular solution, even for a small ai , we can obtain a multivalued solution or the solutions with singularities. Such generalized S-solutions are widely used in the analysis of discontinuity propagation, shock waves, etc. The physical meaning of these solutions should be determined by an appropriate context. Now consider linear system (2). There is a lot of results on applying group analysis to the linear differential equations. The relation between the symmetries of the non-linear system of PDEs and the symmetries of linearized PDEs can be found, for example in Ref. [3]. The linear system always admits the infinite-dimensional symmetry group since we can add any other solution to a given solution. The corresponding infinitesimal generator has the form ∂ ∂ X = ξ(u1 , u2 ) + η(u1 , u2 ) , (6) ∂x ∂y where (ξ, η) is an arbitrary solution of the system (2). This operator generates a oneparameter group of transformations of the form: x′ = x + aξ, y ′ = y + aη,

(7)

where a ∈ R is a group parameter. Let χ1 = (x1 (u1 , u2 ), y1 (u1 , u2 )) and χ2 = (x2 (u1 , u2 ), y2 (u1 , u2 )) be two solutions of the linear system (2), which define implicitly two solutions U 1 and U 2 of quasilinear system (1) respectively. Let us take the coefficients of the operator (6) as the difference of two solutions χ1 and χ2 : ξ = x1 − x2 , η = y1 − y2 , then because of (7) we have: x = x′ (u1 , u2 ) = x2 + aξ = ax1 (u1 , u2 ) + (1 − a)x2 (u1 , u2 ), y = y ′ (u1 , u2 ) = y2 + aη = ay1 (u1 , u2 ) + (1 − a)y2 (u1 , u2 ),

(8)

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that, of course, gives the solution of system (2) as a linear combination of two solutions. But formulas (8) define the S-solution of the form (u1 (x, y, a), u2 (x, y, a)) implicitly. Note that if a = 1, then the S-solution coincides with solution U 1 ; and if a = 0, then S-solution is equal to U 2 . System (1) is an automorphic one with respect to group (7). This means that any nonsingular solution of the system of PDEs under consideration can be transformed to another nonsingular solution of the same system (1) by means of admitted group of point transformations. This fact permits to relate any two solutions U 1 , U 2 of the quasilinear system (1) which can be presented in the form χ1 , χ2 . Let the quasilinear system of PDEs (1) be a hyperbolic one. Then, it has two families of real characteristic curves defined by the following equations: dy dy = A(u1 , u2 ), = B(u1 , u2 ), dx dx where functions A and B are defined by the coefficients of the equations of the system (1) and depend on solution. Under the action of admitted symmetry (3) the above relations define a family of characteristic curves: dy ′ dy ′ ′ ′ = A(u , u ), = B(u′1 , u′2 ) 1 2 dx′ dx′ for the system (1) in therms of transformed variables. Therefore, the characteristic curves are transformed to the characteristic curves. Here the invariance of the equation of characteristic curves is considered as the invariance ”on any solution” of the system of PDEs [9]. If system (1) is a hyperbolic one, the same is valid for the linear system (2). The characteristic curves of the linear system do not depend on the solution. Moreover, the characteristic curves of the system of two independent variables are the plane curves (while a solution is a space surface), therefore it is easier to analyze the action of the symmetries on that kind of curves. It is convenient to observe, changing the value of the group parameter, the evolution of the characteristic curves under the action of the symmetries to look for both suitable boundary conditions and mechanical sense of a corresponding S-solution.

3.

Ideal Plane Plasticity System and Its Known Solutions

Let us consider the classical system of plane ideal plasticity [6] that consists in two equilibrium equations and the Saint-Venant – Mises’ yield criterion that defines condition on the second invariant of the stress tensor: ∂σx ∂τxy ∂τxy ∂σy + = 0, + = 0, ∂x ∂y ∂x ∂y 2 (σx − σy )2 + 4τxy = 4k 2 ,

(9)

where σx , σy , τxy are components of a stress tensor, and k is a constant of plasticity. System (9) describes the stress state of material, which is being plastically deformed.

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Sergey I. Senashov and Alexander Yakhno By means of a change of variables proposed by M. L´evy σx = σ − k sin 2θ, σy = σ + k sin 2θ, τxy = k cos 2θ,

the system (9) is reduced to the quasilinear one   ∂σ ∂θ ∂θ − 2k cos 2θ + sin 2θ = 0, ∂x ∂x ∂y   ∂σ ∂θ ∂θ − 2k sin 2θ − cos 2θ = 0, ∂y ∂x ∂y

(10)

where σ is the hydrostatic pressure, and θ + π/4 is the angle between the first principal direction of a stress tensor and the ox-axis. System (10) is a hyperbolic one and it has two families of characteristic curves (labeled by parameters α and β) given by following relations: σ dy = tan θ, − θ = const = α, dx 2k σ dy = − cot θ, + θ = const = β. dx 2k

(11)

In mathematical theory of plasticity these curves are known as slip-lines. By means of applying hodograph transformation of the form x = x(σ, θ), y = y(σ, θ) to the system (10) one can obtain the corresponding linearized system:   ∂y ∂x ∂x − 2k cos 2θ + sin 2θ = 0, ∂θ ∂σ ∂σ   (12) ∂y ∂x ∂y − 2k sin 2θ − cos 2θ = 0. ∂θ ∂σ ∂σ Passing in (12) to the new dependent variables u, v, originally due to S.G. Mikhlin: x = u cos θ − v sin θ, y = u sin θ + v cos θ,

(13)

for system (12) we obtain ∂u ∂u − v − 2k = 0, ∂θ ∂σ ∂v ∂v + u + 2k = 0, ∂θ ∂σ and finally, taking the curvilinear coordinates α and β from (11) as new independent variables, for the above system we have the following form: ∂u v ∂v u + = 0, + = 0. ∂α 2 ∂β 2

(14)

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The classical system of the plane ideal plasticity has been investigated for many years, but it has few exact solutions: a) Prandtl’s solution; b) the solution for a cavity of circular form, stressed by uniform pressure; c) Nadai’s solution for the stresses in the plastic region around a circular cavity loaded by a constant shear stress in addition to a uniform pressure; d) solution for the channel with straight line borders and e) the spiral-symmetrical solution for the channel with logarithmic spiral borders [2]. The analytical solutions for some boundary problems were constructed in Ref. [13]. All these solutions are widely used for testing numerical calculations, allowing the estimation of an assurance factor of some constructions, etc. Let us consider the simplest solutions only, namely a) and b). 1. The solution of L. Prandtl can be interpreted as a solution to describe stresses of a rectangular block of plastic-rigid material compressed between rigid parallel plates which are assumed to be rough. It is supposed that the block is very wide compared with its height. In therms of the variables σ, θ for the system (10) this solution has the form: r x y2 σ = −p1 − k + k 1 − 2 , h h y = h cos 2θ,

(15)

where 2h = const is a height of the block, that is the straight lines y = ±h are the edges of the plates, p1 = const is a value of the pressure on the plate when x = 0. The corresponding boundary conditions look as: x θ|y=h = πn, n ∈ Z, σ|y=h = −p1 − k . h

(16)

For Prandtl’s solution the families of characteristics curves are cycloids. Its parametric equations have the following form:  p1  , x = h(∓2θ − sin 2θ) − h 2Ci + k (17) y = h cos 2θ, i = 1, 2, where α = const = C1 , β = const = C2 . Each family of cycloids are bounded by its envelopes y = ±h. 2. Another well-known solution [6] has the form π π y + =φ+ , x 4 4 r2 x2 + y 2 = −p + k + k ln , σ = −p2 + k + k ln 2 R2 R2 θ = arctan

(18)

where r, φ are the polar coordinates. This solution describes plastic state around a circular cavity of radius R, situated in an infinite medium loaded by uniformly distributed pressure p2 , with the tangential stress equal to zero, that is the boundary conditions are as follows: π , 4 = −p2 + k.

θ|r=R = φ + σ|r=R

(19)

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Sergey I. Senashov and Alexander Yakhno

As for the corresponding slip-lines, here we have a logarithmic spirals of the form:   p2 − k π + Ci , φ = θ − , r = R exp ±θ + 4 2k

(20)

here C1 = α, C2 = β. It is known [11] that system (10) admits an infinite algebra of generalized (highest) symmetries. Its Lie algebra L of point transformations is formed by the following generators: ∂ ∂ ∂ ∂ ∂ ∂ X1 = x + y , X2 = −y +x + , X3 = , ∂x ∂y ∂x ∂y ∂θ ∂σ ∂ ∂ σ ∂ ∂ + ξ2 (x, y, σ, θ) − 4kθ − , X4 = ξ1 (x, y, σ, θ) (21) ∂x ∂y ∂σ k ∂θ ∂ ∂ X5 = ξ(σ, θ) + η(σ, θ) , ∂x ∂y where

σ σ ξ1 = x cos 2θ + y sin 2θ + y , ξ2 = x sin 2θ − y cos 2θ − x , k k

and (ξ, η) is an arbitrary solution of the linear system (12). The list of non-zero commutators of Lie algebra L is: 1 [X1 , X5 ] = −X5 , [X2 , X4 ] = −4kX3 , [X3 , X4 ] = − X2 , k (2) ∂ (3) ∂ (3) ∂ (2) ∂ +η , [X3 , X5 ] = ξ +η , [X2 , X5 ] = ξ ∂x ∂y ∂x ∂y ∂ ∂ − η (4) , [X4 , X5 ] = −ξ (4) ∂x ∂y

(22)

where the coefficients of the three last generators have the following form: ∂ξ ∂η + η, η (2) = − ξ, ∂θ ∂θ ∂η ∂ξ (3) , η = , ξ (3) = ∂σ ∂σ σ ∂η σ ∂ξ + ξ , η (4) = ξ2 (ξ, η, σ, θ) + 4kθ +η . = ξ1 (ξ, η, σ, θ) + 4kθ ∂σ k ∂σ k ξ (2) =

ξ (4)

(23)

Note, that operator X5 forms an infinite ideal of algebra L, because all
commutators with X5 give particular cases of generator X5 , that is all coefficients ξ (i) , η (i) (i = 2, 3, 4) from (23) are solutions of system (12) due to the closure of L. The groups of point transformations, which correspond to any generator of (21), convert the system (10) to itself. These groups are well known for the generators of L, with the exception of X4 : 1. X1 generates the group of scale (homothety) transformations in the plane xy: x′ = ea1 x, y ′ = ea1 y;

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2. X2 corresponds to the group of rotation: x′ = x cos a2 + y sin a2 , y ′ = −x sin a2 + y cos a2 , θ′ = θ + a2 ; 3. X3 generates the group of translation with respect to the function σ: σ ′ = σ + a3 ; 4. X5 corresponds to generalized translation group in the plane xy: x′ = x + a5 ξ(σ, θ), y ′ = y + a5 η(σ, θ),

(24)

where (ξ(σ, θ), η(σ, θ)) is an arbitrary solution of the system (12) and ai (i = 1, 2..., 5) are sufficiently small group parameters. 5. The one-parameter group of transformation generated by X4 has the following form: x′ = uea4 cos θ′ − ve−a4 sin θ′ ,

y ′ = uea4 sin θ′ + ve−a4 cos θ′ , σ  σ ′ = 2k cosh 2a4 − θ sinh 2a4 ,   2k σ ′ sinh 2a4 − θ cosh 2a4 , θ =− 2k where u and v are the variables from (13): u = x cos θ + y sin θ, v = −x sin θ + y cos θ.

(25)

(26)

Indeed, one can verify the fulfillment of the Lie equations (4) for (25). It is easy to show that transformations (25) act on the variables of the linear system (14) as scale transformations: u′ = ea4 u, v ′ = e−a4 v, α′ = e2a4 α, β ′ = e−2a4 β, so we can call transformations (25) the quasi-scale ones.

4.

Transformation of Solutions

1. As the initial solution U 0 let us take the Prandtl’s solution (15), which may be shown to be an invariant solution with respect to subalgebra X3 + γX5 where the operator X5 has coefficients ξ = 1, η = 0. Indeed, acting by corresponding group of transformations on (15), we shall obtain just another value of an arbitrary constant p1 . The scale transformations corresponding to the operator X1 just change constant h. The application of the transformations of rotation X2 does not produce any significant result from the mechanical point of view, we just obtain rotated parallel plates. Let us consider the action of quasi-scale transformations (25). In the therms of the new variables x′ , y ′ , σ ′ , θ′ the Prandtl’s solution has the same form: s ′ x y′ 2 σ ′ = −p1 − k + k 1 − 2 , (27) h h y ′ = h cos 2θ′ .

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σ − θ = C1 in (17) (see Fig. 1), Let us fix the first family of characteristic curves α = 2k then from (25) we have: σ  σ′ − θ′ = − θ e2a4 = C1 e2a4 , 2k 2k and the equation of characteristic curves for new variables takes the form:  p1  , x′ = −h(2θ′ + sin 2θ′ ) − h 2C1 e2a4 + k y ′ = h cos 2θ′ .

(28)

To obtain reproduced characteristic curves it is necessary to return to the variables x, y, σ σ and θ in (28). Taking into account that = C1 + θ along the fixed curve, we have 2k θ′ = θe−2a4 − C1 sinh 2a4 from (25) and S-curve looks like follows: 
 p1 i  cosh a4 cos(θ′ − θ) − sinh a4 cos(θ′ + θ) 2θ′ + sin 2θ′ + 2C1 e2a4 + k   ′ +h cos 2θ cosh a4 sin(θ′ − θ) − sinh a4 sin(θ′ + θ) , h 
 p1 i  cosh a4 sin(θ′ − θ) + sinh a4 sin(θ′ + θ) y = h 2θ′ + sin 2θ′ + 2C1 e2a4 + k   +h cos 2θ′ cosh a4 cos(θ′ − θ) + sinh a4 cos(θ′ + θ) . (29)

x = −h

h

Figure 1. Initial slip-lines (cycloids) for the solution of Prandtl. σ = C2 − θ, In a similar way one can obtain the S-curve of the second family. Fixing 2k we have θ′ = θe2a4 − C2 sinh 2a4

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Figure 2. Transformed slip-lines. from (25) and S-curve looks as follows: h
  p1 i  x = h 2θ′ − sin 2θ′ − 2C2 e−2a4 + cosh a4 cos(θ′ − θ) − sinh a4 cos(θ′ + θ) k   +h cos 2θ′ cosh a4 sin(θ′ − θ) − sinh a4 sin(θ′ + θ) , h 
 p1 i  cosh a4 sin(θ′ − θ) + sinh a4 sin(θ′ + θ) y = −h 2θ′ − sin 2θ′ − 2C2 e−2a4 + k   ′ +h cos 2θ cosh a4 cos(θ′ − θ) + sinh a4 cos(θ′ + θ) . (30) As for mechanical sense of the field of characteristic curves (29), (30) let us note, that the transformed solution (27) with respect to transformed variables has the same interpretation as the initial solution (15), and the boundary conditions look like this: x′ θ′ y′ =h = πn, n ∈ Z, σ ′ y′ =h = −p1 − k . h

Moreover, the boundary curve y ′ = h will be the envelope for the family of S-curves (29) (in the same way the curve y ′ = −h will be the envelope for family (30)). Let us take θ′ = 0, then from (25) we obtain: x′ = uea4 , y ′ = ve−a4 , σ ′ = σ cosh 2a4 − 2kθ sinh 2a4 , σ sinh 2a4 = 2kθ cosh 2a4 .

If we take the variable σ as a parameter, then σ tanh 2a4 , 2k σ σ′ = , ve−a4 = h, cosh 2a4  
h σ h + p1 . = − σ ′ + p1 = − k k cosh 2a4 θ=

x′ y′ =h

(31)

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From the above two equations we have h v = he , u = − k a4



 σ + p1 e−a4 cosh 2a4

and for the variable x, y we have the parametric equation of the border curve:   σ  σ  σ h + p1 e−a4 cos tanh 2a4 − hea4 sin tanh 2a4 , x=− k cosh 2a4 2k 2k   σ  σ  h σ y=− + p1 e−a4 sin tanh 2a4 + hea4 cos tanh 2a4 . k cosh 2a4 2k 2k

(32)

Finally, we can conclude that the field of the characteristic curves (29), (30) describes the plastic state of the block compressed between rigid plates of the form (32) for the fixed value of the group parameter a4 (see Fig. 2). 2. Now, let us take functions σ and θ from (18) as the initial solution U 0 . It may be shown to be an invariant solution with respect to subalgebra X1 + γX3 . Indeed, acting by corresponding group of transformations on (18), we shall obtain just other values of arbitrary constants p2 and R. The application of the rotation transformations X2 does not produce any significant result from the mechanical point of view, we just obtain rotated circular cavity. Let us consider the action of quasi-scale transformations (25). In the therms of the new variables x′ , y ′ , σ ′ , θ′ the circular solution has the same form: y′ π + , ′ x 4 ′2 + y′ 2 x . σ ′ = −p2 + k + k ln R2 θ′ = arctan

(33)

It is easy to see that v tan θ′ + e−2a4 uea4 sin θ′ + ve−a4 cos θ′ y′ ′ u = a4 = v −2a4 = tan(θ + δ), x′ ue cos θ′ − ve−a4 sin θ′ ′ 1 − tan θ e u v where tan δ = e−2a4 . Then from the first relation of (33) we obtain δ = − π4 , so u v = −ue2a4 .

(34)

Moreover, using the relations (13) we obtain y − tan θ v −x sin θ + y cos θ = = x y = tan(φ − θ) = −e2a4 , u x cos θ + y sin θ 1 + tan θ x and for the function θ we have the explicit formula: θ = φ + arctan e2a4 ,

(35)

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135

where φ is the polar angle. The transformed polar radius r has the form: 2

2

2

r′ = x′ + y ′ = u2 e2a4 + v 2 e−2a4 . From (25), passing to polar coordinates and taking into account (34) and (35) we have:
2r2 e2a4 cos2 arctan e2a4 σ cosh 2a4 = −p2 + k + k ln + 2kθ sinh 2a4 R2 and after some simplifications we have the explicit formula for the function σ: σ=

k r2 −p2 + k + 2k tanh 2a4 (φ + arctan e2a4 ) + ln 2 . cosh 2a4 cosh 2a4 R cosh 2a4

(36)

Finally the S-solution has the form (35), (36). Note, that if a4 = 0, then the S-solution converts to the initial one (18). The S-solution (35), (36) can be interpreted in the following way. For the transformed variables we have the boundary conditions similar to (19): π θ′ r′ =R = φ′ + , 4 σ′ ′ = −p2 + k. r =R

The curve r′ = R looks as


2 r′ = r2 cos2 (φ − θ)e2a4 + sin2 (φ − θ)e−2a4 = R2 ,

but we have φ − θ = arctan e2a4 along this curve, so the boundary line for the S-solution is the circumference of the form r2 = R2 cosh 2a4 . Finally, the S-solution (35), (36) satisfies the following boundary conditions: θ|r=R√cosh 2a4 = φ + arctan e2a4 , −p2 + k σ|r=R√cosh 2a4 = + 2k tanh 2a4 (φ + arctan e2a4 ), cosh 2a4 where the hydrostatic pressure σ depends now on the polar angle φ. 3. Now we shall construct a new analytical solution for the system of plasticity (10) by means of the infinite group (24) of generator X5 . Firstly, let us express solutions (15) and (18) as solutions for linearized system (12). The first one looks as: h h − p1 − h sin 2θ, k k y1 (σ, θ) = h cos 2θ,

x1 (σ, θ) = −σ and the second one has the form x2 (σ, θ) = Re y2 (σ, θ) = Re

p2 −k 2k p2 −k 2k

 π σ cos θ − e 2k , 4  σ π sin θ − e 2k . 4

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Using the relation (8) with a = a5 we obtain the S-solution:    p2 −k π σ h h x = a −σ − p1 − h sin 2θ + (1 − a)Re 2k cos θ − e 2k , k k 4 (37)  p2 −k π σ 2k 2k y = ah cos 2θ + (1 − a)Re e . sin θ − 4 From the first sight, family (37) looks complicated and it seems quite difficult to give some mechanical interpretation. But if we note, that for a = 0 in (37), we have solution (18) with the boundary condition (19) we can seek the boundary curve for S-solution (37) taking σ = −p1 + k, θ = φ + π/4

(38)

and passing to the polar coordinates. Then from the second relation of the S-solution (37) we have: p2 −p1 (39) r = −2ah cos φ + (1 − a)Re 2k ,

while the first relation of (37) is satisfied identically. Therefore, S-solution (37) satisfies boundary conditions (38) along boundary curve (39), which is a limacon of Pascal. This result is similar to the solution obtained in [12].

Figure 3. Initial slip-lines (logarithmic spirals) for the solution for circular cavity. If we pass to the variables α and β (11), then the relations (37) define the parametric equations for the deformed characteristics curves. For example, if we take σ = 2k(α + θ), then the first family of characteristic curves is given by the following equations:    p2 −k π  α+θ p1 x = −ah 2(α + θ) + e , + sin 2θ + (1 − a)Re 2k cos θ − k 4 (40)   p2 −k π α+θ e . y = ah cos 2θ + (1 − a)Re 2k sin θ − 4 In Fig. 3 one can see two families of characteristic curves (20) for the solution (18) with p2 = k for the circular cavity of the radius R = 2. The deformed slip-lines are presented in Fig. 4 for a limacon of Pascal (h = 1, p1 = p2 ).

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Figure 4. Transformed slip-lines for the S-solution for limacon of Pascal.

5.

Conclusion

The chapter is dealing with some applications of group-theoretical methods to the resolution of a hyperbolic system of quasilinear homogeneous PDEs of two independent variables. The principal results consist in using the action of Lie group of point transformation not only over the set of known solutions, but over the families of characteristic curves too. This point of view permits to find out efficiently the suitable boundary conditions for reproduced solutions. Moreover, using of the infinite ideal of admitted Lie algebra of symmetries, as it was shown, permits to construct a new solutions. There are some examples given for the system of the mathematical theory of the plane plasticity. It is necessary to understand quite clearly, that the variation of a group parameter as a parameter of the family of S-solution should be sufficiently small to obtain physically meaningful solutions.

References [1] W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, and S. R. Svirshchevski˘ı, CRC handbook of Lie group analysis of differential equations. Vol. 1, CRC Press, Boca Raton, FL, 1994. Symmetries, exact solutions and conservation laws. [2] B. D. Annin, V. O. Bytev, and S. I. Senashov, Gruppovye svoistva uravnenii uprugosti i plastichnosti, “Nauka” Sibirsk. Otdel., Novosibirsk, 1985 (Russian). [3] George W. Bluman and Sukeyuki Kumei, Symmetries and differential equations, Applied Mathematical Sciences, vol. 81, Springer-Verlag, New York, 1989. [4] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor′kova, I. S. Krasil′shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov, Symmetries and conservation laws for differential equations of mathematical physics,

138

Sergey I. Senashov and Alexander Yakhno Translations of Mathematical Monographs, vol. 182, American Mathematical Society, Providence, RI, 1999. Edited and with a preface by Krasil′shchik and Vinogradov; Translated from the 1997 Russian original by Verbovetsky [A. M. Verbovetski˘ı] and Krasil′shchik.

[5] V. N. Gusyatnikova, A. V. Samokhin, V. S. Titov, A. M. Vinogradov, and V. A. Yumaguzhin, Symmetries and conservation laws of Kadomtsev-Pogutse equations (their computation and first applications), Acta Appl. Math. 15 (1989), no. 1-2, 23–64. Symmetries of partial differential equations, Part I. [6] R. Hill, The Mathematical Theory of Plasticity, Oxford, at the Clarendon Press, 1950. [7] N. H. Ibragimov, A. V. Aksenov, V. A. Baikov, V. A. Chugunov, R. K. Gazizov, and A. G. Meshkov, CRC handbook of Lie group analysis of differential equations. Vol. 2, CRC Press, Boca Raton, FL, 1995. Applications in engineering and physical sciences; Edited by Ibragimov. [8] Peter J. Olver, Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. [9] L. V. Ovsiannikov, Group analysis of differential equations, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. Translated from the Russian by Y. Chapovsky; Translation edited by William F. Ames. [10] B. L. Roˇzdestvenski˘ı and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, RI, 1983. Translated from the second Russian edition by J. R. Schulenberger. [11] S. I. Senashov and A. M. Vinogradov, Symmetries and conservation laws of 2dimensional ideal plasticity, Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 3, 415– 439. [12] S. I. Senashov and A. Yakhno, Reproduction of solutions of bidimensional ideal plasticity, Internat. J. Non-Linear Mech. 42 (2007), no. 3, 500–503. [13] Sergey I. Senashov and Alexander Yakhno, 2-dimensional plasticity: boundary problems and conservation laws, reproduction of solutions, Symmetry in nonlinear mathematical physics. Part 1, 2, 3, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 50, Part 1, vol. 2, Nats¯ıonal. Akad. Nauk Ukra¨ıni ¯Inst. Mat., Kiev, 2004, pp. 231–237. [14] V. V. Sokolovskii, Statics of granular media, Completely revised and enlarged edition. Translated by J. K. Lusher; English translation edited by A. W. T. Daniel, Pergamon Press, Oxford, 1965.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 139-167

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 5

T HE M ODULE S TRUCTURE OF THE I NFINITE -D IMENSIONAL L IE A LGEBRA ATTACHED TO A V ECTOR F IELD Guan Keying∗ Beijing Jiaotong University Beijing, 100044, P.R.China

Abstract Based on a generalized definition on the admittance of a Lie group by a vector field, it is proved that, attached to any given smooth vector field X on a n-dimensional manifold M, there is an infinite-dimensional Lie algebra L(X) formed by infinitesimal generators of all one-parameter Lie groups admitted by X. As a compound module, through its any given basis (X, V1 , V2 , ..., Vn−1 ), L(X) can be treated as a direct sum of two modules L(X) = L<X> ⊕ L where L<X> is generated by X and is a module of rank 1 over the coefficient ring formed by smooth functions, and L is spanned by (V1 , V2 , ..., Vn−1 ) and is a module of rank (n−1) over the coefficient ring formed by all first integrals of the autonomous system determined by X. This module structure is useful in the study of integrating the autonomous system. Based on this structure, examples in seeking exact travelling wave solutions for three famous nonlinear wave equations are given.

1.

Introduction

In the second half of the 19th century Sophus Lie introduced systematically the continuous groups, now known as Lie groups, in order to create a theory of integrating ordinary differential equations similar to Galois theory and Abel’s related works on solving algebraic equations. Since Lie group theory gives fundamental and interlinked rules in and between ∗

E-mail address: [email protected] or [email protected]

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Guan Keying

the basic mathematical sciences–algebra, geometry and analysis, it has had a profound impact on almost all areas of mathematics and mathematically-based science. Roughly speaking, an r-parameter Lie group is a group which is also an r-dimensional differentiable manifold, with the property that the group operations are compatible with the smooth structure. In the practical application of Lie group theory to the differential equations, the applied Lie group is usually locally defined and with complicated form. One of the main method in the research is the infinitesimal method given by Lie, which makes it possible to reduce the study of complicated Lie group G largely to the study of a purely algebraic object, a Lie algebra G. G is an r-dimensional linear vector space spanned by the r infinitesimal generators of G. Besides the ordinary operation on a linear vector space, the operation of Lie bracket between any two elements is closed in G. To every Lie group corresponds a Lie algebra. The particular property of a Lie group and its corresponding Lie algebra is determined by a group of constants, i.e, the structure constants of the Lie group. According to the definition commonly used, a Lie group G is called a symmetry group of a system of differential equation if the system is invariant under the action of the group, or more precisely, this Lie group is said to be admitted by the system. The main results of the modern Lie group theory on ordinary differential equations can be represented as follows (ref. [1]): In the case of ordinary differential equations, invariance under a one-parameter symmetry group implies that we can reduce the order of the equation by one, recovering the solutions to the original equation from those of the reduced equation by a single quadrature. For a single first order equation, this method provides an explicit formula for the general solution. Multi-parameter symmetry groups engender further reductions in order, but, unless the group itself satisfies an additional “solvability” requirement, we may not be able to recover the solutions to the original equation from those of the reduced equation by quadratures alone. Suppose du/dx = F (x, u) is a system of q first order ordinary differential equations, and suppose G is an r-parameter solvable group of symmetries, acting regularly with rdimensional orbits. Then the solutions u=f(x) can be found by quadrature from the solutions of a reduced system dw/dy = H(y, w) of q-r first order equations. In particular, if the original system is invariant under a q-parameter solvable group, its general solution can be found by quadratures alone. No doubt, these results are very important and elegant in both theory and application. However, the requirement that a given system of ordinary differential equations admits a “multi-parameter group”, i.e., the system is invariant under the “multi-parameter group” action, is too strict somehow. In the theory of Lie group, it is well known that every one-parameter Lie group corresponds uniquely to an infinitesimal transformation, which is represented by an infinitesimal generator. If r different one-parameter groups are given through r corresponding different infinitesimal generators, but the r given one-parameter groups, as a whole, may not form a r-parameter group, unless the r infinitesimal generators can form a basis of Lie algebra. Sometimes, for a given system of ordinary differential equations, it is easier to to seek, say r, individual one-parameter groups admitted, than to seek a r-parameter group admitted. Besides, for a given system of nonlinear ordinary differential equation, if a solvable multiple-parameter Lie group admitted by it is known, it is

The Module Structure of the Infinite-Dimensional Lie Algebra...

141

usually still a very difficult task to reduce the system or to seek its first integrals. In the study of integrating an autonomous system of ordinary differential equations through its r independent one-parameter Lie groups, which may not form a r-parameter Lie algebra, we noticed the following facts: I. It is more convenient to accept a generalized definition on the admittance of a Lie group by a vector field X or by its corresponding autonomous system of ordinary differential equations, that is to require only the integral curves of the vector field to be mapped into integral curves of the same vector field instead of requiring the autonomous system is invariant under the action of the Lie group. By this definition, we may prove that a Lie group G is admitted by a given vector X if and only if there is a C ∞ function a(x), such that [V, X]= a(x)X,

(1)

where V is the infinitesimal generator of G. This criterion does not need the help of the prolongation of the Lie group, which is necessary in the traditional theory. II. The infinitesimal generators of all one-parameter Lie groups admitted by a given n-dimensional vector X form a compound module L(X), i.e., through its any basis (X, V1 , V2 , ..., Vn−1 ), L(X) can be decomposed as a direct sum of two modules, L(X)=L<X> ⊕L . where L<X> is spanned by the basis (X) and is a module of rank 1 over the coefficient ring formed by all C ∞ functions, and L is spanned by the basis (V1 , V2 , ..., Vn−1 ) and is a module of rank (n-1) over the coefficient ring formed by all first integrals of the autonomous system determined by X. An element of L<X> is an infinitesimal generator of a one-parameter Lie group admitted trivially by X, and an element of L is an infinitesimal generators of one-parameter Lie groups admitted nontrivially by X. The formation of module L depends on the choice of the basis (V1 , V2 , ..., Vn−1 ). Therefore, for any given basis (X, V1 , V2 , ..., Vn−1 ) of L(X), any element V in L(X) can be expanded uniquely as V= a(x)X+

n−1 X

Ωi (x)Vi ,

i=1

where a(x) ∈ C ∞ , and Ωi (x) is a first integral of the autonomous system or a constant for i=1,2,...,n-1. III. In L(X), the operation of Lie bracket is closed. Especially, for any given basis (X,V1 , V2 , ..., Vn−1 ) of L(X), the Lie bracket between Vi and Vj is expanded uniquely as n−1 X k 0 Cij (x)Vk . [Vi ,Vj ]=Cij (x)X+ k=1

C0ij (x)

C ∞,

where ∈ and the other coefficients Ckij (x) (i, j, k = 1, 2, ..., n − 1) are the first integrals of the autonomous system (some of them may be constant).

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Some of the above facts were posed firstly in [4]. Therefore, for an given vector field X, if a system of n linearly independent infinitesimal generators, X, V1 , V2 , ..., Vn−1 , is given, and if (V1 , V2 , ..., Vn−1 ) is not a basis of a (n − 1)-dimensional Lie algebra, then these infinitesimal generators may provide us more information on integrating the corresponding autonomous system than they span a (n − 1)dimensional Lie algebra, though in the latter case they should satisfy stricter requirements. The rest of this article is arranged as follows. In section 2, we will show that the ndimensional module structure of the infinite-dimensional Lie algebra Vect(M). In section 3, we will introduce the details of the facts I, II, and III mentioned above, and will provide some technique for seeking the first integrals through a given basis (X, V1 , V2 , ..., Vn−1 ) .. Based on the theory obtained, the application examples in seeking the exact travelling wave solutions for 3 famous nonlinear wave equations will be introduced in section 4. In section 5, the conclusion of this article is given.

2.

The Module Structure of Vect(M)

Let M be a n-dimensional C ∞ -smooth (or analytic) manifold, x = (x1 , x2 , ..., xn ) denote the local coordinate, and let Vect(M) be the vector space of all C ∞ -smooth (or analytic) vector fields on M. A vector field V ∈ Vect(M) can be represented in a local domain U ∈ M by the linear partial differential operator: V=V1 (x)

∂ ∂ ∂ + V2 (x) + · · · + Vn (x) , ∂x1 ∂x2 ∂xn

(2)

where Vi (x) ∈ C ∞ (U ) (or analytic function), for i = 1, 2, ..., n. Besides the fundamental operations of vector space, the operation of Lie bracket may also introduced into Vect(M). Let V1 , V2 be any given elements Vect(M), then their Lie bracket is defined as [V1 , V2 ]=V1 V2 −V2 V1 , (3) which is also an element in Vect(M). The Lie bracket satisfies the anti-commutative law [V1 , V2 ]= − [V2 , V1 ], and the Jacobi identity [V1 , [V2 , V3 ]] + [V2 , [V3 , V1 ]] + [V3 , [V1 , V2 ]]=0, where V1 , V2 , V3 are arbitrary elements in Vect(M). Therefore, Vect(M) is a Lie algebra over the real field (or the complex field). Generally speaking, Vect(M) is infinite-dimensional, it has not the conception of structure constants, which may defined only for any finite-dimensional Lie algebra. However, Vect(M) may be treated as a module of finite-rank, then a similar conception ”structure coefficients” can be introduced. In fact, we may let K be the ring of all C ∞ (or analytic) functions on M (note that a function f ∈ K is allowed to be multiple-valued and to have a singularity set which is thin in M), then Vect(M) is a K-module (ref. [2]).

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It is not difficult to see that Vect(M) is of rank n. Assume that the n arbitrary given vector fields, n X ∂ Vi,j (x) Vi = ∈ Vect(M), i = 1, 2, ..., n, (4) ∂xj i=1

are linearly independent in the sense of K-module, i.e., they satisfy   V1,1 (x) V1,2 (x) ... V1,n (x)  V2,1 (x) V2,2 (x) ... V2,n (x)      . . . .  6= 0, ∀x ∈ D, det    . . . .     . . . .

(5)

Vn,1 (x) Vn,2 (x) ... Vn,n (x)

where D is an open and dense subset of M , then any element V ∈ Vect(M) can be represented as a linear combination of the n vector fields: V=

n X

Ai (x)Vi ,

(6)

i=1

where Ai (x) ∈ C ∞ , i = 1, 2, ..., n. So, this system of vector fields is a basis of the module Vect(M). For any analytic manifold M, it is always possible for us to choose a local basis such that, in a given local domain U in M,  1, if i = j , Vi,j (x) = 0, if i 6= j this local basis can be analytically continued to an open and dense subset D of M , while the inequality (5) is kept (note: generally speaking, the local basis can not be continued to the whole manifold). Therefore, we have the following conclusion: Theorem 1. Vect(M) is a K-module of rank n. For any given module basis (V1 , V2 , ..., Vn ) of Vect(M), and for any couple (Vi , Vj ) (i, j = 1, 2, ..., n), the Lie bracket [Vi , Vj ] can be written as a linear combination in terms of this basis, i.e., there exist a series of functions Ckij (x)∈ K, i, j, k = 1, 2, ..., k, such that [Vi , Vj ]=

n X

Ckij (x)Vk ,

i, j = 1, 2, ..., n.

k=1

Obviously, these functions satisfy the following equalities: Ckij (x) = −Ckji (x), ∀ i, j, k = 1, 2, ..., n, and

=

n X

m l m l [Clij (x)Cm lk (x) + Cjk (x)Cli (x) + Cki (x)Clj (x)]

l=1 m m Vi Cm jk (x)+Vj Cki (x)+Vk Cij (x),

∀ i, j, k, m = 1, 2, ..., n

(7)

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Guan Keying

If these functions are all constant, then this basis spans a n-dimensional Lie algebra and generates a n-parameter Lie group G (ref. [1, 3]). In this case, these constants, Ckij , are called the structure constants of the Lie group G. With this background, these functional coefficients, Ckij (x) ∈ K, are called the structure coefficients of Vect(M) with respect to the basis (V1 , V2 , ..., Vn ).. It is an interesting facts that a system of n vector fields (V1 , V2 , ..., Vn ) in Vect(M) may form a basis of a n-dimensional Lie algebra, though these vectors are linearly dependent to each other. See the following example: Example 2. Consider the 3 vector fields on R3 , V1 = z

∂ ∂ −y , ∂y ∂z

V2 = x

∂ ∂ −z , ∂z ∂x

V3 = y

∂ ∂ −x . ∂x ∂y

They form a basis of unsolvable 3-dimensional Lie algebra which generates the rotation group acting on R3 (ref. [6, 7]), but they are linearly dependent in the sense of K-module, for xV1 +yV2 +zV3 = 0.

3.

The Module Structure of L(X)

Let X=

n X

Xi (x)

i=1

∂ . ∂xi

(8)

be a non-zero vector field on a n-dimensional analytic manifold M. It determines an autonomous system of ordinary differential equations: dxi = Xi (x), dt

i = 1, 2, ..., n,

(9)

The basic theorem of the theory of ordinary differential equations guarantees that, in a neighborhood U (⊂ M) of a regular point x of X (X does not vanish at this point), this system has (n − 1) functionally independent local first integrals Ω1 (x), Ω2 (x), ..., Ωn−1 (x), which satisfy XΩi (x) = 0, i = 1, 2, ..., n − 1. We assume that these first integrals can be continued analytically to a common domain D, which is dense in M. Then any other first integral of (9) on D must be a compound function in Ω1 (x), Ω2 (x), ..., Ωn−1 (x) (ref. [5]). The family of integral curves of (9) on D can be represented locally as follows: Ωi (x) = ci , ci ∈ R(orC),

i = 1, 2, ..., n − 1.

(10)

The Module Structure of the Infinite-Dimensional Lie Algebra...

145

Traditionally, a n-th order ordinary differential equation dn y = f(x, y, y′ , ..., y(n−1) ), dxn

(11)

is said to admit a one-parameter Lie transformation (called the point transformation) group G∗ on two variables x and y, generated by V∗ = ξ(x, y)

∂ ∂ + η(x, y) , ∂x ∂y

(12)

if the equation (11) is invariant under the action of the group. In order to check the invariance of the equation, one should also consider how the transformations of the derivatives y′ , y′′ , ..., y(n) depend on the original group G∗ . By taking these related transformations into account, the corresponding transformation group on the n+2 variables (x, y, y1 , ..., yn ) (yi ≡ y(i) ) (called the n-th prolonggation of G∗ , denoted as pr(n) G∗ ) is given by the following infinitesimal generator ∂ ∂ V∗(n) = ξ(x, y) ∂x + η(x, y) ∂y + η (1) (x, y, y1 ) ∂y∂ 1 + ... + η (n) (x, y, y1 , ..., yn ) ∂y∂ n ,

where

η (1) (x, y, y1 ) (2) η (x, y, y1 , y2 )

= ηx + (ηy − ξx )y1 − ξy y12 = ηxx + (2ηxy − ξxx )y1 + (ηyy − 2ξxy )y12 − ξyy y13 + (ηy − 2ξx )y2 − 3ξy y1 y2 ,

(13)

(14)

and the detail forms of η (i) (x, y, y1 , ..., yi ), i = 3, 4, ..., can be found in [7, 1]. Similarly, a system of n first order ordinary differential equations dxi = fi (t, x1 , x2 , ..., xn ), dt

i = 1, 2, ..., n,

is said to admit an one-parameter Lie transformation group G∗ on the n + 1 variables t, x1 , ..., xn , which is generated by V∗ = τ (t, x)

∂ ∂ ∂ + ξ1 (t, x) + ... + ξn (t, x) , ∂t ∂x1 ∂xn

if the system of equations is invariant under the action of the group. Naturally, in order to check the invariance of the system, we should use the corresponding first prolongation of G∗ (ref. [7, 1]). Now for the autonomous system (9), we may generalize the traditional definition on the admittance of a Lie group by a system of ordinary differential equations as follows. Definition 3. A one-parameter Lie group G and its infinitesimal generator V=

n X i=1

Vi (x)

∂ ∂xi

is said to be admitted by the given n dimensional autonomous system (9) and by the corresponding vector field X, if the family (10) of integral curves of (9) in phase space is invariant under the action of the transformation G.

146

Guan Keying

By the suggestive notation (ref. [7, 1]), under the action of a one-parameter Lie group G generated by V, a point x ∈ M is mapped to x∗ =eεV x, and a function Ω(x) is mapped to Ω(x∗ ) = Ω(eεV x) = eεV Ω(x) = Ω(x)+εVΩ(x) + O(ε2 ), where eεV =

∞ k X ε k=0

k!

Vk .

Assume a given autonomous system (9) admits the Lie group G generated by V in the sense of definition 3. If Ω(x) is a first integral of (9), and if C is an arbitrary given integral curve of (9), then C is mapped to the integral curve C ∗ of the same (9) under the action of the group G, where C ∗ depends on the parameter ε. So we have Ω(x) = c, and

∀ x ∈ C,

Ω(x∗ ) = Ω(x)+εVΩ(x) + O(ε2 ) = c∗ (ε), ∀ x ∈ C.

where c is a constant, and c∗ (ε) depends only on the parameter ε and satisfies c∗ (0) = c. It implies that VΩ(x) must be a constant on the integral curve C, i.e., VΩ(x) must also be a first integral of (9). Conversely, for a given Lie group G generated by V, if VΩ(x) is also a first integral of (9) provided Ω(x) is a first integral of (9), then Ψ(ε, x) = eεV Ω(x) is obviously a first integral of (9) for any small parameter ε. Clearly, under the action of this Lie group G, an integral curve of (9) must be mapped to an integral curve of the same (9), i.e., G is admitted by (9) in the sense of definition 3. Therefore, we have obtained the following theorem: Theorem 4. The generalized definition 3 is equivalent to the following analytical requirement to the infinitesimal generator V of G: for any given group of (n−1) functionally independent local first integrals Ω1 (x), Ω2 (x), ..., Ωn−1 (x) of the system (9), the following functions, Pi (x) = VΩi (x) =

n X j=1

Vj (x)

∂Ωi , ∂xj

i = 1, 2, ..., n − 1,

(15)

The Module Structure of the Infinite-Dimensional Lie Algebra...

147

are also first integrals of (9). In [8], it has been proved that Theorem 5. Without loss of generality, assume that Xn (x) 6= 0. The statement that the autonomous system (9) admits a Lie transformation group G generated by V=

n X i=1

Vi (x)

∂ ∂xi

in the sense of definition 3 is equivalent to that the system of (n − 1) first order ordinary differential equations dxi = Xi (x)/Xn (x), i = 1, 2, ..., n − 1, dxn

(16)

admits the same Lie group G in the traditional sense, i.e., the system (16) is invariant under the action of group G. It is well known that the n-th order ordinary differential equation (11) is equivalent to the following system of n first order ordinary differential equations         

dy dx dy1 dx

= y1 = y2 .. .

dyn−1 dx

= f(x, y, y1 , ..., yn−1 )

.

(17)

It is clear that, if the equation (11) is invariant under the action of an one-parameter Lie transformation group G∗ generated by (12), and (13) is the infinitesimal generator of the n-th extension of G, then the corresponding system (17) is also invariant under the action of the Lie group G = pr(n−1) G∗ on the n + 1 variables x, y, y1 , ..., yn−1 , which is generated by ∂ ∂ V = ξ(x, y) ∂x + η(x, y) ∂y + η (1) (x, y, y1 ) ∂y∂ 1 + (18) ... + η (n−1) (x, y, y1 , ..., yn−1 ) ∂y∂n−1 . Therefore, from theorem 5, we may obtain the following corollary immediately: Corollary 6. If the n-th order ordinary differential equation (11) is invariant under the action of the one-parameter Lie transformation group G∗ generated by (12), and (13) is the infinitesimal generator of the n-th prolongation of G∗ , then, by the generalized definition 3, the corresponding (n + 1) dimensional autonomous system  dx  = 1  dt   dy  = y1  dt  dy 1 = y2 , (19) dt  .  .  .     dyn−1 = f (x, y, y , ..., y ) dt

admits the Lie group G generated by (18). The following lemma is obvious:

1

n−1

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Guan Keying

Lemma 7. Let Y be another vector field on M. The autonomous system determined by Y has the same family of integral curves (10) of the given system (9) if and only if there is a non-zero C ∞ function a(x) such that Y= a(x)X.

Theorem 8. An autonomous system (9) admits a Lie group G generated by V in the sense of definition 3, if and only if there is a C ∞ function a(x) such that [V, X]= a(x)X.

(20)

Proof. Assume that (20) is true. Since any first integral Ωi (x) of (9) satisfies that XΩi (x) = 0, then we have a(x)XΩi (x) = 0. On the other hand, [V, X]Ωi (x) = (VX − XV)Ωi (x) =XVΩi (x). From equality (20) we see −a(x)XVΩi (x) = 0. Hence, VΩi (x) must be a first integral Pi (x) of (9), i.e., the equality (15) is satisfied for i = 1, 2, ..., n − 1. So the group G is admitted by (9). Conversely, if the group G is admitted by (9), then (15) is satisfied. This means [V, X]Ωi (x) = 0, for any first integral Ωi (x) of (9), Therefore, by lemma 4, there must be a C ∞ function a(x) such that (20) is satisfied.  Let f(x) be a C ∞ function. It is easy to see that if a group G is generated by V= f(x)X, then it must be admitted by (9). This local Lie group G generated by V = f(x)X is said to be admitted trivially by (9). Clearly, for a trivially admitted Lie group, the right hand side of (15) must be zero for any i = 1, 2, ..., n − 1. If the right hand side of (15) is not zero for some i, then the corresponding Lie group is said to be admitted nontrivially by (9). Let L(X) be the set of infinitesimal generators of all one-parameter Lie groups admitted by the autonomous system determined by X. It is easy to see that L(X) ⊂ Vect(M) and that L(X) is a linear vector space over the number field R or C. Obviously, X∈L(X). Theorem 9. For a given vector field X, there are at least (n − 1) infinitesimal generators, V1 , V2 , ..., Vn−1 , in L(X), such that X, V1 ,V2 , ...,Vn−1 are linearly independent to each other. Proof. Let Ωi (x), i = 1, 2, ..., n − 1 be (n − 1) functionally independent first integrals of the autonomous system (9) determined by X. The equalities XΩi (x) = 0,

i = 1, 2, ..., n − 1,

The Module Structure of the Infinite-Dimensional Lie Algebra...

149

can be explained geometrically as that the (n − 1) gradient vectors, gradΩi (x), i = 1, 2, ..., n − 1, are all perpendicular to the vector X. Since the (n − 1) first integrals are functionally independent, the corresponding gradient vectors are linearly independent. Therefore, the following matrix   X1 (x) X2 (x) ... Xn (x) ∂Ω1 1  ∂Ω1  ... ∂Ω  ∂x1  ∂x2 ∂xn  ∂Ω2 ∂Ω2 ∂Ω2  ...  ∂x1  ∂x2 ∂xn   M= (21) . . . . ,     . . . .     . . . . ∂Ωn−1 ∂Ωn−1 ∂Ωn−1 ... ∂x1 ∂x2 ∂xn is nonsingular. For i = 1, 2, ..., n − 1, let the (n − 1) vectors in matrix form

i = 1, 2, ..., n − 1

Vi = (Vi1 (x), Vi2 (x), ..., Vin (x)), satisfy the following system of linear equations: T MVT i = ei ,

where

(22)

i

z }| { ei = (0, · · · , 0, 1, 0, · · · , 0), {z } | n

T

T

Vi and ei are their transposed representations respectively. From (22), it is easy to see that the corresponding (n − 1) vector fields in operator form, Vi =

n X

Vij (x)

j=1

satisfy Vi Ωj (x) =



1, 0,

if if

∂ , ∂xj

i = 1, 2, ..., n − 1,

i=j , i 6= j

i, j = 1, 2, ..., n − 1.

These equalities imply that, for i = 1, 2, ..., n − 1, the equation (15) is satisfied by Vi , i.e., Vi is an infinitesimal generator of a one-parameter Lie group Gi admitted nontrivially by (9). Obviously, these vector fields are linearly independent. Besides, for any i = 1, 2, ..., n − 1, the first equation of the system (22), i.e., X · Vi =

n X j=1

Xj (x)Vij (x) = 0,

i = 1, 2, ..., n − 1,

can be explained geometrically as that the obtained (n − 1) vector V1 , ..., Vn−1 , are all perpendicular to the vector X. Therefore, the n vectors X, V1 ,V2 , ...,Vn−1 are linearly independent to each other. 

150

Guan Keying Let (X, V1 , V2 , ..., Vn−1 )

(23)

be a system of n linearly independent infinitesimal generators in L(X). We should point out here that it is not necessary to let V1 , V2 , ..., Vn−1 be perpendicular to X, though the obtained V2 , ...,Vn−1 from (22) just be so. Clearly, this system is a basis of the module Vect(M). It is not difficult to prove that the (n − 1) infinitesimal generators , V1 , V2 , ..., Vn−1 , are admitted nontrivially by X and by the corresponding autonomous system (9). The system (23) is also called a basis admitted by X, or is called a basis of L(X). Theorem 10. Let (X, V1 , V2 , ..., Vn−1 ) be a basis admitted by X, then a vector field V belongs to L(X) if and only if it can be represented as V= a0 (x)X+

n−1 X

ai (x)Vi ,

(24)

i=1

where a0 (x) is a C ∞ -function, and ai (x) is a first integral of (9) or a constant for i = 1, 2, ..., n − 1. Proof. Since (X, V1 , V2 , ..., Vn−1 ) is a basis of the module Vect(M), so the vector field V can be represented as V= a0 (x)X+

n−1 X

ai (x)Vi .

i=1

Seeing that all of Vi (i = 1, 2, ..., n − 1) are admitted by X, by theorem 8, there exist (n − 1) C ∞ -functions bi (x) such that [Vi , X]= bi (x)X,

i = 1, 2, ..., n − 1.

(25)

If the vector fields V is also admitted by X, then there is a C ∞ -function b(x) satisfying [V, X]= b(x)X, i.e., [

n−1 X i=1

n−1 X ai (x)bi (x)−Xa0 (x)]X− (Xai (x))Vi = b(x)X. i=1

Since X, V1 , V2 , ..., Vn−1 are linearly independent, we have then [

n−1 X

ai (x)bi (x)−Xa0 (x)] = b(x),

i=1

and Xai (x) = 0, ∀x ∈ U,

i = 1, 2, ..., n − 1.

Therefore ai (x) is a first integrals of (9) or a constant for i = 1, 2, ..., n − 1.

The Module Structure of the Infinite-Dimensional Lie Algebra...

151

Contrarily, if ai (x) is a first integrals of (9) or a constant for i = 1, 2, ..., n − 1, then we have n−1 X [V, X]= [ ai (x)bi (x)−Xa0 (x)]X. i=1

By theorem 8, we see that V is admitted by X.  Theorem 10 has immediately the corollary: Corollary 11. Based on any given basis (X, V1 , V2 , ..., Vn−1 ), the linear space L(X) is a compound module, i.e., it is a direct sum of two modules L<X> and L , where L<X> is spanned by the basis (X) and is a module of rank 1 over the ring formed by all C ∞ -functions, and L is spanned by the basis (V1 , V2 ,..., Vn−1 ) and is a module of rank (n − 1) over the ring formed by all first integrals of (9). Now consider the operation of Lie bracket in L(X). Let Ω1 (x), Ω2 (x), ..., Ωn−1 (x) be n − 1 functionally independent first integrals of (9). For any V1 and V2 in L(X), there must be corresponding first integrals P1i (x) and P2i (x) such that V1 Ωi (x) = P1i (x), V2 Ωi (x) = P2i (x),

∀i = 1, 2, ..., n − 1.

So [V1 , V2 ]Ωi (x) =V1 P2i (x) − V2 P1i (x).

Since V1 P2i (x) and V2 P1i(x) are first integrals of (9), [V1 , V2 ]Ωi (x) must also be a first integral of (9) (or a constant) for i = 1, 2, ..., n − 1. So [V1 , V2 ] is in L(X). Therefore we have obtained Theorem 12. The compound module L(X) is a Lie algebra. Generally speaking, L(X) is an infinite-dimensional Lie algebra. From theorem 10 and 12, we may obtain immediately the following theorem: Theorem 13. For any basis (X, V1 , V2 , ..., Vn−1 ) of L(X), let 0 [Vi , Vj ] = Ci,j (x)X+

n−1 X

Ckij (x)Vk .

(26)

k=1

The following structure coefficients, Ckij (x),

i, j, k = 1, 2, · · · , n − 1,

(27)

are first integrals of (9) or constants. The facts obtained above may provide us useful means in the study of integrating an autonomous system. Example 14. Consider the following 4-dimensional autonomous system:  dx  = y2 + z2   dt    p   dy   dt = −xy − ız x2 + y 2 + z 2 (x, y, z, u) ∈ C 4 , (28) p   dz 2 2 2  = −xz + ıy x + y + z  dt       du = 0 dt

152

Guan Keying √ where ı = −1 is the imaginary unit. It is easy to check that its corresponding vector field X admits the following infinitesimal generators: V1 V2 V3 V4 V5

= = = = =

∂ ∂ z ∂y − y ∂z , ∂ ∂ x ∂z − z ∂x , ∂ ∂ y ∂x − x ∂y , ∂ ∂ ∂ x ∂x + y ∂y + z ∂z ∂ . ∂u

There exist the following interesting facts: (1) Since [V1 , V2 ]=V3 , [V1 , V3 ]= − V2 ,

,

[V2 , V3 ]=V1 ,

V1 , V2 and V3 form a basis of unsolvable 3-dimensional Lie algebra, but they are C ∞ linearly dependent as mentioned in Example 2. So (X, V1 , V2 , V3 ) is not a basis of L(X). (2) (X, V1 , V4 , V5 ) is a basis of L(X). Since [V1 , V4 ]=[V1 , V5 ]=[V4 , V5 ]= 0, V1 , V4 and V5 span a solvable 3-dimensional Lie algebra. But it is not easy to obtain directly three functionally independent first integrals of (28) from this basis. (3) V1 , V2 and X are C ∞ -linearly dependent, i.e. p z −xy − ı z x2 + y2 + z2 V1 − 2 X; V2 = 2 2 y +z y + z2 and V1 , V3 and X are C ∞ -linearly dependent, i.e. p −xz + ı y x2 + y2 + z2 y V3 = V1 + 2 X. 2 2 y +z y + z2 By theorem 10, we have obtained immediately the following two first integrals of (28): p −xy − ı z x2 + y2 + z2 , Ω1 (x, y, z, u) = y 2 + z2 and

p −xz + ı y x2 + y2 + z2 Ω2 (x, y, z, u) = . y 2 + z2

And the fact that the system admits the generator V5 gives directly another first integral: Ω3 (x, y, z, u) = u. Therefore, without using V4 and without using the integral operation, we have obtained easily three functionally independent first integrals of the system. For a given basis (X, V1 , V2 , ..., Vn−1 ), besides the structure coefficient Ckij (x), i, j, k = 1, 2, · · · , n − 1, we may still obtain some other first integrals by the following theorem

The Module Structure of the Infinite-Dimensional Lie Algebra...

153

Theorem 15. For a given basis (X, V1 , V2 , ..., Vn−1 ) of L(X), let Φ1 (x), Φ2 (x), ..., Φn−1 (x) be (n-1) known first integrals or constants, provided that at least one of them is not zero. If these first integrals and constants satisfy Vi Φj (x)−Vj Φi (x) =

n−1 X

Ckij (x)Φk (x),

∀ i, j = 1, 2, ..., n − 1,

k=1

then through solving the following system of linear algebraic equations  XΩ = 0     V Ω = Φ1  1  V2 Ω = Φ2  ..   .    Vn−1 Ω = Φn−1

(29)

(30)

∂Ω we may obtain the n partial derivatives ∂x , i = 1, 2, ..., n of an unknown first integral Ω(x), i and then may obtain Ω(x) through the path integral Z ∂Ω ∂Ω ∂Ω dx1 + dx2 + ... + dxn . (31) Ω(x) = ∂x1 ∂x2 ∂xn

Proof. For convenience, we use the following new notations: V0 =X, Let Vi =

and n X

V0i (x) = Xi (x),

Vij

j=1

∂ , ∂xj

i = 1, 2, ..., n.

i = 1, 2, ..., n − 1.

If there is such an first integral Ω(x) of (9) satisfying (30), then for any i, j = 1, 2, ..., n − 1, the following equality should be satisfied: 0 [Vi , Vj ]Ω(x) = Cij (x)V0 Ω(x) +

n−1 X

k Cij (x)Vk Ω(x).

(32)

k=1

The left hand side of (32) equals [Vi , Vj ]Ω(x) = Vi Vj Ω(x)−Vj Vi Ω(x) = Vi Φj −Vj Φi . And the right hand side of (32) should equal C0ij (x)V0 Ω(x)

+

n−1 X k=1

k Cij (x)Vk Ω(x)

=

n−1 X

k Cij (x)Φk .

k=1

So the equality (32) implies that the condition (29) is necessary for the first integral Ω(x) satisfying (30).

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Guan Keying

Now we prove that the condition (29) is sufficient for that we may obtain an first integral Ω(x) from (30) and (31). The equation system (30) can be written as  ∂Ω ∂Ω ∂Ω V01 ∂x + V02 ∂x +···+ V0n ∂x = Φ0  n 1 2   ∂Ω ∂Ω  V11 ∂Ω + V12 ∂x2 +···+ V1n ∂xn = Φ2 ∂x1 (33) . ..     ∂Ω ∂Ω ∂Ω + V(n−1)2 ∂x + · · · + V(n−1)n ∂x = Φn−1 V(n−1)1 ∂x 1 2 1 where Φ0 = 0. Let

V01 V02 ··· V0n V11 V · · · V 12 1n D = .. .. .. . . . V(n−1)1 V(n−1)2 · · · V(n−1)n

.

For any given i, j, 0 ≤ i < j ≤ n − 1, and any given p, q, 1 ≤ p < q ≤ n, let Vip Viq . Vijpq = Vjp Vjq

(34)

(35)

which is the sub-determinant of order 2 in D, and let Aijpq

(36)

which is the algebraic cofactor of Vijpq in D. The Laplace expansion theorem shows that for any given i, j, 0 ≤ i < j ≤ n − 1, X D= Vijpq Aijpq . (37) 1≤p
Clearly, D 6= 0, for (V0 , V1 , V2 , ..., Vn−1 ) is a basis of L(X). So we can obtain a non-zero solution of (32) ∂Ω ∂Ω ∂Ω ( , , ..., ), (38) ∂x1 ∂x2 ∂xn In order to guarantee that there is a function Ω(x) with the obtained partial derivatives (38) and that Ω(x) can be obtained from the path integral (31), it is necessary and sufficient that the obtained partial derivatives (38) satisfy ∂2Ω ∂2Ω = , ∂xp ∂xq ∂xq ∂xp

∀ p, q,

1 ≤ p < q ≤ n.

(39)

Notice that the operator equality [Vi , Vj ] =

C0ij (x)V0

+

n−1 X

Ckij (x)Vk ,

k=0

is equivalent to the following equality n n−1 X X

l=0 m=1

n−1

(Vim

X ∂Vjl ∂Vil ∂ Ckij (x)Vk . − Vjm ) = C0ij (x)V0 + ∂xm ∂xm ∂xl k=0

(40)

The Module Structure of the Infinite-Dimensional Lie Algebra... Using (40) and (29), i, j, 0 ≤ i < j ≤ n − 1,

155

we may directly calculate [Vi , Vj ]Ω for any given

[Vi , Vj ]Ω = Vi (Vj Ω) − Vj (Vi Ω) P ∂2Ω ∂2Ω = 1≤p
(41)

On the other hand, from (30) we have [Vi , Vj ]Ω = Vi (Vj Ω) − Vj (Vi Ω) = Vi Φj − Vj Φi . Therefore, the following system of X

Vijpq (

1≤p
n(n−1) 2

equalities must be satisfied,

∂2Ω ∂2Ω − ) = 0, ∂xp ∂xq ∂xq ∂xp

∀ i, j,

The system (42) can be treated as a system of n(n−1) unknown variables 2 αpq =

∂2Ω ∂2Ω − , ∂xp ∂xq ∂xq ∂xp

n(n−1) 2

0 ≤ i < j ≤ n − 1.

(42)

homogeneous equations in the

1≤p
. Let



  MV =  

V0112 V0212 .. .

V0113 V0213 .. .

··· ···



V01(n−1)n V02(n−1)n .. .

V(n−2)(n−1)12 V(n−1)(n−2)13 · · · V(n−2(n−2)(n−1)n

which is the coefficient matrix of the equation system (42), and let  A0112 A0212 ··· A(n−2)(n−1)12  A0113 A · · · A(n−2)(n−2)13 0213  M∗V =  .. .. ..  . . .

A01(n−1)n A02((n−1)n · · · A(n−2)(n−2)(n−1)n .

Both MV and M∗V are order

n(n−1) 2

×

n(n−1) . 2



  MV M∗V =  

D 0 .. . 0

From (37) we have  0 ··· 0 D ··· 0   .. ..  . . .  0

··· D



   

  . 

(43)

(44)

156

Guan Keying

So, MV is a nonsingular matrix. It implies that the equalities (39) must be satisfied. Therefore, from the non-zero solution (38) of (33) and the path integral (31) we may obtain the first integral Ω(x) of (9).  Note: If all of Φ1 (x), Φ2 (x), ..., Φn−1 (x) are constants c1 , ..., cn−1 , then the condition (29) can be written as n−1 X

Ckij (x)ck = 0,

k=1

∀ i, j = 1, 2, ..., n − 1.

(45)

In [9], the condition (45) was proved in case of n = 3 and all of Ckij , (i, j, k = 1, 2) are constants. And before that, in [10] the same condition was also proved in a more particular 2 = 0. case: n = 3 and C112 = 1, C1,2

4.

Some Applications

In this section, we will consider more examples of applying the results obtained in section 3 to seek the exact travelling wave solutions for some famous nonlinear wave equation. In this field, the second order ordinary differential equation x ¨ = f(t, x, x) ˙

(46)

appears frequently. In some cases, we may obtain a one-parameter point transformation group G∗ generated by ∂ ∂ V∗ = ξ(t, x) + η(t, x) , (47) ∂t ∂x which is admitted by (46) in the traditional sense that the equation (46) is invariant under the action of this point transformation group, i.e., V∗(2) (¨ x − f(t, x, x)) ˙ = 0, when where

x ¨ = f(t, x, x), ˙

 ∗(2) ∂ ˙ x ¨) ∂¨ V = V∗ + η (1) (t, x, x) ˙ ∂∂x˙ + η (2) (t, x, x,  x   (1) 2 η (t, x, x) ˙ = ηt + (ηx − ξt )x˙ − ξx x˙  η (2) (t, x, x, ˙ x ¨) = ηtt + (2ηtx − ξtt )x˙ + (ηxx − 2ξtx )x˙ 2 −   ξxx x˙ 3 + (ηx − 2ξt )¨ x − 3ξx x¨ ˙x

(ref. [7, 1]). The equation (46) corresponds system  dt  dτ = dx =  dτ dy = dτ

(48)

(49)

the following three dimensional autonomous 1 y f(t, x, y)

(50)

and the vector field

X=

∂ ∂ ∂ +y + f(t, x, y) . ∂t ∂x ∂y

(51)

The Module Structure of the Infinite-Dimensional Lie Algebra...

157

By corollary 6 we see that if the equation (46) admits the one-parameter Lie group G∗ generated by (47) in the traditional sense, then, in the sense of definition 3, the corresponding autonomous system (50) admits the Lie group G generated by V= ξ(t, x)

∂ ∂ ∂ + η(t, x) + η (1) (t, x, x) ˙ . ∂t ∂x ∂ x˙

(52)

Example 16. Consider the Burgers-KdV equation, ut + uux − γuxx + βuxxx = 0.

(53)

This equation is famous for that it has both dissipation term uxx and dispersion term uxxx and that it is not integrable in general case. Its travelling wave u(t, x) = u(x − ct) satisfies the following second order ordinary differential equation −cuθ + uuθ − γuθθ + βuθθθ = 0,

θ = x − ct,

where the constant c is the velocity of the travelling wave. Integrating the both sides of this equation, we obtain 1 (54) −cu + u2 − γuθ + βuθθ = k, 2 where k is an integral constant. The equation (54) can be re-written as the following second order autonomous system,  du  dθ = v (55)  dv 1 1 2 = (k + cu + γv − u ) = f(θ, u, v) dθ β 2 The qualitative research for this system shows that, if k>−

c2 , 2

then this system has two singular points p p A : (c − c2 + 2k, 0), B : (c + c2 + 2k, 0),

and there is a unique heteroclinic orbit connecting these two singular points. This heteroclinic orbit is just corresponding to the non-constant and bounded travelling wave solution of (53) (ref. [11, 12]). In a particular case, c2 + 2k = (

6γ 2 2 ) , 25β

(56)

an exact particular travelling wave solution has been found out by some authors (ref.[13, 14, 15, 16]) through different ways. Theoretically, [17, 18] proved that, if and only if the condition (56) is satisfied, then the autonomous system (55) has an algebraic curve solution, 15625β 3 [−(c − u)3 + 3v2 β] + 37500(c − u)vβ 3 γ ′ −3750(c − u)2 β 2 γ 2 + 9000vβ 2 γ 3 + 900(c − u)βγ 4 + 216γ 6 = 0

(57)

158

Guan Keying

and (55) is Liouville integrable. The equality (57) is in fact a first order ordinary differential equation, for v = du dθ . Solving this equation we obtain the bounded travelling wave solution of (53), γ(ct−x)+d 6γ 2 12γ 2 u(x, t) = c + − (1 + e 5β )−2 . (58) 25β 25β The related proof and calculation is somehow very complicated. We are now applying the theory given in section 3 to the equation (54) and (55) to obtain the same fact in a different way. According to the traditional theory (ref [7, 1]), we may find that1 , if and only if the condition (56) is satisfied, the second order ordinary differential equation (54) admits a two-parameter point transformation group, the corresponding two infinitesimal generators are ∂ ∂ V∗1 = ξ1 (θ, u) ∂θ + η1 (θ, u) ∂u , (59) ∂ ∂ V∗2 = ξ2 (θ, u) ∂θ + η2 (θ, u) ∂u , where ξ1 (θ, u) = 1,

η1 (θ, u) = 0,

γθ 2(25cβ − 25uβ + 6β 2 )γ − 5β e . 2 125β By corollary 6, we see that, corresponding to the second order ordinary differential equation (54), the three dimensional vector field γθ − 5β

ξ2 (θ, u) = e

X=

η2 (θ, u) = −

,

∂ 1 1 ∂ ∂ +u + (k + cu + γv − u2 ) ∂θ ∂u β 2 ∂v

(60)

admits the following two infinitesimal generator V1 = and

∂ , ∂τ

γθ γθ − 5β 2(25cβ−25uβ+6β 2 )γ − 5β ∂ ∂ e 2 ∂θ − ∂u + 125β γθ (375vβ 2 +50cβγ−50uβγ+12γ 3 )γ − 5β ∂ e ∂v . 625β 3

V2 = e

It is easy to see that (X, V1 ,V2 ) forms a basis of the space L(X), and that [V1 , V2 ] = −

γ V2 . 5β

So by theorem 15, through solving the system of equations   XΩ1 = 0 V1 Ω1 = 1  V2 Ω1 = 0

to obtain the three first order partial derivatives of Ω1 and calculating the path integral Z ∂Ω1 ∂Ω1 ∂Ω1 Ω1 (θ, u, v) = dθ + du + dv, ∂θ ∂u ∂v 1

Ref: Liu Minghui and Guan Keying, The Lie group method in the study of integrability of the travelling wave of Burgers-KdV equation, to appear.

The Module Structure of the Infinite-Dimensional Lie Algebra...

159

we may obtain a first integral of the autonomous system determined by X, Ω1 (θ, u, v) =

1 6γ (6θγ

− 5β{ıπ + ln[15625β 3 (−(c − u)3 + 3v2 β)+ 37500(c − u)vβ 3 γ − 3750(c − u)2 β 2 γ 2 + 9000vβ 2 γ 3 + 900(c − u)βγ 4 + 216γ 6 ]}).

By an exponential transformation, this first integral may be reduced to γθ

Φ1 (θ, u, v) = e 5β [15625β 3 (−(c − u)3 + 3v2 β)+ 37500(c − u)vβ 3 γ − 3750(c − u)2 β 2 γ 2 + 1 9000vβ 2 γ 3 + 900(c − u)βγ 4 + 216γ 6 ]− 6 . From this first integral, we see that the algebraic curve (57) is a solution of the autonomous system (55), and from (57) we obtain again the bounded non-trivial travelling wave solution (58) for any parameter β, γ and for any travelling velocity c. In order to obtain the unbounded exact travelling wave solutions in the Liouville integrable case of (56), we may seek another first integral, which is independent of Ω1 or Φ1 through the following schedule: (1) Let V3 = Φ1 V2 ∂ = q(u, v)( ∂θ − where

2γ(25(c−u)β+6γ 2 ) ∂ ∂u 125β 2

+

γ(375vβ 2 +50cβγ−50uβγ+12γ 3 ) ∂ ∂v ) 625β 3

q(u, v) = [15625β 3 (−(c − u)3 + 3v2 β)+ 37500(c − u)vβ 3 γ − 3750(c − u)2 β 2 γ 2 + 1 9000vβ 2 γ 3 + 900(c − u)βγ 4 + 216γ 6 ]− 6 .

From theorem 10 we see that V3 is also admitted by X. (2) Notice that [V1 , V3 ] = 0 and that (X, V1 , V3 ) is also a basis of L(X). Then by theorem 15, through solving the system of equations   XΩ2 = 0 V1 Ω2 = 0 ,  V3 Ω2 = 1

we may obtain the three partial derivatives of another first integral Ω2 of the autonomous system determined by X, ∂Ω2 ∂θ ∂Ω2 ∂u ∂Ω2 ∂v

= 0 2 2 2 2 4 = 125β (625(c−u) β2γ−1250vβ γ−36γ ) q(u, v)5 =

78125vβ 5 q(u, v)5 . γ

Then Ω2 may be obtained by the path integral Z ∂Ω2 ∂Ω2 du + dv. Ω2 (u, v) = ∂u ∂v This integral can be represented through some hypergeometric functions.

160

Guan Keying Therefore, the exact unbounded travelling wave solution can be given by  Φ1 (θ, u, v) = c1 , Ω2 (u, v) = c2

where c1 and c2 are arbitrary constants. Note: The exact unbounded travelling wave solution can also be represented by Weierstrass elliptic function. In fact we may use the following transformation:  u = µU + ν , T = φ(τ ) where µ = −e

2γ τ 5β

,

6γ 2 + c, ν= 25β

√ γ 5 3β 5β φ(τ ) = e τ, 6γ

then (54) may be reduced to the standard form d2 U = 6U2 + S(T), dT2 where S(T) = −

625β 2 24γ 4 T4



2k + c2 −

(61) 36γ 4 625β 2



.

(62)

It is obvious that (61) may be reduced to a Painlev´e equation d2 U = 6U2 , dT2

(63)

if and only if the condition (56) is satisfied. The general solution of (63) is, U = ℘(T + c1 , 0, c2 ), where c1 and c2 are constants, and ℘(T, g2 , g3 ) is the Weierstrass elliptic function, which gives the value of U for which T=

Z

U ∞

1

(4t3 − g2 t − g3 )− 2 dt..

Therefore, we obtain the exact travelling wave solution of (53), √ 2γθ γθ 5 3β 5β 6γ 2 5β − e ℘( e + d1 , 0, d2 ), u(x, t) = c + 25β 6γ

θ = x − ct,

(64)

for any given parameter γ, β and any travelling velocity c. When c2 = 0, it is just the bounded solution (57), and when c2 6= 0 the solution (64) is unbounded. The similar results have been obtained in [19, 20] and [21]. Example 17. Consider the Fisher-type wave equation ut = uxx + u(1 − u)(1 + γu),

(65)

The Module Structure of the Infinite-Dimensional Lie Algebra...

161

where γ is a real parameter. It is a reaction and diffusion equation with an unstable equilibrium state u = 0 and two stable equilibrium states u = − γ1 and u = 1. Its travelling wave solution u = u(x − ct) = u(θ) satisfies the following second order nonlinear ordinary differential equation uθθ + cuθ + u(1 − u)(1 + γu) = 0. (66) When c = 0, it is easy to integrate the equation by quadrature. So we shall not consider this particular case. Liu and Guan proved that2 the equation (65) admits a two-parameter point transformation group in the traditional sense if and only if one of the two conditions (c1 ) and (c2 ) is satisfied, where (c1 ) requires c2 =

9 2

and

γ = 1,

c2 =

25 6

and

γ = 0.

and (c2 ) requires

Under the condition (c1 ), the two infinitesimal generators of the two-dimensional Lie group admitted by the second order ordinary differential equation (66) are V1 = V2 = e

± √1 θ 2

∂ , ∂θ

∂ 1 ± √1 θ ∂ ∓√ e 2 u . ∂θ ∂u 2

Using the similar schedule in example 16, we obtain the first integrals of (66) √

Ω1 (θ, u, v) = e±2

2θ

√ (−u4 + u2 ± 2 2uv + 2v2 ),

(67)

the third infinitesimal generator admitted by (66) 1

V3 =

1

(Ω1 ) 4

V2

1 √ 1 (−u4 +u2 ±2 2uv+2v2) 4

=

∂ ( ∂θ ∓

√u ∂ 2 ∂u

−

√ u±2 2v ∂ 2 ∂v ).

(68)

and another first integral Ω2 (u, v) =

√ √ ± 2u3 ∓ 2u−3v √ 3 du∓ (−u4 +u2√±2 2uv+2x23 ) 4 2v √ 3 dv. (−u4 +u2 ±2 2uv+2x23 ) 4

R

(69)

From (67) we see that the equality √ −u4 + u2 ± 2 2uv + 2v2 = 0 2

(70)

ref. Liu Minghui and Guan Keying, The Lie Group and Integrability of the Fisher Type Travelling Wave Equation, to appear.

162

Guan Keying

gives an algebraic curve solution to the second order autonomous system  du dθ = v , dv dθ = −u(1 − u)(1 + γu) − cv

(71)

which is equivalent to (66). (70) is in fact a first order ordinary differential equation, for v = du dθ . Solving (70), we may obtain the following exact bounded travelling wave solutions: (a) for γ = 1, c = √32 , there are two bounded travelling wave solutions u(x, t) = −

1 1+e

and

x−ct+d √ 2

1

u(x, t) =

1+e

x−ct+d √ 2

,

(72)

,

(73)

where d is an arbitrary constant; (b) for γ = 1, c = − √32 , there are two bounded travelling wave solutions 1

u(x, t) =

√ − x−ct+d

1+e and

,

(74)

2

1

u(x, t) = −

. (75) 2 1+e Under the condition (c1 ), all of the other exact travelling wave solutions are unbounded, they can be represented by  Ω1 (θ, u, v) = d1 , Ω2 (u, v) = d2 √ − x−ct+d

where d1 and d2 are arbitrary constants. Under the condition (c2 ), the two infinitesimal generators of the two-dimensional Lie group admitted by the second order ordinary differential equation (44) are V1 = and

∂ , ∂θ

(76)

∂ 2u ∂ u 3v ∂ ∓√ ∓ (± + √ ) ]. ∂θ 3 6 ∂u 6 ∂v From the similar schedule, we may obtain the first integral √ √ 6 u3 u2 v2 ± 6θ (− + ± uv + ). Ω1 (θ, u, v) = e 3 3 3 2 V2 = e

± √θ

6

[

(77)

(78)

the third admitted infinitesimal generator V3 = =

V2 1

(Ω1 ) 6 3

2

1√

(− u3 + u3 ±

2 1 6 uv+ v2 ) 6 3

∂ ∓ [ ∂θ

2u ∂ √ 6 ∂u

∓ (± u3 +

3v ∂ √ ) ]. 6 ∂v

(79)

The Module Structure of the Infinite-Dimensional Lie Algebra...

163

and another first integral independent to Ω1 Ω2 =

Z

√1 u2 − √1 u ∓ 5 v 6 6 6 √ 2 2 5 3 (− u3 + u3 ± 36 uv + v2 ) 6

du ±

3

(− u3 +

√1 v 6 √ 6 u2 ± 3 3 uv

+

v2 56 2 )

dv.

(80)

Clearly, the equality √ −2u3 + 2u2 ± 2 6uv + 3v2 = 0

(81)

gives the algebraic curve solution of the autonomous system (71) under the condition (c2 ). From this solution, we may obtain the following exact bounded travelling wave solutions: (c) for γ = 0, c = √56 , there is a unique bounded non-trivial travelling wave solution 1

u(x, t) =

1+e

x−ct+d √ 6

!2

;

(82)

(d) for γ = 0, c = − √56 , the unique bounded non-trivial travelling wave solution is 1

u(x, t) =

√ − x−ct+d

1+e

6

!2

.

(83)

As in the case (c2 ), the unbounded non-trivial travelling wave solutions can be given by the two first integrals Ω1 (θ, u, v) and Ω2 (u, v). Example 18. Consider the Klein-Gordon equation n

∂2u X 2 ∂2u − ai 2 = βum , ∂t2 ∂xi

(84)

i=1

where a1 , a2 , ..., an and β are constants. Through the transformation 2

θ=t −

n X x2 i

i=1

a2i

,

(85)

(84) can be reduced to 4θuθθ + 2(n + 1)uθ = βum .

(86)

A non-zero solution u(θ) of (86) gives a centrally symmetric travelling wave solution of (84) n X x2i ). u(t, x1 , x2 , ..., xn ) = u(t2 − a2i i=1

The second order ordinary differential equation (86) is corresponding the three dimensional autonomous system  dθ  dτ = 1 du (87) = v  dτ 1 dv m − 2(n + 1)v) = (βu dτ 4τ

164

Guan Keying

Liu and Guan proved that3 the autonomous system (87) admits a one-parameter Lie group generated by ∂ ∂ ∂ V1 = (m − 1)θ −u − mv . (88) ∂θ ∂u ∂v It implies that the equation (86) is a quasi-homogeneous system, where the degrees of the variables θ and u are (m − 1) and −1 respectively. This fact suggests that (86) may have the following type of particular solution 1

u(θ) = Aθ− m−1 ,

(89)

when m 6= 1. And by a direct checking, we may get A={

1 4m 1 [ − 2(n + 1)]} m−1 . β(m − 1) m − 1

Besides, Liu and Guan proved further that, if and only if n = 15 and

m = 2,

(90)

then the autonomous system (87) admits a two-parameter Lie group generated by ∂ ∂ ∂ −u − 2v . ∂θ ∂u ∂v

(91)

48 ∂ ∂ ∂ − 3(θu + ) − (3u + 5θv) . ∂θ β ∂u ∂v

(92)

V1 = θ and V2 = θ 2 It is easy to see that

[V1 ,V2 ] = V2 . Therefore, we may get a first integral Ω1 =

1 2 6 {8 ln θ − ln(−576 − 24β + β )+ ln[−β 2 θu3 + 576v + 6β(−3u2 + 6θuv

+ θ2 v2 )]},

1 ∂Ω1 ∂Ω1 through solving its partial derivatives ( ∂Ω ∂θ , ∂u , ∂θ ) from   XΩ1 = 0 V1 Ω1 = 1  V2 Ω1 = 0

and calculating the path integral

Ω1 (θ, u, v) =

Z

∂Ω1 ∂Ω1 ∂Ω1 dθ + du + dv. ∂θ ∂u ∂v

Through an exponential transformation of Ω1 , we get the first integral 1

Ψ1 = {θ8 [−β 2 θu3 + 576v + 6β(−3u2 + 6θuv + θ2 v2 )]} 6 , 3

ref. Liu Minghui and Guan Keying, Some exact centrally symmetric travelling wave solutions of KleinGordon equation, to appear

The Module Structure of the Infinite-Dimensional Lie Algebra...

165

which is equivalent to Ω1 . We are now seeking another first integral Ω2 through a schedule different to the one in example 16 and 17. Let Φ1 = 0, Φ2 = Ψ1 . Since V1 Φ2 = Φ2 and V2 Φ2 = 0, we see that the condition (29) is satisfied for Φ1 and Φ2 . According theorem 15, from   XΩ2 = 0 V1 Ω2 = Φ1 ,  V2 Ω2 = Φ2

we may obtain the following partial derivatives ∂Ω2 ∂θ ∂Ω2 ∂u ∂Ω2 ∂v

= = =

βθ1/3 [−8v(4u+θv)+u3 β] 2[576+(24−β)β]1/6 [6θ2 v2 β+36v(16+θuβ)−u2 β(18+θuβ)]5/6 βθ4/3 (−24v+u2 β) 2[576+(24−β)β]1/6 [6θ2 v2 β+36v(16+θuβ)−u2 β(18+θuβ)]5/6 −2βθ4/3 (u+θv) . [576+(24−β)β]1/6 [6θ2 v2 β+36v(16+θuβ)−u2 β(18+θuβ)]5/6

and then get the first integral Ω2 through the path integral Z ∂Ω2 ∂Ω2 ∂Ω2 dθ + du + dv. Ω2 (θ, u, v) = ∂θ ∂u ∂v It turns out that Ω2 is not an elementary function. But from Ψ1 = 0, we may get a particular integral manifold of the autonomous system (87) −β 2 θu3 + 576v + 6β(−3u2 + 6θuv + θ2 v2 ) = 0, which is in fact is a first order ordinary differential equation since v = this equation is 24(θ + 2d) u(θ) = − , β(θ + d)2

(93) du dθ .

The solution of (94)

where d is an arbitrary constant. The corresponding centrally symmetric travelling wave solution of the Klein-Gordon equation (84) is

u(t, x1 , ..., x15 ) = −

5.

24[(t2 − β[(t2 −

P15

x2i i=1 a2 ) i

+ 2d]

x2i i=1 a2 ) i

+ d]2

P15

.

(95)

Conclusion

The definition 3 has generalized the sense of the admittance of a Lie group G by an autonomous system, so that the space L(X) attached to a given vector field X can be uncovered. The theorem 10 and corollary 11 show that L(X) has an interesting compound module structure L(X) = L<X> ⊕ L .

166

Guan Keying

Theorem 13 shows the significance of the structure coefficients. And theorem 15 gives a method for seeking new first integral through a basis of L(X) and through some known first integrals. The examples 14, 16, 17 and 18 show that the module structure of L(X) may provide us important means for seeking the first integrals of a given autonomous system. These facts show that the space L(X) attached to a given vector field X and its structure does exist naturally and intrinsically.

References [1] Peter J. Olver. Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin, 1986. [2] Larry C. Grove. Algebra, Academic Press, New York, 1983. [3] L. V. Ovsiannikov. Group Analysis of Differential Equations, Academic Press, New York, 1982. [4] Guan Ke-ying, Liu Sheng and Lei Jin-zhi. Lie algebra admitted by an ordinary differential equation system, Ann. of Diff. Eqs.,14(2)(1998) 131-142. [5] V. I. Arnold. Ordinary Differential Equations. Massachusetts: MIT Press, 1973. [6] J. T. Cushing. Applied Analytical Mathematics for Physical Scientists, John Wiley & Sons, Inc. New York, 1975. [7] George W. Bluman, Sukeyuki Kumei. Symmetries and Differential Equations, Springer-Verlag, New York, Inc., 1989. [8] Liu Sheng, Lei Jin-zhi and Guan Ke-ying, A new way on identifying symmetry group of differential Equations, Pure and Applied Mathematics, Vol.14, No.4 (1998) 01-06. [9] Liu Hongwei, Guan Keying, Searching for first integrals of 3-th order autonomous system using two one-paramter Lie Groups, Acta Mathematicae Applicatae Sinica, Vol.29, No.3 (2006) 567-573 (in Chinese). [10] Liu Sheng, Guan Keying, A method of constructing first integrals of second order non-autonomous systems, Acta Scientiarun Naturalium Universitatis Neimongol, Vol. 30, No.2 (1999) 135-139 (in Chinese). [11] Jeffrey A., and Kakutani T., Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries Equation, SIAM Review, Vol.14,No.4,1972, 582-643. [12] Guan Keying, Gao Ge, Qualitative research of traveling wave solution of mixed Burgers-KdV equation, Scientia Sinica, A, Vol.17 No.1, (1987) 64-73 (in Chinese). [13] Xiong S L, One class of analytic solution of Burgers-KdV equation, Chinese Sci. Bull., Vol.34, 1158-1162.

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[14] Ma Wenxiu, An Exact Solution to Two-Dimensional Korteweg-de Vries-Burgers Equation, J. Phys. A: Math. Gen. 26 (1993) 117-120 [15] Li Zhibin, Wang Mingliang, Exact solutions for two nonlinear equations, Advances in Mathematics, Vol.26, No.2. April 1997, 127-132. [16] Liu Shida and Liu Shishi, Solitory Wave and Turbulance, Shanghai Scientific and Technical Publisher, Shanghai 1994 (in Chinese). [17] Guan Keying, Lei Jingzhi, Integrability of second order autonomous system, Ann. of Diff. Eqs., 2002, 18: 117-135. [18] Gao Jixin, Lei Jinzhi and Guan Keying, Integrable condition on traveling wave solutions of Burgers-KdV equation, J. Northern Jiaotong Univ. Vol.27, No.3, (2003) 38-42 (in Chinese). [19] Feng Zhaosheng, On explicit exact solutions to the compound Burgers-Korteweg-de Vries equation, Phys. Lett. A 293(2002) 57-66 [20] Feng Zhaosheng, Exact solutions in terms of elliptic functions for the BurgersKorteweg-de Vries equation, Wave Motion 38 (2003) 109-15 [21] Feng Zhaosheng, On travelling wave solutions of the Burgers-Korteweg-de Vries equation, Nonlinearity 20 (2007) 343-356.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 169-186

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 6

L IE G ROUP M ETHODS FOR M ODULUS C ONSERVING D IFFERENTIAL E QUATIONS Jian-Qiang Suna , Hua Weib and Gui-Dong Daic Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China b Department of Science, Liaoning Technical University, Fuxin, 123000, China c Element Department,Beijing Institute of Clothing Technology, Beijing, 100029, China a

Abstract Lie group methods are new geometric numerical methods, which were proposed to solve the Lie group differential equations on manifolds. The famous Lie group methods are the RKMK method and the Magnus method. The Lie group methods can preserve the numerical solutions of the differential equations on the same manifolds. The preservation of the modulus square conserving property is very important for the modulus conserving differential equations, which has good stability. In the article, we applied the Lie group methods, such as the RKMK method and the Magnus method, to the modulus conserving differential equations, such as the ferromagnet equation, the Euler equation of the rigid body problem, the nonlinear Schrodinger equation and the vorticity equation. Numerical results showed that Lie group methods can preserve the modulus square conserving property of the modulus conserving differential equations and have the same accuracy as the classical explicit Runge-Kutta methods. Lie group methods are ideal methods for constructing the explicit square conserving schemes of the modulus conserving differential equations.

1.

Introduction

Geometric integration techniques have become increasingly popular in the modern approach to numerical analysis. In the broad sense, geometric integration refers to numerical solution techniques for differential equations that preserve inherent geometric structures. Geometric integrators include symplectic and multisymplectic integrators that preserve the Hamiltonian or Poisson structure, variational integrators that utilize the variational character of Lagrangian and canonical Hamiltonian system, conservative integrators that preserve

170

Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

first integral or conservational laws, and symmetric integrators that preserve symmetries of the system [1-5]. A geometric integrator will track solution over short time interval as well as a stand scheme of the same order e.g. a Runge-Kutta algorithm. In the case of matrix Lie group, the product is the usual matrix multiplication. Initial value problems for differential equations on a matrix Lie group G can be written in the form ′

Y = A(t, Y )Y,

t ≥ t0 ,

Y (t0 ) = Y0 ,

(1)

where A : R × G → g is a smooth function and Y0 ∈ G. It is well known that the solution of Eq.(1), subject to existence, stays in the Lie group G. Classical numerical methods such as multistep methods and explicit Runge-Kutta methods did not guarantee that the numerical solutions stay in the Lie group [9,14]. The reason for the failure of traditional methods is that matrix Lie groups are nonlinear manifolds, linear combinations of elements in a group need not stay in that group. Hence there is a need for new methods which ensure that the numerical solution stays on the correct manifolds, thereby retaining an important structure feature of the underlying differential system. Many recent authors such as Crouch, Grossman, Munthe-Kaas, Iserles, Marthinsen and Zanna have devoted to the Lie group methods, which can make the numerical solution on the same manifold [6-8, 10-12,15-19].The famous methods in the Lie group methods were the RKMK method and the Magnus method, which were proposed by Munthe-Kaas and Iserles. A major feature of all these methods is that the numerical solution is evaluated locally in the tangent space, which is the Lie algebra g, and the required mapping (for matrical Lie groups) is the matrical exponential. The main purpose of the paper is to solve the modulus conserving differential equations by the Lie group methods. The RKMK method and the Magnus method were introduced in section 2. In section 3, we solved the Schr¨ odinger equation and the vorticity equation by the RKMK method. The Ferromagnet equation and the Euler equation of the rigid body problem were solved by the Magnus methods in section 4. At last, we obtained some conclusions.

2. 2.1.

Lie Group Methods RKMK Method

The ordinary differential equation dY = A(t, Y )Y, dt

Y (0) = Y0 ,

Y ∈ Rn ,

(2)

where A : R+ × O(n) → so(n). Here O(n) is the set of n × n real orthogonal matrices and so(n) is the linear space of n × n real skew symmetric matrices. Our point of departure is the differential equations (2) evolving in the homogeneous space M. We denote the underlying group action by Λ. The function λ maps g × M into X (M ), the set of all vectors fields on M. Since vectors fields describe all possible values of derivatives of flows evolving on a manifolds, we deduce that a generic equation in the homogeneous space M can be always written in the form ′

Y = λ(B(t, Y ))(Y ),

t ≥ 0,

Y (0) = Y0 ∈ M,

(3)

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171

where B : R+ × M → g is sufficiently smooth. The solution of Eq.(3) can be written explicitly in terms of the group action Y (t) = Λ(Q(t), Y0 ),

′

where Q = B(t, Λ(Q, y0 ))Q,

t ≥ 0,

Q(0) = I.

(4)

The main idea is to reduce Eq.(3) to the Lie-group equation (4). instead of approximating Y (t), we seek an approximate group action Q(t) that carries Y0 to Y (t). Thus, provided we can discrete in a Lie group, we can respect the structure of any homogeneous space acted upon by the group in question. Traditionally, Eq.(2) can be solved by the classical Runge-Kutta methods. The set of all elements of gl(n; R) similar to a given matrix. Classical numerical methods such as explicit Runge-Kutta methods and multi-step methods can not preserve the numerical solution on the same manifolds [9]. Classical Runge-Kutta method to solve Eq.(2) is as following  P Vk = Yn + vl=1 ak,l Kl , k = 1, · · · , v. Kk = A(tn + ck h, Vk )Vk , v X bv Kl . (5) Yn+1 = Yn + h l=1

The coefficients ak,l vk,l=1 , bl vl=1 , ck vk=1 satisfy the following left Butcher table. The right Butcher table is the coefficients of the explicit four order Runge-Kutta method. c1 c2 c3 c4

a1,1 a2,1 .. .

a1,2 a2,2 .. .

··· ···

a1,v a2,v .. .

av,1 b1

av,2 b2

··· ···

av,v bv

0 →

1 2 1 2

1

1 2

1 2

0 0

0

1

1 6

2 3

2 3

1 6

In [14], Cooper proved that the coefficient of Runge-Kutta methods satisfy bi aij + bj aji = bi bj ,

for all

i, j = 1, · · · , v,

(6)

Runge-Kutta methods can preserve the numerical solution on the quadratic manifolds. However the Runge-Kutta shceme,which satisfies Eq.(6), is implicit. In general, it can not keep Y in G. However we can change the configuration space from G to g. An appealing feature of Lie algebra is that they are vector spaces. So the Runge-Kutta method still can be applied in approximately solving the transformed equation. Further the smoothness of the transformation map ensures the correct order of the numerical approximation on the homogeneous space. In theory, it can ensure Y in G [1]. The method was well known as the RKMK method. The idea of the RKMK methods was as follow. According to a classical result of Hausdorff [4,13], the solution of Eq.(2) is Y (t) = eσ(t) Y (0),

(7)

where σ : R → g is the solution of the initial value problem ′

σ(t) = dexp−1 σ(t) A(t, Y ),

(8)

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Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

P∞ Bj j ad C. j=0 j! A Taking j is the corresponding number and applying the Runge-Kutta method to Eq.(8), it can construct a kind of RKMK method. ′ Then the corresponding order-p RKMK algorithm for the Lie group equation Y = A(t, Y )Y is obtained as 1 where defining dexp−1 A (C) = C − 2 [A, C] +

Θ=

υ X

1 12 [A, [A, C]]

+ ··· =

ak,l Fl ,

l=1

Fk = dφ−1 (Θk , Ak , p), Ak = hA(tn + ck h, φ(Θk )Yn ), υ X bl Fl , Θ= l=1

Yn+1 = φ(Θ)Yn

for n ∈ N , and it is explicit provided that the underlying RK scheme is explicit. And that φ is a map from g to G, for instance φ = exp, the exponential map, or φ = cay, the Cayley mapping for quadratic Lie groups. The function dφ−1 (B, C, p) is a truncation of dφ−1 B (C) to order p − 1, which is usually sufficient for a method of order p, given that the error is subsumed in the O(hp+1 ) term [6].

2.2.

Magnus Method

The linear ordinary differential equations on matrix Lie group can be written as ′

t ≥ 0,

Y = A(t)Y,

Y (0) ∈ G,

(9)

where A : R → g and A(t) is skew symmetry matrix, A ∈ g and Y ∈ G. The solution of Eq.(9) can be written as Y (t) = exp(Θ(t))Y0 , t ≥ 0, where ′

Θ = dexp−1 Θ A=

∞ X Bk k=0

k!

adkΘ A,

t ≥ 0,

Θ(0) = O,

(10)

{Bk }k∈Z + is Bernoulli number [6]. By Picard iteration, it can get from Eq.(10), Θ0 (t) = O, Z Z t ∞ X Bk t k adΘ[m] (ξ) A(ξ)dξ, m = 0, 1, · · · dexp−1 A(ξ)dξ = Θ[m+1] (t) = Θ[m] (ξ) k! 0 0 k=0 (11) The above can be written as Z t [1] A(ξ1 )dξ1 , Θ (t) = 0 Z t Z Z 1 t ξ1 A(ξ1 )dξ1 − Θ[2] (t) = [ A(ξ2 )dξ2 , A(ξ1 )]dξ1 2 0 0 0 Z ξ1 Z t Z ξ1 1 A(ξ2 )dξ2 , A(ξ1 )]]dξ1 + · · · . A(ξ2 )dξ2 , [ [ + 12 0 0 0

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173

The Picard theorem implies that Θ(t) = limm→∞ Θ[m] (t) exists in a suitably neighborhood of the origin and the above first few iterations indicate that it can be expanded as a linear combination of terms that are composed from integrals and commutators acting recursively on the matrix A [6,10]. This is the M agnus expansion. Θ(t) =

∞ X

Hk (t),

(12)

k=0

where each Hk is a linear combination of terms that include exactly k + 1 integrals. thus Z

Z Z 1 t ξ1 A(ξ1 )dξ1 , H1 (t) = − [ A(ξ2 )dξ2 , A(ξ1 )]dξ1 , 2 0 0 0 Z t Z ξ1 Z ξ1 1 [ H2 (t) = A(ξ2 )dξ2 , [ A(ξ2 )dξ2 , A(ξ1 )]]dξ1 + 12 0 0 0 Z Z Z 1 t ξ1 ξ2 A(ξ3 )dξ3 , A(ξ2 )]dξ2 , A(ξ1 )]dξ1 . [ [ 4 0 0 0 H0 (t) =

t

In order to make the Magnus expansion numerical implementation, consider Z

t

A(ξ)dξ, Z t Z ξ1 [A(ξ2 ), A(ξ1 )]dξ, I2 (t) = 0 0 Z t Z ξ1 Z ξ1 [A(ξ2 , [A(ξ3 ), A(ξ1 )]]dξ, I3 (t) = 0 0 0 Z t Z ξ1 Z ξ2 I4 (t) = [[A(ξ3 ), A(ξ2 )], A(ξ1 )]dξ. I1 (t) =

0

0

0

0

Each Magnus expansion term is of the form Z L(A(ξ1 ), A(ξ2 ), · · · , A(ξs ))dξ, I(h) =

(13)

S

where L is multi-linear form , while S is a polytope of a special form, S = {ξ ∈ Rs : ξ1 ∈ [0, h], · · · , ξl ∈ [0, ξml ], l = 2, 3, · · · , s}, mL ∈ {1, 2, 3, · · · , l − 1}, l = 2, 3, · · · , s. The I(h) is discreted as follows, choose v distinct quadrature points, c1 , c2 , · · · , cv ∈ [0, 1], evaluate Ak = hA(ck h), k = 1, 2, · · · , v and form the quadrature K(h) =

X

k∈Csv

bk L(Ak1 , Ak2 , · · · , Aks ),

(14)

v where of length s from the set {1, 2, · · · , v}, bk = R Qs Cs is the set of all combinations ˜ = {ξ ∈ Rs : ξ1 ∈ [0, 1], ξl ∈ [0, ξml ], l = 2, 3, · · · , s}, l (ξ )dξ, S i k ˜ i=1 i S Q P x − cj ˜ lj (x) = vi=1,i6=j , j = 1, 2, · · · , v,. The A(t) = h−1 vk=1 lk ( ht )Ak in place ci − cj

174

Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

of A(t) in Eq.(9), and carry out the integration explicitly. As a example of this procedure, herewith fourth-order formulae for the first four integrals in the Magnus expansion: Z

Z

h

A(t1 )dt1 ≈

0 h

0

Z

1 h(A1 + A2 ), 2

t1

[A(t1 ), A(t2 )]dt2 dt1 ≈ −

0

√ 3 2 h [A1 , A2 ], 6

√ √ 3 3 3 3 [A(t1 ), [A(t2 ), A(t3 )]]dt3 dt2 dt1 ≈ h [( − )A1 − ( + )A2 , [A1 , A2 ]], 80 48 80 48 0 0 0 √ √ Z h Z t1 Z t1 3 3 3 3 [[A(t1 ), A(t2 )], A(t3 )]dt3 dt2 dt1 ≈ h3 [( + )A1 − ( − )A2 , [A1 , A2 ]]. 80 16 80 16 0 0 0 Z

h

Z

t1

Z

t1

3

where

√ √ 1 1 3 3 A1 = A(( − )h), A2 = A(( + )h). 2 6 2 6 Assembling the above quadrature results in a fourth-order method for the linear equation which respects arbitrary Lie-group structure, √ √ 1 1 3 3 A1 = A(tn + ( − )h), A2 = A(tn + ( + )h), 2 6 √ 2 6 X 1 3 2 1 = h[A1 + A2 ] − h [A1 , A2 ] + h3 [A1 − A2 , [A1 , A2 ]], 2P 12 80 Yn+1 = e Yn . If A(t, y) in place of A(t) in Eq.(9), we replace the function A(t, y) by its Lagrangian interpolating polynomial v X t ˜ y) = h−1 lk ( )Ak , A(t, (15) n k=1

where Ak = hA(ck h, Xk ), k = 1, 2, · · · , v. At each internal stage, we need to calculate quadratures of the form X al;j L(Aj1 , Aj2 , · · · , Ajs ), (16) Kl (h) = j∈Csv

where Csv is the set of all combinations of length s from the set {1, 2, 3, · · · , s}, Z Y s lji (ξi )dξ, ak;j = S˜k i=1

(17)

where S˜k = {ξ ∈ Rs : ξ1 ∈ [0, ck ], ξl ∈ [0, ξml ], l = 2, 3, · · · , s}. is the polytope S˜k scaled to the [0, ck ] cube instead of the unite cube. The weight bj are recovered by substituting ck = 1. So the integral of the Magnus expansion is discreted, we can get the explicit three order Magnus formula of the equations dY = A(t, Y )Y, dt

Y (0) = Y0 ,

A(t, y) = −AT (t, y),

(18)

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175

h h Y˜2 = exp( A1 )Yn , A2 = A(tn + , Y˜2 ), 2 2 ˜ ˜ Y3 = exp(h(−A1 + 2A2 ))Yn , A3 = A(tn + h, Y3 ),

Y˜1 = Yn , A1 = A(tn , Y˜1 ),

2 1 h2 2 1 1 Θ = h( A1 + A2 + A3 ) − ( [A1 − A3 , A2 + A3 ]), 6 3 6 2 15 30 Yn+1 = exp(Θ)Yn .

(19)

Lie group methods are numerical methods for equations evolving on manifolds. The numerical solutions obtained by Lie-group methods evolves on the same manifolds as the analytical solution. The Lie group methods can result in explicit square-conservation schemes for the modulus conserving differential equations. i.e., schemes satisfying kY n k = kY 0 k. k · k denotes the vector norm.We will construct the explicit square conserving schemes to solve the modulus conserving differential equations by the RKMK method and the Magnus method in the Lie group methods.

3. 3.1.

Simulations of Differential Equations by RKMK Method Simulations of the Nonlinear Schr¨ odinger Equation

To the nonlinear Schr¨ odinger equation iψt + ψxx + a|ψ|2 ψ = 0,

(20)

with initial condition ψ(x, 0) = ψ0 (x), x ∈ R, a > 0 is a constant parameter. Eq.(20) is one of the most important integrable models in the theory of solitons. Its application can be found in many areas of physics, including nonlinear optics and plasma physics. Eq.(20) possesses the following electricity and energy conservation property Z ∞ Q= |ψ(x, t)2 |dx = Q0 , (21) −∞ Z ∞ ∂ψ q [| |2 + |ψ|4 ]dx = E0 , E= (22) 2 −∞ ∂x where Q0 and E0 are constant. Sanz-Serna J.M proposed the leap frog scheme and the improved Crank-Nilcolson scheme, using the semi-discrete methods to discuss the conservation and no-conservation scheme. In [26,27], Zhanglu ming and Zhangfei etc proposed new implicit conservation schemes of the nonlinear Schr¨ odinger equation and proved the stability and convergence of the schemes. However all the proposed conservation schemes were implicit and low order. We proposed a high order explicit modulus conserving scheme for the nonlinear Schr¨ odinger equation by the Lie group method. Using ψ = p + iq, Eq.(20) can be written as a pair of real-valued equation pt + qxx + a(p2 + q 2 )q = 0, 2

(23)

2

qt − pxx − a(p + q )p = 0.

(24)

Eq.(20) can be rewritten as dz = JAz, dt

J=



0 1 −1 0



,

A=



D 0 0 D



,

(25)

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Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

where z = (p, q)T , D = −cI − ∂x∂ 2 is a differential operator, c = a(p2 + q 2 ), and I is the identity operator. Eq.(20) possesses the electricity and energy conservation property. It is natural to require a discretization or a semi-discretization to reflect the electricity and energy conservation property of the nonlinear Schr¨ odinger equation. Using second order fi+1 −2fi +fi−1 (x,t) space central difference ∆fi = to discrete ∂f∂x 2 , f = p, q, at xi , We get the (∆x)2 following semi-discrete ordinary differential equation of the nonlinear Schr¨ odinger equation.               

dp1 dt dp2 dt

.. .

dpN dt dq1 dt dq2 dt

.. .

dqN dt





0 0

        0    1    0 =  (∆x)2   −τ1   1     .   ..  0

0 0 0 0 1 −τ2 .. . 0

0 0

0 0

0 0

0 0 0 0 0 ··· 1 ··· .. . . . . 0 ···

0 0 0 0 .. . −τN

τ1 −1 .. . 0 0 0 0 0

−1 0 τ2 −1 .. .. . . 0 0 0 0 0 0 0 0

0 0

··· ··· .. . ··· 0 0 0 0

0 0 .. .



p1 p2 .. .

      τN    pN  0   q1  0    q2  . 0   .. 0 qN



      ,      

(26)

τk = 2 − a(p2k + qk2 )(∆x)2 , k = 1, 2, · · · , N. Y = (p1 , p2 , · · · , pN , q1 , q2 , · · · , qN )T . Eq.(26) is equivalent to dY dt = A(Y )Y , A(Y ) is skew symmetry matrix. It is easy to prove that at any time kY (t)k = constant, k · k denotes the vector norm. Eq.(26) possesses the square conservation property, which reflects the energy conservation property of Eq.(21). Classical explicit numerical methods can not preserve the square conservation property of Eq.(26). Lie group methods are numerical methods for equations evolving on manifolds. The numerical solution obtained by Lie-group methods evolves on the same manifolds as the analytical solution [24]. Applying the RKMK method to Eq.(26) results in arbitrary order square-conservation schemes for the nonlinear Schr¨ odinger equation. Schemes satisfy 0 kY (t)k = kY k for any t = nh, h is time step length. The ordinary equations (26) was solved by the fourth order explicit RKMK methods and the corresponding fourth order Runge-Kutta method. To test whether the RKMK method can preserve the square conservation property of the discrete nonlinear schr¨ odinger equation. The modulus square conservation errors are defined as 2 2 Err(t) = (Y12 (0)+Y22 (0)+· · ·+Y2N (0))−(Y12 (t)+Y22 (t)+· · ·+Y2N (t)),

t = kh. (27)

The following initial conditions are used. When a = 2, the initial condition is ψ(x, 0) = sech(x + 10) exp[2i(x + 10)],

(28)

x ∈ [−15, 15]. When a = 18, the condition is ψ(x, 0) = sech(x).

(29)

x ∈ [−20, 20]. In Fig.1 and Fig.2, the left of Fig.1 and Fig.2 showed the modulus square errors by the fourth order explicit RKMK method. The right of Fig.1 and Fig.2 showed the modulus

Numerical Simulations of the Nonlinear Solitary Waves

177

−12

10

x 10

3

8

2

4

0 1 −4

−8 0

500

100

0 150 0

t

10 t

20

Figure 1. The modulus square errors of the nonlinear Schr¨ odinger equation by the RKMK method and the RK method with a = 2. −12

1

x 10

4.5

0

4

−2

3 −4

2

−5

1

−7

−9 0

60 t

120

0 0

6 t

12

Figure 2. The modulus square errors of the nonlinear Schr¨ odinger equation by the RKMK method and the RK method with a = 18. square errors by the fourth explicit Runge-Kutta method. From Fig.1 and Fig.2, we can see that the explicit RKMK method can preserve the modulus conserving of the nonlinear Schr¨ odinger equation, while the explicit Runge-Kutta method can not preserve the modulus conserving of the nonlinear Schr¨ odinger equation.

3.2.

Simulations of the 2D Vorticity Equation

To the vorticity equation for two-dimension, invicid, incompressible flow ∂ξ + v · ∇ξ = 0, ∂t

(30)

where v = k × ∇ϕ,

ξ = k · ∇ × v = ∇2 ϕ,

(31) (32)

178

Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

ξ is the vorticity, v is the velocity, ϕ is the stream function, ∇ is the two-dimensional operator, and k is a unit vector normal to the plane of motion. Eq.(30) can be rewritten as ∂ξ = J(ξ, ϕ), (33) ∂t where J is the Jacobian operator with respect to the Cartesian coordinates x and y in the plane J(ξ, ϕ) = (∂ξ/∂x)(∂ϕ/∂y) − (∂ξ/∂y)(∂ϕ/∂x), (34) and ξ = ∇2 ϕ.

(35)

There are the following integral constraints, among others, on the Jacobian J(ξ, ϕ) = 0,

(36)

ξJ(ξ, ϕ) = 0,

(37)

where the bar denotes the average over the domain, along the boundary of which ϕ is ¯ the mean constant. From these integral constraints, we can see that the mean vorticity ξ, ¯ 2 square vorticity ξ , are conserved with time. Conservation of these quantities, during the advection process, possesses the important constraints on the statistical properties of the vorticity equations. If we wish to simulate the statistical properties, a finite difference schemes should reflect these quadratic invariant quantities of the vorticity equations [28,29]. So we must use one of such finite difference schemes that approximate conserved these quadratic quantities [22]. Eq.(33) can be discreted into ∂ξ = J(ξ, ϕ), ∂t J = αJ1 + βJ2 + γJ3 , J1 = ∆x ξ∆y ϕ − ∆y ξ∆x ϕ,

J2 = ∆y (ϕ∆x ξ) − ∆x (ϕ∆y ξ),

J3 = ∆x (ξ∆y ϕ) − ∆y (ξ∆x ϕ).

(38) (39) (40) (41) (42)

where ∆x f (x) denotes [f (x + d) − f (x − d)]/2d. ∆y is defined similarly with respect to y, α + β + γ = 1. Eq.(38) is equivalent to dY = A(ϕ)Y. dt where A(ϕ) =

αB(ϕ) + βC(ϕ) + γD(ϕ) , 4∆x∆y

(43)

(44)

When α = β, we can get αB(ϕ) + βC(ϕ) is skew symmetry matrix. We know D(ϕ) is skew symmetry matrix. When we take α = β = 12 , γ = 0 or α = β = γ = 31 or α = β = 0, γ = 1. We get the spacial discreted square-conservation type schemes of the vorticity equation similar to Eq.(2).

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179

−10

8

x 10

3.5 3

4

2

0

1

−4 0

30 t

60

0 0

6 t

12

Figure 3. The modulus square errors of the 2D vorticity equation by the RKMK method and the RK method with α = β = 21 , γ = 0. −10

4

x 10

350 300

0

−4

200

−8 100

−12

−16 0

40

t

80

120

0 0

20 t

40

Figure 4. The modulus square errors of the 2D vorticity equation by the RKMK method and the RK method with α = β = γ = 31 . To the equation ξ = ∇2 ϕ = ϕxx + ϕyy ,

(45)

Using mid difference scheme to Eq.(45), we can get the discreted equations ξi,j = ∆i,j (ϕ)

(46) 2

2

= (ϕi+1,j − 2ϕi,j + ϕi−1,j )/(∆x) + (ϕi,j+1 − 2ϕi,j + ϕi,j−1 )/(∆y) . The discreted equations (46) can be written as the following form Y = EX,

(47) ′

where the vector Y = (ξ11 , ξ21 , · · · , ξm1 , · · · , ξ1n , ξ2n , · · · , ξmn ) , the vector X = ′ (ϕ11 , ϕ21 , · · · , ϕm1 , · · · , ϕ1n , ϕ2n , · · · , ϕmn ) , E is the matrix. The 2D vorticity equation was solved by the fourth order explicit RKMK methods and the corresponding fourth

180

Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

order explicit Runge-Kutta method. The modulus square conserving error was defined as Eq.(27). The global errors at time t = k∆t were defined as error1 = max|Y (k∆t) − Y (t)|,

t = k∆t,

(48)

The global errors at different times of the two methods are listed in the Table.1. The algorithm of the 2D vorticity equation to compute ξ k , ϕk to ξ k+1 , ϕk+1 . The initial value ξ 0 , ϕ0 , t = 0, were given for k = 0, 1, · · · n Step 1: Apply the RKMK method to compute Eq.(43), ξ k+1 was obtained. Step 2 : Compute X k+1 = E −1 Y k+1 , ϕk+1 was obtained. So ξ k+1 , ϕk+1 were obtained. end The experimental function was given

ϕ(x, y, t) =

−gcy Dg + cos(µ(x − at) + ry), 10 10

where µ, r, g, c, a, D are the constants and a = solution of Eq.(30) was given as

ξ=−

gc 10 .

(49)

According to ξ = ϕxx + ϕyy , the exact

Dg 2 (µ + r2 ) cos(µ(x − at) + ry). 10

(50)

In table 1, the numerical solutions of the 2D vorticity equation at different times by the RKMK method and the Runge-Kutta method were given. Numerical results showed that the RKMK method and the Runge-Kutta methods have the same accuracy.

Table 1. Accuracy comparison of the RKMK method and the Runge-Kutta method. t=0.5 t=2 t=6

RKMK method 2.43457 × 10−4 1.147326 × 10−3 1.953256 × 10−3

Runge-Kutta mehod 2.43466 × 10−4 1.147378 × 10−3 1.953278 × 10−3

Error 10−8 10−8 10−8

In Fig.3 and Fig.4, the left of Fig.3 and Fig.4 showed the modulus square errors by the fourth order explicit RKMK method. The right of Fig.3 and Fig.4 showed the modulus square errors by the fourth explicit Runge-Kutta method. From Fig.3 and Fig.4, we can see that the explicit RKMK method can preserve the modulus conserving of the 2D vorticity equation. The explicit Runge-Kutta method can not preserve the modulus conserving of the 2D vorticity equation. It is obvious that the RKMK method has good stability.

Numerical Simulations of the Nonlinear Solitary Waves

4. 4.1.

181

Simulations of Modulus Conserving Differential Equations by Magnus Method Simulations of the Ferromagnet Equation

In 1935, Landau-Lifshitz proposed the Ferro magnetic chain coupled equations Zt = λ1 Z × (∆Z + H) − λ2 Z × (Z × (∆Z + H)), ∂E ▽ ×H = + σE, ∂t ∂Z ∂H −ρ , ▽ ×E = − ∂t ∂t ▽ ·H + β ▽ ·Z = 0, ▽ ·E = 0.

(51) (52) (53) (54) (55)

where λ1 , λ2 , σ, β, ρ are constants, λ2 ≥ 0, σ ≥ 0. Vector function Z(x, t) = (u(x, t), v(x, t), w(x, t))T is intensity of magnetization, H(x, t) = (H1 (x, t), H2 (x, t), H3 (x, t))T is magnetic field , E(x, t) = (E1 (x, t), E2 (x, t), E3 (x, t))T is electric field , H ρ = ∆Z + H valid magnetic ∂u ∂v ∂v T field, ∆Z = (uxx , vxx , wxx )T , ▽ · Z = ( ∂x , ∂x , ∂x ) , × is vector multiplication in R3 , f = f (x, t), f = u, v, w. If H = 0, E = 0, we can get the following Landau-Lifshitz equation with Gilbert component. Zt = λ1 Z × ∆Z − λ2 Z × (Z × ∆Z),

Z(x + L, t) = Z(x, t),

(56)

Z(x, 0) = Z0 (x),

where λ2 ≥ 0 is Gilbert damped component. In 1993, Guo Bo-ling studied the solution of Eq.(56) systematically and found the closed relation of the Riemann map and the solution of Eq.(56). If λ2 = 0, Eq.(56) has the exact solution Z(x, t) = a cos(α) + {b cos(kx − ωt) + c sin(kx − ωt)} sin(α),

(57)

where ω = λ1 k 2 sin(α), (a, b, c) ∈ R3 is right hand unit orthogonal vector. α, k is arbitrary real number. The solution of Eq.(56) is called spin wave[30,31]. Discretizing Eq.(56), we can get ∂t Zj = λ1 Zj × ∆h Zj − λ2 Zj × (Zj × ∆h Zj ) Zj (x + L, t) = Zj (x, t),

where ∆h Zj =

Zj+1 − 2Zj + Zj−1 , (∆h)2

(58)

Zj (x, 0) = Z0 (xj ),

j = 1, 2, · · · , n, and Zj = (uj , vj , wj )T . Taking

Y = (u1 , u2 , · · · , uN , v1 , v2 , · · · , vN , w1 , w2 , · · · , wN )T . The discretizing equation (58) can be written as dY = λ1 A(Y )Y − λ2 B(Y )Y = C(Y )Y, dt

(59)

182

Jian-Qiang Sun, Hua Wei and Gui-Dong Dai −12

10

−7

x 10

x 10

9 8

6

6

4

2 2

−2 0

12 t

24

0 0

1.2

2.4

3.6

t

Figure 5. The modulus square errors of the Ferromagnet equation by the Magnus method and the RK method with λ1 = 2. −12

12

118

x 10

1

10

x 10

0

6 −2

2

−4

−2

−6 0

40 t

80

−6 0

1.25 t

2.5

Figure 6. The modulus square errors of the Ferromagnet equation by the Magnus method and the RK method with λ1 = 5. 

0 1  −W A(Y ) = (∆x)2 V

W 0 −U

 −V U , 0



0 1  ˜ B(Y ) = − W (∆x)2 ˜ V

˜ W 0 ˜ −U

 −V˜ ˜ , U 0

˜ , V˜ ,W ˜ are matrices[20-23, 25]. It is obvious that dY = C(Y )Y is where U , V ,W ,U dt the ordinary equations of n = 3N, where C(Y ) is skew symmetry matrix. The ordinary equations (59) was solved by the third order explicit Magnus method and the corresponding third order Runge-Kutta method. In Fig.5 and Fig.6, the left of Fig.5 and Fig.6 showed the modulus square errors by the third order explicit Magnus method. The right of Fig.5 and Fig.6 showed the modulus square errors by the third explicit Runge-Kutta method. From Fig.5 and Fig.6, we can see that the explicit Magnus method can preserve the modulus conserving of the Ferromagnet equation. The explicit Runge-Kutta method can not preserve the modulus conserving of the Ferromagnet equation. It blew up at small time step. It is obvious that the Magnus method has good stability.

Numerical Simulations of the Nonlinear Solitary Waves −15

6

183

−9

x 10

9

x 10

8 4 6

2 4

0

2

−2 0

0 120 0

60 t

12 t

24

Figure 7. The modulus square errors of the Euler equation by the Magnus method and the RK method with h = 0.001. −15

−7

x 10

2.5

x 10

2

2 1

0

1 −1

−2 0

40 t

80

0 0

25 t

50

Figure 8. The modulus square errors of the Euler equation by the Magnus method and the RK method with h = 0.0025.

4.2.

Simulations of the Euler Equation of a Rigid Body Problem

The Euler equation of a rigid body problem is ′

y = y × M y,

t ≥ 0,

y(0) = y0 ,

(60)

where y ∈ R3 (we assume that ky0 k = 1), the symbol × denotes the classical vector product on R3 and M = diag(m1 , m2 , m3 ) is a diagonal matrix [32]. This system has the Hamiltonian function H(y) = 12 (m1 y22 + m2 y22 + m3 y32 ) and obey kyk2 = 1. It can be represented by means of Lie-group action of SO(3) on R3 by representing the solution y(t) as Q(t)yn , n = 0, 1, 2, · · · , with Q ∈ SO(3), t ∈ [tn , tn+1 ]. Hence, in each interval [tn , tn+1 ], we can solve the differential equation ′

Q (t) = A(y(t))Q(t), t ≥ tn , Q(tn ) = I,

(61)

184

Jian-Qiang Sun, Hua Wei and Gui-Dong Dai

where



 0 −m3 y3 m2 y2 0 −m1 y1  A(y(t)) = −  m3 y3 −m2 y2 m1 y1 0 √ √ √ 2 √ ), m = −( In this numerical experiment,m1 = −1, m2 = −( 2 − 0.51 2− 3 1.51

√ √ 2 ), 1.51

and the initial value y(0) = [0, √12 , √12 ]T . We remark that such action automatically obeys the homogeneous space condition kyk2 = 1, wherever a Lie-group method is applied to Eq.(61). The ordinary equations (60) was solved by the third order explicit Magnus methods and the corresponding third order Runge-Kutta method. The modulus square conserving errors were defined as Eq.(27). In Fig.7 and Fig.8, the left of Fig.7 and Fig.8 showed the modulus square errors by the third order explicit Magnus method. The right of Fig.7 and Fig.8 showed the modulus square errors by the third explicit RK method. From Fig.7 and Fig.8, we can see that the explicit Magnus method can preserve the modulus conserving of the Euler equation. The explicit Runge-Kutta method can not preserve the modulus conserving of the Euler equation. It also proved that the Magnus method can preserve the modulus conserving property of the differential equations.

5.

Conclusion

In the article, the explicit square conserving schemes of the modulus conserving differential equations were proposed by the Lie group methods. The modulus conserving differential equations, such as the nonlinear Schr¨ odinger equation, the ferromagnet equation, the 2D vorticity equation, the Euler equation of the rigid body problem, were solved by the RKMK method and the Magnus method in the Lie group methods. Numercal results showed that the Lie group methods can preserve the modulus conserving property of the modulus conserving differential equations and have the same accuracy as the classical explicit RungeKutta methods. The Lie group methods have also better stability than the classical explicit Runge-Kutta methods. We can concluded that the Lie group methods are ideal methods for the modulus conserving differential equations.

Acknowledgements This work was supported by the National Natural Science Foundation of China (No.10401033).

References [1] P.J.Olver, Applications of Lie group to differential equations, Springer, New York, (1986). [2] Jerrold E.Marsden, Tudor S.Ratiu, Introduction to mechanics and symmetry, SpringerVerlag, New York, (1994).

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[3] Ernst Hairer, Christian Lubich, Gerhard Wanner, Geometric Numerical Integration, Springer-Verlag, Berlin Heidelberg, (2002). [4] K.Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J.Comp.Math, 4, 279-289, (1986). [5] T.J.Bridge, Multi-symplectic structures and wave propagation, Math Proc Cam Phil Soc, 121, 147-190,(1997). [6] A. Iserles, Hans Munthe-Kaas, Lie Group Methods, Acta Numerical, 215-365,(2000). [7] Hans Munthe-Kaas,Runge-Kutta methods on Lie groups, BIT, 38, 92-111, (1998). [8] Hans Munthe-Kaas, High Order Runge-Kutta Methods on Manifolds, Appl. Numer. Math. 29, 115-127,(1999). [9] E.Hairer, S.P.Norsett and G.Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. Second Revised Edition, Springer, Berlin, (1993). [10] A.Iserles, solving linear ordinary differential equations by exponentials of iterated commutators, Numer.Math., 45 ,183-199,(1984). [11] A.Iserles, S.P.Nφrsett and A.F.Rasmussen, Time-symmetry and high order Magnus methods, Technical Report 1998/NA06, DAMTP, University of Cambridge,(1998). [12] A.Iserles, A.Marthinsen and S.P.Nφrsett, On the implementation of the method of Magnus series for linear differential equations, BIT 39, 281-304,(1999). [13] F.Hausdorf, Die symbolische Exponentialformel in der Gruppentheorie, Berichte der S¨ achsischen Akademie der Wissenschaften (Math.Phys.Klasse) 58, 19-48, (1906). [14] G.Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA, J.Numer.Anal, 7, 1-13, (1987). [15] E.Celledoni and A.Iserles, Approximating the exponential from a Lie algebra to a Lie group, Technical Report 1998/NA3, DAMTP, University of Cambridge. [16] G.H.Golub and C.F.Van Loan, Matrix Computation, 3rd edn, Johns Hopkins University Press, Baltimore,(1996). [17] A.Zanna and H. Munthe-Kaas (1997), Iterated commutators, Lie’s reduction method and ordinary differential equations on matrix Lie groups, Foundation of Computational Mathematics, (F.Cucker and M.Shub.eds), springer, 434-441., (1997). [18] M. Calvo, A.Iserles and A.Zanna, Runge-Kutta methods for orthogonal and isospectral flows, Applied Numeric. Math, 22, 153-163,(1996). [19] P.E.Crouch and R.Grossman, Numerical Integration of Ordinary Differential Equations on Manifolds, J.Nonlinear Sci., 3, 1-33,(1993)

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[20] Jian-Qiang Sun, Meng-Zhao Qin, Zhong-Qi Ma, Magnus methods of solving nondamping Landau-Lifshitz equation, Chinese Journal. Numerical Mathematical and Application, 26, 1, 57-64, (2004). [21] Jian-Qiang Sun, Zhong-Qi Ma, Meng-Zhao Qin, RKMK method of solving nondamping LL equations and ferromagnet chain equations, Applied Mathematics and Computation, 157, 407-424, (2004). [22] Jian-Qiang Sun, Meng-Zhao Qin, Zhong-Qi Ma, Lie group method of the vorticity equations for two dimensional incompressible flow, Computational Fluid Dynamics Journal, 12(4),70, 569-579, (2004). [23] Sun Jian-Qiang, Ma Zhong-Qi, Qin Meng-Zhao, Explicit square conserving schemes of Landau-Lifshitz equation with Gilbert component, Applied Mathematics and Mechanics, 26, 1, 73-78, (2005). [24] Jian-Qiang Sun, Zhong-Qi Ma, Wei Hua, Meng-Zhao Qin, New conservation schemes for the nonlinear schr¨ odinger equation, Applied Mathematics and Computation, 177, 446-451, (2006). [25] Jian-Qiang Sun, Xiao-Yan Gu, Zhong Qi Ma, Meng-Zhao Qin, RK-Cayley Fehlberg method on homogeneous manifolds, Communciation in nonlinear science and numerical simulation, 12, 966-975, (2007). [26] Zhang Fei, Victor M. P´ erez-Garcia and Luis V´ azquez, Numerical simulation of nonlinear Schr¨ odinger systems: A new conservative scheme, Appl. Math. Comput, 71, 2-3,165-177, (1995). [27] Zhang Luming, Chang QianShun, A conservation numerical scheme for nonlinear schr¨ odinger equation, Chinese Journal of Computational Physics, 16(16), 661-668, (1999). [28] Arakawa.,Numerical Simulation of Large-Scale Atmospheric Motions, SIAM-AMS proceedings, 11, 24-40, (1970). [29] Arakawa., Computational Design for Long-Term Numerical Integration of the Equation of Fluid Motion: Two-Dimensional Incompressible of Flow. Part I, University of California, Los Angles, Joural of Computational Physics, 1,119-143, (1966). [30] J.A.G.Robert and C.J.Thompson, Dynamical of the classical Heisenberg Spin Chain, Technique Report 11-1987, Mathematics Department, University of Melbourne, Pakville, Victoria 3052, Australia,(1987). [31] Jason Frank, Huang Weizhang, Benediet Leimkuler, eometric integration for classical spin system, GComput Phys, 133(1), 160-172, (1997). [32] McLachlan R.I, Explicit Lie-Poisson integration and the Euler equations, Phys Rev Lett, 71, 3043-3046, (1993).

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 187-198

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 7

S INGULARITIES AND S TABILITY OF A W ORK F UNCTION Jean Lerbet∗ IBISC, FRE CNRS 3190 Universit´e d’Evry Val d’Essone 40 Rue Pelvoux CE 1455 Courcouronnes 91020 Evry Cedex FRANCE

Abstract This work is the beginning of a systematical analysis of singularities and stability conditions of a product of exponential mappings. More precisely, let f be defined as f : θ = (θ1 , . . . , θn ) 7→ f (θ) = exp(θ1 X1 ) . . . exp(θn Xn ) defined from the nspace S n = K1 × . . . × Kn of parameters to a n-dimensional Lie group G where (X1 , . . . , Xn ) is a basis of the Lie algebra G of G and Kk = S 1 or Ik is the 1torus or a compact interval of IR (according to the nature of the corresponding joint in applications). We are looking for conditions (about (X1 , . . . , Xn )) for which f is a stable mapping according to the theory of singularities. This means that the orbit of f under the action of diffeomorphisms in the source and in the target is an open set in the set of differential mappings from (S 1 )n to G. First, we prove that the set Σ1 (f ) of singularities is a (n-1)-dimensional submanifold of (S)n . Secondly, we analyse the conditions so that f is a submersion with fold. Using the fact that f is inf-stable if and only if g = f|Σ1 (f ) is an immersion with normal crossings, we analyse this property and we highlight some consequences. Applications to robotics are suggested.

Key words : tranversality, singularities, stability, robotics, Lie groups, submersion with fold. AMS mathematics subject classifications : 70 - 53 Differential geometry; 70B15 Mechanisms and linkages. ∗

E-mail address: [email protected]

188

Jean Lerbet

Introduction In previous papers ([4],[5] for examples), the regularity of products of exponential mappings of a Lie group has already been analysed. But this treatment aims at the resolution of the equation exp(θ1 X1 ) . . . exp(θn Xn ) = id in the group and for the (local) description of the set of roots in IRn . The results are directly applicable to closed mechanisms and this is why the associated mapping was called a closure function. In this paper we study the singularities of the mentioned exponential mapping when the set of variables has the same dimension as the group. The applications concern open kinematic chains as used in most robots. Indeed, the geometric model of a robot consists in the relations between the articular coordinates and the position and orientation of the end effector. These relations are exactly the elements of closure function that shall be called here the work function. For these applications, we may suppose that the variables θ1 , . . . , θn are in S 1 or in a compact interval of IR because they represent the articular coordinates between two consecutive bodies. Otherwise, the hypothesis of compactness fails and a part of the following study too. The analysis of singularities is carried out in two steps: 1. the first is to describe the set of ’first’ singularities (the 1-class Σ1 (f )): structure of submanifold and tangent bundle 2. the second is to specify the nature of these first singularities: Σ1 (f ) as a set of fold points The last envisaged question shall be : is such a mapping stable? Indeed, stable mappings possess some remarkable properties which ensure mechanical properties of the associated kinematic chain. For example, the play in the joints or unavoidable imprecisions in the realisation of the chain do not generate any problems for describing the topology of the workspace of the robot if the associated mapping is stable because the topology itself remains stable. Finally, two examples of kinematic chains are proposed for illustration and suggest how to continue in this direction.

1.

Some Preliminaries

We shall recall now some standard results about Lie groups, differential geometry and concepts of the theory of singularities which shall be used in the follow up of the study.

1.1.

Lie Groups

We call by G a Lie group of dimension n and by G its Lie algebra. X1 , . . . , Xn shall be a basis of G. In the practice, G is the Lie group D of euclidean displacements or one of its Lie subgroups with n ≤ 6. Moreover, each Xk shall be the infinitesimal generator of a rotation through an axis in the affine space E and G is isomorphic to the well-known space of screws of kinematics. For a in G, we note La : G → G and Ra : G → G the mappings defined by La (g) = ag and Ra (g) = ga.

Singularities and Stability of a Work Function

189

exp : G → G is the classical exponential mapping of G, Ad : G → L(G) its adjoint mapping and [., .] the Lie braket. Definition Let be X1 , . . . , Xn a family of vectors of G. The work function associated to X1 , . . . , Xn noted f(X1 ,...,Xn ) is the mapping f = f(X1 ,...,Xn ) : θ = (θ1 , . . . , θn ) 7→ f (θ) = exp(θ1 X1 ) . . . exp(θn Xn ). It is well known that if (X1 , . . . , Xn ) is a basis of G, then f(X1 ,...,Xn ) is a diffeomorphism from a neighbourhood (nbd) U of 0 onto a nbd V of the unit e of G. For the following of this subsection, f is a work function. Proposition f is an analytic mapping and for each θ in S n , we have: f ′ (θ) : IRn → Tf (θ) G x = (x1 , ..., xn ) 7→

RfT(θ)

n X

k=1

!

xk Yk (θ)

where Yk (θ) = Ad(exp(θ1 X1 ) . . . exp(θk Xk ))Xk . Proof * Let θ in S n and x in IRn . f ′ (θ)(x) = = =

Pn

T T k=1 xk Lexp(θ1 X1 )... exp(θk Xk ) ◦ Rexp(θk+1 Xk+1 )... exp(θn Xn ) (Xk ) −1 RfT(θ) LTexp(θ1 X1 )... exp(θk Xk ) ◦ RT exp(θk+1 Xk+1 )... exp(θn Xn ) (Xk ) ! n X T xk Yk (θ) ∗ Rf (θ) k=1

Because RaT is an isomorphism from Tb G onto Tba G for all a, b in G, the singularities of f shall be studied through those of θ → h(θ)(x) =

n X

k=1

xk Yk (θ) where h(θ) ∈ L(IRn , G).

Especially, the kernel and the rank of f ′ (θ) and h(θ) are the same and Im(f ′ (θ) is isomorphic to Im(h(θ)).

1.2.

Singularities

We now recall only the main concepts which shall be useful in the following. For more details, see for example [2] and [3]. Definition Let M and N be two smooth manifolds, r = min(dim(M ), dim(N )) and f : M → N a differentiable mapping. Σi (f ) = {q ∈ M | rank(f ′ (q)) = r − i} is called the i-class of f . An indispensable step shall be to assign a structure of (sub)manifold to the i-class. It shall be possible with the concept of transversality:

190

Jean Lerbet

Definition Let M and N be two smooth manifolds and f : M → N a smooth mapping. Let W be a submanifold of N and a in M . Then f is transversal to (or intersects transversally) W at a (denoted f ⊤W at a) if either • f (a) ∈ /W • f (a) ∈ W and Tf (a) N = Tf (a) W + f ′ (a)(Ta M ). If A is a subset of M , f is transversal to W on A if f is transversal to W for all a in A. Finally, for A = M , we say f is transversal to W . The main result used is that both the following properties hold: Proposition Let M and N be two smooth manifolds and f : M → N a smooth mapping and W a submanifold of N . If f is transversal to W then f −1 (W ) is a submanifold of M , codim(W ) = codim(f −1 (W ) and if f (a) ∈ W then Ta f −1 (W ) = f ′ (a)−1 Tf (a) W . Remember also that a smooth mapping f : M → N is a submersion (resp. an immersion) at a ∈ M if f ′ (a) : Ta M → Tf (a) N is a surjection (resp. an injection). If it holds for all a ∈ M then f is only called a submersion (resp. an immersion). Obviously, such S mappings f have no singularity ( i∈N Σi (f ) = ∅).

1.3.

Stability

Definition 1. Let f1 and f2 be two elements of C ∞ (M, N ). f1 is equivalent to f2 if there are diffeomorphisms g : M → M and h : N → N such that f2 = h ◦ f1 ◦ g 2. Let f in C ∞ (M, N ). Then f is stable if there is a nbd Vf of f in C ∞ (M, N ) such that each f1 in Vf is equivalent to f . The nbd Vf of f in C ∞ (M, N ) must be understood according to the Whitney C ∞ topology. For details, see example [3]. As we have already noted, the stability is a ’beautiful’ property with important consequences in mechanics but it is very difficult to verify it. Fortunately, there is an infinitesimal point of view and the main result is that both aspects are equivalent (almost for a compact domain). Note also that there are other concepts of stability such as homotopically stability or stability under deformations which (at least for a compact domain) are equivalent. For applications, the most convenient concept is certainly the latter. Let us now describe the infinitesimal stability or inf-stability. Definition Let M and N be two smooth manifolds and f : M → N a smooth mapping. 1. Let πN : T N → N the canonical projection. A smooth mapping w : M → T N is a vector field along f if πN ◦ w = f . 2. f is infinitesimally stable or inf-stable if for all w, vector field along f , there are two vector fields s : M → T M and t : N → T N such that: w = f′ ◦ s + t ◦ f

Singularities and Stability of a Work Function

191

If f = f(X1 ,...,Xn ) is a work function the equation of inf-stability becomes, coming back to the Lie algebra: for all w : S n → G are there t : G → G and x : S n → IRn such that: w(θ) = h(θ)(x(θ)) + t(f (θ))

2.

(1)

Singularities and Stability of a Work Function

In this section, (X1 , . . . , Xn ) is a basis of the Lie algebra G of a Lie group G and f = f(X1 ,...,Xn ) is the associated work function. We shall begin by analysing the 1-class of f .

2.1.

The 1-class Σ1 (f )

Because S n and G have the same dimension, Σ1 (f ) = {θ ∈ T n | h(θ) ∈ Ln−1 (IRn , G)} = h−1 {Ln−1 (IRn , G)} where Lr (E, F ) denotes the set of linear mappings of rank r from IRn to G. If θ belongs to Σ1 (f ), that means that the rank of the family (Y1 (θ), . . . , Yn (θ) is n − 1. We shall suppose in the following that if θ is in Σ1 (f ), then (Y1 (θ), . . . , Yn−1 (θ)) is a free family and Yn (θ) =

n−1 X

αk (θ)Yk (θ) but this assumption is not essential.

k=1

Moreover to verify the transversality condition it shall be necessary to differentiate h. We obtain: Proposition For all θ in S n : h′ (θ) : IRn → L(IRn , G)

y = (y1 , . . . , yn ) 7→ x 7→ BY (θ) (x, y) where BY (x, y) is the bilinear mapping from IRn × IRn to G defined for all sequences Y = (Y1 , . . . , Yn ) of vectors of G by: BY (x, y) =

n X X

xk yl [Yl , Yk ]

k=1 l
For details and properties about this bilinear mapping, see [4]. Two situations may occur: • Y = (Y1 , . . . , Yn ) is a basis of G and we introduce the constants of structure of G: [Yl , Yk ] =

n X

i Clk Yi

i=1

and then BY (x, y) =

n X i=1

 

n X X

k=1 l


i Clk xk yl  Yi

192

Jean Lerbet • Y = (Y1 , . . . , Yn ) is not a basis of G. According to the previous study of the 1-class, we suppose that the n-1 first vectors are independent. Let Z be any vector completing (Y1 , . . . , Yn−1 ) to form a basis of G. We still put: [Yl , Yk ] =

n−1 X

i Z Clk Yi + Clk Z

i=1

Obviously Z depends on Y and the choice of it is not canonical. In this case, we have in a similar way: n−1 X

BY (x, y) =

i=1

 



+

n X X

k=1 l
n X X

k=1 l


i Clk xk yl  Yi



Z Clk xk yl  Z

Put also for all sequences Y = (Y1 , . . . , Yn ) of vectors of G: KY TY

= {x ∈ IRn | = IRn /KY

n X

Sp{Y } = {

FY

=

GY

= G/FY

xk Yk = 0}

k=1 n X

k=1

xk Yk | x ∈ IRn }

Suppose now as above, that the n-1 first vectors are independent. According to previous results we shall make the following identifications: P

TY ≃ {x ∈ IRn | xn = 0} and GY ≃ Sp{Z}

n−1 Indeed, Yn = k=1 αk Yk and x ∈ KY if and only if xk = −αk xn for all k = 1, . . . , n − 1. The summary of the results about the 1-class of is given by:

Proposition h is transversal to Ln−1 (IRn , G) and thus Σ1 (f ) is a (n-1)-submanifold of S n iff there exists k ∈ {1, . . . , n} such that: bk (θ) =

X

Z(θ)

αl (θ)Clk

l
(θ) 6= 0

Moreover, for all θ in Σ1 (f ): Tθ Σ1 (f ) = {x ∈ IRn | (

n X X

Z(θ)

αl (θ)Clk

k=1 l
(θ)xk = 0 }

Proof If θ ∈ / Σ1 (f ), then f ⊤Ln−1 (IRn , G) at θ by definition. Else, let θ ∈ Σ1 (f ). f ⊤Ln−1 (IRn , G) at θ iff the equation h′ (θ)(x) + v = u

(2)

Singularities and Stability of a Work Function

193

is solvable for x ∈ IRn , v ∈ Th (θ)Ln−1 (IRn , G), u ∈ L(IRn , G). A known result (see [3], [4] for example) gives the local description of the submanifold Ln−1 (IRn , G) of L(IRn , G) and allows to obtain here: Th (θ)Ln−1 (IRn , G) = {v ∈ L(IRn , G) | v(x) ∈ FY (θ) ∀ x ∈ KY (θ) } With the above conventions, that means vZ(θ) (x) = 0 if v(x) = vY (x) + vZ (x) with obviously notations. The condition of transversality becomes: Is there x in IRn such that, for all U ∈ GY (θ) , one may find y in KY (θ) with U = h′ (θ)(x)(y)? According to the previous calculations and identifications, this means: find x in IRn such that: (

n X X

Z(θ)

Clk

(θ)xk yl = u

k=1 l
is solvable in y ∈ KY (θ) for all u ∈ IR. Let k be such that (2) and x = IRn . Then the previous equation of transversality is solved. The calculation of Tθ Σ1 (f ) results directly from:

u bk (θ) (δik )i=1,...,n

in

Tθ Σ1 (f ) = h′ (θ)−1 (Th (θ)Ln−1 (IRn , G)) Remarks Z(θ)

1. since θ ∈ Σ1 (f ), there are always k and l such that Clk 6= 0 else FY (θ) = G and f is locally a diffeomorphism which is a contradiction with θ ∈ Σ1 (f ). But it is not equivalent to the transversality condition. 2. the transversality condition is a weak condition. In the practice, it always holds. It should be interesting to compute the set of θ ∈ Σ1 (f ) where it fails.

2.2. f as Submersion with Folds We now study a situation for which the explicit description of singularities is possible and stability can be computed in a simple way. We shall always suppose in the following that the transversality condition holds and f is a work function as above. Definition 1. θ in Σ1 (f ) is a fold point if : KY (θ) + Tθ Σ1 (f ) = IRn and because of the dimensions, the sum is direct. 2. f is a submersion with folds if the singularities of f are only fold points. 3. If f is a submersion with folds, Σ1 (f ) is called the fold locus of f .

194

Jean Lerbet

We have: Proposition θ in Σ1 (f ) is a fold point if and only if Qθ (x) =

n X X

Z(θ)

Clk

(θ)xk xl

k=1 l
is a non degenerated quadratic form on KY (θ) or if n−1 XX

Z(θ)

αk (θ)αl (θ)Clk

(θ) =

n−1 X k=1

k=1 l
αk (θ)bk (θ) 6= 0

(3)

Proof It is necessary and sufficient to prove that KY (θ) ∩Tθ Σ1 (f ) = {0} because dim(KY (θ) ) = 1 and dim(Tθ Σ1 (f )) = n − 1. x belongs to KY (θ) ∪ Tθ Σ1 (f ) if BY (θ) (x, y) ∈ FY (θ) for all x, y are in KY (θ) . Because of the symmetry of BY (θ) on KY (θ) × KY (θ) (see [4] for example), it is equivalent to BY (θ) (x) ∈ FY (θ) for all x in KY (θ) . P

P

Z(θ)

That means that nk=1 l
x2n (

Z(θ)

Clk

(θ)αk (θ)αl (θ)) = 0

k=1 l
that is to say the wanted result. The term

X

x2n (

Z(θ)

Cln (θ)αl (θ))

l
disappears because X

Z(θ)

Cln (θ)xn xl = xn

Now

l=1 Yl (θ)xl

= 0 and [ n X

Pn

l=1 Yl (θ)xl , Yn (θ)]

[Yl (θ)Yn (θ)]xl =

l=1

Z(θ)

Cln (θ)xl

l=1

l
Pn

n X

n X

= 0. We deduce: Z(θ)

Cln (θ)xl = 0

l=1

Another precision may be given about f as submersion with fold: Proposition θ in Σ1 (f ) is never a fold point if rank(Y1 (θ), . . . , Yn (θ)) = n − 1 and Yn (θ) = αi Yi (θ) with i in {1, . . . , n − 1} (whereby n and i can obviously be replaced by any other pair of indices of {1, . . . , n}). Proof

Singularities and Stability of a Work Function

195

Indeed (3) becomes 0 6= 0. In fact it is a direct result of the antisymmetry of the Lie bracket Z(θ) Z(θ) Z(θ) (Clk = −Ckl and Cll = 0) and we have successively: n−1 X

αk (θ)bk (θ) =

k=1

=

n−1 XX

k=1 l
Z(θ)

(θ)αk (θ)αl (θ))

Z(θ)

(θ)αk (θ)αl (θ))

Clk Clk

k=1 l≤k

Z(θ)

= (αi (θ))2 Cii

(θ)

= 0

2.3.

Stability of f

We now suppose that (2) and (3) are both satisfied. Let g the restriction of f to its fold locus Σ1 (f ). g is obviously an immersion from Σ1 (f ) to G and because S n is compact the closed submanifold Σ1 (f ) too (it is the first time the compactness is used). Thus if a = g(θ) then g −1 (a) = {θ1 , . . . , θp } is finite. The stability of f depends of the nature of the crossings at a. More precisely, according to the theory of singularities, let us put: Definition 1. For every set A and s > 1, define A(s) = {(a1 , . . . , as ) ∈ As | ai 6= aj for 1 ≤ i < j ≤ s}. 2. Let φ : M → N be a smooth mapping and φ(s) : M (s) → N s the restriction of φ × . . . × φ : M s → N s to M (s) . Let δN s = {(y, . . . , y) ∈ N s | y ∈ N }. Then φ is a mapping with normal crossings if for every s > 1, φ(s) ⊤δN s . and the main result which shall be used here is:(see [3]) Proposition The work function f is stable if and only if g is an immersion with normal crossings. It now remains to analyse the condition of normal crossings (N-C) of g. So, let θ(s) = (θ1 , . . . , θs ) element of (S n )(s) such that f (θ1 ) = . . . = f (θs ) = w ∈ G. For other points of (S n )(s) , the transversality g (s) ⊤δGs at θ(s) obviously holds and the (N-C) too. T )−1 , the equation of Coming back to the Lie algebra G by the diffeomorphism (Rw transversality becomes: Let (h1 , . . . , hs ) be in G s . Find h in G and x1 , . . . , xs in Tθ1 Σ1 (f ) × . . . × Tθs Σ1 (f ) such that for all i = 1, . . . , s:

hi = h +

n X

xik Yk (θi )

k=1

Because each is a fold point, we can look for xi in IRn . Moreover, this family of equations enables us to obtain: θi

196

Jean Lerbet

Proposition If there are s > n roots of the equation g(θ) = w, then f is not stable. Proof Because g is an immersion and S n compact, g −1 (w) = {θ1 , . . . , θs } is finite. The previous equations of transversality imply that codim(FY (θ1 ) ∩ . . . ∩ FY (θs ) ) = codim(FY (θ1 ) )+. . .+codim(FY (θs ) ) = s and obviously codim(FY (θ1 ) ∩. . .∩FY (θs ) ) ≤ n. More precisely, we may prove that (see [3]): Proposition Let D = FY (θ1 ) ∩ . . . ∩ FY (θs ) . f is stable if and only if there are VY (θ1 ) , . . . , VY (θs ) elements of G such that: G = D ⊕ IRVY (θ1 ) ⊕ . . . ⊕ IRVY (θs ) FY (θi ) = D ⊕

X

FY (θj )

j6=i

G = IRVY (θi ) ⊕ FY (θi ) Therefore, the stability of the work function strongly depends on the intersection D = FY (θ1 ) ∩ . . . ∩ FY (θs ) . We have: Proposition Sp{Y1 (θj ), Yn (θj )} ⊆ D for all j and if the number s of roots of the equation g(θ) = w is such that s > n − 2 , then f is not stable. Proof We have only to prove the first assertion. But Ad(exp(tX))X = X for all X in G which proves that Y1 (θj ) = X1 . Moreover Yn (θj ) = Ad(exp(g(θj )))Xn = Ad(exp(w))Xn for all root θj of g(θ) = w which proves Yn (θi ) = Yn (θj ) for all θi and θj in g −1 (w). QED. We obviously obtain: Corollary For n = 3 (planar motions), the work function f is stable if and only if its restriction g is an injective immersion.

3.

Examples

Here, G is the group of Euclidean displacements D. We begin by giving the family (X1 , ..., Xn ) of 6-vectors. The coordinates of each vector are given with respect to a basis → → − − → − of D which is deduced from a fixed coordinates frame (0; i , j , k ) of E in the following way: → − → −→ − → − → − → −→ − → −→ − i(m) = i , j(m) = j , k(m) = k , ξ(m) = i ∧ 0m, η(m) = j ∧ 0m, ζ(m) = k ∧ 0m for all m in E. Thus the basis of D is (i, j, k, ξ, η, ζ). The studied singularity is 0.

3.1.

First Example

It concerns a chain RRPRRR for which:

Singularities and Stability of a Work Function 

   X1 =    

0 0 0 0 0 1





       , X2 =       



0 0 0 1 0 0



       , X3 =       

0 1 0 0 0 0





       , X4 =       

a 0 0 0 1 0





       , X5 =       

197

0 −b a 1 0 0

The system is singular because the following relation holds:





       , X6 =       

0 0 0 α 0 γ

       

γX1 + αX2 − X6 = 0 Therefore, we have rank(X1 , . . . , X6 ) = 5 that is to say 0 belongs to Σ1 (f ). The calculations prove that G = D and that we may choose FY (0) = Sp(X1 , . . . , X5 ) and G0 = IRη that is to say Z = η. We find: KY (0) = {y ∈ IR6 |y = (−γt, −αt, 0, , 0, t) t ∈ IR} We obtain after some calculations: b1 (O) = 0, b2 (O) = −γ, b3 (O) =

γ b , b4 (O) = 0, b1 (O) = −γ(1 − ), b6 (O) = 0 a a

This proves that the condition of transversality (2) is satisfied. Moreover, Σ1 (f ) is a 6 − (6 − 5)(6 − 5) = 5 dimensional submanifold and we obtain: T0 Σ1 (f ) = {x ∈ IR6 | −γx2 +

b γ x3 − −γ(1 − )x5 = 0} a a

and the condition (3) becomes αγ 6= 0 which is satisfied: 0 is a fold point.

3.2.

Second Example

This example is a robot RRRRRR. The singular configuration which is studied corresponds to the singular situation where the center of the wrist is on the axis of the first link. We obtain: 

   X1 =    

0 0 0 0 0 1





       , X2 =       

0 0 0 1 0 0





       , X3 =       

0 0 −b 1 0 0





       , X4 =       

a 0 0 0 1 0





       , X5 =       

0 −a 0 1 0 0

The system is singular because the following relation holds:





       , X6 =       

αa 0 0 0 α β

       

βX1 + αX4 − X6 = 0 Therefore, we have rank(X1 , . . . , X6 ) = 5 that is to say 0 belongs to Σ1 (f ). The calculations prove that G = D and that we may choose FY (0) = Sp(X1 , . . . , X5 ) and G0 = IRi that is to say Z = i. We find: KY (0) = {y ∈ IR6 |y = (−βt, 0, 0, 0, −αt, t) t ∈ IR}

198

Jean Lerbet For all x, y in IR6 , we obtain after some calculations: BY (0) (x, y) = B FY (0) (x, y) + βay 6 (x2 + x3 )i

with B FY (0) (x, y) element of FY (0) . That proves that the condition of transversality is satisfied (it is sufficient to choose x such that x2 + x3 6= 0). Moreover, Σ5 (f ) is a 6 − (6 − 5)(6 − 5) = 5 dimensional submanifold and we obtain: T0 Σ1 (f ) = {x ∈ IR6 |x2 + x3 = 0}.

The interpretation of this result is easy because the condition x2 = −x3 preserves the alignment of the first link and the center of the wrist. Unfortunately KY (0) ⊂ T0 Σ1 (f )

because if x ∈ KY (0) then x2 = x3 = 0 and x ∈ T0 Σ1 (f ) and 0 is not a fold point (or apply the last proposition of the subsection 2.3).

Conclusion A work function f of a robot or more generally of a kinematical chain is a product of exponential mappings of a Lie group. This mapping comprises all the kinematics of the chain and the analysis of its singularities is an essential point in mechanics. We are concerned with the case where the dimension of the group is equal to the number of variables. First the transversality conditions (2) allowing to describe the set of singularities Σ1 (f ) (structure of manifold and tangent bundle) have been given. Secondly, we have shown that under the condition (3) f is a submersion with fold. Finally, considering the restriction g of f to its fold locus, normal-crossing conditions of g to analyse the stability of f is studied. Examples are given to illustrate the method and interesting extensions may be suggested: stronger singularities (not fold points), points of Σ2 (f ) or of Σ1,1 (f ) could be for example considered.

References [1] J.Bochnak M.Coste M-F.Roy G´eom´etrie alg´ebrique r´eelle Springer-Verlag (1986) [2] V. Arnold A.Varchenko S.Goussein-Zade Singularit´es des applications diff´erentiables Mir (1986) [3] M.Golubitsky V.Guillemin Stable Mappings and Their Singularities Springer-Verlag (1973) [4] J.Lerbet K.Hao Kinematics of Mechanisms to the second Order- Application to the closed Mechanisms Acta Applicandae Mathematicae Vol. 59, pp.1-19 (1999) [5] J.Lerbet Analytic Geometry and singularities of mechanism ZAMM 78,10, pp 687694,(1998) [6] H.Whitney Tangents to an analytic variety Ann.of Math. 81,496-549 (1965)

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 199-265

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 8

T HE C ONFORMAL -A FFINE S TRUCTURE OF OPEN Q UANTUM R ELATIVITY, I TS P HYSICAL R EALIZATION AND I MPLICATIONS G. Basini1 and S. Capozziello2,∗ 1 Laboratori Nazionali di Frascati, INFN, Via E. Fermi C.P. 13, I-0044 Frascati (Roma), Italy 2 Dipartimento di Scienze Fisiche, Universit`a di Napoli ”Federico II” and INFN Sez. di Napoli, Compl. Univ. Monte S. Angelo, Ed.N, Via Cinthia, I-80126 Napoli, Italy

Abstract Beside the post-relativistic theories, Open Quantum Relativity is a gauge theory of interactions based on a nonlinear realization (NLR) of the local Conformal-Affine (CA) group of symmetry transformations. Such a theory, thanks to a covariantsymplectic formulation, succeeds in treating General Relativity and Quantum Mechanics under the same standard. In this Report, we obtain the coframe fields and the gauge connections of the theory while the tetrads and Lorentz group metric are used to induce the spacetime metric. The inhomogenously transforming (under the Lorentz group) connection coefficients serve as gravitational gauge potentials used to define covariant derivatives accommodating the couplings of matter and gauge fields. On the other hand, the tensor valued connection forms serve as auxiliary dynamical fields associated with the dilation and as special conformal and deformation (shear) degrees of freedom inherent in the bundle manifold. As a consequence, the bundle curvature of the theory is determined and the boundary topological invariants are then constructed. They serve as a prototype (source free) gravitational Lagrangian to derive the following dynamics. Finally, the Bianchi identities, covariant field equations and gauge currents are obtained. These mathematical tools give rise to a compact, self-contained approach to physical interactions (in particular gravitation), based on the local gauge invariance. Starting from this general invariance principle, we discuss the global and the local Poincar´e invariance, developing the spinor, vector and tetrad formalisms. This covariant-symplectic approach allows to construct the curvature, torsion and metric tensors starting from the covariant derivative. The resulting theory ∗

E-mail address: [email protected]. (Corresponding author)

200

G. Basini and S. Capozziello describes a spacetime endowed with non-vanishing curvature and torsion, while the gravitational field equations are Yang-Mills-like equations of motion, with the torsion tensor playing the role of the Yang-Mills field strength. Besides other physical consequences and the reliable reproduction of several physical experiments and astrophysical observations described elsewhere [1], such field equations provide, in principle, the theoretical device to achieve Close Time Curves and, consequently, the conceivability of time travels.

1.

Introduction

Quantum theory and Relativity are the two fundamental theories of modern physics. The so-called Standard Model is currently the most successful relativistic quantum field theory both from particle physics and group theory points of view. It is a non-Abelian gauge theory (Yang-Mills theory) associated with the internal symmetry group SU (3) × SU (2) × U (1), in which the SU (3) color symmetry for the strong force in quantum chromodynamics, is treated as exact, whereas the SU (2) × U (1) symmetry, responsible for generating the electro-weak gauge fields, is spontaneously broken. So far as we know, there are four fundamental forces in Nature; namely, electromagnetic force, weak force, strong force and gravitational force. The Standard Model covers the first three, but not the gravitational interaction. In General Relativity, the geometrized gravitational field is described by the metric tensor gµν of pseudo-Riemannian spacetime, and the field equations that the metric tensor satisfies are nonlinear. This nonlinearity is indeed a source of difficulty in quantization of General Relativity. Since the successful Standard Model, in particle physics, is a gauge theory in which all the fields mediating the interactions are represented by gauge potentials, a question arises as to why the fields mediating the gravitational interaction are different from those of other fundamental forces. It is reasonable to expect that there may be a gauge theory in which the gravitational fields stand on the same footing as those of other fields [1]. This expectation has prompted a re-examination of General Relativity from the gauge theoretical point of view. While the gauge groups involved in the Standard Model are all internal symmetry groups, the gauge groups in General Relativity must be associated with external spacetime symmetries. Therefore, the gauge theory of gravity will not be as the usual Yang-Mills theory. It must be one in which gauge objects are not only gauge potentials but also tetrads that relate the symmetry group to the external spacetime. For this reason we have to consider a more complex nonlinear gauge theory where all the interactions are dealt under the same standard [2]. In General Relativity, Einstein took the spacetime metric as the basic variable representing gravity, whereas Ashtekar employed the tetrad fields and the connection forms as the fundamental variables. We also consider the tetrads and the connection forms as the fundamental fields but with the difference that this approach gives rise to a covariant symplectic formalism capable of achieving the result of dealing with physical fields under the same standard [3, 4]. In 1956, Utiyama suggested that gravitation may be viewed as a gauge theory [5] in analogy to the Yang-Mills [6] theory (1954). He identified the gauge potential due to the Lorentz group with the symmetric connection of Riemann geometry, and constructed Einstein’s General Relativity as a gauge theory of the Lorentz group SO(3, 1) with the help

The Conformal-Affine Structure of open Quantum Relativity...

201

of tetrad fields introduced in an ad hoc manner. Although the tetrads were necessary components of the theory, to relate the Lorentz group adopted as an internal gauge group to the external spacetime, they were not introduced as gauge fields. In 1961, Kibble [7] constructed a gauge theory based on the Poincar´e group P (3, 1) = T (3, 1) ⋊ SO(3, 1) (⋊ represents the semi-direct product) which resulted in the Einstein-Cartan theory characterized by curvature and torsion. The translation group T (3, 1) is considered responsible for generating the tetrads as gauge fields. Cartan [8] generalized the Riemann geometry to include torsion in addition to curvature. The torsion (tensor) arises from an asymmetric connection. Sciama [9], and others (Fikelstein [10], Hehl [11, 12]) pointed out that intrinsic spin may be the source of torsion of the underlying spacetime manifold. Since the form and role of the tetrad fields are very different from those of gauge potentials, it has been thought that even Kibble’s attempt is not satisfactory as a full gauge theory. There have been a number of gauge theories of gravitation based on a variety of Lie groups [11, 12, 13, 14, 16, 15, 17]. It was argued that a gauge theory of gravitation corresponding to General Relativity can be constructed with the translation group alone, in the so-called teleparallel scheme. Inomata et al. [18] proposed that Kibble’s gauge theory could be obtained, in a manner closer to the Yang-Mills approach, by considering the de Sitter group SO(4, 1), which is reducible to the Poincar´e group by group-contraction. Unlike the Poincar´e group, the de Sitter group is homogeneous and the associated gauge fields are all of gauge potential type and by the Wigner-In¨onu group contraction procedure, one of the five vector potentials reduces to the tetrad. It is standard to use the fiber-bundle formulation by which gauge theories can be constructed on the basis of any Lie group. Recent work by Hehl et al. [17], on the so-called Metric Affine Gravity (MAG) theory, adopted as a gauge group the affine group A(4, R) = T (4)⋊GL(4, R), which was realized linearly. The tetrad was identified with the nonlinearly realized translational part of the affine connection, on the tangent bundle. In MAG theory, the Lagrangian is quadratic in both curvature and torsion, in contrast to the EinsteinHilbert Lagrangian in General Relativity which is linear in the scalar curvature. The theory has the Einstein limit on one hand and leads to the Newtonian inverse distance potential plus the linear confinement potential, in the weak field approximation, on the other. So, as we have seen above, there are many attempts to formulate gravitation as a gauge theory but currently no theory has been uniquely accepted as the gauge theory of gravity. The nonlinear approach to group realizations was originally introduced by Coleman, Wess and Zumino [19, 20] in the context of internal symmetry groups (1969). It was later extended to the case of spacetime symmetries by Isham, Salam, and Strathdee [21, 22] considering the nonlinear action of GL(4, R) modulus the Lorentz subgroup. In 1974, Borisov, Ivanov and Ogievetsky [23, 24], considered the simultaneous nonlinear realization (NLR) of the affine and conformal groups. They stated that General Relativity can be viewed as a consequence of spontaneous breakdown of the affine symmetry, in the same manner that chiral dynamics, in quantum chromodynamics, is a result of spontaneous breakdown of chiral symmetry. In their model, gravitons are considered as Goldstone bosons associated with the affine symmetry breaking. In 1978, Chang and Mansouri [25] used the NLR scheme employing GL(4, R) as the principal group. In 1980, Stelle and West [26]

202

G. Basini and S. Capozziello

investigated the NLR induced by the spontaneous breakdown of SO(3, 2). In 1982 Ivanov and Niederle considered nonlinear gauge theories of the Poincar´e, de Sitter, conformal and special conformal groups [27, 28]. In 1983, Ivanenko and Sardanashvily [29] considered gravity to be a spontaneously broken GL(4, R) gauge theory. The tetrads fields arise, in their formulation, as a result of the reduction of the structure group of the tangent bundle from the general linear to Lorentz group. In 1987, Lord and Goswami [30, 31] developed the NLR in the fiber bundle formalism based on the bundle structure G (G/H, H) as suggested by Ne’eman and Regge [32]. In this approach, the quotient space G/H is identified with physical spacetime. Most recently, in a series of papers, Lopez-Pinto, Julve, Tiemblo, Tresguerres and Mielke discussed nonlinear gauge theories of gravity on the basis of the Poincar´e, affine and conformal groups [35, 34, 36, 37, 38, 39]. Now, following the prescriptions of General Relativity, the physical spacetime is assumed to be a four-dimensional differential manifold. In Special Relativity, this manifold is the Minkwoski flat-spacetime M4 while, in General Relativity, the underlying spacetime is assumed to be curved in order to describe the effects of gravitation. As we said, Utiyama [5] proposed that General Relativity can be seen as a gauge theory based on the local Lorentz group in the same way that the Yang-Mills gauge theory [53] is developed on the basis of the internal iso-spin gauge group. In this formulation the Riemannian connection is the gravitational counterpart of the Yang-Mills gauge fields. While SU (2), in the Yang-Mills theory, is an internal symmetry group, the Lorentz symmetry represents the local nature of spacetime rather than internal degrees of freedom. The Einstein Equivalence Principle, asserted for General Relativity, requires that the local spacetime structure can be identified with the Minkowski spacetime possessing Lorentz symmetry. In order to relate local Lorentz symmetry to the external spacetime, we need to solder the local space to the external space. The soldering tools can be the tetrad fields. Utiyama regarded the tetrads as objects given a priori while they can be dynamically generated [2] and the spacetime has necessarily to be endowed with torsion in order to accommodate spinor fields. In other words, the gravitational interaction of spinning particles requires the modification of the Riemann spacetime of General Relativity to be a (non-Riemannian) curved spacetime with torsion. Although Sciama used the tetrad formalism for his gaugelike handling of gravitation, his theory fell shortcomings in treating tetrad fields as gauge fields. Following the Kibble approach [7], it can be demonstrated how gravitation can be formulated starting from a pure gauge viewpoint. In particular, the aim of this paper is to show, in details, how a theory of gravitation is a gauge theory which can be obtained starting from some local invariance, e.g. the local Poincar´e symmetry, leading to a given unification scheme. This is the dynamical structure of Open Quantum Relativity [1] by which a gauge theory of gravity, based on a nonlinear realization of the local conformal-affine group of symmetry transformations, can be formulated [4]. Here, we start from a General Invariance Principle (the General Conservation Principle [43]) and consider first the Global Poincar´e Invariance and then the Local Poincar´e Invariance. This approach leads to construct a given theory of gravity as a gauge theory. Such a viewpoint, if considered in detail, can avoid many shortcomings and could be useful to formulate self-consistent schemes for quantum gravity and then the unification of all interactions [2].

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The dynamical consequences of such an approach, beside the reliable reproduction of several experimental results [1], can be extremely interesting because leading to the possibility of conceiving time-travels as straightforward consequence of dynamics, as we will discuss below. After the Introduction, in which we have sketched the current ideas on these topics, the paper is organized in four sections. Sec.II is devoted to the conformal-affine group structure of Open Quantum Relativity. After some mathematical preliminaries, the generalized bundle structure of gravity is presented. Then we describe the conformal-affine Lie algebra, the group actions and the bundle morphisms which we are going to adopt. The introduction of a generalized gauge transformation law enable us the gauging of external spacetime groups. Demanding that tetrads can be obtained as gauge fields requires the implementation of a NLR of the CA group. Such a NLR is carried out over the quotient space CA(3, 1)/SO(3, 1). Then, the transformations of all coset fields, parameterizing this quotient space, are computed while the decomposition of connections and the conformal-affine nonlinear gauge potentials are derived. Besides, the tetrad components of the coframe are used, in conjunction with the Lorentz group metric, to induce a spacetime metric and then we derive the Cartan Structure Equations and the Bianchi identities. Finally, the surface and bulk topological invariants are constructed. The bulk terms (obtained via exterior derivation of the surface terms) provide a tool to ”derive” a prototype (source free) gravitational action (after appropriately distributing the Lie star operators). The covariant field equations and gauge currents are then straightforwardly obtained. After this first formal part, in Sec.III, we discuss the concept of Invariance and Conservation as our starting point and we derive gravity as a local Poincar´e gauge theory as the result of the General Conservation Principle. The differences between global and local Poincar´e invariance are discussed with the aim to show how a local transformation is related to the gauge fields. Spinors, vectors and tetrads, which transform under Lorentz transformations, are then derived in the framework of this covariant-symplectic formalism. In particular we discuss the Fock - Ivanenko connection in the framework of the local Poincar´e transformations. Starting from the Fock - Ivanenko covariant derivative, curvature, torsion and metric tensors are then derived. Field equations for gravity are discussed as a remarkable application of the presented approach. Sec.IV of this Report is devoted to the realization of the above formalism for the Open Quantum Relativity sending the Reader to Reference [1] for further details and applications. We sketch the 5D approach as the minimal unification scheme and then we discuss the role of conservation laws in this framework. Some useful remarks are devoted to the discussion of the geodesic structure of the theory and to the emergence of an extra force term induced by the embedding mechanism from 5D to 4D. The field equations, the masses of the particles and the time-like solutions, emerging from this unified approach are then considered. Conclusions, discussion and perspective are drawn in Sec.V. The most dramatic result of this theoretical construction is the conceivability, in principle, of time-travels. Principles and requirements to achieve themn are discussed at the end. The Appendices are devoted to the adopted notation, to the Maurer-Cartan 1-forms and to the Baker-Campbell-Hausdorff formulas, respectively.

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2.

The Conformal-Affine Structure

2.1.

Fiber Bundles, Gauge Symmetry and Connection Forms

Before considering in details the Conformal-Affine structure of Open Quantum Relativity, let us briefly review the standard bundle approach to gauge theories. We verify that a usual gauge potential Ω is the pullback of 1-form connection ω by the local sections of the bundle. Finally, the transformation laws of the ω and Ω under the action of the structure group G are deduced. Modern formulations of gauge field theories are geometrically expressible in the language of principal fiber bundles. A fiber bundle is a structure hP, M , π; Fi where P (the total bundle space) and M (the base space) are smooth manifolds, F is the fiber space and the surjection π (a canonical projection) is a smooth map of P onto M , π : P → M.

(2.1)

The inverse image π −1 is diffeomorphic to F π −1 (x) ≡ Fx ≈ F,

(2.2)

χi : Ui ×M G → π −1 (Ui ) ∈ P,

(2.3)

S and it is called the fiber at x ∈ M . The partitioning x π −1 (x) = P is referred to as the fibration. Note that a smooth map is one whose coordinatization is C ∞ differentiable; a smooth manifold is a space that can be covered with coordinate patches in such a manner that, a change from one patch to any overlapping patch is smooth [40]. Fiber bundles that admit decomposition as a direct product, locally looking like P ≈M × F, S are called trivial. Given a set of open coverings {Ui } of M with x ∈ {Ui } ⊂ M satisfying α Uα = M , the diffeomorphism map is given by

(×M represents the fiber product of elements defined over the space M ) such that π (χi (x, g)) = x and χi (x, g) = χi (x, (id)G ) g = χi (x) g ∀x ∈ {Ui } and g ∈ G. Here, (id)G represents the identity element of the group G. In order to obtain the global bundle structure, the local charts χi must be glued together continuously. Consider two patches Un and Um with a non-empty intersection Un ∩ Um 6= ∅. Let ρnm be the restriction −1 −1 of χ−1 n to π (Un ∩ Um ) defined by ρnm : π (Un ∩ Um ) → (Un ∩ Um ) ×M Gn . Similarly −1 let ρmn : π −1 (Um ∩ Un ) → (Um ∩ Un ) ×M Gm be the restriction of χ−1 m to π (Un ∩ Um ). The composite diffeomorphism Λnm ∈ G Λmn : (Un ∩ Um ) × Gn → (Um ∩ Un ) ×M Gm ,

(2.4)

−1 Λij (x) ≡ ρji ◦ ρ−1 ij = χi, x ◦ χj, x : F → F

(2.5)

defined as constitutes the transition function between bundle charts ρnm and ρmn (◦ represents the group composition operation) where the diffeomorphism χi, x : F → Fx is written as χi, x (g) := χi (x, g) and satisfies χj (x, g) = χi (x, Λij (x) g). The transition functions

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{Λij } can be interpreted as passive gauge transformations. They satisfy some consistency conditions, i.e. the identity Λii (x), the inverse Λij (x) = Λ−1 ji (x) and the cocycle Λij (x) Λjk (x) = Λik (x). For trivial bundles, the transition function reduces to Λij (x) = gi−1 gj ,

(2.6)

where gi : F → F is defined by gi := χ−1 ei, x , provided the local trivializations {χi } i, x ◦ χ and {e χi } it gives rise to the same fiber bundle. A section is defined as a smooth map s : M → P,

(2.7)

such that s(x) ∈ π −1 (x) = Fx ∀x ∈ M and satisfies π ◦ s = (id)M ,

(2.8)

where (id)M is the identity element of M . It assigns to each point x ∈ M a point in the fiber over x. Trivial bundles admit global sections. A bundle is a principal fiber bundle hP, P/G, G, πi provided that the Lie group G acts freely (i.e. if pg = p then g = (id)G ) on P to the right Rg p = pg, p ∈ P, preserves fibers on P (Rg : P → P), and finally is transitive on fibers. Furthermore, there must exist local trivializations compatible with the G action. Hence, π −1 (Ui ) is homeomorphic to Ui ×M G and the fibers of P are diffeomorphic to G. The trivialization or inverse diffeomorphism map is given by −1 χ−1 (2.9) i : π (Ui ) → Ui ×M G

such that χ−1 (p) = (π(p), ϕ(p)) ∈ Ui ×M G, p ∈ π −1 (Ui ) ⊂ P, where we see from the above definition that ϕ is a local mapping of π −1 (Ui ) into G satisfying ϕ(Lg p) = ϕ(p)g for any p ∈ π −1 (U) and any g ∈ G. Let us observe that the elements of P which are projected onto the same x ∈ {Ui } are transformed into one another by the elements of G. In other words, the fibers of P are the orbits of G and at the same time, they are the set of elements which are projected onto the same x ∈ U ⊂ M . This observation motivates calling the action of the group vertical and the base manifold horizontal. The diffeomorphism map χi is called the local gauge since χ−1 maps π −1 (Ui ) onto the direct (Cartesian) product i Ui ×M G. The action Lg of the structure group G on P defines an isomorphism of the Lie algebra g of G onto the Lie algebra of vertical vector fields on P, tangent to the fiber at each p ∈ P called fundamental vector fields λg : Tp (P) → Tgp (P) = Tπ(p) (P) ,

(2.10)

where Tp (P) is the space of tangents at p, i.e. Tp (P) ∈ T (P). The map λ is a linear isomorphism for every p ∈ P and is invariant with respect to the action of G, that is, λg : (λg∗ Tp (P)) → Tgp (P), where λg∗ is the differential push forward map induced by λg defined by λg∗ : Tp (P) → Tgp (P). Since the principal bundle P (M , G) is a differentiable manifold, we can define tangent T (P) and cotangent T ∗ (P) bundles. The tangent space Tp (P) defined at each point p ∈ P may be decomposed into a vertical Vp (P) and horizontal Hp (P) subspace as Tp (P) := Vp (P) ⊕ Hp (P) (where ⊕ represents the direct sum). The space Vp (P)

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is a subspace of Tp (P) consisting of all tangent vectors to the fiber passing through p ∈ P, and Hp (P) is the subspace complementary to Vp (P) at p. The vertical subspace Vp (P) := {X ∈ T (P) |π (X) ∈ Ui ⊂ M } is uniquely determined by the structure of P, whereas the horizontal subspace Hp (P) cannot be uniquely specified. This result is very important because it makes possible to fix the Cauchy conditions on the dynamics as we will discuss in Sec. V. Thus we require the following condition: when p transforms as p → p′ = pg, Hp (P) transforms as [41], Rg∗ Hp (P) → Hp′ (P) = Rg Hp (P) = Hpg (P) .

(2.11)

Let the local coordinates of P (M , G) be p = (x, g) where x ∈ M and g ∈ G. Let GA denote the generators of the Lie algebra g corresponding to group G satisfying the commutators [GA , GB ] = fABC GC , where fABC are the structure constants of G. Let Ω be i a connection form defined by ΩA := ΩA i dx ∈ g. Let ω be a connection 1-form defined by ∗ ω := ge−1 πPM Ωe g + ge−1 de g

(2.12)

(∗ represents the differential pullback map) belonging to g⊗Tp∗ (P) where Tp∗ (P) is the dual space to Tp (P). In such a case, the differential pullback map, applied to a test function ϕ ∗ ∧ f ∗ β. and p-forms α and β, satisfies f ∗ ϕ = ϕ ◦ f , (g ◦ f )∗ = f ∗ g ∗ and f ∗ (α ∧ β) =  fαβα  If G is represented by a d-dimensional d × d matrix, then GA = [Gαβ ], ge = ge , where α, β = 1, 2, 3,...d. Thus, ω assumes the form

∗ ωαβ = ge−1 αγ de g γβ + ge−1 ργ πPM Ωρσi Gαγ geσβ ⊗ dxi . (2.13)

If M is n-dimensional, the tangent space Tp (P) is (n + d)-dimensional. Since the vertical subspace Vp (P) is tangential to the fiber G, it is d-dimensional. Accordingly, Hp (P) is n-dimensional. The basis of Vp (P) can be taken to be ∂αβ := ∂g∂αβ . Now, let the basis of Hp (P) be denoted by Ei := ∂i + Γαβ i ∂αβ , i = 1, 2, 3, ..n and α, β = 1, 2, 3, ..d

(2.14)

∂ where ∂i = ∂x i . The connection 1-form ω projects Tp (P) onto Vp (P). In order for X ∈ Tp (P) to belong to Hp (P), it has to be X ∈ Hp (P), ωp (X) = hω (p) |Xi = 0. In other words, Hp (P) := {X ∈ Tp (P) |ωp (X) = 0} , (2.15)

from which Ωαβ can be determined. The inner product appearing in ωp (X) = i hω (p) |Xi = 0 is a map h·|·i : Tp∗ (P) × Tp (P) → R defined by hW |V i = 

Wµ V ν dxµ | ∂x∂ ν = Wµ V ν δνµ , where the 1-form W and vector V are given by W = 

Wµ dxµ and V = V µ ∂x∂ ν . Observe also that, dg αβ |∂ρσ = δρα δσβ . A We parameterize an arbitrary group element geλ as ge (λ) = eλ GA = eλ·G , A = 1,..dim (g). The right action Rge(λ) = Rexp(λ·G) on p ∈ P, i.e. Rexp(λ·G) p = p exp (λ · G), defines a curve through p in P. Define a vector G# ∈ Tp (P) by [41] G# f (p) :=

d f (p exp (λ · G)) |λ=0 dt

(2.16)

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where f : P → R is an arbitrary smooth function. Since the vector G# is tangent to P at p, G# ∈ Vp (P), the components of the vector G# are the fundamental vector fields at p which constitute V (P). We have to stress that the components of G# may also be viewed as a basis element of the Lie algebra g. Given G# ∈ Vp (P), G ∈ g,   D E     ∗ ωp G# = ω (p) |G# = ge−1 de g G# + ge−1 πPM Ωe g G# = gep−1 gep

d (exp (λ · G)) |λ=0 , dλ

(2.17)


where use was made of πPM ∗ G# = 0. Hence, ωp G# = G. An arbitrary vector X ∈ Hp (P) may be expanded in a basis spanning Hp (P) as X := β i Ei . By direct computation, one can show D E

∗ ωαβ |X = ge−1 αγ β i Γγβ e−1 αγ πPM Ωρσi β i Gγρ geσβ = 0, ∀β i (2.18) i + g Equation (2.18) yields

from which we obtain

ge−1




αγ

Γγβ e−1 i + g




αγ

∗ πPM Ωρσi Gγρ geσβ = 0,

ρ ∗ γ σβ Γγβ e . i = −πPM Ω σi Gρ g

(2.19) (2.20)

In this manner, the horizontal component is completely determined. An arbitrary tangent vector X ∈ Tp (P) defined at p ∈ P takes the form   ρ ∗ α σβ αβ i (2.21) X = A ∂αβ + B ∂i − πPM Ω σi Gρ ge ∂αβ ,

where Aαβ and B i are constants. The vector field X is comprised of horizontal XH :=
ρ i ∗ α σβ B ∂i − πPM Ω σi Gρ ge ∂αβ ∈ H (P) and vertical XV := Aαβ ∂αβ ∈ V (P) components. Let X ∈ Tp (P) and g ∈ G, then −1 −1 Rg∗ ω (X) = ω (Rg∗ X) = gepg Ω (Rg∗ X) gepg + gepg de gpg (Rg∗ X) ,

(2.22)

Rg∗ ωλ = adg−1 ωλ ,

(2.23)

−1 = g −1 g Observing that gepg = gep g and gegp ep−1 the first term on the RHS of (2.22) reduces to −1 −1 −1 −1 de gepg Ω (Rg∗ X) gepg = g gep Ω (Rg∗ X) gep g while the second term gives gepg gpg (Rg∗ X) = −1 −1 g gep d (Rg∗ X) gep g. We therefore conclude

where the adjoint map ad is defined by adg Y := Lg∗ ◦ Rg−1 ∗ ◦ Y = gY g −1 , adg−1 Y := g −1 Y g.

(2.24)

The potential ΩA can be obtained from ω as ΩA = s∗ ω. To demonstrate this, let Y ∈ Tp (M ) and ge be specified by the inverse diffeomorphism or trivialization map (2.9) with χ−1 eλ ) for p (x) = sλ (x) · geλ . We find s∗i ω (Y ) = ge−1 Ω (π∗ si∗ Y ) ge + λ (p) = (x, g −1 ge de g (si∗ Y ), where we have used si∗ Y ∈ Tsi (P), π∗ si∗ = (id)Tp (M ) and ge = (id)G at si implying ge−1 de g (si∗ Y ) = 0 [41]. Hence, s∗i ω (Y ) = Ω (Y ) .

(2.25)

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To determine the gauge transformation of the connection 1-form ω, we use the fact that Rge∗ X = Xe g for X ∈ Tp (M ) and the transition functions genm ∈ G defined between neighboring bundle charts (2.6). By direct computation we get d d cj (λ (t)) |t=0 = [ci (λ (t)) · geij ] |t=0 dt dt  # −1 = Rgeij ∗ c∗i (X) + geji (x) de gij (X) ,

cj∗ X =

where λ (t) is a curve in M with boundary values λ (0) = m and we obtain the useful result

d dt λ (t) |t=0


# c∗ X = Rge∗ (c∗ X) + ge−1 de g (X) .

(2.26) = X. Thus,

(2.27)

Applying ω to Eq.(2.27), we get

ω (c∗ X) = c∗ ω (X) = adge−1 c∗ ω (X) + ge−1 de g (X) , ∀X.

(2.28)

Ω → Ω′ = adge−1 (d + Ω) = ge−1 (d + Ω) ge.

(2.29)

ω → ω ′ = ge−1 (d + ω) ge.

(2.30)

Hence, the gauge transformation of the local gauge potential Ω reads,

Since Ω = c∗ ω we obtain, from Eq.(2.29), the gauge transformation law of ω

2.2.

A Generalized Bundle Structure for Gravitation

Let us recall the definition of gauge transformations in the context of ordinary fiber bundles. Given a principal fiber bundle P(M , G; π) with base space M and standard G-diffeomorphic fiber, gauge transformations are characterized by bundle isomorphisms λ : P → P exhausting all diffeomorphisms λM on M [44]. This mapping is called an automorphism of P provided it is equivariant with respect to the action of G. This amounts to restrict the action λ of G along local fibers leaving the base space unaffected. Indeed, with regard to gauge theories of internal symmetry groups, a gauge transformation is a fiber preserving bundle automorphism, i.e. diffeomorphisms λ with λM = (id)M . The automorphisms λ form a group called the automorphism group AutP of P. The gauge transformations form a subgroup of AutP called the gauge group G (AutP ) (or G in short) of P. The map λ is required to satisfy two conditions, namely its commutability with the right action of G [the equivariance condition λ (Rg (p)) = λ (pg) = λ (p) g] λ ◦ Rg (p) = Rg (p) ◦ λ, p ∈ P, g ∈ G

(2.31)

according to which fibers are mapped into fibers, and the verticality condition π ◦ λ (u) = π (u) ,

(2.32)

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where u and λ (u) belong to the same fiber. The last condition ensures that no diffeomorphisms λM : M → M given by λM ◦ π (u) = π ◦ λ (u) ,

(2.33)

be allowed on the base space M . In a gauge description of gravitation, one is interested in gauging external transformation groups. This means that the group action on spacetime coordinates cannot be neglected. The spaces of internal fiber and external base must be interlocked in the sense that transformations in one space must induce corresponding transformations in the other. The usual definition of a gauge transformation, i.e. as a displacement along local fibers not affecting the base space, must be generalized to reflect this interlocking. One possible way of framing this interlocking is to employ a nonlinear realization of the gauge group G, provided a closed subgroup H ⊂ G exists. The interlocking requirement is then transformed into the interplay between groups G and one of its closed subgroups H. Let us denote by G a Lie group with elements {g}. Let H be a closed subgroup of G specified by [37, 67] H := {h ∈ G|Π (Rh g) = π (g) , ∀g ∈ G} ,

(2.34)

with elements {h} and known linear representations ρ (h). Here Π is the first of the two projection maps in (2.37), and Rh is the right group action. Let M be a differentiable manifold with points {x} to which G and H may be referred, i.e. g = g(x) and h = h(x). Being that G and H are Lie groups, they are also manifolds. The right action of H on G induces a complete partition of G into mutually disjoint orbits gH. Since g = g(x), all elements of gH = {gh1 , gh2 , gh3 , · · · , ghn } are defined over the same x. Thus, each orbit gH constitutes an equivalence class of point x, with equivalence relation g ≡ g ′ where g ′ = Rh g = gh. A comment is in order at this point: this result directly leads toward the Many Worlds Interpretation [77] of Open Quantum Mechanics as we will see below. By projecting each equivalence class onto a single element of the quotient space M := S G/H, the group G becomes organized as a fiber bundle in the sense that G = i {gi H}. In this manner the manifold G is viewed as a fiber bundle G (M, H; Π) with H-diffeomorphic fibers Π−1 (ξ) : G → M = gH and base space M. A composite principal fiber bundle P(M , G; π) is one whose G-diffeomorphic fibers possess the fibered structure G (M, H; Π) ≃ M× H described above. The bundle P is then locally isomorphic to M × G (M, H). Moreover, since an element g ∈ G is locally homeomorphic to M × H the elements of P are - by transitivity - also locally homeomorphic to M × M × H ≃ Σ × H where (locally) Σ ≃ M × M. Thus, an alternative view of P(M , G; π) is provided by the P-associated H-bundle P(Σ, H; π e) [37]. The total space P may be regarded as G (M, H; Π)-bundles over the base space M or equivalently as H-fibers attached to the manifold Σ ≃ M × M. The nonlinear realization (NLR) technique [19, 20] provides a way to determine the transformation properties of fields defined on the quotient space G/H. The NLR of Diff(4, R) becomes tractable due to a theorem given by V. I. Ogievetsky. According to this theorem [23], the algebra of the infinite dimensional group Diff(4, R) can be taken as the closure of the finite dimensional algebras of SO(4, 2) and A(4, R). Remind that the Lorentz

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group generates transformations that preserve the quadratic form on Minkowski spacetime built from the metric tensor, while the special conformal group generates infinitesimal angle-preserving transformations on Minkowski spacetime. The affine group is a generalization of the Poincar´e group where the Lorentz group is replaced by the group of general linear transformations. As such, the affine group generates translations, Lorentz transformations, volume preserving shear and volume changing dilation transformations. As a consequence, the NLR of Diff(4, R) /SO(3, 1) can be constructed by taking a simultaneous realization of the conformal group SO(4, 2) and the affine group A(4, R) := R4 ⋊ GL(4, R) on the coset spaces A(4, R)/SO(3, 1) and SO(4, 2)/SO(3, 1). One possible interpretation of this theorem is that the conformal-affine group CA (defined below) may be the largest subgroup of Diff(4, R), whose transformations may be put into the form of a generalized coordinate transformation. We remark that a NLR can be made linear by embedding the representation in a sufficiently higher dimensional space. Alternatively, a linear group realization becomes nonlinear when subject to constraints. One type of relevant constraints may be those responsible for symmetry reduction from Diff(4, R) to SO(3, 1) for instance. We take the group CA(3, 1) as the basic symmetry group G. The CA group consists of the groups SO(4, 2) and A(4, R). In particular, CA is proportional to the union SO(4, 2) ∪ A(4, R). We know however (see section Conformal-Affine Lie Algebra) that the affine and special conformal groups have several group generators in common. These common generators reside in the intersection SO(4, 2) ∩ A(4, R) of the two groups, within which there are two copies of Π := D × P (3, 1), where D is the group of scale transformations (dilations) and P (3, 1) := T (3, 1) ⋊ SO(3, 1) is the Poincar´e group. We define the CA group as the union of the affine and conformal groups minus one copy of the overlap Π, i.e. CA(3, 1) := SO(4, 2) ∪ A(4, R) − Π. Being defined in this way we recognize that CA(3, 1) is a 24 parameter Lie group representing the action of Lorentz transformations (6), translations (4), special conformal transformations (4), spacetime shears (9) and scale transformations (1). In this paper, we obtain the NLR of CA(3, 1) modulo SO(3, 1) as a 4D realization of the embedding procedure adopted in Open Quantum Relativity [1, 2].

2.3.

The Conformal-Affine Lie Algebra

In order to implement the NLR procedure, we choose to partition Diff(4, R) with respect to the Lorentz group. By Ogievetsky’s theorem [23], we identify representations of Diff(4, R)/SO(3, 1) with those of CA(3, 1)/SO(3, 1). The 20 generators of affine transformations can be decomposed into the 4 translational PAff µ and 16 GL(4, R) transformaβ β tions Λα . The 16 generators Λα may be further decomposed into the 6 Lorentz generators Lαβ plus the remaining 10 generators of symmetric linear transformation Sαβ , that is, Λαβ = Lαβ + Sαβ . The 10-parameter symmetric linear generators Sαβ can be factored into β 1 β † β the 9-parameter shear (the traceless part of Sαβ ) generator   defined by Sα = Sα − 4 δα D,

and the 1-parameter dilaton generator D = tr Sαβ . Shear transformations generated

by † Sαβ describe shape changing, volume preserving deformations, while the dilaton generator gives rise to volume changing transformations. The four diagonal elements of Sαβ correspond to the generators of projective transformations. The 15 generators of conformal transformations are defined in terms of the set {JAB } where A = 0, 1, 2,..5. The elements

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JAB can be decomposed into translations PConf := J5µ +J6µ , special conformal generators µ ∆µ := J5µ − J6µ , dilatons D := J56 and the Lorentz generators Lαβ := Jαβ . The Lie algebra of CA(3, 1) is characterized by the commutation relations [Λαβ , D] = [∆α , ∆β ] = 0, [Pα , Pβ ] = [D, D] = 0, [Lhαβ , Pµ ] =i ioµ[α Pβ] , [L h αβ , ∆γ ]i= io[α|γ ∆|β] , Λαβ , Pµ = iδµα Pβ , Λαβ , ∆µ = iδµα ∆β ,

[Sαβ , Pµ ] = ioµ(α Pβ) , [Pα , D] = −iP
α , [Lαβ , Lµν ] = −i oα[µ Lν]β − oβ[µ Lν]α
, [Sαβ , Sµν ] = i oα(µ Lν)β − oβ(µ Lν)α
, [Lαβ , Sµν ] = i oα(µ Sν)β − oβ(µ Sν)α , [∆αh, D] = i∆ i α , [S µν , ∆α ] = ioα(µ∆ν) ,

(2.35)

Λαβ , Λµν = i δνα Λµβ − δβµ Λαν , [Pα , ∆β ] = 2i (oαβ D − Lαβ ) ,

where oαβ = diag (−1, 1, 1, 1) is the Lorentz group metric.

2.4.

Group Actions and Bundle Morphisms

Let us now introduce the main ingredients required to specify the structure of the fiber bundle which we adopt, namely the canonical projection, sections etc. We follow the prescriptions in [37] for constructing the composite fiber bundle, but implementing the program for the CA group. The composite bundle P(Σ, H; π e) is comprised of H-fibers, base space Σ (M , M) and a composite map def π e=π eΣM ◦ ΠPΣ : P → Σ → M , (2.36) with component projections

ΠPΣ : P → Σ, π eΣM : Σ → M .

(2.37)

The projection ΠPΣ maps the point (p ∈ P, Rh p ∈ P) into point (x, ξ) ∈ Σ. There is a correspondence between the sections sM Σ : M → Σ and the projection ΠPΣ : P → Σ, in the sense that both maps project their functional argument onto elements of Σ. This is formalized by the relation, ΠPΣ (p) = sM Σ ◦ πPM (p). Hence, the total projection is given by π e := πPM = π eΣM ◦ ΠPΣ . (2.38) Associated with the projections π eΣM and ΠPΣ are the corresponding local sections −1 sM Σ : U → π eΣM (U) ⊂ Σ, sΣP : V → Π−1 PΣ (V) ⊂ P,

(2.39)

π eΣM ◦ sM Σ = (id)M , ΠPΣ ◦ sΣP = (id)Σ .

(2.40)

with neighborhoods U ⊂ M and V ⊂ Σ satisfying

The bundle injection π e−1 (U) is the inverse image of π e (U) and is called the fiber over U. −1 The equivalence class Rh p = pH ∈ π eΣM (U) of left cosets is the fiber of P (Σ, H), while

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each orbit pH through p ∈ P projects into a single element Q ∈ Σ. In analogy to the total bundle projection (2.37), a total section of P is given by the total section composition sM P = sΣP ◦ sM Σ .

(2.41)

Let elements of G/H be labeled by the parameter ξ. Functions on G/H are represented by continuous coset functions c(ξ) parameterized by ξ. These elements are referred to as cosets to the right of H with respect to g ∈ G. Indeed, the orbits of the right action of H −1 on G are the left cosets Rh g = gH. For a given section sM P (x ∈ M ) ∈ πPM , with local coordinates (x, g), one can perform decompositions of the partial fibers sM Σ and sΣP as: sM Σ (x) = e cM Σ (x) · c = Rc′ ◦ e cM Σ (x) ; c = c (ξ) ,

(2.42)

sΣP (x, ξ) = e cΣP (x, ξ) · a′ = Ra′ ◦ e cΣP (x, ξ) ; a′ ∈ H,

(2.43)

e cM P = e cΣP ◦ e cM Σ .

(2.44)

sM P (x) = e cM P (x) · g = Rg ◦ e cM P (x)

(2.45)

with the null sections {e cM Σ (x)} and {e cΣP (x, ξ)} having coordinates (x, (id)M ) and (x, ξ, (id)H ) respectively. A null or zero section is a map that sends every point x ∈ M to the origin of the fiber π −1 (x) over x, i.e. χ−1 c (x)) = (x, 0) in any trivialization. i (e The trivialization map χ−1 is defined in Eq.(2.9). The identity map appearing in the above i trivializations are defined as (id)M : M → M and (id)H : H → H. We assume the total null bundle section be given by the composition law

The images of two sections sΣP and sM Σ over x ∈ M must coincide, implying sΣP (x, ξ) = sM Σ (x). Using Eq. (2.41) with Eqs.(2.42), (2.43) and (2.44), we arrive at the total bundle section decomposition

provided g = c · a and

e cΣP = Rc−1 ◦ e cΣP (x, ξ) ◦ Rc .

The pullback of e cΣP , defined as

(2.46)

e cξ (x) = (s∗M Σ e cΣP ) (x) = e cΣP ◦ sM Σ = e cΣP (x, ξ) ,

(2.47)

e cΣP (x, ξ) = e cM P (x) · c (ξ) .

(2.48)

ensures the coincidence between the images of sections e cξ (x) : M → P and e cΣP (x, ξ) : Σ → P, respectively. With the aid of the above results, we arrive to the important relation which will be widely used below.

2.5.

Nonlinear Realizations and Generalized Gauge Transformations

A generalized gauge transformation is obtained by comparing bundle elements p ∈ P that differ by the left action of elements of the principal group G, Lg∈G . An arbitrary element p ∈ P can be written in terms of the null section with the aid of (2.45), (2.46) and (2.48) as p = sM P (x) = Ra ◦ e cΣP (x, ξ) , a ∈ H.

(2.49)

The Conformal-Affine Structure of open Quantum Relativity...

213

Performing a gauge transformation on p we obtain the orbit λ (p) defining a curve through (x, ξ) in Σ
λ (p) = Lg(x) ◦ p = Ra′ ◦ e cΣP x, ξ ′ ; g (x) ∈ G, a′ ∈ H. (2.50)

Comparison of Eq.(2.49) with Eq.(2.50) leads to


Lg(x) ◦ Ra ◦ e cΣP (x, ξ) = Ra′ ◦ e cΣP x, ξ ′ .

(2.51)

By virtue of the commutability of left and right group translations of elements belonging to G, i.e. Lg ◦ Rh = Rh ◦ Lg , Eq.(2.51) may be recast as
Lg(x) ◦ e cΣP (x, ξ) = Rh ◦ e cΣP x, ξ ′ . (2.52)

where Ra−1 ◦ Ra′ ≡ Ra′ a−1 := Rh and a′ a−1 ≡ h ∈ H (see also [41]). Eq.(2.52) constitutes a generalized gauge transformation. Performing the pullback of Eq.(2.52) with respect to the section sM Σ leads to Lg(x) ◦ e cξ (x) = Rh(ξ, g(x)) ◦ e cξ′ (x) .

(2.53)

Thus, the left action Lg of G is a map that acts on P and Σ. In particular, Lg , acting on fibers defined as orbits of the right action describes, diffeomorphisms that transforming fibers over e cξ (x) into the fibers e cξ′ (x) of Σ, while being simultaneously displaced along H fibers, via the action of Rh . Eq.(2.53) states that nonlinear realizations of G mod H is determined by the action of an arbitrary element g ∈ G, on the quotient space G/H, transforming one coset into another as Lg : G/H → G/H, c(ξ) → c(ξ ′ ) (2.54) inducing a diffeomorphism ξ → ξ ′ on G/H. To simplify the action induced by Eq. (2.53) for calculation purposes we proceed as follows. Starting from (2.47) and substituting sM Σ = Rc ◦ e cM P , we get e cξ (x) = e cΣP ◦ Rc ◦ e cM Σ . (2.55)

Using e cM P ◦ Rc = Rc ◦ e cM P , Eq.(2.55) becomes e cξ (x) = Rc ◦ e cΣP ◦ e cM Σ = Rc ◦ e cM P , where the last equality follows from the use of e cM P = e cΣP ◦ e cM Σ . By analogy, we assume e cξ′ (x) ≡ Rc′ ◦ e cM P . Upon substitution of e cξ′ into Eq.(2.53), we obtain Lg ◦ Rc ◦ e cM P = Rh(ξ, g(x)) ◦ Rc′ ◦ e cM P ,

(2.56)

g·e cM P · c = e cM P · c′ · h.

(2.57)

c′ = g · c · h−1

(2.58)

which, after implementing the group actions, is equivalent to

Operating on Eq.(2.57) from the left by e c−1 c−1 cM P , we get M P and making use of g = e M P ge
−1 ′ cM P · c = c · h, which leads to g · cξ = cξ′ · h, or in short e cM P · g · e where c ≡ cξ and c′ ≡ cξ′ . Observe that the element h is a function whose argument is the couple (ξ, g (x)). The transformation rule (2.58) is, in fact, the key equation to determine the nonlinear realizations of G and specifies a unique H-valued field h(ξ, g (x)) on G/H.

214

G. Basini and S. Capozziello Consider a family of sections {b c (x, ξ)} defined on Σ [38] by b c (x, ξ) := c ◦ e c (x, ξ) = c (e c (x, ξ)) .

(2.59)

ΠPΣ ◦ Lg ◦ b cξ = ΠPΣ ◦ Rh(ξ, g(x)) ◦ b cξ′ = Rh(ξ, g(x)) ◦ e cξ ′ ,

(2.60)

Taking ΠPΣ ◦ Rh ◦ e cΣP = ΠPΣ ◦ e cΣP = (id)Σ into account, we can explicitly exhibit the fact that the left action Lg of G on the null sections, e cΣP : P → Σ induces an equivalence relation between differing elements e cξ , e cξ′ ∈ Σ given by so that

e c′ξ := Rh(ξ, g(x)) ◦ e cξ′ = Lg ◦ e cξ .

(2.61)

e cξ 7−→ e c′ξ = Rh(ξ, g(x)) ◦ e cξ′ ∀h ∈ H.

(2.62)

From Eq.(2.61) we can write

Lg

Eq. (2.62) gives rise to a complete partition of G/H into equivalence classes Π−1 PΣ (ξ) of left cosets [38, 42]  cH = Rh(ξ, g(x)) ◦ c/c ∈ G/H, ∀h ∈ H = {ch1 , ch2 ,..., chn } , (2.63)

where c ∈ (G − H) plays the role of the fibers attached to each point of Σ. The elements −1 chi are single representatives of each equivalence class Rh(ξ, g(x)) ◦ c = cH ∈ π eΣM (U). Thus, any diffeomorphism Lg ◦ e cξ on Σ, together with the H-valued function h (ξ, g (x)), determines a unique gauge transformation e c′ξ = Rh(ξ, g(x)) ◦ e cξ′ . This demonstrates that gauge transformations are those diffeomorphisms on Σ that map fibers over c (ξ) into fibers over c (ξ ′ ) and simultaneously preserves the action of H.

2.6.

Covariant Coset Field Transformations

We now proceed to determine the transformation behavior of parameters belonging to G/H. The elements of the CA and Lorentz groups are respectively parameterized about the identity element as αP

g = eiǫ

α

µν † S µν

eiα

eiβ

µν L

µν

α∆ α

eib

µν L

eiϕD , h = eiu

µν

.

(2.64)

Elements of the coset space G/H are ”coordinatized” by c = e−iξ

αP

α

µν † S µν

eih

eiζ

α∆ α

eiφD .

(2.65)

We consider transformations with infinitesimal group parameters ǫα , αµν , β µν , bα and ϕ. The transformed coset parameters read ξ ′α = ξ α +δξ α , h′µν = hµν +δhµν , ζ ′α = ζ α +δζ α and φ′ = φ+δφ. Note that uµν is infinitesimal. The translational coset field variations reads h i
δξ α = − αβα + ββα ξ β − ǫα − ϕξ α − |ξ|2 bα − 2 (b · ξ) ξ α . (2.66)

For the dilatons we get,

io h n δφ = ϕ + 2 (b · ξ) − uαβ ξ β + ǫα + ϕξ α + bα |ξ|2 − 2 (b · ξ) ξ α ∂α φ.

(2.67)

The Conformal-Affine Structure of open Quantum Relativity...

215

Similarly for the special conformal 4-boosts we find, δζ α = uαβ ζ β + bα − ϕζ α + 2 [(b · ξ) ζ α − (b · ζ) ξ α ] +

(2.68)

io n h − uβλ ξ λ + ǫβ + ϕξ β + bβ |ξ|2 − 2 (b · ξ) ξ β ∂β ζ α .

Observe that the homogeneous part of the special conformal coset parameter ζ α has the same structure as that of the translational parameter ξ α (with the substitutions: ζ α → −ξ α and −ǫα → bα ). For the shear parameters we obtain δrαβ = (αγα + β γα ) rγβ + uβγ rαγ + 2b[α ξ ρ] rρβ ,

(2.69)

αβ

where rαβ := eh . From δrαβ we obtain the nonlinear Lorentz transformation   1 h α −1
β i αβ αβ [α β] µν u = β + 2b ξ − α tanh . ln r µ r ν 2

(2.70)

In the limit of vanishing special conformal 4-boost, this result coincides with that of Pinto et al. [34]. For vanishing shear, the result of Julve et al [35] is obtained. We must stress that all covariant coset field transformations has been determined directly from the nonlinear transformation law (2.58). We observe that the translational coset parameter transforms as a coordinate under the action of G. From the shear coset variation, the explicit form of the nonlinear Lorentz-like transformation has been obtained. From (2.70) it is clear that uαβ contains the linear Lorentz parameter, in addition to conformal and shear contributions via the nonlinear 4-boosts and symmetric GL4 parameters.

2.7.

Decomposition of Connections in πPM : P → M into Components in πPΣ : P → Σ and πΣM : Σ → M

Depending on which bundle is considered, either the total bundle P → M or the intermediate bundles P → Σ, Σ → M , we may construct the corresponding Ehresmann connections for the respective space. With respect to M , we have the connection form ∗ ω = ge−1 (d + πPM ΩM ) ge.

(2.71)

∗ ω = a−1 (d + πPΣ ΓΣ ) a,

(2.72)

The gauge potential ΩM is defined in the standard manner as the pullback of the connection ω by the null section e cM P , ΩM = e c∗M P ω ∈ T ∗ (M ). With regard to the space Σ, an alternative form of the connection is given by where the connection on Σ reads ΓΣ = e c∗ΣP ω. Carrying out a similar analysis and evaluating the tangent vector X ∈ Tp (Σ) at each point ξ along the curve cξ on the coset space G/H (that coincides with the section e c∗ΣP ), we find the gauge transformation law ω → ω ′ = adh−1 (d + ω) .

(2.73)

216

G. Basini and S. Capozziello

∗ Γ = c−1 (d + π ∗ Ω ) c. Taking into Comparison of Eq.(2.71) and Eq.2.72 leads to πPΣ Σ PM M account e c∗ΣP Π∗PΣ = (id)T ∗ (Σ) which follows from ΠPΣ ◦ e cΣP = (id)Σ , we deduce   ∗ ΓΣ = e c∗ΣP c−1 (d + πPM ΩM ) c . (2.74)
By the family of sections pulled back to Σ introduced in Eq.(2.59), we find e c∗ΣP c−1 dc = b c −1 db ∗ ∗ ∗ ∗ ∗ ∗ ∗ −1 ∗ c and e cΣP Rc = Rbc e cΣP . Recalling π ePM = π ePΣ π eΣM , we get c π ePM ΩM c = ∗ Ω . With these results in hand, we obtain the alternative form of the connection Rc∗ π ePM M ΓΣ , ∗ ΓΣ = b c−1 (d + πΣM ΩM ) b c. (2.75)

Completing the pullback of ΓΣ to M , by means of e cM Σ , we obtain, ΓM = e c∗M Σ ΓΣ . By use c∗ξ ω. In terms of the substitution of ΓΣ = e c∗ΣP ω and (2.47), we find ΓM = s∗M Σ e c∗ΣP ω = e b c (x, ξ) → c (x) where c (x) is the pullback of b c (x, ξ) to M , defined as c (x) = s∗M Σ b c= c (e cξ (x)), we arrive at the desired result Γ ≡ ΓM = c−1 (d + ΩM ) c,

(2.76)

which explicitly relates the connection Γ, on Σ pulled back to M , to its counterpart ΩM . The gauge transformation behavior of Γ may be determined directly Eq. (2.29) and the transformation e c′ = ge ch−1 . We calculate

Γ′ = he c−1 g −1 d ge ch−1 + he c−1 Ωe ch−1 + he c−1 dg −1 ge ch−1 . (2.77) Observing however, that



−1 he c−1 g −1 d ge ch−1 = he c−1 g −1 dg e ch + he c−1 de ch−1 + hdh−1 ,

we obtain

 −1 
−1 ch . Γ′ = h e c (d + Ω) e c h−1 + hdh−1 + he c−1 d gg −1 e

(2.78) (2.79)

Thus, we arrive at the gauge transformation law

Γ′ = hΓh−1 + hdh−1 .

(2.80)

According to the Lie algebra decomposition of g into h and c, the connection ΓΣ can be divided into ΓH , defined on the subgroup H and ΓG/H , defined on G/H. From the transformation law (2.80), it is clear that ΓH transforms inhomogeneously as Γ′H = hΓH h−1 + hdh−1 ,

(2.81)

while ΓG/H transforms as a tensor Γ′G/H = hΓG/H h−1 .

(2.82)

In this regard, only ΓH transforms as a true connection. We use the gauge potential Γ to define the gauge covariant derivative ∇ := (d + ρ (Γ))

(2.83)

acting on ψ as ∇ψ = (d + ρ (Γ)) ψ with the desired transformation property

(∇ψ (c(ξ)))′ = ρ (h(ξ, g)) ∇ψ (c(ξ)) ≃ (1 + iu (ξ, g) ρ (H)) ∇ψ (c(ξ))

(2.84)

and leading to δ (∇ψ (c(ξ))) = iu (ξ, g) ρ (H) ∇ψ (c(ξ)) .

(2.85)

The Conformal-Affine Structure of open Quantum Relativity... 2.7.1.

217

Conformal-Affine Nonlinear Gauge Potential in πPM : P → M

The ordinary gauge potential defined on the total base space M reads   T C D GL α α αβ † Ω = −i Γ Pα + Γ ∆α + ΓD + Γ Λαβ .

(2.86)

The horizontal basis vectors, that span the horizontal tangent space H(P) of πPM : P →M are given by Ei = e cM P∗ ∂i − Ωi . (2.87) The explicit form of the connections (2.86) are given by     GL αβ µ ν µ eν αβ ν ν † ∗ ∗ ω = −i VM χ eµ Pν − i iΘ(† Λ) + π χ eα χ eβ Λµν + ϑM βµ ∆ν − ie πPM ΦM D ePM Γ where B)

αβ Θ(† Λ)

=

αβ Θ(L)

αβ + Θ(SY) ,

(2.88) with right invariant Maurer-Cartan forms (see the Appendix

µν µν Θ(L) = iβe[ν|γ dβe|µ]γ − 2idbµ ǫν and Θ(SY) = ie α(ν|γ de α|µ)γ .

(2.89)

The linear connection ΩM varies under the action of G as T

C

D

GL

δΩ = Ω′ − Ω = δ Γ µ Pµ + δ Γ µ ∆µ + δ ΓD + δ Γ where

T

GL

T

D

C

GL

C

D

βν †

Λβν

(2.90)

δ Γ µ = † Dǫµ − Γ α (ααµ + βαµ + ϕδαµ ) − Γǫµ , δ Γ µ = † Dbµ − Γ α (ααµ + βαµ − ϕδαµ ) + Γbµ , GL

δΓ

αβ

=

GL †D

D

ααβ

+

β αβ 

C

δ Γ = dϕ + 2 Γ




  T C [α β] [α β] + Γ b +Γ ǫ ,

αǫ α

T

−Γ

αb α



(2.91)

.

The components of ω on M are identified as spacetime quantities and are determined from the pullback of the corresponding (quotient space) quantities defined on Σ: µ µν ∗ VM = s∗M Σ VΣµ , ϑµM = s∗M Σ ϑµΣ , ΦM = s∗M Σ ΦΣ and Γµν M = sM Σ ΓΣ .

(2.92)

In the following, we depart from the alternative form of the connection ω a−1 (d + Π∗PΣ ΓΣ ) a, ∀ a ∈ H on Σ. 2.7.2.

Conformal-Affine Nonlinear Gauge Potential in πPΣ : P → Σ

The components of ω in P → Σ are oriented along the Lie algebra basis of H   ◦ L L αβ −1 ∗ ω=a d + ie πPΣ Γ Lαβ a = −iω αβ Lαβ , where

=

  L ρσ ∗ eαeβ ω αβ := iΘ(L) + π ePΣ Γρσ [L] β[ρ βσ] .

(2.93)

(2.94)

218 2.7.3.

G. Basini and S. Capozziello Conformal-Affine Nonlinear Gauge Potential on ΠΣM : Σ → M

The components of ω in ΠΣM : Σ → M are oriented [37] along the Lie algebra basis of the quotient space G/H belonging to Σ ∗ ω = −ia−1 (e πΣM VΣν Pν ) a = −iω µ Pµ ,

P

P

(2.95)

∆

∗ ω = −ia−1 (e πΣM ϑνΣ ∆ν ) a = −i ω µ ∆µ ,

∆

(2.96)

D

(2.97)

∗ ω = −ia−1 (e πΣM ΦΣ D) a = −iω[D] D,

  SY ∗ = −ia−1 π eΣM Υαβ Sαβ a = −i ω αβ Sαβ ,

SY

ω

where

∗ ωµ : =π VΣν βeνµ , ω eΣM P

∆µ

SY αβ

∗ ω[D] : = π eΣM ΦΣ , ω

∗ ϑνΣ βeνµ , := π eΣM

∗ α β := π ePΣ Υρσ α e(ρ α eσ) .

By direct computation, we obtain   µ µ αβ ΓCA Σ = −i VΣ Pµ + iϑΣ ∆µ + ΦΣ D + ΓΣ Λαβ .

(2.98)

(2.99) (2.100)

(2.101)

The nonlinear translational and special conformal connection coefficients VΣν and ϑνΣ read    C β ∗ φ β α σ β VΣ = π eΣM e υ (ξ) + r σ Γ Bα (ξ) , (2.102)  i h ∗ ϑβΣ = π eΣM e−φ υ β (ζ) + υ σ (ξ) Bσβ (ζ) ,

with υiβ

(ξ) :=

rσβ

GL † Di ξ σ

D

σ

T

+ Γi ξ + Γ

σ i

!

  , Bαρ (ξ) := |ξ|2 δαρ − 2ξα ξ ρ .

(2.103)

(2.104)

The nonlinear GL4 and dilaton connections are given by b µν + 2ζ [µ ̟ν] , Γµν Σ =Γ

with

and

b µν Γ

  1 ∗ Φ=π eΣM ζβ ̟β − dφ, 2  
GL σβ ν
∗ −1 µ −1 µ σν := π eΣM r Γ rβ − r dr σ σ

(2.105) (2.106)

(2.107)

C

̟ν := υ ν + rνα Γ α .

(2.108)

The Conformal-Affine Structure of open Quantum Relativity...

219

The nonlinear GL4 connection can be expanded in the GL4 Lie algebra according to Γαβ †Λ αβ

◦

= Γ αβ Lαβ + Υαβ † Sαβ , where ◦

b [αβ] + 2ζ [α ̟β] , Υαβ := Γ b (αβ) . Γ αβ Σ Σ := Γ

(2.109)

eµ : = eµi sM Σ∗ ∂i = ∂ξµ − eµi eei ,

(2.110)

The symmetric GL4 (shear) gauge fields Υ are distortion fields describing the difference between the general linear connection and the Levi-Civita connection, one of the differences between General Relativity and Open Quantum Relativity. We define the (group) algebra bases eν and hν dual to the translational and special conformal 1-forms V µ and ϑµ as

hµ : = hµi sM Σ∗ ∂i = ∂ζ µ − hµi e hi ,

(2.111)

with corresponding tetrad-like components   C µ µ φ α σ µ ei (ξ) = e υi (ξ) + r σ Γ i Bα (ξ) ,
hiµ (ξ, ζ) = e−φ υρµ (ζ) + υiσ (ξ) Bσµ (ζ) ,

(2.112) (2.113)

and basis vectors (on M )

    C GL D T µ α µ ν σ µ ν µ eej (ξ) = e cM Σ∗ ∂j − e rµ Γ jα ξ + Γj ξ + Γ j + Γ j r σ Bµ (ξ) ∂ξν φ

(2.114)

and

      GL D T GL C ρ α ρ µ σ α σ σ γ −φ µ e hj (ξ, ζ) = e cM Σ∗ ∂j + e r ρ Γ jα ζ + Γ j + r σ Γ jα ξ + Γj ξ + Γ j Bγ (ζ) ∂ζ µ .

(2.115)

υβ

υβ

Here (ζ) = (ξ → ζ), the orthogonality relations

Bβα (ζ)

=

Bρα (ξ

→ ζ). By definition, the basis vectors satisfy

D E  hj = 0, hV µ |eν i = δνµ , hϑµ |hν i = δνµ . VΣµ |e ej = 0, ϑµΣ |e

(2.116)

We introduce the dilatonic and symmetric GL4 algebra bases

i e fi ♭ := ∂φ − di dei , fµν := ∂αµν − fµν

(2.117)

with auxiliary soldering components di and fiµν , di = fiµν

=

GL † Di ξ ρ

ζσ rσρ r−1


µ

σ

GL

Γ

D

ρ

T

ρ i

C

+ Γi ξ + Γ + Γ

σβ ν i rβ

− r−1


µ

σ

ρ i

∂i rσν .

!

1 − ∂i φ, 2

(2.118)

(2.119)

220

G. Basini and S. Capozziello

The coordinate bases dej and fej read

and

dej (ξ, ζ, φ, h) := e cM Σ∗ ∂j −

fej (ξ, h) := e cM Σ∗ ∂j −



ζσ rσρ

GL † ρ Γ jγ ξ γ


(µ| GL |ν) r−1 σ Γ j σβ rβ

D

ρ

T

ρ j

C

+ Γj ξ + Γ + Γ

−


(µ| r−1 σ

∂j r

!

∂φ ,

(2.120)



∂hµν .

(2.121)

ρ j

σ|ν)

The bases satisfy E E D D E D Φ|dei = 0, Υαβ |fei = 0, hΦ|♭i = I, Υαβ |fµν = δµα δνβ .

(2.122)

With the basis vectors and tetrad components in hand, we observe µ VM := dxi ⊗ eµi , ϑµM := dxi ⊗ hµi ,

(2.123)

ΦM := dxi ⊗ eαi hΦ|eα i = dxi ⊗ di . The symmetric and antisymmetric GL4 connection pulled back to M is given by

µν  µν i i α Υµν M = dx ⊗ ei ΥΣ |eα := dx ⊗ fi , ◦

Γ

µν M

=

dxi

⊗

eαi



◦

Γ

µν Σ |eα



:=

dxi

◦

⊗Γ

µν i .

With the aid of Eq.(2.123) and Eq.(2.124), we determine D E µν Viβ := eiα VΣβ |eα = eiα δαβ = eiβ , ϑβi ≡ hβi , Υµν i ≡ fi , Φi ≡ di . The horizontal tangent subspace vectors in π ePΣ : P → Σ are given by  Int  ◦ αβ b (L) , b Ei = e cM P∗ eei + ie cM Σ∗ Γ |e ei R αβ

and satisfy

 Int  ◦ αβ b (L) , bµ = e E cΣP∗ eeµ + i Γ |e eµ R αβ DL E DL E bj = 0 = ω| bµ . ω|E E

(2.124)

(2.125)

(2.126)

(2.127)

(2.128)

The right invariant fundamental vector operator appearing in (2.126) or (2.127) is given by   ∂ ∂ γ (L) e b + ǫ[µ ν] . (2.129) R µν = i β[µ| ∂ǫ ∂ βe|ν]γ On the other hand, the vertical tangent subspace vector in π ePΣ : P → Σ satisfies DL E DL E b (L) = Lµν = ω|R b (L) , ω|L µν µν

(2.130)

The Conformal-Affine Structure of open Quantum Relativity... where b (L) = iβeγ[µ| L µν

  ∂ ∂ ∂ γ (L) e b , R µν = i β[µ| + ǫ[µ ν] . |ν] ∂ǫ ∂ βe|ν]γ ∂ βeγ

ν and βeµν := eβµ = δµν + βµν + ΠΣM : Σ→M are given by

and satisfy

1 γ ν 2! βµ βγ

221

(2.131)

+ · · ·. The horizontal tangent subspace vectors in ⌣

ej = e ej = e b (D) = e E cΣP∗ eej , H cΣP∗e hj , E cΣP∗ dej , E j = e cΣP∗ fej , i

DP E D E D ⌣ E DD E ej = 0, ∆ e j = 0, SY b (D) = 0. ω|E ω|H ω |E j = 0, ω|E i

(2.132)

(2.133)

The vertical tangent subspace vectors in ΠΣM : Σ→M are given by ⌣

b (SY) , H b (∆) , E b (D) , b (P) , E αβ = e eµ = e eµ = e b (D) = e E cΣP∗ L cΣP∗ L cΣP∗ L cΣP∗ L µ µ αβ

and satisfy DP E D DD E E D ⌣ E eµ = Pµ , ∆ e µ = ∆µ , SY b (D) = D. ω|E ω|H ω |E αβ = † Sαβ , ω|E

(2.134)

(2.135)

The left invariant fundamental vector operators appearing in (2.134) are readily computed, the result being b (P) f νµ ∂ν , e ν ∂ b (∆) = iW L µ = iQ µ ∂ǫν , L µ ∂b (2.136) (SY) ∂ ∂ (D) β b b L αβ = ie αγ(µ| |ν) , L = −iǫ ∂ǫβ , ∂α eγ

αµν

e α := (e fσα := χσα + δσα eϕ ), W = αµν + αµν + 2!1 αµγ αγν + · · ·, Q  α   σα e ασ and W f ασ . Making use of the e −1 f −1 (e χσα + δσα e−ϕ ) satisfying Q = Q = W σ σ transformation law of the nonlinear connection (2.80), we obtain

where α eµν := e

δΓ = δV α Pα + δϑα ∆α + 2δΦD + δΓαβ † Λαβ

where

(2.137)

GL

δV ν = uαν V α , δϑν = uαν ϑα , δΦ = 0, δΓαβ = † ∇uαβ .

(2.138)

GL

From δΓαβ = † ∇uαβ we observe that ◦

δΓ[αβ] = ∇uαβ , δΥαβ = 2uρ (α| Υρ|β) .

(2.139)

According to Eq.(2.138), the nonlinear translational and special conformal gauge fields transform as contravariant vector valued 1-forms under H, the antisymmetric part of Γαβ transforms inhomogeneously as a gauge potential and the nonlinear dilaton gauge field Φ transforms as a scalar valued 1-form. From Eq.(2.139), it is clear that the symmetric part of Γαβ is a tensor valued 1-form. Being characterized as 4-covectors, we identify V ν as ◦

coframe fields. The connection coefficient Γ αβ serves as the gravitational gauge potential. The remaining components of Γ, namely ϑ, Υ and Φ are dynamical fields of the theory. As will be seen in the following subsection, the tetrad components of the coframe are used in conjunction with the H-metric to induce a spacetime metric on M . In open Quantum Relativity, this result is extremely important and directly depends on the intrinsic covariant symplectic structure of the theory [4].

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The Induced Metric

Since the Lorentz group H is a subgroup of G, we inherit the invariant (δoαβ = δoαβ = 0) (constant) metric of H, where oαβ = oαβ = diag (−, + , + , +). With the aid of oαβ and the tetrad components eiα given in (2.112), we define the spacetime metric gij = eiα ejβ oαβ .

(2.140)

GL

Observing † ∇oαβ = −2Υαβ (where we used doαβ = 0) and taking account of the (second) transformation property (2.139), we interpret Υαβ as a sort of nonmetricity, i.e. a deformation (or distortion) gauge field that describes the difference between the general linear connection and the Levi-Civita connection of Riemannian geometry. In the limit of T

vanishing gravitational interactions, we obtain Γ GL

σ

C

∼ Γ

σ

◦

∼ Γ

α β

∼ Υαβ ∼ Φ → 0,

rσβ → δσβ (to first order) and † Dξ σ → dξ σ . Under these conditions, the coframe reduces to V β → eφ δαβ dξ α leading to the spacetime metric     (2.141) gij → e2φ δαρ δβσ (∂i ξ α ) ∂j ξ β oρσ = e2φ (∂i ξ α ) ∂j ξ β oαβ

characteristic of any Weyl geometry.

2.9.

The Cartan Structure Equations

Using the nonlinear gauge potentials derived in (2.103), (2.105), (2.106), the covariant derivative defined on Σ pulled back to M has form ∇ := d − iV α Pα − iϑα ∆α − 2iΦD − iΓαβ † Λαβ .

(2.142)

By use of (2.142), together with the Lie algebra commutators, very relevant for the symplectic structure, we obtain the bundle curvature F := ∇ ∧ ∇ = −iT α Pα − iKα ∆α − iZD − iRαβ † Λαβ .

(2.143)

The field strength components of F are given by the first Cartan structure equations. They are respectively, the projectively deformed, Υ-distorted translational field strength GL

T α := † ∇V α + 2Φ ∧ V α ,

(2.144)

the projectively deformed, Υ-distorted special conformal field strength GL

Kα := † ∇ϑα − 2Φ ∧ ϑα ,

(2.145)

the Ψ-deformed Weyl homothetic curvature 2-form (dilaton field strength) Z := dΦ + Ψ, Ψ = V · ϑ − ϑ · V

(2.146)

and the general CA curvature b αβ + Ψαβ , Rαβ := R

(2.147)

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223

with GL

b αβ := Rαβ + Rαβ , Ψαβ := V [α ∧ ϑβ] . R

(2.148)

The operator † ∇ denotes the nonlinear covariant derivative built from volume preserving ◦

(VP) connection (i.e. excluding Φ) forms. The Υ and Γ-affine curvatures in (2.148) read ◦

Rαβ : = ∇Υαβ + Υαγ ∧ Υγβ , ◦

◦

(2.149)

◦

Rαβ : = dΓ αβ + Γ γα ∧ Γ γβ , ◦

(2.150) ◦

respectively. Operator ∇ is defined with respect to the restricted connection Γ αβ given in (2.109). The field strength components of the bundle curvature have the following group variations δRαβ = uαγ Rβγ − uγβ Rαγ , δZ = 0, δT α = −uβα T β , δKα = −uβα Kβ .

(2.151)

A gauge field Lagrangian is built from polynomial combinations of the strength F defined as F (Γ (Ω, Dξ) , dΓ) := ∇ ∧ ∇ = dΓ + Γ ∧ Γ. (2.152)

2.10.

Bianchi Identities

In what follows, the Bianchi identities (BI) play a central role. As we will see, they constitute the core of the General Conservation Principle. 1a) The 1st translational BI reads, GL

b α ∧ V β + Φ ∧ T a + 2d (Φ ∧ V α ) . ∇T a = R β

(2.153)

1b) Similarly to the case in (1a), the 1st conformal BIs are respectively given by, GL

◦

b α ∧ ϑβ − Φ ∧ Ka − 2d (Φ ∧ ϑα ) , ∇Ka = R β

(2.154)

2a) The Υ and Γ-affine component of the 2nd BI is given by †

GL

GL

∇Rαβ = 2R(α|γ Υγ|β) , † ∇Rαβ = 0,

(2.155)

respectively. Hence, the generalized 2nd BI is given by GL

b α = 2R(α|γ Υγ|ρ) oρβ . ∇R β

(2.156)

∇Ψαβ = † T α ∧ ϑβ + V α ∧ † Kβ ,

(2.157)

†

Since the full curvature Rαβ is proportional to Ψαβ , it is necessary to consider †

GL

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from which we conclude †

GL

∇Rαβ = 2R(α|γ Υγ|β) + † T α ∧ ϑβ + V α ∧ † Kβ .

(2.158)

2c) The dilatonic component of the 2nd BI is given by GL

GL

GL

∇Z = dZ + ∇ (V ∧ ϑ) = ∇Ψ + Φ ∧ Ψ,

(2.159)

From the definition of Ψ, we obtain ∇Ψ = T α ∧ ϑα + Vα ∧ Kα + Φ ∧ (Vα ∧ ϑα ) .

(2.160)

Defining Σµν := Bµν + Ψµν , Bµν := B µν + B µν , B µν := V µ ∧ V ν , B µν := ϑµ ∧ ϑν , (2.161) and asserting V α ∧ ϑα = 0, we find Σµν ∧ Σµν = 0. Using this result,we obtain ∇Ψ = T α ∧ ϑα + Vα ∧ Kα .

2.11.

(2.162)

Action Functional and Field Equations

We seek for an action for a local gauge theory based on the CA (3, 1) symmetry group. We consider the 3D topological invariants Y of the non-Riemannian manifold derived from CA connections. Our objective is the 4D boundary terms B, obtained by means of exterior differentiation of these 3D invariants, i.e. B = dY. The Lagrangian density of CA gravity is modelled after B, with appropriate distribution of Lie star operators, so as to re-introduce the dual frame fields. The generalized CA surface topological invariant reads     b a + 1 Aab ∧ A c ∧ Aca + θA Aab ∧ R b b 3 1   Y=− 2 (2.163) , 2l α −θV Va ∧ T + θΦ Φ ∧ Z where Tα := T α + Kα . The associated total CA boundary term is given by,   bβα ∧ Bβα + Σ[βα] ∧ Σ[βα] − R b αβ ∧ R bαβ − Z ∧ Z+ R    1  α α α α B= 2 +Kα ∧ K + Tα ∧ T − Φ ∧ (Vα ∧ T + ϑα ∧ K ) +    2l  
α β α β −Υαβ ∧ V ∧ T + ϑ ∧ K .

(2.164)

Using the boundary term (2.164) as a guide, we choose [48, 51, 54, 56, 66] an action of form   b αβ ∧ Σ⋆αβ + B⋆αβ ∧ Bαβ + Ψ⋆αβ ∧ Ψαβ + η⋆αβ ∧ η αβ  d (V α ∧ Tα ) + R       Z    1 µν α α I= − 2 (R⋆µν ∧ R + Z ∧ ⋆Z) + T⋆α ∧ T + K⋆α ∧ K +  M      
  ⋆α ⋆α α ⋆β α ⋆β −Φ ∧ (T ∧ Vα + K ∧ ϑα ) − Υαβ ∧ V ∧ T + ϑ ∧ K . (2.165)

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Note that the action integral (2.165) is invariant under Lorentz rather than CA transformations. The Lie star ⋆ operator is defined as ⋆Vα = 3!1 ηαβµν V β ∧ V µ ∧ V ν . The field equations are obtained from variation of I with respect to the independent gauge potentials. It is convenient to define the functional derivatives GL

V

GL

ϑ

δLgauge δV α

:= − ∇Nα + Tα ,

δLgauge δϑα

:= − ∇Mα + Tα ,

Zαβ := where

δLgauge bα δΓ β

(2.166)

GL

c αβ + E b αβ . = − † ∇M

V ϑ c α := − ∂Lgauge , E b αβ := ∂Lgauge , Tα := ∂Lgauge , Tα := ∂Lgauge , Θ := ∂Lgauge . M β β α bα b α ∂V ∂ϑα ∂Φ ∂Γ ∂R β (2.167) The gauge field momenta are defined by

Nα := −

∂Lgauge ∂T α ,

Mα := −

∂Lgauge ∂Kα ,

Ξ := −

∂Lgauge ∂Z ,

∂Lgauge c c[αβ] := Nαβ = −o[α|γ ∂Lgauge M |β] , M(αβ) := Mαβ = −2o(α|γ |β) . ∂Rγ

(2.168)

∂Rγ

Furthermore, the shear (gauge field deformation) and hypermomentum current forms are given by

b(αβ) := Uαβ = −V(α ∧ Mβ) + Nβ) − Mαβ , E b[αβ] := Eαβ = −V[α ∧ Mβ] + Nβ] , E (2.169) The analogue of the Einstein equations read GL

V

Gα + Λb ηα + † ∇T⋆α + Tα = 0, with Einstein-like three-form   Gα = Rβγ + Υ[β|ρ ∧ Υ|γ]ρ ∧ (ηβγα + ⋆ [Bβγ ∧ ϑα ]) ,

(2.170)

(2.171)

coupling constant Λ and mixed three-form ηbα = ηα + ⋆ (ϑα ∧ Vβ ) ∧ V β . Observe that Gα includes symmetric GL4 (Υ) as well as special conformal (ϑ) contributions. The gauge V

field 3-form Tα is given by V

Tα =

D E 

Lgauge |eα + hZ|eα i ∧ Ξ + T β |eα ∧ Nβ +

D E D E 1D β E γ Rγ |eα M β , + Kβ |eα ∧ Mβ + Rγβ |eα ∧ N γβ + 2

(2.172)

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We remark that to interpret (2.171) as the gravitational field equation analogous to the Einstein equations, we must transform from the Lie algebra index α to the spacetime basis index k, by contracting over the former (α) with the CA tetrads eαk . D E D E V Tα = Tα [T ] + Tα [K] + Tα [R] + Tα [Z] − T β |eα ∧ Nβ − Kβ |eα ∧ M(2.173) β + D E b αβ + − Rγβ |eα ∧ N γβ − hZ|eα i ∧ Ξ + Ψ⋆αβ ∧ ϑβ + hΣ⋆γβ |eα i ∧ R D E D E b γβ |eα + + Υγβ ∧ (Vγ ∧ T⋆β + ϑγ ∧ K⋆β ) |eα + Σ⋆γβ ∧ R D E B⋆γβ ∧ B γβ |eα + hB⋆γβ |eα i ∧ Bγβ + hΨ⋆γβ |eα i ∧ Ψγβ

respectively, with

Tα [R] = 21 a1 (Rργ ∧ hR⋆ργ |eα i − hRργ |eα i ∧ R⋆ργ ) , Tα [T ] = 21 a2 (Tγ ∧ hT ⋆γ |eα i − hTγ |eα i ∧ T ⋆γ ) , Tα [K] =

1 2 a3 (Kγ

∧

hK⋆γ |eα i

− hKγ |eα i ∧

(2.174)

K⋆γ ) ,

Tα [Z] = 21 a4 (dΦ ∧ h⋆dΦ|eα i − hdΦ|eα i ∧ ⋆dΦ) . From the variation of I with respect to ϑα we get ϑ

GL

Gα + Λb ωα + † ∇K⋆α + Tα = 0, where, in analogy to Eq.(2.171), we have   Gα = hαi Rβγ + Υ[β|ρ ∧ Υ|γ]ρ ∧ (ωβγα + ⋆ [Bβγ ∧ Vα ]) , ϑ

(2.175)

(2.176)

ϑ

with ω bα = ωα + ⋆ (ϑα ∧ Vβ ) ∧ ϑβ . The quantity Ti = hαi Tα is similar to (2.172) but with the algebra basis eα replaced by hα and the CA tetrad components eαi replaced by hαi . The two gravitational field equations (2.171) and (2.176) are P − ∆ symmetric. We may say that they exhibit P − ∆ duality symmetry invariance. ◦

From the variational equation for Γ αβ we obtain the CA gravitational analogue of the Yang-Mills-torsion type field equation,   ◦ ◦ ∇ ⋆ Rαβ + ∇ ⋆ Σαβ + V β ∧ T⋆α + ϑβ ∧ K⋆α = 0.

(2.177)

∇ ⋆ Σαβ − Υ(α|γ ∧ Σ⋆γ|β) + V(α ∧ T⋆β) + ϑ(α ∧ K⋆β) = 0.

(2.178)

Variation of I with respect to Υαβ leads to ◦

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227

Finally, from the variational equation for Φ, the gravi-scalar field equation is given by d ⋆ dΦ + Vα ∧ T ⋆α + ϑα ∧ K⋆α = 0.

(2.179)

The field equations of CA gravity were obtained in this section. The analogue of the Einstein equation, obtained from variation of I with respect to the coframe V , is characterized by an Einstein-like 3-form that includes symmetric GL4 as well as special conformal contributions. Moreover, the field equation in (2.171) contains a non-trivial torsion contribution. Performing a P − ∆ transformation ( i.e. V → ϑ, T → K, D → −D) on (2.171) we obtain (2.176). This result may also be obtained directly by varying I with respect ϑ. A mixed CA cosmological constant term arises in (2.171), (2.176)) as a consequence of the structure of the 2-form Rαβ . The field equation (2.177) is a Yang-Mills-like equation, that represents the generalization of the Gauss torsion-free equation ∇ ⋆ B αβ = 0. In our case, we considered a mixed volume form involving both V and ϑ leading to the substitution B αβ → Σαβ . Additionally, ◦

even in the case of vanishing T ρ = ∇V ρ , the CA torsion depends on the dilaton potential Φ, which, in general, is non-vanishing. A similar argument holds for the special conformal quantity Kρ . Admitting the quadratic curvature term Rβα ∧ ⋆Rαβ in the gauge Lagrangian, it becomes clear how we draw the analogy between (2.177) and the Gauss equation. Eq. (2.178) follow from similar considerations as (2.177), the significant differences being the ◦

◦

lack of a ∇ ⋆ Rαβ counterpart to ∇ ⋆ Rαβ since ⋆Rαβ = 0. Finally, (2.179) involves both T ρ and Kρ in conjunction with a term that resembles the source-free Maxwell equations, with the dilaton potential playing a similar role to the electromagnetic vector potential.

3.

The Physical Realization

3.1.

The Invariance Principle

The above formalism, can be physically realized starting from the Invariance Principle. Our goal is to show that Open Quantum Relativity can be naturally realized starting from the invariance and conservation laws. In this context, the minimal unification scheme, as we will show below, is achieved in 5D. As it is well-known, the field equations and conservation laws can be obtained from a least action principle. The same principle is the basis of any gauge theory, so we start from it to develop our considerations. In this sense, Open Quantum Relativity is a straightforward unification scheme. Let us start from a least action principle and the Noether theorem. Let χ(x) be a multiplet field defined at a spacetime point x and L{χ(x), ∂j χ(x); x} be the Lagrangian density of the system. The action integral of the system, over a given spacetime volume Ω, is defined by Z I(Ω) =

Ω

L{χ(x), ∂j χ(x); x} d4 x.

(3.1)

Now let us consider the infinitesimal variations of the coordinates xi → x′i = xi + δxi ,

(3.2)

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and the field variables χ(x) → χ′ (x′ ) = χ(x) + δχ(x). Correspondingly, the variation of the action is given by Z Z Z  ′ ′  ′ ′ 4 ′ 4 L (x )||∂j x′j || − L(x) d4 x. δI = L (x ) d x − L(x) d x = Ω′

Ω

(3.3)

(3.4)

Ω

Since the Jacobian for the infinitesimal variation of coordinates becomes ||∂j x′j || = 1 + ∂j (δxj ), the variation of the action takes the form, Z   δL(x) + L(x) ∂j (δxj ) d4 x δI =

(3.5)

(3.6)

Ω

where

δL(x) = L′ (x′ ) − L(x).

(3.7)

For any function Φ(x) of x, it is convenient to define the fixed point variation δ0 by, δ0 Φ(x) := Φ′ (x) − Φ(x) = Φ′ (x′ ) − Φ(x′ ).

(3.8)

Expanding the function to first order in δxj as Φ(x′ ) = Φ(x) + δxj ∂j Φ(x),

(3.9)

we obtain δΦ(x) = Φ′ (x′ )−Φ(x) = Φ′ (x′ )−Φ(x′ )+Φ(x′ )−Φ(x) = δ0 Φ(x)+δxj ∂j Φ(x), (3.10) or δ0 Φ(x) = δΦ(x) − δxj ∂j Φ(x).

(3.11)

The advantage to have the fixed point variation is that δ0 commutes with ∂j : δ0 ∂j Φ(x) = ∂j δ0 Φ(x).

(3.12)

δχ = δ0 χ + δxi ∂i χ,

(3.13)

δ∂i χ = ∂i (δ0 χ) − ∂(δxj )∂i χ.

(3.14)

For Φ(x) = χ(x), we have and Using the fixed point variation in the integrand of (3.6) gives Z   δ0 L(x) + ∂j (δxj L(x)) d4 x. δI =

(3.15)

δL + L∂j (δxj ) = δ0 L + ∂j (Lδxj ) = 0.

(3.16)

Ω

If we require that the action integral, defined over any arbitrary region Ω, be invariant, i.e. δI = 0, then we must have

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229

If ∂j (δxj ) = 0, then δL = 0, that is, the Lagrangian density L is invariant. In general, however, ∂j (δxj ) 6= 0, and L transforms like a scalar density. In other words, L is a Lagrangian density unless ∂j (δxj ) = 0. For convenience, let us introduce a function h(x) that behaves like a scalar density, namely δh + h∂j (δxj ) = 0. (3.17) We further assume L(χ, ∂j χ; x) = h(x)L(χ, ∂j χ; x). Then we see that δL + L∂j (δxj ) = hδL.

(3.18)

Hence the action integral remains invariant if δL = 0.

(3.19)

The newly introduced function L(χ, ∂j χ; x) is the scalar Lagrangian of the system. Let us calculate the integrand of (3.15) explicitly. The fixed point variation of L(x) is a consequence of a fixed point variation of the field χ(x), δ0 L =

∂L ∂L δ0 χ + δ0 (∂j χ) ∂χ ∂(∂j χ)

(3.20)



(3.21)

which can be cast into the form, δ0 L = [L]χ δ0 χ + ∂j where

∂L [L]χ ≡ − ∂j ∂χ



 ∂L δ0 χ ∂(∂j χ)

∂L ∂(∂j χ)



.

Consequently, we have the action integral in the form   Z  ∂L j k δχ − Tk δx d4 x, δI = [L]χ δ0 χ + ∂j ∂(∂ χ) j Ω where T j k :=

∂L ∂k χ − δkj L ∂(∂j χ)

(3.22)

(3.23)

(3.24)

is the canonical energy-momentum tensor density. If the variations are chosen in such a way that δxj = 0 over Ω and δ0 χ vanishes on the boundary of Ω, then δI = 0 gives us the Euler-Lagrange equation,   ∂L ∂L = 0. (3.25) [L]χ = − ∂j ∂χ ∂(∂j χ) On the other hand, if the field variables obey the Euler-Lagrange equation, [L]χ = 0, then we have   ∂L j k δχ − T k δx = 0, (3.26) ∂j ∂(∂j χ) which gives rise, considering also the Noether theorem, to conservation laws. These very straightforward considerations are at the basis of our following discussion.

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The Global Poincar´e Invariance

As standard, we assert that our spacetime in the absence of gravitation is a Minkowski space M4 . The isometry group of M4 is the group of Poincar´e transformations (PT) which consists of the Lorentz group SO(3, 1) and the translation group T (3, 1). The Poincar´e transformations of coordinates are PT

xi → x′i = ai j xj + bi ,

(3.27)

where aij and bi are real constants, and aij satisfy the orthogonality conditions aik akj = δji . For infinitesimal variations, we have δx′i = χ′ (x′ ) − χ(x) = εi j xj + εi

(3.28)

where εij + εji = 0. While the Lorentz transformation forms a six parameter group, the Poincar´e group has ten parameters. The Lie algebra for the ten generators of the Poincar´e group is [Ξij , Ξkl ] = ηik Ξjl + ηjl Ξik − ηjk Ξil − ηil Ξjk ,

(3.29)

[Ξij , Tk ] = ηjk Ti − ηik Tj , [Ti , Tj ] = 0, where Ξij are the generators of Lorentz transformations, and Ti are the generators of fourdimensional translations. Obviously, ∂i (δxi ) = 0 for the Poincar´e transformations (3.27). Therefore, our Lagrangian density L, which is the same as L with h(x) = 1 in this case, is invariant; namely, δL = δL = 0 for δI = 0. Suppose that the field χ(x) transforms under the infinitesimal Poincar´e transformations as 1 δχ = εij Sij χ, (3.30) 2 where the tensors Sij are the generators of the Lorentz group, satisfying Sij = −Sji , [Sij , Skl ] = ηik Sjl + ηjl Sik − ηjk Sil − ηil Sjk .

(3.31)

Correspondingly, the derivative of χ transforms as 1 δ(∂k χ) = εij Sij ∂k χ − εi k ∂i χ. 2

(3.32)

Since the choice of infinitesimal parameters εi and εij is arbitrary, the vanishing variation of the Lagrangian density δL = 0 leads to the identities, ∂L ∂L Sij χ + (Sij ∂k χ + ηki ∂j χ − ηkj ∂i χ) = 0. ∂χ ∂(∂k χ) We also obtain the following conservation laws   ∂j Tkj = 0, ∂k S k ij − xi T k j + xj T k i = 0,

(3.33)

(3.34)

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∂L Sij χ. ∂(∂k χ)

231

(3.35)

These conservation laws imply that the energy-momentum and angular momentum Z Z
  0 (3.36) Pl = Tl0 d3 x, Jij = S ij − xi T 0 j − xj T 0 i d3 x,

are conserved. This means that the system, invariant under the ten parameter symmetry group, has ten conserved quantities. This is an example of Noether symmetry. The first term of the angular momentum integral corresponds to the spin angular momentum while the second term gives the orbital angular momentum. The global Poincar´e invariance of a system means that, for the system, the spacetime is homogeneous (all spacetime points are equivalent) as dictated by the translational invariance and is isotropic (all directions about a spacetime point are equivalent) as indicated by the Lorentz invariance. It is interesting to observe that the fixed point variation of the field variables takes the form δ0 χ = where

1 j ε k Ξj k χ + εj Tj χ, 2

  Ξj k = Sj k + xj ∂k − xk ∂j , Tj = −∂j .

(3.37)

(3.38)

We remark that Ξj k are the generators of the Lorentz transformation and Tj are those of the translations.

3.3.

The Local Poincar´e Invariance

As next step, let us consider a modification of the infinitesimal Poincar´e transformations (3.28) by assuming that the parameters εjk and εj are functions of the coordinates and by writing them altogether as δxµ = εµ ν (x) xν + εµ (x) = ξ µ ,

(3.39)

which we call the local Poincar´e transformations (or the general coordinate transformations). In order to make a distinction between the global transformation and the local transformation, we use the Latin indices (j, k = 0, 1, 2, 3) for the former and the Greek indices (µ, ν = 0, 1, 2, 3) for the latter. The variation of the field variables χ(x) defined at a point x is still the same as that of the global Poincar´e transformations, 1 δχ = εij S ij χ. 2

(3.40)

The corresponding fixed point variation of χ takes the form, δ0 χ =

1 εij S ij χ − ξ ν ∂ν χ. 2

(3.41)

Differentiating both sides of (3.41) with respect to xµ , we have 1 1 δ0 ∂µ χ = εij Sij ∂µ χ + (∂µ εij ) S ij χ − ∂µ (ξ ν ∂ν χ). 2 2

(3.42)

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By using these variations, we obtain the variation of the Lagrangian L, 1 δL + ∂µ (δxµ )L = hδL = δ0 L + ∂ν (Lδxν ) = − (∂µ εij ) S µ ij − ∂µ ξ ν T µν , 2

(3.43)

which is no longer zero unless the parameters εij and ξ ν become constants. Accordingly, the action integral, for the given Lagrangian density L, is not invariant under the local Poincar´e transformations. We notice that while ∂j (δxj ) = 0 for the local Poincar´e transformations, ∂µ ξ µ does not vanish under local Poincar´e transformations. Hence, as expected L is not a Lagrangian scalar but a Lagrangian density. As mentioned earlier, in order to define the Lagrangian L, we have to select an appropriate non-trivial scalar function h(x) satisfying δh + h∂µ ξ µ = 0.

(3.44)

Now we consider a minimal modification of the Lagrangian so then to make the action integral invariant under the local Poincar´e transformations. It is rather obvious that if there is a covariant derivative ∇k χ which transforms as δ(∇k χ) =

1 ij ε Sij ∇k χ − εi k ∇i χ, 2

(3.45)

then a modified Lagrangian L′ (χ, ∂k χ, x) = L(χ, ∇k χ, x), obtained by replacing ∂k χ of L(χ, ∂k χ, x) by ∇k χ, remains invariant under the local Poincar´e transformations, that is δL′ =

∂L′ ∂L′ δχ + δ(∇k χ) = 0. ∂χ ∂(∇k χ)

(3.46)

To find such a k-covariant derivative, we introduce the gauge fields Aij µ = −Aji µ and define the µ-covariant derivative 1 ∇µ χ := ∂µ χ + Aij µ Sij χ, 2

(3.47)

in such a way that the covariant derivative transforms as 1 δ0 ∇µ χ = Sij ∇µ χ − ∂µ (ξ ν ∇ν χ). 2

(3.48)

The transformation properties of Aabµ are determined by ∇µ χ and δ∇µ χ. Making use of 1 1 1 1 ij ε ,µ Sij χ + εij Sij ∂µ χ − (∂µ ξ ν ) ∂ν ψ + δAij µ Sij χ + Aij µ Sij εkl Skl χ 2 2 2 4 (3.49) and comparing with (3.47) we obtain,  1  ij kl δAij µ Sij χ + εij ,µ Sij χ + A µ ε − εij Aklµ Sij Skl χ + (∂µ ξ ν ) Aij ν Sij χ = 0. (3.50) 2 δ∇µ χ =

Using the antisymmetry in ij and kl to rewrite the term in parentheses on the RHS of (3.50) as [Sij , Skl ] Aij µ εkl χ, we see the explicit appearance of the commutator [Sij , Skl ]. Using the expression for the commutator of Lie algebra generators 1 [ef ] [Sij , Skl ] = c [ij][kl] Sef , 2

(3.51)

The Conformal-Affine Structure of open Quantum Relativity...

233

[ef ]

where c [ij][kl] (the square brackets denote anti-symmetrization) is the structure constants of the Lorentz group (deduced below), we have [Sij , Skl ] Aij µ εkl =


1 ic j i Aµ εc − Acj µ εc Sij . 2

(3.52)

The substitution of this equation and the consideration of the antisymmetry of εcb = −εbc enables us to write δAij µ = εi k Akj µ + εj k Aik µ − (∂µ ξ ν )Aij ν − ∂µ εij .

(3.53)

We require the k-derivative and µ-derivative of χ to be linearly related as ∇k χ = Vk µ (x)∇µ χ,

(3.54)

where the coefficients Vk µ (x) are position-dependent and behave like a new set of field variables. From (3.54) it is evident that ∇k χ varies as δ∇k χ = δVkµ ∇µ χ + Vkµ δ∇µ χ.

(3.55)

Comparing with δ∇k χ = 12 εab Sab ∇k χ − εjk ∇j χ we obtain, Vαk δVkµ ∇µ χ − ξ ν ,α ∇ν χ + Vαk εjk ∇j χ = 0.

(3.56)

δVk µ = Vk ν ∂ν ξ µ − Vi µ εi k .

(3.57)


Then exploiting δ Vαk Vkµ = 0, we find that the quantity Vk µ transforms according to

It is also important to recognize that the inverse of det(Vk µ ) transforms like a scalar density as h(x) does. For our minimal modification of the Lagrangian density, we utilize this available quantity for the scalar density h; namely, we let h(x) = [det(Vk µ )]−1 .

(3.58)

When we consider Poincar´e transformations, that are not spacetime dependent, we have Vk µ → δkµ so that h(x) → 1. This is a desirable property, because we replace the Lagrangian density L(χ, ∂k χ, x), invariant under the global Poincar´e transformations, by a Lagrangian density L(χ, ∂µ χ; x) → h(x)L(χ, ∇k χ), (3.59) the action integral, with this modified Lagrangian density, remains invariant under the local Poincar´e transformations. Since the local Poincar´e transformations δxµ = ξ µ (x) are nothing else but generalized coordinate transformations, the newly introduced gauge fields Viλ and Aij µ can be interpreted, respectively, as the tetrad (vierbein) fields which set the local coordinate frame and as a local affine connection, with respect to the tetrad frame (see also [3]).

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Spinors, Vectors and Tetrads

Let us consider first the case where the multiplet field χ is the Dirac field ψ(x) which behaves like a four-component spinor under the Lorentz transformations and transforms as ψ(x) → ψ ′ (x′ ) = S(Λ)ψ(x),

(3.60)

where S(Λ) is an irreducible unitary representation of the Lorentz group. Since the bilinear form v k = iψγ k ψ is a vector, it transforms according to v j = Λjk v k ,

(3.61)

where Λji is a Lorentz transformation matrix satisfying Λij + Λji = 0.

(3.62)

The invariance of v i (or the covariance of the Dirac equation) under the transformation ψ(x) → ψ ′ (x′ ) leads to (3.63) S −1 (Λ)γ µ S(Λ) = Λµν γ ν , where the γ ′ s are the Dirac γ-matrices satisfying the anticommutator, γi γj + γj γi = ηij 1.

(3.64)

Furthermore, we notice that the γ-matrices have the following properties: 
† 0 2 = (γ )2 = −1, γ = −γ 0 and γ γ 0 = 1  (γ ) = −γ , γ 0 0 0 0 0     
2 (γk )† = γk , γ k = (γk )2 = 1; (k = 1, 2, 3) and γk = γ k       (γ5 )† = −γ5 , (γ5 )2 = −1 and γ 5 = γ5 .

(3.65)

µν

The transformation S(Λ) can be put into the form S(Λ) = eΛµν γ . Expanding S(Λ) about the identity and only retaining terms up to the first order in the infinitesimals and finally expanding Λµν to the first order in εµν Λµν = δµν + εµν , εij + εji = 0,

(3.66)

we get

1 S(Λ) = 1 + εij γij . (3.67) 2 In order to determine the form of γij , we substitute (3.66) and (3.67) into (3.63) to obtain 1 h ij k i εij γ , γ = η ki εji γ j . 2

(3.68)

 1  η ki εji γ j = εij η ki γ j − η kj γ i , 2

(3.69)

Rewriting the RHS of (3.68) using the antisymmetry of εij as

The Conformal-Affine Structure of open Quantum Relativity... it yields

i h γ k , γ ij = η ki γ j − η kj γ i .

235

(3.70)

Since the solution has the form of an antisymmetric product of two matrices, we obtain γ ij :=

1  i j γ ,γ . 2

(3.71)

If χ = ψ, the group generator Sij appearing in (3.31) is identified with 1 Sij ≡ γij = (γi γj − γj γi ). 2

(3.72)

To be explicit, the Dirac field transforms under Lorentz transformations (LT) as 1 δψ(x) = εij γij ψ(x). 2

(3.73)

The Pauli conjugate of the Dirac field denoted as ψ and defined by

transforms under LTs as,

ψ(x) := iψ † (x) γ0 , i ∈ C,

(3.74)

1 δψ = −ψ εij ψγij . 2

(3.75)

Now, under local LTs, εab (x) becomes a function of spacetime, and, unlike ∂µ ψ(x), the derivative of ψ ′ (x′ ) is no longer homogenous, due to the occurrence of the term γ ab [∂µ εab (x)] ψ(x) in ∂µ ψ ′ (x′ ), which is non-vanishing unless εab is constant. When going from locally flat to curved spacetime, we must generalize ∂µ to the covariant derivative ∇µ to compensate for this extra term, allowing to gauge the group of LTs. Thus, by using ∇µ , we can preserve the invariance of the Lagrangian for arbitrary local LTs at each spacetime point ∇µ ψ ′ (x′ ) = S(Λ(x))∇µ ψ(x). (3.76) To determine the explicit form of the connection belonging to ∇µ , we study the derivative of S(Λ(x)). The transformation S(Λ(x)) is given by 1 S(Λ(x)) = 1 + εab (x)γ ab . 2

(3.77)

Since εab (x) is only a function of spacetime for local Lorentz coordinates, we express this infinitesimal LT in terms of general coordinates only by shifting all spacetime dependence of the local coordinates into tetrad fields as εab (x) = Va λ (x)V νb (x)ελν . Substituting this expression for εab (x), we obtain i h ∂µ εab (x) = ∂µ Va λ (x)V νb (x)ελν .

(3.78)

(3.79)

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However, since ελν has no spacetime dependence, this reduces to ∂µ εab (x) = Va λ (x)∂µ Vbλ (x) − Vb ν (x)∂µ Vaν (x).

(3.80)

ωµba := Vb ν (x)∂µ Vaν (x),

(3.81)

Being the first and second terms in Eq.(3.80) become Vaλ (x)∂µ Vbλ (x) Vbν (x)∂µ Vaν (x) = 21 ωµba respectively. Using the identification

=

1 2 ωµab

∂µ εab (x) = ωµab ,

and

(3.82)

we write

1 ∂µ S(Λ(x)) = − γ ab ωµab . 2 According to (3.47), the covariant derivative of the Dirac spinor is 1 ∇µ ψ = ∂µ ψ + Aij µ γij ψ. 2

(3.83)

(3.84)

Correspondingly, the covariant derivative of ψ¯ is given by 1 ¯ ij . ∇µ ψ = ∂µ ψ − Aij µ ψγ 2

(3.85)

¯ we can show that Using the covariant derivatives of ψ and ψ, ∇µ vj = ∂µ vj − Ai jµ vi .

(3.86)

The same covariant derivative should be used for any covariant vector vk under the Lorentz transformation. Since ∇µ (vi v i ) = ∂µ (vi v i ), the covariant derivative for a contravariant vector v i must be ∇µ v i = ∂µ v i + Ai jµ v j . (3.87)

Since the tetrad Vi µ is a covariant vector under Lorentz transformations, its covariant derivative must transform according to the same rule and using ∇a = Vaµ (x)∇µ , the covariant derivatives of a tetrad in local Lorentz coordinates read ∇ν Vi µ = ∂ν Vi µ − Ak iν Vk µ , ∇ν V i µ = ∂ν V i µ + Ai kν V k µ .

(3.88)

The inverse of Vi µ is denoted by V i µ and satisfies V i µ Vi ν = δ µ ν , V i µ Vj µ = δ i j .

(3.89)

Now, to allow the transition to curved spacetime, we take account of the general coordinates of objects that are covariant under local Poincar´e transformations. Here we define the covariant derivative of a quantity v λ , which behaves like a contravariant vector under the local Poincar´e transformation. Namely Dν v λ ≡ Vi λ ∇ν v i = ∂ν v λ + Γλ µν v µ , Dν vµ ≡ V i µ ∇ν vi = ∂ν vµ − Γλ µν vλ ,

(3.90)

The Conformal-Affine Structure of open Quantum Relativity...

237

where Γλ µν := Vi λ ∇ν V i µ ≡ −V i µ ∇ν Vi λ .

(3.91)

The definition of Γλ µν implies

Dν Vi λ = ∇ν Vi λ + Γλ µν Vi µ = ∂ν Vi λ − Ak iν Vk λ + Γλ µν Vi µ = 0,

(3.92)

Dν V i µ = ∇ν V i µ − Γλ µν V i λ = ∂ν V i µ + Ai kν V k µ − Γλ µν V i λ = 0. From (3.92) we find, Ai kν = V i λ ∂ν Vk λ + Γλ µν V i λ Vk µ = −Vk λ ∂ν V i λ + Γλ µν V i λ Vk µ .

(3.93)

or, equivalently, in terms of ω defined in (3.81), Ai kν = ω iνk + Γλ µν V i λ Vk µ = −ωkνi + Γλ µν V i λ Vk µ .

(3.94)

Using this in (3.84), we may write ∇µ ψ = (∂µ − Γµ )ψ,

(3.95)

where

 1 i ω jµ − Γλ µν V i λ Vj ν γi j , (3.96) 4 which is known as the Fock-Ivanenko connection. We now study the transformation properties of Aµab . Recall ωµab = Va λ (x)∂µ Vβλ (x) and since ∂µ ηab = 0, we write Γµ =

Λaa ηab ∂µ Λb b = Λaa ∂µ Λab .

(3.97)

Note that barred indices are equivalent to the primed indices used above. Hence, the spin connection transforms as Aabc = Λaa Λbb Λcc Aabc + Λaa Λcc V µa (x)∂µ Λbc .

(3.98)

To determine the transformation properties of Γabc = Aabc − [V µa (x)∂µ V νb (x)] Vνc (x), we consider the local LT of [Va µ (x)∂µ V νb (x)] Vνc (x) which is, i h V µa (x)∂µ V νb Vνc (x) = Λaa Λb b Λcc [Aνab Vνc (x)] + Λaa Λc c V µa (x)∂µ Λcb .

(3.99)

(3.100)

From this result, we obtain the following transformation law, Γabc = Λaa Λb b Λcc Γabc .

(3.101)

We now explore the consequence of the antisymmetry of ωabc in bc. Recalling the equation for Γabc , exchanging b and c and adding the two equations, we obtain Γabc + Γacb = −V µa (x) [(∂µ V νb (x)) Vνc (x) + (∂µ V νc (x)) Vνb (x)] .

(3.102)

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We know however, that ∂µ [V νb (x)Vνc (x)] = Vνc (x)∂µ V νb (x) + Vλb (x)∂µ V λc (x) + Vb ν (x)V λc (x)∂µ gλν . (3.103) Letting λ → ν and exchanging b and c, we obtain ∂µ [V νb (x)Vνc (x)] = −Vb λ (x)V νc (x)∂µ gνλ ,

(3.104)

Γabc + Γacb = Va µ (x)Vb λ (x)Vc ν (x)∂µ gνλ .

(3.105)

so that, finally, This, however, is equivalent to Γabc + Γa cb = V µa (x)V λb (x)V νc (x)∂µ gνλ ,

(3.106)

Γµλν + Γµνλ = ∂µ gνλ ,

(3.107)

and then which we recognize as the general coordinate connection. It is known that the covariant derivative for general coordinates is ∇µ Aνλ = ∂µ Aνλ + Γλµσ Aνσ − Γσµν Aσλ .

(3.108)

In a Riemannian manifold, the connection is symmetric under the exchange of µν, that is, Γλµν = Γλνµ . Using the fact that the metric is a symmetric tensor we can now determine the form of the Christoffel connection by cyclically permuting the indices of the general coordinate connection equation (3.107), yielding Γµνλ =

1 (∂µ gνλ + ∂ν gλµ − ∂λ gµν ) . 2

(3.109)

Since Γµνλ = Γνµλ is valid for general coordinate systems, it follows that a similar constraint must hold for local Lorentz transforming coordinates as well, so we expect Γabc = Γbac . Recalling the equation for Γabc and exchanging a and b, we obtain   ωabc − ωbac = Vνc (x) V µa (x)∂µ V νb (x) − V µb (x)∂µ V νa (x) . (3.110) We now define the objects of anholonomicity as   Ωcab := Vνc (x) V µa (x)∂µ V νb (x) − V µb (x)∂µ V νa (x) .

(3.111)

Using Ωcab = −Ωcba , we permute indices in a similar manner as was done for the derivation of the Christoffel connection above yielding, ωabµ =

1 [Ωcab + Ωbca − Ωabc ] V cµ ≡ ∆abµ . 2

(3.112)

For completeness, we determine the transformation law of the Christoffel connection. Making use of Γλµν eλ = ∂µ eν where ∂µ eν = X µµ X νν ∂µ eν + X µµ (∂µ X νν ) eν ,

(3.113)

The Conformal-Affine Structure of open Quantum Relativity... we can show

239

Γλµ ν = X µµ X νν Xλλ Γλµν + X µµ Xν λ X νµν ,

(3.114)

X νµν ≡ ∂µ ∂ν xν .

(3.115)

where In the light of the above considerations, we may regard infinitesimal local gauge transformations as local rotations of basis vectors, belonging to the tangent space [16, 57] of the manifold. For this reason, given a local frame on a tangent plane to the point x on the base manifold, we can obtain all other frames on the same tangent plane, by means of local rotations of the original basis vectors. Reversing this argument, we observe that by knowing all frames residing in the horizontal tangent space to a point x on the base manifold, it enables us to deduce the corresponding gauge group of symmetry transformations.

3.5.

Curvature, Torsion and Metric

From the definition of the Fock-Ivanenko covariant derivative, we can find the second order covariant derivative   1 1 Dν Dµ ψ = ∂ν ∂µ ψ + Scd ψ∂ν Aµcd + Aµcd ∂ν ψ + Γρµν Dρ ψ + Sef Aνef ∂µ ψ 2 2 1 ef cd + Sef Scd Aν Aµ ψ. (3.116) 4 Recalling Dν V cµ = 0, we can solve for the spin connection in terms of the Christoffel connection Aµcd = −V dλ ∂µ V cλ − Γµcd . (3.117) The derivative of the spin connection is then   ∂µ Acdν = −V dλ ∂µ ∂ν V cλ − ∂ν V cλ ∂µ Vλ d − ∂µ Γcdν .

(3.118)

Relabelling running indices, we can write   1 1 Sef Scd Aefν Acdµ − Aefµ Acdν ψ = [Scd , Sef ] Aefµ Acdν ψ. 4 4

(3.121)

Noting that the Christoffel connection is symmetric and partial derivatives commute, we find h  i 1 h  i 1 [Dµ , Dν ] ψ = Scd ∂ν Acdµ − ∂µ Acdν ψ + Sef Scd Aefν Acdµ − Aefµ Acdν ψ , 2 4 (3.119) where ∂ν Acdµ − ∂µ Acdν = ∂µ Γcdν − ∂ν Γcdµ . (3.120)

Using {γa , γb } = 2ηab to deduce

{γa , γb } γc γd = 2ηab γc γd , we find that the commutator of bi-spinors is given by i 1h ηce δda δfb − ηde δca δfb + ηcf δea δdb − ηdf δea δcb Sab . [Scd , Sef ] = 2

(3.122)

(3.123)

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Clearly the terms in brackets on the RHS of (3.123) are antisymmetric in cd and ef and also antisymmetric under the exchange of pairs of indices cd and ef . Since the alternating spinor is antisymmetric in ab, it must be the terms in brackets: this means that the commutator does not vanish. Hence, the term in brackets is totally antisymmetric under interchange of indices ab, cd and ef and exchange of these pairs of indices. We identify this as the structure constant of the Lorentz group [58] h i ηce δda δfb − ηde δca δfb + ηcf δea δdb − ηdf δea δcb = c[cd][ef ] [ab] = c[ab] [cd][ef ] , (3.124)

with the aid of which we can write

where

h i 1 1 [Scd , Sef ] Aefµ Acdν ψ = Sab Aaeν Aebµ − Ab eν Aaeµ ψ, 4 2

(3.125)

Aaeν Aebµ − Ab eν Aaeµ = Γaνe Γebµ − Γbνe Γeaµ .

(3.126)

Combining these results, the commutator of two µ-covariant differentiations gives 1 [∇µ , ∇ν ]χ = − Rij µν Sij χ, 2

(3.127)

Ri jµν = ∂ν Ai jµ − ∂µ Ai jν + Ai kν Ak jµ − Ai kµ Ak jν .

(3.128)

where Using the Jacobi identities for the commutator of covariant derivatives, it follows that the field strength Ri jµν satisfies the Bianchi identity ∇λ Ri jµν + ∇µ Ri jνλ + ∇ν Ri jλµ = 0.

(3.129)

Permuting indices, this can be put into the cyclic form εαβρσ ∇β Rijρσ = 0,

(3.130)

where εαβρσ is the Levi-Civita alternating symbol. Furthermore, Rij µν = η jk Ri kµν is antisymmetric with respect to both pairs of indices, Rij µν = −Rji µν = Rji νµ = −Rij νµ .

(3.131)

This condition is known as the first curvature tensor identity. To determine the analogue of [∇µ , ∇ν ]χ in local coordinates, we start from ∇k ψ = V µk ∇µ ψ. From ∇k ψ we obtain,
(3.132) ∇l ∇k ψ = V νl ∇ν V µk ∇µ ψ + V νl V µk ∇ν ∇µ ψ. Permuting indices and recognizing

Vµ a ∇ν V µk = −Vk µ ∇ν V aµ ,
Vµa Vkµ = 0), we arrive at

(which follows from ∇ν


V νl ∇ν V µk ∇µ ψ − V µk (∇µ V νl ) ∇ν ψ = V µl V νk − V µk V νl ∇ν Vµ a ∇a ψ.

(3.133)

(3.134)

The Conformal-Affine Structure of open Quantum Relativity... Defining


C akl := V µk V νl − V µl V νk ∇ν Vµ a ,

241 (3.135)

the commutator of the k-covariant differentiations takes the final form [7] 1 [∇k , ∇l ]χ = − Rij kl Sij χ + C i kl ∇i χ, 2

(3.136)

Rij kl = Vk µ Vl ν Rij µν .

(3.137)

where As done for Ri jµν using the Jacobi identities for the commutator of covariant derivatives, we find the Bianchi identity in Einstein-Cartan spacetime [63, 56] εαβρσ ∇β Rijρσ = εαβρσ Cβρλ Rijσλ .

(3.138)

The second curvature identity

leads to,

Rk[ρσλ] = 2∇[ρ Cσλ]k − 4C[ρσb Cλ]bk

(3.139)

εαβρσ ∇β Cρσk = εαβρσ Rkjρσ V βj .

(3.140)

Γλµν = Vi λ ∇ν V iµ = −Vµ i ∇ν V λi ,

(3.141)

Notice that if then


Γλµν − Γλνµ = Viλ ∇ν V iµ − ∇µ V iν .

(3.142)

Contracting by Vkµ Vlν , we obtain [7],

C akl = Vk µ Vl ν Vλ a



 Γλµν − Γλνµ .

(3.143)

We therefore conclude (and stress it) that C akl is related to the antisymmetric part of the affine connection (3.144) Γλ[µν] = Vµ k Vν l Va λ C akl ≡ T λµν , which is usually interpreted as spacetime torsion T λµν . Considering ∆abµ defined in (3.112), we see that the most general connection, in the Poincar´e gauge approach to gravitation, is Aabµ = ∆abµ − Kabµ + Γλ νµ Vaλ Vb ν , (3.145) where

  Kabc = − T λ νµ − Tνµλ + Tµ λν Vaλ Vb ν Vc µ ,

(3.146)

Rρ σµν = ∂ν Γρσµ − ∂µ Γρσν + Γρ λν Γλ σµ − Γρ λµ Γλ σν .

(3.147)

ρ is the contorsion tensor [54]. Now, the quantity Rσµν = Vi ρ Ri σµν may be expressed as

Therefore, we can regard Rρ σµν as the curvature tensor with respect the affine connection Γλ µν . By using the inverse of the tetrad, we define the metric of the spacetime manifold by gµν = V i µ V j ν ηij .

(3.148)

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From (3.92) and the fact that the Minkowski metric is constant, it is obvious that the metric so defined is covariantly constant, that is, Dλ gµν = 0.

(3.149)

The spacetime thus specified by the local Poincar´e transformation is said to be metric. It is not difficult to show that √

−g = [det V i µ ] = [det Vi µ ]−1 ,

where g = det gµν . Hence we may take

3.6.

√

(3.150)

−g for the density function h(x).

Field Equations for Gravity

Finally, we are now able to deduce the field equations for the gravitational field. This application is particularly useful in view of the forthcoming considerations for Open Quantum Relativity. From the curvature tensor Rρ σµν , given in (3.147), the Ricci tensor follows Rσν = Rµ σµν .

(3.151)

and the scalar curvature L

R = Rν ν = R + ∂i Kaia − Ta bc Kbca

(3.152)

L

where R denotes the usual Ricci scalar of General Relativity. Using this scalar curvature R, we choose the Lagrangian density for free Einstein-Cartan gravity   L 1 √ ia bc a LG = −g R + ∂i Ka − Ta Kbc − 2Λ , 2κ

(3.153)

where κ is a gravitational coupling constant, and Λ is the cosmological constant. Observe that the second term in the brackets is a divergence and may be ignored. The field equation can be obtained from the total action, Z  S= Lfield (χ, ∂µ χ, Vi µ , Aij µ ) + LG d4 x, (3.154) where the matter Lagrangian density is taken to be Lfield =


 1 a ψγ Da ψ − Da ψ γ a ψ . 2

(3.155)

Taking into account the Christoffel symbols, spin connection and contorsion contributions so then to operate on general spinorial arguments, we have   L 1 Γµ = gλσ ∆σµρ − Γ σρµ − K σρµ γ λρ . 4

(3.156)

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243

It is important to keep in mind that ∆σµρ act only on multi-component spinor fields, while L

Γ σρµ act on vectors and arbitrary tensors. The gauge covariant derivative for a spinor and adjoint spinor is then given by Dµ ψ = (∂µ − Γµ ) ψ, Dµ ψ = ∂µ ψ − ψΓµ .

(3.157)

The variation of the field Lagrangian is δLfield = ψ (δγ µ Dµ + γ µ δΓµ ) ψ. We know that the Dirac gamma matrices are covariantly vanishing, so i h b κ = 0. Dκ γι = ∂κ γι − Γµικ γµ + γι , Γ

(3.158)

(3.159)

b κ are real matrices, used to induce similarity transformations on quanThe 4 × 4 matrices Γ tities with spinor transformation [62] properties, according to b −1 γi Γ. b γi′ = Γ

b κ leads to, Solving for Γ

b κ = 1 [(∂κ γι ) γ ι − Γµ γµ γ ι ] . Γ ικ 8 bκ , Then, taking the variation of Γ bκ = δΓ =

  1 (∂κ δγι ) γ ι + (∂κ γι ) δγ ι − (δΓµικ ) γµ γ ι −Γµικ ((δγµ ) γ ι + γµ δγ ι ) 8 1 [(∂κ δγι ) γ ι − (δΓµικ ) γµ γ ι ] . 8

(3.160)

(3.161)

(3.162)

Since we require the anticommutator condition on the gamma matrices {γ µ , γ ν } = 2g µν to hold, the variation of the metric gives 2δg µν = {δγ µ , γ ν } + {γ µ δγ ν }.

(3.163)

One solution to this equation is, δγ ν =

1 γσ δγ σν . 2

(3.164)

With the aid of this result, we can write 1 (∂κ δγι ) γ ι = ∂κ (γ ν δgνι ) γ ι . 2

(3.165)

Finally, exploiting the anti-symmetry in γµν we obtain bκ = δΓ

 1 gνσ δΓµκσ − gµσ δΓνκσ γ µν . 8

(3.166)

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The field Lagrangian, defined in the Einstein-Cartan spacetime, can be written [59, 12, 63, 60, 56] explicitly in terms of its Lorentzian and contorsion components as    n o L L ~c 1 (3.167) Lfield = Dµ ψ γ µ ψ − ψγ µ Dµ ψ − Kµαβ ψ γ µ , γ αβ ψ. 2 8 Using the following relations   1 βα γ µ ψ − 1 K µ αβ  − 4 Kµαβ ψ γ µ , γ αβ ψ = 14 Kµαβ ψγ 4 µαβ ψγ γ ψ, µ νλ γ µ γ ν γ λ εµνλσ = 3!γσ γ5 ,  µ= νλγ , γ [µ ενµνλσ  γ ,γ = γ γ γ λ] ,

(3.168)

we obtain

n o
1 Kµαβ ψ γ µ , γ αβ ψ = Kµαβ εαβµν ψγ5 γν ψ . 2i Here we define the contorsion axial vector Kν :=

1 αβµν ε Kαβµ . 3!

Multiplying by the axial current jν5 = ψγ5 γν ψ, we obtain
ψγ5 γν ψ εαβµν Kµαβ = −6ijν5 K ν .

Thus, the field Lagrangian density becomes    L L 1 3i~c µ µ Lfield = Dµ ψ γ ψ − ψγ Dµ ψ + Kµ j5µ , 2 8

(3.169)

(3.170)

(3.171)

(3.172)

and the total action reads Z Z √ √ δI = δ LG −gd4 x + δ Lfield −gd4 x Z √ = (δLG + δLfield ) −gd4 x.

(3.173)

Writing the metric in terms of the tetrads g µν = V iµ V νi , we observe

By using we are able to deduce


√ 1√ δ −g = − −g δV iµ Vµi + Vνi δV νi . 2
δV νi = δ η ij V jν = η ij δV jν , √ √ δ −g = − −gVµi δVi µ .

(3.174)

(3.175) (3.176)

µ

For the variation of the Ricci tensor Riν = Vi Rµν , so we have L

L

L

δ Riν = δVi µ Rµν + Vi µ δ Rµν .

(3.177)

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In an inertial frame, the Ricci tensor reduces to L

L

L

Rµν = ∂ν Γ ββµ − ∂β Γ βνµ , so that

L

µ

L

δ Riν = δVi Rµν + Vi

µ



L

∂ν δ Γ

β βµ

(3.178) L

− ∂β δ Γ

β νµ



.

(3.179)

The second term can be converted into a surface term, so it may be ignored. Collecting our results, we have  δg µν = −g µρ g νσ δgρσ ,  √ √ √    δ −g = − 21 −ggµν δg µν = − −gVµi δVi µ ,     L L λρ λρ ρ λ , δ R = δV µ R (3.180) δR = g ∇ δΓ − ∇ δΓ + T δΓ ν µν ρµ ν µν iν λ ρν i λ λµ      L L L    δR = R µν δgµν + g µν ∇λ δ Γ λ µν − ∇ν δ Γ λµλ − Ta bc δKbca .

From the above results, we obtain
  Z Ri µ − 21 Vi µ R − Vi µ Λ δV µi + 2g ρλ Tµλσ δΓµρσ √   1 L L   −gd4 x. (3.181) δIG = µν λ λ +g ∇λ δ Γ µν − ∇ν δ Γ µλ 16π

The last term in the action can be ignored being a surface term. Using the four-current v µ introduced earlier, the action for the matter fields read [62] Z h i b µ ψ √−gd4 x ψδγ µ ∇µ ψ + ψγ µ δ Γ (3.182) δIfield =   1 µν   Z  2 g ψγi (∇ν ψ) + T µρσ Ti ρσ − δiµ Tλρσ T λρσ δV µi    √ L L = −gd4 x.  + 81 (g ρν v µ − g ρµ v ν ) gµσ δ Γ σνρ − gνσ δ Γ σµρ 

Removing the derivatives of variations of the metric appearing in δΓσνρ via partial integration, and equating to zero the coefficients of δg µν and δT σνρ in the variation of the action integral, we obtain     1 1 1 1 ψγν ∇µ ψ − ∇µ vν Rµν − gµν R − gµν Λ + (3.183) 0 = 16π 2 2 4 +∇σ Tµνσ + Tµρσ Tνρσ − gµν Tλρσ T λρσ

and Tρσλ = 8πτρσλ .

(3.184)

Eqs.(3.183) have the form of Einstein equations Gµν − gµν Λ = 8πΣµν ,

(3.185)

where the Einstein tensor and non-symmetric energy-momentum tensors are 1 Gµν = Rµν − gµν R, 2

(3.186)

246

G. Basini and S. Capozziello Σµν = Θµν + Tµν ,

(3.187)

respectively. Here we identify Θµν as the canonical energy-momentum Θµν =

∂Lfield ∇ν χ − δ µν Lfield , ∂(∇µ χ)

(3.188)

while Tµν is the stress-tensor form of the non-Riemannian manifold. For the case of spinor fields, being considered here the explicit form of the energy-momentum components [61], are (after symmetrization of corresponding canonical source terms in the Einstein equation), 

 Θµν = − ψγµ ∇ν ψ − ∇ν ψ γµ ψ + ψγν ∇µ ψ − ∇µ ψ γν ψ

(3.189)

and by using the second field equation (3.184), we determine

Tµν = ∇σ Tµνσ + Tµρσ τνρσ − gµν Tλρσ τ λρσ ,

(3.190)

where τµνσ is the so-called spin - energy potential [12, 63] τµνσ :=

∂Lfield γµν χ. ∂(∇σ χ)

(3.191)

Explicitly, the spin energy potential reads τ µνσ = ψγ [µ γ ν γ σ] ψ. The equation of motion obtained from the variation of the action with respect to ψ reads [12, 63] 3 γ µ ∇µ ψ + Tµνσ γ [µ γ ν γ σ] ψ = 0. 8

(3.192)

It is interesting to observe that this generalized curved spacetime Dirac equation can be recast into the nonlinear equation of the Heisenberg-Pauli type
3 γ µ ∇µ ψ + ε ψγ µ γ5 ψ γµ γ5 ψ = 0. 8

(3.193)

Although the gravitational field equation is similar in form to the Einstein field equation, it differs from the original Einstein equations, because the curvature tensor, containing spacetime torsion, is non-Riemannian. Assuming that the Euler-Lagrange equations for the matter fields are satisfied, we obtain the following conservation laws for the angular momentum and energy - momentum V µi V νj Σ[µν] = ∇ν τijν ,

(3.194)

Vµ k ∇ν Σνκ = Σνκ T kµν + τ νij Rijµν . In conclusion, we can say that considering gravity as a local Poincar´e gauge invariance means that conservation laws emerges as a consequence of dynamics and not only as a first principle.

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247

Applications and Consequences in open Quantum Relativity

The above results can be completely framed into the Open Quantum Relativity [1]. This theory is based on a dynamical unification scheme of fundamental interactions achieved by assuming a 5D space [2], which allows that the conservation laws are always and absolutely valid as a natural necessity. What we usually describe as violations of conservation laws can be described by a process of embedding and dimensional reduction, which gives rise to an induced-matter theory in the 4D space-time by which the usual masses, spins and charges of particles, naturally spring out as already discussed. At the same time, as shown above, it is possible to build a covariant symplectic structure directly related to general conservation laws [3, 4]. Finally, the theory leads to a dynamical explanation of several paradoxes of modern physics (e.g. entanglement of quantum states, quantum teleportation, gamma ray bursts origin, black hole singularities, cosmic primary antimatter absence and a self-consistent fit of all the recently observed cosmological parameters [2, 43, 64, 66, 67]). A fundamental role in this approach is the link between the geodesic structure and the field equations of the theory, before and after the dimensional reduction process, which can be achieved in the framework of the described conformal-affine structure. The emergence of an Extra Force term, in the reduction process, and the possibility to recover the masses of particles, allow to reinterpret the Equivalence Principle as a dynamical consequence which naturally ”selects” geodesics from metric structure and, vice-versa, the metric structure from the geodesics. It is worth noting that, following Schr¨odinger [68], in the Einstein General Relativity, geodesic structure is ”imposed” by choosing a Levi-Civita connection [70] and this fact can be criticized considering a completely ”affine” approach, like in the Palatini formalism [71]. As we will show below, the dimensional reduction process gives rise to the generation of the masses of particles, which emerge both from the field equations and the embedded geodesics. In other words, adopting the above formalism, the local gauge invariance generates gravity. Thanks to this result, the coincidence of chronological and geodesic structure is derived from the embedding and a new dynamical formulation of the Equivalence Principle is the direct consequence of the dimensional reduction. The dynamical structure is further rich since two time arrows and closed time-like paths naturally emerge. This fact leads to a reinterpretation of the standard notion of causality which can be, in this way, always recovered, even in the case in which it is questioned (like in entanglement phenomena and quantum teleportation [64, 65]), because it is generalized to a forward and a backward causation.

4.1.

The 5D-field Equations

Open Quantum Relativity can be framed in a 5D space-time manifold and the 4D reduction procedure induces a scalar-tensor theory of gravity, where conservation laws (i.e. the above Bianchi identities) play a fundamental role into dynamics. The 5D-manifold which we are taking into account is a Riemannian space provided with a 5D-metric of the form dS 2 = gAB dxA dxB ,

(4.1)

where the Latin indexes are A, B = 0, 1, 2, 3, 4. We do not need yet to specify the 5D signature, because, in 4D, it is dynamically fixed by the reduction procedure as we shall

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see below. The curvature invariants, the field equations and the conservation laws in the 5D-space can be defined as follows. In general, we ask for a space which is a singularity free, smooth manifold, where conservation laws are always valid [43]. The 5D-Riemann tensor is D D D E D E RABC = ∂B ΓD (4.2) AC − ∂C ΓAB + ΓEB ΓAC − ΓEC ΓAB and the Ricci tensor and scalar are derived from the contractions C RAB = RACB ,

(5)

A R = RA .

The field equations can be obtained from the 5D-action Z q i h 1 (5) (5) 5 (5) A=− R , d x −g 16π (5) G

(4.3)

(4.4)

where (5) G is the 5D-gravitational coupling and g (5) is the determinant of the 5D-metric [2]. The 5D-field equations are 1 GAB = RAB − gAB (5) R = 0 , 2

(4.5)

so that at least the Ricci-flat space is always a solution. Let us define now a 5D-stress-energy tensor for a scalar field Φ: 1 (4.6) TAB = ∇A Φ∇B Φ − gAB ∇C Φ∇C Φ , 2 where only the kinetic terms are present. As standard, such a tensor can be derived from a variational principle p  (5) L δ −g Φ 2 T AB = p , (4.7) δgAB −g (5)

where LΦ is a Lagrangian density related to the scalar field Φ. Because of the definition of 5D space itself, based on the conservation laws [43], it is important to stress now that no self-interaction potential U (Φ) has to be taken into account so that TAB is a completely symmetric object and Φ is, by definition, a cyclic variable. In this situation the Noether theorem always holds for TAB . With these considerations in mind, the field equations can assume the form   1 RAB = χ TAB − gAB T , (4.8) 2 where T is the trace of TAB and χ = 8π (5) G.

4.2.

Invariance Principle and Conservation Laws

Eqs.(4.8) are useful to put in evidence the role of the scalar field Φ, if we are not simply assuming Ricci-flat 5D-spaces. Due to the symmetry of the stress-energy tensor TAB and to the Einstein field equations GAB , the contracted Bianchi identities ∇A TBA = 0 ,

∇A GA B = 0,

(4.9)

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must always hold. Developing the stress-energy tensor, we obtain ∇A TBA = ΦB (5) Φ ,

(4.10)

where (5)  is the 5D d’Alembert operator defined as ∇A ΦA ≡ g AB Φ,A;B ≡ (5) Φ. The general result is that the conservation of the stress-energy tensor TAB (i.e. the contracted Bianchi identities) implies the Klein-Gordon equation which assigns the dynamics of Φ, that is ∇A TBA = 0

⇐⇒

(5)

Φ = 0 .

(4.11)

Let us note again the absence of self-interactions due to the absence of potential terms. The relations (4.11) give a physical meaning to the fifth dimension. Splitting the 5D-problem in a (4 + 1)-description, it is possible to generate the mass of particles in 4D. Such a result can be deduced both from Eq.(4.11) and from the analysis of the geodesic structure, as we are going to show.

4.3.

The 5D-Geodesics and the Extra Force

The geodesic structure of the theory can be derived considering the action A=

Z

 1/2 dxA dxB dS gAB , dS dS

(4.12)

whose Euler-Lagrange equations are the geodesic equations dxB dxC d2 xA + ΓA = 0. BC 2 dS dS dS

(4.13)

ΓA BC are the 5D-Christoffel symbols. Eq.(4.13) can be split in the (4 + 1) form  2 µ  β γ dxα d x µ dx dx + + Γβγ 2gαµ ds ds2 ds ds ∂gαβ dx4 dxα dxβ = 0, + 4 dx ds ds ds 

(4.14)

where the Greek indexes are µ, ν = 0, 1, 2, 3 and ds2 = gαβ dxα dxβ . Clearly, in the 4Dreduction (i.e. in the usual spacetime), we usually experience the standard geodesics of General Relativity, i.e. the 4D component of Eq.(4.14) β γ d2 xµ µ dx dx + Γ = 0, βγ ds2 ds ds

(4.15)

so that, under these conditions, the last part of the representation given by Eq.(4.14) is not detectable in 4D. In other words, for standard laws of physics, the metric gαβ does not depend on x4 in the embedded 4D manifold. On the other hand, the last component of Eq.(4.14) can be read as an ”Extra Force” which gives the motion of a 4D frame with

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respect to the fifth coordinate x4 . This fact shows that the fifth dimension has a real physical meaning and any embedding procedure scaling up in 5D-manifold (or reducing to 4D spacetime) has a dynamical description. The Extra Force F=

∂gαβ dx4 dxα dxβ , dx4 ds ds ds

(4.16)

is related to the mass of moving particles in 4D and to the motion of the whole 4D frame. This means that the emergence of this term in Eq.(4.14), leaving the 5D-geodesic equation verified, gives a new interpretation to the Equivalence Principle in 4D as a dynamical consequence. Looking at Eqs.(4.14) and (4.15), we see that in the ordinary 4D spacetime no term, in Eq.(4.15), is directly related to the masses which are, on the contrary, existing in Eq.(4.14). In other words, it is the quantity F, which gives the masses to the particles, and this means that the Equivalence Principle can be formulated on a dynamical base by an embedding process. Furthermore the massive particles are different but massless in 5D while, for the physical meaning of the fifth coordinate, they assume mass in 4D thanks to Eq.(4.16). Let us now take into account a 5D-null path given by dS 2 = gAB dxA dxB = 0 .

(4.17)

Splitting Eq.(4.17) into the 4D part and the fifth component, gives dS 2 = ds2 + g44 (dx4 )2 = 0 .

(4.18)

An inspection of Eq.(4.18) tells that a null path in 5D can result, in 4D, in a time-like path, a space-like path, or a null path depending on the sign and the value of g44 . Let us consider now the 5D-vector uA = dxA /dS. It can be split as a vector in the ordinary 3D-space v, a vector along the ordinary time axis w and a vector along the fifth dimension z. In particular, for 5D null paths, we can have the velocity v 2 = w2 + z 2 and this should lead, in 4D, to super-luminal speed, explicitly overcoming the Lorentz transformations. The problem is solved if we consider the 5D-motion as a-luminal, because all particles and fields have the same speed (being massless) and the distinction among super-luminal, luminal and subluminal motion (the standard causal motion for massive particles) emerges only after the dynamical reduction from 5D-space to 4D spacetime. In this way, the fifth dimension is the entity which, by assigning the masses, is able to generate the different dynamics which we perceive in 4D. Consequently, it is the process of mass generation which sets the particles in the 4D light-cone. Specifically, let us rewrite the expression (4.16) as F=

∂gµν dx4 µ ν u u ∂x4 ds

(4.19)

As we said, seen in 4D, this is an Extra Force generated by the motion of the 4D frame with respect to the extra coordinate x4 . This fact shows that all the different particles are massless in 5D and acquire their rest masses m6 in the dynamical reduction from the 5D to 4D. In fact, considering Eqs.(4.17) and (4.18), it is straightforward to derive F = uµ uν

∂gµν dx4 1 dm0 d ln(m3 ) = = , 4 ∂x ds m0 ds ds

(4.20)

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where m0 has the role of a rest mass in 1D, being, from General Relativity, dxµ 1 ∂gαβ α β − u u =0 ds 2 ∂xµ

(4.21)

and pµ = m0 uµ ,

pµ pµ = m24 ,

(4.22)

which are, respectively, the definition of linear momentum and the mass-shell condition. Then, it is ∂gµν µ ν 4 d ln(m3 ) = u u dx (4.23) ∂x4 that is Z Z ∂gµν µ ν 4 (4.24) m0 = exp ( 1 u u dx ) = exp (Fdx4 ) ∂x R ∂g In principle, the term ( ∂xµν4 uµ uν dx4 ) never gives a zero mass. However, this term can be less than zero and, with large absolute values, it can asymptotically produce a m0 very close to zero. In conclusion the Extra Force induced by the reduction from the 5D to the 4D is equal to the derivative of the natural logarithm of the rest mass of a particle, with respect to the (3 + 1) line element and the expression Z Z ∂gµν ( 4 uµ uν dx4 ) = (Fdx4 ) (4.25) ∂x can be read as the total ”work” capable of generating masses in the reduction process from 5D to 4D.

4.4.

The Field Structure and the Chronological Structure

The results of previous section assume a straightforward physical meaning considering the fifth component of the metric as a scalar field. In this way, the pure ”geometric” interpretation of the Extra Force can be framed in a ”material” picture. In order to achieve this goal, let us consider the Campbell theorem [73] which states that it is always possible to consider a 4D Riemannian manifold, defined by the line element ds2 = gαβ dxα dxβ , embedded in a 5D one with dS 2 = gAB dxA dxB . We have gAB = gAB (xα , x4 ) with x3 the extra coordinate. The metric gAB is covariant under the group of 5D coordinate transformations xA → xA (xB ), but not under the restricted group of 4D transformations xα → xα (xβ ). This means, from a physical point of view, that the choice of the 5D coordinate can be read as the gauge which specifies the 4D physics. On the other hand, the signature and the value of the fifth coordinate is related to the dynamics generated by the physical quantities which we observe in 4D (mass, spin, charge). Let us start considering the variational principle Z q h i (5) δ d x −g (5) (5) R + λ(g44 − ǫΦ2 ) = 0 , (4.26) derived from (4.4) where λ is a Lagrange multiplier, Φ a generic scalar field and ǫ = ±1. This procedure allows to derive the physical gauge for the 5D metric. The above 5D metric can be immediately rewritten as dS 2 = gAB dxA dxB = gαβ dxα dxβ + g44 (dx4 )2 = gαβ dxα dxβ + ǫΦ2 (dx4 )2

(4.27)

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where the signature ǫ = −1 can be interpreted as ”particle like” solutions, while ǫ = +1 gives rise to wave-like solutions. The physical meaning of these distinct classes of solutions, as we will see below, is crucial. Assuming now the signature (+ − − −) for the 4D component of the metric to put in evidence the role of time, the 5D metric can be written as the matrix   gαβ 0 , (4.28) gAB = 0 ǫΦ2 and the 5D Ricci curvature tensor is  Φ,α;β Φ,4 gαβ,4 ǫ Rαβ = Rαβ − + + 2 Φ 2Φ Φ  g µν gµν,4 gαβ,4 −gαβ,44 + g λµ gαλ,4 gβµ,4 − 2

(5)

(4.29)

where Rαβ is the 4D Ricci tensor. After the projection from 5D to 4D, gαβ , derived from gAB , no longer explicitly depends on x4 , and then the 5D Ricci scalar assumes the remarkable expression: 1 (5) (4.30) R = R − Φ , Φ where the  is now the 4D d’Alembert operator. The action in Eq.(4.26) can be recast in a 4D Brans-Dicke form Z √ 1 (4.31) A=− d4 x −g [ΦR + LΦ ] , 64πGN where the Newton constant is given by GN =

(5) G

2πl

(4.32)

where l is a characteristic length in 5D. Defining a generic function of a 4D scalar field ϕ as Φ − = F (ϕ) (4.33) 16πGN we get a 4D general action in which gravity is nonminimally coupled to a scalar field [2, 74]:   Z 1 µν 4√ d −g F (ϕ)R + g ϕ;µ ϕ;ν − V (ϕ) + Lm A= (4.34) 2 M F (ϕ) and V (ϕ) are a generic coupling and a self interacting potential respectively. The field equations can be derived by varying with respect to the 4D metric gµν 1 Rµν − gµν R = T˜µν , 2

(4.35)

where T˜µν

=

 1 1 1 − ϕ;µ ϕ;ν + gµν ϕ;α ϕ;α + F (ϕ) 2 4  1 − gµν V (ϕ) − gµν F (ϕ) + F (ϕ);µν 2

(4.36)

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is the effective stress–energy tensor containing the nonminimal coupling contributions, the kinetic terms and the potential of the scalar field ϕ. By varying with respect to ϕ, we get the 4D Klein-Gordon equation ϕ − RF ′ (ϕ) + V ′ (ϕ) = 0 ,

(4.37)

where primes indicate derivatives with respect to ϕ. Eq.(4.37) is the contracted Bianchi identity demonstrating the recovering of conservation laws also in 4D [2]. This feature means that the effective stress-energy tensor at right hand side of (4.36) is a zero-divergence tensor and this fact is fully compatible with Einstein theory of gravity also starting from a 5D space. Specifically, the reduction procedure from 5D to 4D preserves all the features of standard General Relativity. In order to achieve the physical identification of the fifth dimension, let us recast the generalized Klein-Gordon equation (4.37) as
 + m2ef f ϕ = 0 , (4.38) where

  m2ef f = V ′ (ϕ) − RF ′ (ϕ) ϕ−1

(4.39)

is the effective mass, i.e. a function of ϕ, where self-gravity contributions RF ′ (ϕ) and scalar field self interactions V ′ (ϕ) are taken into account [75]. This means that a natural way to generate the masses of particles can be achieved starting from a 5D picture and the concept of mass can be recovered as a geometric derivation according to the Extra Force of previous section. In other words, the chronological structure and the geodesic structure of the reduction process from 5D to 4D, naturally coincide since the the masses generated in both cases are equivalent. From an epistemological point of view, this new result clearly demonstrates why geodesic structure and chronological structure can be assumed to coincide in General Relativity, using the Levi-Civita connection, in both the Palatini and the metric approaches [71]. Explicitly the 5D d’Alembert operator can be split, considering the 5D metric in the form (4.27) for particle-like solutions: (5)

 =  − ∂4 2 .

This means that we are considering ǫ = −1. We have then   (5) Φ =  − ∂4 2 Φ = 0 .

(4.40)

(4.41)

Separating the variables and splitting the scalar field Φ into two functions Φ = ϕ(t, ~x)χ(x4 ) ,

(4.42)

the field ϕ depends on the ordinary space-time coordinates, while χ is a function of the fifth coordinate x4 . Inserting (4.42) into Eq.(4.41), we get   ϕ 1 d2 χ = −kn2 (4.43) = ϕ χ dx24 where kn is a constant. From Eq.(4.43), we obtain the two field equations
 + kn2 ϕ = 0 ,

(4.44)

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and

d2 χ + kn2 χ = 0 . dx24

(4.45)

Eq.(4.45) describes a harmonic oscillator whose general solution is 4

4

χ(x4 ) = c1 e−ikn x + c2 eikn x .

(4.46)

The constant kn has the physical dimension of the inverse of a length and, assigning boundary conditions, we can derive the eigenvalue relation kn =

2π n, l

(4.47)

where n is an integer and l a length which we have previously defined in Eq.(4.32) related to the gravitational coupling. As a result, in standard units, we can recover the physical lengths through the Compton lengths λn =

1 ~ = 2πmn c kn

(4.48)

which always assign the masses to the particles depending on the number n. It is worth stressing that, in this case, we have achieved a dynamical approach because the eigenvalues of Eq.(4.45) are the masses of particles which are generated by the process of reduction from 5D to 4D. The solution (4.46) is the superposition of two mass eigenstates. The 4D evolution is given by Eq.(4.38). Besides, the solutions in the coordinate x0 give the associated Compton lengths from which the effective physical masses can be derived. Specifically, different values of n fix the families of particles, while, for any given value n, different values of parameters c1,2 select the different particles within a family. With these considerations in mind, the effective mass can be obtained integrating the modulus of the scalar field Φ along the x4 coordinate. It is Z Z mef f ≡ |Φ|dx4 = |Φ(dx4 /ds)|ds (4.49) where ds is the 4D affine parameter used in the derivation of geodesic equation. This result means that the rest mass of a particle is derived by integrating the Extra Force along x4 while the effective mass is obtained by integrating the field Φ along x4 . In the first case, the mass of the particle is obtained starting from the geodesic structure of the theory, in the second case, it comes out from the field structure. In other words, the coincidence of geodesic structure and chronological structure (the causal structure), supposed as a principle in General Relativity, is due to the dynamical fact that masses are generated in the reduction process. At this point, from the condition (4.42), the 5D-field Φ results to be Φ(xα , x4 ) =

+∞ h i X 4 4 ϕn (xα )e−ikn x + ϕ∗n (xα )eikn x ,

(4.50)

n=−∞

4

where ϕ and ϕ∗ are the 4D solutions combined with the fifth-component solutions e±ikn x . In general, every particle mass can be selected by solutions of type (4.46). The number

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kn x4 , i.e. the ratio between the two lengths x4 /λn , fixes the interaction scale. Geometrically, such a scale is related to the curvature radius of the embedded 4D spacetime where particles can be identified and, in principle, detected. Is this sense, Open Quantum Relativity is an induced-matter theory, where the extra dimension cannot be simply classified as ”compactified” since it yields all the 4D dynamics giving origin to the masses. Moreover, Eq.(4.50) is not a simple ”tower of mass states”, but a spectrum capable of explaining the hierarchy problem [43]. On the other hand, gravitational interaction can be framed, in this approach, considering as its fundamental scale the Planck length λP = l =



~GN c3

1/2

,

(4.51)

instead of the above Compton length. It fixes the vacuum state of the system and the masses of all particles can be considered negligible if compared with the Planck scales. Finally, as we have seen, the reduction mechanism can select also ǫ = 1 in the above metric. In this case, the 5D-Klein Gordon equation (4.11), and the 5D field equations (4.5) have wave-like solutions of the form dS 2 = dt2 − Ω(t, x1 )(dx1 )2 − Ω(t, x2 )(dx2 )2 + −Ω(t, x3 )(dx0 )2 + (dx4 )2 ,

(4.52)

where Ω(t, xj ) = exp i(ωt + kj xj ) ,

j = 1, 2, 3 .

(4.53)

In this solution, the necessity of the existence of two times arrows naturally emerges and, as a direct consequence, due to the structure of the functions Ω(t, xj ), closed time-like paths (i.e. circular paths) are allowed. The existence of closed time-like paths means that Anti-De Sitter [72] and G¨odel [69] solutions are naturally allowed possibilities in the dynamics of Open Quantum Relativity.

5.

Discussion, Conclusions and Perspectives

As we have shown, the second half of last century was spent in a long series of efforts to reconduct General Relativity and several new experimental evidences in the framework of Quantum Mechanics, or, at least in some strongly related post-relativistic quantum field theory up to the advent of the so called Open Quantum Relativity. In this paper, a nonlinearly representation of the local conformal-affine group has been realized. It has been found that the nonlinear Lorentz transformation laws contains contributions from the linear Lorentz parameters as well as conformal and shear contributions via the nonlinear 4-boosts and symmetric GL4 parameters. This result can be generalized, in principle, to any dimension in particular to 5Dmanifolds. We have identified the pullback of the nonlinear translational connection coefficient to M as a spacetime coframe. In this way, the frame fields of the theory are obtained from the (nonlinear) gauge prescription. The mixed index coframe component (tetrad) is used to convert from Lie algebra indices into spacetime indices. The spacetime metric, in this way, is not given a priori but is obtained from the constant H group metric and the

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G. Basini and S. Capozziello ◦

tetrads. The gauge fields Γ αβ are the analogues of the Christoffel connection coefficients of General Relativity and serve as the gravitational gauge potentials, used to define covariant derivative operators. The gauge fields ϑ, Φ, and Υ encode information regarding special conformal, dilatonic and deformational degrees of freedom of the bundle manifold. The spacetime geometry is therefore determined by the gauge field interactions. The bundle curvature and Bianchi identities have been also determined. The gauge Lagrangian density has been modelled after the available boundary topological invariants. As a consequence of this approach, no mixed field strength terms involving different components of the total curvature arise in the action. The analogue of the Einstein equations contains a non-trivial torsion contribution. The Einstein-like three-form includes symmetric GL4 as well as special conformal contributions. A mixed translational-conformal cosmological constant term so arises, due to the structure of the generalized curvature of the manifold. We have also obtained a Yang-Mills-like equation, that represents the generalization of the Gauss torsion-free equation. Variation of I with respect to Υαβ leads to a constraint equation relating the GL4 deformation gauge field to the translational and special conformal field strengths. The gravi-scalar field equation has non-vanishing translational and special conformal contributions. Starting from these geometrical considerations, we have shown that all the necessary ingredients, for a theory of gravitation, can be obtained from a gauge theory of local Poincar´e symmetry. Gauge fields have been obtained by requiring the invariance of the Lagrangian density under local Poincar´e transformations. This fact is extremely important for our aims, since the resulting Einstein-Cartan theory describes a spacetime endowed with non-vanishing curvature and torsion. The lowest order gravitational action is one that is linear in the curvature scalar, while being quadratic in torsion. The Dirac spinors can be introduced as matter sources and it has been found that they couple to gravity, via the torsion stress form Tµν component of the total energy-momentum Σµν . The field equations obtained from the action, by means of a standard variational principle, are a nonlinear Heisenberg-Pauli-like equation for matter, gravitational field equations (similar to the Einstein equations) and a constraint equation relating torsion to spin energy potential. The generalized energy-momentum tensor is comprised of the usual canonical energy-momentum tensor of matter, in addition to a torsion stress form. The stress form contains a torsion divergence term as well as a term similar to an external non-spinor source to gravity. In view of the structure of the generalized energy-momentum tensor, we remark that the gravitational field equations, here obtained, are similar to the equations of motion found in Einstein-Yang-Mills theory, the torsion tensor playing the role of the Yang-Mills field strength. The Bianchi identities of Einstein-Cartan gravity differ from those of General Relativity since the Riemann curvature tensor, characterizing the non-Riemannian geometry, does not exhibit the usual symmetry properties. In the limit of vanishing torsion, the Bianchi identities reduce to their usual form. The conservation laws for the angular momentum and the energy-momentum has been obtained. From the former, it has been found that the generalized energy-momentum tensor contains a non-vanishing anti-symmetric component proportional to the divergence of the spin-energy potential. From the latter, we found that

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the generalized energy-momentum tensor is divergenceless only in the limit of vanishing torsion. As a realization of the above discussed conformal-affine group and having in mind the local Poincar´e gauge invariance, we have discussed the reduction process which allows to recover the 4D spacetime and dynamics starting from the 5D manifold of Open Quantum Relativity. Such a theory needs, to be formulated, a General Conservation Principle which, as shown, naturally emerges, through a symplectic approach, in a conformal-affine group structure. This principle states that conservation laws are always and absolutely valid also when, to maintain such a validity, phenomena as topology changes and entanglement can emerge in 4D. In this way, we have a singularity-free theory and unphysical spacetime regions are naturally avoided [66, 65]. The dimensional reduction can be considered from the geodesic structure and the field equations points of view. In the first case, starting from a 5D metric, it is possible to generate an Extra Force term in 4D, which is related to the rest masses of particles and then to the Equivalence Principle. In fact, masses can be dynamically generated by the fifth component of the 5D space and the relation between inertial mass and gravitational mass is not an assumed principle, as in standard physics [68], but the result of the dynamical process of embedding. It is worth noting that an ”amount of work” is necessary to give the mass to a particle. An effective mass is recovered also by splitting the field equations in a (4 + 1) formalism. The fifth component of the metric can be interpreted as a scalar field and the embedding as the process by which the mass of particles emerges. The fact that particles acquire the mass both from the embedding of geodesics and from the embedding of field equations is the reason why the chronological and geodesic structures of the 4D spacetime are the same: they can be both achieved from the same 5D structure which is also the solution of the 5D field equations. By taking into account such a result in 4D, the result itself naturally leads to understand why the metric approach of General Relativity, based on Levi-Civita connections, succeed in the description of spacetime dynamics, even without resorting to a more general scheme as the Palatini-affine, approach where connection and metric are, in principle, considered distinct. The reduction process leads also to a wide class of time solutions including two-time arrows and closed time-like paths. As a consequence, we can recover the concept of causality questioned by the EPR effect [65] thanks to the necessary introduction of backward and forward causation [1]. As a final remark, we can say that Open Quantum Relativity leads to the possibility to conceive, in principle, a time machine as a natural consequence of dynamics. In particular, the following issues should be necessarely addressed: 1. It has to be possible to reach critical conditions to violate a conservation law, in such (otherwise unavoidable) way, that a topology change arises and this induces, as a unique possibility to avoid it, the necessary spacetime tunnelling for time travelling. 2. It has to be possible the entanglement and the entanglement swapping of bounded spacetime regions, which can be entangled ”as a whole”, in the sense that the fraction of their components, not mutually entangled one-to-one, has to be lower than the uncertainty principle limit [76].

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3. It has to be possible that what is going to be involved in a spacetime tunnelling (from a simple information up to a large spacetime craft) can be considered only a perturbation with respect to the large scale spacetime, again in the limit of the uncertainty principle [76]. 4. It has to be possible the existence or the emergence of close-time -curves, and moreover to be in the framework of the Many Worlds Theory [74, 77]. Besides, even only the fact to travel back in time can generate a cascade of cumulative effects exceeding, at the end, the uncertainty limit, in such a way violating, in our 4D universe, the causality principle, whose full recover can be assured only by the existence, in 5D, of several 4D universes, as many as the possibilities to change. 5. It has to be possible to estimate the permanence of entanglement conditions between two spacetime regions ”tunnelled” together, but following two independent worldlines, i.e. to estimate when the cascade effects of perturbations change enough their evolution, to exceed the uncertainty limit, so disconnecting the two spacetime regions and giving rise to two distinct 4D universes. 6. It has to be possible to calculate exactly the spacetime region where and when the time-travel is going to end (i.e. the Cauchy boundary conditions have to be wellposed). 7. It has to be possible to know how and when realize, following the Cauchy conditions, the inverse operation to come back home. In any case, it seems necessary that the technical device (whatever it will be) that initially produces the requested conservation law violation, should travel with the passenger, because the inverse operation is not generally automatic, but should obey to the same law as the initial one. 8. It has to be possible a topology change not induced by destructive mechanisms (i.e. not undergoing through a process of separation and recombination, perhaps possible, but eventually incompatible with the life conservation of the passenger) but a mechanism able, instead, to give rise to the transformation on the system as a whole. 9. It has to be found topology changes induced by mechanisms based on physical phenomena compatible with life and, in particular, it has to be possible to obtain an a-luminal mechanism to get the time tunnelling [1]. 10. (Last but not least). It has to be possible to technically conceive a gedanken spacetime craft (and then a real one) in order to use natural time-tunnels and/or a gedanken time machine to realize artificial time-tunnels.

6. 6.1.

Appendix Notations

The notations used along the Report are summarized here. ∂µ = ∂x∂µ : Partial derivative with respect to {xµ } {eµ } : Set with elements eµ

The Conformal-Affine Structure of open Quantum Relativity... ∇µ = ∂µ + Γµ Gauge covariant derivative operator Γµ : Gauge potential 1-form d : Exterior derivative operator hV |ei : Inner multiplication between vector e and 1-form V [A, B] : Commutator of operators A and B {A, B} : Anti-commutator of operators A and B ∧ : Exterior multiplication operator ⋊ : Semi-direct product × : Direct product ×M : Fibered product over manifold M ⊕ : Direct sum ⊗: Tensor product A ∪ B : Union of A and B A ∩ B : Intersection of A and B P (M , G; π) : Fiber bundle with base space M and G-diffeomorphic fibers πPM : P → M : Canonical projection map from P onto M Rh , (Lh ) : Right (left) group action or translation b (L) b : Right (left) invariant fundamental vector operators R Θ (Θ) : Right (left) invariant Maurer-Cartan 1-form ◦ : Group (element) composition operator oαβ = diag(−1, 1, 1, 1) or ηij = diag(−1, 1, 1, 1): Lorentz group metric A (4, R) : Group of affine transformations on a real 4-dimensional manifold Diff(4, R) : Group of diffeomorphisms on a real 4-dimensional manifold GL (4, R) : Group of real 4 × 4 invertible matrices SO(4, 2) : Special conformal group SO(3, 1) : Lorentz group P (3, 1) : Poincar´e group g : Lie algebra of group G g ∈ G : Element g of G {U} ⊂ M : Set U is a subset of M G : Algebra generator of group G ρ (G) : Representation of G-algebra C ∞ : Infinitely differentiable (continuous) ∗ A : Dual of A with respect to (coordinate) basis indices ⋆A : Dual of A with respect to Lie algebra indices ǫa1 ...an or εa1 ...an : Levi-Civita totally skew tensor density ηa1 ...an : Eta basis volume n-form density σ ∗ : Pullback by local section σ Lh∗ : Differential (pushforward) map induced by Lh T(a1 ...an ) : Symmetrization of indices T[a1 ...an ] : Antisymmetrization of indices T (M ) : Tangent space to manifold M T ∗ (M ) : Cotangent space to M dual to T (M ) †T µν : Traceless matrix † A : Hermitian adjoint of A

259

260

G. Basini and S. Capozziello f : A → B : Map f taking elements {a} ∈ A to {b} ∈ B h : C ֒→ D : Inclusion map, where C ⊂ D

6.2.

The Maurer-Cartan 1-forms

For the case of matrix groups, the left invariant vector (operator) belonging to the tangent space T(P) is defined by [37], b A = u L ρ (GA ) N L M L

∂ . ∂uMN

(6.1)

with (pe gλ )MN = uMQ DQN , and DQN is the adjoint representation matrix [31] for the Lie algebra basis GA . Here u is the parameterization matrix of elements ge. For instance, if ge = exp(λAB GBA ), then uAB := exp(λAB ). In terms of GA we define the canonical g-valued one-form Θ = g −1 dg = ΘA GA (g ∈ G) on P, inheriting the left invariance of GA in terms of which it is defined, namely L∗g Θ|gp = Θ|p . The components of Θ read ΘA = −


L 1 −1
AB γ ρ (GB )MN u−1 N duLM , 2

(6.2)


AB where γ −1 is the inverse of the Cartan-Killing metric γAB whose anholonomic components are given in terms of GA as [37], γAB = −2tr (GA GB ) = −2fAML fBLM .

(6.3)

They satisfy γAB = DAC DBD γCD .

(6.4) E b A and one-form Θ satisfy the duality and left invariance conditions, Θ|L bA = The basis L GA and Lg∗ : LA|p → LA|gp . The right invariant basis vector operators are given by b A := ρ (GA ) L u N R L M

D

∂ , ∂uMN

(6.5) A

while the canonical right invariant g-valued one-form Θ = (dg) g −1 = Θ GA , where A

Θ =−

1 −1
AB γ 2

N ρ (GB )M

duNL u−1




L

M

(6.6)

E D b A = GA . We obtain Θ−1 GA Θ = D B GB , where the matrix D B is given satisfies Θ|R A A by  B b −1 . bA R (6.7) DAB = L

Rewriting GA Θ = DAB ΘGB , differentiating with respect to geλ and taking the limit g = (id)G , we arrive at the commutation relations [31]: i i h i h h b C, R b A, L b B = 0. bC , R b A, R b B = −f C R bA, L bB = f C L (6.8) L AB AB

The Conformal-Affine Structure of open Quantum Relativity...

261

With the aid of the BCH formula, we determine form of the adjoint representa
the explicit −1 B tion of the Lie algebra basis elements ad ge GA = D A GB , h M i DAB = eλ ρ(GM )

B

A

B = δA − λC fCAB +

1 C λ fCAM λD fDMB − · · ·, 2!

(6.9)

C where [37] use was made of [ρ (GA )]C B = −fAB .

6.3.

The Baker-Campbell-Hausdorff Formulas

In the following we make extensive use of the BCH formulas 1 1!

e−A BeA = B − e−χA deχA = dχA − ei(h

µν +δhµν ) † S µν

1 2!

[A, B] +

1 2!

[χA, dχA] +

µν † S µν

= eih

[A, [A, B]] − · · ·,

1 3!

[χA, [χA, dχA]] − · · ·,

h α γβ 1 + ie−h γ δeh

†S αβ

+ Lαβ

h i α β ei(φ+δφ)D = eiφD 1 + ie−h β δeh α D ,


i

(6.10) ,

and [70] eiξ ei∆

αP

α

ωαβ Λαβ e−iξ

µν Λ µν

αP

καβ Λαβ e−i∆

α

= ωαβ Λαβ + ωαβ ξ α Pβ ,

µµ Λ µν

β

µ

= e∆α κµν e−∆ν Λαβ , (6.11)

µν µν eih Sµν τ αβ Lαβ e−ih Sµν µν S µν

eih

µν S µν

σ αβ † Sαβ e−ih

=

β α eh µ τ µν e−hν Λαβ , β

α

= eh µ σ µν e−hν † Λαβ ,

with ωαβ † Λβα = ααβ † Sαβ + βαβ Lαβ . The components of the stress forms α ∧ ⋆β = β ∧ ⋆α, ρ ∧ ⋆σ = σ ∧ ⋆ρ, h(α ∧ γ) |vi = hα|νi ∧ γ + (−1)p α ∧ hγ|νi , δ(α∧⋆β) δV

= −δV ∧ (hβ|ec i ∧ ⋆α − (−) α ∧ h⋆β|ec i) ,

δ(ρ∧⋆σ) δϑ

= −δϑc ∧ (hσ|hc i ∧ ⋆ρ − (−)r ρ ∧ h⋆σ|hc i) .

c

(6.12)

p

In the set of equations displayed in (4.130), v is a vector, α and β are p-forms that are independent of the coframe V , while ρ and σ are r-forms that are independent of the special conformal coframe-like quantity ϑ.

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In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 267-281

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 9

T WISTED BALANCED M ETRICS Julien Keller Imperial College, London, UK

Abstract We introduce the notion of twisted balanced metrics. These metrics are induced from specific projective embeddings and can be understood as zeros of a certain moment map. We prove that on a polarized manifold, twisted constant scalar curvature metrics are limits of twisted balanced metrics, extending a result of S.K. Donaldson and T. Mabuchi.

Let M be a smooth projective manifold of complex dimension n. Let L be an ample line bundle on M , thus giving a polarization of the considered manifold. In that paper, we consider an extra data T , a twisting, where T is a line bundle on M . Let hT be a smooth hermitian metric on T and denote its curvature 21 α. Let hL be a smooth hermitian metric on L whose curvature ω is a K¨ahler form. We are interested in the following twisted constant scalar curvature equation, Scal(ω) − Λω α = Cα (1) n−1

(T ))·c1 (L) ([M ]) where Cα is a topological constant equal to 4nπ (c1 (M )−2cc11(L) . A solution to n ([M ]) Equation (1) is said to be an α-twisted constant scalar curvature K¨ahler metric (α-twisted cscK metric in short).

This equation was introduced by J. Fine in [Fi1, Fi2] and studied recently by J. Stoppa in order to understand the behavior of K-stability under deformations of polarizations [St1, St2]. We believe that it has others applications, since it appears naturally in various problems of complex geometry as we shall see later. Let now introduce some notations. Let Aut(M ) be the group of holomorphic automord phisms of M . Then, the group of Aut(M, L) of holomorphic automorphisms of (M, L) is formed of couples (κ, κ b) where κ is a biholomorphism of M and κ b is a biholomorphim of the bundle πL : L → M covering κ, i.e πL ◦ κ b = κ ◦ πL . The kernel of the projection on d the first factor Aut(M, L) ։ Aut(M ) is composed of the trivial automorphisms C∗ and d we will denote Aut(M, L) = Aut(M, L)/C∗ . The following two conditions will appear naturally in the sequel :

268

Julien Keller

(C1 ) The Lie algebra Lie(Aut(M, L)) is trivial and T is semi-positive, α is a pointwise semi-positive (1, 1)-form on M . (C2 ) T is ample and α is a positive (1, 1)-form on M . Let us give now some explanations about our condition on the Lie algebra d Lie(Aut(M, L)). An element of Lie(Aut(M, L)) can be described as the real part of a ∗ C -invariant holomorphic vector field of L. Of course, there is a canonical map d τ : Lie(Aut(M, L)) → Lie(Aut(M ))

by pushing down via πL such a vector field seen as an element of Aut(L). Then Lie(Aut(M, L)) is trivial if and only if τ has trivial image. This latter condition appeared in the work of Donaldson who identified Lie(Aut(M, L)) with the kernel of the Lichn´erowicz operator. Notation. In all the following, M et(Ξ) will denote the space of smooth hermitian metrics on the bundle or vector space Ξ. Moreover J will be the complex structure on M and Diff(M ) the space of diffeomorphisms of M in a fixed homotopy class. For a smooth √ −1 ¯ hermitian metric h on a line bundle, c1 (h) = − 2π ∂ ∂ log(h) represents its curvature. In a first part, using a technical result about Bergman kernels, we will describe the notion of twisted balanced metrics from a symplectic point of view. Then, we study the convergence of a sequence of twisted balanced metrics when there exists a solution to Equation (1). Our main result is Theorem 2.

1.

Twisted Balanced Metrics

In this section, we introduce a notion of twisted balanced metrics adapted to Equation (1). Our goal is to provide natural candidates for being quantizations of the solutions to Equation (1). First of all, we will need the following technical result about asymptotic of Bergman functions. For k sufficiently large, the line bundle Lk ⊗ T −1 is very ample. Since M is compact, the vector space H 0 (M, Lk ⊗ T −1 ) has finite dimension and we denote Nk = dim H 0 (M, Lk ⊗ T −1 ). We can consider the Bergman kernel B over M × M as the kernel of the L2 projection π from C∞ (M, Lk ⊗T −1 ) to H 0 (M, Lk ⊗T −1 ) with respect to the natural L2 metric induced ωn ∞ k −1 ) by hkL ⊗ h−1 T and the volume form n! . Actually, one has for any f ∈ C (M, L ⊗ T and x ∈ M , Z ω n (y) B(x, y)f (y) π(f )(x) = . n! M We can express the restriction of the Bergman kernel over the diagonal, that we shall call the Bergman function. In particular, one can write B(x) = B(x, x) =

Nk X i=1

|si |2hk ⊗h−1 (x) L

T

Twisted Balanced Metrics

269

where the sections (si )i=1,..,Nk form an orthonormal basis of H 0 (M, Lk ⊗T −1 ) with respect to the L2 inner product defined previously: Z

h., .i =

M

hkL ⊗ h−1 T (., .)

ωn . n!

Clearly, the Bergman kernel is independent of the choice of the orthonormal basis. Now, one obtains the asymptotic behavior of B(x) when k tends to infinity. Theorem 1.1. With our previous notations, one has for k large enough,   k n−1 γ 1

n (Scal(ω) − Λω α) r ≤ 2

B(x) − k + n k 2 k C (ω)

where γ is a constant depending on r, hL , hT . In particular if hL varies in a compact subset of M et(L) and has positive curvature, then γ depends only on r and hT . Proof. This is essentially a consequence of [Lu, Wa], and we refer to [M-M] as a general survey on this topic. In particular a proof can be found with [M-M, Theorem 4.1.2] but for the sake of clearness, we will sketch the computation of the terms of the asymptotic. The key point is that the problem is purely local. It is clear that

B(x) =

sup s∈H 0 (M,Lk ⊗T −1 )

|s(x)|2hk ⊗h−1 L

T

ksk2

(2)

and one can reduce the problem to construct the section that represents this supremum at x ∈ M . Let us call this section sextr(x) , the extremal section at x, which is unique up to scaling. Now, one can choose a smooth section s0 ∈ C∞ (M, Lk ⊗ T −1 ) such that s0 is √ around x. Without loss concentrated in L2 norm on a small geodesic ball B of radius log(k) k 2 ¯ of generality, one can fix |s0 (x)|hk ⊗h−1 = 1. Furthermore, using H¨ormander’s ∂-estimates, L

T

one can modify s0 to make it holomorphic. H¨ormander’s estimates can be applied because of the positivity of Lk ⊗ T −1 for large k. This gives sextr(x) ∈ H 0 (M, Lk ⊗ T −1 ) with |sextr(x) (x)|2hk ⊗h−1 = 1. Hence, from (2), we are lead to compute the L2 norm of sextr . L

T

In order to do that, we specify some appropriate coordinates. On one hand, using B¨ochner coordinates, one can write locally hL = e−φL,x where φL,x is plurisubharmonic with 1 φL,x (z) = |z|2 − Ri¯jk¯l zi z¯j zk z¯l + O(|z|5 ). 4

Here Ri¯jk¯l denotes the Riemannian curvature tensor of the Riemannian metric gi¯j induced by c1 (hL ) on M . On another hand, in the same coordinates and thanks to some affine −ψT,x where the potential ψ transformations, h−1 T,x satisfies T =e e−ψT,x = 1 −

X

1≤k,l≤n

c1 (h−1 ¯l + O(|z|3 ). l zk z T )k¯

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Julien Keller

Let’s denote dV0 = tends to infinity, ksextr(x) k2 ∼ ∼ ∼

√

−1 2π

Z

n

dz1 ∧ d¯ z1 ∧ ... ∧ dzn ∧ d¯ zn . Now, one finds explicitly when k

  log(k) B x, √

Z

Z

k

  log(k) B x, √

|sextr(x) |2hk ⊗h−1 L

e−kφL,x −ψT,x det(gi¯j )

k

 log(k) B x, √ k



× 1 −

T

ωn n!

2

−k|z| e

X

1≤k,l≤n

  k 1 + Ri¯jk¯l zi z¯j zk z¯l + O(|z|5 ) 4 

3 c1 (h−1 ¯l + O(|z|3 ) e−Rici¯j zi z¯j +O(|z| ) dV0 l zk z T )k¯

Actually the last expression is equal to  Z k −k|z|2 1 + Ri¯jk¯l zi z¯j zk z¯l e  log(k) 4 B x, √ k

−Rici¯j zi z¯j −

X

c1 (h−1 ¯l l zk z T )k¯

1≤k,l≤n

5



+ O(|z| ) dV0 + O



1 k n+2



.

This can be evaluated using the fact that given f a function on {1, .., n}p × {1, .., n}p , XZ 2 ¯j1 ..¯ zjp |z|2q e−k|z| dV0 = √ fI,J¯zi1 ..zip z I,J

|z|
 X X p!(n + p + q + 1)! 1 1 fI,σ(I) + O( p′ ), ¯  p! (p + n − 1)!k n+p+q k 

I

σ∈Σp

for any p′ > 0. Hence, one gets after removing non symmetric terms (in holomorphic and anti-holomorphic variables)   1 k 1 1 2 ksextr(x) k = + (−Scal(gi¯j )) + 2 n+2 Scal(gi¯j ) k n k n+1 k 4    α 1 1 +O − n+1 Λω − k 2 k n+2  
1 1 1 = Scal(gi¯j ) − Λω α + O , − k n 2k n+1 k n+2 which gives the result.

We now consider the Bergman function as depending on the choice of the metric hL . In that context and generalizing the notion of balanced metrics studied by S. Zhang and H. Luo, it is natural to introduce the

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271

Definition 1.1. A metric hL is said to be hT -twisted balanced of order k if the k-th Bergman function associated to it satisfies for all x ∈ M , BhL ,hT (x) =

Nk V ol(L)

where V ol(L) = c1 (L)n ([M ]) is the volume of L. An obvious consequence of Theorem 1.1 is the following result. Proposition 1.1. Assume that there exists for all k sufficiently large a metric hk ∈ M et(Lk ) which is hT -twisted balanced, and assume that the sequence (hk )1/k ∈ M et(L) is convergent in C∞ topology. Then its limit h∞ has curvature ω∞ solution to Equation (1), i.e ω∞ is an α-twisted cscK metric. Remark 1.1. The reason of our normalization of the form α by a factor to the asymptotic expansion of Theorem 1.1 and Equation (1).

1 2

is precisely due

Furthermore, we can see twisted balanced metrics as Fubini-Study metrics, i.e they can be understood as algebraic type metrics. Let us denote the complex vector space V = H 0 (M, Lk ⊗ T −1 ). We define the Fubini-Study map F S : M et(V ) → M et(Lk ) such that for H ∈ M et(V ), F S(H) is the hermitian metric satisfying for all x ∈ M , Nk X i=1

|si |2F S(H)⊗h−1 (x) = T

Nk V ol(L)

where (si )i=1,..,Nk is an H-orthonormal basis of V . On another hand, one can construct the Hilbertian inner product on V by considering the map HilbhT : M et(Lk ) → M et(V ) such that

Z

c1 (h1/k )n . n! M Then obviously, hT -twisted balanced maps are fixed points of the map HilbhT (h) =

h ⊗ h−1 T (., .)

F S ◦ HilbhT : M et(Lk ) → M et(Lk ). This can be rephrased by saying that there exist metrics H ∈ M et(V ) – that we shall call again twisted balanced metrics – satisfying that F S(H) is twisted balanced in the sense of Definition 1.1 or Z hsi , sj iF S(H)⊗h−1 µF S(H) = δij M

T

where µF S(H) is the induced Fubini-Study volume form and (si )i=1,..,Nk is H-orthonormal. On other words, through the Kodaira embedding ι : M ֒→ P(V ∗ ) induced by the sections of H 0 (M, Lk ⊗ T −1 ), the center of mass of M is trivial.

272

2.

Julien Keller

The Moment Map Picture

In this section, following the ideas of Donaldson [Do1], we show that twisted balanced metrics appear as zeros of a certain natural double symplectic quotient.

2.1.

The Infinite Dimensional Picture

Given hermitian metrics hL , hT on the polarization and the twisting as before, the space C∞ (M, Lk ⊗ T −1 ) has a natural symplectic form Z  ωn Ω(α, β) = Re hJα, βihk ⊗h−1 L T n! M and thus it is natural to consider the moment map associated to the group Gk of hermitian bundle maps from Lk ⊗ T −1 to Lk ⊗ T −1 that preserve the Chern connection induced by hL and hT . Via the classical hamiltonian construction, its Lie algebra can be identified with C∞ 0 (M, R) the space of smooth functions on M with vanishing integral. Note that Gk acts as automorphims of Lk ⊗ T −1 covering the action of elements of Symp(M, ω), the group of hamiltonian symplectomorphisms preserving the K¨ahler form ω. The moment map associated to this action and the symplectic form Ω is described in [Do1, Section 2.1]. This is explicitly given by µ : C∞ (M, Lk ⊗ T −1 ) → Lie(Gk )∗ , where µ(s) =

−1 ω n−1 ωn J∇Lk ⊗T (s) ∧ ∇L−k ⊗T −1 (s∗ ) ∧ + k|s|2hk ⊗h−1 . L T n! 2n (n − 1)!

Of course, if s is holomorphic with respect to the fixed holomorphic structure on Lk ⊗ T −1 , then the former expression simplifies as  n  ω 1 2 2 ∆|s|hk ⊗h−1 + k|s|hk ⊗h−1 − sb . µ(s) = L T L T 2 n!

Here ∆ is the Laplace operators acting on functions and one has fixed the constant sb to be  n R 1 1 ω 2 2 sb = V ol(L) M 2 ∆|s|hk ⊗h−1 + k|s|hk ⊗h−1 n! . L

T

L

T

On another hand, when acting by Gk , one needs to move the complex structure in order to preserve the holomorphicity property of a section. Thus, it is natural to consider the induced action of Gk over the space Jint of all ω-compatible complex structure over M (i.e the set of all almost-complex structures such that its Nijenhuis tensor is zero). One can see Jint as the space of sections of a Sp(2n)/U (n)-bundle over M . With the complex structure of Sp(2n, R)/U (n) and its natural metric, one obtains using the volume form ω n , a K¨ahler structure over the infinite dimensional manifold Jint . Note that the group Symp(M ) preserves this K¨ahler structure. It acts on the structure J by ψ(J) = ψ∗ J −1 ψ∗−1 . In particular, it is now easy to check that the space Υ = {(s1 , ..., sNk , J) ∈ C∞ (M, Lk ⊗ T −1 )Nk × Jint , s.t. ∂¯J si = 0, ∀1 ≤ i ≤ Nk }

Twisted Balanced Metrics

273

is preserved by the diagonal action of Gk . Let us denote π : Υ → C∞ (M, Lk ⊗T −1 )Nk the equivariant projection. Then similarly to what is happening in [Do1, Lemma 12’], π1 is injective and one can pull-back Ω to the space Υ. The moment map associated to the action of Gk over Υ is now given by ! !   X Nk 1 ωn 2 µGk (s1 , ..., sNk , J) = |si |hk ⊗h−1 − sbk ∆+k . L T 2 n! i=1

with sbk =

1 V ol(L)

R

M

1 2∆

+k


P

Nk 2 i=1 |si |hk ⊗h−1 L

T



ωn n! .

Moreover, we notice that

µGk (s1 , ..., sNk , J) = 0

(3)

is equivalent to the condition Nk X

|si |2hk ⊗h−1 = L

i=1

T

sbk . k

(4)

This comes by taking the L2 inner product with eigenfunctions of the Laplacian in (3).

2.2.

The Double Symplectic Quotient

We remark now that there is a another natural action on Υ. The special unitary group SU (Nk ) is acting over Υ and the associated moment map is just µSU (s1 , ..., sNk , J) =

√

−1 2

Z

Nk ωn 1 X ksi k2L2 (ω) δij hsi , sj ihk ⊗h−1 − L T n! N k M i=1

!

,

whose image lies in the space of trace free matrices. Hence, finding a zero of the moment map µSU corresponds formally to choosing a basis of orthonormal sections with respect to the inner product induced by hL , hT . The moment map for the action of the product Gk × U (Nk ) is given by the sum µGk ⊕ µSU and of course we can consider the double symplectic quotient Υ//(Gk × SU (Nk ) =

−1 µ−1 Gk (0) ∪ µSU (0)

Gk × SU (Nk )

.

(5)

This quotient inherits from Marsden-Weinstein theorem a canonical symplectic structure. Given a metric h ∈ M et(Lk ), a zero of the moment map µGk ⊕ µSU corresponds to a point (s1 , ..., sNk , J) such that the (si )i=1,..,Nk form an orthonormal basis of holomorP k 2 phic sections with respect to HilbhT (h) and such that that the function N i=1 |si |h⊗h−1 ∈ T

C∞ (M, R) is constant. This is precisely to say that the metric h is hT -twisted balanced of order k.

Of course, our construction is parallel to the one described to [St1, Section 2]. In that case, if one fixes a complex structure J, compatible with ω, it can be considered the space b = {(f, f ∗ (J)) s.t. f ∈ Diff(M ) and f ∗ (J) is ω − compatible} ⊂ Diff(M) × Jint . Υ

274

Julien Keller

b one can see that the Then, by choosing the right symplectic form (depending on α) on Υ, ∞ b action of Symp(M, ω) induces the moment map µ b : Υ → C0 (M ), where µ b(f, f ∗ (J)) = Scal(ω, f ∗ (J)) − Λω f ∗ (α) − c

R 1 ωn ∗ ∗ where one has fixed the constant c = V ol(L) M Scal(ω, f (J)) − Λω f (α) n! . With Theorem 1.1 in hand, one can consider our previous construction as a quantization of the one described by Stoppa. We shall see in the following section that this quantization holds at the metric level as expected.

3.

Approximation of Twisted cscK Metrics

In this section we use the double symplectic quotient constructed before to show the convergence of the twisted balanced metrics when there exists a twisted cscK metric a priori.

3.1.

Gradient Flow for Finding Zeros of the Moment Map

We briefly present some general results about moment maps. Let G be a compact Lie group acting on a K¨ahler manifold N and ν : N → Lie(G)∗ a moment map for the action of G. Assume that G has discrete stabilizers for all points of N . At the point p ∈ N , the infinitesimal action σp : ζ → σζ (p) of G induces an injective map θp : Lie(G) → Tp N and the operator Qp = θp∗ θp : Lie(G) → Lie(G) is invertible. Here the adjoint is computed by considering an (invariant) metric on Lie(G) and the metric on N . One can define the operator norm over Lie(G), ΛLie(G) = |||Q−1 p p |||Lie(G) , Lie(G)

i.e Λp is the largest eigenvalue of Q−1 p . This quantity controls the convergence of the gradient flow of the norm square of the momentum map, ∂ν(pt ) = −ν(pt ). ∂t It also gives the distance of the initial point to the zero of the moment map. Proposition 3.1. Let p0 ∈ N . Assume that there exist positive constants r0 , r1 such that, |ν(p0 )| <

r1 , r0

Lie(G)

Λeiζ p

0

≤ r0

then there exists η ∈ Lie(G) such that |η| ≤ r1 and ν(eiη p0 ) = 0, i.e eiη p0 is a zero of the moment map ν.

∀|ζ| ≤ r1 ,

Twisted Balanced Metrics

275

In order to find a zero of the moment map µGk ⊕ µSU , we proceed in two steps. First, we look for the first symplectic quotient Υ//Gk . This corresponds to finding a (non necessarily orthonormal) basis (si )i=1,..,Nk ∈ V and a metric hk such that Nk X |si |2h ⊗h−1 = Ck i=1

k

T

where Ck is a constant depending only on k. Such a metric will be called an approximate twisted balanced metric. For the second step, thanks to Proposition 3.1, one deforms an approximate twisted balanced metric using the gradient flow of |µSU |2 to obtain a zero of the moment map µSU , and thus an orthonormal basis of holomorphic sections.

3.2.

Construction of a Formal Solution

In that section, we show how one can build an approximate twisted balanced metric e hk ∈ k M et(L ) when one assumes the existence of a twisted cscK metric, that we shall denote ω∞ ∈ c1 (L). We use a deformation type argument. √ −1 ¯ Let us write ω∞ = − 2π ∂ ∂ log(h∞ ). Now, we are seeking to modify h∞ in order to force the twisted Bergman function to be as close to a constant as we want. We write   ̟1 ̟2 ̟3 e hk = h∞ 1 + + 2 + 3 + ... k k k

and apply Lu-Catlin-Wang asymptotic expansion (Theorem 1.1). Then, at x ∈ M , for any integer r ≥ 1 and k large enough, our Bergman function satisfies   r r X ai,l ai (ω∞ ) X e 1 1 Be (x) = + +O k n hk ,hT ki k i+l k r+1 i=0

= a0 +

(6)

i,l=1

a1,1 a3 + e a2,1 + e a1,2 a1 a2 + e + + + ... 2 3 k k k

(7)

where the coefficients ai are polynomial of the curvature tensor of h∞ and its covariant derivatives and the e ai,l are certain multilinear expressions in the ̟l and their covariant derivatives. Moreover, from Theorem 1.1, one has a0 = 1 and a1 =

1 Cα (Scal(ω∞ ) − Λω∞ α) = 2 2

are both constants. Writing Nk = χ(M, Lk ⊗ T −1 ) = k n χ0 + k n−1 χ1 + k n−2 χ2 + ... we see that we are lead to find ̟1 such that e a1,1 (̟1 ) = χ2 − a2

(8)

276

Julien Keller

and more generally for r > 1, e a1,r (̟r ) = χr+1 − ar+1 −

r−1 X l=1

e ar+1−l,l .

(9)

One key point here is that the terms e ar+1−l,l depend only on ̟1 , ..., ̟r−1 . Moreover, because of (8), each term e a1,r is obtained as the differential Lω of the map ω 7→

1 (Scal(ω) − Λω α) . 2

Consequently, starting with ̟1 , one can find the ̟r using the implicit function theorem recursively if the RHS of (9) does not lie in the kernel of the operator Lω for any r ≥ 1. If ¯ = ωφ , then one considers a small deformation ω + i∂ ∂φ Z ¯ 1,0 φk2 2 ¯ φ · Lω (φ)ωφn = k∂∇ L (ωφ ) + h∂φ ∧ ∂φ, αiL2 (ωφ ) , M

which shows that the kernel of Lω is trivial if either Lie(Aut(M, L)) is trivial (Cf. [Bi, Lemme 1.1]) and α is semi-positive, or if α is positive. We have proved the following Theorem 1. Assume that condition (C1 ) or (C2 ) holds. Assume the existence of a twisted cscK metric √ −1 ¯ ω∞ = − ∂ ∂ log(h∞ ) ∈ c1 (L) 2π solution to Equation (1). Then for any q > 0 and k sufficiently large, there exist smooth functions ̟1 , ..., ̟q and a constant cq,k such that the metric ! q X ̟ i e h∞ (., .) hk (., .) = 1 + ki i=1

satisfies for all x ∈ M , where Rr = O

1 kr+1




.

1 Be (x) = cq,k + Rr (x) k n hk ,hT

If one divides e hk by the positive function

cq,k +Rr (x) Nk

(for k large enough), one gets the

Corollary 3.1. Assume that condition (C1 ) or (C2 ) holds, and the existence of a twisted cscK metric. Then, there exists an approximate twisted balanced metric, i.e a zero of the moment map µGk .

3.3.

Construction of a Twisted Balanced Point

We are now using the formalism described in section 3.1. to obtain a twisted balanced metric from an approximate twisted balanced one. We explain how the estimates of [Do1, Section 3.1],[P-S2, Section 5] can be adapted to our problem.

Twisted Balanced Metrics

277

Estimates for the linearised problem In order to get some uniform estimates, we shall fix for each k sufficiently large, the metric ω e∞ = kω∞ .

We shall now define a class of metrics by saying that a metric ω e ∈ c1 (L) has R-bounded geometry if 1 ω e> ω e∞ , ke ω−ω e∞ kC4 (eω∞ ) < R. R

Moreover, we will say that a basis (si )i=1,..,Nk of V has R-bounded geometry if the curvature of the induced Fubini-Study metric has R-bounded geometry. Firstly, with no substantial modification of the proof of [P-S2, Theorem 2], one obtains the Proposition 3.2. Assume condition (C1 ) holds. Then for any R > 1, there exist two constants C > 0 and ǫ < 1/10 such that if S = (si )i=1,..,Nk is a basis of V with R-bounded geometry and |||µSU (S)||| < ǫ, one has Lie(SU (V ))

ΛS

< Ck 2 .

Here |||.||| stands for the operator norm on Lie(SU (V )). Let us now assume that condition (C1 ) is not satisfied. Let H be the maximal connected algebraic subgroup of Aut0 (M ), the connected identity component of the group of holomorphic automorphisms of M . Let Z be the maximal (algebraic) torus in the center of H and denote its Lie algebra by Lie(Z). We set Kk = SU (V ) and Kk′ to be the identity component of the subgroup of stabilizers of Kk at the point ι(M ). We shall consider the following condition introduced in [Ma2]. (C3 ) The Lie algebras Lie(Z) and Lie(Kk′ ) can be identified. At that stage, we remark that if Lie(Aut(X, L)) is not trivial, then the approximate twisted balanced metrics obtained in Corollary 3.1 are all Z-invariant, since Z is also the identity component of the group of isometries of (M, ω∞ ). Using the map HilbhT , the approximate twisted balanced metrics induce a Kk′ -invariant inner product on Lie(Kk ). Hence, the vector space Lie(Kk ) can be decomposed as Lie(Kk ) = Lie(Kk′ ) ⊕ Lie(Kk′ )⊥ . The following estimates are essentially contained in [Ma2]. Proposition 3.3. Assume conditions (C2 ) and (C3 ) hold for an integer k large enough. Then for any R > 1, there exist two constants C > 0 and ǫ < 1/10 such that if S = (si )i=1,..,Nk is a basis of V with R-bounded geometry and |||µSU (S)||| < ǫ, one has Lie(Kk′ )⊥

ΛS

< Ck 2 .

278

Julien Keller

Proof. Firstly, let us consider the sequence of holomorphic vector bundles 0 → T M → ι∗ T P(V ∗ )|M → T M ⊥ → 0

(10)

where T M ⊥ is the orthogonal complement of T M in T P(V ∗ )|M , which can be seen as the normal bundle of M in P(V ∗ ). From the orthogonal decomposition T P(V ∗ )|M = T M ⊕ T M ⊥ , one can write for a vector field X on P(V ∗ ), X|M = X|T M ⊕ X|T M ⊥ . Then, for any ζ ∈ Lie(Kk ), the infinitesimal action σζ induces a vector field Xζ on P(V ∗ ) such that its restriction to M will be denoted Xζ,|M . Now, from [Bi, Lemme 2.3] the inequality to prove is just equivalent to Z |Xζ,|T M ⊥ |2hF S µ eF S |ζ|2 ≤ Ck 2 M

for all ζ ∈ Lie(Kk′ )⊥ . Here µ eF S is the volume form of the K¨ahler metric c1 (hF S ) ∈ kc1 (L) induced by S using the Fubini-Study map. Obviously, this inequality can be deduced from the following three inequalities: |ζ|2 ≤ γ1 kkXζ,|M k2L2 (eµF S )

kXζ,|M k2L2 (eµF S ) kXζ,|T M k2L2 (eµF S )

≤ ≤

(11)

kXζ,|T M k2L2 (eµF S ) + kXζ,|T M ⊥ k2L2 (eµF S ) γ2 kkXζ,|T M ⊥ k2L2 (eµF S )

(12) (13)

where γ1 , γ2 are constants independent of k. The arguments of [P-S2, Theorem 2] can be applied with no change in order to get (11) and (12). Moreover, from the exact sequence (10), one can derive the following estimate (see [P-S2, (5.16)]) 2 ¯ kXζ,|T M ⊥ k2L2 (eµF S ) ≥ γ3 k∂X ζ,|T M ⊥ kL2 (e µF S ) .

(14)

For ∂¯ seen as acting on smooth (0, 1)-form on M with values in T M , one considers the ∗ 1/k ¯ As it is pointed out in [Ma2], the first eigenvalue λ1 of  1/k operator  1/k = ∂¯ hF S ∂. hF S

hF S

is bounded from below independently of k since S has R-bounded geometry. Moreover, since ζ ∈ Lie(Kk′ )⊥ , Xζ,T M is orthogonal to the projection on T M of any holomorphic vector field on ι(M ) by condition (C3 ). Thus, one has Z

M

1/k

¯ ζ,T M |2 1/k |∂X hF S

c1 (hF S )n ≥ λ1 n!

Z

M

1/k

|Xζ,T M |2 1/k hF S

c1 (hF S )n . n!

¯ ζ,T M + X But now, ∂(X ζ,T M ⊥ ) = 0 and thus 2 ¯ k∂X ζ,T M ⊥ kL2 (e µF S ) ≥

λ1 kXζ,T M k2L2 (eµF S ) . k

Finally both (15) and (14) imply Inequality (13). This is our main result.

(15)

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279

Theorem 2. Assume that either condition (C1 ) holds or both conditions (C2 ) and (C3 ) hold for a sequence of strictly increasing integers kj > k0 . Assume that there exists a twisted cscK metric solution to Equation (1) in the class c1 (L). Then, • For k0 large enough, there exists a hT -twisted balanced metric ωkj ∈ kj c1 (L), • The sequence k1j ωkj is convergent when j → +∞ towards the twisted cscK metric in C∞ -topology. Proof. Under condition (C1 ), the proof of the Theorem is a consequence of Corollary 3.1, Proposition 3.2 and Proposition 3.1 together with the double symplectic quotient picture. The convergence is Cr topology (for any r) is obtained by the fact that one can choose, up to any order, an approximate twisted balanced metric in Theorem 1 (see [Do1, Proof of Theorem 3]). Finally, the uniqueness of the twisted cscK metric is a consequence of [St1], and one could recover this result in the projective setting by studying the uniqueness of twisted balanced metric up to SU (V ) action. The convergence of the twisted balanced metric is clear by construction. Let us now assume conditions (C2 ) and (C3 ). The main difference with previous case is that one has to check that the gradient flow of |µSU |2 is still converging with the estimate obtained from Proposition 3.3. From [Ma2, Lemma 3.4] and [Ma3, Theorem 3.2], it is sufficient to obtain a zero of the moment map by considering the one parameter subgroups perpendicular to the subgroup of stabilizers (one could also invoke [Si, Proposition 9]). Now, Proposition 3.3 shows the convergence of the gradient flow when one restricts the moment map to Lie(Kk′ )⊥ and by condition (C3 ), we can conclude.

4.

Further Directions

Let us discuss some examples of twisted cscK metrics in the literature (see also [Fi1, Fi2]). • Let M → CP1 be an elliptically fibred K3 surface with 24 singular fibres of type I1 . Then, there is a Weil-Petersson metric ωW P induced from the fibres on CP1 . In [S-T1], it is proved that the K¨ahler-Ricci flow converges to the McLean’s metric satisfying the twisted csck equation Ric(ω) = ωW P . • Let us consider an almost K¨ahler-Einstein Fano manifold M [Ba]. By definition, it carries for any 0 ≤ t < 1 a K¨ahler metric ωt ∈ c1 (M ) such that Ric(ωt ) = tωt + (1 − t)ω0 If condition (C3 ) holds, one can modify our arguments to construct a convergent sequence of twisted balanced metrics for any 0 ≤ t < 1. We don’t know if in general such a manifold is balanced in the sense of Zhang-Luo. • Let us consider M an algebraic manifold with semi-ample canonical line bundle. From the minimal model program, we know that M admits an algebraic fibration

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Julien Keller τ : M → Mcan over its canonical model Mcan . We assume that 0 < dim Mcan < dim M , Mcan is non singular and the fibre τ −1 (p) is non singular for any p ∈ Mcan . Thus, each fibre τ −1 (p) is a smooth Calabi-Yau manifold. The L2 metric on the moduli space of Calabi-Yau manifolds induces a semi-positive Weil-Petersson (1, 1)form ωW P on Mcan . Then, the main result of [S-T2] proves the convergence of the K¨ahler-Ricci flow in that context and identifies its limit. In that case, it satisfies the following twisted cscK equation on Mcan , Ric(ω) = −ω + ωW P .

(16)

In a forthcoming paper, we shall study the dynamical system HilbhT ◦ F S in order to construct numerical approximations of the solution to Equation (16) for a minimal elliptic surface, by finding twisted balanced metrics. Finally, one could consider a slightly more general framework. Assume that T = (Tj ) is a finite family of twistings such that 12 αTj are the curvature of the hermitian metrics hTj ∈ M et(Tj ). Then, one can consider the T -twisted balanced metrics ω solution to Scal(ω) −

X

Λω αTj = C

j

where C is a constant. Then, in view of a generalization of Theorems 1 and 2, conditions (C1 ) and (C2 ) can be replaced respectively by (C′1 ) The Lie algebra Lie(Aut(M, L)) is trivial and the Tj are semi-positive i.e for all j, αTj is a pointwise semi-positive (1, 1)-form on M . (C′2 ) There exists j0 such that Tj0 is a ample and αTj0 is a positive (1, 1)-form on M . For all j 6= j0 , Tj is semi-positive, αTj is pointwisely semi-positive.

Acknowledgements The author is very grateful to Prof. T. Mabuchi, J. Stoppa and G. Sz´ekelyhidi for stimulating discussions.

References [Ba]

S. Bando, The K-energy map, almost Einstein K¨ahler metrics and an inequality of the Miyaoka Yau type, Tohoku Math. J. 39 (1987)

[Bi]

O. Biquard, M´etriques K¨ahl´eriennes a` courbure scalaire constante : unicit´e, stabilit´e, S´eminaire Bourbaki, 938 (2004)

[Do1] S.K. Donaldson, Scalar curvature and projective embeddings I, J. Differential Geom. 59 (2001) [Do2] S.K. Donaldson, Scalar curvature and projective embeddings II, Quaterly Jour. Math 56 (2005)

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[GF]

M. Garcia Fern´andez, Ph.D thesis in preparation (2007)

[Fi1]

J. Fine, Constant scalar curvature K¨ahler metrics on fibred complex surfaces, J. Differential Geom. 68 (2004)

[Fi2]

J. Fine, Fibrations with constant scalar curvature K¨ahler metrics and the CM-line bundle, Math. Res. Lett. 14 (2007)

[Lu]

Z. Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math 122 (2000)

[M-M] X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, Birkhauser Verlag, (2007) [Ma1] T. Mabuchi, Extremal metric and stabilities on polarized manifolds, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zurich, (2006) [Ma2] T. Mabuchi, An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I, Invent. Math. 159 (2005) [Ma3] T. Mabuchi, Stability of extremal K¨ahler manifolds, Osaka J. Math. 41 (2004) [Ma4] T. Mabuchi, An obstruction to asymptotic semistability and approximate critical metrics, Osaka J. Math. 41 (2004) [P-S1] D. H. Phong and J. Sturm, Stability, energy functionals, and K¨ahler-Einstein metrics , Comm. Anal. Geom. 11 (2003) [P-S2] D.H. Phong and J. Sturm, Scalar curvature, moment maps, and the Deligne pairing, Amer. J. Math. 126 (2004) [Th]

R. Thomas, Notes on G.I.T and symplectic reduction for bundles and varieties, Surveys in Differential Geometry 10 (2006)

[Si]

G. Sz´ekelyhidi, The K¨ahler-Ricci flow and K-stability, arXiv:0803.1613 (2008)

[S-T1] J. Song and G. Tian, The K¨ahler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007) [S-T2] J. Song and G. Tian, Canonical measures and Kahler-Ricci flow, arXiv:0802.2570 (2008) [St1]

J. Stoppa, Twisted cscK metrics and K¨ahler slope stability, arXiv:0804.0414 (2008)

[St2]

J. Stoppa, K-stability of constant scalar curvature K¨ahler manifolds, arXiv:0803.4095 (2008)

[Wa]

X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13, (2005)

[Zh]

S. Zhang, Heights and reductions of semi-stable varieties, Compos. Math. 104, (1996)

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 283-306

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 10

R EDUCTION , H YDRODYNAMICS AND C ONTROL FOR G EODESICS OF L EFT- OR R IGHT I NVARIANT M ETRICS ON L IE G ROUPS Mikhail V. Deryabin∗ Mads Clausen Instituttet, Syddansk Universitet, Grundtvigs All´e 150, DK-6400 Sønderborg.

Abstract In contrast to the Euler-Poincar´e reduction of geodesic flows of left- or rightinvariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid with a constant pressure. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. We give explicit general expressions for the reduced vector field and its symmetry fields, provide examples of such reduction and discuss two applications of this approach. As the first application, we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group, and assume that the mass geometry of the system may change under the action of internal control forces. Such system can also be reduced to the Lie group. With no controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and thus its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions, under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. The hydrodynamic interpretation of the system both provides a convenient ”language” and sharpens the controllability results: we show that by changing the mass geometry, one can bring one vortex manifold to any other vortex manifold. As an example we consider the n-dimensional Euler top. The other application is the reduction for the Euler equations of an ideal fluid, that describe the geodesics of a right-invariant metric on a Lie group SDiff(M ) of the volume-preserving diffeomorphisms of a Riemannian manifold M , to the group ∗

E-mail address: [email protected]

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Mikhail V. Deryabin SDiff(M ). For a typical coadjoint orbit we find all symmetry fields of a reduced flow, and, as a corollary, we get a simple proof for nonexistence of new invariants of coadjoint orbits, which are the integrals of local densities over the flow domain.

Keywords: Ideal hydrodynamics, Lie groups, control theory, invariants of coadjoint orbits.

1.

Introduction

For the Euler top, the Hamiltonian vector field on the cotangent bundle T ∗ SO(3) can be uniquely projected onto the Lie algebra so(3) – this is a classical reduction, known in the general case as the Euler-Poincar´e reduction. In the 30’s, E.T.Whittaker suggested an ”alternative” reduction procedure for the Euler top: by fixing values of the Noether integrals, the Hamiltonian vector field can be uniquely projected from T ∗ SO(3) onto the group SO(3) [31]. The Whittaker reduction is valid for any Hamiltonian system on a cotangent bundle T ∗ G to a Lie group G, provided the Hamiltonian is invariant under the left (or right) shifts on the group G. An important example of such Hamiltonian systems is a geodesic flow of a left-(right-)invariant metric on a Lie group. If we reduce a Hamiltonian system to the Lie group G, and then factorize the reduced vector field by the orbits of its symmetry fields, then, by the Marsden-Weinstein theorem, we get the same Hamiltonian system on a coadjoint orbit on the dual algebra g∗ , as if we first reduced the system to the dual algebra g∗ , and then to the coadjoint orbit (see also [1], Appendix 5). Thus, the Whittaker reduction can be regarded as a part of the MarsdenWeinstein reduction of Hamiltonian systems with symmetries [26]. In contrast to the Marsden-Weinstein reduction, it has not been payed much attention to the Whittaker reduction alone. However, it is itself worth studying. It turns out that a vector field, reduced to a Lie group G has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid, that flows on the group G (viewed as a Riemannian manifold), and is incompressible with respect to some left-(or right-)invariant measure on G, see [9, 18, 19, 20] for details. The reduction to a Lie group is also useful for a series of applications, which include stability theory, noncommutative integration of Hamiltonian systems, discretization, differential geometry of diffeomorphism groups and control theory, see, e.g., [20, 21, 8, 12]. In this chapter we consider two applications of this approach, mostly following [9, 10]. We first review the Whittaker reduction and its hydrodynamic essence, and provide an explicit expression for the reduction of a geodesic flow of a left- or right-invariant metric onto a Lie group. For any Lie group we find both the reduced vector field and its ”symmetry fields”, i.e., left- or right-invariant fields on the group that commute with our reduced vector field. These fields have also a hydrodynamical meaning: these are the vortex vector fields for our stationary flow (i.e., they annihilate the vorticity 2-form), cf. [20], [9]. The distribution of the vortex vector fields in always integrable, thus they define a manifold, that we call the vortex manifold. Typically, these manifolds are tori. Next, we consider the following control problem. We study mechanical systems, whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group, and we assume that the mass geometry of the system may change under the action of internal control forces. Such systems can also be reduced to the Lie group, and

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they also have an interesting hydrodynamic interpretation: the reduced vector field is the velocity of a stationary flow of an electron gas (with no controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and thus its reduced flow is a stationary ideal fluid flow). Notice that without relating to hydrodynamics, controlled systems on Lie groups were studied in many works, see, e.g., [7] and references therein. A control problem for such systems would be to find the conditions, under which the system can be brought from any initial position on the Lie group to another preassigned position by changing its mass geometry. We show how the hydrodynamic interpretation of the system can sharpen the controllability results. An immediate consequence is that under the standard controllability conditions, one can bring the whole vortex manifold to any other preassigned vortex manifold by changing the mass geometry. A simple necessary controllability condition can be obtained by studying the Bernoulli surfaces: roughly speaking, if an (analytic) system is controllable, then it is always possible to change the controls such that on every finite time interval, the phase curves of the controlled system intersect the Bernoulli surface only in a finite number of points (but not necessarily transversally). Another application of this method would be to studying mechanical systems with damping, which reduction to the Lie group leads to ideal fluid flows with external friction. As an example, we consider the n-dimensional Euler top. We write down the reduced controlled system explicitly, find the vortex manifolds, which typically (when the momentum matrix has the maximal rank) are tori, and show that, by changing the mass geometry, every such vortex manifold can be transformed to any other vortex manifold. We write down the necessary controllability conditions, based on studying the Bernoulli surface, which, in the 3-dimensional case, also turn out to be sufficient. We also write down controllability conditions for the Euler top with damping. As the second application, we define the ”secondary hydrodynamics”, i.e., we study the reduction for the Euler equations of ideal fluid, that describe the geodesics of a rightinvariant metric on a Lie group SDiff(M ) of the volume-preserving diffeomorphisms of a Riemannian manifold M , to the group SDiff(M ). To get a reduced vector field, we fix a coadjoint orbit. In [27], certain special coadjoint orbits were identified with pointvortex dynamics. This identification can also be thought of as a reduction to the group for these special coadjoint orbits (the particle velocity is explicitly defined for the point vortices system). Notice that in this case, the ”reduced” system (i.e., the system of point vortices in 2D) is a finite-dimensional system, while the Whittaker reduction on the Lie group SDiff(M ) in the general case is an infinite-dimensional system. We will pay most attention to the symmetry fields of the reduced flow. For a ”typical” coadjoint orbit we find all symmetry fields of a reduced flow, which are the variational derivatives of the invariants of coadjoint orbits, found in [17]. The term ”typical” means that a certain vector field does not admit any symmetries. Coadjoint orbits with no extra symmetries were also considered in [14] in relation with topology of steady even-dimensional flows. Now, as an invariant of a coadjoint orbit generates a symmetry field for the reduced flow, we conclude readily that there are no new invariants of coadjoint orbits for ideal fluid flows, linearly independent of the classical ones on these ”typical” orbits (these invariants should be the functionals on the dual algebra, such that the variational derivative of such functional is a smooth vector field from the Lie algebra SVect(M ) of divergence-free vec-

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tor fields on M ). If we can take for granted the fact that the ”typical” case are everywhere dense in the space of all 1-forms on M in some appropriate topology, then we readily conclude that there are no invariants of coadjoint orbits, which are integrals of local densities over the flow domain, which are linearly independent of the classical ones. Thus, there are no new invariants of coadjoint orbits for ideal fluid flows in the above class of functionals. The results on the absence of new integral invariants complement the theorem on local invariants, i.e., on the local description of isovorticed fields existence at a generic point in ideal hydrodynamics [3]. This theorem states that at a generic point, in the evendimensional case, there is only one local invariant, which is the vorticity function, while in the odd-dimensional case, there are no local invariants at all. In the Appendix we study the Whittaker reduction for nonholonomic systems and formulate and discuss the standard controllability conditions together with their extensions.

2.

Reduction of a Geodesic Flow to a Lie Group

We start with some basic facts on coadjoint representations, inertia operators on Lie algebras and the Euler equations (see, e.g., [3]). Let G be an arbitrary Lie group, g be its Lie algebra, and g∗ be the corresponding dual algebra. The group G may be infinitedimensional, and not necessarily a Banach manifold, but we assume that the exponential map exp : g → G exists. Any vector g˙ ∈ Tg G and any covector m ∈ Tg∗ G can be translated to the group unity by the left or the right shifts. As the result we obtain the vectors ωc , ωs ∈ g and the momenta mc , ms ∈ g∗ : ωc = Lg−1 ∗ g, ˙

ωs = Rg−1 ∗ g, ˙

mc = L∗g m,

ms = Rg∗ m.

The following relation plays the central role in the sequel: mc = Ad∗g ms ,

(2.1)

Ad∗g : g∗ → g∗ being the group coadjoint operator. Let us fix the ”momentum in space” ms . Then relation (2.1) defines a coadjoint orbit. The Casimir functions are the functions of the ”momentum in the body” mc , that are invariants of coadjoint orbits. For example, for the Euler top, the Casimir function is the length of the kinetic momentum. Let A : g → g∗ be a positive definite symmetric operator (inertia operator) defining a scalar product on the Lie algebra. This operator defines a left- or right-invariant inertia operator Ag (and thus a left- or right-invariant metric) on the group G. For example, in the left-invariant case, Ag = L∗g −1 A Lg−1 ∗ . Let the metric be left-invariant. The geodesics of this metric are described by the Euler equations m˙ c = ad∗A−1 mc mc ,

(2.2)

Here ad∗ξ : g∗ → g∗ is the coadjoint representation of ξ ∈ g. Given a solution of the Euler equations ωc = A−1 mc , the trajectory on the group is determined by the relation Lg−1 ∗ g˙ = ωc .

(2.3)

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The Euler equations follow from the fact that ”the momentum in space” ms is constant, whereas ”the momentum in the body” mc is obtained from ms by (2.1), see [3]. Remark. Strictly speaking, in the infinite dimensional case the operator A is invertible only on a regular part of the dual algebra g∗ . In our case this means, that some natural restriction on values of ms (or mc ) have to be imposed (see [3]). The Euler equations can be considered as Hamilton’s equations on the dual algebra, where the Hamiltonian equals H = 12 (A−1 m, m), m ∈ g∗ , and the Poisson structure is defined by the following Poisson brackets. For two functions F (m) and G(m) on the dual algebra g∗ , {F, G} = (m, [dF (m), dG(m)]) , where dF (m), dG(m) ∈ g are the differentials of functions F and G, and [ξ, η] = adξ η is the commutator (adjoint action) on the Lie algebra g. Let now 1 H = (A−1 mc , mc ) + (λ, mc ), 2 where λ ∈ g is a constant vector. Then Equation (2.2) becomes m˙ c = ad∗A−1 mc +λ mc ,

(2.4)

and the velocity ωc = A−1 mc + λ. In case of a right-invariant metric mc is constant, the Euler equations read m˙ s = −ad∗A−1 ms +λ ms , and the trajectory on the group is determined by the equation Rg−1 ∗ g˙ = ωs . The result of the reduction onto the group is a vector field v(g) ∈ T G such that the trajectory on the group is defined by the equation g˙ = v(g). The field v(g) will be referred to as reduced. Proposition 2.1. (The Whittaker reduction) For the case of the left-invariant or the rightinvariant metric, the vector field v(g) has the form v(g) = Lg∗ (A−1 Ad∗g ms + λ)

(2.5)

v(g) = Rg∗ (A−1 Ad∗g−1 mc + λ).

(2.6)

and, respectively Here ms , respectively mc , is constant. Notice that in Proposition 2.1, to find the reduced vector field we do not need the Hamiltonian equations on T ∗ G and the explicit expression for the Noether integrals. We only need the Lie group structure and the inertial operator. This is important for generalizations to the infinite-dimensional case. Unlike for the Marsden-Weinstein reduction, we do not have to assume nondegeneracy conditions on the momenta ms or mc . Proof. We consider only the case of the left-invariant metric; for the right-invariant case the proof is similar. Relation (2.1) determines the function mc = mc (ms , g) on the group

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G depending on ms as a parameter. From the equality ωc = A−1 mc + λ and Lg−1 ∗ g˙ = ωc follows that for any g ∈ G, Lg−1 ∗ g˙ = A−1 mc (ms , g) + λ which implies (2.5). 2 In Appendix A we consider the case, when the inertia operator is not left- or rightinvariant, i.e, A = A(g). Some nonholonomic systems have this form. It turns out that system of equations (2.3), (2.4) can still be reduced to the group G (although now Equation (2.4) cannot be separated). Even if λ = 0, the reduced vector fields (2.5), (2.6) are is in general neither left- nor right-invariant. An important exception is when the inertia operator defines a Killing metric on the Lie algebra. However, the reduced covector fields are always right- or left-invariant. Proposition 2.2. Let λ = 0. If the metric is left-invariant, then the reduced covector field m(g) = Ag v(g) is right-invariant. Proof. m(g) = L∗g −1 ALg−1 ∗ v(g) = L∗g −1 AA−1 L∗g Rg∗ −1 ms = Rg∗ −1 ms . 2 Let w(g) ∈ T G be a right-invariant vector field on the group G, which is defined by a vector ξ ∈ g: w(g) = Rg∗ ξ. We fix a momentum ms . Theorem 2.3. For the momentum ms fixed, the vector field w(g) on G is a symmetry field of the reduced system v(g) if and only if the vector ξ satisfies the condition ad∗ξ ms = 0.

(2.7)

In the finite-dimensional case this means that the flows of the vector fields v(g), w(g) on the group commute. In the infinite-dimensional case one should be more accurate: the equation g˙ = v(g) is a partial integral-differential equation, rather than an ordinary differential equation, hence, strictly speaking, it is not clear if it has a solution. On the other hand, equation g˙ = Rg∗ ξ always has a solution, which is a one-parametric family of the left shifts on the group G: g → (exp τ ξ)g, see, for example, [30], as we have assumed that the exponential map exists. Notice also that, in view of Proposition 2.2, under the assumption of Theorem 2.3, the Lie derivative Lw(g) m(g) = 0. Proof of Theorem 2.3. The vector fields w(g) and Lg∗ λ commute, as right-invariant fields always commute with left-invariant fields. Thus, it is sufficient to show that v((exp τ ξ)g) = L(exp τ ξ)∗ v(g) if and only if the condition of the theorem is fulfilled. Indeed, v((exp τ ξ)g) = L(exp τ ξ)g∗ A−1 Ad∗(exp τ ξ)g ms = Lexp τ ξ∗ Lg∗ A−1 Ad∗g (Ad∗exp τ ξ ms ).

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The last term equals L(exp τ ξ)∗ v(g) for any g ∈ G if and only if Ad∗exp τ ξ ms = ms for all values of the parameter τ . Differentiating the last relation by τ we arrive at the statement of the theorem. 2 Let again λ = 0, and consider a subspace F of functions on the dual algebra g∗ , such that the variational derivatives of these functions belong to the Lie algebra g. Proposition 2.4. A function F (m) ∈ F is constant on a coadjoint orbit Ad∗g ms only if its differential generates (by the right shifts) a symmetry field to the reduced flow with the momentum ms . Proof. Let F be constant on the coadjoint orbit Ad∗g ms : F (Ad∗g ms ) = const for any g ∈ G. Then,   δF (ms ) ∗ , ada ms = 0 = (a, ad∗δF (ms )/δm ms ) δm (ms ) is a symmetry field to the reduced for any a ∈ g, thus ad∗δF (ms )/δm ms = 0 and Rg∗ δFδm flow with the momentum ms . 2 This Proposition is an important tool in finding the invariants of coadjoint orbits. Suppose that the differentials of the known invariants constitute all the symmetry fields. Then this automatically leads to nonexistence of additional invariants of coadjoint orbits among the functionals from F. In the finite-dimensional case, finding invariants of coadjoint orbits reduces to an analysis of some algebraic equations. In the infinite-dimensional case, it reduces to studying a system of ordinary differential equations. At last, we consider the Euler equations with dissipation:

m ˙ = ad∗ω m − νm,

(2.8)

where ν is a positive constant, or, in more general case, a non-negative function of time. Such system can also be reduced to the Lie group: in the previous formulas, the constant ms should be replaced by   Z t ν(s)ds ms (0). ms (t) = exp − 0

Indeed, from (2.1) one can see that the covector ms (t) should satisfy the equation m ˙ s = −ν(t)ms . All the subsequent results remain true, we only have to be careful with Theorem 2.3, as the vector fields (v(g, t), 1) and (w(g), 1) do not commute in the extended phase space G × R (but the fields (v(g, t), 1) and (w(g), 0) obviously do commute).

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Ideal Flows on Lie Groups

We now formulate some results on the hydrodynamics character of the reduced vector fields from the previous section. Consider first the Euler equations for an ideal incompressible fluid, that flows on a Riemannian manifold M : ∂v + ∇v v = −∇p, div v = 0, ∂t where ∇v v is the covariant derivative of the fluid velocity vector v by itself with respect to the Riemannian connection and p is a pressure function. Consider a geodesic vector field u on the manifold M . Locally it always exists, but it may not be defined globally on M – take a two-sphere as a simple example. Then u is a stationary flow of the ideal fluid with a constant pressure. Indeed, as u is a geodesic vector field, its derivative along itself is zero: ∇u u = 0. Remark. The converse is of course not true: there are stationary flows that are not geodesics of the Riemannian metric. The stationary flows with constant pressure form a background for hydrodynamics of Euler equations on Lie groups. Consider a Hamiltonian system on a finite-dimensional Lie group G, with a left-invariant Hamiltonian, which is quadratic in the momenta (in terms of Section 2., vector λ = 0). This Hamiltonian defines a left-invariant metric on the Lie group G. As we reduce this system to the group, the reduced vector field is globally defined on G, and is a geodesic vector field of the Riemannian metric, defined by the left-invariant Hamiltonian, and it defines a stationary flow of an ideal fluid on G. Thus, the reduced vector field (2.5) (and (2.6)) is the velocity vector field for a stationary flow on the Lie group G with left- (right-) invariant metric. An immediate corollary of Proposition 2.2 is Proposition 3.1. There is an isomorphism between the stationary flows with constant pressure, defined by a left-invariant metric on a finite-dimensional Lie group G, and the space of right-invariant covector fields on this group. Remark. Stationary flows with constant pressure play an important role in studying the differential geometry of diffeomorphism groups, see [5, 16, 28]: they define asymptotic directions on the subgroup of the volume-preserving diffeomorphisms of the group of all diffeomorphism. Proposition 3.1 is a generalization of [29], where it was shown that every left-invariant vector field on a compact Lie group equipped with a bi-invariant metric is asymptotic: if a Hamiltonian defines the bi-invariant metric on the Lie algebra, then the reduced vector field (2.5) is itself left-invariant. Moreover, its flow (which are right shifts on the Lie group G) are isometries of this metric (see, e.g., [11]). Recall now that the reduced covector field is right-invariant (Proposition 2.2). Thus, the condition ad∗η ms = 0 is equivalent to Lη(g) m(g) = 0, where m(g) is the right-invariant 1-form (being equal to ms at g = id), and η(g) = R∗g η is the right-invariant symmetry field. By the homotopy formula, 0 = Lη(g) m(g) = iη(g) dm(g) + d(η(g), m(g)) = iη(g) dm(g), as (η(g), m(g)) = (η, ms ) = const for all g (both vector and covector fields are rightinvariant).

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We now define a vortex vector field, as an annihilator of the vorticity 2-form. Then the condition iη(g) dm(g) = 0 is exactly the definition of a vortex field. Thus, we have proved Proposition 3.2. Any symmetry field to the reduced vector field is a vortex vector field. Consider now ideal hydrodynamics on a Riemannian manifold M . The Euler equations of an ideal fluid are the geodesics of a right-invariant metric on a Lie group SDiff(M ) of volume-preserving diffeomorphisms of the manifold M . Thus, we can apply the above procedure, and reduce the geodesic flow to the group SDiff(M ). The reduced ”vector field” can be thought of as a stationary flow of an ”ideal fluid”, which flows on the Lie group SDiff(M ). This justifies the term ”secondary hydrodynamics” in the infinite-dimensional case (cf. [20]). Remark. The stationary flows on Lie groups that we study are typically degenerate: in the even-dimensional case, the rang of the vorticity 2-form dm(g) is not maximal. Thus, the standard topological results on even-dimensional stationary flows (see, e.g., [3]) cannot be applied. Vortex vectors, i.e., vectors ξ ∈ g that satisfy condition (2.7), are the isotropy vectors. We now review some classical results on the isotropy vectors and the Casimir functions, see, e.g., [2] for details, and adapt them to our case. Proposition 3.3. The distribution of the isotropy vectors is integrable. The Proposition says that if vectors ξ1 , ξ2 ∈ g satisfy condition (2.7), then the vector [ξ1 , ξ2 ] = adξ1 ξ2 also satisfies this condition, which is a simple consequence of the Jacobi identity. The integrable distribution of the isotropy vectors defines a manifold (at least locally), that we, following [20], call a vortex manifold. Notice that for compact Lie groups, vortex manifolds are closed surfaces. The isotropy vectors ξ ∈ g form a Lie subalgebra h ⊂ g, called an isotropy algebra for the coadjoint orbit m = Ad∗g ms . If the differentials of the Casimir functions form a basis of the isotropy algebra h, then h is Abelian. In general, an isotropy algebra is not necessarily Abelian. A very simple example is G = SO(3): if the ”momentum in space” ms = 0, then h = so(3). However, in the finite-dimensional case isotropy algebras are Abelian on an open and dense set in g∗ (the Duflo theorem). Thus, the corresponding vortex manifolds (that pass through the group unity) are commutative subgroups of the Lie group G. Notice that in the infinite-dimensional case, vortex fields cam still define a certain commutative subgroup, which can also be referred to as ”vortex manifold”. Vortex manifolds have always the dimension of the same parity as the Lie group dimension. This is a simple corollary of the fact that coadjoint orbits are always even-dimensional (also the degenerate ones), see, e.g., [2]. One can show that if the Hamiltonian has also terms, linear in the momenta (in the other words, if λ 6= 0), then the reduced field has the following hydrodynamic sense: it is the velocity of the stationary flow for the electron gas, which satisfies an ”infinite conductivity equation”, again, with a constant pressure, see [3]. For the reduced system, one can define the Bernoulli functions: these are the functions, which are invariants both of the flow velocity field and the vorticity fields. For example, a function 1/2(v, v), where v is the reduced vector field, and (·, ·) is the Killing metric, is a Bernoulli function.

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We consider now systems with damping, or dissipation, as described at the end of the previous section. These systems describe flows with ”external friction” (which are no longer stationary), given by ∂v + ∇v v = −νv, ∂t

(3.1)

where ν is a positive constant. Such systems experience a ”complete vortex diffusion” (see [20]) in the following sense. We rewrite Equation (3.1) as ∂[u] + Lv [u] = −ν[u], ∂t where u is the 1-form, obtained from v by lowering the indices, and [u] is the coset of 1-forms. Let Z u I(t) = gvt γ

for some closed curve γ (obviously the value of I does not depend on the choice of a 1-form u from the coset [u]). Then the function I(t) satisfies the equation I˙ = −νI, and thus an integral of the 1-form u over any closed cycle tends to zero exponentially fast.

4.

Controllability, Bernoulli Functions and Flows with External Friction

Consider a Lagrangian system on a tangent bundle T G to a Lie algebra G, with the Lagrangian, which is left-invariant under the action of the Lie group G. In order to introduce the controls in our system, we consider Lagrangians on T G of the following form: ! k k X X 1 A(ω + ui λi ), ω + u i λi , L(ω, u) = 2 i=1

i=1

where ω ∈ g is the system velocity, λi ∈ g are constant vectors, ui (t) ∈ R are controls, and A : g → g∗ is the inertia operator. Notice that the dimension k of the control vector u(t) may be lower than the dimension of the Lie algebra. We assume that there is a positive constant ǫ, such that ku(t)k ≤ ǫ, i.e., our controls are always bounded. Physically, these controls mean that we can change the system mass geometry by internal forces. The Euler equations (2.4) are: m ˙ = ad∗ω m, P where the momentum m = A(ω + ki=1 ui λi ) ∈ g∗ . The system, reduced to the group G, is (cf. (2.5): ! k X −1 ∗ g˙ = Lg∗ A Adg ms − (4.1) ui λi = vλ (g). i=1

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From Theorem 2.3 and Proposition 3.3 follows the following result. Suppose that System (4.1) is controllable (we formulate some standard conditions in the Appendix B.), and we assume that the controls u(t) are piecewise constant functions. We fix the ”momentum in space” ms : with ms fixed, so are the vortex manifolds. Theorem 4.1. By applying controls u(t), one can transform any vortex manifold H1 to any other prescribed vortex manifold H2 , such that the following diagram is commutative: H1 s ↓ gw H1

gvt

→λ

gvt

→λ

H2 s ↓ gw

(4.2)

H2 ,

s being its phase where by w we denote vortex vector fields for the given momentum ms , gw flow, and gvt λ is the phase flow of System (4.1).

This theorem is a reflection of a well-known fact that vortex lines are frozen into the flow of an ideal fluid. Proof. By Theorem 2.3, the vector fields vλ (g) and the vortex fields w(g) commute (the vortex fields are right-invariant, while the vectors Lg∗ λi are left-invariant, and we have also assumed that u(t) is piecewise constant). Pick up the controls (i.e., functions u(t)), that s h to send a point h1 ∈ H1 to a point h2 ∈ H2 . Then the same controls send a point gw 1 s gw h2 , due to commutativity, which proves the theorem. 2 A simple corollary is that all vortex manifolds, that correspond to the same value of the momentum ms , are homotopic to each other. Another observation is that an electron gas, flowing on a Lie group, can be controlled by changing an external electro-magnetic field. The condition that u(t) are piece-wise constants can be dropped: the vortex manifolds will still be transformed into vortex manifolds, but, strictly speaking, the diagram (4.2) will no longer be valid, as there is no commutativity in the extended phase space G × R, cf. Remark in Section 2.. In the sequel, we will consider the general case, when u(t) is not necessarily piece-wise constant. Let us now define the Bernoulli surface as the intersection of the levels of all independent Bernoulli functions (see the previous section). There is always one ”classical” Bernoulli function
(4.3) α(g) = 1/2(v(g), v(g)) = 1/2 A−1 Ad∗g ms , Ad∗g ms . (at least for semi-simple Lie groups, where the Killing metric is nondegenerate).

Proposition 4.2. If System (4.1) is controllable, then there is a control vector λ ∈ g, such that no trajectory g(t) of System (4.1) belongs to the surface α = const. In particular, for any stationary solution Ad∗g(t) ms = const = mc , the coadjoint action ad∗λ mc 6= 0. This condition simple (and the proof is straight-forward), but it is easy to check in applications, and in the 3-dimensional case, it opens the possibility of controlling the system by only one control vector (provided the momentum is not zero). Indeed, for the 3-dimensional fluid flows on compact Lie groups, such that the vorticity field is not parallel to the fluid velocity field, the Bernoulli surfaces, defined by the Bernoulli function (4.3) are 2-dimensional

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tori, and one control is enough to jump both from a torus to another torus, and from one trajectory on a torus to another one on the same torus. Example. For controllability of a reduced Euler top on SO(3) it is enough to have only one control, provided all the principal axes of the inertia ellipsoid are different, and the momentum is the space ms 6= 0 is not directed along any of the principal axes [25]. We consider now control problem for systems with dissipation. Suppose that the dissipation coefficient ν is a constant (see the previous section). A control system with external friction takes the form ! k X (4.4) u i λi . g˙ = Lg∗ A−1 Ad∗g e−νt ms − i=1

First, we formulate the sufficient conditions for controllability of System (4.4). Proposition 4.3. Let the minimal Lie algebra, containing the control vectors λi , coincide with the algebra g if the Lie group G. Then System (4.4) is controllable for any ν ≥ 0. The proof is the application of the Rashevsky-Chow theorem, see Appendix B.. Suppose now that the minimal Lie algebra, containing λi , does not coincide with the Lie algebra g. Proposition 4.4. Suppose that there exists time T > 0, such that at ν = 0, any point on G can be brought to any other point on the time interval [0, T ]. Then System (4.4) is controllable for ν < 1/T . Proof. Consider a time transformation: τ=

1 1 − exp(−νt). ν ν

Under this time transformation, System (4.4) becomes: ′

g = Lg∗

−1

A

Ad∗g ms

−

k X i=1

u ˜ i λi

!

.

(4.5)

This is the original system, which, by the condition of the Proposition, is controllable on time interval τ ≤ T . But sup τ = 1/ν, thus we get 1/ν < T . 2 Remark. This condition can be strengthened by noticing that, as t → ∞, the multiplier exp(νt) → ∞, thus the controls u become unbounded.

5. n-dimensional Euler Top As an example, we consider the control problem for an n-dimensional rigid body with a fixed point in Rn (n-dimensional top). We follow the reduction procedure, suggested in [8]. Let so(n) be the Lie algebra of SO(n), R ∈ SO(n) be the rotation matrix of the top, Ωc = R−1 R˙ ∈ so(n) be its angular velocity in the moving axes, and Mc ∈ so∗ (n) be its angular momentum with respect to the fixed point of the top, which is also represented in the moving axes.

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The angular momentum in space Ms = Ad∗R−1 Mc ≡ RMc R−1 is a constant matrix, and the Euler equations have the following matrix form generalizing the classical Euler equations of the rigid body dynamics M˙ c + [Ωc , Mc ] = 0.

(5.1)

We assume, that the inertia operator of A : so(n) → so∗ (n) is defined by the relation Ωc = A−1 M = U M + M U , where U is any constant nondegenerate operator. Thus the system (5.1) is a closed system of n(n − 1)/2 equations, which was first written in an explicit form by F. Frahm (1874) [13]. As was shown in [24] (for n = 4, in [13]), with the above choice of the inertia tensor, the system ( 5.1) is a completely integrable Hamiltonian system on the coadjoint orbits of the group SO(n) in so∗ (n). Now we fix the angular momentum Ms (and, therefore, the coadjoint orbit) and assume that rank Ms = k ≤ n (k is even). Then, according to the Darboux theorem (see, e.g., [2]), there exist k mutually orthogonal and fixed in space vectors x(l) , y (l) , l = 1, . . . , k/2 such, that |x(l) |2 = |y (l) |2 = hl , hl =const, and the momentum can be represented in the form Ms =

k/2 X l=1

x(l) ∧ y (l) ,

that is Ms = X T Y − Y T X ,

(5.2)

where X T = (x(1) · · · x(k/2) ), Y T = (y (1) · · · y (k/2) ), x(l) ∧ y (l) = x(l) ⊗ y (l) − y (l) ⊗ x(l) , and ( )T denotes transposition. Under these conditions on x(l) , y (l) the set of k × n matrices Z = (x(1) y (1) · · · x(k/2) y (k/2) )T forms the Stiefel variety V(k, n) (see, for example, [11]). The momentum in the body Mc has the same expression as (5.2), but here the components of matrices X , Y are taken in a frame attached to the body, see (2.1). Since the above vectors are fixed in space, in the moving frame they satisfy the Poisson equations, which are equivalent to matrix equations X˙ = X Ωc ,

Y˙ = YΩc .

(5.3)

Now we set Ωc = U Mc + Mc U and substitute this expression into (5.3). Then taking into account (5.2), we obtain the following dynamical system on V(k, n) X˙ = X [U (X T Y − Y T X ) + X T YU ], Y˙ = Y[U (X T Y − Y T X ) − Y T X U ].

(5.4)

Notice that in the case of maximal rank k (k = n or k = n − 1), the Stiefel variety is isomorphic to the group SO(n), and the components of vectors x(1) y (1) · · · x(k/2) y (k/2) form redundant coordinates on it. Thus the system (5.4) describes required reduced flow (2.5) on SO(n). The representation (5.2) is not unique: rotations in 2-planes spanned by the vectors (l) x , y (l) in Rn (and only they), leave the angular momentum M invariant (in the case of the maximal rank). As a result, the system (5.4) on SO(n) has k/2 vortex vector fields w1 (g), . . . , wk/2 (g), which are generated by the right shifts of vectors ξ l ∈ so(n), such that

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ad∗ξl Ms ≡ [ξ l , Ms ] = 0, cf. Section 3.. In the redundant coordinates the fields take the form ˙ = −(y (l) , y (l) )x(l) , l = 1, . . . , k/2. x˙ (l) = (x(l) , x(l) )y (l) , y (l) One can easily see that in the case of maximal rank of the momentum matrix, the corresponding vortex manifolds are k/2-dimensional tori. This is a general fact: if a Lie group is compact, then the vortex manifolds are compact manifolds, and, by the Duflo theorem, for a dense set of the momenta ms , the vortex manifolds are tori (in our case, this dense set is determined by the condition that the momentum rank is maximal). The torus, that passes through the group unity, is called the maximal torus for the Lie group; maximal tori play an important role in classification of compact Lie groups. One can furthermore show that if the rank of the momentum is not maximal, the vortex manifolds would be products of a torus and a certain SO(m) Lie group. Consider the Poisson bracket on V(k, n) obtained as the Dirac restriction of the standard bracket in Rkn . Proposition 5.1. System (5.4) is Hamiltonian with respect to the above bracket. Proof. Direct calculation, see [12]. 2 Theorem 5.2. The Bernoulli surfaces for flow (5.4) are tori, and the flow is quasi-periodic on these tori. Proof. The reduced system on the coadjoint orbit of the group SO(n) is integrable, and its generic invariant manifolds are tori of dimension half of the dimension of the orbit. On the other hand, the pre-image of a generic point M of the orbit V(k, n) is a k/2-fold product of circles S 1 × · · · × S 1 . This implies that the original system on V(k, n) has generic invariant tori of dimension half of the dimension of the symplectic manifold V(k, n). Hence, the original system (5.4) is also integrable, the Bernoulli surfaces are tori, and the flow is quasi-periodic on these tori. 2 We now introduce the controls in System (5.4) by the above scheme. Using Equation (4.1) and the fact that any left-invariant vector field on the Lie group SO(n) in our redundant coordinates can be written as X˙ = X Λ,

Y˙ = YΛ,

Λ ∈ so(n),

we get at once the following controlled system on the group: P X˙ = X [U (X T Y − Y T X ) + X T YU ] − X ( i ui Λi ) , P Y˙ = Y[U (X T Y − Y T X ) − Y T X U ] − Y ( i ui Λi ) .

(5.5)

This system describes an n-dimensional rigid body with ”symmetric flywheels”, which is a direct generalization of the Liouville problem of the rotation of a variable body [23]. Proposition 5.3. On can choose two vectors Λ1 and Λ2 , such that for any choice of the inertia operator U , one can transform any vortex manifold to any other vortex manifold for any momentum in space Ms , using the corresponding two control functions u1 (t) and u2 (t).

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Proof. First, we notice that System (5.4) preserves volume in the phase space of the redundant variables X , Y (this can be checked by the direct computation, but the general result of the existence of an invariant measure for a reduced system (2.5) or (2.6) with λ = 0 follows from [20]). As the Lie algebra so(n) is semi-simple, controllability of System (5.5) follows from Corollary 2.1, Appendix B.. Proposition 5.3 follows now from Theorem 4.1. 2

6.

Reduction of the Euler Equations for Ideal Incompressible Fluid

Let M be a Riemannian manifold of dimension n. Consider a Lie group SDiff(M ) of diffeomorphisms of this manifold, which preserve volume µ on M and, if boundary ∂M is nonempty, transform the boundary to itself. The corresponding Lie algebra g = SVect(M ) is the space of zero divergence vector fields on M that are tangent to the boundary ∂M . The dual algebra g∗ is the quotient Ω1 /dΩ0 of differential 1-forms on M modulo exact 1-forms. The action of coadjoint operator on g∗ coincides with the standard action of diffeomorphisms on cosets: Ad∗g [u] = g ∗ [u], [u] ∈ Ω1 /dΩ0 , and the operator of the coadjoint representation is the Lie derivative: ad∗ξ [u] = Lξ [u] [3]. The energy quadratic form Z 1 E= (v, v)µ, (6.1) 2 M

defines the right-invariant metric on the group SDiff(M ). Here (v, v) is the scalar product determined by the Riemannian metric on M and µ is the volume form. The inertia operator A : g → g∗ is specified by the condition Z (v, w)µ hAv, wi = M

v, w ∈ g being zero divergence vector fields on M that are tangent to the boundary. According to the principle of stationary action, motions of the ideal fluid on the manifold M are geodesics of the right-invariant metric (6.1). The latter are described by the Euler equations on the dual algebra g∗ , ∂[u] = −ad∗A−1 [u] [u] = −Lv [u] , ∂t

(6.2)

where the “momentum” [u] ∈ Ω1 /∂Ω0 , Av = [u]. For a particular choice of a 1-form u ∈ [u] we get the classical equation ∂u = −Lv u + df , ∂t see [3]. Now we fix the ”body momentum” uc = u0 and apply our general results on the reduction of a geodesic flow on a group.

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According to Proposition 2.1, the reduced field v(g) has the form (2.6), which can be interpreted in the following way. At each point g we act on the coset [u0 ] by diffeomorphisms g −1 , then we apply the inverse operator A−1 . As a result, we obtain a zero divergence vector field on M . The vector v(g) represents a vector field on M which is obtained from the zero divergence field A−1 Ad∗g−1 [u0 ] by applying a differential of the right shift, i.e., by the ”particle relabeling”. Let a diffeomorphism g ∈ SDiff(M ) map initial particles’ positions x0 to x(t, x0 ). As g is invertible, one can express x0 = x0 (x, t). The ”space momentum” 1-form us = us (x, t) is expressed through the ”body momentum” uc (x0 ) as us (x, t) = Ad∗g−1 uc (x0 ) =

∂xi0 uci (x0 (x, t))dxj . ∂xj

Then, vs = A−1 us . Suppose for simplicity, that the metric on M is Euclidean. Let function f be the solution to equation ∆f = δus , such that on the boundary ∂M of the domain M the following relation holds: ∗(us − df )|∂M = 0. Here δ = ∗d∗, where ∗ is the Hodge operator. This solution exists, and one can write it down explicitly, which, for example, can be done if one finds the Green’s function. The explicit computation of the reduced field thus relies on the possibility of finding the Green’s function. Now, vs = us − grad f Indeed, div vs = div us − div(grad f ) = 0 and v || ∂M , as ∗(us − df ) = iv µ (µ is the Euclidean volume form). The equation on the group can be written for an inverse mapping g −1 as   ∂x0 ∂x0 ∂x0 d −1 . (g ) = −Lg−1 ∗ Rg−1 ∗ g; ˙ =− vs x0 , dt ∂t ∂x ∂x Let w(g) be a left-invariant field on the group SDiff(M ), which is defined by a vector ξ ∈ SVect(M ): w(g) = Lg∗ ξ. Theorem 6.1. The phase flow of the vector field w(g) on the group SDiff(M ) is a symmetry of the reduced field v(g) if and only if the vector ξ satisfies the condition iξ du0 = df,

(6.3)

where df is the differential of some function f on M . Proof. This theorem is a corollary of Theorem 2.3 and the homotopy formula Lv ω = iv dω + div ω. Indeed, the condition ad∗ξ [u0 ] = 0 means that Lξ [u0 ] = [Lξ u0 ] = 0. Hence, Lξ u0 = dF . On the other hand, Lξ u0 = iξ du0 + diξ u0 . 2

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Remark. It is interesting to note that since the isotropy vectors form a subalgebra, their distribution is integrable. Indeed, at any point the commutator (that is minus Poisson bracket of the vector fields) of any two vectors belongs to the same distribution. The structure of the symmetry fields in even and odd dimensions is the following. Let dim M = 2n. For ideal fluid flows on an even-dimensional manifold there always exists the vorticity function λ = dun /µ, where [u] ∈ g∗ [3]. Proposition 6.2. Let ξ ∈ SVect(M ) satisfy the condition of Theorem 6.1. Then ξ commutes with the vector field η, defined by iη µ = dλ ∧ (du0 )(n−1) . Proof. As the vector field ξ satisfies the condition of Theorem 6.1, 0 = Lξ (du0 )n = Lξ λµ = (Lξ λ)µ + λLξ µ. In the above relation, Lξ µ = 0, as ξ ∈ SVect(M ) and thus is divergence-free. Thus, Lξ λ = iξ dλ = 0. Now, take i[ξ,η] µ = Lξ iη µ − iη Lξ µ = Lξ iη µ = diξ (dλ ∧ (du0 )(n−1) ) = 0, as iξ dλ = 0 by above, and iξ du0 = df by Theorem 6.1. 2 There are many such vector fields ξ (at least if the boundary of M is empty): for any function f (λ), a field ξ, defined by iξ µ = df (λ) ∧ (du0 )n−1 is divergence-free and commutes with η. Notice that all of them are linearly dependent. Let now the dimension of the manifold M be odd: dim M = 2n+1. Then, the vorticity lines are integral curves of the vorticity field η, which is set by the following condition [3]: iη µ = (du)n . Proposition 6.3. Let ξ ∈ SVect(M ) satisfy the condition of Theorem 6.1. Then ξ commutes with the vorticity field η : iη µ = (du0 )n . If the manifold M has no boundary, then the vorticity field η itself is a symmetry field (as in this case it belongs to the Lie algebra SVect(M )). Proof. We use following identity: i[ξ,η] µ = Lξ iη µ − iη Lξ µ = 0 : indeed, Lξ µ = 0, as ξ is divergence-free, and Lξ (du0 )n = 0 by the condition of Theorem 6.1. 2 Thus, the question of symmetry fields for the reduced vector field v(g), being defined on the Lie group SDiff(M ), is reduced to the question of existence of symmetry fields for a usual vector field on M . Let the boundary of M be empty. Then Propositions 6.2 and 6.3 describe all the symmetry fields in the ”typical” situation: in the even-dimensional case, all such fields are defined by iξ µ = df (λ) ∧ (du0 )n−1 , while in the odd-dimensional case there is essentially one such field, defined by iξ µ = (du0 )n . The term ”typical” means that the above vector fields do not admit extra symmetries.

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7.

Symmetry Fields and Invariants of Coadjoint Orbits

For the Lie group SDiff(M ), we get from Propositions 6.2 and 6.3, that in the ”typical” case there are no other symmetries than described above. We will show below that those symmetry fields are the differentials of the Casimir functions, found in [17]. Thus, by Proposition 2.4, from the absence of new symmetry fields we readily conclude the nonexistence of new invariants of coadjoint orbits from the class of functionals on the dual algebra, such that their variational derivatives are smooth vector fields from the Lie algebra SVect(M ). Consider again the Lie group SDiff(M ) of volume-preserving diffeomorphisms of a manifold M . We assume that M has no boundary. Then the integrals Z Z n I([u]) = u ∧ (du) , If ([u]) = f (λ)µ, λ = (du)n /µ, (7.1) M

M

are Casimir functions for the Euler equations in odd- and even-dimensional case respectively [17], [3]. As above, here [u] is the coset of a 1-form u, µ is an invariant volume form, and the function f : R → R is arbitrary. Proposition 7.1. Let dim(M ) = 2n + 1. Then the variational derivative of the Casimir function I is a vorticity vector (n + 1)η, where iη µ = (du)n . Proof. Since both du and dw are 2-forms, the wedge product du ∧ dw = dw ∧ du is commutative and the derivative d n (du + sdw) = ndw ∧ (du)n−1 . ds s=0

Thus

d ds

Z (u + sw) ∧ (du + sdw) w ∧ (du)n + u ∧ ndw ∧ (du)n−1 = M M s=0 Z w ∧ (du)n = ([w], (n + 1)η), = (n + 1)

Z

n

M

as the boundary of the manifold M is empty. Here Z w ∧ iη µ ([w], η) = M

is the pairing of the dual algebra element [w] and the vector η ∈ SVect(M ). 2 Proposition 7.2. Let dim(M ) = 2n. Then the variational derivative of the Casimir function If equals ξf ∈ SVect(M ), which is a divergence-free vector field on M , linearly dependent with the vorticity vector ξ, defined by condition iξ du = dλ. The proof is similar to that of Proposition 7.1. We now will need the following assumption: we will here assume that there exist a Green’s function, i.e., the inertia operator A−1 can be written as   Z A−1 [u](x) = A˜−1 u(x) + d G(x, y)δu(y)dn y , M

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where A˜−1 is the standard ”lifting of indices”, and δ = ∗d∗. With this assumption, from Proposition 2.4, together with Propositions 7.1 and 7.2 we readily get the following result. Theorem 7.3. Let the flow domain M have no boundary, and let F be an invariant of coadjoint orbits, which is an integral over flow domain M of a smooth function of the fluid velocity v and a finite number of its derivatives. Then, on a ”typical” coadjoint orbit, the variational derivative δF/δ[u] equals the variational derivative of the Casimir functions (7.1), up to a constant multiplier in the odd-dimensional case. Remark. This is a stronger fact than linear dependence of vector fields on M . Proof. In Section 2. we defined a subspace F, which in our case are functions on the dual algebra with the Lie group SDiff(M ), such that the variational derivatives of these functions belong to SVect(M ). To prove the theorem, we only have to show that the functions on the dual algebra to SVect(M ), that are integral invariants of local densities, i.e., the integrals over M of smooth functions of v and a finite number of its derivatives, belong to F. Indeed, if this is the case, then by Propositions 7.1 and 7.2, we get that for a ”typical” coadjoint orbit, there are no other symmetries than the above ones. By Proposition 2.4, these are differentials of the Casimirs (7.1). We show now that the variational derivative of any integral invariant of local densities is a smooth divergence-free vector field on M . Let the flow domain M have no boundary. Consider an integral Z Z F (v([u]))dn x,

F (v)dn x =

M

M

A−1 [u]

where v([u]) = – it should be defined as a functional on cosets [u]. To calculate its variational derivative, one should replace u by u + ǫw, and differentiate with respect to ǫ at ǫ = 0. One can see that the result is a sum of the following terms: Z ([i]) F˜ij wj dn x, M

where by w([i]) we have denoted partial derivatives with the multiindex i: w([i]) =

∂ [i] w ∂xi11 . . . ∂xinn

,

i = (i1 , . . . , in ),

[i] = i1 + · · · + in .

The functions F˜ij contain both the derivatives of F , and the derivatives of the Green’s functions. Every such term can be integrated by parts (the boundary of M is empty), reducing each summand to Z Z n ˜ w ∧ αj . wj Fj d x = M

M

We note that integration by parts and studying variational derivatives is a standard tool in finding first integrals and determining stability in 2-dimensional ideal hydrodynamics, see, e.g., [3]. Thus the whole expression can be written as: Z w∧α M

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for some (n − 1)-form α. The form α is closed, as the original functional (and thus its variational derivative) is well-defined on cosets. Thus, this variational derivative is a vector from SVect(M ), and the theorem is proved. 2 If we can take for granted the fact that the ”typical” case, when our vortex fields do not admit extra symmetries, is everywhere dense in the space of all 1-forms on M in some appropriate topology, then from Theorem 7.3 follows readily that there are no invariants of coadjoint orbits, which are the integrals of local densities, that are linearly independent of the integrals (7.1). Remark. The similar assumption, formulated for continuous first integrals may not be true. Consider for example a perturbed smooth 2-dimensional Hamiltonian system. One can always construct a continuous first integral of such systems, by setting it to constants between the Kolmogorov tori, see, e.g., [4]. It may be possible to imitate this type of construction (for low-dimensional hydrodynamics) using, for example, the Casimir functions of the type Z C

u ∧ du,

where C is a subset of the flow domain M , which is an invariant set of the vorticity vector field for the instantaneous velocity. Above results were proved in case when the manifold M has no boundary. The case of flows on manifolds with boundaries is discussed in [9].

8.

Conclusion

We considered the reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the group. The reduced vector field has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid, the the right- or left-invariant symmetry fields of the reduced field being vortex vector fields, i.e., they annihilate the vorticity 2-form. The distribution of the vortex fields is always integrable, thus it defines a manifold (at least locally), that we call a vortex manifold. Typically, the vortex manifolds are tori. We studied the following control problem. Consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group, and assume that the mass geometry of the system may change under the action of internal control forces. Such system can also be reduced to the Lie group; with no controls, it describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and thus its reduced flow is a stationary flow of an ideal fluid. The control problem for such system is to find the conditions, under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We showed that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold. As an example, we considered the n-dimensional Euler top. We wrote down the reduced controlled system explicitly, showed that the vortex manifolds are tori, and proved that, by changing the mass geometry, every such torus can be transformed to any other torus.

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The reduction that we use admits a straight-forward infinite-dimensional generalization. Consider the Euler equations on a Lie group of volume-preserving diffeomorphisms SDiff(M ) of a Riemannian manifold M , and perform the above reduction. As an invariant of a coadjoint orbit generates a symmetry field for the reduced flow, and in a ”typical” case one can find all such symmetry fields, we get readily a simple proof for nonexistence of new invariants of coadjoint orbits for ideal fluid flows in the class of functionals on the dual algebra, such that the variational derivative of such functional is a smooth vector field from the Lie algebra SVect(M ), if we can take for granted the fact that the ”typical” case is everywhere dense in the space of all 1-forms on M in some appropriate topology.

Appendix A.

Reduction to the Lie Group for Nonholonomic Systems

Consider the following equations, that we will refer to as the generalized Euler equations (the left-invariant case): m˙ c = ad∗A(g)mc mc ,

(1.1)

Lg−1 ∗ g˙ = A(g)mc .

(1.2)

Here A(g) : g∗ → g is positive definite symmetric operator. Example. Consider the Chaplygin problem of a rigid ball rolling on a horizontal plane. The equations of motion are: M˙ = M × ω,

γ˙ = γ × ω,

M = Iω + Dγ × (ω × γ),

where M is the ball momentum with respect to the moving axes, fixed in the ball (momentum in the body), ω is the angular velocity, γ is the unit vertical vector, also written with respect to the moving axes, matrix I is the inertia tensor and D is a constant. One can see that these equations are of the form (1.1). Theorem 1.1. (The Euler theorem) The momentum in space ms is constant for the generalized Euler equations (1.1-1.2). Proof. Differentiate relation (2.1) by time and apply (1.2), cf. [3]. 2 Proposition 2.1 relied only on relation (2.1), which turns out to be true also for this case. Thus, reduction to the group is possible, and the reduced vector field is v(g) = Lg∗ A(g)Ad∗g ms

(1.3)

v(g) = Rg∗ A(g)Ad∗g−1 mc .

(1.4)

correspondingly in the left- or right-invariant case. Here ms , respectively mc , is constant. It would be interesting to find hydrodynamic description of the reduced field. Introducing dissipation is another way to get coupled equations of the Euler type.

304

B.

Mikhail V. Deryabin

Controllability Conditions

As system (4.1) has the standard form x˙ = f (x) +

X i

ui gi (x),

|u(t)| ≤ ǫ

(2.1)

of a classical control system, one can apply general theorems to it. Theorem 2.1. Let the Lie group G be compact. Then system (4.1) is controllable for all ǫ > 0, if the minimal Lie subalgebra of vector fields on G, which contains both vector fields Lg∗ λi and v(g), spans the tangent space Tg G at any g ∈ G. From Theorem 2.1 follows, that the minimal number of controls is mainly defined by the Lie algebra structure. This minimal number of controls should not necessarily be equal to the number of the degrees of freedom of the system – unless the Lie algebra is commutative. If the dimension of the Lie algebra g is greater than 1, and we are interested in controllability for all inertia operators and all momenta ms , then the minimal number of controls should necessarily be greater than 1. Proof. We only have to check that the vector field v(g) = Lg∗ A−1 Ad∗g ms is Poisson-stable (i.e., almost all trajectories come back to the vicinity of the initial conditions infinitely many times). Indeed, if the minimal Lie subalgebra, which contains vectors Lg∗ λi and v(g), spans the tangent space Tg G at any g ∈ G, then for each point g0 ∈ G, the set of points g(t, g0 , u(t)), accessible by the controls u(t) for 0 < t < T , form an open set (the point g0 itself may belong to the boundary). Under the condition of the Poisson stability, these sets can be joined together to get the necessary trajectory, see, e.g., [22] for details. The Poisson stability follows from the existence of a smooth invariant measure of the reduced system g˙ = v(g), as we have assumed that the group G is compact. But this is exactly the case: if the Lie group G is compact, the reduced system always preserves a bi-invariant Haar measure on G, see [20]. 2 Corollary 2.1. Under conditions of Theorem 2.1, let the Lie algebra g be real and semisimple. Then two controls is sufficient for controllability for all values of the inertia operator and of the momentum ms . Proof. It is well known that a real semisimple Lie algebra is generated by 2 elements, see, e.g., [6]. 2 Remark. If the momentum ms = 0, then under conditions of Theorem 2.1, system (4.1) is controllable even if the Lie group is noncompact – this is the classical RashevskyChow theorem, see, e.g., [15] The condition of Theorem 2.1 is usually referred to as the Lie algebra rank condition (see, e.g., [25]). In real systems, it may be difficult to check it directly, as, in principle, the number of commutators one has to take is not bounded from above. One can suggest using a ”transversality” condition, see [10].

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References [1] Arnold V.I. Mathematical methods in classical mechanics. Springer-Verlag. [2] Arnold V.I., Givental A.B. Symplectic Geometry. Dynamical Systems 4, SpringerVerlag. [3] Arnold V.I., Khesin B.A. Topological Methods in Hydrodynamics. Springer-Verlag. 1998. [4] Arnold V.I., Kozlov V.V., Neishtadt A.I. Mathematical aspects of classical and celestial mechanics. Springer-Verlag, 1988. 291 p. [5] Bao D., Ratiu T. On the geometrical origin and the solutions of a degenerate Monge– Amp`ere equation. Proc. Symp. Pure Math. AMS, Providence 54 (1993), 55–68. [6] N. Bourbaki, Groupes et alg`ebres de Lie, Hermann, 1975, Chapitres 7 et 8. [7] Cardetti, F., Mittenhuber, D. Local controllability for linear systems on Lie groups. J. Dyn. Contr. Sys., 11, 3, July 2005, 353-373. [8] Deryabin M.V., Fedorov Yu.N. On Reductions on Groups of Geodesic Flows with (Left-) Right-Invariant Metrics and Their Fields of Symmetry. Doklady Mathematics. Interperiodica Translation, 68, 1, (2003) 75–78. [9] Deryabin M.V. Ideal hydrodynamics on Lie groups. Physica D, 221 (2006), 84–91. [10] Deryabin M.V. Control of mechanical systems on Lie groups and ideal hydrodynamics. J.Math.Sci. (2008). In print. [11] Dubrovin B.A., Novikov S.P., Fomenko A.T. Modern geometry. Vol. 2. Springer. [12] Fedorov Yu. N. Integrable flows and B¨acklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3), J. Nonlinear Math. Phys. 12 (suppl. 2) (2005) 7794. ¨ [13] Frahm F. Uber gewisse Differentialgleichungen. Math. Ann. 8 (1874), 35–44 [14] Ginzburg V.L., Khesin B.A. Topology of steady fluid flows. Topological aspects of the dynamics of fluids and plasmas, eds. H.K.Moffat et al. (1992). Kluwer Acad. Publ., Dordrecht, 265–272. [15] Hermann, R. (1968). Accessibility problems for path systems (2nded .). In Differential geometry and the calculus of variations (pp. 241257). Brookline, MA: Math Sci Press. [16] Khesin B., Misiołek G. Asymptotic Directions, Monge-Amp`ere Equations and the Geometry of Diffeomorphism Groups. Journal of Math. Fluid Mech. 7 (2005) 365375. [17] Khesin B.A., Chekanov Yu.Y. Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in d dimensions. Phys. D. 40 (1989), 119–131.

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[18] Kozlov V.V. Hydrodynamics of Hamiltonian systems. Vestn. Moskov. Univ., Ser. I. Mat. Mekh., No. 6 (1983) 10–22 (Russian) [19] Kozlov V.V. The vortex theory of the top. Vestn. Moskov. Univ., Ser. I. Mat. Mekh. No. 4 (1990) 56–62 (Russian) [20] Kozlov V.V. Dynamical systems X. General vortex theory. Springer-Verlag, 2003. [21] Kozlov V.V. Dynamics of variable systems and Lie groups. J. Appl. Math. Mech. 68 (2004), 803–808 [22] Lian, K. Y., Wang, L. S., and Fu, L. C. Controllability of spacecraft systems in a central gravitational field. IEEE Transactions on Automatic Control, 39 (12), (1994), 2426-2441. [23] Liouville, J., Developpements sur un chapitre de la ”Mechanique” de Poisson. J. Math. Pares et Appl., (1858), 3, 1-25. [24] Manakov S.V. Note on the integration of Euler’s eqations of the dynamics of an ndimensional rigid body. Funkts. Anal. Prilozh. 10 (1976), 93–94. English transl. in: Funct. Anal. Appl. 10 (1976), 328–329. [25] Manikondaa V., Krishnaprasad P.S. Controllability of a class of underactuated mechanical systems with symmetry. Automatica 38 (2002), 1837-1850. [26] Marsden J.E., Weinstein A. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5 (1974), 120–121. [27] Marsden J., Weinstein A. Coadjoint orbits, Vortices, and clebsch Variables for Incompressible Fluids. Physica D, 7 (1983), 305–323. [28] Misiołek G. Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms, Indiana Univ. Math. J. 42 (1993), 215-235. [29] B. Palmer. The Bao-Ratiu equations on surfaces. Proc. Roy. Soc. London Ser. A 449 (1995), no. 1937, 623-627. [30] Warner F.W. Foundations of differentiable manifolds and Lie groups. Springer-Verlag, 1983. [31] Whittaker E.T. A treatise on analytical dynamics. 4-d ed. , Cambridge Univ. Press, Cambridge 1960

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 307-323

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 11

Some Approximation Theorems for Quasimetric, Induced by C 1 -smooth Non-commutative Vector Fields A.V. Greshnov 1 Institute of Mathematics of Siberian Branch of Russian Academy of Science. pr. Akad. Koptuga, 4., 630090, Novosibirsk

Abstract

On some domain O ⊂ RN we consider some collection of C 1 -smooth non-commutative vector fields X = {Xi }i=1,...,N such that rank(X1 , . . . , XN )(g) = N , for every g ∈ O, equipped with the graduation. Let us denote by θg the canonical (exponential) mapping induced by X in some neighbourhood O of g, acting on some neighbourhood of origin to O. We suppose that the vector fields {(θg−1 )∗ Xi }i=1,...,N are satisfying some special conditions of homogeneity. Using the properties of the mapping θg and the conditions of homogeneity we define some anisotropic metric (quasimetric) dX cc which agrees with our graduation and consider the metric spaces (quasispases) (O, dX cc ), X

X

(Og , dccg ), where (O, dccg ) is the local homogeneous approximation of (O, dX cc ) with respect to the action of the homogeneous operator of dilatation which agrees with

b

b

X

our graduation in some neighbourhood of g ((O, dccg ) is some analogue of so-called

b X b (O, d g ),

nilpotent tangen cone). For the quasispaces (O, dX cc cc ) we develope some technique, which help us to get the local approximation theorem for quasimetric. As the consequence, we get some results for quasispaces induced by the collection of C 1 basis canonical non-commutative vector fields, by the collection of C 2 basis non-commutative vector fields. 1 E-mail address: [email protected]. Work partially supported by RFBR (grant 06-0100735-a).

308

A.V. Greshnov

Keywords: Existence and uniqueness of the solutions of the ordinary differential equations, Theorem of Ascoli — Arzela, quasimetric, tangent cone, local homogeneous approximation, vector field

Introduction Let X = {Xi }i=1,...,N be a collection of basis vector fields defined on some domain O ⊂ RN , i. e. rank(X1 , . . . , XN )(g) = N, ∀g ∈ O ⇐⇒ det X(g) 6= 0 ∀g ∈ O. Hereinafter symbol X will be denote (depending on context) both the collection of vector fields and the N × N -matrix i-column of which is coinciding with Xi . Also let us consider some increasing finite sequence of numbers hi , i = 1, . . . , M , such that hM = N , M ≤ N . Give to every number i = 1, . . . , N its own formal degree deg i = k, where hk−1 < i ≤ hk , h0 = 0. Set of numbers hi and degrees deg i we call graduation of the collection X. We suppose that X ∈ C 1 , and the elements of the matrix X expressing (locally) in canonical coordinate system satisfy the ¡¡homogeneity¿¿ conditions (2.1), see Proposition 2.1, Theorem 2.1. One of hte goals of our work is the comparison of the anisotropic riemannian distance (quasidistance) dX cc , which is induced by X and agreed with our graduation, bg see § 2, (2.2), with the quasidistance dX cc , which is induced by the collection of b g = {X b g }i=1,...,N is the homogeneous local approximation of the vector fields X i b g , see collection X, defined in some neighbourhood of fixed point g ∈ O, and X g b definition 3.1. These collections of vector fields X and X appeared naturally in different parts of analysis, non-holonomic geometry and the theory of partial differential equations, see for example, [1–11]. The main results of our work are the Theorems 3.1, 4.1 and the Corollaries 3.3, 4.4, which are the local approximation theorems for corresponding quasimetrics. Interest to such problems were initiated, from one point, by well-known Gromov’s work [1], where the local approximation theorem for Carnot-Cara´eodory metrics was proved, from other point — by works [2, 3], where the theorem of differentiability of the quasiconformal mappings on Carnot-Careth´eodory spaces equipped by quietly smooth horizontal distributions were proved. In ¡¡smooth¿¿ case we can construct the (nilpotent) tangent cone and prove corresponding local approximation theorem with the help of the formal Campbell-Hausdorff series, see for example [4, 5]. But when X is C 1 only, such method is useless. In [1] it was stated the idea of the construction of collecb g , which was (in some way) the ¡¡basis of tangent space¿¿ of tion of vector fields X (local) nilpotent tangent cone (or homogeneous nilpotent approximation, see [6]) for X in some neighbourhood O of the fixed point g on condition that the vector fields Xi ∈ X ∈ C 1 , i = 1, . . . , N , satisfied the following ¡¡table of commutators¿¿ [Xi , Xj ](u) =

X

Cijk (u)Xk (u),

u ∈ O,

k

where Cijk (u) = 0 for deg i + deg j < deg k. For the realization of this idea Gromov used some technique which based on the ¡¡dilatation¿¿ by the factor 1/ε of ε-scaling vector fields εdeg i Xi expressing in the terms of coordinate system of the second kind

Some Approximation Theorems

309

(ε-1/ε-scaling method) and study the corresponding system of ordinary differential equation for the vector fields εdeg i Xi , see [1, 1.4]. In a different way nilpotent homogeneous approximation for C 1 -smooth ¡¡canonical¿¿ collections of vector fields X was obtained in terms of canonical coordinate system of the first kind in [7]; also we want to refer [8], where reader can find all necessary mentions and comments closely connected with here discussing questions. Simplest analysis, see the Chapter 1 of the present paper, shows us that it is necessary (in some sense) to hold the ¡¡homogeneity¿¿ conditions, see the Proposition 1.1, (1.5), for the validation of the ε-1/ε-scaling method in the case of arbitrary collection of vector fields X ∈ C 1 defined in some neighbourhood of origin; on the other hand if the ¡¡homogeneity¿¿ conditions hold (2.1) for X (in terms of corresponding coordinate system) when b g for X obviously. Using ¡¡homothere is a local homogeneous approximation X geneity¿¿ conditions we estimate the deviation of the integral lines corresponding b g without the help of the Campbell-Hausdorff vector fields belonging to X and X series, see the Theorem 3.1 (in the Theorem 4.1 we obtained more precise estimate), from what we got local approximation theorem for the quasimetrics for our considerations, see the Corollaries 3.3, 4.3. Note that estimate o(ε) in local approximation theorem follows directly from the convergence the vector fields from X to the ¡¡approximating¿¿ vector fields (theorem 3.1); more precise estimates (Theorem 4.1) don’t follow directly from the convergence. Example 4.1 of the present paper touches upon the question about uniqueness of the definition of the (nilpotent) tangent cone, see [1, 2, 4, 8]. Namely, let us consider all integral lines of the vector b g with starting points belonging to some neighbourhood fields from the collection X of the fixed point g as ¡¡abstract¿¿ curves of ¡¡tangent¿¿ space. Example 4.1 shows that from the ¡¡abstract¿¿ point of view we can find many algebras Lie Vg which are ¡¡tangent¿¿ to considering class of the curve, and local approximation theorem type estimates hold for quasimetric induced by the collections of the vector V fields X and quasimetric of every quasispace (Gg , dccg ) corresponding to Vg , see the Vg Corollaries 2.2, 2.3. At that quasimetrics dcc are not equivalent to each other in general, see the Theorem 2.1 and Corollary 2.1. One of the possible methods to get Vg the ¡¡tangent cone¿¿ for (O, dX cc ) among all (Gg , dcc ) is convergence by nets under action of corresponding operator of dilatation, see the Theorem 4.3 and also [5, § 6]. This method are very close to the Gromov-Hausdorff convergence of abstract compact metric spaces, see [5], and it can be useful to study differential properties of mappings on anisotropic riemannian spaces (quasispaces).

§ 1. ε-scaled Vector Fields and <> Systems of Differential Equations In present paper we will use some definitions and notions from [4, 5]. Let |x|∞ = |(x1 , . . . , xN )|∞ = max |xi |, i=1,...,N

x = (x1 , . . . , xN ),

and δji is Kroncker symbols, ei is usual basis vector of Euclidean space RN . Let us define nonhomogeneous operator of dilatation in RN δt corresponding to choosing graduation as δt (x) = δt (x1 , . . . , xN ) = (tx1 , . . . , tM xN ). For every N -dimensional

310

A.V. Greshnov

multiindex β = (β1 , . . . , βN ) let us suppose |β|h =

N P

βi deg i; also

i=1

Boxecc (0, ε) = {x ∈ RN | max |xi |1/ deg i < ε}. i=1,...,N

Let X = {Xi }i=1,...,N be the collection of continuous basis vector fields in RN . Then for every sufficient small numbers αi ∈ R the following Cauchy problem (1.1)

z(s) ˙ =

N X

αi Xi (z(s)),

s ∈ [0, 1],

z(0) = z0 ∈ U ⊂ RN ,

i=1

has some solution z(s). Definition 1.1. Collection of continuous vector fields X from (1.1) is homogeneous with respect to the action of the operator δt , if the curve zt (s) = δt z(s) is the solution of the following Cauchy problem (1.2)

z˙t (s) =

N X

αi tdeg i Xi (zt (s)),

s ∈ [0, 1],

zt (0) = δt z0 ∈ U.

i=1

i Let us denote Xi = (f1i , . . . , fN ), εdeg i Xi = Xiε .

Proposition 1.1. Suppose that the families of the functions {εdeg i−deg k fki (δε x)},

deg k > deg i,

i = 1, . . . , N,

are uniformly bounded and equicontinuous on Boxecc (0, 1). Then there is the sequence of numbers εl → 0 such that vector fields (δ1/εl )∗ Xiεl , i = 1, . . . , N , conbi , such that the collection {X bi } verge uniformly to some continuous vector fields X is homogeneous with respect to the action of the operator δt . Proof. Let us consider the following Cauchy problem for ε-scaling vector fields Xiε :  deg i i f1 (xε ),   x˙ ε,1 = ε (1.3) ...   i (xε ), xε (0) ∈ Boxecc (0, ε). x˙ ε,N = εdeg i fN Change the variables xεi = ε− deg i xε,i . Then the system (1.3) looks like  f i (εdeg 1 xε1 ,...,εdeg N xεN ) i  = F1,ε (xε ), x˙ ε1 = 1  εdeg 1−deg i        

... x˙ εN =

i fN (εdeg 1 xε1 ,...,εdeg N xεN ) εdeg N −deg i

i = FN,ε (xε ),

δ1/ε xε (0) = xε (0) ∈ Boxecc (0, 1).

Because we consider (1.3) on Boxecc (0, const ·ε), then we have ( f i (εdeg 1 xε ,...,εdeg N xε ) k 1 N @ >> ε → 0 > 0, εdeg k−deg i fki (εdeg e1 xε1 ,...,εdeg N xεN ) @ εdeg k−deg i

>> ε → 0 >

aik

=

fki (0)

= const,

deg k < deg i, deg k = deg i.

Some Approximation Theorems

311

Using well-known Theorem of Ascoli — Arzela we get that there is the sequence of i numbers εl → 0 such that ¡¡limit¿¿ functions Fbki (ˆ x) = lim Fk,ε (xεl ), deg k > deg i, l εl →0

are continuous. Denote lim xεl = x ˆ. We can get that εl →0

Fbki (δt x ˆ) = tdeg k−deg i Fbki (ˆ x).

(1.4)

i Indeed, we have Fbki (ˆ x) = lim Fk,ε (xεl ), but from other side l εl →0

i i Fk,ε (xεl ) = tdeg k−deg i Fk,tε (xtεl ), l l i and lim Fk,tε (xtεl ) = Fbki (δ1/t x ˆ), so tdeg k−deg i Fbki (δ1/t x ˆ) = Fbki (ˆ x), from what we l tεl →0

have requiring ¡¡homogenety¿¿ (1.4). It follows from (1.4), that Fbki (0) = 0. So, we proved, that for every i = 1, . . . , N vector fields (δ1/εl )∗ Xiεl uniformly converge to bi such, that vector fields X bi = (0, . . . , 0, aih X , . . . , aihdeg i , Fbhi deg i +1 (ˆ x), . . . , FbNi (ˆ x)), deg i−1 +1 (1.5)

ail = const,

Fbki (δε x) = O(εdeg k−deg i ),

Fbki (0) = 0.

Let us consider following Cauchy problem for vector fields (1.5) (1.6)

x(s) ˙ =

N X

bi (x(s)), αi X

αi = const,

s ∈ [0, 1],

x(0) = x0 .

i=1

Let x(s) is the solution of the problem (1.6); denote xt (s) = δt x(s). Then using (1.4) we get (1.7)

(δt )∗ (x(s)) ˙ = x˙ t (s) =

N X

bi (δ1/t xt (s)) = αi · (δt )∗ X

i=1

N X

bi (xt (s)). αi tdeg i X

i=1

So the Proposition 1.1 has proved.

§ 2.

Quasimetrics

Let A = (an,m ) denotes some N × N -matrix, where n is number of the line of A, m is number of the column of A. So, expression an,m ∈ A means that an,m is the element of matrix A which is situated on the intersection of n-line and m-column of A. Also denote Ai,j , 1 ≤ i, j ≤ M , the part of matrix A, which is the rectangular matrix consists of all elements an,m such that hi−1 < n ≤ hi , hj−1 < m ≤ hj . Lemma 2.1. Let the elements of matrix A = A(ε) satisfy following conditions of ¡¡homogemety¿¿  m n ≤ m,   δn + o(1), (2.1)

an,m ∈ Ai,i , an,m ∈ / diag Ai,i ,

o(1),

an,m =

 

O(ε

i−j

),

an,m ∈ Ai,j , i > j,

312

A.V. Greshnov

where o(1)@ >> ε → 0 > 0 uniformly. Then the elements of matrix A−1 also satisfy conditions of ¡¡homogemety¿¿ (2.1). Proof. Lemma 2.1 can be proved with the help of the method of proof of the Lemma 4.1 from [5]. On some domain O ⊂ RN , where diam O is sufficiently small, let us consider some collections of C 1 -smooth basis vector fields X = {Xn }n=1,...,N , Y = {Yn }n=1,...,N . Using well-known theorems about existence and uniqueness solutions of ordinary differential equations, see, for example, [12], we can get that for arbitrary points N N P P u, v ∈ O there are unique vector fields pi Xi , qi Yi , where pi , qi = const, i=1

such that exp

N P

i=1

N

P pi Xi (u) = v, exp qi Yi (u) = v; hereinafter the expression

i=1

i=1

exp(aA)(g) denotes the endpoint of integral line of vector field A having starting point g, and length of this integral line is equal to 1. Recall that the mapping N  X xi Zi (u), θu,Z : (x1 , . . . , xN ) → exp

u ∈ RN ,

Z = {Zi }i=1,...,N ,

i=1

is the diffeomorphism of class C k between some (sufficiently small) neighbourhood of origin and some neighbourhood of u, if vector fields Zi ∈ C k , i = 1, . . . , N , are basis in some neighbourhood of u, see, for example, [13]. So the mapping θu,Z induces the normal coordinate system or coordinate system of the first kind in neighbourhood of origin. Let us denote by the symbols θg,X , θg,Y , g ∈ O ⊂ RN , the diffeomorphisms induce the coordinate systems of the first kind connected with corresponding collections X, Y ∈ C 1 of basis vector fields defined in O, and let

(2.2)

m,X −1 (θg,X )∗ Xm = (xm,X 1,g , . . . , xN,g ),

m,X m,X −1 (θg,X )∗ Ym = (y1,g , . . . , yN,g ),

m,Y −1 (θg,Y )∗ Xm = (xm,Y 1,g , . . . , xN,g ),

m,Y m,Y −1 (θg,Y )∗ Ym = (y1,g , . . . , yN,g ).

−1 e g . Let us consider the Remark 2.1. Denote α = (α1 , . . . , αN ), (θg,X )∗ Xi = X i following Cauchy problem

(2.3)

x˙ =

N X

e g (x(s)), αi X i

x(0) = b,

s ∈ [0, s0 ],

i=1

where α, b belong to some neighbourhood U of origin, and diam U is sufficiently e g , i = 1, . . . , N , are a priori only continuous small. Because the vector fields X i we can not guarantee the uniqueness of the solution of (2.2), but for well-known Peano’s Theorem, see, for example, [14], the solutions for (2.2) always exist. One −1 of these solutions is the curve θg,X (θu,X (sα)), where u = θg,X (b), Later when we will consider Cauchy problem (2.3) we will intend just this solution. −1 Remark 2.2. We have θg,X (exp(

N P

αi Xi )(g)) = sα; so the line sα is the solution

i=1

of the problem (2.3), where b = 0. So, (α1 , . . . , αN ) =

N P i=1

e g }i=1,...,N is canonical, see [15]. that collection {X i

e g (sα). This means αi X i

Some Approximation Theorems

313

Let 1/ deg i dX }, cc (u, v) = max {|pi | i=1,...,N

dYcc (u, v) = max {|qi |1/ deg i }, i=1,...,N

where coefficients pi , qi are defined above, and let X BoxX cc (g, ε) = {v ∈ O | dcc (v, g) < ε}.

Recall, see, for example [4, 5], that function d : A × A → R+ ∪ 0, defined on some set A, is called quasimetric, if: 1) d(u, v) ≥ 0, and d(u, v) = 0 ⇔ u = v; 2) d(u, v) ≤ c1 d(v, u) for some constant c1 > 0, which doesn’t depend on u, v ∈ A; 3) d(u, v) ≤ Q(d(u, w) + d(w, v)) for some constant Q > 0, which doesn’t depend on u, v, w ∈ A. Pair (A, d) we call quasispace. m,Y m,X Proposition 2.1. Suppose that estimates (2.1) hold for am n = xn,g , yn,g , see (2.2), −1 on θg,X BoxX cc (g, const ε) uniformly with respect to g ∈ O and ε ∈ [0, ε0 ). Then the Y functions dX cc (u, v), dcc (u, v) are quasimetrics on O.

Proof. Taking into account the Remarks 2.1, 2.2, we can use the arguments of the Theorem 4.1 from [7]. For example, let us consider dX cc (u, v). It is obvious that dX cc (u, v) = 0 ⇔ u = v. P   P  N N X If u = exp yj Xj (v), then v = exp − yj Xj (u), so dX cc (u, v) = dcc (v, u). j=1

i=1

Let us consider points g, u, v ∈ O such, that N N N X  X  X  exp yj Xjε ◦ exp xj Xj (g) = exp yj Xjε (u) = v, j=1

j=1

j=1

where |x|∞ = |y|∞ = 1, y = (y1 , . . . , yN ), x = (x1 , . . . , xN ). Using well-known theorems from the theory of ordinary differential equations we get that there is P  N N P unique vector field pj Xj , pj = const, such that v = exp pj Xj (g). We want i=1

j=1

to prove that there is a constant Q > 0, such that Q doesn’t depend on points g, u, v and dX cc (v, g) ≤ Q(ε + ).

(2.4) We have

N N N   X  X X g deg Xj e g (0), e e g (0) = exp deg Xj xj X ◦ exp ε y X pj X θg−1 (v) = exp j j j j j=1

j=1

j=1

314

A.V. Greshnov

N e g = (θ−1 )∗ Xj . Let θ−1 (v) = p = (p1 , . . . , pN ), Ye ε = P εdeg Xj yj X eg. where X g g j j j=1

Then p = η(1), where η(s) ˙ = Ye ε (η(s)),

s ∈ [0, 1],

η(0) = δ x,

so Z1 (2.5)

p = δ x +

Ye ε (η(s)) ds.

0

From (2.1), (2.5) we get X

pj = deg Xj xj + εdeg Xj yj +

cjp,q p εq ,

deg Xj >p,q>0 p+q=deg Xj

where cjp,q = cjp,q (x, y, , ε, g) are some continuous functions on A = Be (0, κ) × Be (0, κ) × [0, ε0 ] × [0, ε0 ] × O, which and uniformly bounded by some c0 = const. Then it is obviously that there is some constant c˜ = c˜(c0 ) such that |pj | ≤ ( + c˜ε)j ,

(2.6)

|pj | ≤ (ε + c˜)j .

So (2.4) is proved and Q = 2˜ c. m,X m,Y m,Y m,X Theorem 2.1. Suppose that estimates (2.1) hold for am n = xn,g , yn,g , xn,g , yn,g e e −1 −1 on sets θg,X Boxcc (0, ν · ε), θg,Y Boxcc (0, ν · ε) uniformly with respect to g ∈ O, ε ∈ [0, ε0 ), and ν > 1 is some constant, which doesn’t depend on g, ε. Then

(2.7)

N N n X X

Y

o t Y Xit (uε ), uε max dX (u ), u , d exp exp ε ε i cc cc i=1

i=1



≤ C1 · max

i=1,...,N

tdeg i +

deg i−1 X

1/ deg i , tdeg i−k · O(εk )

C1 = const,

k=1

Y where uε ∈ BoxX cc (g, ν · ε) ∩ Boxcc (g, ν · ε); in particular, Y X BoxX cc (g, c2 · ε) ⊂ Boxcc (g, c3 · ε) ⊂ Boxcc (g, c4 · ε)

(2.8)

for some positive constants c2 , c3 , c4 , which don’t depend on ε. Proof. Using (2.1), we have (2.9)

Yit

=

N X

i ηj,Y

i ηj,Y

Xj ,

 =

j=1

ci,j,Y · tdeg i , deg i

ci,j,Y · t

1 ≤ j ≤ hi ,

· O(ε

deg j−deg i

),

j > hi ,

where ci,j,Y is uniformly bounded on their domains on definition. From (2.9) it follows that (2.10)

N X i=1

t

deg ei

Yi =

N X i=1

ζi Xi ,

ζi = c˜i,Y t

deg i

+

deg i−1 X k=1

c˜i,k,Y tdeg i−k O(εk ),

Some Approximation Theorems

315

where c˜i,Y , c˜i,k,Y are uniformly bounded. In the terms of θu−1 let us consider the ε ,X N P e uε , p(0) = 0, s ∈ [0, 1]. Denote p(1) = vε . following Cauchy problem p(s) ˙ = ζi X i i=1 1 N R P e uε ds = |fu |, where Using (2.1), (2.10), we get |vε | ≤ ζi · X ε i 0 i=1

(2.11) fuε = (f1,uε , . . . , fN,uε ),

fi,uε = Ci,uε ,Y tdeg i +

deg i−1 X


cˆi,k,Y tdeg i−k · O(εk ) ,

k=1 Y and Ci,uε ,Y , cˆi,k,Y are uniformly bounded. From (2.11) we get (2.7) for dX cc ; for dcc estimate (2.7) can be proved by the same way; (2.8) follows from (2.7). Y Corollary 2.1. Quasimetrics dX cc , dcc are not equivalent. Y Corollary 2.2. If t = O(ε), then it holds |dX cc (uε , vε ) − dcc (uε , vε )| = O(ε) uniX formly with respect to uε , vε ∈ Boxcc (g, νε).

Proof. Corollary 2.2 follows from (2.3); at that O(ε) can’t be o(ε) in general because Ci,uε ,Y , cˆi,k,Y from (2.11) are not equal Ci,uε ,X , cˆi,k,X in general. Y Corollary 2.3. If t = o(ε), then |dX cc (uε , vε ) − dcc (uε , vε )| = o(ε) holds uniformly X with respect to uε , vε ∈ Boxcc (g, νε).

§ 3.

Local Homogeneous Approximations of Collections of Vector Fields Fix some point g ∈ O and sufficiently small number ε0 > 0 such that BoxX cc (g, ε0 ) ⊂ e g,ε , X e g,ε = O. For collection X from § 2 let us consider vector fields (δ1/ε )∗ X i i g e , i = 1, . . . , N , on Boxecc (0, ε0 ) = θ−1 (BoxX εdeg i X cc (g, ε0 )). We suppose that i g,X collection of vector fields X such that the following uniform convergences take place on Boxecc (0, ε0 ) (3.1)

e g,ε ⇒ε→0 X b 0,g , (δ1/ε )∗ X i i

i = 1, . . . , N.

Then, taking into account remark 2.2, from (3.1) we get (2.1) for vector fields belong eg. to X 0,g bm Property 3.1. Vector fields X , m = 1, . . . , N , are 10 basis, 20 homogeneous with respect to the action of the operator δt , 30 segment of the straight line (β1 s, . . . , βN s), s ∈ [0, 1], bi = const, is solution of the following Cauchy problem

y(s) ˙ =

N X

b 0,g (y(s)), βi X m

y(0) = 0,

s ∈ [0, 1].

m=1

Proof. Item 10 follows from (2.1), (3.1), item 20 follows from the Proposition 1.1. Let us prove item 30 . For this let us consider segment of the straight line (α1 s, . . . , αN s),

316

A.V. Greshnov

s ∈ [0, 1], which is solution of (2.3), see remark 2.2. Let αi = εdeg i βi , |βi | ≤ const. N P e g (δε y ε (s)), Then we change the variables y ε = δ1/ε x in (2.3) and get y˙ ε (s) = (δ1/ε )∗ εdeg i βi X i m=1

y ε (0) = 0, s ∈ [0, 1]. It is not difficult to see that y ε (s) = (β1 s, . . . , βN s), and then, using (3.1), we get item 30 . g 0,g g bm bm b g = {X bm Definition 3.1. Let us denote X = (θg )∗ X , X }m=1,...,N . We say g b is local homogeneous approximation of the that the collection of vector fields X collection of the vector fields X with respect to the action under operator ∆g,X = ε −1 θg,X ◦ δε ◦ θg,X .

 P N b g (u), u ∈ BoxX Consider the mapping θu,Xb g : (x1 , . . . , xN ) → exp xi X cc (g, ε0 ). i i=1

b g ∈ C, so the mapping θˆ b g may be false generally, beCertainly we have X g,X cause an ordinary differential equation with continuous right side may have more b g ∈ C 1 . In than one solution. So later we will suppose (for simplicity) that X ˆ this case we can guarantee that the mapping θu,Xb g is diffeomorphism. Then from item 30 of the Property 3.1 it follows that θg,X (x1 , . . . , xN ) = θg,Xb g (x1 , . . . , xN ), so bg

bg

X X BoxX cc (g, ε) = Boxcc (g, ε). Let dcc be quasimetric induced in O by the collection bg X g b , agreed with the graduation. Note that quasimetrics dX of vector fields X cc , dcc satisfy the conditions of the Theorem 2.1. bg

bg

bg

X g,X X g,X Property 3.2. Let u, v ∈ BoxX cc (g, ε0 ). Then dcc (∆ε u, ∆ε v) = εdcc (u, v).

Proof. There is unique collection of numbers ai , i = 1, . . . , N , such that N  X b g (u). ai X v = exp i i=1

 P N b g (∆g,X ai εdeg i X Using the arguments of the proof of (1.7), we get ∆g,X ε u). ε v = exp i i=1

bg

Then, using definition of quasimetric dX cc , we get the Property 3.2. Let Wε,ω =

N P i=1

N c g = P ωi εdeg i X b g , where ε ≤ ε0 , ω = (ω1 , . . . , ωN ), ωi εdeg i Xi , W ε,ω i i=1

ωi = const, |ω| < c = const. Theorem 3.1. The following estimate 
Xb g
cg cg max dX = o(ε) cc exp(Wε,ω )(uε ), exp(Wε,ω )(uε ) , dcc exp(Wε,ω )(uε ), exp(Wε,ω )(uε ) holds uniformly with respect to uε ∈ BoxX cc (g, ε). Proof. From (3.1) we get (3.2)

ε bg (∆g,X 1/ε )∗ Xi ⇒ε→0 Xi ,

i = 1, . . . , N,

uniformly on Boxcc (g, ε0 ). Let us consider a sequence of points uε ∈ BoxX cc (g, ε) g,X X g 1 b such that ∆1/ε (uε ) = u ∈ Boxcc (g, ε0 ). Because X ∈ C , solution of the following

Some Approximation Theorems N P

Cauchy problem x(s) ˙ =

m=1

317

b g (x(s)), x(0) = u, s ∈ [0, 1], is unique for every ωi X m

point u. So, using (3.2), we have the following uniform convergence, see [13, Theorem 2.4], cg ∆g,X s ∈ [0, 1], 1/ε exp(sWε,ω )(uε ) ⇒ε→0 exp(sW1,ω )(u),
bg g,X cg from what it follows dX cc ∆1/ε exp(Wε,ω )(uε ), exp(W1,ω )(u) = rε,ω,u = o(1), where rε,ω,u is uniform as o(1) with respect to u, ε, ω. Then, using Property 3.2, we get
bg cg dX cc exp(Wε,ω )(uε ), exp(Wε,ω )(uε ) = o(ε), from what, using the method of proof of Theorem 2.1, we get dX cc -estimate. Corollary 3.2. The following estimate
Xb g cg cg max{dX cc exp(Wλ,ω )(gε ), exp(Wλ,ω )(gε ) , dcc exp(Wλ,ω )(gε ), exp(Wλ,ω )(gε ))} = o(ε) holds uniformly with respect to gε ∈ BoxX cc (0, ε) and λ ∈ [0, ε]. Lemma 3.1. There is a constant κ > 0 such, that for all g ∈ O and sufficiently small 1 , 2 ∈ (0, ε), ε ∈ [0, ε0 ] the following inclusion holds [ X BoxX cc (v, 2 ) ⊂ Boxcc (g, 1 + κ2 ). v∈BoxX cc (g,1 )

Proof. Taking into account Remark 2.1, Remark 2.2, Lemma 3.1 follows from the estimates (2.6). Corollary 3.3. Inclusions bg

bg

X BoxX cc (gε , λε) ⊂ Boxcc (gε , λε + o(ε))

X BoxX cc (gε , λε) ⊂ Boxcc (gε , λε + o(ε)),

hold uniformly with respect to gε ∈ BoxX cc (0, ε) and λ ∈ [0, 1]. Proof. Corollary 3.3 follows from Corollary 3.2 and Lemma 3.1, see also Corollary 3.2 from [5]. Remark 3.1. From Corollary 3.3 it follows bg

X |dX cc (uε , vε ) − dcc (uε , vε )| = o(ε)

uniformly with respect to uε , vε ∈ BoxX cc (0, ε) and ε ∈ (0, ε0 ].

§ 4.

Some Applications

4.1. Consider some C 1 -smooth basis canonical (see Remark 2.2) collection of vector e = {X ei }i=1,...,N in some neighbourhhod of origin O ⊂ RN such that fields X X ei , X ej ] = ek , (4.1) [X Cijk (x)X k

318

A.V. Greshnov

where Cijk (x) = 0 if deg ei + deg ej < deg ek , and C ∞ -smooth basis canonical b 0 = {X b 0 }i=1,...,N , such that collection of vector fields X i X bi0 , X bj0 ] = bijk X bk0 , [X C deg ei +deg ej =deg ek

bijk = Cijk (0) = const. In [§ 2, 7] we proved that X b 0 is local homogeneous where C e with respect to the action of the operator of dilatation nilpotent approximation of X b = (X b0 , . . . , X b 0 ), δτ agreed with choosing graduation. Denote A = (X1 , . . . , XN ), A 1 N e −1 b −1 b . Let x ∈ Box (0, ε0 ), w = w(t) = tV (tδε x)hej i, w = V = A , V = A cc {wk }k=1,...,N , for some fixed j = 1, . . . , N . In [§ 2, 7] we also proved that w satisfies the following Cauchy problem: (4.2)

∂t w = ej + Ctδε x hw, δε xi,

t ∈ [0, 1].

w(0) = 0,

In the terms of coordinate function the equalities (4.2) look like X (4.3) ∂t wk = δjk ej + Cilk · wi · εdeg el xl deg ek =deg ei +deg el

X

+

Cilk · wi · εdeg el xl .

deg ek <deg ei +deg el

By induction we proved in [7] the following estimates (but it is not difficult to get these estimates without assistance) ( i δj + ε˜ cjg,i (xε ), i ≤ hdeg ej , wi (1) = εdeg ei −deg ej c˜jg,i (xε ), i > hdeg ej , for every i = 1, . . . , N and some continuous functions c˜jg,i . Then we can write (4.3) as (4.4)  δ k e + O(ε), k ≤ hdeg ej ,   j jP deg ek −deg ej +1 deg ek −deg ej xl + O(ε ), k > hdeg ej . Cilk · wi · ε ∂t wk =   deg ek =deg ei +deg el

Also in [7] we proved that w ˆ = w(t) ˆ = tVb (tδε x)hej i, w ˆ = {w ˆk }k=1,...,N , for some fixed j = 1, . . . , N is solution of the following Cauchy problem (4.5)

b0 hw, ∂t w ˆ = ej + C ˆ xi,

w(0) ˆ = 0,

t ∈ [0, 1],

b0 is defined by the identities where action of (structural) operator C X b0 hei , ej i = bijk ek , C bijk = Cijk (0) = const . C C deg ei +deg ej =deg ek

In terms of coordinate function the equalities (4.5) look like X bilk · w (4.6) ∂t w ˆk = δjk ej + C ˆi · εdeg el xl . deg ek =deg ei +deg el

Some Approximation Theorems

319

e is such that the functions Cilk (x) Suppose that the collection of vector fields X satisfy by the conditions |Cilk (δε x) − Cilk (0)| ≤ const εα

(4.7)

e ∈ C 1+α . Then one can for some 1 > α > 0. Note that (4.7) are right in the case X writes the identities (4.4) as (4.8)  k k ≤ hdeg ej ,  δj ej + O(ε),  P deg e −deg e deg e −deg e +α i j k k bilk · wi · ε ∂t wk = C xl + O(ε ), k > hdeg ej .   deg ek =deg ei +deg el

Theorem 4.1. We have wk (δε x) = w ˆk (δε x) + O(ε) for k ≤ hdeg ej , wk (δε x) = w ˆk (δε x) + O(εdeg ek −deg ej +α ) for k > hdeg ej . Proof. Step 1. Case k ≤ hdeg ej obviously follows from (4.6), (4.8). Step 2. Consider the case when hdeg ej < k ≤ hdeg ej +1 . From conditions of adding deg ek = deg ei + deg el it follows that deg ek > deg ei , so i ≤ hdeg ej . Then using identities wi = w ˆi + O(ε), i ≤ hdeg ej , and (4.6), we get ∂t wk − ∂t w ˆk = O(εdeg ek −deg ej +α ),

(4.9)

t ∈ [0, 1],

then (4.10) wk = w ˆk + O(εdeg ek −deg ej +α ) = w ˆk + O(ε1+α ),

hdeg ej < k ≤ hdeg ej +1 .

Step 3. Consider the case hdeg ej +1 < k ≤ hdeg ej +2 . From conditions of adding it follows that i ≤ hdeg ej +1 , because if hdeg ej < i, then, using (4.10), we get (4.9); if hdeg ej ≥ i, then, using step 1, we also get (4.9). So we have wk = w ˆk + O(εdeg ek −deg ej +α ) = w ˆk + O(ε2+α ),

hdeg ej +1 < k ≤ hdeg ej +2 .

It is clear that we shell get necessary estimates in the rest of cases by the same way. Corollary 4.1. We have V (δε x) = Vb (δε x)+B(δε x), where B(δε x) = (bi,j )i,j=1,...,N ,    O(ε), hk < i, j ≤ hk+1 , k = 0, . . . , M, bi,j = O(ε), i ≤ j,   O(εdeg ei −deg ej +α ) = O(εl−k+α ), hl < i ≤ hl+1 , hk < j ≤ hk+1 , k − l ≥ 1. Lemma 4.1 [7]. Let x ∈ Boxecc (0, ε0 ) and ej (δε x) = A(δε x)hej i, (aj1 , . . . , ajN )(δε x) = X b 0 (x) = A(x)he b (ˆ aj1 , . . . , a ˆjN )(x) = X j i. j Then ( ajk (δε x)

=

δkj + O(ε), ε

deg ek −deg ej

k ≤ hdeg ej , P k · Fbα,e · xβ + o(εdeg ek −deg ej ), j

k > hdeg ej ,

k where we add by the conditions β > 0, |β + ej |h = deg ek , Fbβ,e = const, and O(ε), j e deg ek −deg ej o(ε ) are uniform on Boxcc (ε);  j δ , i ≤ hdeg Xj ,   i P j i β a ˆi (x) = Fbβ,ej · x i > hdeg ej .   |α+ej |h =deg ei , α>0

320

A.V. Greshnov

N N P b 0 = P ηj X e e b 0 , where Corollary 4.2. We have X ηˆij X j i i i , Xj = i=1

ηij , ηˆij =

(4.11)



i=1

δij + O(ε),

1 ≤ j ≤ hj ,

O(εdeg ei −deg ej +α ),

j > hj .

ηij )i=1,...,N , j = 1, . . . , N . Then Proof. Let us denote η j = (ηij )i=1,...,N , ηˆj = (ˆ bj , ηˆj = Vb Xj for all j = 1, . . . , N , and (4.11) follows from Corollary 4.1, ηj = V X Lemma 2.1 and Lemma 4.1. 4.2. Here we consider collection of vector fields X from Introduction such that X ∈ C 2 ; so (θg−1 )∗ X ∈ C 1 and |Cilk (θg (δε x)) − Cilk (θg (0))| ≤ const ε

(4.12)

uniformly with respect to g ∈ O. Theorem 4.2. We have the following uniform (with respect to gε ∈ BoxX cc (g, ε)) estimate  bg X cg cg max dX cc (exp(Wε,ω )(gε ), exp(Wε,ω )(gε )), dcc (exp(Wε,ω )(gε ), exp(Wε,ω )(gε )) 1

= O(ε1+ M ), where Wε,ω =

N P j=1

N c g = P ωj X b g,ε , ω = (ω1 , . . . , ωN ), |ω| ≤ c = const, ωj X ε , W ε,ω j j=1

b g }j=1,...,N = X b g is local homogeneous approximation of X with respect to and {X j the action under operator of dilatation ∆g,X ε . Proof. Consider the case then ω = ej for some fixed j = 1, . . . , N , so ωj = 1, ωi = 0 b 0,gε = (a1 , . . . , aN ) = (θ−1 )∗ X b gε , and θ−1 = (t1 , . . . , tN ) = t. for i 6= j. Denote X gε j j gε ,X g b ε }j=1,...,N is local homogeneous approximation by X Because the collection {X j

(but in some neighbourhood of gε !) the coefficients ai satisfy the following properties (compare with the Lemma 4.1):

(4.13)

ai =

 j δ ,   i  

1 ≤ i ≤ hj , gε Cβ,i,j tβ ,

P

i > hj ,

gε Cβ,i,j = const .

|β+ej |h =deg i, β>0, β6=kej ∀k∈N

e gε ,ε (x(s)), x(0) = 0, Let us consider the following Cauchy problems: x(s) ˙ = X j s ∈ [0, 1], b g,ε = (4.14) z(s) ˙ = (θg−1 ) X j ε ,X ∗

N X i=1

e gε = εj ηjg,i · X i

N X i=1

εdeg i ηjg,i ·

N X

b 0,gε , ηigε ,k · X k

i=1

z(0) = 0,

s ∈ [0, 1],

Some Approximation Theorems

321

where ηjg,i , ηkgε ,i is defined by the Corollary 4.2. Using (4.11), we can write (4.14) as (4.15)

z(s) ˙ =

N X

b 0,gε , εdeg j φij · X i

s ∈ [0, 1],

z(0) = 0,

i=1

where functions φij satisfy the estimates (4.11). Let x(s) = (x1 (s), . . . , xN (s)), z(s) = (z1 (s), . . . , zN (s)). We have xi (s) = δij εdeg i s, so x˙ i (s) = δij εdeg i , i = 1, . . . , N . Then, using (4.11), (4.13), (4.15), we get z˙i (s) − x˙ i (s) = O(εj+1 ) for i = 1, . . . , hj , from what zi (s) = δij εdeg i s + O(εj+1 ),

(4.16)

i = 1, . . . , hj .

After that let us consider zi (s) for i = hj + 1, . . . , hj+1 . Using the conditions of the adding from (4.13), and also (4.11), (4.15), (4.16), we get z˙i (s)− x˙ i (s) = O(εj+2 ) for i = hj + 1, . . . , hj+1 , from what it follows zi (s) = O(εj+2 ), i = hj + 1, . . . , hj+1 . Using the same arguments, by induction we get zi (s) = O(εdeg i+1 )

(4.17)

∀i > hj .

Let p(s) is the line segment, connecting the points x(1) = p(0) and z(1) = p(1). Then, using (4.16), (4.17) and Lemma 2.1, Lemma 4.1, we get

(4.18) p(s) ˙ =

N X

(zi (1) − xi (1))ei =

i=1

N X

(zi (1) − xi (1))

N X

i=1

j=1

hj

=

X

 gε e gε ·X vi,j i

e gε + O(εj+1 ) · X i

i=1

N X

e gε . O(εdeg i+1 ) · X i

i=hj +1

−1 Considering (4.18) in the terms of coordinates θv,X , where v = θgε ,X (x(1)), we 1 g,ε ε 1+ M b ). One can get another (exp( X )(g ), exp(X )(g )) = O(ε get the estimate dX ε ε cc estimate by the same manners. One can get general case by the same manners also, but it needs to pay in attention the following equalities

cg = W ε,ω

Wε,ω =

N X

ωj εdeg ej ·

N X

j=1

i=1

N X

N X

j=1

ωj εdeg ej ·

N  X ηij Xi = φˆj Xj ,

φˆj = ωj εdeg ej + O(εdeg j+1 ),

j=1 N  X bg = φj Xj , ηˆij X i

φj = ωj εdeg ej + O(εdeg j+1 ).

j=1

i=1

Corollary 4.3. There is a constant κ ˜ > 0 such that the following inclusions 1

X BoxX ˜ ε1+ M ), cc (gε , λε) ⊂ Boxcc (gε , λε + κ e

bg

1

X BoxX ˜ ε1+ M ) cc (gε , λε) ⊂ Boxcc (gε , λε + κ

hold uniformly with respect to gε ∈ BoxX cc (g, ε) and λ ∈ [0, 1].

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A.V. Greshnov

Proof. Corollary 4.3 follows from the Theorem 4.2 and the Lemma 3.1, see also Corollary 3.2 from [5]. c g )(u), u ∈ BoxX 4.3. Let us consider the curves exp(sW cc (g, ε0 ), from the Theo1,ω Y rem 3.1 as an elements of some quasispace (BoxX cc (g), dcc ), where the collection of b vector fields Y is homogeneous under the action of the operator ∆g,X . Then, using t b c g )(gε )) = the Theorem 2.1 and Property 3.2, we can get that the estimates dYcc (exp(Vε )(gε ), exp(W 1,ε b

eg

X o(ε) which are uniform with respect to gε ∈ BoxX cc (g, ε); note, that quasimetrics dcc , b b Y dYcc can be not equivalent, but BoxX cc (g, ε) = Boxcc (g, ε).

Example 4.1. Consider on Boxecc (0, ε0 ) ⊂ R3 the following smooth collections of e = {X ei }i=1,2,3 , X e1 = (1 + f1 , f2 , 2y + f3 ), canonical vector fields Yb = {ei }i=1,2,3 , X e e X2 = (g1 , 1 + g2 , −2x + g3 ), X3 = (h1 , h2 , 1 + h3 ), where f1 (εx, εy, ε2 z) = O(ε),

f2 (εx, εy, ε2 z) = O(ε),

f3 (εx, εy, ε2 z) = o(ε),

g1 (εx, εy, ε2 z) = O(ε),

g2 (εx, εy, ε2 z) = O(ε),

g3 (εx, εy, ε2 z) = o(ε)

h2 (εx, εy, ε2 z) = O(ε),

h3 (εx, εy, ε2 z) = O(ε),

(4.19) h1 (εx, εy, ε2 z) = O(ε),

for every u = (x, y, z) ∈ Boxecc (0, ε0 ), and [X1 , X2 ] = C1 X1 + C2 X2 + (4 + C3 )X3 for some smooth functions Ci . Then e1ε ; X e2ε ; X e3ε } = {(1, 0, 2y) = X b1 ; lim (δ1/ε )∗ {X

b2 ; (0, 1, −2x) = X

ε→0

b3 }. (0, 0, 1) = X

b1 ; X b2 ] = 4X b3 , and [ei , ej ] = 0 ∀i, j = 1, 2, 3. We have Note that [X 3 X  b ε (δε u) (4.20) wε = exp ωi X i i=1


= exp ω1 eε1 + ω2 eε2 + (2ω1 y − 2ω2 x + ω3 )eε3 (δε u) = (wε1 , wε2 , wε3 ). Apart, dYcc (δε u, δε v) = εdYcc (u, v) for every suitable u, v ∈ Boxecc (0, ε0 ). Denote b

b

3  X eiε (δε u) = (v 1 , v 2 , v 3 ). ωi X vε = exp ε ε ε i=1

Using the Theorem 3.1, see also [5], we have dX cc (vε , wε ) = o(ε), and from (4.19), (4.20) we get b dYcc (vε , wε ) = max {|vεi − wεi |1/ deg i } = o(ε). b

i=1,2,3

Y X Y Obviously we have BoxX cc (0, ε) = Boxcc (0, ε), but quasimetrics dcc , dcc are not equivalent: it is sufficient to compute the corresponding distances between points (x, y, 0) and (x + ε, y, 2yε), where ε is sufficiently small, and y 6= 0. b

b

b

4.4. Let us consider the quasimetric tdX cc on the domain O, where X from § 3. Denote Bt (u, r){v ∈ O | tdX cc (u, v) < r}.

Some Approximation Theorems

323

Theorem 4.3. Let us consider the sequence of compact quasispaces (B tk (g, r), tk dX cc ), where tk → ∞, r < ε0 . Then for every number  > 0 and every finite g,X  dense net Γk ⊂ (B tk (g, r), tk dX Γk is ck -dense net for quasispace cc ) the set ∆t bg X

(Boxcc (g, r), dX cc ), and ck @ >> k → ∞ > 1. e

Proof. Using the Theorem 3.1 and the Property 3.2, one can prove Theorem 4.3 with the method of the proof of the Theorem 6.1 from [5]. Remark 4.2. In is not difficult to see, using the Property 3.2, that the Theorem 4.3 e X b g = Yb , where X, e Yb are from the Example 4.1. is not correct, if X = X, References [1]. Gromov M., Carnot-Caratheodory spaces seen from within, Sub-Reimannian geometry, Basel: Birkh¨ auser, 1996, pp. 79–323. [2]. Margulis G. A., Mostow G. D., The differential of quasi-conformal mapping of a Carnot–Carath´ eodory spaces, Geometric and Functional Analysis 5 (1995), no. 2, 402–433. [3]. Vodopyanov S. K., Greshnov A. V., On differentiability of mappings of CarnotCarath´ eodory spaces, Dokl. Ross. Akad. Nauk. 389 (2003), no. 5, 592–596. (Russian) [4]. Greshnov A. V., Metrics and Tangent Cones of Uniformly Regular CarnotCarath´ eodory Spaces, Sib. Maht. Journal. 47 (2006), no. 2, 209–238. (English) [5]. Greshnov A. V., Local Approximation of Uniformly Regular Carnot-Carath´ eodory Quasispaces by Their Tangent Cones, Sib. Math. Joural. 48 (2007), no. 2, 229– 248. (English) [6]. Bell¨ aiche A., The tangent space in sub-Riemannian geometry, Sub-Reimannian geometry, Basel: Birkh¨ auser, 1996, pp. 1–78. [7]. Greshnov A. V., Application of the methods of group analysis ot differential equations for some collections of C 1 -smooth non-commuting vector fields, Sib. Math. Journal. (2008), (to appear). [8]. Vodopyanov S. K., Differentiability of Mappings in Geometry of Carnot Manifolds, Sib. Math. Journal. 48 (2007), no. 2, 197–213. (English) [9]. Mitchell J., On Carnot-Carath´ eodory metrics, J. Differential Geometry 21 (1985), 35–45. [10].Metivier G., Fonction spectrale et valeurs proposes d’une classe d’operateurs, Comm. Partial Differential Equations 1 (1976), 479–519. [11].Rothchild L. P., Stein E. S., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320. [12].Pontryagin L. S., Ordinary Differential Equations, Fizmatgiz, Moscow, 1961. [13].Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962. [14].Hartman P., Ordinary Differential Equations, John Wiley& Sons, New YorkLondon-Sydney, 1964. [15].Ovsyannikov L. V., Group Analysis of Differential Equation, Nauka, Moscow, 1978.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 325-349

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 12

L IE T HEORY IN P HYSICS Gabriela P. Ovando∗ CONICET y ECEN-FCEIA, Universidad Nacional de Rosario, Pellegrini 250, 2000 Rosario, Santa Fe, Argentina

Abstract The purpose of this material is to review the Adler Kostant Symes scheme as a theory which can be developped succesfully in different contexts. It was useful to describe some mechanical systems, the so called generalized Toda, and now it was proved to be a tool for the study of the linear approach to the motion of n uncoupled harmonic oscillators. The complete integrability of these systems has an algebraic description. In the original theory this is related to ad-invariant functions, but new examples show that new conditions should be investigated.

(2000) Mathematics Subject Classification: 53C15, 53C55, 53D05, 22E25, 17B56

1.

Introduction

In this work we are interested in the use of Lie theory to understand some Hamiltonian systems. For the study of completely integrable systems one needs to identify the following: i) the symplectic structure, which gives the system its Hamiltonian character, ii) first integrals or constants of motion, iii) action angle variables, and the computation of their evolution. Indeed this is a very difficult approach but it is possible for systems related to certain Lie groups. To this end there are several methods with a common idea: the realization of canonical equations on Lie algebras, or on orbits of a certain action or on symmetric spaces. These ideas appeared in the 70’s and were developped, under other by several authors as Adler, Fomenko, Kostant, Mischenko, Olshanetsy, Perelomov, Trofimov, Symes, etc. (see for instance [Ad1] [F-M1] [F-M2] [F-T] [Ko1] [O-P] [P] and [Sy] and their references). In any method all of the above steps i) ii) iii) are reflected by algebraic circumstances. One needs a way for imbedding a certain Hamiltonian system into a Lie algebra, effective ∗

E-mail address: [email protected]. The author GO was partially supported by CONICET, ANPCyT and SECyT-UNC

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Gabriela P. Ovando

methods for constructing sets of involution and the proof of the full integrability of a wide family of functions in involution. In this chapter we are concerned with so called Adler-Kostant-Symes scheme, which brings together a mathematical framework with Lie theory but also consequences in the dynamics of the Hamiltonian system. This method was successful when studying some mechanical systems such as the rigid body or the generalized Toda lattice [Ad2] [Ko2] [Sy] [R2]. In this setting the phase space of the Hamiltonian systems become coadjoint orbits represented on a Lie algebra and the functions in involutions are presented as ad-invariant functions. On the one hand for this kind of functions, the corresponding Hamiltonian systems become a Lax equation and on the other hand they are in involution on the orbits. Whenever studying Poisson commuting conditions the ad-invariance property can be replaced by a weaker one as in [R1]. In the framework of this theory what we need is a Lie algebra with an ad-invariant metric, a splitting of this Lie algebra into a direct sum as vector subspaces of two subalgebras and a given function. These algebraic tools were used with semisimple Lie algebras, where the Killing form is the natural candidate for the ad-invariant metric. However there are more Lie algebras admitting an ad-invariant metric. We shall examplify here how can be applied the theory for semisimple Lie algebras, and also for other ones, such as the solvable ones. For the general case one should see that any Lie algebra with an ad-invariant metric can be constructed by a double extension procedure, whose more simple application follows from Rm . In this way one gets a solvable Lie algebra g, that results a semidirect extension of the 2n+1-dimensional Heisenberg Lie algebra hn and that can be endowed with an ad-invariant metric which is an extension of a non degenerate bilinear form on R2n . But for other cases the resulting Lie algebras could be no semisimple and no solvable. In any case, the Lie algebra g splits naturally as a direct sum of vector spaces of two subalgebras. Looking at the coadjoint orbits of one of the Lie subalgebras, one gets Hamiltonian systems on these orbits and one can identify the original Hamiltonian system with one of these. In particular for the restriction of the quadratic corresponding to the ad-invariant metric we obtain a Hamiltonian system that becomes a Lax equation, whose solution can be computed with the Adjoint representation. As example we work out the Toda lattice and the linear equation of motion of nuncoupled harmonic oscillators. The first one corresponds to a semisimple Lie algebra, and the second one is associated to a solvable one. Furthermore it is proved that the Hamiltonian for the last one is completely integrable on all maximal orbits. We notice that the functions in involution we are making use, are not ad-invariant and they do not satisfy the involution conditions of [R1]. The setting for the second example applies for quadratic hamiltonians. The Poisson commutativity conditions we get for some polynomials can be read off in the Lie algebra sp(n) of derivations of the Heisenberg Lie algebra of dimension 2n+1 hn . In particular for the case of the motion of n-uncoupled harmonic oscillators we need a abelian subalgebra in the Lie algebra of isometries of the Heisenberg Lie group Hn , endowed with its canonical inner product. This is not surprising if we consider that symplectic automorphisms of the Heisenberg Lie group produce symplectic symmetries of p-mechanical, quantum and classical dynam-

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ics for more general systems than the linear ones (see [Ki2]). The appearence of the Heisenberg Lie algebra related to the motion of n-uncoupled harmonic oscillators is not so surprising. In fact, it is known that in quantum mechanics a good approach to the simple harmonic oscillator is through the Heisenberg Lie algebra. In dimension three this is the Lie algebra generated by the position operator Q = multiplication d by x, the momentum operator P = −i dx and 1 with the only non trivial commutation relation [Q, P ] = 1 These operators evolve according to the Heisenberg equations dP = −Q dt

dQ =P dt

An attempt to relate the classical mechanical system of the linear approximation of the motion of n-uncoupled harmonic oscillators was presented by the theory of p-mechanics, which makes use of the representation theory of the Heisenberg Lie group to show that both quantum and classical mechanics can be derived from the same source (see for instance [Ki1] [Ki2]). This theory contructs a more general setting that unifies both quantum and classical mechanics. The starting point for p-mechanics is the method of orbit of Kirillov [K1] [K2], which says that the orbits of the coadjoint representation of the Heisenberg Lie group parametrise all unitary irreducible representations [F]. Thus non commutative representations are known to be connected with quantum mechanics. In the contrast commutative representations are related to classical mechanics in the observation that the union of one dimensional representations naturally acts as the classical phase space in p-mechanics. In this theory the time evolution of both quantum and classical mechanics observables can be derived from the time evolution of p-observables, choosen as particular functions or distributions on the Heisenberg Lie group. These considerations allow to suppose that new applications of the Adler Kostant Symes scheme are possible and maybe it comes a new time to understand old mechanical systems with new tools, which should be developped for these purposes. As an introduction to the topic one can find exceptional ideas in the books of Arnold, Abraham and Marsden, Ratiu and Marsden, etc. all of them classics in the literature concerning classical mechanics. The chapter is organised as follows: in the first part we present basic ideas concerning symplectic geometry. The second part is devoted to the Adler-Kostant-Symes scheme and the third part to the examples: on the one hand the Toda lattice with generalization in ([Ko2] and [Sy]), and on the other hand the systems corresponding to quadratic Hamiltonians on R2n .

2.

Basic Notions on Symplectic Manifolds

In this section we present the basic elements to work with symplectic geometry. Some texts concerning this topic are [L-M] [CdS]. Let M denote a differentiable manifold. Definition 2.1. A 2-form on M , ω is called a symplectic form if dω = 0 and ωp is non degenerate for every p ∈ M .

328

Gabriela P. Ovando The pair (M, ω) is a symplectic manifold. It follows that the dimension of M must be even.

Example 2.2. Let R2n be the usual euclidean space equipped with global coordinates x1 , . . . , xn , y1 , . . . , yn . The 2-form given by ω=

n X i=1

dyi ∧ dxi

defines a symplectic form on R2n . Note that if ( , ) denotes the canonical inner product on R2n and J the canonical complex structure   0 −I J= I 0 where I is the identity n × n matrix, then ω(X, Y ) = (X, JY ) Example 2.3. If (M1 , ω1 ) and (M2 , ω2 ) are symplectic manifolds, the direct product M1 × M2 is a symplectic manifold. Example 2.4. Coadjoint orbits. Let G denote a Lie group with Lie algebra g and let g∗ be the dual space of g. The coadjoint action of G on g∗ is defined as: g · ϕ = ϕ ◦ Ad(g −1 )

g ∈ G, ϕ ∈ g∗ .

Notice that the orbit throught ϕ is the set G · ϕ = {g · ϕ : g ∈ G} and the isotropy subgroup at ϕ is Gϕ = {g ∈ G : ϕ ◦ Ad(g −1 ) = ϕ}; thus as usual one has G · ϕ = G/Gϕ . The action of G on g∗ induces an action of g on g∗ as X · ϕ = −ϕ ◦ ad(X)

X ∈ g, ϕ ∈ g∗

that cames from the derivative d exp(tX) · ϕ = −ϕ ◦ ad(X), dt |t=0 in other words ˜ X(ϕ) = −ϕ ◦ ad(X)

is the infinitesimal generator induced by X ∈ g at ϕ ∈ g∗ . Any coadjoint orbit G · ϕ is a symplectic manifold with the 2-form ˜ Y˜ ) = −ϕ([X, Y ]), ωϕ (X,

ϕ ∈ g∗ , X, Y ∈ g.

called the Kirillov-Kostant-Souriau symplectic structure. Locally any symplectic manifold looks like the example (2.2) above. This is a classical result of Darboux.

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Theorem 2.5. Darboux. Let (M, ω) be a symplectic manifold. For every p ∈ M there exists coordinate system (U, (x1 , . . . , xn , y 1 , . . . , y n )) such that p ∈ U and ω|U = P n i=1 dyi ∧ dxi .

The symplectic 2-form ω bilds a isomorphism on Tx M for x ∈ M . In fact, since ωx : Tx M × Tx M is non degenerate, there is a linear isomorphism Kx : Tx M → Tx∗ M defined by Kx (v)(u) = ωx (u, v) = (−iv ω)(u). Let H : M → R be a differentiable function, since its differential belongs to T ∗ M , via the isomorphism above we get a vector field XH given by XH (x) = Kx (dHx ),

(1)

that is, XH is the vector field on M satisfying v(H) = dH(v) = ω(v, XH ) and this is called the Hamiltonian vector field associated to the Hamiltonian function H. Example 2.6. For the standard symplectic structure on R2n the isomorphism Kx is given by Kx v = −Jv, where J denotes the canonical complex structure on R2n . Let H ∈ C ∞ (R2n ), its associated Hamiltonian vector field is XH (m) = J(∇H) =

X ∂H ∂ ∂H ∂ ( − ), ∂yi ∂xi ∂xi ∂yi i

where ∇H is the gradient of H, with respect to the canonical inner product. Definition 2.7. The Hamiltonian system for a Hamiltonian H ∈ C ∞ (M, ω) is x′ (t) = XH (x(t)).

(2)

Example 2.8. Let H be a smooth function on R2n , the Hamiltonian equation is the classical one ∂f x′i = ∂y i ∂f yi′ = − ∂x i where xi is actually xi (t), that is it depends on t, for all i (and also for any yi ). Example 2.9. On R2n a quadratic Hamiltonian is a smooth function as 1 H(x) = (Ax, x) 2

for

A symmetric linear map,

which yields the Hamiltonian system x′ = JAx

(3)

In classical mechanic this system describes “small oscillations”, that is, it approximates the motion of a particle on Rn or equivalently the motion of n uncoupled particles on R, near an equilibrium position.

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Gabriela P. Ovando

For instance the motion of n-uncoupled harmonic oscillators near an equilibrium position can be approximated with H a quadratic Hamiltonian as above by taking A = I; therefore (3) becomes x′i (t) = yi (t) (4) yi′ (t) = −xi (t) where x(t) = (x1 (t), . . . , xn (t), y1 (t), . . . , yn (t)). In classical mechanics it is usual to name the coordinates as xi the position coordinates and yi as the velocity coordinates for every i = 1, . . . , n. Definition 2.10. A diffeomorphism φ on a symplectic manifold (M, ω) is symplectic if φ∗ ω = ω. Recall that the Lie derivative on a smooth manifold M given as LX T =

d ψ ∗ (T ) dt |t=0 t

where X is a vector field on M with one parameter group ψt and T a tensor, satisfies the following identities LX LX iY

= iX d + diX = iLX Y + iY LX = i[X,Y ] + iY LX

For a proof see for instance [Wa]. Definition 2.11. A vector field X on a symplectic manifold (M, ω) is symplectic if LX ω = 0. Proposition 2.12. A vector field X ∈ χ(M ) is symplectic if and only if the one parameter subgroup ψt generated by X is symplectic. Proof. If ψt is symplectic, using the definition of LX it is easy to see that LX ω = 0. Conversely assume LX ω = 0, then d d ψ∗ω = ψ ∗ ψ ∗ ω = LX ψs∗ ω = ψs∗ LX ω = 0 dt |t=s t dt |t=0 t s hence ψt∗ ω is constant. But ψ0 = Id and so ψt∗ ω = ω. Corollary 2.13. i) If ω is a symplectic form then LX ω = diX ω. ii) A vector field on (M, ω) is symplectic if and only if iX ω is closed. Definition 2.14. A vector field X on a symplectic manifold (M, ω) is Hamiltonian if and only if −iX ω is exact. The vector field associated to a Hamiltonian function H defined in (1) is Hamiltonian. Notice that the fact of being X Hamiltonian says that there is H ∈ C ∞ (M ) such that dH = −iX ω, therefore X is symplectic. On the other hand for any p ∈ M there always exists local solutions to dH = −iX ω for any X ∈ χ(M ). For global solutions we must ask extra conditions as below.

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Proposition 2.15. Let (M, ω) be a symplectic manifold such that H 1 (M, R) = 0. Every symplectic vector field on M is Hamiltonian. A symplectic 2-form ω on a symplectic manifold M induces a Poisson bracket { , } on C ∞ M by: {f, g}(p) = ωp (Xf , Xg ) = Xf (g) = −Xg (f )

for any f, g ∈ C ∞ M.

Proposition 2.16. Let C ∞ (M ) is a Lie algebra under the Poisson bracket defined above and f → Xf is a Lie algebra anti-homomorphism of C ∞ (M ) into χ(M ). Proof. Since Kx : Tx M → Tx∗ M is a linear isomorphism, the map f → Xf is linear. Now we should prove that [Xf , Xg ] = X{f,g} . Using the properties of LX one gets LXf iXg ω = i[Xf ,Xg ] ω + iXg LXf ω. Since LXf ω = 0, one gets LXf iXg ω = i[Xf ,Xg ] ω. The Lie derivative on 1-forms follows LX θ = iX dθ + diX θ. Taking iXg ω = dg and applying above it holds LXf iXg ω = LXf dg = iXf d2 g + diXf dg = d(Xf (g)) = d{f, g}. Therefore i[Xf ,Xg ] ω = d{f, g} Thus the left side of the equality above i[Xf ,Xg ] ωx coincides with −Kx (X{f,g} , and since Kx is an isomorphism [Xf , Xg ] = −X{f,g} . Recall that a Poisson structure is a bracket { , } on a associative algebra A, such that • { , } is a Lie bracket on A and • f {g, h} = {f g, h} + {g, f h} for all f, g, h ∈ A. the last one is called the Leibnitz rule. In [Sy] a such structure is called Hamiltonian. The space of smooth functions on a differentiable manifold is a associative Lie algebra, hence a natural space to be endowed with a Poisson structure. The Proposition we already proved says that whenever (M, ω) is a symplectic manifold, C ∞ (M ) has a Poisson structure induced by ω: the Poisson bracket { , } is a Lie bracket and the Leibnitz rule holds, since any vector field is a derivation on C ∞ (M ). Example 2.17. On R2n , the Poisson structure associated to the standard symplectic form is given by X ∂f ∂g ∂f ∂g {f, g} = (∇f, J∇g) = − . (5) ∂xi ∂yi ∂yi ∂xi i

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Gabriela P. Ovando

Example 2.18. Let g be a Lie algebra and g∗ its dual. As usual one identifies g∗ with its tangent space. Given a function F : g∗ → R, we define the gradient of F at α ∈ g∗ , denoted by ∇F (α), as an element ∇F (α) ∈ g such that hβ, ∇F (α)i = dFα (β) for any β ∈ g∗ , where h , i denotes the evaluation map. The Kirillov’s Poisson bracket on g∗ is given by {f, h}(α) = halpha, [∇f (α), ∇h(α)]i. Proposition 2.19. If {f, g} = 0 then g is constant on the integral curves of Xf .

Proof. Assume x′ (t) = Xf (x(t)) then

d g(x(t)) = dg(x′ (t)) = Xf (g)(x(t)) = {f, g}(x(t)) = 0. dt Thus g is called a constant of motion of the flow defined by Xf . Since { , } is skew symmetric, g is a constant of motion of Xf if and only if f is a constant of motion of Xg . Constant of motion always exist, in fact f is a constant of motion of Xf . Definition 2.20. A function f on a 2n-dimensional Poisson manifold (M, { , }) is completely integrable if there exist n functions f1 , . . . , fn ∈ C ∞ M such that: i) {f, fi } = 0, {fi , fj } = 0 for all 1 ≤ i, j ≤ n, ii) The differentials df1 , . . . , dfn are linearly independent on a open set invariant under the flow of Xf . Two functions f, g : M → R such that {f, g} = 0 are said to be in involution or Poisson commute. A subset N ⊂ M is invariant under the flow of Xf if the solution x for the Hamiltonian system (2) corresponding to the Hamiltonian f lies on N if x(0) ∈ N .

Example 2.21. On R2n for H(x) = 21 (x, x) the polynomials

1 fi (x) = (p2i + qi2 ) i = 1, . . . , n 2 shows that H is completely integrable. In fact it is easy to check that {H, fi } = 0 = {fi , fj } for all i = 1, . . . , n. Let F = (f1 , . . . , fn ), then F −1 (c) is a torus which is invariant under the flow generated by XH . Let (θ1 , . . . , θn ) denote the angle variable on the torus F −1 (c). Then (f1 , . . . , fn , θ1 , . . . , θn ) is a local coordinate on R2n . With these coordinates, the Hamiltonian equation becames fi′ = 0 θi′ = −1 and the coordinate functions satisfy

{fi , fj } = {θi , θj } = 0,

{fi , θj } = δij ,

therefore the flow Xh is linear on F −1 (c) for c ∈ Rn . Moreover since the level sets {x ∈ R2n : H(x) = c} are compact we have action angle coordinates (see Liouville Theorem below).

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Generally m Poisson commuting functions f1 , f2 , . . . , fm on a symplectic manifold (M, ω) give rise to an action of Rm on M . Let (ψi )t be the one parameter subgroup generated by Xfi . Then (t1 , . . . , tm ) · p = (ψ1 )t1 (ψ2 )t2 . . . (ψm )tm (p). defines a RM action on M . Since {fi , fj } = 0 for all i, j, the set N = {x ∈ M, : fi (x) = ci } is invariant under the Rm -action, for constants c1 , . . . , cm . If N is compact, the Rm action descends to a torus action on N . When m = 1/2 dim M , one gets the Liouville theorem. Theorem 2.22 (Liouville). Let f be a completely integrable function on M , with dim M = 2n, and assume f1 := f, f2 , . . . , fn are commuting Hamiltonians which are linearly independent and let F = (f1 , . . . , fn ) : M → Rn be proper. Then F −1 (c) is invariant under the Rn action and it descends to a torus T n -action. Let θ1 , . . . , θn denote the angle coordinates on the invariant tori. Then {fi , fj } = {θi , θj } = 0 and {fi , θj } = cij (F ) for some functions cij : Rn → R. In particular, the flow of Xf in coordinates (f1 , . . . , fn , θ1 , . . . , θn ) is linear. Coordinates as above, are called action-angle variables for the Hamiltonian system of f .

3.

Symplectic Actions: The AKS-Scheme

Let M denote a differentiable manifold and let G be a Lie group. An differentiable action of G on M is a differentiable map η : G × M → M , η : (g, m) → η(g, m) := g · m such that i) e·m = m for all m ∈ M and ii) (gh) · m = g · (h · m) for all m ∈ M, g, h ∈ G. Notice that if η is an action, the applications ηg : M → M given by ηg (m) = g · m are diffeomorphisms of M . In fact, ηg are differentiable for any g and they are diffeomorphisms since the inverse of any ηg is ηg−1 (see ii) above). Therefore an action of a Lie group on M induces a representation of G on Dif f (M ) the diffeomorphisms of M , given by g → ηg .

Example 3.1. Let GL(n, R) denote the Lie group of non singular transformations of Rn . This acts on Rn as evaluation: A · v = v for A ∈ GL(n, R) and v ∈ Rn . It is easy to verify that this is in fact an action. Example 3.2. Let H be a Lie subgroup of a Lie group G, then H acts on G by conjugation, H × G → G, (h, x) = h−1 xh, for any h ∈ H, x ∈ G. If H is a normal subgroup, one can consider the action of G on H by conjugation. Example 3.3. Let G be a Lie group with Lie algebra g, then G acts on g by the Adjoint action, G × g → g, (g, X) = Ad(g)X, for any g ∈ G, X ∈ g. Recall that Ad(g) = dI(g)e where Ig denotes the conjugation by g (see the previous example). It is easy to see that Igh = Ig ◦ Ih for all g, h ∈ G, hence the map G → GL(g) is a representation of G, called the Adjoint representation. This has a correlative at the Lie algebra level, the adjoint representation: g × g → g given by X · Y = [X, Y ] for all X, Y ∈ g.

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Recall that in (2.4) we defined the coadjoint action of a Lie group G on the space g∗ , the dual of the Lie algebra g · ϕ = ϕ ◦ Ad(g −1 )

g ∈ G, ϕ ∈ g∗

and also we gave the corresponding action of g on g∗ by X · ϕ = −ϕ ◦ ad(X)

X ∈ g, ϕ ∈ g∗ .

The orbit of an action of a Lie group G on a set M is G · m = {g · m : g ∈ G} and the isotropy or stabilizer group of the action at the point m is the closed subgroup of G given by Gm = {g ∈ G such that g · m = m}. It is known that the orbit at m is diffeomorphic to the quotient space of G and the isotropy group, G · m ≃ G/Gm (see [Wa] for instance). Thus any curve at the orbit G · m ˜ at Tm (G · m) through m is γ(t) = exp tX · m and this generates the infinitesimal vector X by d ˜ X(m) = exp tX · m. dt |t=0 Hence the tangent space of a G-orbit at m is ˜ Tm (G · m) = {X,

X ∈ g}

being g the Lie algebra of G. Assume M and N are two differentiable maps on which a given Lie group G acts. A map F : M → N is called equivariant if F (g · m) = g · F (m) for all m ∈ M , g ∈ G. The condition is also expressed as F intertwines the two G-actions. Definition 3.4. Let (M, ω) be a symplectic manifold. An action η of a Lie group G on M is called symplectic if the diffeomorphisms ηg are symplectic maps for any g ∈ G, that is ηg∗ ω = ω. The coadjoint orbits are examples of symplectic manifolds. Recall that they are endowed with the 2-form given by: ˜ Y˜ ) = −β([X, Y ]), ωβ (X,

β ∈G·µ

which is symplectic. In fact, it is closed since for X1 , X2 , X3 ∈ g one has ω([X˜1 , X˜2 ], X˜3 ) = −ϕ([[X1 , X2 ], X3 ]]), hence dω(X˜1 , X˜2 , X˜3 ) = −ϕ([[X1 , X2 ], X3 ]]) − ϕ([[X2 , X3 ], X1 ]]) − ϕ([[X3 , X1 ], X2 ]]) = 0 where the last equality holds after Jacobi for [·, ·].

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The 2-form ω is non degenerate on a orbit: let ϕ ∈ g∗ and let X ∈ g such that ˜ Y˜ ) = 0 for all Y ∈ g. ω(X, Then −ϕ([X, Y ]) = 0 for all Y ∈ g, says that X · ϕ = 0 implying that X ∈ L(Gϕ ). In fact exp tX ∈ Gϕ if and only if exp tX · ϕ = ϕ for t near 0. Thus taking derivative at t = 0 we have X · ϕ = 0, and this is the set corresponding to the Lie algebra of Gϕ . Since ˜ = 0. the tangent space of the orbit at ϕ is Tϕ (G · ϕ) = g/L(Gϕ ), one gets X Definition 3.5. An ad-invariant metric on g is a bilinear map h , i : g × g → R, which is a non-degenerate symmetric and such that ad(X) is skew symmetric for any X ∈ g, that is h[X, Y ], Zi + hY, [X, Z]i = 0

for all X, Y, Z ∈ g.

This ad-invariant metric gives rise to a bi-invariant pseudo Riemannian metric on a connected Lie group G with Lie algebra g; bi-invariant means that the maps Ad(g) are isometries for all g ∈ G, that is hAd(g)Y, Ad(g)Zi = hY, Zi

for all Y, Z ∈ g, g ∈ G,

and conversely any bi-invariant pseudo Riemannian metric on G induces an ad-invariant metric on its Lie algebra, just by taking derivative of the last equality at t = 0 with g = exp tX. Examples of Lie algebras with ad-invariant metrics are: a) semisimple Lie algebras with the Killing form; b) semidirect products g ⋉coad g∗ with the canonical neutral metric h(x1 , ϕ1 ), (x2 , ϕ2 )i = ϕ1 (x2 ) + ϕ2 (x1 ) An ad-invariant metric h , i induces a diffeomorphism between the adjoint orbit G · X and the coadjoint orbit G · ℓX where ℓX (Y ) = hX, Y i. In fact g · ℓX (Y ) = hX, Ad(g −1 )Y i = hAd(g)X, Y i

for all X, Y ∈ g, g ∈ G,

implying that the map ℓ : X → ℓX is equivariant. Thus the adjoint orbits become symplectic manifolds with the 2-form: ˜ = hX, [Y, Z]i ωX (Y˜ , Z)

for X, Y, Z ∈ g.

We shall consider these ideas to construct Hamiltonian systems on orbits that are included on Lie algebras. Recall that given a metric h , i on g the gradient of a function f : g → R at the vector X ∈ g is defined by h∇f (X), Y i = dfX (Y )

Y ∈ g.

Suppose g+ , g− are Lie subalgebras of the Lie algebra g such that g = g+ ⊕ g−

(6)

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as a direct sum of linear subspaces, that is (g, g+ , g− ) is a product structure on g. The Lie algebra g also splits as ⊥ g = g⊥ + ⊕ g− , and g⊥ ±

g∗∓ .

is isomorphic as vector spaces to

This follows from the isomorphism ℓ : g → g∗ . In fact, let X ∈ g⊥ + maps to ℓX . Since ⊥ ∗ ℓX (Y ) = 0 for all Y ∈ g+ , the image of ℓ(g+ belongs to g− , and the isomorphism follows from dimensions. Let G− denote a subgroup of G with Lie algebra g− . Then the coadjoint action of G− ⊥ on g∗− induces an action of G− on g⊥ + : for g− ∈ G− , X ∈ g+ , Y ∈ g− one has: g− · ℓX (Y ) = hX, Ad(g −1 )Y i = hAd(g)X, Y i = πg⊥ (Ad(g− )X), Y i, +

where πg⊥ denotes the projection of g on g⊥ + ; therefore the action is given as +

g− · X = πg⊥ (Ad(g− )X), +

∗ and ℓ : g⊥ + → g− is equivariant. The infinitesimal generator corresponding to Y− ∈ g− is

d Y˜− (X) = exp tY− · X = πg⊥ ([Y− , X]) + dt |t=0

X ∈ g⊥ +.

The orbit G− · Y becomes a symplectic manifold with the symplectic structure given by ωX (U˜− , V˜− ) = hX, [U− , V− ]i

for U− , V− ∈ g− , X ∈ G− · Y

which is induced from the Kostant-Kirillov-Souriau symplectic form on the coadjoint orbits in g∗− . Consider a smooth function f : g → R and restrict it to an orbit G− · X := M ⊂ g⊥ +. Then the Hamiltonian vector field of the restriction H = f|M is the infinitesimal generator corresponding to −∇f− , that is XH (Y ) = −πg⊥ ([∇f− (Y ), Y ]) +

(7)

where Z± denotes the projection of Z ∈ g with respect to the decomposition g = g+ ⊕ g− . In fact for Y ∈ g⊥ + , V− ∈ g− we have ωY (V˜− , XH ) = dHY (V˜− ) = h∇f (Y ), πg⊥ ([V− , Y ])i = h∇f− (Y ), [V− , Y ]i + = hY, [∇f− (Y ), V− ]i = ωY (∇f−˜(Y ), V˜− ). Since ω is non degenerate, one gets (7). Therefore the Hamiltonian equation for x : R → g follows x′ (t) = −πg⊥ ([∇f− (x), x]). +

(8)

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In particular if f is ad-invariant then 0 = [∇f (Y ), Y ] = [∇f− (Y ), Y ] + [∇f+ (Y ), Y ]. ⊥ Since the metric is ad-invariant [g+ , g⊥ + ] ⊂ g+ , in fact ⊥ h[g+ , g⊥ + ], g+ i = hg+ , [g+ , g+ ]i = 0.

Hence the equation (8) takes the form x′ (t) = [∇f+ (x), x] = [x, ∇f− (x)],

(9)

that is, (8) becomes a Lax equation, that is, it can be written as x′ = [P (x), x]. If we assume now that the multiplication map G+ × G− → G, (g+ , g− ) → g+ g− , is a diffeomorphism, then the initial value problem  dx = [∇f+ (x), x] dt (10) x(0) = x0 can be solved by factorization. In fact if exp t∇f (x0 ) = g+ (t)g− (t), then x(t) = Ad(g+ (t))x0 is the solution of (10). R EMARK . If the multiplication map G+ × G− → G is a bijection onto an open subset of G, then equation (8) has a local solution in an interval (−ε, ε) for some ε > 0. The theory we already exposed shows the application of Lie theory to the study of ODE’s as in equation (9). Even when it is possible to give the solution, one need more information. This can be obtained from involution conditions. They help in some sense to control the solutions. A first step in the construction of action angle variables is to search for functions which Poisson commute. The Adler-Kostant-Symes Theorem shows a way to get functions in involution on the orbits M. We shall formulate it in its classical Lie algebra setting. Theorem 3.6 (Adler-Kostant-Symes). Let g be a Lie algebra with an ad-invariant metric h , i. Assume g− , g+ are Lie subalgebras such that g = g− ⊕ g+ as direct sum of vector subspaces. Then any pair of ad-invariant functions on g Poisson commute on g⊥ + (resp. on ⊥ g− ). Sometimes the ad-invariant condition is too strong, so the following version of the previous Theorem given by Ratiu [R1] asks for a weaker condition. Theorem 3.7. Let g be a Lie algebra carrying an ad-invariant metric h , i. Assume it admits a splitting into a direct sum as vector spaces g = g+ ⊕ g− , where g+ is an ideal and g− is a Lie subalgebra. If f, h are smooth Poisson commuting functions on g, then the restrictions ⊥ of f and h to g⊥ + are in involution in the Poisson structure of g+ . Remark. This theorem was used in [R2] to prove the involution of the Manakov integrals for the free n-dimensional rigid body motion.

4.

Applications of the Adler-Kostant-Symes-scheme to Classical Mechanics

In this section we show the explicit use of the theory above in some Lie groups and Lie algebras. The first example is done with semisimple Lie algebras, and it is known as the Toda Lattice.

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Gabriela P. Ovando

The Toda Lattice

The Toda lattice is the mechanical system which describes the motion of n particles on a line with an exponential restoring force, that is the Hamiltonian function on R2n is n

n−1

i=1

i=1

1 X 2 X xi −xi−1 H(x, y) = e . yi + 2 The phase space is R2n which is a symplectic manifold with its canonical symplectic structure. It follows that the Hamiltonian equation is x′k = yk yk′ = exk−1 −xk − exk −xk+1

(11)

and with ex0 −x1 = 0 = exn −xn+1 . Flaschka considered a change of coordinates (called Falschka transform) as follows φ : R2n → R2n , where

ak = − 12 yk 1 bk = 21 e 2 xk −xk+1 xn bn = 21 e 2

φ(x, y) = (a, b) 1 ≤ k ≤ n, 1≤k ≤n−1

Therefore the equation (11) yields a′k = 2(b2k − b2k−1 ) 1≤k≤n b′k = bk (ak+1 − ak ) 1 ≤ k ≤ n P P P xi = yi = 0 and let with an+1P = 0 = b0 . P Notice that i yi′ = 0. Assume V = {(x, y)/ i xi = 0 = i yi }, then the system above becomes a′k = 2(b2k − b2k−1 ) 1≤k ≤n−1 b′k = bk (ak+1 − ak ) 1 ≤ k ≤ n − 1,

(12)

Consider g the semisimple Lie algebra of traceless real matrices sl(n, R equipped with the ad-invariant metric hx, yi = tr(x, y) for all x, y ∈ sl(n, R). Let g+ = so(n) the Lie subalgebra of skew symmetric real matrices and g− the Lie algebra of upper triangular matrices of trace zero. ⊥ Then g⊥ + is the space of real symmetric matrices in sl(n, R) and g− ) is the space of strictly upper triangular matrices in sl(n, R). Pn−1 The coadjoint orbit M = G− ·x0 for x0 = i=1 ei,i+1 +ei+1,i is the set of tri-diagonal real symmetric matrices n X i=1

ai Ei,i +

n−1 X i=1

bi (Ei,i+1 + Ei+1,i )

X i

ai = 0,

bi > 0 ∀i.

where Ei,j denotes the matrix with a 1 at the place i, j and 0 in the others components.

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Let f : sl(n, R) → R be the function given by f (X) = 12 hX, Xi = 12 tr(XX). It is easy to see that the gradient of f at X is X, and hence applying the theory of the previous section we get (9) x′ = [x+ , x] for x+ ⊂ g+ a curve in g+ . Writing the last system in terms of coordinates (a, b) we get the system (12). A generalization of this system can be read in [Sy], where also aplications of the theory to other differential equations are explained.

4.2.

The Motion of n Uncoupled Harmonic Oscillators

Recall that the motion of n-uncoupled harmonic oscillators near an equilibrium position can be approximated with H the quadratic Hamiltonian as 21 (x, x) where ( , ) is the canonical inner product in R2n . Let ω the canonical symplectic structure, the corresponding Hamitonian system follows x′i (t) = yi (t) (13) yi′ (t) = −xi (t) where x(t) = (x1 (t), . . . , xn (t), y1 (t), . . . , yn (t)). The associated Poisson structure on R2n is given as follows {f, g} = (∇f, J∇g) =

X ∂f ∂g ∂f ∂g − . ∂xi ∂yi ∂yi ∂xi

(14)

i

for smooth functions f, g on R2n . Thus with P respect to the Lie bracket { , } the subspace over R generated by the functions H = 21 i (x2i + yi2 ), the coordinates xi , yi , and 1 form a solvable Lie algebra of dimension 2n+2, which is a semidirect extension of the Heisenberg Lie algebra spanned by the functions xi , yi , 1 i=1, . . .,n. In fact they obey the following non trivial rules {xi , yj } = δij {H, xi } = −yi {H, yi } = xi . In order to simplify notations let us rename these elements identifying Xn+1 with H, Xi with xi , Yi with yi and X0 with the constant function 1 1 xi yi H

↔ ↔ ↔ ↔

X0 Xi Yi Xn+1

and set g denotes the Lie algebra generated by these vectors with the Lie bracket [·, ·] derived from the Poisson structure. This Lie algebra is known as a oscillator Lie algebra. The Lie algebra g splits into a vector space direct sum g = g+ ⊕ g− , where g± denote the Lie subalgebras g− = span{X0 , Xi , Yj }i,j=1,...n , hn .

g+ = RXn+1 .

(15)

Notice that g− is isomorphic to the 2n+1-dimensional Heisenberg Lie algebra we denote

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The quadratic form on g which for X = x0 (X)X0 + xn+1 (X)Xn+1 is given by f (X) =

P

i (xi (X)Xi

+ yi (X)Yi ) +

1X 2 (xi + yi2 ) + x0 xn+1 2 i

induces an ad-invariant metric on g denoted by h , i. It is easy to show that the gradient of f at a point X is ∇f (X) = X. The restriction of the quadratic form to v := span{Xi , Yj } i, j=1, . . ., n, coincides with the canonical one ( , ) on R2n ≃ v. The metric induces a decomposition of the Lie algebra g into a vector subspace direct ⊥ sum of g⊥ + and g− where g⊥ − = span{X0 }

g⊥ + = RXn+1 ⊕ span{Xi , Yj }i,j=1,...,n ,

and it also induces linear isomorphisms g∗± ≃ g⊥ ∓ . Let G denote a Lie group with Lie algebra g and G± ⊂ G is a Lie subgroup whose Lie algebra is g± . Hence the Lie subgroup ⊥ G− acts on g⊥ + by the “coadjoint” representation; which in terms of U− ∈ g− and V ∈ g+ is given by P ad∗U− V = xn+1 (V ) i (yi (U )Xi − xi (U )Yi ) (16)

It is not difficult to see that the orbits are 2n-dimensional if xn+1 (V ) 6= 0 and furthermore V and W belong to the same orbit if and only if xn+1 (V ) = xn+1 (W ), hence the orbits are parametrized by the xn+1 -coordinate; so we denote them by Mxn+1 . They are topologically like R2n . In fact Mxn+1 = G− · V ≃ Hn /Z(Hn ), where Hn denotes the Heisenberg Lie group with center Z(Hn ). Equipp these coadjoint orbits with the canonical symplectic structure, that is for U− , V− ∈ g− take ˜− , V˜− ) = hY, [U− , V− ]i = xn+1 (Y ) ωY (U

n X i=1

(xi (U− )yi (V− ) − xi (V− )yi (U− )).

Indeed on the orbit M1 the coordinates xi , yj , i, j = 1, . . . n, are the canonical symplectic coordinates and one can identify this orbit with R2n in a natural way. This says that the identification is a symplectomorphism between R2n with the canonical symplectic structure and the orbit with the Kirillov-Kostant-Souriau symplectic form. Consider H, the restriction to a orbit Mxn+1 of the function f . Since f is ad-invariant the Hamiltonian system of H = f|Mx reduces to n+1

dx dt

= [xn+1 Xn+1 , xv + xn+1 Xn+1 ] (17) x(0) = x0 P where x0 = x0v + x0n+1 X0 and x0v = i (x0i Xi + yi0 Yi ). For xn+1 ≡ x0n+1 ≡ 1 this system is that one we get on R2n . The trajectories x(t) with coordinates xi (t), yj (t), x0n+1 are parametrized circles of angular velocity x0n+1 , for all i,j, that is

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xi (t) = x0i cos(x0n+1 t) + yi0 sin(x0n+1 t) yj (t) = −x0j sin(x0n+1 t) + yj0 cos(x0n+1 t) xn+1 (t) = x0n+1 This solution coincides with that computed in the previous section, when we considered systems on coadjoint orbits. In fact it can be written as x(t) = Ad(exp tx0n+1 Xn+1 )x0 , and one verifies that the flow at the point X 0 ∈ g⊥ + is ∆t (X 0 ) =

P

0 0 i [(xi cos(xn+1 t)

+ yi0 sin(x0n+1 t))Xi + (−x0i sin(x0n+1 t)+ (18)

yi0 cos(x0n+1 t))Yi ] + x0n+1 Xn+1 By taking L and M the following matrices: 

0 xn+1 0 0 −xn+1 0 0 0   0 0 0 x n+1   0 0 −xn+1 0   M =       0 0 ... 0 0 ...

..

.

0

0 xn+1 −xn+1 0



0 xn+1 0 0 −xn+1 0 0 0   0 0 0 xn+1   0 0 −x 0 n+1   .. L= .   xn+1   −xn+1 0  1 1 1 1 1  − 1 y1 x − y x . . . − y 1 2 2 n 2 2 2 2 2 2 xn 0 0 0 0 ... 0 0

we get L′ = [M, L] = M L − LM , the Lax pair equation.

5.

 0 0  0  0  ..  .  0 0  0 0  0 0 0 0 0 0 0 0 .. .

.. . 0 0 0 0

 x1 y1   x2   y2   ..  .   xn   yn   0 0

Quadratic Hamiltonians and Coadjoint Orbits

In this section we shall prove that Hamiltonian systems corresponding to quadratic Hamiltonians in R2n of the form H(x) = 12 (Ax, x) where A is a non singular symmetric map, can be described using the scheme of Adler-Kostant-Symes on a solvable Lie algebra.

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Let us consider the linear system of one degree of freedom on R2n with Hamiltonian given by: 1 H(x) = (Ax, x) 2 where x = (q1 , . . . , qn , p1 , . . . , pn ) is a vector in R2n written in a symplectic basis and A is a non singular symmetric linear operator with respect to the canonical inner product ( , ). This yields the following Hamiltonian equation

′

(3)

x = JAx,

with J =

 0 −Id Id 0



and being Id the identity. The phase space for this system is R2n . We shall construct a solvable Lie algebra that admits an ad-invariant metric on which the system (3) can be realized as a Hamiltonian system on coadjoint orbits. Moreover it can be written as a Lax pair equation. Let b denote the non degenerate bilinear form on R2n = span{Xi , Yj }ni,j=1 given by b(X, Y ) = (AX, Y ). In our terms, b defines a metric on R2n but it is not necessary definite. Note that the linear JA is non singular and skew symmetric with respect to b, where J is the canonical complex structure on v ≃ R2n as above: b(JAX, Y ) = (AJAX, Y ) = (JAX, AY ) = −(AX, JAY ) = −b(X, JA). Let g denote the Lie algebra g which as vector space is the diract sum g = RX0 ⊕ v ⊕ RXn+1 where v = R2n and with the Lie bracket given by the non trivial relations [U, V ] = b(JAU, V )X0

[Xn+1 , U ] = JAU

for all U ∈ v.

(19)

Thus in this way one defines a structure of a solvable Lie algebra on g. Note that A = Id is the particular case we considered in the previous subsection. This Lie algebra g can be equipped with the ad-invariant metric defined by hx10 X0 + U 1 + x1n+1 Xn+1 , x20 X0 + U 2 + x2n+1 Xn+1 i = b(U 1 , U 2 ) + (x10 x2n+1 + x20 x1n+1 ). (20) Thus if h , iv denotes the restriction of the metric of g to v = span{Xi , Yj }i,j=1,...,n , then clearly h , i is a generalization of the non degenerate symmetric bilinear map b of R2n . Moreover g admits a orthogonal splitting g = span{X0 , Xn+1 } ⊕ v. Denote by g± the Lie subalgebras g+ = RXn+1 ,

g− = RX0 ⊕ span{Xi , , Yi }.

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They induce the splitting of g into a vector space direct sum g = g+ ⊕ g− , which by the ⊥ ad-invariant metric gives the following linear decomposition g = g⊥ + ⊕ g− , direct sum as vector spaces, for g⊥ − = RX0

g⊥ + = span{Xi , Yi }i=1,...,n ⊕ RXn+1 .

Note that g− is an ideal of g isomorphic to the 2n+1-dimensional Heisenberg Lie algebra hn . Let G denote a Lie group with Lie algebra g, set G− ⊂ G the Lie subgroup with Lie subalgebra g− . As we already explained G− acts on g⊥ + by the coadjoint action g− · X = πg⊥ (Ad(g− )X) +

g− ∈ G− ,

X ∈ g⊥ +,

where πg⊥ is the projection of g on g⊥ + , which in infinitesimal terms gives the following +

action of g− on g⊥ + ad∗U V := U · V

= xn+1 (V )JAXv(U )

for U ∈ g− , V ∈ g⊥ +.

(21)

being Xv(U ) the projection of U onto v with respect to the orthogonal splitting g = span{X0 , Xn+1 } ⊕ v. The orbits are 2n-dimensional if xn+1 (V ) 6= 0 and furthermore V and W belong to the same orbit if and only if xn+1 (V ) = xn+1 (W ), and therefore one parametrizes the orbits by the xn+1 -coordinate and one enotes them by Mxn+1 . The orbits are topologically like R2n since they are diffeomorphic to the quotient Hn /Z(Hn ), if Z(Hn ) = RX0 is the center of the Heisenberg subgroup. Endow the orbits with the canonical symplectic structure of the coadjoint orbits, that is for X ∈ g⊥ + , U− , V− ∈ g− set ωX (U˜− , V˜− ) = hX, [U− , V− ]i = xn+1 (X)b(JAUv, Vv).

Consider f : g → R the ad-invariant function given by

1 hX, Xi. 2 The gradient of the function f at a point X is the so called position vector f (X) =

∇f (X) = X. Since f is ad-invariant the Hamiltonian system of H = f|Mx , the restriction of f to the n+1 orbit Mxn+1 , given by (9) becomes dx dt

= [∇f+ (x), x] = [xn+1 Xn+1 , xv + xn+1 Xn+1 ] = xn+1 JAxv x(0) = X 0

(22)

where X 0 ∈ g⊥ +. Thus this Hamiltonian system written as a Lax pair equation is equivalent to (3) for xn+1 = x0n+1 = 1. The solution X(t) for the initial condition X 0 ∈ g⊥ + can be computed via the Adjoint map on G, that is, X(t) = Ad(exp tx0n+1 Xn+1 )X 0 . The previous explanations prove the following result.

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Theorem 5.1 ([O2]). Let H(X) = 21 (AX, X) be a quadratic Hamiltonian on R2n with corresponding Hamiltonian system (3). Then H can be extended to a quadratic function f on a solvable Lie algebra g containing the Heisenberg Lie algebra as a proper ideal. The function f induces a Hamiltonian system on coadjoint orbits of the Heisenberg Lie group, that can be written as a Lax pair equation and which is equivalent to (3). Moreover the trajectories on R2n for the initial condition V 0 can be computed with help of the Adjoint map on g. Explicitely they are the curves x(t) = exptJA V 0 , where exp denotes the usual exponential map of matrices. If we take L, M ∈ M (2n + 2, R) as   xn+1 JA 0 z 0 0 M = 0 0 0 0

  xn+1 JA 0 z 0 0 L =  i 12 z T 0 0 0

where z T = (x1 , x2 , · · · , xn , y1 , y2 , . . . , yn ) then the Hamiltonian equation can be written in the following way L′ = [M, L]. Example 5.2 (The motion of n-uncoupled inverse pendula). As example of the previous construction consider the linear approximation of the motion of n uncoupled inverse pendula. This corresponds to the Hamiltonian H(x) = 12 (Ax, x) with   Id 0 . A= 0 −Id This yields the Hamiltonian system x′ = JAx, which in coordinates takes the form dxi dt dyi dt

= yi = xi

(23)

As we said the phase space is R2n . In the setting of the AKS scheme we can construct coadjoint orbits M of the Heisenberg Lie group, that are included in a solvable Lie algebra g with Lie bracket (19) and ad-invariant metric (20). The Hamiltonian system for the restriction to the orbits of the ad-invariant function on g, f (X) = 12 hX, Xi, can be written as dx = [xn+1 Xn+1 , xv + xn+1 Xn+1 ] dt (24) x(0) = X 0 P 0 0 0 where X 0 = i (xi Xi + yi Yi ) + xn+1 Xn+1 . The Hamiltonian system above on the coadjoint orbit M1 written in coordinates is clearly equivalent to (23). P (x (t)X The trajectories on g⊥ , x = i + yi (t)Yi ) + xn+1 Xn+1 are parametrized by + i i xi (t) = x0i cosh(x0n+1 t) + yi0 sinh(x0n+1 t) yi (t) = x0i sinh(x0n+1 t) + yi0 cosh(x0n+1 t) xn+1 (t) = x0n+1

The flow at the point X 0 ∈ g⊥ + is P 0 0 0 0 ∆t (X 0 ) = i [(xi cosh(xn+1 t) − yi sinh(xn+1 t)Xi + 0 0 0 0 +(xi sinh(xn+1 t) + yi cosh(xn+1 t)Yi ] + x0n+1 Xn+1

(25)

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The system (24) is a Lax pair equation L′ = [M, L] = M L − LM , and has a matricial representation by choosing L and M the following matrices in M (2n + 2, R):   0 xn+1 0 0 0 0 xn+1 0 0 0 0 0    0 0 0 xn+1 0 0    0 0 xn+1 0 0 0    .. ..  .. M = . . .    0 x 0 0 n+1     0 x 0 0 0 n+1    0 0 ... 0 0 0 0 ... 0 0   0 xn+1 0 0 x1  xn+1 0 0 0 y1     0 0 0 xn+1 x2     0 0 xn+1 0 y2      . . . . . . L= . . .      x 0 x n+1 n    xn+1 0 0 yn    − 1 y1 1 x1 − 1 y2 1 x2 . . . − 1 yn 1 xn 0 0  2

0

2

0

2

0

2

0

...

2

0

2

0

0

0

Now we shall investigate involution conditions on the coadjoint orbits of the Heisenberg Lie group for the restrictions of the quadratic functions f (X) = 12 hX, Xi, where h , i denotes the ad-invariant metric on the solvable Lie algebra g. Let gi , gj be two quadratics on R2n that are realted to the symmetric maps Ai , Aj : v → v respectively, that is 1 gi (X) = (Ai X, X) 2

1 gj (X) = (Aj X, X). 2

Consider quadratic functions on the solvable Lie algebra g, which are extensions of gi , gj to RX0 ⊕ RXn+1 , for instance as 1 gi (X) = (Ai Xv, Xv) + x0 xn+1 2

1 gj (X) = (Aj Xv, Xv) + x0 xn+1 . 2

For the following results these extensions are not unique. For instance extending them trivially we get the same conclusions. Let Hi , Hj denote the restrictions of gi , gj to the orbits Mxn+1 and let X ∈ Mxn+1 ⊂ g⊥ . + The symplectic structure on the orbits induces a Poisson bracket which for the functions Hi , Hj follows: {Hi , Hj }(X) = hX, [∇gi − (X), ∇gj − (X)]i By computing one can see that the gradients of gi and gj are

∇gi (X) = A−1 Ai Xv+x0 X0 +xn+1 Xn+1

∇gj (X) = A−1 Aj Xv+x0 X0 +xn+1 Xn+1 .

Thus we are ready to prove the following result.

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Gabriela P. Ovando

Theorem 5.3 ([O2]). The functions Hi , Hj are in involution on the orbits Mxn+1 if and only if [JAi , JAj ] = 0 (26) where J is the canonical complex structure on R2n . Proof. Let X ∈ Mxn+1 ⊂ g⊥ + . For the functions Hi , Hj the Poisson bracket on the orbit Mxn+1 follows: {Hi , Hj }(X) = hX, [Ai Xv, Aj Xv]i = hxn+1 [Xn+1 , A−1 Ai Xv], A−1 Aj Xvi = xn+1 hJAi Xv, A−1 Aj Xvi = xn+1 (JAi Xv, Aj Xv) Therefore {Hi , Hj }(X) = 0 if and only if (Aj JAi Xv, Xvi = 0 which is equivalent to Aj JAi = Ai JAj , if and only if JAj JAi = JAi JAj , that is [JAi , JAj ] = 0. The natural question is what is the meanning of (26)? Fix h , i′ the inner product on hn defined so that the vectors Xi , Yj , X0 are orthonormal for all i,j=1,. . ., n. The metric is an extension of the canonical one on R2n . The Lie bracket on hn = RX0 ⊕ v where R2n ≃ v = span{Xi , Yj }i,j=1,...,n is expressed as h[X, Y ], x0 X0 i′ = x0 hJX, Y i′

with J as in (3)

and note that h , i|v×v = ( , ). A derivation D of hn acting trivially on the center must satisfy [DU, V ] = −[U, DV ] for all U, V ∈ v. Equivalently in terms of h , i′ , we have that a map D in hn is a derivation acting trivially on the center of hn if and only if the restriction of D to v (denoted also D) satisfies (JDU, V ) = −(JU, DV )

for all U, V ∈ v,

where we replaced h , i′v by ( , ) since they coincide on v ≃ R2n . Denote by d the set of derivations on hn acting trivially on the center of hn . Theorem 5.4. There is a bijection between the set of derivations of hn acting trivially on the center and the set so(n) of symmetric linear maps on R2n . This correspondence is given by D ∈ d → JD ∈ so(n), where J is the complex structure as in (3). Corollary 5.5. If there exists an n-dimensional abelian subalgebra on z(JA)d, where z(JA)d = {D ∈ d such that [D, JA] = 0} then the Hamiltonian function H restriction of the function f (X) = pletely integrable on the orbits Mxn+1 for xn+1 6= 0.

1 2 (AX, X)

is com-

Proof. The previous theorem says that the restrictions to the orbit Mxn+1 of the functions gi , gj are in involution if their corresponding derivations commute in d. In particular for gi and f , we have that H and Hi Poisson commute on the orbit if and only if JAi belongs to the centralizer of JA in d, z(JA)d. Since the complete integrability requires of n linearly independent functions, this can be done with a basis of an n-dimensional abelian subalgebra of z(JA)d, finishing the proof.

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A linear map t is a derivation of hn acting trivially on the center z(hn ) if and only if Jt + t∗ J = 0, if and only if t ∈ sp(n). The derivations of nilpotent Lie algebras of H-type were computed in ([Sa]). In the case of the motion of n-uncoupled harmonic oscillators, we can see that the corresponding derivation is an element of a Cartan subalgebra of sp(n).

References [AM]

Abraham, R., Marsden, J., Foundations of Mechanics, Second edition. The Benjamin Cummings publishing company, (1985).

[Ad1]

Adler, M., A new integrable system and a conjecture by Calogero, (1975).

[Ad2]

Adler, M., On a trace for formal pseudodifferential operators and the symplectic structure for the KdV type equations, Invent. Math., 50, 219-248, (1979).

[Ar]

Arnold, V. I., Mathematical methods of classical mechanics, Springer Verlag, (1980).

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Cannas da Silva, A., Lectures on symplectic geometry, Lecture notes in Math. 1764, Springer Verlag, (2001).

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Favre,G., Santharoubane L.J., Symmetric, invariant, non-degenerate bilinear form on a Lie algebra, J. Algebra 105, 451–464 (1987).

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A. Fomenko, A. Mischenko, Euler equations on finite-dimensional Lie groups, Izv. Akad.Nauk SSSR Ser Mat. 42, (1978), 396-415 [rus]; english: Math. USSR Izv.12, (1978), 371 –389.

[F-M2]

Fomenko, A., Mischenko, A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. and its Applic., 12, 113 -121, (1978).

[F-T]

Fomenko, A., Trofimov, V., Integrable systems in Lie algebras and symmetric spaces, Gordon and Breach Sc. Publ., (1988).

[G]

Guest, M, Harmonic Maps, Loop Groups and Integrable Systems. (London Math. Soc. Student Texts; 38). New York: Cambridge University Press (1997).

[G-S]

Guillemin, V., Sternberg, S., Symplectic techniques in physics. Cambridge New York Port Chester Melbourne Sydney: Cambridge University Press (1991).

[Ka]

Kac V., Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1985.

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Kirillov, A. A., Elements of the theory of representations, Springer-Verlag, (1976).

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[K2]

Kirillov, A. A., Merits and demerits of the orbit method, bULL. AMS (N.S.), 36 4, 433-488, (1999).

[Ki1]

Kisil, V. V., Plain mechanics: classical and quantum, J. Natur. Geom., 9 1, 1-14, (1996).

[Ki2]

Kisil, V. V., p-mechanics as a physical theory: an introduction, J. Physics, 37 1, 183-204, (2004) (arXiv:quant-ph/0212101).

[Ko1]

Kostant, B., Quantization and Representation Theory, in: Representation Theory of Lie groups, Proc. SRC/LMS Res. Symp., Oxford 1977. London Math. Soc. Lecture Notes Series, 34, 287-316, (1979).

[Ko2]

Kostant, B., The solution to a generalized Toda lattice and representation theory, Advances in Math., 39, 195 - 338, (1979).

[L-M]

Libermann, P., Marle C.M. , Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, 1987.

[M-R]

Medina, A., Revoy, Ph., Alg`ebres de Lie et produit scalaire invariant, Ann. scient. ´ Norm. Sup., 4e s´erie, t. 18, 391 - 404, (1985). Ec.

[O-P]

Olshanetsky, M.A., Perelomov, A.M., Completely integrable hamiltonain systems connected with semisimple Lie algebras, Inventiones math. 37, 93–108 (1976).

[O1]

Ovando, G., Estructuras complejas y sistemas hamiltonianos en grupos de Lie solubles, Tesis Doctoral, Fa.M.A.F. Univ. Nac. de C´ordoba,( Marzo 2002).

[O2]

Ovando, G., Small oscillations and the Heisenberg Lie algebra, J. Phys. A: Math. Theor. 40, 2407–2424 (2007).

[P]

A. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, vol. I, Birkh¨auser Verlag, Basel - Boston - Berlin, (1990).

[Ra]

Raghunathan, M., Discrete subgroups of Lie groups, Springer, New York,(1972).

[R1]

Ratiu, T., Involution theorems, Geometric methods in Math. Phys., Lect. Notes in Math., 775, Procedings, Lowell, Massachusetts 1979, Springer Verlag, (1980).

[R2]

Ratiu, T., The motion of the free n-dimensional rigid body, Indiana Univ. Math. Journal, 29, 609 - 629, (1980).

[Sa]

Saal, L. The automorphism group of a Lie algebra of Heisenberg type Rend. Sem. Mat. Univ. Pol. Torino, 54 2, (1996).

[Sy]

Symes, W., Systems of Toda type, inverse spectral problems and representation theory, Invent. Math., 59, 13 - 53, (1978).

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Varadarajan, V., Lie groups, Lie algebras and their representations, Springer, (1984).

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F. Warner, Fundations of differentiable manifolds and Lie groups, Springer Verlag, New York (1983).

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Woodhouse, N. M., Geometric quantization, Oxford Math. Monographs, The Clarendon Press Oxford Univ. Press, New York 1992, Oxford Science Publication.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 351-383

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 13

L E´ VY P ROCESSES IN L IE G ROUPS AND H OMOGENEOUS S PACES Ming Liao∗† Department of Mathematics, Auburn University, Auburn, AL 36849, USA

Abstract A L´evy process in a Lie group is a process that possesses independent and stationary multiplicative increments. The theory of such processes is not merely an extension of L´evy processes in Euclidean spaces. Because of the unique structures possessed by non-commutative Lie groups, these processes exhibit certain interesting properties which are not present for their counterparts in Euclidean spaces. The concept of L´evy processes may be extended to include Markov processes in a homogeneous space that are invariant under the group action. More generally, we will also study processes in Lie groups and homogeneous spaces that possess independent, but not necessarily stationary, increments, called nonhomogeneous L´evy processes. These processes appear naturally when studying a decomposition of a general Markov process in a manifold invariant under a group action. In these notes, we will provide an introduction to L´evy processes in Lie groups and homogeneous spaces, and present some selected results in this area. The reader is referred to the literature for the most of proofs, but some explanation will be given to the results not yet published.

2000 Mathematics Subject Classification Primary 60J25, Secondary 58J65. Key words and phrases homogeneous spaces, L´evy processes, Lie groups, Markov processes.

1.

An Informal Review of Lie Groups and Homogeneous Spaces

A d-dimensional (smooth) manifold is a second countable Hausdorff topological space M that locally may be identified with an open subset of the d-dimensional Euclidean space Rd . ∗

E-mail address: [email protected] These notes were prepared for talks at Institute of Mathematics, Academia Sinica, Taipei, Taiwan in May 2008. The author wishes to thank Professor Tzuu-Shuh Chiang for arranging the visit †

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Ming Liao

When two neighborhoods on M are so identified with open subsets of Rd , their intersection induces a map from (part of) Rd into Rd , which is required to be smooth (C ∞ ). Thus, a manifold M is locally a Euclidean space. Near any point in M , local coordinates may be introduced. The smoothness of a function on M or a function from M into another manifold may be defined. As for the Euclidean space Rd , tangent vectors at a point x in M are just directional derivatives at x. They together form a vector space, called the tangent space at x and is denoted by Tx M . A (smooth) vector field is just a smooth assignment of a tangent vector to each point in M . For any vector field X and smooth function f on M , Xf is also a smooth function on M . If F : M → N is a smooth map between two manifolds, then for x ∈ M , it induces naturally a linear map Df : Tx M → Tf (x) N , called the differential of f at x. A topological group is a group and a topological space such that both the product map: G × G ∋ (g, h) 7→ gh ∈ G and the inverse map: G ∋ g 7→ g −1 ∈ G are continuous. A Lie group is a group and a manifold such that these two maps are smooth. It is well known that a Lie group is in fact analytic in the sense that the underlying manifold structure together with the product and inverse maps are analytic. The Lie algebra g of a Lie group G is the tangent space Te G of G at the identity element e of G. For g ∈ G, let lg : G ∋ h 7→ gh ∈ G and and rg : G ∋ h 7→ hg ∈ G be respectively the left and right translations on G. A vector field Y on G is called left invariant if Dlg (Y ) = Y for any g ∈ G. It is determined by its value at e, X = Y (e) ∈ g, as Y (g) = Dlg (X), and is denoted by X l . Similarly, we defined a right invariant vector Y using Drg and denote it by X r for X = Y (e). We may write gX for X l (g) and Xg for X r (g) more suggestively. For any X and Y in g, [X l , Y l ] = X l Y l − Y l X l is a left invariant vector field. The Lie bracket [X, Y ] is the its value at e, that is, [X, Y ]l = [X l , Y l ]. It is linear in X and Y , is anti-symmetric in the sense that [Y, X] = −[X, Y ], and satisfies the Jacobi identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0. The exponential map exp: g → G is defined by exp(X) = ψ(1), where ψ(t) is the solution of the ordinary differential equation (d/dt)ψ(t) = ψ(t)X satisfying the initial condition ψ(0) = e, and is a diffeomorphism of an open neighborhood of 0 in g onto an open neighborhood of e in G. We may write eX for exp(X). For g ∈ G, the conjugation map cg : G ∋ h 7→ ghg −1 ∈ G is a Lie group isomorphism (a product-preserving differeomorphism). Its differential map: g ∋ X 7→ gXg −1 ∈ g, denoted by Ad(g) and called the adjoint map, is a Lie algebra isomorphism (a linear bijection preserving Lie bracket). It can be shown that [X, Y ] = (d/dt)Ad(etX )Y |t=0 for Y ∈ g. An important example of Lie groups is the matrix group of all n × n real invertible matrices, denoted by GL(n, R) and called the general linear group on Rn . Its Lie algebra, denoted by gl(n, R), is the space of all n × n real matrices with Lie bracket [A, B] = AB The exponential map is given by the usual matrix exponentiation eX = I + P∞− BA. k −1 is given by usual matrix product. k=1 X /k! and the adjoint map Ad(g)X = gXg The matrix product can mean either the product of group elements or the translation of Lie algebra element.

L´evy Processes in Lie Groups and Homogeneous Spaces

353

Another useful example of Lie groups is the special orthogonal group SO(n), the group of n × n orthogonal matrices of determinant 1, also called a rotation group because each matrix represents a rotation on Rn . Its Lie algebra is the space o(n) of n×n skew-symmetric matrices (A′ = −A, where ′ denotes matrix transpose). An action of a Lie group G on a manifold M is a smooth map F : G × M → M satisfying F (gh, x) = F (g, F (h, x)) and F (e, x) = x for g, h ∈ G and x ∈ M . For simplicity, we may write gx for F (g, x). The subset Gx = {gx; g ∈ G} of M is called an orbit of G on M . The action of G on M will be called transitive if any orbit of G is equal to M . Let H be a closed subgroup of G. The set of left cosets gH for g ∈ G is denoted by G/H and is called a homogeneous space of G. It is equipped with the quotient topology. By [15, II.Theorem 4.2], there is a unique manifold structure on G/H under which the natural action of G on G/H, defined by g ′ H 7→ gg ′ H, is smooth. Suppose a Lie group G acts on a manifold M . Fix p ∈ M . Let H = {g ∈ G; gp = p}. Then H is a closed subgroup of G, called the isotropy subgroup of G at p. By Theorem 3.2 and Proposition 4.3 in [15, Chapter II], if the action of G on M is transitive, then the map: gH 7→ gp is a diffeomorphism from G/H onto M , therefore, M may be identified with G/H. As an example, the rotation group SO(n) acts on the unit sphere S n−1 in Rn transitively. The isotropy subgroup at the “north pole” p = (1, 0, . . . , 0) is diag{1, SO(n−1)} ≡ SO(n−1). Thus, S n−1 may be identified with the homogeneous space SO(n)/SO(n−1).

2.

L´evy Processes in Lie Groups

Let G be a Lie group with identity element e. A process gt in G with rcll paths (right continuous paths with left limits) is called a L´evy process if it possesses independent and stationary multiplicative increments, that is, for s < t, the increment gs−1 gt is independent of the process up to time s, and has the same distribution as g0−1 gt−s . Let gte = g0−1 gt . Then gte is a L´evy process in G starting at e and is independent of g0 . In these notes, a process is always assumed to have an infinite life time except when explicitly stated otherwise. In the sequel, a measure µ on a topological space X is always understood to be defined on the Borel σ-algebra B(X) of XR unless when explicitly stated otherwise, and for any (Borel) function f on X, µ(f ) = f (x)µ(dx). The convolution of two measures µ and RR ν on G is the measure µ ∗ ν on G defined by µ ∗ ν(f ) = f (gh)µ(dg)ν(dh) for any function f on G. A family of probability measures µt on G, t ∈ R+ = [0, ∞), is called a convolution semigroup on G if µs+t = µs ∗ µt . It is called continuous if µt → µ0 weakly as t → 0. The relation between L´evy processes and convolution semigroups is summarized below. Theorem 1 If gt is a L´evy process in G with g0 = e, then its distribution µt is a continuous convolution semigroup with µ0 = δe (unit point mass at e). Conversely, given a continuous convolution semigroup µt with µ0 = δe , there is a unique (in the sense of distribution) L´evy process gt with g0 = e and distribution µt .

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It is easy to show that a L´evy process gt is a Markov process with transition semigroup Pt given by Z Pt f (g) = E[f (ggte )] =

f (gh)µt (dh)

(1)

G

for f ∈ B(G)+ . By this we mean that gt has the following Markov property: E[f (gt+s ) | Ft ] = Ps f (gt )

(2)

almost surely for s < t and f ∈ Cb (G) (the space of bounded continuous functions on G), where the left hand side is the conditional expectation given the σ-algebra Ft generated by the process up to time t, and for t ≥ 0, Pt is a probability kernel from G to G, that is, Pt (x, ·) is a probability measure on G for x ∈ G and Pt (x, R B) is measurable in x for measurable B ⊂ G such that P0 (x, ·) = δx and Pt+s (x, ·) = Pt (x, dy)Ps (y, ·). A Markov process in G with semigroup Pt is called a Feller process if for f ∈ C0 (G), Pt f ∈ C0 (G) and Pt f → f uniformly on G as t → 0, where C0 (G) is the space of continuous functions on G vanishing at infinity (under the one-point compactification topology). The distribution of a Feller process is completely determined by its generator L defined by Lf = limt→0 (1/t)Pt f with domain D(L) consisting of f ∈ C0 (G) for which the limit exists uniformly on G. A Feller process is called a diffusion process if its generator restricted to Cc∞ (G) is a second order differential operator (with no constant term). Such a process is necessarily continuous. By (1), a L´evy process is a Feller process and its semigroup Pt is left invariant in the sense that Pt (f ◦lg ) = (Pt f )◦lg for any g ∈ G. A Markov process in G with a left invariant semigroup is called a left invariant Markov process. Theorem 2 A L´evy process gt in G is a left invariant Feller process. Conversely, a left invariant Markov process gt in G is a L´evy process. The L´evy process defined above may be called a left L´evy process. Similarly, one may define a right L´evy process using the increment gt gs−1 instead of gs−1 gt . Then it is a Feller process in G with transition semigroup Pt f (g) = E[f (gte g)], where gte = gt g0−1 , and is invariant under the right translation rg . The left and right L´evy processes are in natural duality. In the following, a L´evy process will mean a left L´evy process unless explicitly stated otherwise. The results of this section hold in fact for L´evy processes in a locally compact and second countable Hausdorff topological group G.

3.

Generators and Stochastic Integral Equations

Hunt [18] obtained an explicit expression for the generator L of a L´evy process gt in a Lie group G, see also Heyer [17] and Liao [24]. We need some preparation before presenting this result. Let g be the Lie algebra of G. Recall that for X ∈ G, X l and X r are respectively the left and right invariant vector fields on G induced by X. Let C02,l (G) be the space of functions f ∈ C0 (G) such that X l f ∈ C0 (G) and X l Y l f ∈ C0 (G) for any X, Y ∈ G. Similarly, the function space C02,r (G) is defined replacing X l by X r .

L´evy Processes in Lie Groups and Homogeneous Spaces

355

Let {X1 , . . . , Xd } be a basis of G, where d = dim(G). A set of functions x1 , . . . , xd ∈ (the space of smooth functions on G with compact supports) will be called coordinate functions associated to the above basis if xi (e) = 0 and Xi xj = δij . These functions form localPcoordinates near e. Note that the coordinate functions xi may be chosen to satisfy g = exp[ di=1 xi (g)Xi ] for g near e, and then they will be called exponential coordinates. P Let |x|2 = di=1 x2i . Cc∞ (G)

Theorem 3 Let gt be a L´evy process in a Lie group G. Then the domain D(L) of its generator L contains C02,l (G) and for f ∈ C02,l (G), Lf (g) = +

Z

d 1 X aij Xil Xjl f (g) + X0l f (g) 2 i,j=1

[f (gh) − f (g) −

X

xi (h)Xil f (g)]Π(dh),

(3)

i

where aij are some constants forming a non-negative definite symmetric matrix, X0 ∈ G and Π is a measure on G satisfying Z |x|2 dΠ < ∞ and Π(U c ) < ∞ (4) Π({e}) = 0, U

for any neighborhood U of e. Conversely, given aij , X0 and Π as above, there is a L´evy process in G, unique in distribution, whose generator restricted to C02,l (G) is given by (3). Any measure Π on G satisfying (4) is called a L´evy measure. In the case of a L´evy process gt , Π is the characteristic measure of the Poisson random measure N on R+ × G that counts the jumps of gt , that is, −1 N ([0, t] × B) = #{s ∈ (0, t]; gs− gs ∈ B}

(5)

and Π(B) = E[N ((0, 10 × B)]. Hence, gt is continuous if and only if Π = 0. It is clear that Π is independent of the choice of the basis of g and the associated coordinate functions. The differential operator LD = (1/2)

d X

aij Xil Xjl

(6)

i,j=1

is called the diffusion part of the generator L, also independent of the basis of g and associated coordinate functions. However, the drift vector X0 ∈ g in (3) may depend on this choice. It is shown in Applebaum-Kunita [2] (see also [24]) that a L´evy process gt in G can be characterized by a stochastic integral equation driven by a Brownian motion Bt and the Poisson random measure N , corresponding to the L´evy -Itˆo representation in Euclidean ˜ be the compensated random measure of N defined by N ˜ (dt dg) = N (dt dg) − case. Let N dtΠ(dg).

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Theorem 4 There is a d-dim Brownian motion Bt with covariance matrix {aij } (E(Bti ) = 0 and E(Bti Btj ) = aij t) and independent of N such that for f ∈ C02,l (G), f (gt ) = f (g0 ) + +

Z tZ 0

+

i=1

t

0

Xil f (gs− )

◦

G

˜ (ds dh) [f (gs− h) − f (gs− )]N

G

[f (gs h) − f (gs ) −

Z tZ 0

d Z X

d X

dBsi

+

Z

t 0

X0l f (gs )ds

xi (h)Xil f (gs )]dsΠ(dh).

(7)

i=1

Conversely, given a triple of independent G-valued random variable g0 , Brownian motion Bt and Poisson random measure N on R+ × G with characteristic measure Π being a L´evy measure, and X0 ∈ g, there is a unique rcll process gt in G satisfying (7) for any f ∈ C02,l (G) and it is a L´evy process. Note that the first integral in (7) is a Stratonovich stochastic integral, R andR the integral ˜ exists and is finite because of (4). The convention t = with respect to N 0 (0, t] is used here. R If the L´evy measure Π has a finite first moment, that is, if |x| dΠ < ∞, then both (3) and (7) simplify as Lf (g) =

Z d 1 X aij Xil Xjl f (g) + Y0l f (g) + [f (gh) − f (g)]Π(dh) 2 G

(8)

i,j=1

and f (gt ) = f (g0 ) + +

Z tZ 0

d Z X i=1

G

t 0

Xil f (gs− )

◦

dBsi

+

Z

0

t

Y0l f (gs )ds

[f (gs− h) − f (gs− )]N (dsdh),

(9)

R P where Y0 = X0 − di=1 ( xi dΠ)Xi . If Π is finite, then there are iid (independent and identically distributed) stopping times τn of exponential distribution of mean 1/Π(G), and an independent sequence of iid Gvalued random variables σn of distribution Π/Π(G), such that the L´evy process gt may be obtained by solving the stochastic differential equation dgt =

d X i=1

Xil (gt ) ◦ dBti + Y0l (gt )dt

(10)

P together with jump conditions g(Tn ) = g(Tn −)σn , where Tn = ni=1 τi . All these results hold also for a right L´evy process with suitable changes, for example, 2,l C0 (G), X l and gh in (3) should be replaced by C02,r (G), X r and hg.

L´evy Processes in Lie Groups and Homogeneous Spaces

357

Because a general Lie group G does not have a natural linear structure, the stochastic integral equation for the L´evy process gt can only be given in a functional form as in (7) and (9). If G is a matrix group, then it is possible to write down a stochastic integral equation directly for the process gt . Now let G = GL(d, R) be the general linear group of the d × d real invertible matrices. Its Lie algebra g = gl(d, R) is the space of all d × d real matrices. 2 We may identify g = gl(d, R) with the Euclidean space Rd and G = R) with a PGL(d, 2 2 2 d d dense open subset of R . For any X = {Xij } ∈ R , let |X| = ( i,j Xij )1/2 be its Euclidean norm. Let Eij be the matrix that has 1 at place (i, j) and 0 elsewhere. Then Eij , i, j = 1, 2, . . . , d, form a basis of g. A set x = {xij } of associated coordinate functions may be chosen such that x(g) = g − Id in matrix form for g close to e = Id (the d × d identity matrix). 2 For g ∈ G = GL(d, R), the tangent space Tg G can be identified with Rd , therefore, any element X of Tg G can be represented by a d × d real matrix {Xij } in the sense that P 2 Xf = di,j=1 Xij (∂/∂gij )f (g), where gij are the standard coordinates on Rd . It can be shown that for g, h ∈ GL(d, R) and X ∈ Te G = gl(d, R), Dlg ◦ Drh (X) is represented by the matrix product gXh, where X is identified with its matrix representation {Xij }. Therefore, we may write gXh for Dlg ◦ Drh (X). Thus, X l (g) = gX and X r (g) = Xg. Let gt be a L´evy process in G = GL(d, R). Then it satisfies the stochastic integral equation (7) for any f ∈ C02,l (G). Let f be the matrix-valued function on G defined by f (g) = g for g ∈ G. Although f is not contained in C02,l (G), at least formally, (7) leads to the following stochastic integral equation in matrix form: gt = g0 + +

Z tZ 0

d Z X

i,j=1 0

G

t

dBsij

gs− Eij ◦

˜ (ds dh) + gs− (h − Id )N

Z tZ 0

G

+

Z

t

gs X0 ds

0

gs [h − Id − x(h)]dsΠ(dh),

(11)

where X0 ∈ g = gl(d, R), Bt = {Btij } is a d2 -dim Brownian motion and N is an independent Poisson random measure on R+ × G with characteristic measure Π being a L´evy measure. See Section 1.5 in [24] for the proof of the following result. Theorem 5 Assume 2

E[|g0 | ] < ∞ and

Z

G

|h − Id |2 Π(dh) < ∞.

(12)

Then there is a unique rcll process gt in G = GL(d, R) that satisfies the equation (11). Moreover, gt is a L´evy process and for any t > 0, E[ sup |gs |2 ] < ∞.

(13)

0≤s≤t

Conversely, any L´evy process gt in G satisfying (12) is the unique solution of a stochastic integral equation of the form (11).

358

4.

Ming Liao

L´evy Processes in Compact Lie Groups

Let G be a compact Lie group. A Lie group homomorphism U from G into the group U (n) of n × n unitary matrices is called a unitary representation of G, which may be regarded as a linear action of G on Cn . It is called nontrivial if U 6≡ I and irreducible if it has no nontrivial invariant subspace of Cn . Two representations U1 and U2 are called equivalent if U2 = BU1 B −1 for some invertible matrix B, that is, if they differ only by a change of basis on Cn . The set Irr(G)+ of equivalence classes of non-trivial irreducible unitary representations of G is countable. For δ ∈ Irr(G)+ , let U δ ∈ δ with dimension n = dδ . For any matrix A, let A′ denote its transpose and A¯ its complex conjugate, and write A∗ = A¯′ . The following standard result can be found in Sections II.4 and III.3 in Br¨ocker and tom Dieck [5]. Let ρG denote the normalized Haar measure on G. √ Theorem 6 (Peter-Weyl) The set of matrix elements Uijδ , multiplied by dδ , form a complete orthonormal system in L2 (ρG ). Thus, any f ∈ L2 (ρG ) has a Fourier series expansion: f = ρG (f ) +

X

δ∈ Irr(G)+

dδ

dδ X

¯ijδ )Uijδ = ρG (f ) + ρG (f U

i,j=1

X

dδ Trace(Aδ U δ ) (14)

δ∈ Irr(G)+

R

in L2 sense, where Aδ = ρG (f U δ ∗ ) = f (g)U δ ∗ (g)ρG (dg). Moreover, if f is continuous, then its Fourier series converges uniformly on G. For G = S 1 , the unit circle. All irreducible representations are 1-dimensional, given by S 1 ∋ θ 7→ enθ for n = 0, ±1, ±2, . . .. Theorem 6 becomes the usual Fourier series expansion. In this section, we will study the Fourier expansion of the distribution density of a L´evy process gt in a compact Lie group G, and from it to obtain the exponential convergence of the distribution to ρG as t → ∞ under total variation norm. The results of this section are taken from Liao [23], see also [24, Chapter 4]. More generally, Fourier method has been applied to the convolution products of probability measures on locally compact groups in Heyer [17] and Siebert [33]. It has also been used to study the convergence of random walks to Haar measure on finite and some special compact groups in Diaconis [7] and Rosenthal [32]. Let {X1 , . . . , Xd } be a basis of g and let {aij } be the coefficient matrix in (6). The L´evy process gt is saied to have a non-degenerate diffusion part if the matrix {aij } is nondegenerate and it is said to have a hypoelliptic diffusion part if the vectors d X

aij Xi ,

j = 1, 2, . . . , d

i=1

generate the Lie algebra G. The hypoelliptic condition is weaker than the non-degeneracy. A continuous L´evy process satisfying this condition is a hypoelliptic diffusion process in the usual sense. It is well known that such a process possesses a smooth distribution density pt = dµt /dρG for t > 0. For a general L´evy process gt , it is shown in [23] that if it has a non-degenerate diffusion part and also a finite L´evy measure, then it has an L2 distribution density pt for t > 0. We will see that the existence of pt holds also under hypoellipticity if the L´evy process gt possesses additional invariance property.

L´evy Processes in Lie Groups and Homogeneous Spaces

359

Theorem 7 Let gt be a L´evy process with generator L and g0 = e. Assume it has an L2 distribution density pt = dµt /dρG for t > 0. Then for g ∈ G, X pt (g) = 1 + dδ Trace{exp[t L(U δ ∗ )(e)] U δ (g)}. (15) δ∈ Irr(G)+

The series converges absolutely and uniformly for (t, g) in [ε, ∞)×G for any ε > 0, hence, (t, g) 7→ pt (g) is continuous on (0, ∞) × G. Moreover, if gt has a hypoelliptic diffusion part, then the eigenvalues of the matrix L(U δ ∗ )(e) have negative real parts. Consequently, pt → 1 uniformly on G as t → ∞. Let x1 , . . . , xd ∈ Cc∞ (G) be Pexponential coordinates associated P to the basis i xi (g)Xi for g near e. {X , . . . , X } of g, that is, g = e Then n i xi Ad(h)Xi = P1 i (xi ◦ ch )Xi near e for all h ∈ G. Because G is compact, xi may be suitably modified so that this holds on G. A vector X ∈ g is called Ad(G)-invariant ifPAd(g)X = X for all g ∈ G. A symmetric matrix aij is called Ad(G)-invariant if aij = p,q apq [Ad(g)]ip [Ad(g)]jq for all g ∈ G, where [Ad(g)P , Xd }, that is, ij is the matrix representing Ad(g) under the basis {X1 , . . . P l Ad(g)Xj = i [Ad(g)]ij Xi . Note that the vector field X and the operator i,j aij Xil Xjl are conjugate invariant if and only if X and aij are Ad(G)-invariant. The following result, which holds on a general Lie group, may be derived from the generator formula (3). Theorem 8 A L´evy process gt with g0 = e has a conjugate invariant distribution µt for all t > 0 if and only if in the generator formula (3) under exponential coordinates chosen as above, Π is conjugate invariant, and aij and X0 are Ad(G)-invariant. Let L2ci (ρG ) be the space of conjugate invariant functions in L2 (ρG ). The character χδ of an irreducible unitary representation U δ , defined by χδ = Trace(U δ ), is conjugate invariant. A version of Peter-Weyl theorem says that the irreducible characters χδ form an orthonormal basis of L2ci (ρG ). Thus, for f ∈ L2ci (ρG ), X f = ρG (f ) + ρG (f χδ )χδ (16) δ∈ Irr(G)+

in L2 -sense, and the convergence is uniform on G if f is continuous. It can be shown that the character χδ is positive definite in the sense that k X

i,j=1

χδ (gi gj−1 )ξi ξ¯j ≥ 0

(17)

for any finite set of gi ∈ G and ξi ∈ C. From this it is easy to show that |χδ | ≤ χδ (e) = dδ . Let ψδ = χδ /dδ (normalized character). Let kf k2 and kf k∞ be respectively the usual L2 -norm and the L∞ -norm of a function f (under ρG ). The total variation norm of a signed measure ν is defined by kνktv = sup|f |≤1 |ν(f )|. By Theorem 7, if gt is hypoelliptic, then its distribution µt converges to the normalized Haar measure ρG under the total variation norm: kµt − ρG ktv → 0 as t → ∞.

360

Ming Liao

Theorem 9 Let gt be a L´evy process with g0 = e, a hypoelliptic diffusion part, and conjugate invariant distribution µt . Then it has an L2 distribution density pt for t > 0 and for g ∈ G, X pt (g) = 1 + dδ et Lψδ (e) χδ (g). (18) δ∈ Irr(G)+

The series converges absolutely and uniformly for (t, g) in [ε, ∞) × G for any ε > 0, and Lψ δ (e) = (1/2)

d X

i,j=1

aij Xil Xjl ψδ (e) +

Z

(Re ψδ − 1)dΠ,

δ ∈ Irr(G)+ ,

(19)

has a negative upper bound −λ, and for any ε > 0, there are constants C > c > 0 such that for t > ε, kpt −1k∞ ≤ Ce−λt , ce−λt ≤ kpt −1k2 ≤ Ce−λt , ce−λt ≤ kµt −ρG ktv ≤ Ce−λt . (20) Remark In Theorem 9, the hypoelliptic condition on the diffusion part may be replaced by an asymptotic condition on the L´evy measure Π as to be described below. Two nonnegative functions φ and ψ are called asymptotically equal at a point x0 , denoted as φ ≍ ψ at x0 , if there are positive constants c1 < c2 such that c1 φ ≤ ψ ≤ c2 φ in a neighborhood of x0 . Similarly, two measures ξ and η are called asymptotically equal at x0 if c1 ξ ≤ η ≤ c2 ξ in a neighborhood of x0 . This is equivalent to the asymptotic equality of their densities if those exist. We will say ξ to be asymptotically larger than η at x0 if ξ dominates a measure that is asymptotically equal to η at x0 . Let the compact Lie group G be equipped with a left invariant Riemannian metric and let r(g) be the distance between e and g ∈ G. It can be shown that if gt is a L´evy process in G with g0 = e and a conjugate invariant distribution µt for all t > 0, and if its L´evy measure Π is asymptotically larger than rβ dρG for some β ∈ (d, d + 2), where d = dim(G), then all the conclusions of Theorem 9 hold. This can be proved by the arguments in [27, section 7]. Because G is compact, there is an Ad(G)-invariant inner product on its Lie algebra g, which induces a bi-invariant (left and right invariant) Riemannian metric on G. We may assume the basis {X1 , . . . , Xd } of g is orthonormal under P the Ad(G)-invariant inner product. Then the Laplace operator ∆ on G is given by ∆ = i Xil Xil . A Lie algebra g is called simple if it does not contain any ideal (a sub Lie algebra i satisfying [i, g] ⊂ i) except {0} and g, and is called semisimple if it does contain any abelian ideal ([i, i] = 0) except {0}. A Lie group is called simple or semisimple if its Lie algebra is so. If G is simple, then up to a constant factor, the Ad(G)-invariant inner product on g is unique and hence so is the bi-invariant Riemannian metric on G. In this case, it can be shown that any second order bi-invariant differential operator is a constant multiple of the Laplace operator ∆. In particular, if gt is a continuous L´evy process in G, then its generator is given by L = a∆ for some constant a ≥ 0. Example: The rotation group G = SO(3) is a simple Lie group. Any matrix in SO(3) is conjugate to a unique rotation about x1 -axis by angle θ ∈ [0, π]. Therefore, any conjugate invariant function on G may be regarded as a function of θ ∈ [0, π]. By [5, section II.5],

L´evy Processes in Lie Groups and Homogeneous Spaces 361 P2n i(n−j)θ the irreducible characters (including the trivial one) are given by χn (θ) = j=0 e = sin[(n + 1/2)θ]/ sin(θ/2) of dimension 2n + 1 for n = 0, 1, 2, . . .. By choosing a bi-invariant Riemannian metric on G and an orthonormal basis X1 , X2 , X3 of g = Te G, we may write ∆ = X1l X1l + X2l X2l + X3l X3l . We may assume X1l = ∂/∂θ. By the conjugate invariance of characters, ∆χn (e) = 3χ′′n (e). A direct computation yields ∆ψn (e) = ∆χn (e)/(2n + 1) = −n(n + 1). Let gt be a continuous L´evy process in G = SO(3) with g0 = e and generator L. If it is hypoelliptic and conjugate invariant, then L = a∆ for some constant a > 0, and hence it has an L2 distribution density pt for t > 0 given by pt (θ) =

∞ X

n=0

5.

(2n + 1)e−an(n+1)t

sin[(n + 1/2)θ] . sin(θ/2)

(21)

L´evy Processes in Homogeneous Spaces

Let M be a manifold and let G be a connected Lie group acting on M . A Markov process xt in M with transition semigroup Qt is called G-invariant if Qt is G-invariant in the sense that Qt (f ◦ g) = (Qt f ) ◦ g for f ∈ C0 (M ) and g ∈ G. A G-invariant Markov process xt in a homogeneous space G/K, under the natural action of G on G/K, will be called a L´evy process in G/K because when K = {e}, xt becomes a L´evy process in G. The main results in this section are taken from Section 2.2 in [24]. We will assume K is a compact subgroup of G in the rest of this section. Let o = eK be the origin in G/K. It is easy to show that if gt is a L´evy process in G with g0 = e and if it is K-conjugate invariant in the sense that its transition semigroup Pt satisfies Pt (f ◦ ck ) = (Pt f )◦ck for f ∈ C0 (G) and k ∈ K, then xt = gt o is a L´evy process in G/K with x0 = o. Later we will see that any L´evy process in G is obtained in this way. We now show that the convolution of measures may be naturally defined on a homogeneous space G/K, and L´evy processes in G/K are associated to convolution semigroups just as in a Lie group G. A measure µ on M is called K-invariant if kµ = µ for any k ∈ K, where kµ denote the measure obtained from µ via the action of k on G/K, defined by kµ(B) = µ(k −1 (B)) for B ⊂ G/K, or equivalently by kµ(f ) = µ(f ◦ k) for function f on G/K A section map on G/K is a (Borel measurable) map S: G → G/K satisfying π ◦ S = idG/K , where π: G → G/K is the natural projection and idG/K is the identity map on G/K. For two K-invariant measures µ and ν on M , their convolution is the measure defined by Z f (S(x)y)µ(dx)ν(dy) for any function f on G/K µ ∗ ν(f ) = (G/K)×(G/K)

with the choice of a section map S. By the K-invariance of µ and ν, µ ∗ ν is independent of the choice of S and is K-invariant. Moreover, (µ ∗ ν) ∗ λ = µ ∗ (ν ∗ λ) for three K-invariant measures µ, ν and λ. Thus, the n-fold convolution µ∗n = µ ∗ · · · ∗ µ is well defined. The following result is easy to prove.

362

Ming Liao

Theorem 10 The distributions νt of a L´evy process xt in G/K with x0 = o form a continuous convolution semigroup of K-invariant probability measures on G/K with ν0 = δo . Conversely, given such a convolution semigroup νt on M , there is a L´evy process xt in G/K, unique in distribution, with distribution νt and x0 = o. Moreover, its transition R semigroup Qt is given by Qt f (x) = f (S(x)y)νt (dy) with the choice of a section map S.

Because a section map S may be chosen to be continuous near any point in G/K, it is easy to show from Theorem 10 that a L´evy process in G/K is a Feller process. We have mentioned Hunt’s formula for the generator of a L´evy process in G. We will now state an analog of this result for L´evy process in G/K, obtained also by Hunt [18], in the form presented in [24, Theorem 2.1]. A differential operator T on G/K is called G-invariant if T (f ◦ g) = (T f ) ◦ g for f ∈ C ∞ (G/K) and g ∈ G. Such an operator is completely determined by T f (o) for f ∈ C ∞ (G/K). It will be called a G-invariant diffusion generator if it is the generator of a G-invariant diffusion process in G/K. Because K is compact, there is an Ad(K)-invariant subspace p of the Lie algebra g of G that is complementary to the Lie algebra k of K. Let X1 , . . . , Xd be a basis of g such that X1 , . . . , Xn ∈ p and Xn+1 , . . . , Xd ∈ P k. Local coordinates y1 , . . . , yn on n G/K around o may be chosen to satisfy x = exp[ i=1 yi (x)Xi ]o for x near o. Then Pn Pn −1 i=1 yi (x)Ad(k)Xi = i=1 yi (kxk )Xi for all k ∈ K and for x near o. As functions in Cc∞ (G/K), yi may be suitably extended such that this holds for all x ∈ G/K. Theorem 11 Let L be the generator of a L´evy process xt in G/K with x0 = o. Then its domain D(L) contains Cc∞ (M ), and with local coordinates y1 , y2 , . . . , yn on G/K near o chosen as above, for any f ∈ Cc∞ (G/K), Lf (o) = T f (o) +

Z

G/K

[f (x) − f (o) −

n X

yi (x)

i=1

∂ f (o)]Π(dx), ∂yi

(22)

where T is a G-invariant diffusion generator on G/K, called the diffusion part of L, and Π is a K-invariant measure on G/K satisfying Π({o}) = 0,

n X yi2 ) < ∞ and Π( i=1

Π(U c ) < ∞

for any neighborhood U of o, called the L´evy measure of process xt . Conversely, given T and Π as above, there is a L´evy process xt in G/K with x0 = o, unique in distribution, such that its generator L at o, restricted to Cc∞ (G/K), is given by (22). Recall that if gt is a K-conjugate invariant L´evy process in G with g0 = e, then xt = gt o is a L´evy process in G/K. Theorem 11 may be used to prove the converse of this statement. Theorem 12 If xt is a L´evy process in G/K with g0 = o, then there is a K-conjugate invariant L´evy process gt in G with g0 = e such that xt = gt o (in distribution as processes).

L´evy Processes in Lie Groups and Homogeneous Spaces

363

Theorem 12 implies that any continuous convolution semigroup νt on G/K with ν0 = δo is the projection πµt of a continuous convolution semigroup µt of K-conjugate invariant probability measures on G with µ0 = δe . Now let µt be a continuous convolution semigroup on G. Then µ0 ∗ µ0 = µ0 and by Theorem 1.2.10 in Heyer [17], µ0 is the normalized Haar measure ρK on some compact subgroup K of G. Because µ0 ∗ µt = µt ∗ µ0 = µt , each µt is bi-K-invariant (invariant under both left and right K-translations). The projection νt = πµt is a continuous convolution semigroup on G/K with ν0 = o, while νt is also the projection of a K-conjugate invariant convolution semigroup µet on G with µe0 = δe . It is then easy to show µt = µ0 ∗ µet . This is summarized in the following theorem. Theorem 13 Let µt be a continuous convolution semigroup on a Lie group G. Then there is a compact subgroup K of G and a K-conjugate invariant continuous convolution semigroup µet on G with µe0 = δe such that µt = ρK ∗ µet = µet ∗ ρK , where ρK is the normalized Haar measure on K.

6.

L´evy -Khinchin Formula

Let G be a Lie group and K be a compact subgroup. Recall that lg and rg are respectively the left and right translations on G by g ∈ G. A function f on G is called left (resp. right) invariant if f ◦ lg = f (resp. f ◦ rg = f ) for any g ∈ G. It is called bi-invariant if it is both left and right invariant. It is called left (resp. right, resp. bi-) K-invariant if g ∈ G is replaced by k ∈ K in the above definitions. A function f on G/K is called G-invariant (resp. K-invariant) if f ◦ g = f for g ∈ G (resp. g ∈ K). An operator T on G with domain Dom(T ), a space of functions on G, is called left invariant if for f ∈ Dom(T ), f ◦ lg ∈ Dom(T ) and T (f ◦ lg ) = (T f ) ◦ lg for any g ∈ G. The right invariant, left and right K-invariant operators on G and G-invariant operators on G/K are defined similarly. The domain of a differential operator is automatically the space of smooth functions unless when explicitly stated otherwise. Let D(G) denote the space of left invariant differential operators on G, let DK (G) be the subspace of D(G) consisting of those which are also right K-invariant, and let D(G/K) be the space of G-invariant differential operators on G/K. A complex valued smooth function φ on G is called a spherical function if φ(e) = 1, if it is bi-K-invariant and if it is a common eigenfunction of the operators in DK (G), that is, ∀T ∈ DK (G),

T φ = β(T, φ) φ

for some constant β(T, φ).

(23)

Let dk be the normalized Haar measure on K. By IV.Proposition 2.2 in [16], a nonzero complex valued continuous function φ on G is spherical if and only if Z φ(xky)dk = φ(x)φ(y). (24) ∀x and y ∈ G, K

Because of its right K-invariance, a spherical function φ on G may be naturally regarded as a function on G/K. Thus, a function φ on G/K will be called spherical if φ ◦ π is a spherical function on G. Equivalently, a complex valued smooth function φ on G/K is

364

Ming Liao

spherical if φ(o) = 1, if it is K-invariant and if it is a common eigenfunction of the Ginvariant differential operators on G/K, that is, if (23) holds with DK (G) replaced by D(G/K). Let νt be a continuous convolution semigroup of K-invariant probability measures on G/K with ν0 = δo . Its generator L is defined to be the generator of the associated L´evy process. Let φ be a bounded spherical function on G/K. By (24), it can be shown that νs+t = νs (φ)νt (φ), and then νt (φ) = ety with y = (d/dt)νt (φ) |t=0 = Lφ(o). This implies the following result in Liao and Wang [27], based on ideas in Applebaum [1]. Theorem 14 Let xt be a L´evy process in G/K starting from o with distribution νt and generator L given by (22). For any bounded spherical function φ on G/K, Z n X ∂ tLφ(o) yi (x) νt (φ) = e = exp{t[β(T, φ) + (φ(x) − 1 − φ(o))Π(dx)]}, (25) ∂yi G/K i=1

where β(T, φ) is the eigenvalue of T for eigenfunction φ as given in (23) . When G = Rn (additive group) and K = {0}, G/K = Rn . Then the bounded √ spherical functions are the exponentials φy (x) = ei(x·y) for x, y ∈ Rn , where i = −1 and (x · y) is the usual inner product on Rn . In this case, (25) becomes the classical L´evyKhinchin formula and hence it may be regarded as a L´evy-Khinchin type formula on a general homogeneous space G/K.

7.

Symmetric Spaces

As before, let G be a Lie group and let K be a compact subgroup. Their Lie algebras are denoted respectively as g and k. Because K is compact, there is an Ad(K)-invariant inner product h·, ·i on g. Fix such an inner product and let p be the orthogonal complement of k in g under this inner product. A bijective map Θ: G → G that preserves the product structure, and satisfies Θ 6= idG and Θ2 = idG is called a Cartan involution. The pair (G, K) together with a Cartan Θ Θ involution Θ is called a symmetric pair if GΘ 0 ⊂ K ⊂ G , where G is the fixed point set of Θ and GΘ 0 is its identity component. Then the differential DΘ of Θ at e has eigenvalues ±1 with k and p as eigenspaces of +1 and −1 respectively. The direct sum g = k⊕p is called a Cartan decomposition of g. The homogeneous space G/K is called a symmetric space which plays an important role in differential geometry, see [15, 16] for more information. We now assume G/K is a symmetric space. By Theorem 3.1 and Proposition 3.8 in [16, chapter IV], the convolution product on G/K is commutative, that is, µ∗ν = ν ∗µ, and any K-invariant finite measure µ on G/K is completely determined by its spherical transform φ 7→ µ(φ) with φ ranging over all bounded spherical functions. Note that the two results from [16] cited here are stated for functions on G, but from which the above statements for measures may be derived. Therefore, the L´evy-Khinchin formula (25) provides a complete characterization of a continuous convolution semigroup νt on a symmetric space G/K. The Killing form of the Lie algebra g is defined by B(X, Y ) = Trace[ad(X)ad(Y )] (a symmetric bilinear form), where ad(X) is the linear map: g → g given by ad(X)Y = [X, Y ] (Lie bracket). The Lie group G is semisimple if and only if B is nondegenerate.

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The symmetric space G/K is said to be of compact type if the Killing form B is negative definite, it is said to be of noncompact type if B is negative definite on k and positive definite on p, and it is said to be of Euclidean type if p is an abelian ideal of g. For a compact (resp. noncompact) type symmetric space G/K, G is semisimple, and both G/K and G are compact (resp. noncompact). In general, a symmetric space G/K with G simply connected is a direct product of symmetric spaces of these three basic types. A Euclidean space Rn is a trivial example of a Euclidean type symmetric space with G = Rn (additive group) and K = {0}. A less trivial example is Rn = Mn /SO(n), where Mn is the group of Euclidean motions on Rn and SO(n) is the subgroup of rotations that fixes the origin. The n-dimensional sphere S n = SO(n + 1)/SO(n) is a compact type symmetric space. The n-dimensional hyperbolic space Hn = SO(1, n)+ /SO(n) is a noncompact type space, where SO(1, n)+ is the connected Lorentz group on Rn+1 . The positive definiteness of a function in a Lie group G is defined by (17). A function φ on G/K is called positive definite if φ ◦ π is so on G. Such a function is bounded with |φ| ≤ φ(o). The following result is proved in [27]. Theorem 15 Let G/K be a symmetric space. Assume there is no nonzero Ad(K)-invariant element of p (that is, X ∈ p with Ad(k)X = X for any k ∈ K implies X = 0, which holds if G is semi-simple). Let φ be a positive definite spherical function on G/K. Then (∂/∂yi )φ(o) = 0 in (25). Consequently, the L´evy-Khinchin formula (25) takes the following simpler form: Z νt (φ) = exp{t[β(T, φ) +

(φ − 1)dΠ]}.

(26)

Moreover, β(T, φ) ≤ 0.

The L´evy -Khinchin type formula (26) on symmetric spaces was obtained in Gangolli [13]. With t = 1, it in fact characterizes all infinite divisible K-invariant distributions ν1 ∗n for some K-invariant ν on a symmetric space G/K (ν1 = ν1/n 1/n and all integer n > 0). The spherical transform of a finite measure µ: φ 7→ µ(φ) with φ varying over positive definite spherical functions, plays the same role as the Fourier transform on a Euclidean space. On each of three basic types of symmetric spaces G/K, using (26) and the inverse spherical transform, we can prove that the distribution νt of a L´evy process xt in G/K has a density qt (x), smooth in (t, x) ∈ (0, ∞) × (G/K), with respect to a G-invariant reference measure ρ on G/K for t > 0 under either of the following two conditions: (a) the diffusion part T of the generator L is non-degenerate, or (b) the L´evy measure Π is asymptotically larger (see definition in section 4) than rβ dρ for some β ∈ (n, n + 2), where n = dim(G/K) and r is the distance from o, and obtain a representation of qt in terms of spherical functions, see [27] for more details.

Example: On the unit sphere S 2 = SO(3)/SO(2) in R3 , the spherical functions are φn (r) = Pn (cos r), where Pn is the nth degree Legendre polynomial and r is the geodesic distance on S 2 from the north pole o (fixed by K), and the Laplace operator ∆ on S 2 has eigenvalue −n(n + 1) corresponding to φn . A continuous L´evy process xt in S 2 = SO(3)/SO(2) is just a time rescaled Brownian motion because its generator is L = a∆ for some a ≥ 0, due to the irreducibility of SO(3)/SO(2). Assume x0 = o and let νt be

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the distribution of xt . Then νt (φn ) = e−an(n+1)t . If a > 0, then νt has a smooth density qt with respect to the uniform distribution on S 2 given by qt (r) = 1 +

∞ X

(2n + 1)e−an(n+1)t Pn (cos r).

(27)

n=1

8.

Limiting Properties on Noncompact Type Symmetric Spaces

The limiting properties of Brownian motions and random walks in semi-simple Lie groups or symmetric spaces of noncompact type, both of which are examples of L´evy processes, have been studied by many people. Dynkin [8] in 1961 studied the limiting properties of the Brownian motion in a special type of symmetric space in connection with Martin boundary. Part of Dynkin’s result was extended by Orihara [30] to a general symmetric space. Limiting properties of Brownian motion in a general semi-simple Lie group was obtained by Malliavin-Malliavin [28]. See also Norris, Rogers and Williams [29], Taylor [34, 35], Babillot [4], and Liao [21] for some of subsequent and related study. In a different direction, Furstenberg and Kesten [11] in 1960 studied the limiting properties of products of iid matrix or Lie group valued random variables. Such processes may be regarded as random walks or discrete time L´evy processes in Lie groups. This study was continued in Furstenberg [10], Tutubalin [36], Virtser [37] and Raugi [31]. In Guivarc’h and Raugi [14], the limiting properties of random walks on semi-simple Lie groups of non-compact type were established under a very general condition. In Liao [22], these results were extended to general L´evy processes and were applied to study the dynamical properties of certain stochastic flows on homogeneous spaces. We will briefly describe the basic structure theory of semi-simple Lie groups of noncompact type. The reader is referred to [15] for more details. Let (G, K) be a symmetric pair of noncompact type with the Cartan involution Θ and the Cartan decomposition g = k ⊕ p of the Lie algebra g of G as defined before. We will assume that G is connected and has a finite center. Then K is a maximal compact subgroup of G. Let a be a maximal abelian subspace of p. A linear functional α on a is called a root if the space gα = {X ∈ g; ad(H)X = α(H)X for H ∈ a}, called the root space of α, is nonzero. Fix an Ad(K)-invariant inner product on g. We have an orthogonal direct sum decomposition X gα , (28) g = g0 ⊕ α

where g0 = a ⊕ m with m = {Z ∈ k; ad(Z)H = 0 for H ∈ a}. The subspaces of a determined by the equations α = 0 divide a into several open convex conic regions, called the Weyl chambers. Fix a Weyl chamber a+ . A root α is called positive if α > 0 on a+ . Any root α is either positive or negative, that is, equal to −α for some positive root α. Let X X n+ = gα and n− = g−α , (29) α>0

α>0

where the summations are taken over all positive roots. Both are nilpotent Lie algebras. Let A, N + and N − be respectively the subgroups of G generated by a, n+ and n− . Let A+ = exp(a+ ) and let A+ be its closure.

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Let M = {k ∈ K; Ad(k)H = H for H ∈ a} and M ′ = {k ∈ K; Ad(k)a ⊂ a}, (30) called respectively the centralizer and the normalizer of a in K. The former is a normal subgroup of the latter, but has the same Lie algebra m. The quotient group W = M ′ /M is finite and is called the Weyl group. It acts on a via the map: W × a ∋ (s, H) 7→ sH = Ad(ks )H ∈ a with s = ks M ∈ W for ks ∈ M ′ . The group G possesses the Cardan decomposition G = KA+ K in the sense that any g ∈ G may be written as g = ka+ h with k, h ∈ K and a unique a+ ∈ A+ . Although the choices for (k, h) are not unique, when a+ ∈ A+ , they are given by (km, m−1 h) for m ∈ M . An element g of G is called regular if a+ ∈ A+ . Regular elements form an open dense subset G′ of G, which projects to an open dense subset (G/K)′ of G/K. The Cartan decomposition G = KA+ K induces the polar decomposition on G/K: Any point x in G/K can be written as x = ka+ o for kM ∈ K/M and a+ ∈ A+ . When x ∈ (G/K)′ , a+ and kM are unique and are called respectively the radial and angular components of x. The group G also has the Iwasawa decomposition G = N − AK in the sense that the map: N − × A × K ∋ (n, a, k) 7→ g = nak ∈ G is a diffeomorphism. There are other versions of Iwasawa decompositions such as G = KAN + . A typical example of a semi-simple Lie group of noncompact type is G = SL(d, R), the group of d × d real matrices of determinant one, called the special linear group. Its Lie algebra is g = sl(d, R), the space of d × d traceless real matrices. The Cartan involution is given by Θ(g) = g ′−1 with K = SO(n) and p being the space of d × d traceless real symmetric matrices. In this case, the subspace a of p formed by traceless diagonal matrices is a maximal abelian subspace of p. The roots are αij given by αij (H) = Hi − Hj for H = diag(H1 , . . . , Hd ) ∈ a and i 6= j. The root space gαij is one-dimensional and is spanned by the matrix Eij that has 1 at place (i, j) and 0 elsewhere. One may take a+ = {diag(H1 , . . . , Hd ) ∈ a; H1 > H2 > · · · > Hd } to be the chosen Weyl chamber. Then positive roots are αij with i < j. The nilpotent Lie algebras n+ and n− are respectively the spaces of upper triangular and lower triangular matrices of zero diagonal. The groups A, N + and N − are respectively the subgroups of G consisting of diagonal matrices of positive diagonal, upper triangular matrices of unit diagonal, and lower triangular matrices of unit diagonal. The group M is discrete with Lie algebra m = {0} and consists of diagonal matrices with ±1 along diagonal (with an even number of −1’s), the group M ′ is formed by permutation matrices, and the Weyl group W = M ′ /M acts on a by permuting the diagonal elements of H ∈ a. For G = SL(d, R), the Cartan decomposition G = KA+ K can be established by diagonalizing the symmetric matrix gg ′ for g ∈ SL(d, R). The Iwasawa decomposition G = N − AK (resp. G = KAN + ) is a consequence of Gram-Schmidt orthogonalization applied to the rows (resp. columns) of a matrix in SL(d, R). Now return to a general G. Let gt be a L´evy process in G and let µt be the distribution of gte = g0−1 gt . Let Tµ be the closed semigroup generated by the supports supp(µt ) of measures µt for t ∈ R+ , that is, Tµ is the closed subset of G containing all supp(µt ) and satisfying x, y ∈ Tµ =⇒ xy ∈ Tµ . Let Gµ be the closed subgroup of G generated by all supp(µt ).

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Following Guivarc’h and Raugi [14], a subset H of G will be called totally irreducible if there do not exist g1 , . . . , gk , x ∈ G such that H⊂

k [

gi (N − M AN + )c x,

i=1

where the superscript c denotes the complement in G. It will be called totally right irreducible if gi (N − M AN + )c x is replaced by x(N − M AN + )c gi . Note that N − M AN + is an open subset of G whose complement is lower dimensional, that is, the complement is contained in a union of finitely many lower dimensional sub-manifolds of G. Therefore, if H is not a lower dimensional subset of G, then it is totally (right) irreducible. Note that exp: a → A is a bijection, so its inverse log: A → a is well defined. A sequence gj in G is called contracting if in the Cartan decomposition gj = ξj a+ j ηj , + α(log aj ) → ∞ as j → ∞ for any positive root α. For g ∈ G with Cartan decomposition g = ka+ h, let kgk = k log a+ k. Theorem 16 Let gt be a L´evy process in G with the Cartan decomposition gt = ξt a+ t ηt and − the Iwasawa decomposition gt = nt at kt of GR= N AK. Assume Gµ is totally irreducible and Tµ contains a contracting sequence, and G kgk Π(dg) < ∞. Then almost surely, ξt M converges in K/M and nt converges in N − as t → ∞, and H + = lim

t→∞

1 1 log a+ lim log at t = t→∞ t t

(31)

exists, is non-random and is contained in a+ . The convergence of (1/t) log a+ t and ξt U in the above theorem are often referred to as the radial and angular convergences. See section 6.6 in [24] for some sufficient conditions which guarantee the hypotheses of Theorem 16. Theorem 16 holds also for a right L´evy process gt in G with the following changes: the total irreducibility should be replaced by the total right irreducibility, the convergence of ξt M in K/M by that of M ηt in M \K (right coset space), and the decomposition gt = nt at kt of G = N − AK by gt = kt at nt of G = KAN + .

9.

Dynamical Aspect

The limiting properties of L´evy processes may also be studied from a dynamic point of view. Let Ω be the underlying probability space. A collection of maps θt : Ω → Ω, t ∈ R+ , is called a semigroup of time-shift operators if each θt preserves the probability measure P on Ω, and they together form a semigroup in t in the sense that θt θs = θs+t and θ0 = idΩ . Let X be a (smooth) manifold and let Diff(X) be the group of the diffeomorphisms X → X. A dynamical system, or a stochastic flow, on X is a stochastic process φt in Diff(X) with φ0 = idX together with a semigroup of time shift operators θt such that the following co-cycle property holds: ∀s, t ∈ R+ and ω ∈ Ω,

φs+t (ω) = φs (θt ω)φt (ω).

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See Arnold [3] for a comprehensive treatment of the dynamical theory of such systems. We now describe the Lyapunov exponents and the associated stable manifolds of a stochastic flow φt on a manifold X equipped with a Riemannian metric {k · kx ; x ∈ X}. The family of maps Φt : X × Ω → X × Ω given by (x, ω) 7→ (φt (ω)x, θt ω), t ∈ R+ , is a semigroup in t and is called the skew-product flow associated to φt . A probability measure ν on X is called a stationary measure of φt if E(φt ν) = ν. Under a certain condition, there is a unique stationary measure ν. Moreover, there are constants λ1 > λ2 > · · · > λr , a subset Γ of X × Ω invariant under Φt , in the sense that Φ−1 t (Γ) = Γ, with ν × P (Γ) = 1, and for any (x, ω) ∈ Γ, the subspaces of the tangent space Tx X: Tx X = V1 (x, ω) % V2 (x, ω) % · · · % Vr (x, ω) % Vr+1 (x, ω) = {0} such that ∀v ∈ [Vi (x, ω) − Vi+1 (x, ω)],

1 log kDφt (ω)vkφ(ω)x = λi t→∞ t lim

for i = 1, 2, . . . , r, where Vi (x, ω) − Vi+1 (x, ω) is the set difference. The numbers λi are called the Lyapunov exponents. The random subspace Vi (x, ω) is called the subspace of Tx X associated to the exponent λi , and di = dim[Vi (x, ω) − Vi+1 (x, ω)] is independent of (x, ω) and is called multiplicity of λi . The Lyapunov exponents λi are the limiting exponential rates at which the lengths of tangent vectors on X are stretched or contracted under the stochastic flow φt , and together with Vi (x, ω), they are independent of the Riemannian metric on a compact manifold X. A connected sub-manifold X′ of X is called a stable manifold of a negative Lyapunov exponent λi at (x, ω) ∈ Γ if X′ ⊂ {y ∈ X; (y, ω) ∈ Γ}, Tx X′ = Vi (x, ω) and ∀y ∈ X′ ,

lim sup t→∞

1 log dist(φt (ω)x, φt (ω)y) ≤ λi , t

where dist denotes the Riemannian distance on X. Roughly speaking, the distance between any two points in X′ tends to zero exponentially fast at the negative exponential rate λi under the stochastic flow φt . A stable manifold of λi at (x, ω) is called maximal if it contains any stable manifold of λi at (x, ω). The local existence of stable manifolds is proved in Carverhill [6]. Intuitively, one would expect that the maximal stable manifolds of a negative exponent form a foliation of a random open dense subset of X, but such a global theory under a general setting can be quite complicated, see [3, chapter 7]. If a Lie group G acts on a manifold X, then a right L´evy process gt in G with g0 = e may be regarded as a stochastic flow on X when Ω is taken to be the canonical sample space of the process with natually defined time shift θt . Let G be a semi-simple Lie group of noncompact type with a finite center. We will continue to use the notation introduced in the previous section. Let Q be a closed subgroup of G with Lie algebra q. Assume Q contains AN + . Then the homogeneous space X = G/Q is compact. For G = SL(d, R), such homogeneous spaces include the sphere S d−1 , the special orthogonal group SO(d) and several other interesting spaces. Let gt be a right L´evy process in G with g0 = e. We will describe explicitly, in terms of the group structure, the Lyapunov exponents and the associated stable manifolds of gt

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regarded as a stochastic flow on X = G/Q, as well as a clustering pattern of this stochastic flow. See Liao [24, Chapter 8] for more detail. Let gt = ξt a+ t ηt = kt at nt be respectively the Cartan decomposition G = KA+ K and the Iwasawa decomposition G = KAN + of gt , and for g ∈ G, let gt g = ξtg atg+ ηtg = ktg agt ngt be the corresponding decompositions of gt g. Assume the process gt satisfies the hypotheses in the version of Theorem 16 for right L´evy processes. Then for any g ∈ G, almost surely, ngt and M ηtg converge as t → ∞, and the limit H + in (31) hold with g+ g + ∈ a is a+ + t and at replaced by at and at respectively. Moreover, the non-random H independent of g ∈ G. By choosing the K-components in the Cartan decomposition of gt g properly, one may assume that ηtg converges as t → ∞. For H ⊂ X × Ω, let H(x) = {ω ∈ Ω; (x, ω) ∈ H}, called the x-section of H at x ∈ X, and let H(ω) = {x ∈ X; (x, ω) ∈ H}, called the ω-section of H at ω ∈ Ω. Let π: G → X = G/Q be the natural projection. For g ∈ G, X ∈ g and v ∈ Tx X, we may write gX for Dlg (X) ∈ Tg G and gv for Dg(v) ∈ Tgx X. By [24, Proposition 8.5], there is a subset Γ of X × Ω invariant under the skew-product flow associated to the stochastic flow gt such that P (Γ(x)) = 1 for all x ∈ X and Γ(ω) = gng∞ (ω)−1 π(N − M AN + )

(32)

for (x, ω) ∈ Γ and g ∈ π −1 (x). Note that Γ(ω) is a dense open subset of X. Theorem 17 Let α be a negative root or zero. For any (x, ω) ∈ Γ, g ∈ π −1 (x) and Y ∈ Ad(ng∞ (ω)−1 )[gα − (gα ∩ q)], we have

1 log kDφt (ω)Dπ(gY )kφt (ω)x = α(H + ), t where [gα − (gα ∩ q)] is the set difference. Consequently, the Lyapunov exponents of the stochastic flow φt on X = G/Q are given by α(H + ), where α ranges over all negative roots and zero with gα 6⊂ q. Therefore, all the exponents are non-positive, and they are all negative if and only if m ⊂ q. lim

t→∞

Let λ1 > λ2 > · · · > λr be the set of all the distinct Lyapupov exponents, and let X gα (Lie subalgebra) and Ni = exp(ni ) (Lie subgroup). (33) ni = α(H + )≤λi

Theorem 18 Let (x, ω) ∈ Γ and g ∈ π −1 (x). Then for 1 ≤ i ≤ r, Vi (x, ω) = Dπ[gng∞ (ω)−1 ni ] is the subspace of Tx X associated to the exponent λi , and if λi is negative, then Xi (x, ω) = π[gng∞ (ω)−1 Ni ] is the maximal stable manifold of λi at (x, ω).

A family of sub-manifolds {Hσ } of a manifold H, each of dimension k, is said to be a foliation of H if any x ∈ H has a coordinate neighborhood V with coordinates x1 , . . . , xd such that each subset of V determined by xk+1 = c1 , xk+2 = c2 , . . . , xd = cd−k is equal to Hσ ∩ V for some σ, where c1 , c2 , . . . , cd−k are arbitrary constants. The submanifolds Hσ form a disjoint union of H and are called the leaves of the foliation.

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Theorem 19 Let λi < 0. Then the family of stable manifolds Xi (x, ω) of λi is a foliation of Γ(ω). Moreover, if i < r, then each Xi (x, ω) is foliated by {Xi+1 (y, ω); y ∈ Xi (x, ω)}, the family of the stable manifolds of the exponent λi+1 contained in Xi (x, ω). Any X ∈ g induces a vector field X ∗ on X defined by X ∗ f (x) = (d/dt)f (etX x) |t=0 for f ∈ C 1 (X). Since K is compact, the Riemannian metric on X may be chosen under which K acts isometrically on X. The limiting property under the Cartan decomposition gt = ξt a+ t ηt implies that for large t, the stochastic flow gt is approximately composed of the following three transformations: a fixed random isometric transformation η∞ , the non-random flow of the vector field (H + )∗ and a “moving” random isometric transformation ξt . Because an isometric transformation preserves the geometry on X, the asymptotic behavior of the stochastic flow gt is largely determined by the flow of the single vector field (H + )∗ . In general, a point x ∈ X is called a stationary point of a vector field X on X if X(x) = 0. This is equivalent to saying that x is a fixed point of the flow ψt of X. A stationary point x is said to attract a subset W of X if ∀y ∈ W , ψt (y) → x as t → ∞. A subset W of X is called invariant under the flow ψt if ψt (W ) ⊂ W . A stationary point x of X is called attracting if there is an open neighborhood V of x that is a disjoint union of positive dimensional sub-manifolds Vα such that each Vα is invariant under ψt and contains exactly one stationary point that attracts Vα . These definitions may not be standard and do not include all possible patterns of stationary points, but they are sufficient for our purpose here. The stochastic flow gt on X exhibits the following clustering pattern at large time t: gt (ω) sweeps Γ(ω), an open dense subset of X, into a collection of “moving” points, and these points form a subset of X that is an isometric image of the set of attracting stationary points of (H + )∗ . Therefore, it is important to know the set of attracting stationary points of (H + )∗ . This information is provided below. Theorem 20 The set of stationary points of (H + )∗ on X is π(M ′ ) and the set of attracting stationary points is π(M ).

10.

Nonhomogeneous L´evy Processes in Lie Groups

In the definition of a L´evy process in a Lie group, if one drops the requirement of stationary increments, one obtains a more general process, called a nonhomogeneous L´evy process. Thus, a process xt in a Lie group G with rcll paths is called a nonhomogeneous L´evy process if for s < t, its increment x−1 s xt is independent of process up to time s. The distributions µs,t of the increments x−1 s xt , s ≤ t, form a two-parameter convolution semigroup in the sense that for s < t < u, µs,t ∗ µt,u = µs,u , which is continuous in the sense that µs,t → µs,s = δe weakly as t ↓ s. In fact, a nonhomogeneous L´evy process in G may be defined as a process xt with rcll paths such that for any 0 = t0 < t1 < t2 < · · · < tn and f ∈ Cb (Gn+1 ), Z E[f (xt0 , xt1 , xt2 . . . , xtn )] = f (x0 , x0 x1 , x0 x1 x2 , . . . , x0 x1 · · · xn ) µ0 (dx0 )µ0,t1 (dx1 )µt1 ,t2 (dx2 ) · · · µtn−1 ,tn (dxn(34) )

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for a probability measure µ0 (initial distribution) and a continuous two-parameter convolution semigroup µs,t on G with µs,s = δe . By the classical L´evy -Itˆo representation (see for example Theorem 15.4 in [19]), a nonhomogeneous L´evy process in Rn with no fixed jumps is a sum of a continuous drift b(t), a continuous Gaussian process with independent increments, and an independent jump process driven by a Poisson random measure N . The distribution of the L´evy process is determined by b(t), the covariance matrix function Aij (t) of G(t) and the characteristic measure Π of N . A similar representation is obtained by Feinsilver [9] for nonhomogeneous L´evy processes in a general Lie group in the form of a martingale representation. Let ξ1 , . . . , ξn be a basis of the Lie algebra g of G and let φ1 , . . . , φn ∈ Cc∞ (G) be associated exponential coordinates. A covariance function A is a continuous n × n symmetric matrix valued function such that A(0) = 0 and for s < t, A(t) − A(s) is nonnegative definite. A L´evy measure function Π(t, ·) is a measure valued function on G such that Π(0, ·) = 0, Π(t, {e}) = 0 and for f ∈ Cb∞ (G) with f (e) = ξi f (e) = 0, Π(t, f ) is finite and continuous in t. Let xt be a nonhomogeneous L´evy process in G with x0 = e. It is called stochastic continuous (or to have no fixed jumps) if xt = xt− almost surely for each fixed t. Note that a (homogeneous) L´evy process is automatically stochastic continuous. The following result is proved in [9], see also [25]. Theorem 21 Let xt be a stochastic continuous nonhomogeneous L´evy process in a Lie group G with x0 = e. Then there are unique G-valued continuous function bt with b0 = e, covariance function A and L´evy measure function Π, such that xt = zt bt and for f ∈ Cc∞ (G), f (zt ) − −

Z tZ 0

Z

0

t

G

{f (zs bs τ b−1 s ) − f (zs ) −

X

φi (τ )[Ad(bs )ξi ]l f (zs )}Π(ds, dτ )

i

1X [Ad(bs )ξi ]l [Ad(bs )ξj ]l f (zs ) dAij (s) 2

(35)

i,j

is a martingale. Moreover, given (b, A, Π) as above, there is a rcll process xt = zt bt in G with x0 = e such that (35) is a martingale for f ∈ Cc∞ (G). Furthermore, such a process xt is unique in distribution and is a stochastic continuous nonhomogeneous L´evy process in G. Proof We provide an outline of the proof, see [9] for the details. Fix T > 0. Let 0 = t0 < t1 < · · · < tn ≤ T be a partition of [0, T ]Rwith ti+1 − ti = 1/n and let µni = µti−1 ,ti for 1 ≤ i ≤ n. Define bni ∈ G by φi (bni ) = φi dµni . By the stochastic continuity of xt , bni is uniformly small (close to e) in i as n → ∞. For each n > 1, define a G-valued function by bn (t) = bn1 bn2 · · · bn[nt] for 0 < t ≤ T with bn (0) = e, where [nt] is the integer part of nt, a measure function Πn by Πn (t, ·) = P[nt] i=1 µni for 0 < t ≤ T with Π(0, ·) = 0, and a matrix valued function An (t, U ) of time t

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and measurable U ⊂ G by An (0, U ) = 0 and for 0 < t ≤ T , [An (t, U )]ij =

[nt] Z X k=1

U

[φi (x) − φi (bnk )][φj (x) − φj (bnk )]µnk (dx).

Let xni , 1 ≤ i ≤ n, be independent random variables in G with distributions µni . De−1 −1 fine xn (t) = xn1 xn2 · · · xn[nt] , yn (t) = xn1 b−1 n1 · · · xn[nt] bn[nt] and zn (t) = xn (t)bn (t) . Then as n → ∞, the process xn (t) converges in distribution to xt , and it can be shown that bn (t) converges uniformly to a continuous function bt , Πn converges to a L´evy measure function Π in the sense that for any f ∈ Cb (G) vanishing near e, Πn (t, f ) → Π(t, f ) uniformly for 0 ≤ t ≤ T , and for Rany neighborhood U of e which is a continuity set of Π(T, ·), [An (t, U )]ij → Aij (t) + U φi (x)φj (x)Π(t, dx) uniformly for 0 ≤ t ≤ T , for some covariance function Aij (t). Moreover, yn (t) converges in distribution to a stochastic continuous process yt such that for any f ∈ Cc∞ (G), f (yt ) − −

Z tZ 0

G

[f (ys y) − f (ys ) −

Z 1 tX 2

0

X

φj (y)ξjl f (ys )]Π(ds, dy)

j

ξil ξjl f (ys )dAij (s)

(36)

i,j

is a martingale, and zn (t) converges in distribution to a stochastic continuous process zt for which (35) is a martingale. Thus, (b, A, Π) are the parameters in the representation of xt .  Note that if the Lie group G is commutative such as G = Rn , then yt = zt and the martingale representation (35) takes the simpler form (36). In this case, bt is the continuous drift, Aij (t) is the covariance of the continuous Gaussian process and Π is the characteristic measure of the Poisson process of jumps, that is, for any t ≥ 0 and B ⊂ G, Π(t, B) is the expected number of jumps contained in B by time t. On a general Lie group G, bt may be regarded as a drift and Aij (t) as governing the continuous part of the L´evy process xt , and Π is still the characteristic measure of the Poisson random measure that counts the jumps of the process. In particular, xt is continuous if and only if Π = 0.

11.

Nonhomogeneous L´evy Processes in Homogeneous Spaces

Let G be a Lie group and H be a compact subgroup. As before, o = eH is the origin of G/H and π: G → G/H is the natural projection. Recall a Borel measurable map S: G/H → G is called a section map if π ◦ S = idG/H , and the convolutions of Hinvariant measures on G/H are defined using a section map S but are independent of S. A point b ∈ G/H or a subset B of G/H is called H-invariant if hb = b or hB = B for all h ∈ H. For x ∈ G/H, and H-invariant b ∈ G/H and B ⊂ G/H, xb = S(x)b and xB = S(x)B are well defined because they are independent of S. Note that g ∈ G with go H-invariant is characterized by g −1 Hg ⊂ H, and hence by g −1 Hg = H. Therefore, the set of H-invariant points in G/H is the natural projection of a closed subgroup

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of G containing H as a normal subgroup, and hence has a natural group structure with product b1 b2 = S(b1 )b2 and inverse b−1 = S(b)−1 o (independent of S). In general, the product x1 x2 · · · xn−1 xn = S(x1 )S(x2 ) · · · S(xn−1 )xn is not well defined because it depends on the choice of S. However, if x1 , x2 , . . . , xn are independent random variables and x2 , . . . , xn have H-invariant distributions, then the distribution of the product x1 x2 · · · xn and also that of the sequence yi = x1 x2 · · · xi for i = 1, 2, . . . are independent of S, and hence such a product or sequence is meaningful in the sense of distribution. Note that for H-invariant finite measures µ and ν on G/H, an integral like Z Z f (xy, xyz)µ(dy)ν(dz) = f (S(x)y, S(x)S(y)z)µ(dy)ν(dz)

R is well defined (independent of choice of section map S). So is f (xbyb−1 )µ(dy) if b is an H-invariant point in G/H. A process xt in G/H with rcll paths will be called a nonhomogeneous L´evy process if there is a continuous two-parameter convolution semigroup µs,t of H-invariant probability −1 measures on G/H such that (34) holds. Then for s < t, x−1 s xt = S(xs ) xt has distribution µs,t (independent of choice for section map S) and is independent of the process up to time s. Thus, a nonhomogeneous L´evy process in G/H may be characterized as a rcll process with independent increments. Because H is compact, there is a subspace p of g that is complementary to the Lie algebra h of H and is Ad(H)-invariant in the sense that Ad(h)p = p for h ∈ H. Choose ∞ a basis ξ1 , . . . , ξm of Ppmand local coordinates φ1 , . . . , φm ∈ Cc (G/H) around o on G/H such that x = exp( i=1 φi (x)ξi )o for x near o. Then by (2.16) in [24], ∀h ∈ H,

m X i=1

φi Ad(h)ξi =

m X i=1

(φi ◦ h)ξi

(37)

near o. The functions φi may be suitably extended so that (37) holds globally on G/H. Any ξ ∈ g is a left invariant vector field on G. If ξ is Ad(H)-invariant, it may also be d regarded as a vector field on G/H given by ξf (x) = dt f (xetξ o) |t=0 for f ∈ C ∞ (G/H) tξ and x ∈ G/H (note that e o is H-invariant), which is G-invariant in the sense that ξ(f ◦ g) = (ξf ) ◦ g for g ∈ G. In fact, any G-invariant vector field on G/H is given by a unique Ad(H)-invariant ξ ∈ p. Note that if ξ ∈ g is Ad(H)-invariant and b ∈ G/H is H-invariant, then Ad(b)ξ = Ad(S(b))ξ is Ad(H)-invariant independent of section map S. By R and is P (37), Rfor any H-invariant measure µ on G/H, µ(dx) i φi (x)ξi is Ad(H)-invariant, and R P P so is µ(dx) i φi (x)Ad(b)ξi = Ad(b) µ(dx) i φi (x)ξi . Let ξ, η ∈ g. With a choice of section map S, ξη may be regarded as a second order 2 differential operator on G/H by setting ξηf (x) = ∂t∂ ∂s f (S(x)etξ esη o) |t=s=0 . As in [9], P 2 ij it can be shown that ξi ξj f (x) = ∂t∂∂s f (S(x)etξi +sξj o) |t=s=0 + m k=1 ρk ξk f (x) with ji ∞ ρij k = −ρk . Thus, if aij is a symmetric matrix, then for f ∈ C (G/H), m X

i,j=1

aij ξi ξj f (x) =

m X

i,j=1

aij

Pm ∂2 f (S(x)e p=1 tp ξp o) |t1 =···=tm =0 . ∂ti ∂tj

(38)

P The matrix aij is called Ad(H)-invariant if aij = p,q apq [Ad(h)]ip [Ad(h)] Pjq for h ∈ H, where [Ad(h)]ij is the matrix representing Ad(h), that is, Ad(h)ξj = i [Ad(h)]ij ξi .

L´evy Processes in Lie Groups and Homogeneous Spaces 375 P Then the operator i,j aij ξi ξj is independent of section map S and is G-invariant. In fact, any second order G-invariant differential operator on G/H without constant term is such an vector field. Note that if b ∈ G/H is H-invariant, then P operator plus a G-invariantP i,j aij [Ad(S(b))ξi ][Ad(S(b))ξj ] is a G-invariant operator i,j aij [Ad(b)ξi ][Ad(b)ξj ] = on G/H (independent of S). A covariance function A and a L´evy measure function Π on G/H are defined as on G with the additional requirements that A(t) is Ad(H)-invariant and Π(t, ·) is H-invariant. By the preceding discussion, the expression in (35) is meaningful on G/H and is independent of the choice of section map S in Ad(bs ) = Ad(S(bs )). The following result is an extension of Feinsilver’s martingale representation to nonhomogeneous L´evy processes in G/H. Theorem 22 Let xt be a stochastic continuous nonhomogeneous L´evy process in G/H with x0 = o. Then there is a unique triple (b, A, Π) of a continuous function bt with b0 = o, taking H-invariant values in G/H, a covariance function A and a L´evy measure function Π on G/H such that xt = zt bt and (35) is a martingale for f ∈ Cc∞ (G/H). Moreover, given (b, A, Π) as above, there is a rcll process xt = zt bt in G/H with x0 = o and represented by (b, A, Π) as above. Furthermore, such a process xt is unique in distribution and is a stochastic continuous nonhomogeneous L´evy process in G/H. Proof Proceed as in the proof of Theorem 21, but note that µni is H-invariant and hence bni is H-invariant. The proof in [9] on G can be suitable modified to work on G/H, such as properly interpreting the product on G/H as discussed here and using H-invariant sets on G/H for various neighborhoods used in [9].  Remark 1 (L´evy measure): The L´evy measure Π is associated to the jumps of the L´evy process xt in G/H as in G, and hence xt is continuous if and only if Π = 0. Remark 2 (drift): Because the natural action of H on G/H fixes o, it induces an action on the tangent space To (G/H) at o. If there is no nonzero H-invariant tangent vector in To (G/H), then in a neighborhood of o, there is no H-invariant point except o. In this case, the drift bt in Theorem 22 must be trivial, that is, bt = o for all t ≥ 0. Note that the existence of nonzero H-invariant vector in To (G/H) is equivalent to the existence of nonzero Ad(H)-invariant element in p. For example, on the sphere S n−1 = SO(n)/SO(n − 1), there is no nonzero Ad(H)-invariant vector in p, and hence a stochastic continuous L´evy process in S n−1 can only have trivial drift. A measure µ on G is called H-conjugate invariant if ch µ = µ, where ch : G → G is the conjugation map x 7→ hxh−1 . A G-valued random variable is called H-conjugate invariant if its distribution is H-conjugate invariant. Let gt be a nonhomogeneous L´evy process in P G with g0 = e. Recall the coordinate functions P to satisfy P φi on G are chosen x = exp[ i φ(x)ξi ] for x near e. Then for h ∈ H, i (φi ◦ ch )ξi = i φi Ad(h)ξi near e. Because H is compact, φi may be chosen so that this holds globally on G. It can now be shown that gt has H-conjugate invariant increments gs−1 gt for s < t if and only if in the representation (b, A, Π), bt and Π are H-conjugate invariant, and A(t) is Ad(H)invariant. Then by (34) it is easy to show that xt = gt o is a nonhomogeneous L´evy process

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in G/H with x0 = o. A converse of this statement may be obtained as a consequence of Theorem 22. This is similar to Theorem 12 for (homogeneous) L´evy processes. Corollary 1 Let xt be a stochastic continuous nonhomogeneous L´evy process in G/H with x0 = o. Then there is a stochastic continuous L´evy process gt with g0 = e and H-conjugate invariant increments such that the two processes xt and gt o are identical in distribution. Proof Let (b, A, Π) be the representation of xt in TheoremR22. There is a continuous Gvalued function b′t in G with b′0 = e and bt = b′t o. Let ˆbt = H hb′t h−1 dh, where dh is the normalized Haar measure on H. Then ˆbt is a continuous H-conjugate invariant function in G with ˆb0 = e and bt = ˆbt o. The basis ξ1 , . . . , ξm of p may be extended to be a basis ξ1 , . . . , ξn of g such that ξm+1 , . . . , ξn form a basis of h. The covariance function Aij (t) on G/H may be regarded as covariance function on G P with Aij (t) = 0 if either i > m or j > m. Let S be a section map satisfying S(x) = exp[ m i=1 φi (x)ξi ] for x near o, and let R −1 ˆ ˆ ˆ ·) is Π(t, ·) be defined by Π(t, f ) = H f (hS(x)h )dhΠ(t, dx) for f ∈ Cb (G). Then Π(t, a H-conjugate invariant L´evy measure function on G. The nonhomogeneous L´evy process ˆ satisfies xt = gt o in distribution because gt in G with g0 = e and representation (ˆb, A, Π) ˆ ˆ (b, A, Π) on G project to (b, A, Π) on G/H, and has H-conjugate invariant increments ˆ ·) are H-conjugate invariant, and A(t) is Ad(H)-invariant.  because both ˆbt and Π(t,

12.

A Decomposition of a Markov Process

Under the spherical polar coordinates, a Brownian motion xt in Rn (n ≥ 2) may be expressed in terms of its radial part rt = |xt | and angular part θt = xt /rt . It is well known that rt is a Feller process in R+ , called a Bessel process, and θt is a process in the unit sphere S n−1 and is a time changed spherical Brownian motion. This is sometimes called the skew-product decomposition of Brownian motion in Rn . This decomposition is naturally related to the action of the rotation group SO(n), as the Brownian motion xt has an SO(n)-invariant distribution with its radial part rt transversal to the orbits of SO(n) and angular part θt contained in an orbit, namely the unit sphere S n−1 . More generally, it is shown in Galmarino [12] that a continuous Markov process in Rn with an SO(n)-invariant distribution is a skew product of its radial motion and an independent spherical Brownian motion with a time change. In this section, we will consider a general Markov process xt in a smooth manifold X that has a distribution invariant under the smooth action of a Lie group K. Given a submanifold Y transversal to the orbits of K, the radial and angular parts of xt are respectively its projections to Y and to a typical K-orbit. It is easy to show that the radial part is a Markov process in Y . Our main purpose is to study the conditioned angular process given a radial path, and as an application we will obtain an extension of Galmarino’s result to a more general setting by a conceptually more transparent proof. See Liao [25] for more details. Let xt be a Markov process in X with rcll paths and transition semigroup Pt . It is allowed to have a finite life time and thus Pt (x, X) may be less than 1. We will assume the Markov process xt or equivalently its transition semigroup Pt is K-invariant in the sense that ∀f ∈ Cb (X) and k ∈ K, Pt (f ◦ k) = (Pt f ) ◦ k. (39)

L´evy Processes in Lie Groups and Homogeneous Spaces

377

This means that for k ∈ K, kxt is the same Markov process starting at kx0 . Let Y be a submanifold of X, possibly with a boundary, that is transversal to the action of K in the sense that it intersects each orbit of K at exactly one point, that is, ∀y ∈ Y,

(Ky) ∩ Y = {y}

and X = ∪y∈Y Ky.

(40)

Let J: X → Y be the projection map J(x) = y for x ∈ Ky, which is continuous if K is compact. Note that J ◦ k = J for k ∈ K. The following result is easy to prove. Theorem 23 yt = J(xt ) is a Markov process in Y with transition semigroup Qt given by Qt f (y) = Pt (f ◦ J)(y),

y ∈ Y and f ∈ Cb (Y ).

(41)

Moreover, for a compact K, if xt is a Feller process in X with generator L, then so is yt in Y with generator LY given by (LY f ) ◦ J = L(f ◦ J) with domain D(LY ) = {f ∈ C0 (Y ), f ◦ J ∈ D(L)}. The process yt = J(xt ) in Theorem 23 will be called the radial part of process xt (relative to K and Y ). Note that for a diffusion process xt with generator L, the generator LY of yt is the radial part of the differential operator L as defined in [16]. For x ∈ X, let Kx = {k ∈ K; kx = x} be the isotropy subgroup of K at x. Let Y ◦ be Y minus its boundary. We will assume that Ky is the same compact subgroup M of K as y varies over Y ◦ . This assumption is often satisfied when the transversal submanifold Y is properly chosen. We will now strengthen the transversality condition (40) by assuming that ∀y ∈ Y ◦ ,

Ty X = Ty (Ky) ⊕ Ty Y

(direct sum),

(42)

where Ty X is the tangent space of X at y, see Lemma 3.3 in [16, Chapter II]. Then the union of the K-orbits through Y ◦ , denoted by X ◦ , is an open dense subset of X, and X ◦ = Y ◦ × (K/M ) as a product manifold. All our assumptions are satisfied in the following examples. Example 1: We have mentioned earlier that the radial part of a Brownian motion xt in X = Rn (n ≥ 2), under the action of K = SO(n), is a Bessel process in a fixed ray Y from the origin. We may take Y to be the positive half of x1 -axis, which is transversal to K with boundary containing only the origin. Then M = diag{1, SO(n − 1)}. Example 2: Let X be the space of n×n real symmetric matrices (n ≥ 2) with K = SO(n) acting on X by conjugation. The set Y of all n × n diagonal matrices with non-ascending diagonal elements is a submanifold of X transversal to SO(n), its boundary consists of diagonal matrices with at least two identical diagonal elements, and M is the finite subgroup of SO(n) consisting of diagonal matrices with ±1 along diagonal. The map J: X → Y maps a symmetric matrix to the diagonal matrix of its eigenvalues in non-ascending order. Note that X = GL(n, R)/SO(n). Example 3: Let Y be a manifold and K be a Lie group with a compact subgroup M , and let X = Y × (K/M ) as a product manifold. Then K acts on X as its natural action

378

Ming Liao

on K/M , Y is transversal to K and M is the isotropy subgroup of K at all y ∈ Y . For example, X = Rn+m = Y × K with Y = Rn , K = Rm (additive group) and M = {0}. Example 4: Let X = S n be the n-dimensional sphere, regarded as the unit sphere in Rn+1 , under the natural action of K = diag{1, SO(n)}. The half circle Y connecting two poles (±1, 0, . . . , 0), given by (cos t, sin t, 0, . . . , 0) for 0 ≤ t ≤ π, is transversal to K, and M = diag{1, 1, SO(n − 2)}. Example 5: Let X = G/K be a symmetric space of noncompact type. Using the notation in Section 8 with a+ as the fixed Weyl chamber with closure a+ and boundary ∂a+ . Y = exp(a+ ) is a submanifold of X transversal to the action of K on G/K with boundary exp(∂a+ ), and the isotropy subgroup M of K at any y ∈ Y ◦ = exp(a+ ) is the centralizer M of a in K. The exit time ζ of process xt from X ◦ is the stopping time when xt together with its left limit first leaves X ◦ or reaches its life time ξ. More precisely, it is defined by ζ = inf{t > 0;

xt 6∈ X ◦ , xt− 6∈ X ◦ or t ≥ ξ},

(43)

with inf of an empty set defined to be ∞. The exit time of yt from Y ◦ is also denoted by ζ. Fix T > 0. Because the process xt has rcll paths, it may be regarded as a random variable in the space DT (X) of rcll maps: [0, T ] → X, equipped with Skorohod topology. Let Px be the distribution on DT (X) associated to the process xt starting at x ∈ X. Its total mass may be less than 1 because xt may have a finite life time. For y ∈ Y ◦ and z ∈ K/M , zy = S(z)y ∈ X ◦ is well defined and is independent of choice of section map S: K/M → K. Let xt = zt yt be the decomposition of the process xt with x0 ∈ X ◦ and t < ζ. Then yt is the radial part as defined before, and zt is a process in K/M with rcll paths and will be called the angular part of xt . Recall J is the projection map X ∋ x 7→ y ∈ Y . Let J2 be the projection map X ◦ ∋ x 7→ z ∈ K/M associated to the decomposition x = zy. We will also use J and J2 to denote the maps J: DT (X) ∋ x(·) 7→ y(·) ∈ DT (Y ) and J2 : DT (X ◦ ) ∋ x(·) 7→ z(·) ∈ DT (K/M ) respectively given by the decomposition x(·) = z(·)y(·). Y Let F0,T = σ{yt ; 0 ≤ t ≤ T } be the σ-algebra generated by the radial process yt for 0 ≤ t ≤ T , which may be regarded as a σ-algebra on DT (Y ) and induces the σY ) on D (X). By the existence of regular conditional distributions (see algebra J −1 (F0,T T y(·)

for example [19, chapter 5]), there is a probability kernel Rz from DT (Y ◦ ) × (K/M ) to DT (K/M ) such that for any x ∈ X ◦ and measurable F ⊂ DT (K/M ), J[x(·)]

Y RJ2 (x) (F ) = Px [J2−1 (F ) | J −1 (F0,T )] for Px -almost all x(·) in [ζ > T ] ⊂ DT (X ◦ ). (44) y(·) The probability measure Rz is the conditional distribution of the angular process zt given a radial path y(·) in DT (Y ◦ ) and z0 = z.

Theorem 24 Fix T > 0. Almost surely on [ζ > T ], given a radial path yt for 0 ≤ t ≤ T , the conditioned angular process zt is a nonhomogeneous L´evy process in K/M . More precisely, this means that for y ∈ Y ◦ , z ∈ K/M , and JPy -almost all y(·) in [ζ > T ] ⊂ y(·) DT (Y ◦ ), the angular process zt is a nonhomogeneous L´evy process under Rz .

L´evy Processes in Lie Groups and Homogeneous Spaces

379

Proof For x ∈ X ◦ , let P˜t (x, B) = Px {[xt ∈ B]∩[ζ > t]} for measurable B ⊂ X ◦ . By the Markov property of xt , it is easy to show that P˜t is the transition semigroup of the Markov ˜ t be the transition semigroup process xt for t < ζ and it is K-invariant. Similarly, let Q ˜ t (y, ·) are respectively sub-probability kernels from of yt for t < ζ. Then P˜t (x, ·) and Q ◦ ◦ ◦ X to X = Y × (K/M ) and from Y ◦ to Y ◦ . By the existence of a regular conditional distribution, there is a probability kernel Rt (y, y1 , ·) from (Y ◦ )2 to K/M such that for ˜ t (y, dy1 )Rt (y, y1 , dz1 ). The K-invariance of P˜t implies y ∈ Y ◦ , P˜t (y, dy1 × dz1 ) = Q ˜ t (y, ·)-almost all y1 . Modifying Rt on that the measure Rt (y, y1 , ·) is M -invariant for Q ˜ t (y, ·)-measure, we may assume Rt (y, y1 , ·) is M -invariant for an exceptional set of zero Q ◦ all y, y1 ∈ Y . Therefore, for z ∈ K/M , it is meaningful to write Rt (y, y1 , z −1 dz1 ) = Rt (y, y1 , S(z)−1 dz1 ) because it is independent of choice of section map S. We then have ∀y ∈ Y ◦ and z ∈ K/M,

˜ t (y, dy1 )Rt (y, y1 , z −1 dz1 ). P˜t (zy, dy1 × dz1 ) = Q

(45)

It follows that for 0 < s1 < s2 < · · · < sk < ∞, y ∈ Y ◦ , z ∈ K/M and f ∈ Cb ((K/M )k ), Z Ezy [f (zs1 , . . . , zsk ) | ys1 , . . . , ysk ] = Rs1 (y, ys1 , dz1 )Rs2 −s1 (ys1 , ys2 , dz2 ) · · ·

Rsk −sk−1 (ysk−1 , ysk , dzk )f (zz1 , zz1 z2 , . . . , zz1 · · · zk )](46)

on [ζ > sk ]. Let Γ be the set of dyadic numbers i/2m for integers i ≥ 0 and m > 0. May assume T ∈ Γ. For s, t ∈ Γ with s < t ≤ T , let s = s1 < s2 < · · · < sk be a partition of [0, T ] spaced by 1/2m with s = si and t = sj , and let µm s,t = Rsi+1 −si (ysi , ysi+1 , ·) ∗ Rsi+2 −si+1 (ysi+1 , ysi+2 , ·) ∗ · · · ∗ Rsj −sj−1 (ysj−1 , ysj , ·). By (46), M -invariance of Pt (y, y1 , ·) and the measurability of µm s,t in ysi , . . . , ysj , −1 −1 µm s,t (f ) = Ezy [f (zs zt ) | ys1 , . . . , ysk ] = Ezy [f (zs zt ) | ysi , . . . , ysj ] on [ζ > T ] (47)

for f ∈ Cb (K/M ), which is independent of the choice for section map S to represent Y and zs−1 zt = S(zs )−1 zt . By the right continuity of yt , as m → ∞, σ{ys1 , . . . , ysk } ↑ F0,T Y , it follows that as m → ∞, almost surely, µm → µ σ{ysi , . . . , ysj } ↑ Fs,t s,t weakly for s,t some M -invariant probability measure µs,t on K/M such that Y Y µs,t (f ) = Ezy [f (zs−1 zt ) | F0,T ] = Ezy [f (zs−1 zt ) | Fs,t ] on [ζ > T ]. (48) Y Note that µs,t is an Fs,t -measurable random measure independent of starting point zy. Because Γ is countable, the exceptional set of probability zero in the above almost sure convergence may be chosen simultaneously for all s < t in Γ. Moreover, for t1 < t2 < · · · < tn of [0, T ] in Γ, it can be shown from (46) and by choosing a partition s1 < s2 < · · · < sk of [0, T ] from Γ containing all ti , spaced by 1/2m , that almost surely on [ζ > T ], for f ∈ Cb ((K/M )n ),

∀f ∈ Cb (K/M ),

Y Ezy [f (zt1 , . . . , ztn ) | F0,T ] = lim Ezy [f (zt1 , . . . , ztn ) | ys1 , . . . , ysk ] m→∞ Z m m = lim f (zz1 , zz1 z2 , . . . , zz1 · · · zn )µm 0,t1 (dz1 )µt1 ,t2 (dz2 ) · · · µtn−1 ,tn (dzn ). m→∞

380

Ming Liao

This implies that almost surely on [ζ > T ], for 0 ≤ t1 < · · · < tn ≤ T in Γ, Y Ezy [f (zt1 , . . . , ztn ) | F0,T ] Z = f (zz1 , zz1 z2 , . . . , zz1 · · · zn )µ0,t1 (dz1 )µt2 ,t1 (dz2 ) · · · µtn−1 ,tn (dzn ). (49)

In particular, µs,t for s < t in Γ form a two-parameter convolution semigroup on K/M . It can be extended to all real s < t ≤ T and for which (49) holds for real 0 ≤ t1 < · · · < tn ≤ T . See [25] for more details.  Remark It is clear that given a constant radial path yt ≡ y, the angular process zt becomes a (homogeneous) L´evy process in K/M under the conditional distribution. One may obtain a L´evy process in K/M by forcing the process xt to run in the orbit Ky. More precisely, starting at y ∈ Y ◦ , run the process xt for a small time ε > 0 and project paths to the orbit Ky via J2 (using the natural identification of Ky with K/M ), then at the end of each projected path, run xt again for ε and project to Ky. Repeat the procedure to obtain a process ztε in Ky ≡ K/M . It can be shown that as ε → 0, the process ztε converges to a L´evy process zt in K/M in the sense of finite dimensional distribution, see [26] for more details. It is interesting to compare this “forced” process with the conditioned angular process given a constant radial path y. It turns our that they are not always equal in distribution. Because the natural action of M on K/M fixes o, it induces an action on the tangent space To (K/M ) at o. The homogeneous space K/M will be called irreducible if the action of M on To (K/M ) is irreducible (that is, it has no nontrivial invariant subspace). In this case, there is no nonzero M -invariant tangent vector in To (K/M ), and hence a nonhomogeneous L´evy process in K/M has only trivial drift, see Remark 2 in Section 11. Among the examples mentioned earlier, K/M is irreducible in Examples 1 and 4, and in Example 3 if it is chosen to be so, and in Example 5 if the symmetric space G/K is of rank 1 (see [16]). If K/M is irreducible, then, up to a constant multiple, there is a unique M -invariant inner product on To (K/M ) (see for example Appendix 5 in [20]). By choosing a Kinvariant Riemannian metric on K/M , which is unique up to a constant factor, any second order K-invariant differential operator on K/M is a multiple of the Laplace operator ∆K/M on K/M . The following result is an extension of Galmarino’s result mentioned earlier. Theorem 25 Assume K/M is irreducible. If xt is a continuous K-invariant Markov process in X with radial part yt in Y , then there are a Brownian motion B(t) in K/M under a K-invariant Riemannian metric, independent of process xt , and a real continuous nonY -measurable for s < t, such that decreasing process at , with a0 = 0 and at − as being Fs,t the two processes xt and B(at )yt , t < ζ, are identical in distribution. Proof By Theorem 24, given a radial path yt for 0 ≤ t ≤ T with [ζ > T ], the conditioned angular process zt is a continuous nonhomogeneous L´evy process in K/M . Let (b, A, Π) be its representation in Theorem 22. PBy the irreducibility of K/M , the drift bt = o. Because zt is continuous, Π = 0. Because i,j Aij (t)ξi ξj is a K-invariant second order differential operator on K/M , it must be equal to at ∆K/M for a continuous non-decreasing function Y -measurable. Then from the construction of A(t) from at with a0 = 0. By (48), µs,t is Fs,t

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Y -measurable. Let B(t) be a µs,t in the proof of Theorem 22, it is seen that at − as is Fs,t Brownian motion in K/M independent of process xt , and hence independent of process at . It is enough to show that the conditioned process zt is equal to the Brownian R t time-changed 1 motion B(at ) in distribution. For f ∈ Cb (K/M ), f (B(t)) − 0 2 ∆K/M f (B(s))ds is a Rt martingale, and hence, f (B(at )) − 0 21 ∆K/M f (B(as ))das is a martingale (here at is nonrandom given a radial path). On the other hand, for the conditioned process zt , f (zt ) − Rt 1 ∆ 0 2 K/M f (zs )das is a martingale. The uniqueness of the process in distribution with the given representation (b, A, Π) implies that zt = B(at ) in distribution. 

References [1] Applebaum, D., “Compound Poisson processes and L´evy processes in groups and symmetric spaces”, J. Theo. Probab. 13, 383-425 (2000). [2] Applebaum, D. and Kunita, H. (1993) “L´evy flows on manifolds and L´evy processes on Lie groups”, J. Math. Kyoto Univ. 33, 1105-1125. [3] Arnold, L. (1998) “Random dynamical systems”, Springer-Verlag. [4] Babillot, M. (1991) “Comportement asymptotique due mouvement Brownien sur une vari´et´e homog`ene a` courbure n´egative ou nulle”, Ann. Inst. H. Poincar´e (prob et Stat) 27, 61-90. [5] Br¨ocker, T. and Dieck, T. (1985) “Representations of compact Lie groups”, SpringerVerlag. [6] Carverhill, A.P. (1985) “Flows of stochastic dynamical systems: ergodic theory”, Stochastics 14, 273-317. [7] Diaconis, P. (1988) “Group representations in probability and statistics”, IMS, Hayward, CA. [8] Dynkin, E.B. (1961) “Nonnegative eigenfunctions of the Laplace-Betrami operators and Brownian motion in certain symmetric spaces”, Dokl. Akad. Nauk SSSR 141, 1433-1426. [9] Feinsilver, P., “Processes with independent increments on a Lie group”, Trans. Amer. Math. Soc. 242, 73-121 (1978). [10] Furstenberg, H. (1963) “Noncommuting random products”, Trans. Am. Math. Soc. 108, 377-428. [11] Furstenberg, H. and Kesten, H. (1960) “Products of random matrices”, Ann. Math. Statist. 31, 457-469. [12] Galmarino, A.R., “Representation of an isotropic diffusion as a skew product”, Z. Wahr 1, 359-378 (1963).

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[13] Gangolli, R. (1964) “Isotropic infinitely divisible measures on symmetric spaces”, Acta Math. 111, 213-246. [14] Guivarc’h, Y. and Raugi, A. (1985) “Fronti`ere de Furstenberg, propri´et´es de contraction et convergence”, Z. Wahr. Gebiete 68, 187-242. [15] Helgason, S. (1978) “Differential geometry, Lie groups, and symmetric spaces”, Academic Press. [16] Helgason, S. (2000) “Groups and geometric analysis”, Amer. Math. Society. [17] Heyer, H. (1977) “Probability measures on locally compact groups”, Springer-Verlag. [18] Hunt, G.A. (1956) “Semigroups of measures on Lie groups”, Trans. Am. Math. Soc. 81, pp 264-293. [19] Kallenberg, O. (2002) “Foundations of modern probability, second edition”, Springer. [20] Kobayashi, S. and Nomizu, K. (1963) “Foundations of differential geometry, vol I”, Interscience Publishers. [21] Liao, M. (1994) “The Brownian motion and the canonical stochastic flow on a symmetric space”, Trans. Amer. Math. Soc. 341, 253-274. [22] Liao, M. (1998) “L´evy processes in semi-simple Lie groups and stability of stochastic flows”, Trans. Amer. Math. Soc. 350, 501-522. [23] Liao, M. (2004) “L´evy processes and Fourier analysis on compact Lie groups”, Ann. Probab. 32, 1553-1573. [24] Liao, M. (2004) “L´evy processes in Lie group”, Cambridge Univ. Press. [25] Liao, M. (2008) “A decomposition of Markov processes via group actions”, to appear in J. Theo. Probab. [26] Liao, M. (2008) “Markov processes invariant under a Lie group action”, to appear in Stoch. Processes and their Appl. [27] Liao, M. and Wang, L. (2007) “Levy-Khinchin formula and existence of densities for convolution semigroups on symmetric spaces”, Potential Analysis 27, 133-150. [28] Malliavin, M.P. and Malliavin, P. (1974) “Factorizations et lois limites de la diffusion horizontale audessus d’un espace Riemannien symmetrique”, Lecture Notes Math. 404, 164-271. [29] Norris, J.R., Rogers, L.C.G. and Williams, D. (1986) “Brownian motion of ellipsoids”, Trans. Am. Math. Soc. 294, 757-765. [30] Orihara, A. (1970) “On random ellipsoids”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17, 73-85.

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[31] Raugi, A. (1997) “Fonctions harmoniques et th´eor`emes limites pour les marches gl´eatoires sur les groupes”, Bull. Soc. Math. France, m´emoire 54. [32] Rosenthal, J.S. (1994) “Random rotations: characters and random walks on SO(N)”, Ann. of Probab. 22, pp 398-423. [33] Siebert, E. (1981) “Fourier analysis and limit theorems for convolution semigroups on a locally compact groups”, Adv. in Math. 39, 111-154. [34] Taylor, J.C. (1988) “The Iwasawa decomposition and the limiting behavior of Brownian motion on a symmetric space of non-compact type”, in Geometry of random motion, ed. by R. Durrett and M.A. Pinsky, Contemp. Math. 73, Am. Math. Soc., 303-332. [35] Taylor, J.C. (1991) “Brownian motion on a symmetric space of non-compact type: asymptotic behavior in polar coordinates”, Can. J. Math. 43, 1065-1085. [36] Tutubalin, V.N. (1965) “On limit theorems for a product of random matrices”,Theory Probab. Appl. 10, 25-27. [37] Virtser, A.D. (1970) “Central limit theorem for semi-simple Lie groups”, Theory Probab. Appl. 15, 667-687.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 385-399

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 14

S YMMETRY C LASSIFICATION OF D IFFERENTIAL E QUATIONS AND R EDUCTION T ECHNIQUES Giampaolo Cicogna∗ Dipartimento di Fisica “E.Fermi” dell’Universit`a di Pisa and Istituto Nazionale di Fisica Nucleare, Sez. di Pisa Largo B. Pontecorvo 3, Ed. B-C, I-56127, Pisa, Italy

Abstract The symmetry classification of differential equations containing arbitrary functions can be a source of several interesting results. We study two particular but significant examples: a nonlinear ODE and a linear PDE (the 1-dimensional Schr¨odinger equation). We provide first of all a necessary, but very restrictive, simple condition involving the arbitrary functions in order that the given equation does admit Lie symmetries. In the first example, we show that some symmetry appears only if a precise numerical relation between the involved parameters is satisfied. In the case of Schr¨odinger equation, we see that only for a very limited class of potential functions some symmetry is admitted, and that the Lie generators of these symmetries are precisely recursion operators and are related to the Dirac step up - step down operators, well known in Quantum Mechanics. In connection with all these symmetries, we also discuss the important problem of the reduction of the differential equations, in both the different contexts of ODE’s and of PDE’s.

1.

Introduction

The symmetry analysis of differential equations containing arbitrary functions, i.e. the problem of discovering how the symmetry properties of the given equation depend on the choice of these functions, and thus classifying all possible cases, is in general a not easy task [9, 17, 21, 26, 34, 35, 44]. It can be also a source of several interesting, sometimes surprising, results. A striking and well known example is provided by the nonlinear Laplace ∗

E-mail address: [email protected]

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equation ∇2 u = g(u), with u = u(x, y) (or also the equation uxx − uyy = g(u)), where one finds an infinite dimensional algebra of Lie point-symmetries if g(u) = exp(±u) (the Liouville equation), and only relatively trivial symmetries (essentially, scaling symmetries), or no symmetry at all, for any other choice of g(u) [11, 13, 18, 23, 37]. In the following, we shall study two particular but significant examples: a nonlinear ODE, which is related to several problems of mathematical and physical interest, and respectively a linear PDE, the 1-dimensional Schr¨odinger equation. We shall provide first of all a necessary, but very restrictive, simple condition involving the arbitrary functions in order that the given equation does admit Lie symmetries. In our first example, where arbitrary functions of both the independent and the dependent variables are considered, we shall show that some symmetry appears only if some precise numerical relations between the involved parameters are satisfied. In the case of Schr¨odinger equation, we shall see that only for a very limited class of potential functions some symmetry is admitted, and that the Lie generators of these symmetries are precisely recursion operators [29] and are related to the Dirac step up - step down operators [12], well known in Quantum Mechanics. In connection with all these symmetries, we shall also discuss the important problem of the reduction of the differential equations, in both the (substantially) different contexts of ODE’s and of PDE’s. It should be stressed that we will not consider here the possible presence of discrete symmetries of our equations: although they may “interact” with Lie symmetries and produce interesting consequences [16,19], their determination requires in general some specific procedures. Therefore, for the sake of definiteness, we prefer to restrict here our attention only to continuous (Lie) symmetries.

2.

Case 1: A Nonlinear ODE

We consider the following relatively simple (but far from trivial, as we shall see) example of a quasi-linear ODE for the unknown function u = u(x) uxx + f (x)g(u) = 0

(1)

and look for the Lie point-symmetries which are admitted by this equation depending on the choice of the two functions f = f (x) and g = g(u). It can be worth pointing out that more general equations of the form a urr + ur + f (r)g(u) = 0 r

,

u = u(r),

a = const

can be transformed into the above equation (1) simply putting x = |r|1−a if a 6= 1, and x = log |r| if a = 1 (notice that usually in these equations, as suggested by the notation, r is a radial variable, r ≥ 0); as an example, we quote the classical Bratu equation [6]: 1 urr + ur + rk exp u = 0 r

,

k = const .

Similarly, the ODE’s which are obtained when looking for radial solutions u = u(r) to several PDE’s are of the above form; this is the case for instance of the Grad-Schl¨uter-

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Shafranov equation [8, 41] 1 urr − ur + uzz − r2 g(u) = 0 r

,

u = u(r, z)

which is well known in plasma physics, or other similar equations (see e.g. [38])   1 urr − ur + uzz + r2 exp r2 + u + F (r, z) = 0 . r

Considering then our equation (1), and denoting the generators of the Lie pointsymmetries in the usual form (see e.g. [3, 5, 15, 20, 29, 32, 39]) X = ξ(x, u)

∂ ∂ + ϕ(x, u) ∂x ∂u

one obtains, first of all, from the determining equations that the coefficient functions ξ, ϕ must be of the form ξ = ξ(x) ,

ϕ = A(x) + u B(x) ,

1 with B = (ξx + b), 2

b = const

(2)

with clear notations, whereas there is only one equation which involves the two functions f and g, which is Axx + u Bxx − Bf g + 2ξx f g + ξfx g + Af gu + Bf u gu = 0 . It is convenient to write this equation in the form of a scalar product in R5 (P, G) :=

5 X

Pi (x)Gi (u) = 0

(3)

i=1

with P ≡ (Axx , Bxx , −Bf + 2ξx f + ξfx , A f, B f ) G ≡ (1, u, g, gu , u gu ) .

(4)

Differentiating repeatedly (3) with respect to u, one has Pi Gi = Pi Gi,u = . . . = 0, and one easily concludes that either all Pi (x) = 0, or the functions Gi (u) must be linearly dependent. In the first case, one obtains from (4) and (2) the quite obvious result f (x) =

1 x2

and

X = x

∂ . ∂x

(5)

We can then state the following1 Proposition 1. If f = 1/x2 , equation (1) admits the scaling symmetry X = x ∂/∂x for any g(u). Otherwise, a necessary condition in order that equation (1) does admit Lie pointsymmetries is that there are five constants λi , not all zero, such that some linear conditions λ i Gi = 0 1

(6)

With a little and commonly accepted abuse of language, we will denote by X both the symmetry and its Lie generator.

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exist between the functions Gi defined in (4). The functions Pi which determine the admitted symmetries, span the subspace orthogonal in R5 to the subspace spanned by the Gi . This means that equation (1) can admit symmetries, apart from the trivial case (5) (see also below), only if g(u) satisfies the very simple linear first-order ODE’s given in (6). Notice that g(u) may satisfy one or more equations of this form: e.g., if g = exp(u) or if g = u2 , then respectively one or two conditions as in (6) are satisfied. A condition of linear dependence similar to (6) holds of course also for the functions Pi (x), but this would be less convenient to handle because it involves more than one “unknown” variable (i.e. ξ and f ). Actually, it is simpler to start from (6), and impose the orthogonality condition (3) to each given solution g(u) to (6); observing that Pi depend only on x and Gi only on u (see also [9]), one then obtains directly the Pi , and therefore all possible symmetries of our equation (1). Notice also that not all choices for the constants λi in (6) are to be considered: e.g., λ3 = λ4 = λ5 = 0 is not admitted; similarly, the case λ1 = λ2 = λ3 = 0 would imply g = const, the choice λ4 = λ5 = 0 corresponds to the (nearly trivial) case where g is a linear function of u. Before enumerating the list of functions f and g with the corresponding admitted symmetries, let us recall the related and extremely relevant property which involves the reducibility of the given ODE to an equation of lower order (cf. e.g. [22]): i) if a given second order ODE admits just one symmetry X, then it can be reduced to a locally equivalent first order ODE introducing “symmetry-adapted” variables y and w, where y is X-invariant, and w is the coordinate “along the flow” of X: Xy = 0

,

Xw = 1;

(7)

these are to be chosen respectively as new independent variable and dependent one: w = w(y); ii) if the ODE admits two symmetries X1 , X2 such that (this will be precisely our case) [X1 , X2 ] = X1

(8)

and with X1 ∨ X2 := ξ1 ϕ2 − ξ2 ϕ1 6= 0, then choosing the new variables y and w according to the conditions X1 y = 0 ,

X1 w = 1

,

X2 y = y

,

X2 w = w

(9)

the equation is reduced to a locally equivalent equation of the form y wyy = F (wy )

(10)

which can be solved by quadratures. The full list of all cases and subcases of the possible symmetries admitted by equation (1) may be probably annoying and scarcely interesting. We will give only the most significant examples, useful for illustrating the present discussion. The interested reader will have no difficulty at all to complete our list with the few remaining possibilities.

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1.1) Let us start with the simplest and before mentioned case, where f = 1/x2 , with arbitrary g(u). Keeping fixed f = 1/x2 , this would correspond to symmetries of the kernel group of the full group, see [9, 32]. Thanks to the symmetry X = x∂/∂x, introducing the variables y = u, w = log(|x|), equation (1) is transformed into the equation (of the first order in z(y) := wy , as expected) wyy + wy2 = wy3 g(y) . No other symmetry is present in this case for generic g(u), apart from the following exception. 1.2) Let as before f = 1/x2 ; another symmetry arises if g(u) has precisely the special form 2(s + 1) g(u) = us + u (s + 3)2 where s is any constant (s 6= −3). This new symmetry is generated by X = xm+1

m+1 ∂ ∂ + xu ∂x 2 ∂u

,

m=

1−s . 3+s

For instance, with m = 2, let us write the two admitted symmetries as X1 = x3

3 ∂ ∂ + x2 u ∂x 2 ∂u

,

1 ∂ X2 = − x 2 ∂x

in such a way that the commutation rule holds in the form (8). Performing the change of variables as indicated by (9), one has y = u4/3 x−2 , w = y − 1/(2x2 ) and the equation is reduced to the form 12ywyy = 16(wy − 1)3 + 3(wy − 1) in agreement with (10). The special case s = −1, i.e. g(u) = 1/u, may be of interest: the new variables are y = u/x, w = (u − 1)/x and the reduced equation is y wyy = (wy − 1)3 . 2.1) Let now f = 1/xr with r 6= 2. Our equation admits one symmetry if g = us , for any s; its generator is (for s 6= 1: we do not consider the linear case, whose symmetries are standard) ∂ r−2 ∂ X = x + u . ∂x s − 1 ∂u 2.2) With f = 1/xr , r 6= 2, a second symmetry appears if the two quantities r and s are related by the condition s = r − 3. For instance, with r = 6, s = 3 we have the two generators satisfying (8) X1 = x2

∂ ∂ + xu ∂x ∂u

,

X2 = −x

∂ ∂ − 2u ∂x ∂u

and the equation uxx + u3 /x6 = 0 becomes, introducing symmetry-adapted variables according to (9), ywyy = 3 − 5wy − wy3 + 3wy2 .

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3.1) Another interesting case occurs with g(u) = exp(s u). A single symmetry given respectively by X = x

∂ r+2 ∂ − ∂x s ∂u



or

X = x+

is admitted with the particular choices f (x) = xr

1 ∂ 2(r − 1) ∂ − r ∂x rs ∂u

f (x) = (1 + r x)−2/r .

or

3.2) More interestingly, still with g(u) = exp(s u), if f (x) = exp(r x) then two symmetries are admitted, given by ∂ r ∂ − ∂x s ∂u

X1 =

,



X2 = x −

2 ∂ r ∂ − x . r ∂x s ∂u

Choosing e.g. r = s = 1, the reduced equation takes in this case the form 2ywyy = 2(2 − wy ) − (wy − 1)3 . 4) As a final example, with the singular choice for f (x) f (x) = exp(1/x) x−s−3 one has the symmetry X = x2

and

g(u) = us

 1  ∂ ∂ u + x+ . ∂x s−1 ∂u

Few other possibilities are left. Even considering the transformations of the equivalence group [32] (nearly trivial in the present case), no new interesting case is found. In conclusion, we have seen that nontrivial symmetries are admitted for very special choices of g(u), and that – correspondingly – the function f (x) must satisfy very restrictive conditions. In particular, the birth of a second symmetry, particularly important since it allows a reduction of the initial equation into an equation solvable by quadratures, is possible in general only when precise conditions between the involved coefficients are satisfied.

3.

Case 2: A Linear PDE: The Schr¨odinger Equation

We now consider the case of the linear 1-dimensional Schr¨odinger equation for a particle moving in a potential V (x) i

∂u 1 ∂2u = − + V (x) u ∂t 2 ∂x2

,

u = u(x, t)

(11)

and we want to look for the appearance of (nontrivial) Lie point–symmetries X = ξ(x, t, u)

∂ ∂ ∂ + τ (x, t, u) + ϕ(x, t, u) ∂x ∂t ∂u

(12)

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depending on the choice of the function V (x) (see [27] and references therein for more general Schr¨odinger-type equations, possibly with time-dependent potentials). Standard calculations show that, for any V (x), the coefficients ξ, τ, ϕ in (12) must be of the form ξ = a(t) + x b(t) ,

τ = τ (t)

,

ϕ = A(x, t) + u B(x, t)

(13)

where the functions a, b , c and B must be related by the condition i B = iat x + bt x2 + ic(t) 2

(14)

whereas the only equation containing the function V (x) is x2 i (a + bx)Vx + 2bV − bt + ct + x att + btt = 0 . 2 2

(15)

Exactly with the same arguments as in the case of ODE’s (Sect. 2), it is easy to conclude that Schr¨odinger equation (11) can admit some symmetry only if V (x) verifies the very restrictive linear equation (λ1 + λ2 x)Vx + λ3 V + λ4 + λ5 x + λ6 x2 = 0

(16)

where λi are constants, not all zero. On a closer inspection, eq.s (13–16) actually show that, quite disappointingly, the only cases where some symmetry is admitted by eq. (11) is when V (x) =

k 2 γ x + 2 2 2x

(k, γ = const) .

(17)

The case k ≤ 0 is of quite limited interest from the physical point of view, and will not be considered here. It is then not restrictive to put k = 1, and we find that the admitted symmetries are  ∂ ∂ 1 ∂  + i + (x2 − )u X+ = exp(−2it) x ∂x ∂t 2 ∂u (18)  ∂ ∂ 1 ∂  X− = exp(2it) x − i − (x2 + )u ∂x ∂t 2 ∂u and, in addition if γ = 0 (which corresponds to the case of the quantum harmonic oscillator, see also [27])  ∂

Y+ = exp(−it)

∂x

+ xu

∂  ∂u

,

 ∂

Y− = exp(it)

∂x

− xu

∂  . ∂u

(19)

We just mention for completeness the trivial symmetries, i.e. the time translation and the symmetries following from the linearity of the equation. In particular, the term A(x, t) appearing in (13) turns out to be any solution to equation (11) and says the obvious fact that if u(x, t) is a solution to the equation, so is u + A. It is therefore not restrictive to choose A = 0. We also disregard the transformations of the equivalence symmetry group [32], which amount in this case to the quite obvious transformations V (x) → V (x)+const. and x → x+const.

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It is more convenient from now on to introduce and use the notion of evolutionary operator, related to any vector field X and essentially equivalent to it [29], and which will be denoted by XQ : XQ := Q

∂ ∂u

where

Q := −ξux − τ ut + ϕ .

(20)

We are considering in this Section only the case of a linear PDE, then it is known [3,29] that its symmetries are such that Q depends linearly on u and its derivatives; we can then introduce a “linear” differential operator Q (i.e. an operator not depending on u and its derivatives) defined by (cf. also (13), with A = 0 as said before) Q = −ξ Dx − τ Dt + B

(21)

where Dx , Dt denote the total derivatives, in such a way that Q(u) ≡ Q .

(22)

An important property of this notion (and the corresponding notation as well) is that it can be extended naturally to include generalized symmetries (see e.g. [29]). Let us now recall the two following fundamental results [3, 29]. Proposition 2. Let ∆ = 0 be a linear PDE and XQ one of its symmetries (possibly generalized); then whenever u0 is a solution to ∆ = 0, so is u1 := Q(u0 ) where Q is the operator defined in (21). As a consequence, the same is true for un := Qn (u0 ), for any n = 1, 2, . . .. Proposition 3. With the same assumptions as before, one has that Q is a “recursion operator”: i.e., given any symmetry XQ0 = Q0 ∂/∂u, then also XQ1 := Q1

∂ ∂u

where

Q1 := Q(Q0 (u))

is a symmetry for the PDE. It follows, in particular, that also Q1 (u0 ) = Q(Q0 (u0 )) solves the PDE, and that ∂ ∂ Q2 (u) , . . . , Qn (u) , . . . ∂u ∂u are (generalized) symmetries for the PDE ∆ = 0. Let us now introduce the operators Q± related to the vector fields X± given in (18), where Q± (u) = Q± and XQ± = Q± ∂/∂u, as in (20–22). In order to obtain a solution to our Schr¨odinger equation (11) with the potential (17) and with γ 6= 0, we start looking for the invariant solution u0 = u0 (x, t) under the symmetry generator X− . This solution is obtained from the corresponding equation Q− (u0 ) = 0, which becomes  1 xu0,x + i u0,t + x2 + u0 = 0 2

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and substituting into the initial equation (11); in this way the equation is transformed into a reduced ODE, which is easily solved to get u0 = xα exp(−x2 /2) exp(−it(α + 1/2))

(23)

√ where α = (1 + 1 + 4γ)/2. √ There is also a similar solution with the exponent α replaced by α− = (1− 1 + 4γ)/2, but – as before – we decide to restrict to solutions which are interesting for physics (and Quantum Mechanics). Indeed, imposing the condition of self-adjointness in L2 (R) of the Schr¨odinger operator −d2 /dx2 + V (x), one has to require Re α > 1/2, which excludes the solution with exponent α− and gives also the condition γ > −1/4. For the same physical reasons, we do not consider the other solution which could be obtained as invariant solution under X+ , which is in fact exponentially divergent for x → ±∞. Using the above Propositions, we then obtain starting from the solution (23) and using the recursion operator Q+ , an infinite family of solutions: u0 (x, t), u1 (x, t) = Q+ (u0 ) = xα exp(−x2 /2)(1 + 2α − 2x2 ) exp(−it(α + 5/2)), u2 (x, t) = xα exp(−x2 /2)(4x4 − 12x2 − 8αx2 + 4α2 + 8α + 3) exp(−it(α + 9/2)) and in general, for n = 0, 1, 2, . . ., un (x, t) = Qn+ (u0 ) = xα exp(−x2 /2)Pn (x) exp(−it(α + 1/2 + 2n)) where Pn (x) is a 2n-degree polynomial. Let us now recall that if u(x, t) is a solution to the Schr¨odinger equation which is an eigenfunction of the operator i∂/∂t, then the corresponding eigenvalue is interpreted in Quantum Mechanics as the energy of that solution. We can then say that the recursion operators Qn+ produce solutions un (x, t) with increasing and equally spaced eigenvalues of the energy, which are given in this case by En = α + 1/2 + 2n . This property is a particular case of the following general and simple Lemma. Let u(x, t) be an eigenfunction of i∂/∂t with eigenvalue λ; if X is any vector field of the form X = exp(iβt) X0 where X0 does not depend on t, then v(x, t) := Q(u) is eigenfunction of i∂/∂t with eigenvalue λ − β. It is enough indeed to remark that [i∂/∂t, X] = −βX. Then in our case, Q+ increases the energy by 2, Q− decreases by the same quantity. In particular, one has (also in agreement with Proposition 3) Q+ Q− (un ) = c(+) n un (±)

where cn

,

Q− Q+ (un ) = c(−) n un

= −En2 ± 2En − 3/4 + γ, and, e.g., Q2− (u1 ) = 0, and so on.

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The above property, which is shared also by the other symmetry operators (19), generalizes the notion of step-up/step-down Dirac operators [12] well known in Quantum Mechanics. Few words, to conclude, for the case γ = 0 of the quantum harmonic oscillator. Writing the two symmetry generators Y± given in (19) in evolutionary form (we use here the notation R instead of Q to avoid confusion) YR± = R±

∂ ∂u

and introducing as before the corresponding linear operators R± := exp(∓it)(−Dx ± x) ,

R± (u) = R±

one easily sees that  1  R+ (R+ (u)) = exp(−2it)(uxx −u−2xux +x2 u) = 2 exp(−2it) −iut −xux +(x2 − )u 2

thanks also to equation (11); a similar result holds for the symmetry vector field Y− . In conclusion, one has R2+ = 2 Q+ , R2− = 2 Q− .

Therefore, it is enough to consider only the two symmetries Y± (or equivalently the operators R± ). It is now an easy exercise to check that the solution u0 (x, t) to the Schr¨odinger equation which is invariant under Y− (i.e. R− (u0 )) is the 0-th order Hermite function u0 = exp(x2 /2 − it/2), and that using the recursion operator R+ one obtains all the well known solutions un = Hn (x) exp(x2 /2−it(n+1/2)), where Hn are the n-degree Hermite polynomials, which are eigenfunctions of i∂/∂t with energy eigenvalue n + 1/2. In this case, the recursion operators R± are exactly the Dirac step up/down operators.

4.

On the Reduction Techniques

As clearly suggested by the examples considered above, the existence of Lie pointsymmetries is deeply connected with the problem of obtaining some reduction of the original differential equation; this is actually one of the most important applications of the theory. It is also clear that the situation is completely different when the reduction is applied to ODE’s with respect to the case where this technique is applied to PDE’s. Very schematically, in the first case one obtains a lower order equation (and possibly a supplementary equation) which is locally equivalent to the initial one, in the case of PDE’s one typically obtains an equation which provides only particular solutions (the invariant solutions under the symmetry). Let us briefly recall what are the reasons of this difference, and examine some of its consequences. When a single independent variable x and a single dependent one u are involved, as in the case of ODE’s, one easily verifies that, for any vector field X = ξ(x, u)

∂ ∂ + ϕ(x, u) ∂x ∂u

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the following identity [25, 29] [ X (k) , Dx ] = −Dx (ξ) Dx

(24)

holds for any k-th prolongation X (k) of X. From this, one can easily deduce that, if y is an invariant quantity under X, i.e. X y = 0, and β a first order differential invariant, X (1) β = 0, then Dx β dβ ≡ (25) Dx y dy is a second order invariant under X (2) , and so on; thus, all higher order differential invariants can be obtained in this way. Then, using these invariants as new variables, the order of the ODE is lowered. For instance, let us consider the vector field X1 which is a symmetry generator for the equation examined under the item 2.2) in Sect. 2. Putting y = u/x, β = u − x ux , it is easy to verify that dβ/dy = x3 uxx /(u − x ux ) is indeed (2) invariant under X1 , and – more interestingly – that our ODE uxx + u3 /x6 = 0 becomes a first order equation with the manifestly invariant form dβ β + y3 = 0 dy (with the additional equation β = u − x ux ). The same happens if one considers the second vector field X2 for the same case 2.2) as above: now, with y = u/x2 , β = ux /x, the equation uxx + u3 /x6 = 0 becomes in this case dβ (β − 2y) + β + y 3 = 0 . dy Clearly, in Sect. 2 we preferred to choose the new variables according to the rule (9), in order to obtain the more convenient standard (and integrable by quadratures) form (10) of the equation; it may be interesting, however, to compare the different forms assumed by the same equation when written in the different coordinates. In the presence of p > 1 independent variables xi , i = 1, . . . , p, as in the case of a PDE, writing the vector field (sum over repeated indices) X = ξi ·

∂ ∂ +ϕ ∂xi ∂u

we find the rule, instead of (24), [ X (k) , Dxi ] = −(Dxi ξj ) Dxj

(26)

which does not allow, in general, to reach the same conclusions about the higher order differential invariants as in the case of ODE’s. But an even more severe restriction comes from the fact that it is not granted (in contrast to the single variable case) that it is possible to replace the initial variables u and xi with some new invariant variables β and yi in such a way that the higher order differentials uxi xj (and so on) can be expressed as functions of the differentiated invariants βyi yj (and so on) [30]. An example will clearly illustrate the

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situation. Let us consider the case of p = 2 independent variables x1 , x2 , and the vector field ∂ ∂ +u . (27) X = x1 ∂x1 ∂u The invariants of order ≤ 1 of X are u x1

,

x2

,

ux1

,

u x2 . u

Choose as new variables e.g. y1 = u/x1 , y2 = x2 , β = ux1 (we stress that any other choice would lead to analogous results). Thanks to the particular form of the functions ξi ≡ (x1 , 0) in the above vector field X, the rule (26) takes a simpler form which does allow in this case to conclude that Dx1 β/Dx1 y1 is still invariant under X (2) , and the same for all quantities of this type. Nevertheless, it is impossible to express the second order differentials uxi xj in terms of the new variables. Notice also that, instead of (25), one has here Dx 1 β ∂β ∂β Dx1 y2 = + D x 1 y1 ∂y1 ∂y2 Dx1 y1 and so on; in the present case one obtains ∂β x21 ux1 x1 = ∂y1 x1 ux1 − u

;

∂β x1 ux1 x1 = u x1 x2 − u x2 . ∂y2 x1 ux1 − u

In conclusion, only if the given second order PDE, admitting the symmetry (27), happens to be a function of the following quantities x21 ux1 x1 x1 ux1 x1 u , x2 , ux1 , , ux1 x2 − ux2 x1 x1 ux1 − u x1 ux1 − u then it can be reduced to a locally equivalent first order PDE depending on y1 , y2 , β, ∂β/∂y1 , ∂β/∂y2 . Otherwise, one can look for those solutions which are invariant under the above vector field (27). The most general second order PDE admitting this symmetry must be a function of the quantities u ux ux x , x2 , ux1 , 2 , ux1 x2 , x1 ux1 x1 , 2 2 ; x1 u u writing now y = x2 , v = u/x1 the invariants under X, this equation is reduced to an ODE containing only y, v, vy , vyy , and then just the invariant solution v = v(y) is provided in this way, as expected.

5.

Conclusion

We started our study from one of the most interesting applications of the theory of Lie symmetries, namely the symmetry classification of differential equations containing arbitrary functions. Two different cases have been considered in detail, and the main conclusion was that only special choices of these functions can allow the presence of some symmetry. In turn, the presence of symmetries is strictly connected to the likewise relevant problem of

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the reduction and the integrability (possibly by quadratures) of the differential equations, as we have seen. Our present approach was essentially based on the direct determination of Lie pointsymmetries. Actually, many generalizations of the notion of Lie point-symmetries have been introduced, and also several parallel procedures have been invented. Apart from the notion of generalized symmetries, which played some role also in our discussion in Sect. 3, let us just mention only the notion of conditional symmetry [4,24] (see also [14,40]) and the one of λ-symmetry [25], which are indeed particularly useful for finding invariant solutions and in the reduction problem even in the absence of standard symmetries. More specifically, for what concerns the general problem of the reduction procedure, see for instance [28, 31, 33, 36, 42, 43, 45] and the references therein. We can refer, e.g., to [3, 15, 20], and also to the more recent papers [7,9,10] for larger (although unavoidably incomplete) lists of works devoted to the above mentioned ideas and their countless applications and generalizations. Let us also mention, finally, that other more sophisticated and/or more geometrically oriented approaches have been proposed, mainly concerned with the problem of integrability. We refer in particular to procedures based on the concept of solvable structures in the context and the language of differential forms [1, 2, 30]. Let us only point out – to conclude – that one of the peculiar results of this procedure relates the presence of a solvable algebra of symmetry vector fields with the integrability by quadratures: we just notice that our case ii) in Sect. 2 is precisely a special case of this situation.

References [1] Barco, M.A.; Prince, G.E., Acta Appl. Math. 66, 89 (2001), and Appl. Math. Comp. 124, 169 (2001) [2] Basarab-Horwath, P., Ukr. Math. J. 43, 1236 (1991) [3] Bluman, G.W.; Anco, S.C., Symmetry and integration methods for differential equations, Springer: New York, 2002 [4] Bluman, G.W.; Cole, J.D., J. Math. Mech. 18, 1025 (1969) and Similarity methods for differential equations, Springer: Berlin, 1974 [5] Bluman, G.W.; Kumei, S., Symmetries and differential equations, Springer: Berlin, 1989 [6] Bratu, G., Bull. Soc. Math. France 42, 113 (1914) [7] Cicogna, G.; Laino, M., Rev. Math. Phys. 18, 1 (2006) [8] Cicogna, G.; Ceccherini, F.; Pegoraro, F., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2, Paper 017 (2006) [9] Cicogna, G., Nonlinear Dynamics 51, 309 (2008) [10] Cicogna, G., Phys. Lett. A 372, 3672 (2008)

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[11] Crowdy, D.G., Int. J. Engn. Sci. 35, 141 (1997) [12] Dirac, P.A.M., The principles of quantum mechanics, Clarendon Press: Oxford, 1958 [13] Fushchych, W.I.; Serov, N.I., J. Phys. A: Math. Gen. 16, 3645–3658 (1983) [14] Fushchych, W.I., in Modern group analysis: advanced analytical and computational methods in mathematical physics, Ed. by N.H. Ibragimov, M. Torrisi and A. Valenti (Kluwer, Dordrecht 1993) p. 231–239 [15] Gaeta, G., Nonlinear symmetries and nonlinear equations, Kluwer: Dordrecht, 1994 [16] Gaeta, G.; Rodr´ıguez, M. A., J. Phys. A: Math. Gen. 29, 859 (1996) [17] G¨ung¨or, F.; Lahno, V.I.; Zhdanov, R.Z., J. Math. Phys. 45, 2280 (2004) [18] Gusyatnikova, V.N.; Samokhin, A.V.; Titov, V.S.; Vinogradov, A.M.; Yamaguzhin, V.A., Acta Appl. Math. 15, 23 (1989) [19] Hydon, P.E., Symmetry methods for differential equations, Cambridge University Press: Cambridge, 2000 [20] Ibragimov, N.H. (Ed.), CRC Handbook of Lie group analysis of differential equations, CRC Press: Boca Raton, 1994, Vol.1; 1995, Vol.2; 1996, Vol. 3 [21] Ibragimov, N.H., Sov. Math. Dokl. 9, 1365 (1968) [22] Ibragimov, N.H., Elementary Lie group analysis and ordinary differential equations, J. Wiley & Sons: Chichester, 1999 [23] Kiselev, A.V., Acta Appl. Math. 72, 33 (2002) [24] Levi, D.; Winternitz, P., J. Phys. A: Math. Gen. 22, 2915 (1989) [25] Muriel, C.; Romero, J.L., IMA J. Appl. Math. 66, 111 (2001) and ibid 66, 477 (2001) [26] Nikitin, A.G.; Popovych, R.O., Ukr. Math. J. 53, 1255–1265 (2001) [27] Nucci, M.C.; Leach, P.G.L., arXiv:nlin.SI/0709.3389 [28] Nucci, M.C.; Clarkson, P.A., Phys. Lett. A 164, 49 (1992) [29] Olver, P.J., Application of Lie groups to differential equations; Springer: Berlin, 1993, second Edition [30] Olver, P.J., Equivalence, invariants, and symmetry, Cambridge Univ. Press: Cambridge, 1995 [31] Olver, P.J.; Rosenau, Ph., Phys. Lett. A 114, 107 (1986) and SIAM J. Appl. Math. 47, 263 (1987)

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[32] Ovsjannikov, L.V.: Group properties of differential equations, Siberian Acad. of Sciences: Novosibirsk, 1962 and Group analysis of differential equations, Academic Press: New York, 1982 [33] Popovych, R.O., Proc. Inst. Math. N.A.S. Ukr. 2, 437 (1997) [34] Popovych, R.O.; Ivanova, N.M., J. Phys. A: Math. Gen. 37, 7547–7565 (2004) [35] Popovych, R.O.; Yehorchenko, I.A., Ukr. Math. J. 53 1841–1850 (2001) [36] Pucci, E; Saccomandi, G., J. Phys. A: Math. Gen. 35, 6145 (2002) [37] Pucci, E.; Salvatori, M.C., Int. J. Nonlinear Mech. 21, 147 (1986) [38] Rostoker, N.; Qerushi, A., Phys. Plasmas 9, 3057 (2002) [39] Stephani, H., Differential equations. Their solution using symmetries, Cambridge University Press, Cambridge, 1989 [40] Vorob’ev, E.M., Sov. Math. Dokl. 33, 408 (1986), and Acta Appl. Math. 23, 1 (1991), and ibid 26, 61 (1992) [41] Wesson, J.: Tokamaks, The Oxford Engineering Series 48, Clarendon: Oxford, 1997, 2nd Edition [42] Winternitz, P., in Group theoretical methods in physics (XVIII ICGTMP), Ed. by V.V. Dodonov and V.I. Man’ko (Springer, Berlin 1991) p. 298–322 [43] Zhdanov, R.Z., Nonlinear Dynamics, 28, 17 (2002) [44] Zhdanov, R.Z.; Lahno, V.I., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 1, paper 009 (2005) [45] Zhdanov, R.Z.; Tsyfra I.M.; Popovych, R.O., J. Math. Anal. Appl. 238, 101 (1999)

Reviewed by Giuseppe Gaeta, Dipartimento di Matematica, Universit`a di Milano Via Saldini 50, I–20133 Milano (Italy)

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 401-446

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 15

D EFORMATION AND C ONTRACTION S CHEMES FOR N ON - SOLVABLE R EAL L IE A LGEBRAS UP TO D IMENSION E IGHT R. Campoamor-Stursberg1,∗ and J. Guer´on2,† 1 I.M.I., Universidad Complutense de Madrid, Plaza de Ciencias 3, E-28040 Madrid 2 Intituto de Astronom´ıa y F´ısica del Espacio, UBA-CONICET, CC 67 sucursal 26, 1428 Capital Federal

1.

Introduction

Contractions and deformations of Lie algebras and their relations have played an important role in many fields since their introduction in the 1950’s, and many progress has been done in understanding their structural and geometrical properties. Although being a rather active research field, there remain various important problems concerning contractions and deformations that have still not be satisfactorily solved. The notion of contraction appeared first in physical context by Segal [1], and was soon recognized to have important consequences, like the possibility of switching off interactions, or analyzing the precise effect of some physical quantities when others are disregarded. The formal introduction of contractions, done by In¨on¨u and Wigner [2], was soon defined more generally by Saletan and Kupczy´nski [3], in order to cover other limiting processes observed in symmetry groups used in Physics, like the transition from relativistic to non-relativistic physics. Other, more or less specifical, types of contractions have been introduced in the literature since, and their structural properties analyzed [4, 5, 6, 7, 9, 10]. In addition, the contractions among Lie algebras of fixed dimension have been studied in detail [11, 12, 13, 14, 15, 16], as well as important classes of algebras, like those of kinematical groups [17, 18]. However, the lack of complete classifications for Lie algebras from dimension six onwards is an important ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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R. Campoamor-Stursberg and J. Guer´on

obstruction that motivated different approaches to the problem. The relation of deformation theory, a formalism born in Differential Geometry, with contractions of Lie algebras, was first observed in [5], and has offered a kind of “inverse” procedure to study contractions. This point of view also suggested a geometrical interpretation of contractions in terms of orbits in a manifold, the points of which correspond to Lie algebras [19]. An advantage of this approach is a definition of contractions that includes all special types used in the literature, and that allow to establish different sufficiency criteria for the existence of contractions. Moreover, this motivated the application of specific techniques like cohomology of Lie algebras, which have proven to be an essential tool in many problems [20, 21, 22, 23]. Starting from this geometrical interpretation, the deformation and contraction problem naturally leads to the computation of the irreducible components of the manifold of Lie algebra structure tensors. The latter is deeply related to the notion of stability, which corresponds to Lie algebras not admitting non-trivial deformations. In low dimensions results on these components exist, although generally they have not been focused from the contraction classification problem. The objective of this chapter is to analyze in detail the deformation → − and contraction problem for solvable real Lie algebras g ⊕ R r having a non-trivial decomposition, up to the eight dimensional case. These algebras, being a semidirect product of semisimple and solvable algebras, are of great interest and importance in applications. We approach their contractions using the notion of linear deformations, which turns out to be sufficient for our purpose. This completes recent work concerning the contractions of simple Lie algebras [24], that points out the similarities of the contraction classification and the embedding problem for semisimple Lie algebras and the branching rules of representations. In fact, Levi subalgebras of Lie algebras possess an interesting stability property that allows to control, up to some extent, how the deformations and contractions behave [25]. In contrast to solvable algebras, the representation of the Levi part describing the semidirect product constitutes a first criterion to decide whether contractions among two given Lie algebras can exist or not. Using the reversibility of contractions, an important structural result, the deformation and contraction trees for these Lie algebras are established. The problem of integrability of infinitesimal deformations and its relation to the stability of systems described by these Lie algebras studied in connection with their invariant theory. This further provides additional information concerning other specific properties, like the existence of non-degenerate metrics associated to non-Abelian Yang-Mills theories and their behavior with respect to deformation and contraction patterns. Unless otherwise stated, any Lie algebra g considered in this work is defined over the field R of real numbers. We convene that non-written brackets are either zero or obtained by antisymmetry. We also use the Einstein summation convention. Abelian Lie algebras of dimension n will be denoted by the symbol nL1 .

2.

Levi Decomposition of Lie Algebras

As follows from the general theory, 1 the classification of Lie algebras is essentially reduced by means of the Levi decomposition theorem, which states that any algebra is formed from a semisimple Lie algebra s, called the Levi subalgebra, and a maximal solvable ideal r called 1

For this and other basic facts on Lie algebras the reader is referred to [8].

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403

the radical. The Levi subalgebra s, which is unique up to conjugacy, acts on the radical r in two possible ways, namely: [s, r] = 0,

(1)

[s, r] 6= 0.

(2)

The first possibility implies that the algebra is decomposable, s ⊕ r, whereas the second implies the existence of a representation R of s which describes the action, i.e., such that [x, y] = R (x) .y, ∀x ∈ s, y ∈ r.

(3)

The latter equations means that the Levi subalgebra acts by derivations on the solvable al→ − gebra r. We will use the notation ⊕ R to describe semidirect products. From equation (3), structural restrictions on the possible radicals are expected, while for direct sums any solvable Lie algebra is suitable. Real Lie algebras with non-trivial Levi decomposition 2 have only been classified up to dimension nine [26, 27], essentially because of the non-existence of classifications for seven dimensional solvable Lie algebras. However, the structure of the representation R provides valuable information on the radical r, as obtained in [27]: → − Proposition 1 Let s ⊕ R r be a Levi decomposition of a Lie algebra g. 1. If R is an irreducible representation, then the radical r is Abelian. 2. If the representation R does not possess a copy of the trivial representation D0, then the radical r is a nilpotent Lie algebra. This result establishes a method to classify algebras having a specific Levi decomposition. Fixed a semisimple Lie algebra f raks and a representation R, the radical is either solvable or nilpotent according to the decomposability of the representation. The action of the Levi subalgebra on the radical further tells that the Lie algebra of derivations of r must contain the semisimple algebra s, the action being given exactly by R. This procedure has been show to be effective in low dimensions, and also for important types of algebras, like isotropical Lie algebras [26, 28] or Abelian Lie algebras, where it is known that the classi→ − fication of Lie algebras s ⊕ R nL1 is reduced to classify the semisimple subalgebras of the special affine algebra sl(n, C) and their real forms.

3.

Deformations and Xohomology of Lie Algebras

Taking into account the action of the general linear group GL(n, R) on a given Lie algebra g, the latter can be seen as a pair g = (V, µ) formed by a vector space V and a bilinear skew-symmetric tensor µ : V × V → V that satisfies the Jacobi identity. For any fixed k of basis of V , the coordinates of this tensor are identified with the structure constants Cij g. In this sense, the set of real Lie algebra laws µ over V forms a manifold Ln embedded into R 2

n3 −n2 2

[19]. Coordinates of points correspond to the structure tensor of an algebra g.

By this we means that neither the Levi subalgebra nor the radical reduce to zero, and that the representation R does not reduce to copies of the trivial representation D0 .

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The orbits O(g) of a point g (i.e., Lie algebra) by the action of the linear group GL(n, R) are formed by all Lie algebras isomorphic to g. The study of the structure of these orbits naturally leads to consider deformations of Lie algebras. More specifically, this leads to the analysis of neighborhoods of a given Lie algebra, as well as the intersection properties of orbits. A special place is reserved to stable Lie algebras, i.e., algebras the orbits O(g) of which are open in the manifold [25]. In the context of physical applications, these are the desirable algebras, because small perturbations of the system do not change the symmetry group [21]. In addition to the geometrical methods to study orbits of Lie algebras [29], more specifical techniques like the adjoint cohomology of Lie algebras have been shown to be a powerful tool [19]. In this work we will restrict ourselves to the adjoint cohomology groups, for being the relevant object to study the orbits, although cohomologies can be defined on arbitrary modules.3 A n-cochain ϕ on a Lie algebra g = (V, µ = [., .]) is a multi-linear skew-symmetric map ϕ : V × .n . × V → V , where V is the adjoint g-module. Observe that by the identification of g with the pair (V, µ), we can suppose that the Lie bracket [., .] is given by [X, Y ] = µ(X, Y ) for all X, Y ∈ V . By means of the coboundary operator dϕ(X1, .., Xn+1) =

i=1

X 1≤i,j≤n+1

i h b i, .., Xn+1) + (−1)i+1 Xi, ϕ(X1, .., X

n+1 X

  b i, .., X b j, ..Xn+1 (−1)i+j ϕ [Xi , Xj ] , X1, .., X

(4)

 we obtain a cochain complex d : C n (V, V ) → C n+1 (V, V ), n ≥ 0 , i.e., the condition d ◦ d = 0 holds. An element ϕ ∈ C n (V, V ) is called n-cocycle if dϕ = 0, and a ncoboundary if there exists σ ∈ C n−1 (V, V ) such that dσ = ϕ. The spaces of cocycles and coboundaries are denoted by Z n (V, V ), respectively B n (V, V ). By equation (4), we have the inclusion relation B n (V, V ) ⊂ Z n (V, V ) for all n, and the quotient space H n (V, V ) = Z n (V, V )/B n (V, V )

(5)

is called n-cohomology space of g for the adjoint representation [20]. For practical purposes, the most important cohomology spaces correspond to the values n = 0, 1, 2, 3 [25, 21, 30, 31]. For n = 0 it is straightforward to verify that H 0(g, g) coincides with the center of g. For n = 1 we have Z 1 (g, g) = {f : g −→ g | df = 0} . Since the coboundary condition implies that df (X, Y ) = [f (X) , Y ] + [X, f (Y )] − f [X, Y ], it follows that the space Z 1 (g, g) is nothing but the Lie algebra of derivations of g:4 Z 1 (g, g) = Derg. 3 4

A module is nothing but the linear space underlying a representation of g, i.e., the representation space. Recall that a derivation is a linear map f : g → g satisfying [f (X) , Y ] + [X, f (Y )] = f [X, Y ].

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405

Evaluating the coboundaries we find the relation: B 1 (g, g) = {adX, X ∈ g} . Therefore the space H 1 (g, g) can be interpreted as the set of the outer derivations of the Lie algebra g. The second and third cohomology groups have not such an obvious interpretation in terms of the usual invariants of Lie algebras. They gain a specific sense when we consider paths in the manifold Ln of n-dimensional Lie algebras that start from a certain point. This geometrical notion can be immediately translated into an algebraic expression by means of cohomological tools. A formal one-parameter deformation gt of a Lie algebra g = (V, [., .]) is given by a deformed commutator: [X, Y ]t := [X, Y ] + ψm (X, Y )tm ,

(6)

where t is a parameter and ψm : V × V → V is a skew-symmetric bilinear map. If we impose that these formal brackets satisfy the Jacobi identity up to quadratic order of t,5 we obtain the following expression:     Xi , [Xj , Xk ]t t + Xk , [Xi , Xj ]t t + [Xj , [Xk , Xi]t]t   2 1 = tdψ1 (Xi, Xj , Xk ) + t (7) [ψ1, ψ1] + dψ2 (Xi, Xj , Xk ) + O(t3 ), 2 (8)

where dψl is the trilinear map of (4) for n = 2 and [ψ1, ψ1] is defined by 1 [ψ1 , ψ1 ] (Xi, Xj , Xk ) := ψ1 (ψ1 (Xi , Xj ), Xk ) + ψ1 (ψ1 (Xj , Xk ), Xi ) + ψ1 (ψ1 (Xk , Xi ), Xj ) . 2

For the special case where equation (7) vanishes, we get the conditions dψ1(Xi , Xj , Xk ) = 0, 1 [ψ1, ψ1] (Xi, Xj , Xk ) + dψ2(Xi, Xj , Xk ) = 0. 2

(9) (10)

Equation (9) shows that ψ1 is a 2-cocycle in H 2(g, g), implying that deformations are generated by 2-cocycles. More specifically, the linear term of the deformation is a cocycle. On the other hand, equation (10) implies that the deformation satisfies a so-called integrability condition. Additional integrability conditions are obtained if the deformed bracket is developed up to higher orders of t [19, 30]. In particular, if for some ψ1 ∈ Z 2 (g, g) we have [ψ1, ψ1] = 0, then the cocycle is called integrable and the linear deformation g+tψ1 defines a Lie algebra. It can be shown that the integrability conditions of linear deformations are codified by the third cohomology space H 3 (g, g). If the latter vanishes, then any cocycle can be taken as the linear term of a deformation [19]. If the deformed algebra gt is isomorphic to g, we say that the deformation gt is trivial. It is not difficult to show that whenever this happens, we can find a non-singular map ft : 5

It can be developed up to an arbitrary order, which provides additional conditions to be satisfied.

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V → V such that ft ([X, Y ]t ) = [ft X, ftY ] for all X, Y ∈ V . This means that ψ1 = dft , and the cocycle is trivial, i.e., a coboundary. Thus trivial deformations are generated by 2-coboundaries B 2 (V, V ) [21]. It can therefore be expected that the vanishing of H 2 (g, g) implies some important property concerning the corresponding Lie algebra. Actually, this is the contents of an important structural theorem due to Nijenhuis and Richardson [19]: Theorem 1 Let g be a (real) Lie algebra such that H 2 (g, g) = 0. Then the orbit O (g) of g is an open set in the manifold Ln . Roughly speaking, this result means that any Lie algebra g0 close to g is isomorphic to g [23, 10, 21], i.e., that the algebra has no non-trivial deformations. We therefore define Definition 1 A Lie algebra g = (V, µ) is called stable 6 if its orbit O(µ) is open. As known, the Whitehead lemmas imply that H 2 (s, s) = 0 for any semisimple Lie algebra. Therefore these algebras provide very important examples of rigid algebras. This result was generalized in [32], where the stability of any parabolic subalgebra was shown. However, the stability theorem of Nijenhuis and Richardson only constitutes a sufficient, but not necessary condition for a Lie algebra to be stable [25]. An interesting structural result concerning stable Lie algebras was obtained in [33]: Theorem 2 Any rigid Lie algebra g satisfies one of the following conditions: 1. The radical r7 is not nilpotent and satisfies dim Der (g) = dim g. If moreover codimg [g, g] > 1, then g is complete 8. 2. The radical is nilpotent and verifies one of the following constraints: (a) g is perfect (i.e., g = [g, g]), (b) g is the direct sum of K and one perfect stable algebra, the derivations of which are all inner, (c) g is not perfect, has no direct non-zero Abelian factor and satisfies te (g) = 0, where te (g) denotes the common dimension of Abelian subalgebras of Der(g) generated by outer semisimple derivations. In general, the effective computation of the cohomology of Lie algebras is a difficult task. However, for the case of Lie algebras having a non-trivial Levi decomposition, there exists a useful reduction, called the Hochschild-Serre spectral sequence [20]. If g has the → − Levi decomposition g = s ⊕ R r, where s denotes the Levi subalgebra, r the radical of g and R a representation of s that acts by derivations on the radical [26], then the adjoint cohomology H p(g, g) admits the following decomposition: X H p (g, g) ' H i (g, R) ⊗ H j (r, g)g , (11) i+j=p 6

Other authors also use the word rigid. The radical of a Lie algebra g is defined as the largest solvable ideal. 8 A Lie algebra g is called complete if Z(g) = H 1 (g, g) = 0. 7

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

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where H j (r, g)g is the space of g-invariant cocycles. These are the multilinear skewsymmetric maps ϕ ∈ C j (r, s) that satisfy the coboundary operator (4) and such that j   X ϕ [X, Yi ] , Y1, , .Ybi, .., Yj = 0, (Xϕ)(Y1, .., Yj ) = [X, ϕ(Y1, .., Yj )] − i=1

∀X ∈ s, Y1 , .., Yj ∈ r. For the particular case p = 2 the formula simplifies to H 2 (g, g) ' H 2 (r, g)s .

(12)

This result suggests that Levi subalgebras are stable in some sense, and that deformations are determined by appropriate modification of the brackets in the radical. This idea actually constitutes the germ of the important strong stability theorem of Page and Richardson [25], which will be essential in our analysis. This result makes a precise statement about stability of Levi subalgebras. Theorem 3 Let L = (V, µ) be a Lie algebra, s a semisimple subalgebra of L and r the complementary subspace of s in V . There exists a neighborhood U µ ∈ Ln of µ such that if µ1 ∈ U µ , then the algebra L1 = (V, µ1) is isomorphic to a Lie algebra L0 = (V, µ0) that satisfies the conditions 1. µ(X, X 0) = µ0 (X, X 0), ∀X, X 0 ∈ s, 2. µ(X, Y ) = µ0 (X, Y ), ∀X ∈ s, Y ∈ r. In essence, this stability theorem establishes that if the Lie algebra g has a semisimple subalgebra s, then its deformations will have some subalgebra isomorphic to s, and that the action of s on the remaining generators is preserved, that is, the representation describing the semidirect product is preserved. Taking into account the Hochschild-Serre spectral sequence, this means that the main information about deformations of semidirect products is codified in the radical of the algebra. As a consequence of this theorem, some properties on the cohomology of Lie algebras can be easily derived: Proposition 2 Let g = s ⊕ r be the direct sum of a semisimple Lie algebra s and an arbitrary algebra r. Then H 2 (g, g) ' H 2 (r, r). Proof. By the Hochschild-Serre spectral sequence, formula (12) holds. As an r-module, the space H 2 (r, g)g is trivial [20], and this implies that H 2 (r, g)s ' H 2 (r, g)g . It suffices therefore to consider the s-invariance. Now, for any ϕ ∈ H 2 (r, g)s and X ∈ s, Y, Z ∈ r we have (Xϕ) (Y, Z) = [X, ϕ (Y, Z)] − ϕ ([X, Y ] , Z) − ϕ (Y, [X, Z]) = [X, ϕ (Y, Z)] = 0. (13)

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because the sum is direct. Now ϕ (Y, Z) ∈ g, and by the decomposition of g we can rewrite it as (14) ϕ (Y, Z) = W1 + W2 , W1 ∈ s, W2 ∈ r. Since s is semisimple, for any X there exists X 0 ∈ s such that [X, X 0] 6= 0. By the invariance condition (13) we must have W1 = 0, thus ϕ (Y, Z) ∈ r for all Y, Z ∈ r. This proves that any invariant cochain is actually a 2-cochain of the radical, from which the assertion follows by imposing the coboundary condition. Thus for direct sums s ⊕ r of a semisimple and an arbitrary Lie algebra r, the deformations are exclusively those of r. It should be remarked that if none of the direct summands is semisimple, this results does not longer hold. Some interesting criteria were found in [34], in connection with the problem of space-time and internal symmetry of elementary particle physics.

4.

Contractions of Lie Algebras

Contractions of Lie algebras were first consider by Segal in a physical context [1], analyzing sequences of Lie groups, the structure constants of which converged to some nonisomorphic group. This result was later developed by In¨on¨u and Wigner, establishing the beginning of contraction theory. The first connection between contractions and deformations was observed in [5], basing on the important class of Saletan contractions [3]. 9 Classically, a contraction is defined as follows: Let g be a Lie algebra and Φt ∈ End(g) a family of non-singular linear maps of g, where t ∈ [1, ∞). For any X, Y ∈ g, the bracket over the transformed basis has the form [X, Y ]Φt := Φ−1 t [Φt (X), Φt(Y )] .

(15)

[X, Y ]∞ := lim Φ−1 t [Φt (X), Φt(Y )]

(16)

If the limit t→∞

exists for any X, Y ∈ g, then equation (16) defines a Lie algebra g0 called the contraction of g by Φt ). It is called non-trivial if g and g0 are non-isomorphic Lie algebras. With this definition it is obvious that contractions are transitive, i.e., given two contractions g g0 and g0 g00, we obtain the contraction g g00.10 Taking into account the expressions of the coboundary operator d used in cohomology, it is not difficult to see that the infinitesimal version of equation (16) is generated by a coboundary [21]. In fact, if we consider a trivial cocycle ψ ∈ B 2 (g, g), let σ be the 1-cochain such that dσ = ψ. Using the exponential map we obtain the linear transformation ft = exp(−tσ), and expressing the brackets over the transformed basis {ft (Xi)}, we get [X, Y ]t = ft−1 [ft (X), ft(Y )] . 9

(17)

An interesting review about the evolution of contractions and their relation to deformations can be found in [6] and [35]. 10 Of course, the contractions in the intermediary algebra g0 have to be expressed in the same basis before conposition.

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Therefore a contraction can be obtained by taking limits in (17). This is the main point to relate contractions of Lie algebras with deformations. An important result states that for any contraction of Lie algebras g → g0, there is a deformation of g0 that reverses it [5]. However, it should be remarked that a formal deformation is not necessarily related to a contraction [9, 40]. In this context, it is worthy to be mentioned that nontrivial cocycles generate nontrivial deformations, while coboundaries generate trivial deformations. In particular, this implies that the infinitesimal version of equation (16) is generated by a trivial cocycle. This allows us to interpret contractions as finite deformations generated by a trivial cocycle [31]. Using this fact, we can define a contraction by pure geometrical means: Definition 2 Let g be a real Lie algebra. Then g0 is a contraction of g if g0 ∈ O(g). Otherwise stated, contractions correspond to points in the closure O(g) of the orbit of the Lie algebra g. This definition is completely independent on the particular form of the endomorphims Φt used, and comprises all different types of contractions considered in the literature, like simple In¨onu-Wigner, Saletan, L´evy-Nahas or generalized In¨on¨u-Wigner contractions [1, 2, 3, 5, 6, 9]. From this representative free definition, the notion of equivalence of contractions follows also at once. In particular, it implies that the orbit O(g0 ) is contained in the orbit of g. The analysis of many important types of Lie algebras, like classical kinematical algebras [17], have shown explicitly the close relation between deformations and contractions commented above, and may suggest that those contractions and deformations appearing in physical applications are inverse procedures. Although it is not globally true, since there are deformations not related to contractions [36], any contraction is actually related to a deformation [9]. Definition 3 A deformation gt (0 ≤ t ≤ 1) is called of plateau type if g0 6' g1 and gt ' g1 for all t ∈ (0, 1]. The problem of which deformations are related to a contraction is solved in the following result: Theorem 4 For any contraction g g0 there exists a plateau deformation g0 → g inverse to the contraction. Conversely, for any deformation of plateau type there exists a contraction inverse to it. As a consequence, non-invertible deformations are not of plateau type, i.e., for different values of the parameter the deformed Lie algebras are pairwise non-isomorphic. In particular, this implies that a stable Lie algebra can never appear as the contraction of a non-isomorphic algebra. It should however be observed that there exist Lie algebras which are not contractions, and nevertheless they are not stable. Further, this result gives a hint for which classes of Lie algebras the invertibility of deformations can fail, namely families of Lie algebras with some parameter that acts as a scaling factor on some of its generators. Resuming these observations, it follows that contractions of Lie algebras can be studied either directly, as has been done in low dimensions (see [15, 14, 9] and references therein), or applying cohomological tools [5]. More specifically, in the latter case the deformations

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of Lie algebras are computed, and those being invertible provide the searched contractions [31, 36]. This procedure seems to be more effective for algebras in high dimensions, or having a Levi subalgebra, due to the successive simplifications provided by results like the Hochschild-Serre reduction. This approach also allows us to find some criteria that simplify the study of contractions. Proposition 3 Let g be an indecomposable Lie algebra with non-trivial Levi subalgebra s. Then g cannot contract onto a direct sum s0 ⊕ r of a semisimple Lie algebra s0 with an arbitrary Lie algebra r. This result is a direct consequence of Theorem 2 [24]. It states that deformations of reductive Lie algebras s ⊕ nL1 are always decomposable, i.e, splittable into direct sums. Moreover, they cannot appear as contractions of indecomposable Lie algebras having a nontrivial Levi decomposition or semisimple Lie algebras. As is well known, any Lie algebra s ⊕ t, t being an arbitrary n-dimensional algebra, contracts onto the reductive algebra s⊕nL1 of the same dimension. The preceding result does not exclude the possibility that an indecomposable Lie algebra contracts onto a non-solvable decomposable algebra, it merely states that none of the ideals intervening in the decomposition can be semisimple. Large classes of Lie algebras having this type of contractions exist, like semidirect products of semisimple and Heisenberg Lie algebras [37, 38]. Since any Lie algebra contracts onto the abelian Lie algebra nL1 of the same dimension, in some sense contractions of Lie algebras can be thought of as an “Abelianizing” operator. This suggests the analysis of criteria based on quantities which are either invariant or semi-invariant by contraction. Under invariant we understand some numerical invariant which is preserved, while semi-invariance means the existence of inequality between the corresponding quantities of initial and contracted algebras. These criteria can be used to establish a first approach to the existence of a given contraction, even if the number of invariants is not complete, and in many cases a direct analysis is required. There are large lists of quantities that are either preserved or increase (decrease) by contraction, but in this work only a small number will be used. Denoting by r the maximal solvable ideal (radical) and by n the maximal nilpotent ideal of a Lie algebra g, and κ(g) the Killing form, and taking into account that the latter is completely determined by the number of positive, zero and negative eigenvalues, we establish the following Theorem 5 If the Lie algebra g0 is a contraction of g, the following inequalities hold: 1. dim [g, g] ≥ dim [g0, g0] 2. dim H 1 (g, g) < dim H 1 (g0 , g0), 3. dim H k (g, g) ≤ dim H k (g0, g0) , k 6= 1, 4. dim r0 ≥ dim r,, 5. dim n0 ≥ dim n,     6. κ (g0 )+ , κ (g0 )0 , κ (g0 )− ≤ κ (g)+ , κ (g)0 , κ (g)− .

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Observe that, as a consequence of the interpretation of the lowest dimensional cohomology groups, the second inequality states that any contraction has a strictly higher dimensional algebra of derivations. This is the only strict inequality to be known for contractions, although recent developments on generalized derivations are suitable candidates to find additional properties of this kind [39]. As follows from reversal of contractions, any plateau deformation reverses the preceding inequalities, although it is known that an arbitrary deformation can even preserve them [40]. Another important property concerns the number N (g) of generalized Casimir invariants [41, 42]. Given a basis {X1, .., Xn} of g and the n o k structure tensor Cij , then g can be realized in the space C ∞ (g∗ ) by means of the differential operators: bi = C k xk ∂ , (18) X ij ∂xj k X (1 ≤ i < j ≤ n) and {x , .., x } is a dual basis of {X , .., X }. where [Xi , Xj ] = Cij k 1 n 1 n The invariants of g (in particular, Casimir operators) are the solutions of the system of partial differential equations:

bi F = 0, X

1 ≤ i ≤ n.

(19)

The number N (g) of functionally independent solutions is obtained from the classical criteria for differential equations, and equals:   k N (g) := dim g − rank Cij xk , (20)   kx where A(g) := Cij k is the matrix associated to the commutator table of g over the given basis. It is known (see e.g. [36]) that for a contraction g following inequality must be satisfied
N (g) ≤ N g0 .

g0 of Lie algebras, the (21)

That is, contractions may generate additional independent invariants for the coadjoint representation. By Theorem 4, any deformation of plateau type reverses the preceding inequality. For completeness in the exposition, the generalized Casimir invariants of all indecomposable non-solvable real Lie algebras up to dimension eight are given in Tables 4 and 5.

5.

Contractions of Simple Lie Algebras

Simple and semisimple Lie algebras are without discussion the most important case of Lie algebras, and therefore their contractions are of essential interest in applications. We have seen before that these algebras cannot appear as contractions, for being stable. For the case of non-solvable contractions, the Levi part imposes several conditions that are deeply related to the embeddings of semisimple Lie algebras and the branching rules of representations. Therefore the inspection of the Levi decomposition often provides information to decide whether a given Lie algebra can appear as contraction of a semisimple Lie algebra [24]. We recall the following result for non-solvable contractions of semisimple Lie algebras:

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→ − Proposition 4 Let g = s ⊕ Rr be a contraction of a semisimple Lie algebra s0 . Then the following holds: 1. there exists some semisimple subalgebra s1 of s0 isomorphic to s, 2. identifying s with s1 via an isomorphism, the adjoint representation of s0 decomposes as ad(s0 )|s = ad(s) ⊕ R with respect to the embedding s ,→ s0. 3. g has at least rank(s0 ) independent Casimir operators. This result constitutes a slight variation of the Page-Richardson stability theorem seen → − before. If g = s ⊕ R r is a contraction of s0, then there exists some deformation of g reversing the contraction [9]. By the stability theorem, this deformation has some subalgebra that is isomorphic to the Levi part of g, and acts the same way on the generators of the radical. Therefore the embedding of semisimple Lie algebras s ,→ s0 induces a branching rule for representations, and the quotient algebra s0/s, seen as an s-module, is isomorphic to the representation R, that is, ad(s0)|s = ad(s) ⊕ R. This proves (i) and (ii). Finally, the third condition follows from the properties of contractions of invariants [36]. Corollary 1 Let s be a semisimple Lie algebra of a semisimple algebra s0 , and R be a representation of s. If ad(s0)|s 6= ad(s) ⊕ R, then no Lie algebra with Levi decomposition → − s ⊕ Rr ( r solvable) can arise as a contraction of s0 . The problem of analyzing the non-solvable contractions of semisimple Lie algebras s0 is therefore reduced to analyze the deformations of Lie algebras having Levi decomposition → − s ⊕ Rr, where s is some semisimple subalgebra of s0 , R is obtained from the branching rules with respect to the embedding s ,→ s0 and r is a solvable Lie algebra. In view of the Hochschild-Serre reduction theorem, whether such a deformation onto a semisimple algebra is possible or not depends essentially on the structure of the radical r. In general, → − the following cases can appear when studying the deformations gt of s ⊕ R r: 1. s is a maximal semisimple subalgebra of s0, and either gt is isomorphic to s0 or there → − exists a solvable Lie algebra r0 such that gt ' s ⊕ R r0. 2. s is not a maximal semisimple subalgebra of s0 . In this case, a deformation gt that is → − not semisimple is either isomorphic to a semidirect product s ⊕ Rr0 with r0 solvable, or there exists a semisimple subalgebra s1 of s0 and a representation R1 of s1 such → − that gt ' s1 ⊕ R1 r0 for some solvable Lie algebra r0. If the latter holds, then we have the chain s ,→ s1 ,→ s0 of semisimple Lie algebras, and the branching rule ad(s1) ⊕ R1 = ad(s) ⊕ R is satisfied. Case 2. is typical for double inhomogeneous Lie algebras, and has also appeared in the classification of kinematical Lie algebras [17, 35, 44, 43]. The first possibility has been used to establish the stability of certain semidirect products of simple Lie algebras with Abelian algebras such that the describing representation is irreducible [25, 45].

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

6.

413

Detailed Analysis of Deformations and Contractions. The Lie Algebra L8,14

In this section, as example of the procedure, we analyze in detail the (linear) deformations and contractions of an eight dimensional indecomposable Lie algebra. For the remaining Lie algebras in this work, independently on their dimension or being indecomposable or not, the procedure is the same. For our analysis we choose the algebra → − L8,14 = sl (2, R) ⊕ 2D 1 ⊕D0 A5,1, which presents many features but without involving a te2 dious number of subcases to be analyzed separately. First of all, following the previous results concerning the stability theorems, any deformation of this algebra is either semisimple → − or has the Levi decomposition sl (2, R) ⊕ 2D 1 ⊕D0 r, where r is a five dimensional solvable 2 algebra. In principle, any of the algebras having this describing representation can appear as deformation of L8,14, since deformations do not have to preserve most of the main invariants of Lie algebras [40]. As follows from Table 3, the cocycles classes can be chosen as ϕ1 (X4 , X8) = X4, ϕ1 (X5 , X8) = X5, ϕ2 (X4 , X8) = X6, ϕ2 (X5 , X8) = X7, ϕ3 (X6 , X7) = X8. Let us consider a generic linear deformation L8,14 (ε1, ε2 , ε3) = L8,3 + ε1 ϕ1 + ε2ϕ2 + ε3 ϕ3 , where εi ∈ R are real parameters. It follows from the stability theorems that L8,14 (ε1 , ε2, ε3) is a semisimple Lie algebra or isomorphic to one of the Lie algebras having the describing representation 2D 1 ⊕ D0 of sl (2, R) in its Levi decomposition. For this 2 Lie algebra it can be shown that dim H 3 (L8,14, L8,14) = 3, which implies that the Jacobi condition will be satisfied under constraints of the parameters εi , i.e., constraints on the integrability condition are expected. In fact, these are given by ε2 ε3 = 0, ε1 ε3 = 0. Therefore two different cases, according to ε3 = 0 or ε3 6= 0, must be analyzed separately. 1. Let ε3 = 0. In this case, the deformed brackets are11 : [X4, X8] = ε1 X4 + ε2 X6, [X7, X8] = X5.

[X5, X8] = ε1 X5 + ε2 X7,

[X6 , X8] = X4,

For any values of ε1 and ε2, it is straightforward to verify that the deformation L8,14 (ε1 , ε2, 0) satisfies the condition N (L8,3 (ε1 , ε2, 0)) = 0,

(22)

if and only if ε1 6= 0. 11

Since both the brackets of the Levi subalgebra and the action on the radical remain unchanged by the deformation, we skip its explicit presentation.

414

R. Campoamor-Stursberg and J. Guer´on (a) Let ε1 6= 0 and ε2 = ε3 = 0. In L8,14 (ε1 , ε2, 0) we consider the change of basis12 −1 −1 0 0 X60 = X6 − ε−1 1 X4 , X7 = X7 − ε1 X5 , X8 = ε1 X8 .

Over the new basis {X1 , .., X5, X60 , X70 , X80 } the brackets of L8,14 (ε1 , ε2, 0) are transformed to [X4, X80 ] = X4,

[X5 , X80 ] = X5,

[X60 , X80 ] = 0,

[X7, X8] = 0,

proving that L8,14 + ε1 ϕ1 ' L08,17. Since the deformation does not depend on the particular value of ε1 , this deformation is of plateau type, and therefore reversible. In order to obtain the corresponding contraction L8,14, L08,17 we first consider the change of basis X60 = X4 + X6, X70 = X5 + X7. Over this new basis, the brackets of L08,17 are determined by [X40 , X8] = X40 , [X70 , X8] = X50 .

[X50 , X8] = X50 ,

[X60 , X8] = X40 ,

Now consider the family of linear isomorphisms X400 =

1 0 1 1 1 1 X , X500 = 3 X50 , X600 = X60 , X700 = X70 , X800 = 2 X80 , t2 4 t t t t

where t ∈ R. The brackets are h 00 i X4 , X800 = t12 X400, [X500, X800] =

1 X 00, t2 5

[X600, X800] = X400,

[X700, X800] = X500, and it is easily verified that for t → ∞, the resulting algebra is identical with L8,14. (b) Let ε2 6= 0 and ε1 = ε3 = 0. As follows from the structure of the deformation, for any value of ε2 the deformation L (0, ε2, 0) satisfies the condition N (L (0, ε2, 0)) = 2,

(23)

therefore L (0, ε2, 0) must be isomorphic to one of the following Lie algebras: 0 Lε8,13, L8,15, L−1 8,17 , L8,18 . Further, dim [L (0, ε2 , 0) , L (0, ε2, 0)] = 7, which excludes the Lie algebras Lε8,13 and L8,15, since these are perfect, i.e., they 12

Again, the invariant generators are skipped for brevity.

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

415

coincide with the commutator ideal. If ε2 6= 0, we can normalize it to ±1. In fact, consider the change of basis X40 = γβX4, X50 = γβX5, X60 = βX6, X70 = βX7 , X80 = γX8. Then [X40 , X80 ] = γ 2 ε2 X60 ,

[X50 , X80 ] = γ 2 ε2 X70 ,

[X60 , X80 ] = X40 ,

[X70 , X80 ] = X50 .

and thus γ 2ε2 = ±1 depending whether ε2 is positive or negative. (a) If ε2 = 1. In this case, consider the change of basis X40 = αX4 + αX6 , X50 = αX5 + αX7 , X60 = βX4 − βX6 , X70 = βX5 − βX7, where αβ 6= 0. It follows at once that [X40 , X80 ] = X40 ,

[X50 , X80 ] = X50 ,

[X60 , X80 ] = −X60 ,

[X70 , X80 ] = −X70 ,

showing that L8,14 + ε2 ϕ2 ' L−1 8,17 . >0

Defining the linear isomorphisms on L−1 8,17 determined by X40 = t−3 X4, X70 = t−2 X5 + t−1 X7 ,

X50 = t−3 X5, X80 = 12 t−1 X8 ,

X60 = t−2 X4 + t−1 X6 ,

we obtain the transformed brackets 1 X40 , [X40 , X80 ] = 2t 1 0 0 [X7, X8] = X50 − 2t X70 ,

[X50 , X80 ] =

1 0 2t X5 ,

[X60 , X80 ] = X40 −

1 0 2t X6 ,

which shows that for t → ∞ we recover the brackets of L8,14, thus the contraction L−1 8,17

L8,14,

2. If ε2 = −1, the same change of basis as before shows that L8,14 + ε2 ϕ2 ' L08,18. <0

In this case, the contraction of L08,18 onto L8,14 is determined by the transformations X40 = t−1 X4 , X70 = X7,

X50 = t−1 X5, X80 = t−1 X8.

X60 = X6

Let ε1 ε2 6= 0. Since ε1 6= 0, the deformation L (ε1 , ε2, 0) has no invariants, and must therefore be isomorphic to one of the following Lie algebras =−1 =0 , Lp68,18 . L8,16, Lp68,17

We distinguish two cases, according to the constraint ε1 + ε2 = 1.

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1. If ε1 = 2, ε2 = −1, then the change of basis X40 = αX4 − αX6 , X50 = αX5 − αX7 , X60 = βX4 + (α − β) X6 , X70 = βX5 + (α − β) X7 shows that [X40 , X80 ] = X40 ,

[X50 , X80 ] = X50 ,

[X60 , X80 ] = X40 + X60 ,

[X70 , X80 ] = X50 + X70 ,

thus L8,14 + 2ϕ1 − ϕ2 ' L8,16. The contraction reversing this deformation is given by the transformations X40 = t−1 X4, X70 = X7 ,

X50 = t−1 X5 , X80 = t−1 X8 .

X60 = X6

2. Let ε2 6= −1 and ε1 + ε2 = 1. For the change of basis X40 = αX4 + αε2 X6 , X50 = αX5 − αε2 X7, X60 = βX4 − βX6 , X70 = βX5 − βX7 with αβ 6= 0 shows that 2 L8,14 + ε1ϕ2 + ε2 ϕ2 ' L−ε 8,17 .

Define X40 = t−1 X4 , X70 = t−1 X5 + X7 ,

X50 = t−1 X5, X80 = t−1 X8.

X60 = t−1 X4 + X6

for the deformed algebra. It is straightforward to verify that this family defines a con2 L8,14. traction L−ε 8,17 3. Finally, if ε1 + ε2 6= 1, the change of basis X40 = αX4 + βX6, X50 = αX5 + βX7, X60 = γX4 + δX6 , X70 = γX5 + δX7 with ε1 γ α=δ+ , β=− 2



 ε21 ε1 +1 γ− δ 4 2

leads to the deformation ε /4

1 L8,14 + ε1 ϕ2 + ε2 ϕ2 ' L8,18 .

It should be remarked that, in contrast with the previous deformations, this is not reversible. In fact, the structure of the 3-dimensional subalgebras of Lp8,18 and L8,14 generated by the elements {X4 , X6, X8} and {X5 , X7, X8} prevent the existence of a contraction onto L8,14 [13].

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras =0 Lp68,18

-

6

↓ ↓

  L8,14 



- L8,15





L8,16

=−1 Lq68,17

L08,18



417



-

L−1 8,17

L08,17

Figure 1. Deformation and contraction diagram of L8,14. ε3 6= 0, ε1 = ε2 = 0. Taking the scaling transformations X40 = α3 ε3 X4, X50 = α3 ε3 X5, X60 = αX6 , X70 = αX7 , X80 = α2 ε3 X8 we obtain the brackets [X60 , X80 ] = X40 ,

[X70 , X80 ] = X50 ,

[X60 , X70 ] = X80 ,

and therefore L8,14 + ε3 ϕ3 ' L8,15. The contraction L8,15

L8,14

is easily obtained considering the following linear isomorphisms of L8,15 : Xi0 =

1 Xi, i = 4, 5, 6, 7. t

Over the transformed basis {X1 , .., X3, X40 , .., X70 , X8} the structure tensor reads [X60 , X80 ] = X40 ,

[X70 , X80 ] = X50 ,

[X60 , X70 ] =

1 X0, t2 8

and the latter commutator vanishes for t → ∞. Graphically, the deformations and contractions obtained can be represented in Figure 1 13 :

7.

Contractions and Deformations of Non-solvable Lie Algebras in Dimension n ≤ 7

In this paragraph we analyze systematically the deformation problem in dimensions at most seven. These cases, being rather easy, show however some properties that will be the common rule for dimension eight. Following our previous analysis, there are some cases that 13

The arrows with two endings indicate that the deformation can be reversed, while arrow with only one indicate that only deformation occurs.

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must be distinguished to obtain a complete study of deformations and contractions of nonsolvable Lie algebras of dimension n. The most important case corresponds to indecomposable algebras. In view of the stability theorems, their deformations depends essentially on the structure of the maximal solvable ideal r. If g is decomposable, there are two possibilities: either g is the direct sum of an indecomposable algebra g0 with Levi subalgebra and an arbitrary algebra r, or g = s ⊕ r is the direct sum of a semisimple and a solvable algebra. The latter case, as commented, reduces the problem to analyze exclusively the deformations and contractions of solvable algebras. These have been analyzed up to dimension four, and partial studies for higher dimensions exist [14, 15]. For notations and structure constants, we have followed strictly the classification of [26] (see Tables 1 and 4 for details).

7.1.

Dimension 5

There is only one indecomposable Lie algebra in this dimension having a non-trivial Levi decomposition, the semidirect product of sl (2, R) and the irreducible representation D 1 . 2 This algebra is obviously stable, and any contraction is necessarily solvable. The remaining algebras having a Levi subalgebra are the direct sums so (3) ⊕ r and sl (2, R) ⊕ r, with r a two dimensional Lie algebra. For these, the deformation problem is trivial, since there are only two Lie algebras in dimension two.

7.2.

Dimension 6

Six is the lowest dimension where we find non-solvable Lie algebras appearing as contractions of semisimple algebras. Any six dimensional real Lie algebra with Levi subalgebra is either semisimple, indecomposable (isomorphism classes L6,1 − L6,4 in Table 4), isomorphic to L5,1 ⊕ R or the direct sum of a simple algebra of rank one and a three dimensional solvable Lie algebra. For the latter class, the deformation problem again reduces to the analysis of deformations and contractions in dimension three, which is well known. The semisimple algebras so (1, 3), so (4), so∗ (4) and so (2, 2), as well as the algebras L6,2 and L6,3 are stable, and therefore have no deformations. The Lie algebras L6,1 and L6,4 , being inhomogeneous, appear as contractions of semisimple algebras, while the decomposable algebra L5,1 ⊕ L1 deforms onto L6,2 and L6,3. The situation can be resumed in the following Proposition 5 Following contractions hold: 1. L6,1 is a contraction of so (1, 3) and so (4) , 2. L6,4 is a contraction of so (1, 3) and so (2, 2) , 3. L5,1 ⊕ L1 is a contraction of L6,2 and L6,3.

7.3.

Dimension 7

Seven dimensional algebras with non-trivial Levi decomposition were partially studied in [24] and [40], where some interesting patterns were observed in their behavior with respect to contractions and deformations. Any non-solvable seven dimensional algebra is either reductive (i.e, the direct sum of a semisimple and an Abelian algebra), indecomposable

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

419

and isomorphic to L7,1 − L7,7 (see Table 4), isomorphic to L6,1 ⊕ L1, L6,2 ⊕ L1, L6,3 ⊕ L1 , L6,4 ⊕ L1 , L5,1 ⊕ 2L1 or L5,1 ⊕ r2 , or the direct sum of a rank one simple Lie algebra and a four dimensional solvable Lie algebra. The interesting fact about this dimension is that almost any indecomposable non-solvable algebra is stable [24]. Proposition 6 The reductive Lie algebras so (1, 3) ⊕ L1 , so (4) ⊕ L1, so∗ (4) ⊕ L1 , so (2, 2) ⊕ L1 and the indecomposable algebras L7,1, L7,2, L7,4, L7,5, L7,6 and L7,7 are stable. In the remaining cases, many of the possibilities are straightforward generalizations of the deformations and contractions observed in lower dimension. Proposition 7 For the non-stable algebras, following relations hold: 1. Lp7,3 deforms into Lq7,3, no contraction exists, 2. L27,3 is a contraction of L7,4, 3. L6,1 ⊕ L1 appears as contraction so (1, 3) ⊕ L1 , so (4) ⊕ L1 and L7,1, 4. L6,2 ⊕ L1 appears as contraction of L7,4 , 5. L6,3 ⊕ L1 deforms into Lp7,3, no contraction exists, 6. L6,4 ⊕ L1 appears as contraction so (1, 3) ⊕ L1 , so (2, 2) ⊕ L1 and L7,5, p

7. L5,1 ⊕ r2 deforms into L7,3, but no contraction exists, 8. L5,1 ⊕ 2L1 appears as contraction of L6,2 ⊕ L1, L6,3 ⊕ L1, Lp7,3, L7,4 and L5,1 ⊕ r2 . Only the Lie algebras L7,6 and L7,7 have no non-solvable contraction, due to the structure of their defining representation R.

8.

Deformations of Indecomposable Eight Dimensional Algebras with Levi Subalgebra

In this paragraph we analyze the complete deformation and contraction problem for the remaining Lie algebras in dimension eight with nontrivial Levi decomposition. According to the Page-Richardson stability theorem and its variant, the cases to be analyzed correspond to the different representations R describing the semidirect product. In the following we adopt systematically the following notation: If g is the Lie algebra and {ϕ1, .., ϕk} a basis of H 2 (g, g), a formal deformation is denoted as follows: g (ε1 , .., εk) = g + ε1ϕ1 + .. + εk ϕk .

(24)

The corresponding bases of the cohomology space, as well as other important characteristic like the dimension of the commutator and derivation algebras, number of invariants etc. are given in Table 2. The structure constants of the Lie algebras are given in Table 1.

420

8.1.

R. Campoamor-Stursberg and J. Guer´on

R = adso (3) ⊕ 2D0

→ − There is only one algebra having this representation, namely Lp8,1 = so (3) ⊕ R A1,1,p 5,27 (p 6= 0). This case is trivial, and any deformation is easily seen to belong to the same family  p+ε L8,1 , p + ε 6= 0 Lp8,1 (ε) ' L7,1 ⊕ L1 , p + ε = 0 None of these deformations gives rise to a contraction since Lp8,1 and L7,1 ⊕ L1 have a nine dimensional derivation algebra (see Table 2).

8.2.

R = R4 ⊕ D0

Three types of isomorphism classes are found for this describing representation, one of them parameterized continuously. 8.2.1.

L8,2

→ − The algebra L8,2 has Levi decomposition L8,2 = so (3) ⊕ R A5,4 and a one dimensional cohomology space. Let L8,2() = L8,2 + ϕ be a linear deformation of L8,2. For any value of  the deformed commutator satisfies the Jacobi identity, thus defines a Lie algebra. Computing the Killing metric tensor κ over the basis {X1 , .., X8}, we obtain the matrix   −3 0 0 0 0 0 0 0 3  0 −2 0  0 0 0 0 2    0 0 −3 0 0 0 0 0     0  0 0 −6 0 0 0 0  κ=  0 0 0 0 −6 0 0 0     0 0 0 0 0 −6 0 0     0 0 0 0 0 0 −6 0  3 0 0 0 0 0 −92 0 2 We have det(κ) = 2238 76 6= 0 for  6= 0, and therefore the deformation is a semisimple Lie algebra. To identify to which real form g is isomorphic, we compute the spectrum of κ and obtain s ) (  7 9 2 1 3 3 + . (25) 92 − Spec(κ) = −3, 3, (−6) , −  − 1 ± 2 2 2 4 q
Since 92 + 2 > 92 − 32 + 74 for any , the two last roots of Spec(κ) are always negative, and the signature σ of κ is given by  −8, ε > 0 σ (κ) = . 0, ε < 0 For σ = −8 we obtain the compact Lie algebra su(3), while for σ = 0 we get the pseudo-unitary algebra su(2, 1) [46]. Finally, starting from the deformed bracket, applying formula (16) to the family of linear maps defined by ft (Xi ) = Xi, (i = 1, 2, 3), ft (Xi) = t−1 Xi , (i = 4, ..., 7) and ft (X8) = t−2 X8, we obtain the contraction of su(2, 1) onto L8,2.

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras 8.2.2.

421

L8,3

→ − The Lie algebra L8,2 = so (3) ⊕ R A1,1,1 5,7 has no invariants, it can thus never appear as the contraction of a semisimple Lie algebra, in view of proposition 3. For a formal deformation L8,3 (ε1 , ε2, ε3) = L8,3 + ε1 ϕ1 + ε2 ϕ2 + ε3 ϕ3 there is no obstruction to the Jacobi identity, therefore for any values of εi it defines a Lie algebra. Moreover, independently of the values of the deformation parameters, we have N (L8,3 (ε1 , ε2, ε3)) = 0. In view of Table 2, this means that there is only one possibility for the deformations, namely L8,3 (ε1 , ε2, ε3) ' Lp8,4. It follows from the structure of the three dimensional Lie algebras spanned by {X4, X6, X8} and {X5, X7, X8} that no contraction of Lp8,4 onto L8,3 can exist. 8.2.3.

Lp8,4

→ − For the family Lp8,4 = so (3) ⊕ R A1,p,p 5,17 two cases have to be distinguished, according to p being zero or not. 1. If p 6= 0, then there is only one cocycle class ϕ, and it follows at once that Lp8,4 + εϕ ' Lε+p 8,4 , which cannot give rise to a contraction for being a member in the same family.
2. For p = 0, we have dim H 2 L08,4, L08,4 = 2. A deformation L08,4 (ε1 , ε2) = L08,4 + ε1 ϕ1 + ε2 ϕ2 has to satisfy the integrability condition ε1 ε2 = 0. It is straightforward to verify that the linear deformation L08,4(0, 2) = L08,4 + 2 ϕ2 has a codimension one derived ideal, and cannot be semisimple. Indeed, 2 L08,4 (0, ε2) ' Lε8,4 ,

and no reversal is possible because both algebras have the same number of derivations. Considering the remaining deformation L08,4 (1, 0) and computing the spectrum of the Killing tensor κ, we obtain that  Spec(κ) = (−3)3 , −4, (−61)4 . Thus σ (κ) =



−8, 1 > 0 , 0, 1 < 0

which means that L08,4 + 1 ϕ1 ' su(3) if 1 > 0 and L08,4 + 1 ϕ1 ' su(2, 1) if 1 < 0. Defining on L08,4(1 , 0) the linear maps ft (Xi ) = Xi , (i = 1, 2, 3, 8); ft (Xi ) = t−1 Xi , (i = 4, ..., 7), it follows that the contraction defined by them for t → ∞ is isomorphic to L08,4, showing the invertibility of the deformations.

422

8.3.

R. Campoamor-Stursberg and J. Guer´on

R = R5

→ − Since L8,5 = so (3) ⊕ R5 5L1 is an inhomogeneous algebra, any non-trivial deformation must be a semisimple Lie algebra. For L8,5() = L8,5 + ϕ the spectrum of the Killing form κ is given by  Spec(κ) = (−12)2, −8, −4, −6, (−24)2, −72 , thus σ(κ) = −8 if  > 0 and σ(κ) = 2 if  < 0. This proves that L8,5() ' su(3) if  > 0 and L8,5() ' sl(3, R) otherwise. The contraction of su (3) respectively sl (3, R) onto L8,5 are defined by the transformations ft (Xi) = Xi , (i = 1, 2, 3); ft (Xi) = t−1 Xi , (i = 4, ..., 8).

8.4.

R = D 12 ⊕ 3D0

This representation is the first where a considerable number of parameterized families with very similar properties appear. This fact is a consequence of the high number of copies of the trivial representation that R contains. We also remark that this case has no counterpart for the Levi subalgebra so (3), because the half-spin representation D 1 of sl (2, R) is of 2 second class. 8.4.1.

L8,6

→ − For the Lie algebra L8,6 = sl (2, R) ⊕ R h2 the integrability condition of a formal deformation L8,6 (ε1 , ε2, ε3 , ε4) is given by the non-linear system ε3 (2ε3 + ε4 ) = 0, ε3 (ε2 − 2ε1) = 0, ε1 (2ε3 + ε4 ) = 0, ε1 (ε2 − 2ε4 ) = 0, ε1 ε4 + ε2 ε3 = 0. The general solution of this system can be represented as {(0, ε2, 0, ε4) , (ε1 , 2ε1, ε3, −2ε3)} . In the first case, we obtain L8,6 (0, ε2, 0, ε4) ' L6,2 ⊕ r2, thus the deformation is a decomposable algebra. Moreover, it is invertible by means of the linear transformations ft (Xi ) = Xi , (i = 1, 2, 3); ft (Xi ) = t−1 Xi, (i = 4, ..., 7); ft (X8) = t−2 X8 of L6,2 ⊕ r2 . On the other hand, the deformation L8,6 (ε1 , 2ε1, ε3, −2ε3) leads also to a decomposable Lie algebra, namely L8,6 (ε1, 2ε1, ε3, −2ε3 ) ' L7,4 ⊕ L1. Also in this case, the inverse contraction is determined by the transformations ft (Xi) = Xi , (i = 1, 2, 3); ft (Xi) = t−1 Xi , (i = 4, ..., 7).

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

423

It may wonder why the remaining eight dimensional algebras with the same R do not appear as deformations of L8,6. This is easily explained observing that L8,6 (ε1 , ε2, ε3, ε4) has a one dimensional center, which happens only for L08,8. However, this algebra cannot appear because of the structure of the radical. 8.4.2.

L8,7

→ 1,p,q − The algebra Lp,q 8,7 = sl (2, R) ⊕ R A5,7 is the first of the two families to depend on two parameters. As expected, for this class of algebras the cohomology depends heavily on the values of p and q, and gives rise to a high number of subcases (see Table 3). Here we also find the highest dimensional cohomology space. 1. p 6= q, p 6= 2, q 6= 2. There are no integrability conditions for the deformation Lp,q 8,7 (ε1 , ε2 ). Lp,q 8,7 (ε1 , ε2 )

'



1 ,q+ε2 , (p + ε1 ) (q + ε2) 6= 0, Lp+ε 8,7 p L7,3 ⊕ L1, (p + ε1 ) (q + ε2) = 0.

1 ,q+ε2 cannot contract onto Lp,q The Lie algebra Lp+ε 8,7 8,7 because of the dimension of the algebra of derivations, while Lp7,3 ⊕ L1 cannot contract because of the dimension of the commutator subalgebra.

2. p 6= q, p = 2, q 6= 2 [equivalent  to p 6= q, q = 2, p 6= 2] 2,q 2 For these values the space H L2,q 8,7 , L8,7 has dimension three. The integrability 2,p

condition for a deformation L8,7 (ε1 , ε2, ε3) is ε1 ε3 = 0. For ε3 = 0 we immediately obtain that 2+ε1 ,q+ε2 L2,q , 8,7 (ε1 , ε2, 0) ' L8,7

and no contraction is possible. If ε3 6= 0, then ε1 = 0 and k L2,p 8,7 (0, ε2, ε3 ) ' L8,10

with k = (q + ε2 ) ε−1 3 . For any value of k we can construct the contraction Lk8,10

L2,q 8,7

by means of the transformations ft (Xi) = Xi (i = 1, 2, 3, 8); ft (Xi ) = t−1 Xi (i = 4, 5, 6, 7) . 3. p = q For these values of p and q, a formal deformation depends on four parameters if p 6= 2, and on six for the equality. For the case p = 2 the integrability condition is ε1ε5 + ε4 ε6 = 0, ε2 ε6 + ε2 ε5 = 0.

424

R. Campoamor-Stursberg and J. Guer´on It follows at once from this that for ε5 = ε6 = 0, no further condition is required. This implies that the deformations of Lp,p 8,7 for p 6= 2 are very similar to those of p = 2 for the vanishing of the parameters ε5 and ε6 . It therefore suffices to analyze this case and later extract those deformations that do not depend on the two last parameters. Since the high number of the ε0i s leads to an enormous number of equivalent cases, as happens for the solutions {ε1 = ε3 = ε6 = 0} and {ε2 = ε4 = ε5 = 0} of the integrability condition, only those giving rise to non-equivalent deformations will be considered (the other being deduced by means of a change of basis). The possible deformations are listed below:  2+ε2 ε2 ε5 6= 0, ε1 = ε3 = ε6 = 0 L8,10 ,      L8,11, ε5 6= 0, ε1 = ε2 = ε3 = ε6 = 0   2   L ⊕ L , ε 1 2 = −2, ε1 = ε3 = ε5 = ε6 = 0  7,3  p   ε2 6= 0, εi = 0, i 6= 2  L8,7, 2 . L (ε , ε , ε , ε , ε , ε ) ' , ε4 6= 0, εi = 0, i 6= 4 L2,2 8,8 8,7 1 2 3 4 5 6  p   ε1 = −2, εi = 0, i 6= 1  L8,7,  p,q   , ε1 = −2, ε2 6= 0, εi = 0, i 6= 1, 2 L  8,9    L6,3 ⊕ 2L1, ε1 = ε2 = −2, εi = 0, i 6= 1, 2    p ε1 6= −2, ε3 6= 0, εi = 0, i 6= 1, 3 L8,8, Among these possibilities, only two of the deformations can be reversed to give a contraction onto L2,2 8,7 . The contraction L8,11

L2,2 8,7

is determined by the transformations ft (Xi ) = t−1 Xi (i = 4, 5, 6) ; ft (X7 ) = t−2 X7, ft (Xi ) = Xi , (i = 1, 2, 3, 8), while the contraction L28,8

L2,2 8,7

is determined by the linear isomorphisms ft (X6) = t−1 X6, ft (X7 ) = t−2 X7 , ft (Xi ) = Xi , (i =6= 6, 7) . As for the deformations of Lp,p 8,7 for p 6= 2, these are easily recovered from the studied case, and equal  L27,3 ⊕ L1 , ε2 = −2, ε1 = ε3 = 0      ε2 6= 0, εi = 0, i 6= 2 Lp8,7 ,   p   L , ε4 6= 0, εi = 0, i 6= 4  8,8 p p,p L8,7 , ε1 = −2, εi = 0, i 6= 1 L8,7 (ε1 , ε2, ε3, ε4) ' .  q,r   , ε = −2, ε = 6 0, ε = 0, i = 6 1, 2 L 1 2 i  8,9    L ⊕ 2L , ε = ε = −2, ε = 0, i 6= 1, 2 6,3 1 1 2 i    Lp , ε = 6 −2, ε = 6 0, ε 1 3 i = 0, i 6= 1, 3 8,8 The different criteria for contractions show that none of these deformations is of plateau type, and cannot thus be reversed.

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras 8.4.3.

425

Lp8,8

This family presents the particularity that the radical depends on the values of the parameter p. Specifically, the Levi decomposition equals ( , p 6= 0 Ap−1,p−1 → − 5,9 . Lp8,8 = sl (2, R) ⊕ R 1 p=0 A5,8, The cohomology also depends on the values of p. 1. p = 0 The integrability condition for a deformation L08,8 (ε1 , ε2 , ε3) reads ε2 ε3 = 0. The obtained deformations are, according to this constraint:  L6,3 ⊕ r2, ε2 = 0    p,q  4ε2 + ε21 < 0, ε3 = 0   L8,9, ε1 /2 L08,8 (ε1 , ε2, ε3) ' , 4ε2 + ε21 = 0, ε3 = 0 . L8,8  p,q   4ε2 + ε21 > 0, ε3 = 0  L8,7,   Lp ⊕ L , ε 6= 0, ε = ε = 0 1 1 2 3 7,3 Among these deformations, only the first leads to a contraction L6,3 ⊕ r2

L08,8.

The contraction is defined by the transformations ft (X6 ) = t−1 X6 , ft (X7) = t−1 X7 , ft (Xi) = Xi, (i 6= 6, 7) . 2. p = 2 The integrability conditions for L28,8 (ε1 , ε2, ε3) are ε1 ε3 = ε2 ε3 = 0. As expected, the deformations follow a similar pattern to the previous case:   L8,11, ε3 6= 0    Lp,q , 4ε + ε2 < 0 2 1 8,9 L28,8 (ε1 , ε2 , ε3) ' ε1 /2 2 =0 .  , 4ε + ε L 2 1  8,8  p,q  L8,7, 4ε2 + ε21 > 0 Only the first can be reversed. The linear isomorphisms ft of L8,11 defined by ft (Xi ) = t−1 Xi (i = 4, 5, 6, 7) , ft (Xi ) = Xi , (i 6= 4, 5, 6, 7) lead to the contraction L8,11

L28,8.

426

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3. p 6= 0, 2 As follows from the structure of the cohomology space, the deformations of Lp8,8 for general p are special cases of the special values analyzed before. The deformations are therefore  p,q 2   L8,9, 4ε2 + ε1 < 0 p ε1 /2 L8,8 (ε1 , ε2 , ε3) ' L8,8 , 4ε2 + ε21 = 0 .   Lp,q , 4ε + ε2 > 0 2 1 8,7 None of them can be reversed. 8.4.4.

Lp,q 8,9

→ 1,p,q − Although the Lie algebras Lp,q 8,9 = sl (2, R) ⊕ RA5,13 depend on two parameters p and q 6= 0, its cohomology space is quite simple (see Table 2), and for all values of p, q the cocycle classes coincide. This absence of special cases makes the analysis of deformations quite easy. There is no integrability condition to be satisfied. In this case, it can be shown that p,q any formal deformation Lp,q 8,9 (ε1, ε2 ) is equivalent to a deformation of the type L8,9 (ε1 , 0). For the latter, the possibilities are  p+ε1 /2, 21 q 0   L8,9 , q 02 = 4q − ε21 > 0 p,q L8,9 (ε1 , 0) ' q 02 = 4q − ε21 = 0 . Lp±q ,   8,8 r,s L8,7, q 02 = 4q − ε21 < 0 The dimension of the corresponding algebras of derivations show that none of these deformations can be reversed to a contraction onto Lp,q 8,9 . 8.4.5.

Lp8,10

→ − The algebras Lp8,10 = sl (2, R) ⊕ R A2,p 5,19 must also be analyzed separately for some values of the parameter p. 1. p 6= 2 For these values of p, there is only one cocycle class, which turns out to be integrable. The corresponding deformation Lp8,10 (ε1 ) is easily seen to belong to the same family or be decomposable. More precisely, ( p(1+ε)−1 L p , p + ε 6= 0 8,10 L8,10 (ε1 ) ' L6,2 ⊕ 2L1 , p + ε = 0 None of these deformations can be reversed. For the first it is obvious from the dimension of the derivation algebra, while for the second, it follows from the fact that L6,2 ⊕ 2L1 has non-trivial center, while Lp8,10 has not. 2. p = 2 For L28,10 (ε1 , ε2) there is no integrability condition, thus for any values of ε1 and ε2

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

427

they define Lie algebras. If ε1 6= 0 and ε2 = 0, the deformations are exactly those previously obtained for p 6= 2. Taking L28,10 (0, ε2), a change of basis proves that L28,10 (0, ε2) ' L8,11. In this case, there is a contraction of L8,11 onto L28,10 determined by the linear isomorphisms ft (X6 ) = t−1 X6, ft (X7 ) = t−2 X7 , ft (Xi ) = Xi (i 6= 6, 7) . For arbitrary non-zero values of ε1 and ε2 the deformations are the following L28,10 (ε1 , ε2)

'

(

2(1+ε )−1

6 0 , L8,10 1 , 1 + ε1 = L6,2 ⊕ r2, 1 + ε1 = 0

and none of them can be reversed. 8.4.6.

L8,11

→ − The only cocycle class of L8,11 = sl (2, R) ⊕ R A25,20 is integrable, and any deformation L8,11 (ε) satisfies the constraint dim [L8,11 (ε) , L8,11 (ε)] = 7, which excludes L8,6 to be reached. Further, for any value of ε we obtain that dim Der (L8,11 (ε)) = 9, which implies that none of the deformations can be reverted to give a contraction. The detailed analysis gives ( 2(1+ε)−1 , 1 + ε 6= 0 . L 8,10 L8,11 (ε) ' L6,2 ⊕ r2, 1 + ε = 0

8.5.

R = D1 ⊕ 2D0

There is only one algebra having this describing representation, namely Lp8,12 = → − p sl (2, R) ⊕ RA1,1,p 5,7 . For any value of ε, the deformation L8,12 + εϕ defines a Lie algebra. It is trivial to verify that Lp8,12 (ε) ' Lε+p 8,12 , and the dimension of the derivations prevents this deformation to be reverted.

8.6.

R = 2D 1 ⊕ D0 2

This representation provides the second largest group of Lie algebras, although only two of them depend on a continuous parameter. Some of these Lie algebras turn out to have the same complexification as the algebras having the compact Levi subalgebra so (3) and describing representation R4 ⊕ D0 . This is also the group where most of the non-solvable contractions of simple Lie algebras appear. The algebra L8,14, studied in detail in the previous section, has been omitted from this list.

428 8.6.1.

R. Campoamor-Stursberg and J. Guer´on Lε8,13 (ε = ±1)

→ − For Lε8,13 = sl (2, R) ⊕ R A5,4 we consider the deformations Lε8,13 (µ) = L8,13 + µϕ. For any nonzero µ, the spectrum of the Killing form is given by o n Spec (κ) = −6, 6, 12, (−12µ)2 , (12µ)2 , −36µ2 . The signature is σ (κ) = 0 for  = 1 and 2 for  = −1, proving that L18,13 (µ) is isomorphic to su (2, 1) and L−1 8,13(µ) is isomorphic to sl(3, R). In both cases, the deformations can be reversed, and the corresponding contraction is obtained from the changes of basis in L8,13(µ) defined by ft (Xi) = Xi, (i = 1, 2, 3); ft (Xi) = t−1 Xi , (i = 4, ..., 7); ft (X8) = t−2 X8. 8.6.2.

L8,15

→ − Let L8,15 (ε) = L8,15 + εϕ be a formal deformation of L8,15 = sl (2, R) ⊕ R A5,3. The computation of the Killing form gives det (κ) = 21438ε5 6= 0 for nonzero ε, and the spectrum is given by o n Spec (κ) = −6, 6, 12, (−12ε)3 , (12ε)2 , thus σ (κ) = 0 for positive ε and σ (κ) = 2 for ε < 0. We obtain the deformations  su (2, 1) , ε > 0 L8,15 + εϕ ' . sl (3, R) , ε < 0 The deformations are reversed considering the linear maps ft (Xi) = Xi , (i = 1, 2, 3); ft (Xi) = t−3 Xi , (i = 4, 5); ft (Xi ) = t−1 Xi, (i = 6, 7); ft (X8) = t−2 X8. 8.6.3.

L8,16

The third of the algebras not depending on continuous parameters is L8,16 = → − sl (2, R) ⊕ RA15,15. The single cohomology class is not subjected to integrability conditions, and in any case the deformation satisfies the constraint N (L8,16 (ε)) = 0. In addition, any of the possible target Lie algebras has a nine dimensional derivation algebra, which proves that no algebra can contract onto L8,16. Specifically, we have  p L8,17, ε 6= −1 , L8,16 (ε) ' L−1 8,18 , ε = −1 where

1 p= 2



3ε − 1 +

q

2



(ε − 1) + 4 (1 + ε)−1 .

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras 8.6.4.

429

Lp8,17

→ − The family Lp8,17 = sl (2, R) ⊕ R A1,p,p 5,7 , originally studied in [40], presents some interesting properties with respect to deformations. 1. p = 1 In this case there is no integrability condition, thus L18,17 (ε1 , ε2, ε3 ) satisfies the Jacobi identity for all values of εi . Ignoring the cases which lead to equivalent deformations, the possibilities are the following

L18,17 (ε1 , ε2, ε3) '

    

ε1 +1 L8,17 , L8,16,

ε1 6= 0 (0, ε2, 0) , (0, 0, ε3)

1

(1−ε2 ε3 )  , L2    8,18q L8,17,

ε2 ε3 ± 1 = 0 ε2 ε3 ± 1 6= 0

.

Among these deformations, only L8,16 provides a contraction onto L−1 8,17, determined by the transformations ft (X6 ) = t−1 X6, ft (X7 ) = t−1 X7 , ft (Xi ) = Xi (i 6= 6, 7) . 2. p 6= ±1 For any value of ε we immediately obtain Lp8,17 (ε) ' Lp+ε 8,17 , which cannot be reversed because of the dimension of the algebras of derivations. 3. p = −1 This case is the most interesting, since it corresponds to a singular point of the family that deforms onto a semisimple Lie algebra [40]. The integrability condition for a generic deformation L−1 8,17 (ε1 , ε2 ) is ε1 ε2 = 0. Let ε1 6= 0. We note that L−1 8,17 + ε1 ϕ1 is semisimple by considering the Killing form κ. The spectrum of κ equals σ (κ) = 2 for any nonzero values of ε1 , therefore obtain the deformation L−1 8,17 + ε1 ϕ1 ' sl (3, R). The corresponding contraction is determined by the linear maps ft (Xi) = Xi , (i = 1, 2, 3, 8); ft (Xi ) = t−1 Xi, (i = 4, ..., 7). For ε2 6= 0, it is straightforward to verify that ε2 −1 L−1 8,17 (0, ε2) ' L8,17 ,

which does not lead to a contraction.

430 8.6.5.

R. Campoamor-Stursberg and J. Guer´on Lp8,18

→ − This family, Lp8,18 = sl (2, R) ⊕ R A1,p,p 5,7 , similar to the previous, has the particularity of having a value for which it appears as the contraction of a simple Lie algebra. 1. p = 0 In this case we find two independent cocycle classes. The integrability condition for the deformation L08,18 (ε1 , ε2) = L08,18 + ε1 ϕ1 + ε2 ϕ2 is ε1 ε2 = 0. Considering the first possibility L08,18 + ε1 ϕ1 and computing the Killing tensor, we obtain det (κ) = 21437ε41 6= 0 and the spectrum o n Spec (κ) = −6, −4, 6, 12, (−12ε1 )2 , (12ε1 )2 , and in any case σ (κ) = 0, showing that L08,18 + ε1 ϕ1 ' su (2, 1). To obtain the contraction, we consider on the deformations the transformations ft (Xi ) = Xi , (i = 1, 2, 3, 8); ft (Xi ) = t−1 Xi , (i = 4, ..., 7).
If ε1 = 0, then N L08,18 (0, ε2) = 0 for any value of ε2 . Having in mind the dimension of the algebra of derivations of the different target algebras (see Table 2), this means that the deformations L08,18 (0, ε2) do never lead to a contraction onto L08,18. The precise deformation  √ −1  ±ε 4−ε22   L 2 , |ε2 | < 2 8,18 L08,18 (0, ε2) ' |ε2 | = 2 , L8,16,    Lp8,17, |ε2 | > 2 where p=

ε2 ± ε2 ∓

p p

ε22 − 4 ε22 − 4

.

2. p 6= 0 First of all, we compute the number of invariants of a formal deformation Lp8,18 (ε) and find    2, 2p + ε = 0 p N L8,18 (ε) = . 0, 2p + ε 6= 0   Since additionally dim D1 Lp8,18 (ε) = 7, the algebras L±1 8,13 and L8,15 cannot be reached by deformation. For the condition 2p + ε = 0 we obtain the deformations  −1  L8,17, |p| > 1 Lp8,18 (ε) ' L , p=1 .  8,14 L08,18, |p| < 1

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

431

Obviously none of these can be reversed. The detailed analysis for 2p + ε 6= 0 is schematically resumed as follows:  L8,16, p = 2, ε = −2 p , L8,18 (ε) ' q L8,17, otherwise
where q = p2 − p − 1 (p − 1)−1 6= 1.

8.7.

R = D2

There is only one algebra with this describing representation. Since the Lie algebra L8,21 = → − sl (2, R) ⊕ R5L1 is inhomogeneous and dim H 2 (L8,21, L8,21) 6= 0, it follows from [25] that L8,21 is the contraction of a semisimple Lie algebra. Considering L8,21 + εϕ, the spectrum of the Killing form κ is given by Spec (κ) = {−24, 24, 48, −48ε, −12ε, 12ε, 48ε, 72ε} . The signature is σ (κ) = 2 for ε > 0, σ (κ) = 0 for ε < 0. We thus obtain that  su (2, 1) , ε < 0 . L8,21 + εϕ ' sl (3, R) , ε > 0 In both cases, the contraction follows at once from the linear maps ft (Xi) = Xi , (i = 1, 2, 3); ft (Xi) = t−1 Xi , (i = 4, ..., 8).

8.8.

R = D 1 ⊕ D1 2

→ − The only semidirect product having this R, L8,22 = sl (2, R) ⊕ R 5L1 is also inhomogeneous, but, in contrast to the previous case, R is not irreducible. Therefore a deformation is not necessarily semisimple. A formal deformation L8,22 (ε) satisfies the Jacobi condition for any ε. The Killing form κ of L8,22 (ε) turns out to be degenerate, thus L8,22 (ε) cannot be semisimple. However, the subalgebra generated by the elements {X1, .., X6} is semisimple and isomorphic to sl (2, R) ⊕ sl (2, R) for any value of ε. In view of this, it is not difficult to show that L8,22 (ε) ' sa (2, R) ⊕ sl (2, R) . The contraction reversing the deformation is given by the linear maps ft (Xi) = t−1 Xi (i = 4, 5, 6) ; ft (Xi ) = Xi (i 6= 4, 5, 6) .

9.

Deformations of Decomposable Eight Dimensional Algebras with Levi Subalgebra

We finally determine the deformations and contractions of decomposable eight dimensional Lie algebras with non-trivial Levi decomposition. As expected, many of the existing contractions and deformations are a direct consequence of the lower dimensional cases. The

432

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Lie algebras g = s ⊕ r being a direct sum of a rank one and a five dimensional solvable algebra are left out, since the deformation problem is reduced exclusively to the solvable radical. On the other hand, no detailed study has been yet done for the contractions of solvable Lie algebras in this dimension. One of the reasons is the high number of parameterized families, as well as the difficulty of determining the cohomologies for these Lie algebras, where no reduction criteria exist. We also ignore the Lie algebras L5,1 ⊕ r, r being a three dimensional algebra. With two exceptions, algebras of this type deform onto algebras with the same decomposition, and are therefore not of great interest in our analysis. We first of all determine those algebras that do not have deformations and therefore are not contractions. Proposition 8 The Lie algebras so (1, 3) ⊕ r2 , so (4) ⊕ r2, so (2, 2) ⊕ r2, so∗ (4) ⊕ r2 and L6,3 ⊕ r2 are stable. As a direct consequence of this result, we immediately get the contractions s ⊕ r2

s ⊕ 2L1,

where s = so (1, 3) , so (4) , so (2, 2) or so∗ (4). There are also no other possibilities for the deformations of these direct sums with the two dimensional Abelian algebra. For the remaining, non-stable algebras we study the problem separately, as done for the indecomposable case. Up to some special cases, the endomorphisms that give rise to the contractions are skipped, for being formally very similar to the contractions already studied.

9.1.

L6,1 ⊕ 2L1

For this Lie algebra we find dim H 2 (L6,1 ⊕ 2L1 , L6,1 ⊕ 2L1 ) = 5, generated by the cocycles ϕ1 (X5, X6) = X1, ϕ1 (X4, X6) = −X2 , ϕ1 (X4, X5) = X3, ϕ2 (X4, X7) = X4, ϕ2 (X5, X7) = X5, ϕ2 (X6, X7) = X6. ϕ3 (X4, X8) = X4, ϕ3 (X5, X8) = X5, ϕ3 (X6, X8) = X6, ϕ4 (X7, X8) = X7, ϕ5 (X7, X8) = X8, A formal deformation L6,1 ⊕ 2L1 + εi ϕi is subjected to the constraints ε1 ε3 = ε2 ε3 = 0, ε2 ε4 + ε3 ε5 = 0. We discard those solutions to the integrability conditions which determine equivalent deformations. For the remaining cases, we get the scheme  so (1, 3) ⊕ r2,    so (1, 3) ⊕ 2L1, L6,1 ⊕ 2L1 (ε1 , .., ε5) ' L  6,1 ⊕ r2,   p L8,1,

ε1 6= 0, ε4 ε5 6= 0 ε1 6= 0, εi = 0, i 6= 1 . ε4 6= 0 or ε5 6= 0, εi6=4,5 = 0 ε2 ε5 6= 0

From any of these deformation we can easily construct the corresponding contraction onto L6,1 ⊕ 2L1, which shows that all of them can be reversed.

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

9.2.

433

L6,1 ⊕ r2

A basis of the cohomology space is given by the cocycles ϕ1 (X5, X6) = X1, ϕ1 (X4, X6) = −X2 , ϕ1 (X4, X5) = X3, ϕ2 (X4, X7) = X4, ϕ2 (X5, X7) = X5, ϕ2 (X6, X7) = X6. The integrability condition for L6,1 ⊕ r2 (ε1 , ε2) = L6,1 ⊕ r2 + ε1 ϕ1 + ε2ϕ2 is given by ε1 ε2 = 0. Therefore only two cases must be analyzed. It follows that ( so (1, 3) ⊕ r2 , ε2 = 0 . L6,1 ⊕ r2 (ε1 , ε2) ' −ε−1 L8,12 , ε1 = 0 Only the first deformation is of plateau type and defines a contraction (see the case in dimension six), while the second cannot be reversed because of the dimension of the algebra of derivations.

9.3.

L6,2 ⊕ 2L1

For this Lie algebra, we find that dim H 2 (L6,2 ⊕ 2L1, L6,2 ⊕ 2L1) = 5 and that a basis can be taken as ϕ1 (X4 , X7) = X4, ϕ1 (X5, X7) = X5, ϕ1 (X6 , X7) = X6 , ϕ2 (X4 , X8) = X4, ϕ2 (X5, X8) = X5, ϕ2 (X6 , X8) = X6 , ϕ2 (X7 , X8) = X6, ϕ4 (X7, X8) = X7, ϕ5 (X7 , X8) = X8 . There is only one integrability condition for a deformation, namely ε1 ε4 + ε2 ε5 = 0. The resulting non-equivalent deformations are        L6,2 ⊕ 2L1 (ε1 , ε2, ε3, ε4, ε5) '      

L27,3 ⊕ L1 , (ε1 , 0, ε3, 0, 0) ε ε−1

5 1 L8,10 , L8,11, L6,2 ⊕ r2 , L8,6,

(ε1 , 0, ε3, 0, ε5) (ε1 , 0, ε3, 0, 2ε1) . (0, 0, ε3, 0, 0, ε5) (0, 0, ε3, 0, 0)

It is straightforward to verify that all these deformation can be reversed and give rise to contractions onto L6,2 ⊕ 2L1 .

9.4.

L6,2 ⊕ r2

The Lie algebra L6,2 ⊕ r2 has a one dimensional H 2 (L6,2 ⊕ r2 , L6,2 ⊕ r2 ) generated by the cocycle class

cohomology

ϕ (X4, X7) = X4, ϕ (X5, X7) = X5, ϕ (X6 , X7) = 2X6 .

space

434

R. Campoamor-Stursberg and J. Guer´on

There is obviously no integrability condition, and for any nonzero value of ε we obtain the deformation −1 L6,2 ⊕ r2 + εϕ ' L−ε 8,10 .   Since dim Der Lp8,10 ≥ 9 and dim Der (L6,2 ⊕ r2) = 9, this deformation does not give rise to a contraction.

9.5.

L6,3 ⊕ 2L1

For the decomposable algebra L6,3 ⊕ 2L1 the cohomology space is six dimensional, generated by the cocycles ϕ1 (X6 , X7) = X7, ϕ2 (X6, X8) = X7, ϕ3 (X7 , X8) = X7 , ϕ4 (X6 , X7) = X8, ϕ5 (X6, X8) = X8, ϕ6 (X7 , X8) = X8 . In any case, a deformation L6,3 ⊕ 2L1 (ε1 , .., ε6) satisfies the constraint dim [L6,3 ⊕ 2L1 (ε1 , .., ε6) , L6,3 ⊕ 2L1 (ε1 , .., ε6)] = 7. The integrability conditions for the deformation parameters are ε3 ε5 − ε2 ε6 = 0, ε1 ε6 − ε3 ε4 = 0.   L6,3 ⊕ r2,     Lp7,3 ⊕ L1,    L0 , 8,8 L6,3 ⊕ 2L1 (ε1 , .., ε6) ' p,q L  8,7 ,   p   L  8,8 ,   Lp,q , 8,9

ε4 = ε5 = ε6 = 0, ε3 6= 0 ε3 = ε4 = ε5 = ε6 = 0, ε1 ε2 6= 0 ε1 = ε2 = ε4 = ε5 = ε6 = 0 . ε1 ε5 6= 0, ε3 = ε6 = 0 ε2 6= 0, ε1 = ε5 , ε3 = ε6 = 0 ε1 = ε5 , ε2 = −ε4 , ε3 = ε6 = 0

Among these deformations, it is not difficult to see that the only contractions that can exist are L6,3 ⊕ r2

L6,3 ⊕ 2L1,

L08,8

L6,3 ⊕ 2L1,

the other being forbidden by the properties of contractions or the structure of three dimensional subalgebras [14].

9.6.

L6,4 ⊕ 2L1

For this Lie algebra we can find five independent cocycles, given respectively by ϕ1 (X4, X6) = X1, ϕ2 (X4, X8) = X4, ϕ3 (X4, X7) = X4, ϕ4 (X7, X8) = X7,

ϕ1 (X4 , X5) = −2X2, ϕ1 (X5, X6) = 2X3, ϕ2 (X5 , X8) = X5, ϕ2 (X6, X8) = X6, ϕ3 (X5 , X8) = X7, ϕ3 (X6, X7) = X6, ϕ5 (X7 , X8) = X8.

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

435

The only integrability conditions that a deformation depending on the ε0i s has to satisfy are ε1 ε2 = ε1 ε3 = 0. As a consequence of a case by case inspection, the deformation scheme, avoiding equivalent cases, can be resumed as:  so (1, 3) ⊕ 2L1,     so (2, 2) ⊕ 2L1,      so (1, 3) ⊕ r2 , so (2, 2) ⊕ r2 , L6,4 ⊕ 2L1 (ε1 , ε2, ε3, ε4, ε5) '   L6,2 ⊕ r2 ,      L7,5 ⊕ L1 ,   p L8,12,

ε1 < 0, εi = 0, i 6= 4, 5 ε1 > 0, εi = 0, i 6= 4, 5 ε1 < 0, ε4 ε5 6= 0, ε1 > 0, ε4 ε5 6= 0, . ε4 ε5 6= 0, εi = 0, i 6= 4, 5 ε2 6= 0, εi = 0, i 6= 2 ε5 = pε2 6= 0, εi = 0, i 6= 5

Basing on the deformations studied in dimensions n ≤ 7 and the structure of this algebra, it is routinary to verify that all these deformations admit an inverse, and therefore any of the deformed algebras contracts onto L6,4 ⊕ 2L1.

9.7.

L6,4 ⊕ r2

The cohomology satisfies dim H 2 (L6,4 ⊕ 2L1 , L6,4 ⊕ 2L1) = 2, spanned by ϕ1 (X4, X6) = X1, ϕ1 (X4 , X5) = −2X2, ϕ1 (X5, X6) = 2X3, ϕ2 (X4, X8) = X4, ϕ2 (X5 , X8) = X5, ϕ2 (X6, X8) = X6, with the integrability condition ε1 ε2 for a generic deformation L6,4 ⊕ r2 + ε1 ϕ1 + ε2 ϕ2 . Taking into account the deformations of L6,4 in dimension six, we obtain the deformations   so (1, 3) ⊕ r2, ε1 < 0 so (2, 2) ⊕ r2, ε1 > 0 . L6,4 ⊕ r2 + ε1 ϕ1 + ε2 ϕ2 '  ε2 ε2 6= 0 L8,12, As follows from the lower dimensional analysis, the two first deformations can be reversed and provide a contraction, while the third cannot be reversed because of the dimensions of the corresponding derivation algebras.

9.8.

L7,1 ⊕ L1

→ − 2 For L7,1 ⊕L1 = so (3) ⊕ adso(3)⊕D0 A1,1 4,5 ⊕L1 we find dim H (L7,1 ⊕ L1 , L7,1 ⊕ L1 ) = 1, generated by ϕ (X7, X8) = X8. In this case, the deformation problem is trivial, and for any value of ε we find that L7,1 ⊕ L1 + εϕ ' Lε8,1,   which cannot be reversed since dim Der (L7,1 ⊕ L1) = dim Der Lp8,1 = 9.

436

9.9.

R. Campoamor-Stursberg and J. Guer´on

L7,2 ⊕ L1

This algebra provides the highest dimensional cohomology space among the decomposable algebras, with dim H 2 (L7,2 ⊕ L1, L7,2 ⊕ L1 ) = 7. A basis of cocycles can be chosen as ϕ1 (X4 , X5) = X8, ϕ3 (X4 , X6) = X8, ϕ4 (X6 , X8) = X6, ϕ5 (X6 , X8) = X4, ϕ7 (X4 , X8) = −X6,

ϕ1 (X6 , X7) = −X8, ϕ3 (X5 , X7) = X8, ϕ4 (X7 , X8) = X7, ϕ5 (X7 , X8) = X5, ϕ7 (X5 , X8) = −X7.

ϕ2 (X4, X7) = X8 , ϕ4 (X4, X8) = X4 , ϕ5 (X4, X8) = −X6 , ϕ6 (X4, X8) = X4 ,

ϕ2 (X5, X6) = −X8 , ϕ4 (X5, X8) = X5, ϕ5 (X5, X8) = −X7 , ϕ6 (X5, X8) = X5,

In this case, the integrability conditions are rather complicated, but fortunately, many of them lead to equivalent deformations, since there are only three possible target algebras, namely L8,2, L8,3 and Lp8,4. Ignoring those cases which are equivalent, the possibilities for a formal deformation of L7,2 ⊕ L1 are:   L8,2, ε1 6= 0, εi = 0, i 6= 1 L8,3, ε4 = 6 0, εi = 0, i 6= 1 . εi ϕi ' L7,2 ⊕ L1 +  p j=1 L8,4, pε6 + ε4 = 0. 7 X

A simple computation (see Table 1) shows that any of these deformations can be inverted, thus give rise to the contractions L8,2 L7,2 ⊕L1, L8,3 L7,2 ⊕L1 and Lp8,4 L7,2 ⊕L1 for any p.

9.10. Lp7,3 ⊕ L1 with (p 6= 0) Since this is the only decomposable algebra depending on parameters, it is expected that the cohomology space will vary with respect to them. Indeed, the precise dimensions are    3, p 6= 2 dim H 2 Lp7,3 ⊕ L1 , Lp7,3 ⊕ L1 = . 4, p = 2 For p 6= 2 the cocycles classes can be taken as ϕ1 (X4, X7) = X4, ϕ1 (X5, X7) = X5 , ϕ2 (X4 , X8) = X4, ϕ2 (X5, X8) = X5 , [p = 2]. ϕ3 (X7, X8) = X8, ϕ4 (X4 , X5) = X6 while the integrability condition is ε2 ε3 = 0 in the general case and the additional constraints ε2 ε4 = ε1 ε4 if p = 2. The complete deformation scheme resulting for these Lie algebras is  −pε−1   L5,1 ⊕ A3,5 3 , ε3 6= 0, 1 + ε1 = 0   −1 −1  p(1+ε ) ,−ε (1+ε )  1 3 1  L8,7 , ε3 6= 0, 1 + ε1 6= 0   p L6,3 ⊕ r2, 2ε2 = −1 − ε1 6= 0 L7,3 ⊕ L1 (ε1 , ε2 , ε3) ' .   ⊕ r ⊕ L ε = 6 0, ε = ε = 0 L 5,1 2 1 1 2 3    3  L−ε ε4 6= 0, p = 2  8,10 ,   Lq ⊕ L ε3 6= 0, εi = 0, i 6= 3 1 7,3

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras Of these possibilities, only two contractions arise. The contraction L6,3 ⊕ r2 determined by the endomorphisms

437 Lp7,3 is

ft (X7) = t−1 X7, ft (X8 ) = pX7 + X8, ft (Xi ) = Xi , i 6= 7, 8. The other contraction L08,10 ' L7,4 ⊕ L1 L27,3 ⊕ L1 is the same as in proposition 6, where the additional generators remains unchanged.

9.11. L7,4 ⊕ L1 The cocycle ϕ (X7, X8) = X8 generates the cohomology for the Lie algebra L7,4 ⊕ L1 . Since there is no integrability condition, for any ε the deformation L7,4 ⊕ L1 + εϕ defines a Lie algebra, which is easily seen to be isomorphic to Lε8,10. By the dimension of the corresponding derivation algebras, the deformation cannot be reversed and there is no contraction.

9.12. L7,5 ⊕ L1 The Lie algebra L7,5 ⊕ L1 has also an adjoint cohomology space generated by a unique cocycle, which can be chosen as ϕ (X7 , X8) = X8 , as before. For any value of the parameter to the indecomposable algebra ε we find that the deformation L7,5 ⊕ L1 + εϕ isisomorphic 

Lε8,12. Since dim Der (L7,5 ⊕ L1) = dim Der Lp8,12 = 9, no contraction of Lp8,12 onto L7,5 ⊕ L1 can exist.

9.13. L7,6 ⊕ L1 Although separately the Lie algebras L7,6 and L1 are stable14 , their direct sum is not, and in fact it satisfies the equality dim H 2 (L7,6 ⊕ L1 , L7,6 ⊕ L1) = 2, generated by the cocycles ϕ1 (X4 , X7) = X8 , ϕ1 (X5, X6) = −3X8, ϕ2 (X4 , X8) = X4 , ϕ2 (X7 , X8) = X7 . ϕ2 (X5 , X8) = X5 , ϕ2 (X6, X8) = X6, For a formal deformation (L7,6 ⊕ L1 ) (ε1 , ε2) the integrability condition is ε1 ε2 = 0. It is trivial to verify that  L8,19, ε1 6= 0 , L7,6 ⊕ L1 + ε1 ϕ1 + ε2 ϕ2 ' L8,20, ε2 6= 0 and that both deformations are of plateau type, thus are reversible to give the contractions L8,19 L7,6 ⊕ L1 and L8,20 L7,6 ⊕ L1 .

9.14. L7,7 ⊕ L1 This Lie algebra, having the same complexification as L7,2 ⊕ L1 , has also a seven dimensional cohomology space. A basis of cocycles can be taken as ϕ1 (X4, X8) = X4, ϕ1 (X5 , X8) = X5, ϕ2 (X6 , X8) = X4, ϕ2 (X7, X8) = X5, ϕ3 (X4, X8) = X6, ϕ3 (X5 , X8) = X7, ϕ4 (X6 , X8) = X6, ϕ4 (X7, X8) = X7, ϕ5 (X4, X5) = X8, ϕ6 (X4 , X7) = X8, ϕ6 (X5, X6) = −X8 , ϕ7 (X6, X7) = X8. 14

This happens because L7,6 , although stable, is not semisimple.

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For a generic deformation L7,7 ⊕ L1 (ε1 , .., ε7), the integrability constraints are the most complicated of the studied algebras ε1 ε5 + ε3 ε6 = 0, ε4 ε7 + ε2 ε6 = 0, ε3 ε6 − ε4 ε5 = 0, ε4 ε6 − ε3 ε7 = 0, ε1 ε6 + ε3 ε7 + ε2 ε5 + ε4ε6 = 0.

ε1 ε6 − ε2 ε5 = 0, ε2 ε6 − ε1 ε7 = 0,

After cumbersome computations, and ignoring equivalent cases, the possibilities for deformations are resumed in the following scheme:  ±1  L8,13, ε5 ε7 6= 0, εi = 0, i 6= 5, 7    L±1  ε6 6= 0, εi = 0, i 6= 6  8,13    L ,  8,14 ε2 6= 0, εi = 0, i 6= 2 L7,7 ⊕ L1 (ε1 , .., ε7) ' . L8,15, ε2 ε7 6= 0, εi = 0, i 6= 2, 7   L  , ε ε ε = 6 0, ε = 0, i = 6 1, 2, 4 8,16 1 2 4 i     Lp8,17, ε1 ε4 6= 0, εi = 0, i 6= 1, 4    Lp , ε = ε 6= 0, ε = −ε 6= 0 1 4 2 3 8,18 As could be expected from the structure of L7,7 ⊕L1 , all these deformations can be reversed, we thus obtain a contraction of any of the target algebras onto the decomposable Lie algebra L7,7 ⊕ L1.

10. Conclusion We have determined all deformations of eight dimensional real Lie algebras with a nontrivial Levi decomposition. Once these deformations were obtained, those being invertible were computed, and the corresponding contractions of Lie algebras obtained. Due to the transitivity properties of contractions, the analysis provides the contractions among this class of Lie algebras. Decomposable algebras were also considered, which provided additional possibilities and confirmed some properties of contractions and deformations already commented in [40]. It should be remarked that the same problem for the nine dimensional case, as classified in [27], is much more complicated, not only because of the high number of isomorphism classes, but also to the presence of many continuous parameters in several families. For a satisfactory approach to this dimension, alternative criteria that allow to reduce the number of cases to be analyzed should be developed. In contrast to dimension eight, however, the analysis of contractions of semisimple algebras is muss less interesting, due to the non-existence of simple algebras in dimension nine. Therefore, determining their contractions essentially reduces to the lower dimensional cases. In this context, it is worthy to recall those Lie algebras that have appeared as a contraction of an eight dimensional semisimple Lie algebra, decomposable or not. The contraction diagram, following the results already obtained in [24], are reproduced in Figure 2.

Acknowledgment During the preparation of this work, the first author (RCS) was financially supported by the research project MTM2006-09152 of the M.E.C. and the project and CCG07-UCM/ESP2922 of the U.C.M.-C.A.M.

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

439

Table 1. Turkowski classification in dimension eight. Algebra L8,1 L8,2

L8,3 Lp8.4

L8,5

L8,6 Lp,q 8,7 pq 6= 0 Lp8,8 Lp,q 8,9 q 6= 0 Lp8,10 L8,11 L8,12 Lε8,13 L8,14 L8,15 L8,16 Lp8,17 Lp8,18

L8,19 L8,20 L8,21 L8,22

Structure constants 1 3 2 6 5 6 4 5 C23 = 1, C12 = 1, C13 = −1, C15 = 1, C16 = −1, C24 = −1, C26 = 1, C34 = 1, j 4 7 C35 = −1, Cj8 = 1 (4 ≤ j ≤ 7) , C78 = p. 1 3 2 7 6 5 4 5 C23 = 1, C12 = 1, C13 = −1, C14 = 12 , C15 = 12 , C16 = − 12 , C17 = − 12 , C24 = 12 , 1 1 1 1 1 1 1 4 7 6 6 7 4 5 8 C25 = 2 , C26 = 2 , C27 = − 2 , C34 = 2 , C35 = − 2 , C36 = − 2 , C37 = 2 , C45 = 1, 8 C67 = −1. 1 3 2 7 6 5 4 5 C23 = 1, C12 = 1, C13 = −1, C14 = 12 , C15 = 12 , C16 = − 12 , C17 = − 12 , C24 = 12 , 1 1 1 1 1 1 1 4 7 6 6 7 4 5 4 C25 = 2 , C26 = 2 , C27 = − 2 , C34 = 2 , C35 = − 2 , C36 = − 2 , C37 = 2 , C48 = 1, 5 6 7 C58 = 1, C68 = 1, C78 = 1. 1 3 2 7 6 5 4 5 C23 = 1, C12 = 1, C13 = −1, C14 = 12 , C15 = 12 , C16 = − 12 , C17 = − 12 , C24 = 12 , 1 1 1 1 1 1 1 4 7 6 6 7 4 5 48 C25 = 2 , C26 = 2 , C27 = − 2 , C34 = 2 , C35 = − 2 , C36 = − 2 , C37 = 2 , C48 = p, 5 6 7 6 7 4 5 C58 = p, C68 = p, C78 = p, C48 = −1, C58 = −1, C68 = 1, C78 = 1. 1 1 1 3 2 7 6 5 8 4 C23 = 1, C12 = 1, C13 = −1, C14 = 2 , C15 = − 2 , C16 = 2, C16 = −1, C17 = −2, 1 1 6 6 7 4 5 8 7 5 C18 = 3, C24 = 2 , C25 = 2 , C26 = −2, C27 = −2, C27 = −1, C28 = 3, C34 = 2, 4 7 6 C35 = −2, C36 = 1, C37 = −1. 2 3 1 4 5 4 5 8 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C67 = 1. 2 3 1 4 5 4 5 4 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C48 = 1, 5 6 7 C58 = 1, C68 = p, C78 = q. 2 3 1 4 5 4 5 4 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C48 = 1, 5 6 6 7 C58 = 1, C68 = p, C78 = 1, C78 = p. 2 3 1 4 5 4 5 4 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C48 = 1, 5 6 7 7 7 C58 = 1, C68 = p, C68 = −q, C78 = q, C78 = p. 2 3 1 4 5 4 5 4 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C48 = 1, 5 6 7 C58 = 1, C68 = 2, C78 = p. 2 3 1 4 5 4 5 4 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C48 = 1, 5 6 6 7 C58 = 1, C68 = 2, C78 = 1, C78 = p. 2 3 1 4 6 4 5 4 C12 = 2, C13 = −2, C23 = 1, C14 = 2, C16 = −2, C25 = 2, C26 = 1, C34 = 1, 5 4 5 6 C35 = 2, C48 = 1, C58 = 1, C68 = 1, 2 3 1 4 5 4 5 6 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C16 = 1, 7 6 7 8 8 C17 = −1, C27 = 1, C36 = 1, C45 = 1, C67 = ε. 2 3 1 4 5 4 5 6 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C16 = 1, 7 6 7 4 5 C17 = −1, C27 = 1, C36 = 1, C48 = 1, C58 = 1. 2 3 1 4 5 4 5 6 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C16 = 1, 7 6 7 8 4 5 C17 = −1, C27 = 1, C36 = 1, C67 = 1, C68 = 1, C78 = 1. 2 3 1 4 5 4 5 6 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C16 = 1, 7 6 7 4 5 4 6 5 7 C17 = −1, C27 = 1, C36 = 1, C48 = 1, C58 = 1, C68 = 1, C68 = 1, C78 = 1, C78 = 1. 2 3 1 4 5 4 5 6 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C16 = 1, 7 6 7 4 5 6 7 C17 = −1, C27 = 1, C36 = 1, C48 = 1, C58 = 1, C68 = p, C78 = p. 2 3 1 4 5 4 5 6 C12 = 2, C13 = −2, C23 = 1, C14 = 1, C15 = −1, C25 = 1, C34 = 1, C16 = 1, 7 6 7 4 6 5 7 4 C17 = −1, C27 = 1, C36 = 1, C48 = p, C48 = −1, C58 = p, C58 = −1, C68 = 1, 6 5 7 C68 = p, C78 = 1, C78 = p. 2 3 1 4 5 6 7 4 C12 = 2, C13 = −2, C23 = 1, C14 = 3, C15 = 1, C16 = −1, C17 = −3, C25 = 3, 5 6 5 6 7 8 8 C26 = 2, C27 = 1, C34 = 1, C35 = 2, C36 = 3, C47 = 1, C56 = −3. 2 3 1 4 5 6 7 4 C12 = 2, C13 = −2, C23 = 1, C14 = 3, C15 = 1, C16 = −1, C17 = −3, C25 = 3, 5 6 5 6 7 4 5 6 7 C26 = 2, C27 = 1, C34 = 1, C35 = 2, C36 = 3, C48 = 1, C58 = 1, C68 = 1, C78 = 1. 2 3 1 4 5 7 8 4 C12 = 2, C13 = −2, C23 = 1, C14 = 4, C15 = 2, C17 = −2, C18 = −4, C25 = 4, 5 6 7 5 6 7 8 C26 = 3, C27 = 2, C28 = 1, C34 = 1, C35 = 2, C36 = 3, C37 = 4. 2 3 1 4 6 7 8 4 C12 = 2, C13 = −2, C23 = 1, C14 = 2, C16 = −2, C17 = 1, C18 = −1, C25 = 2, 5 7 5 6 8 C26 = 1, C28 = 1, C34 = 1, C35 = 2, C37 = 1.

440

R. Campoamor-Stursberg and J. Guer´on Table 2. Fundamental characteristics of the Lie algebras L8,k

g

dim Der (g)

N (g)

dim [g, g]

dim H 2 (g, g)

Lp8,1

9

2

7

1

L8,2

9

2

8

1

L8,3

11

0

7

Lp8,4

9

L8,5

9

2

8

L8,6

11

2

6

2

7

Lp,q 8,7

(

12, p = q

Lp8,10

(

0, p 6= 0

7

2, p = 0

10, p 6= q

Lp8,8 Lp,q 8,9

(

(

6, p = 0

10

2

10

2

7

2

7

9,

p 6= 2

10, p = 2

(

3 1, p 6= 0 2, p = 0 1

4   2, p = 6 q, p 6= 2, q 6= 2     3 p 6= q, p = 2 or q = 2  4 p = q, p 6= 2     6 p=q=2 ( 3, p = 0, 2

7, p 6= 0

2 (

p 6= 0, 2 2

1, p 6= 2 2, p = 2

L8,11

9

2

7

1

Lp8,12 Lε8,13

9

2

7

1

9

2

8

1

L8,14

10

2

7

3

L8,15

9

2

8

1

0

7

9

L8,16 Lp8,17

(

9,

p 6= 1

(

11, p = 1

0, p 6= −1 2, p = −1

(

0, p 6= −1

7

1     2, p = −1

1, p 6= ±1    3, p = 1 ( 1, p 6= 0

Lp8,18

9

L8,19

8

2

8

0

L8,20

8

0

7

0

L8,21

9

2

8

1

L8,22

10

2

8

1

2, p = −1

7

2, p = 0

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

441

Table 3. Adjoint cohomology spaces for indecomposable eight dimensional algebras g L8,1 L8,2

L8,3

Lp8,4 L08,4

L8,5 L8,6 Lp,q 8,7 Lp,p 8,7 (p 6= 2) 2,2 L8,7 L08,8 L28,8 Lp8,8 (p 6= 0, 2) Lp,q 8,9 Lp8,10 L28,10 L8,11 L8,12 Lε8,13 L8,14 L8,15 L8,16 =±1 Lp68,17 −1 L8,17 L08,18

L8,21

Cocycle basis of H 2 (g, g) ϕ (X7 , X8 ) = X7 . ϕ (X4 , X5 ) = X2 , ϕ (X4 , X6 ) = X3 , ϕ (X4 , X7 ) = X1 , ϕ (X4 , X8 ) = − 32 X5 , ϕ (X5 , X6 ) = X1 , ϕ (X5 , X7 ) = −X3 , ϕ (X5 , X8 ) = 32 X4 , ϕ (X6 , X7 ) = X2 , ϕ (X6 , X8 ) = 32 X7 , ϕ (X7 , X8 ) = − 32 X6 . ϕ1 (X4 , X8 ) = X6 , ϕ1 (X5 , X8 ) = X7 , ϕ1 (X6 , X8 ) = −X4 , ϕ1 (X7 , X8 ) = −X5 , ϕ2 (X4 , X8 ) = X5 , ϕ2 (X5 , X8 ) = −X4 , ϕ2 (X6 , X8 ) = −X7 , ϕ2 (X7 , X8 ) = X6 , ϕ3 (X4 , X8 ) = X7 , ϕ3 (X5 , X8 ) = −X6 , ϕ3 (X6 , X8 ) = X5 , ϕ3 (X7 , X8 ) = −X4 . ϕ (X4 , X8 ) = X4 , ϕ (X5 , X8 ) = X5 , ϕ (X6 , X8 ) = X6 , ϕ (X7 , X8 ) = X7 . ϕ1 (X4 , X5 ) = X2 , ϕ1 (X4 , X6 ) = X3 + 32 X8 , ϕ1 (X4 , X7 ) = X1 , ϕ1 (X5 , X6 ) = X1 , ϕ1 (X5 , X7 ) = −X3 + 32 X8 , ϕ1 (X6 , X7 ) = X2 . ϕ2 (X4 , X8 ) = X4 , ϕ2 (X5 , X8 ) = X5 , ϕ2 (X6 , X8 ) = X6 , ϕ2 (X7 , X8 ) = X7 . ϕ (X4 , X5 ) = X3 , ϕ (X4 , X6 ) = X2 , ϕ (X4 , X7 ) = X1 , ϕ (X5 , X6 ) = −X1 , ϕ (X5 , X7 ) = X2 , ϕ (X6 , X7 ) = 2X3 , ϕ (X6 , X8 ) = −6X1 , ϕ (X7 , X8 ) = −6X2 . ϕ1 (X4 , X5 ) = 2X6 , ϕ1 (X4 , X7 ) = X4 , ϕ1 (X5 , X7 ) = X7 ; ϕ2 (X6 , X7 ) = X6 , ϕ3 (X4 , X5 ) = −2X7 , ϕ3 (X4 , X6 ) = X4 , ϕ3 (X5 , X6 ) = X5 ; ϕ4 (X6 , X7 ) = X7 . ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X7 , X8 ) = X7 . [p 6= q, p 6= 2, q 6= 2] ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X7 , X8 ) = X7 ; ϕ3 (X4 , X5 ) = X6 . [p 6= q, p = 2 or q = 2] ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X7 , X8 ) = X7 ; ϕ3 (X6 , X8 ) = X7 ; ϕ4 (X7 , X8 ) = X6 . ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X7 , X8 ) = X7 ; ϕ3 (X6 , X8 ) = X7 ; ϕ4 (X7 , X8 ) = X6 , ϕ5 (X4 , X5 ) = X6 ; ϕ6 (X4 , X5 ) = X7 . ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X6 , X8 ) = X7 ; ϕ3 (X6 , X7 ) = X6 . ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X6 , X8 ) = X7 ; ϕ3 (X4 , X5 ) = X6 . ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X6 , X8 ) = X7 . ϕ1 (X6 , X8 ) = X6 ; ϕ2 (X6 , X8 ) = X7 . ϕ1 (X4 , X8 ) = X4 , ϕ1 (X5 , X8 ) = X5 , ϕ1 (X6 , X8 ) = 2X6 . ϕ1 (X4 , X8 ) = X4 , ϕ1 (X5 , X8 ) = X5 , ϕ1 (X6 , X8 ) = 2X6 ; ϕ2 (X7 , X8 ) = X6 . ϕ (X4 , X8 ) = X4 , ϕ (X5 , X8 ) = X5 , ϕ (X6 , X8 ) = 2X6 . ϕ (X7 , X8 ) = X7 . ϕ (X4 , X6 ) = −2X2 , ϕ (X4 , X7 ) = X1 , ϕ (X4 , X8 ) = −3εX6, ϕ (X5 , X6 ) = X1 , ϕ (X5 , X7 ) = 2X3 , ϕ (X5 , X8 ) = −3εX7 , ϕ (X6 , X8 ) = 3X4 , ϕ (X7 , X8 ) = 3X5 . ϕ1 (X4 , X8 ) = X4 , ϕ1 (X5 , X8 ) = X5 ; ϕ2 (X4 , X8 ) = X6 , ϕ2 (X5 , X8 ) = X7 ; ϕ3 (X6 , X7 ) = X8 . ϕ (X4 , X5 ) = 3X8 , ϕ (X4 , X6 ) = −2X2 , ϕ (X4 , X7 ) = X1 , ϕ (X4 , X8 ) = −3X6 , ϕ (X5 , X6 ) = X1 , ϕ (X5 , X7 ) = 2X3 , ϕ (X5 , X8 ) = −3X7 . ϕ (X4 , X8 ) = X6 , ϕ1 (X5 , X8 ) = X7 . ϕ1 (X6 , X8 ) = X6 , ϕ1 (X7 , X8 ) = X7 . ϕ1 (X4 , X6 ) = −2X2 , ϕ1 (X4 , X7 ) = X1 − 3X8 , ϕ1 (X5 , X6 ) = X1 + 3X8 , ϕ1 (X5 , X7 ) = 2X3 ; ϕ2 (X6 , X8 ) = X6 , ϕ2 (X7 , X8 ) = X7 . ϕ1 (X4 , X5 ) = 3X8 , ϕ1 (X4 , X6 ) = −2X2 , ϕ1 (X4 , X7 ) = X1 , ϕ1 (X5 , X6 ) = X1 , ϕ1 (X5 , X7 ) = 2X3 , ϕ1 (X6 , X7 ) = 3X8 . ϕ2 (X6 , X8 ) = X6 , ϕ2 (X7 , X8 ) = X7 . ϕ (X4 , X7 ) = −2X2 , ϕ (X4 , X8 ) = X1 , ϕ (X5 , X6 ) = 6X2 , ϕ (X5 , X7 ) = −2X1 , ϕ (X5 , X8 ) = 2X3 , ϕ (X6 , X7 ) = −6X3 .

442

R. Campoamor-Stursberg and J. Guer´on Table 4. Structure constants of indecomposable algebras with non-trivial Levi decomposition dim g ≤ 7 [26]

Algebra Structure tensor L5,1 L6,1

L6,2 L6,3 L6,4

L7,1

L7,2

L7,3

L7,4

L7,5

L7,6

L7,7

2 C12 4 C25 1 C23 5 C16 4 C35 2 C12 4 C25 2 C12 4 C25 2 C12 6 C16 6 C35 1 C23 5 C16 4 C35 1 C23 6 C15 4 C25 7 C35 2 C12 5 C15 5 C57 2 C12 5 C15 4 C47 2 C12 6 C16 5 C35 2 C12 5 C15 5 C26 7 C36 2 C12 5 C15 6 C16

= = = =

Invariants

3 2, C13 5 1, C34 3 1, C12

1 4 = −2, C23 = 1, C14 = 1, 5 = 1, C15 = −1. 2 6 = 1, C13 = −1, C15 = 1, 6 4 5 −1, C24 = −1, C26 = 1, C34 = 1,

I1 = x3 x24 − x1 x4 x5 − x2 x25 I1 = x24 + x25 + x26 I2 = x1 x4 + x2 x5 + x3 x6

= −1. 3 1 4 = 2, C13 = −2, C23 = 1, C14 = 1,

= = = = =

5 1, C34 3 2, C13 5 1, C34 3 2, C13

5 1, C15

6 = = −1, C45 1 4 = −2, C23 = 1, C14 j 5 = 1, C15 = −1, Cj7 1 4 = −2, C23 = 1, C14 4 5 5 −2, C25 = 2, C26 = 1, C34

I1 = 2x2 x3 x6 + x25 x2 + x1 x4 x5 − x3 x24 + 12 x21 x6

= 1.

I2 = x6

= 1,

—

= 1.(j = 5, 6) = 2,

I1 = x25 − 4x4 x6

= 1,

I2 = x1 x5 + 2x2 x6 − 2x3 x4

= 2. 3 2 6 = 1, C12 = 1, C13 = −1, C15 = 1,


I1 = x24 + x25 + x26 (x1 x4 + x2 x5 + x3 x6 )−2

6 4 5 = −1, C24 = −1, C26 = 1, C34 = 1, j = −1, Cj7 = 1 (4 ≤ j ≤ 7) . 3 2 7 = 1, C12 = 1, C13 = −1, C14 = 12 ,

I1 = x24 + x25 + x26 + x27

5 4 5 = 12 , C16 = − 12 , C17 = − 12 , C24 = 12 , 7 6 6 = 12 , C26 = 12 , C27 = − 12 , C34 = 12 , 4 5 = − 21 , C36 = − 12 , C37 = 12 . 3 1 4 = 2, C13 = −2, C23 = 1, C14 = 1,

= = = = = = = = = = =

4 5 4 −1, C25 = 1, C34 = 1, C47 = 1, 6 1, C67 = p (p 6= 0) . 3 1 4 2, C13 = −2, C23 = 1, C14 = 1, 4 5 6 −1, C25 = 1, C34 = 1, C45 = 1, 5 6 1, C57 = 1, C67 = 2. 3 1 4 2, C13 = −2, C23 = 1, C14 = 2, 4 5 4 −2, C25 = 2, C26 = 1, C34 = 1, j 2, Cj7 = 1 (j =, 4, 5, 6) . 3 1 4 2, C13 = −2, C23 = 1, C14 = 3, 6 7 4 1, C16 = −1, C17 = −3, C25 = 3, 6 5 6 2, C27 = 1, C34 = 1, C35 = 2,

I1 = x3 x24 − x1 x4 x5 − x2 x25

= =

4 −1, C25 7 1, C17 =

6 5 = 1, C27 = 1, C34 7 −1, C36 = 1.

= 1,

x26

I1 = (x25 x2 + x1 x4 x5 − x3 x24 )x−1 6 + 2x2 x3 + + 12 x21 I1 = (x1 x5 + 2x2 x6 − 2x3 x4 )2 x25 − 4x4 x6

I1 = 27x24 x27 − 18x4 x5 x6 x7 − x25 x26 +
+4 x36 x4 + x7 x35

= 3. 3 1 4 = 2, C13 = −2, C23 = 1, C14 = 1,


−p

I 1 = x 4 x 7 − x5 x 6


−1

Deformation and Contraction Schemes for Non-solvable Real Lie Algebras

443

Table 5. Generalized Casimir invariants in dimension eight Algebra L8,1 L8,2

Lp8.4

L8,5 L8,6 Lp,q 8,7 pq 6= 0 Lp8,8 p 6= 0 L08.8 Lp,q 8,9 q 6= 0 Lp8,10 L8,11 L8,12 Lε8,13 ε = ±1 L8,14 L8,15 L−1 8,17 L08,18 L8,19

L8,21 L8,22

Non-constant invariants
p I1 = x24 + x25 + x26 x−2 7 p I2 = (x1 x4 + x2 x5 + x
3x6) x−1 7
2 2 2 2 I1 = 16 x1 + x2 + x3 x8 + x44 + x45 + x46 + x47
2 2 2 2 2 2 +2 x24x25 + x24x26 + x24 x27 + x
5x6 + x5 x7 + x6x7 2 2 2 2 −8x2 x8 x7 + x6 − x4 − x5 − 16 (x4x6 − x5x7) x1 x8 + 16x3x8 (x4x7 + x5 x6) . I2 = x8 2 2 I1 = x24 + x25 +
x6 + x7
2 2 I2 = x4 + x6 (x8 − 2x3) + x25 + x27 x8+
+4 (x2x4 x7 − x1 x4x5 − x2 x5x6 − x
1x6 x7) + 2x3
x25 + x27 I1 = 29 x38 + x8 x
27 + x26 − x25 −
8x24 + 6 x26 − x27 x5 − 12x4x6x7 I2 = 12 x24 + x25 + 3 x26 + x27 + x28 I1 = x8 I2 = 2x1 x4x5 − 2x3x24 + 2x2
x25 + 4x2x3x8 + x21 x8 p I1 = x3x24 − x1 x4x5 − x2x25 x−2 6
q I2 = x3x24 − x1 x4x5 − x2x25
x−2 7 p I1 = x3x24 − x1 x4x5 − x2x25 x−2 6 −1 I2 = (p ln x7 − x6 ln x6) (px6) I1 = x6
I2 = 2x7 − x6 ln x3 x24 − x1x4 x5 − x2x25 p−iq p+iq I1 = (x7 − ix6 ) (x7 + ix6 )
p2 +q2 2(iq−p) I2 = x3x24 − x1 x4x5 − x25x2 (x7 − 6)
ix−1 2 2 I1 = 2x2x3x6 + x5 x2 + x1x4x5 − x3 x4 x6 + 12 x21 I2 = x27x−p 6
1 2 I1 = 2x2x3x6 + x25 x2 + x1x4x5 − x3 x24 x−1 6 + 2 x1 x6 −1 I2 = (2x7 − x6 ln
x6 ) x6 p I1 = x25 − 4x4x6 x−2 7 p I2 = (x1 x5 − 2x3x4 + 2x2 x6 ) x−2 7 I1 = x8


I2 = εx28 4x2x3 + x21 + 2x2x8 x27 + εx25 − 2x3x8 εx24 + x26 +2x6 x7 (x1 x8 + x4x5 ) − x24 x27 I1 = x4x7 − x5 x6 I2 = x1x4x5 + x5x6x8 − x4 x7x8 + x2x25 − x3 x24 I1 = x4x7 − x5 x6 − 12 x28 I2 = x1x4x5 + x5x6x8 − x4 x7x8 + x2x25 − 13 x38 − x3x24 I1 = x4x7 − x5 x6 I2 = x1 (x4x7 + x5 x6) + 2x2x5 x7 − 2x3x4x6 + x8 (x4x7 − x5x6 ) I1 = x4x7 − x5 x6
I2 = x1 (x4x7 + x5 x6) + x2 x25 + x27 − x3 x24 + x8 (x5 x6 − x4x7 ) − x3 x26 I1 = x8

I2 = 12 x2 x3x28 + x2x5 x7x8 − x3x4 x6x8 + 4 x4 x36 + x3 x25x8 + x35x7 − x2x26x8 +18 (x1 x4x7x8 − x4x5x6 x7) + 27x24x27 + 3x21x28 − 2x1x5x6 x8 − x25x26 I1 = 3x5 x7 − x26 − 12x4 x8 I2 = −2x36 + 9x5x6x7 + 72x4 x6 x8 − 27x4 x27 − 27x25x8 I1 = x25 − 4x4x6 I2 = x4x28 + x6 x27 − x5 x7x8

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L7,2 ⊕ L1

L8,2 

-

-

L8,5

-

?

L08,4  - L8,15 

L−1 8,17

- ? - L8,14

L8,21

--

-

-





-

?

L−1 8,13

su(2, 1)

 

sl(3, R)

?

L18,13

L08,18



L7,7 ⊕ L1

Figure 2. Non-solvable contractions of eight-dimensional simple Lie algebras [24].

References [1] I. E. Segal, Duke Math. J 18, 221 (1951) [2] E. In¨on¨u and E. P. Wigner, Proc. Natl. Acad. Sci. U.S. 39, 510 (1953) [3] E. Saletan, J. Math. Phys. 2, 1 (1961); M. Kupczy´nski, Comm. Math. Phys. 13, 154 (1969) [4] G. Berendt, Acta Phys. Austriaca 25, 207 (1967) [5] M. Levy-Nahas, J. Math. Phys. 8, 1211 (1967) [6] Ya. Kh. Lykhmus, Predel’nye gruppy Li, (Institut Fiziki AN Estonskoi’ SSR, Tartu, 1969) [7] C. P. Boyer, Rev. Mexicana F´ıs. 23, 99 (1974) [8] R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications , (Wiley, N.Y., 1974)

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[9] E. Weimar-Woods, Rev. Math. Phys. 12, 1505 (2000) [10] J. F. Cari˜nena, J. Grabowski and G. Marmo, J. Phys. A: Math. Gen. 34, 3769 (2001) [11] C. W. Conatser, J. Math. Phys. 13, 196 (1972) [12] P. L. Huddleston, J. Math. Phys. 19, 1645 (1978) [13] E. Weimar-Woods, J. Math. Phys. 32, 2028 (1991) [14] M. de Montigny, A. Fialowski, J. Phys. A: Math. Theor. 39, 6335 (2006) [15] I. Nesterenko and R. O. Popovych, J. Math. Phys. 47, 123515 (2006) [16] A. Fialowski and M. Penkava, Int. J. Theor. Phys. 47, 561 (2008) [17] H. Bacry and J.-M. Levy-L´eblond, J. Math. Phys. 9, 1605 (1968) [18] J. Figueroa-O’Farrill, J. Math. Phys. 30, 2375 (1989) [19] A. Nijenhuis and R. W. Richardson, Bull. Amer. Math. Soc. 72, 1 (1966) [20] G. Hochschild and J.-P. Serre, Ann. Math. 57, 591 (1953) [21] R. Vilela Mendes, J. Phys. A: Math. Gen. 27, 8091 (1994) [22] J. A. de Azc´arraga, J. M. Izquierdo, M. Pic´on and O. Varela, Int. J. Theor. Phys. 46, 2738 (2007) [23] J. A. de Azca’rraga, J. M. Izquierdo, J. C. Pe´rez Bueno, Rev. R. Acad. Cienc. Exactas Fi´s. Nat. Ser. A Mat. 95, 225 (2001) [24] R. Campoamor-Stursberg, J. Phys. A: Math. Theor. 40, 14773 (2007) [25] R. W. Richardson and S. Page, Trans. Amer. Math. Soc. 127, 302 (1967) [26] P. Turkowski, J. Math. Phys. 29, 2139 (1988) [27] P. Turkowski, Linear Alg. Appl. 171, 191 (1992) [28] H. Bacry and J. Nuyts, J. Math. Phys. 27, 2455 (1986) [29] A. A. Kirillov, Elements of Representation Theory (Springer Verlag, N.Y., 1976) [30] J. A. de Azc´arraga and J. M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and some Applications to Physics , (Cambridge Univ. Press, Cambridge, 1995) [31] C. Chryssomalakos and E. Okon, Int. J. Mod. Phys. D 13, 1817 (2004) [32] A. K. Tolpygo, Mat. Zametki 42, 251 (1972) [33] R. Carles, Ann. Inst. Fourier 34, 65 (1984) [34] V. D. Lyakhovsky, Comm. Math. Phys. 11, 131 (1968)

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[35] J. Lˆohmus and R. Tammelo, Hadronic J. 20, 361 (1997) [36] R. Campoamor-Stursberg, Acta Physica Polonica B 34, 3901 (2003) [37] C. Quesne, J. Phys. A: Math. Gen. 21, L321 (1988) [38] R. Campoamor-Stursberg, J. Phys. A: Math. Gen. 38, 4187 (2005) [39] P. Novotn´y and J. Hrivn´ak, Bulg. J. Phys. 33, 321 (2006) [40] R. Campoamor-Stursberg, Physics Letters A 362, 360 (2007) [41] L. Abellanas and L. Martinez Alonso, J. of Math Phys 16, 1580 (1975) [42] J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, J. of Math. Phys. 17, 986 (1976) [43] F.J. Herranz, J.C. Perez-Bueno and M. Santander, J. Phys. A: Math. Gen. 31, 5327 (1998) [44] R. Campoamor-Stursberg, J. Phys. A: Math. Gen. 39, 2325 (2004) [45] Y. Folly, Rend. Sem. Fac. Sci. Univ. Cagliari 67, 1 (1997) [46] A. L. Onishchik, Lectures on real semisimple Lie algebras and their representations , (ESI, Z¨urich, 2004)

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 447-483

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 16

T HE AUTOMORPHISM G ROUPS OF S OME G EOMETRIC S TRUCTURES ON O RBIFOLDS A.V. Bagaev∗ and N.I. Zhukova† Nizhny Novgorod State University after N. I. Lobachevsky Nizhny Novgorod, Russia

Abstract The Ehresmann’s theorem about a Lie structure in the hole automorphism group of a finite type G-structure on manifold is generalized to orbifolds. Estimates for dimension of such Lie group are established, depending on stratifications of orbifolds. Particular attention is devoted to affine connected, pseudoRiemannian and Riemannian orbifolds. The content is illustrated by examples.

Key Words: Lie group of transformations, G-structure, automorphism, isometry, orbifold, pseudo-Riemannian structure, Loretzian structure, Riemannian structure AMS Subject Classification: 53C10, 53C15, 53C21, 53C50, 57R55.

Introduction Orbifold can be regard as a manifold with singularities. The topological space of ndimensional orbifold is locally homeomorphic to a quotient space of Rn by a finite group Γ of diffeomorphisms of Rn . The group Γ is not fixed and can be changed by passing from the one chart of an orbifold to an other chart. Orbifolds were introduced by Satake [29]. They were named V -manifolds by him. Orbifolds appear naturally in many branches of mathematics and mathematical physics. A theory of deformation quantization on symplectic orbispaces, which include symplectic orbifolds, is developed in [28]. For example, symplectic reduction often gives rise to orbifolds [22]. Orbifolds are used in string theory [13]. Orbifolds arise in foliation theory as ∗ †

E-mail address: an [email protected] E-mail address: [email protected]

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A.V. Bagaev and N.I. Zhukova

“good“ spaces of leaves [36]. Famous result s of Thurston [33] on classification of closed 3-manifolds use the classification of 2-dimensional orbifolds. The problem of a finding of conditions guaranteeing existence of Lie structure for transformation group is one of the central problems of differential geometry [19]. Myers and Steenrod [26] proved that the group of all isometries of a Riemannian manifold is a Lie group. Kobayashi [18] demonstrated that the group of all conformal transformations of a Riemannian manifold is a Lie group. The theorem that the automorphism group of a finite type G-structure on a manifold admits a Lie group structure is due to Ehresmann [14]. Nomizu [27] proved that the group of all affine transformations of a complete affinely connected manifold is a Lie group. Later Hano and Morimoto [17] have received this result without the assumption of completeness. The isometry groups of pseudo-Riemannian and Lorentzian manifolds is devoted numerous paper of authors Zimmer [38], D’Ambra and Gromov [12], Adams and Stuck [1, 2], Zeghib [34, 35]. We generalize the Ehresmann’s theorem, mentioned above, and prove that the automorphism group A(N ) of a G-structure of finite type and order m on a smooth n-dimensional orbifold is a Lie group of dimension at most d(n, g) := n+dim g+dim g1 +. . .+dim gm−1 , where gi is the ith prolongation of the Lie algebra g of G; moreover, the group A(N ) admits a unique topology and a unique smooth structure that makes it into a Lie group (Theorem 3.1). The presence of orbifold points is shown to sharply decrease the dimension of the transformation group A(N ) of an orbifold, with the equality dim A(N ) = d(n, g) is possible only in the case when N is a smooth homogeneous space with transitive action of A(N ) (Theorem 3.1). We investigate an influence of the existence of k-dimensional stratum ∆k of N on the dimension of the automorphism group A(N ). It is shown that a G-structure on an orbifold N inducts G-structure on each connected component ∆ck of the stratum ∆k (subsection 2.5). We observe that the subgroup A(N , ∆ck ) consisting from automorphisms of A(N ) which preserve the connected component ∆ck is an open-closed Lie subgroup of the Lie group A(N ) (Proposition 3.1). Hence, A(N , ∆ck ) has the same dimension as the Lie group A(N ). Using this observation we get some estimates of the dimension of the Lie group A(N ) (Theorems 3.3 and 3.4). The specific character of automorphism groups of G-structures on good orbifolds is indicated (Theorem 3.2). Some classes of geometric structures on orbifolds are considered: affine connected, pseudo-Riemannian and Riemannian orbifolds (Section 4). In particular, we demonstrate that a pseudo-Riemannian metric of signature (p, q) on a n-dimensional orbifold N induces pseudo-Riemannian metrics on each connected component ∆ck and on the closure ∆ck of ∆ck (Proposition 2.9). It is shown that the induced pseudo-Riemannian metric on ∆cs can have arbitrary signature (k, l) where 0 ≤ k ≤ p, 0 ≤ l ≤ q, s = k + l < n, in general (Examples 4.1 and 4.2). Estimates are established for the dimension of the isometry group of pseudo-Riemannian orbifolds, depending on the types of orbifold points (Theorem 4.2). For each n ≥ 3 we construct n-dimensional compact Lorentzian orbifolds N with noncompact isometry groups I(N ) and nonproper actions of I(N ) on N (Example 4.3). In Sections 4 we present some results about the Lie groups of automorphisms of affine connected orbifolds and the isometry groups of Riemannian orbifolds belong to the au-

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thors [5, 6, 7]. The content of the article is illustrated by examples.

1. 1.1.

An Introduction to Orbifolds The Category of Orbifolds

Throughout this article we understand by smoothness the smoothness of class C ∞ . Given some smooth mapping of manifolds f : M → N, denote by f∗ and f ∗ the differential and codifferential of f . Recall the definition of a smooth orbifold [6, 14]. Let N be a connected Hausdorff topological space with a countable base, let U be an open subset of N , and let n be a fixed natural number. An orbifold chart on N is a triple (Ω, Γ, p), consisting of a connected open subset Ω of the n-dimensional arithmetic space Rn , a finite group Γ of diffeomorphisms of Ω, and the composition p : Ω → N of the quotient mapping r : Ω → Ω/Γ and a homeomorphism q : Ω/Γ → U of the quotient space Ω/Γ onto U. The subset U is called a coordinate neighborhood of (Ω, Γ, p). Note that, unlike Satake [6], we do not require the dimension of the fixed-point set FixΓ of Γ to be smaller than n − 1. Let U and U ′ be coordinate neighborhoods of orbifold charts (Ω, Γ, p) and (Ω′ , Γ′ , p′ ), with U ⊂ U ′ . An embedding φ : Ω → Ω′ such that p′ ◦ φ = p. is called an embedding of the orbifold chart (Ω, Γ, p) into the orbifold chart (Ω′ , Γ′ , p′ ) corresponding to the inclusion U ⊂ U ′ . It is known [15] that each embedding φ induces a (unique) monomorphism of groups ψ : Γ → Γ′ for which φ ◦ γ = ψ(γ) ◦ φ ∀γ ∈ Γ, and if φ is a diffeomorphism then ψ is an isomorphism between the groups Γ and Γ′ . Two orbifold charts (Ω1 , Γ1 , p1 ) and (Ω2 , Γ2 , p2 ) with coordinate neighborhoods U1 and U2 are called compatible if in the case U1 ∩ U2 6= ∅ for each point x ∈ U1 ∩ U2 there exist: (a) an orbifold chart (Ω, Γ, p) with coordinate neighborhood U such that x ∈ U ⊂ U1 ∩ U2 ; (b) embeddings of orbifold charts φ1 : Ω → Ω1 and φ2 : Ω → Ω2 , corresponding to inclusions U ⊂ U1 and U ⊂ U2 . A set A = {(Ωi , Γi , pi ) | i ∈ J} of orbifold charts is called an orbifold atlas if the family {Ui := pi (Ωi ) | i ∈ J} is an open covering of N and each pair of orbifold charts in A is compatible. An orbifold atlas A is called maximal if A coincides with every orbifold atlas that includes it. A maximal orbifold atlas is called the structure of a smooth n-dimensional orbifold on N . A pair (N , A), where A is a maximal orbifold atlas on N , is called a smooth n-dimensional orbifold. Note that each orbifold atlas is included in a unique maximal orbifold atlas, and thus defines the structure of a smooth orbifold. Henceforth we assume all orbifolds N to be smooth and denote by A = {(Ωi , Γi , pi ) | i ∈ J} the maximal atlas of N . The embedding φij of an orbifold chart (Ωi , Γi , pi ) into an orbfold chart (Ωj , Γj , pj ) corresponding to the inclusion of coordinate neighborhoods Ui ⊂ Uj , is called an embedding of orbifold charts and is denoted by φij : Ωi → Ωj , i, j ∈ J. Note that the coordinate neighborhood U of an orbifold chart (Ω, Γ, p) which is homeomorhic to Ω/Γ belongs to the orbifolds. Orbifolds U are called elementary. For each point x ∈ N there exists an orbifold chart (Ω, Γ, p) ∈ A, such that Ω is a n-dimensional arithmetic space Rn , p(0) = x, with 0 = (0, . . . , 0) ∈ Rn , and Γ is a finite

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group of orthogonal transformations of Rn . Such an orbifold chart (Rn , Γ, p) is called a linearized chart at x. For orbifold charts (Ω, Γ, p) and (Ω′ , Γ′ , p′ ) in A with coordinate neighborhoods containing x ∈ N , the isotropy subgroups Γy and Γ′z of the points y ∈ p−1 (x) and z ∈ p′−1 (x) are respectively isomorphic. Therefore, to each point x of N there corresponds a unique (up to a group isomorphism) abstract group Γy called the orbifold group of x. A point x is called regular if its orbifold group is trivial. Singular we call a point that is not regular. If an orbifold N has a singular point, then N is called proper. A continuous mapping f : N → N ′ of an orbifold (N , A) into an orbifold (N ′ , A′ ) is called orbifold mapping or smooth mapping if for each point x ∈ N there exist: (a) an orbifold chart (Ω, Γ, p) ∈ A with coordinate neighborhood U ∋ x; (b) an orbifold chart (Ω′ , Γ′ , p′ ) ∈ A′ with coordinate neighborhood U ′ such that f (U ) ⊂ U ′ ; (c) a smooth mapping f˜: Ω → Ω′ of Ω into Ω′ such that p′ ◦ f˜ = f |U ◦ p. In this case the smooth mapping f˜ is called a local lift of f. The category of orbifolds is the category whose morphisms are given by the orbifold mappings of orbifolds and the composition of morphisms is the composition of orbifold mappings. We denote this category by Orb. The category of smooth manifolds with smooth mappings of manifolds as morphisms is a full subcategory of Orb. A bijection f : N → N ′ of orbifolds is called a diffeomorphism (or isomorphism) if f and f −1 is a smooth mapping in the category Orb. Are known some other more refined notions of morphism between orbifolds, and they give rise to the same isomorphisms as in the category Orb. The orbifold mapping f : N → R is called a smooth function. The algebra of all smooth functions on N is denoted by F(N ). Remark that f ∈ F(N ) iff for any orbifold chart (Ω, Γ, p) ∈ A the composition f ◦ p : Ω → R is a smooth mapping of manifolds. Let (N ′ , A′ ) and (N , A) be two smooth orbifolds. A smooth mapping π : N ′ → N is called a submersion if each representative π ˆ : Ω′ → Ω of π in charts (Ω′ , Γ′ , p′ ) ∈ A′ and (Ω, Γ, p) ∈ A with coordinate neighborhoods U ′ and U such that π(U ′ ) ⊂ U, is a submersion from the manifold Ω′ onto the manifold Ω. The correctness of this definition, i. e. independence from a choice of charts follows from compatible charts of atlases. An orbifold (N , A) is called oriented if for all i ∈ J the manifolds Ωi are oriented so that each transformation γ ∈ Γi as well as each embedding φij : Ωi → Ωj , i, j ∈ J, preserves orientation. E XAMPLE 1.1. Let G be a discrete group acting properly on a manifold M. Then the orbit space M/G is a smooth orbifold. E XAMPLE 1.2. If (M, F) is a foliation with compact leaves and finite holonomy groups then the leaf space M/F is a smooth orbifold. E XAMPLE 1.3. If all leaves of transversally complete Riemannian foliation (M, F) are embedded submanifolds of M then the leaf space M/F is a smooth orbifold [36]. E XAMPLE 1.4. Define the action of the generator f : Rn → Rn of the group Γ = hf | f 2i ∼ = Z2 by the equality f (x) := −x ∀x ∈ Rn where n ≥ 3. The quotient space N := n R /Γ is a smooth n-dimensional orbifold with the unique singular point a := p(0), where 0 is the origin, and p : Rn → Rn /Γ is the quotient mapping. It is easy to check that the

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underlying topological space of N is not locally Euclidean. This example shows that, in contrast to manifolds and 2-dimensional orbifolds, for n ≥ 3 the underlying topological spaces of n-dimensional orbifolds may be not locally Euclidean.

1.2.

The Stratification of Orbifolds

Let (N , A) be a n-dimensional orbifold. If there are charts at points x and y of N with coordinate neighborhoods isomorphic in the category Orb then x and y are said to have the same orbifold type. The subspace of points of the same orbifold type with the induced topology has a natural smooth manifold structure, with this subspace is disconnected in general. The manifolds of points of different orbifold types may have the same dimension. Denote by ∆k the union of such manifolds of dimension k. It is possible that ∆k = ∅, k ∈ {0, . . . , n − 1}. The family ∆(N ) = {∆k }k∈{0,...,n} is called the stratification of the orbifold N , and ∆k themselves are called strata. The proof of the following theorem is held in [6]. T HEOREM 1.1. Let N be a n-dimensional orbifold and let ∆(N ) = {∆k }k∈{0,...,n} be the stratification of N . Then:

(i) Each connected component ∆ik of stratum ∆k is formed by orbifold points of same orbifold type.

(ii) The closure ∆ik of ∆ik is naturally endowed by smooth k-dimensional orbifold structure for which ∆ik is a set of regular points. (iii) The stratum ∆n is a connected open and everywhere dense smooth n-dimensional manifold consisting of the all regular points of N .

E XAMPLE 1.5. The n-dimensional orbifold N of Example 1.45 has the stratification ∆(N ) = {∆0 , ∆n }, with ∆0 = {a}.

E XAMPLE 1.6. Consider the action on the Euclidean space En , n ≥ 3, of the finite group Γ generated by two isometries α and β, where α and β is given by matrixes     cos 2π sin 2π 0 1 0 0 p p  − sin 2π cos 2π 0  ,  0 1 0  , p p 0 0 −E 0 0 E

where E is the identity matrix. Then Γ =< α, β | αp , β 2 >∼ = Zp ⊕ Z2 . The orbit space n N := E /Γ is a smooth n-dimensional orbifold. It easy to see that the orbifold N has stratification ∆(N ) = {∆0 , ∆2 , ∆n−2 , ∆n }, and ∆0 consists from one point. E XAMPLE 1.7. Let Γ be a finite group generated by isometries αi : En → En of the Euclidean space En , n ≥ 1, αi (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) := (x1 , . . . , xi−1 , −xi , xi+1 , . . . , xn ),

where (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) ∈ En , i = 1, . . . , n. Then Γ ∼ = (Z2 )n and the n orbit space N = E /Γ is a smooth n-dimensional orbifold with stratification ∆(N ) = {∆0 , . . . , ∆n }, and ∆i 6= ∅ ∀i = 0, . . . , n.

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A.V. Bagaev and N.I. Zhukova

Fiber Bundles over Orbifolds

Recall that an antihomomorphism of some group Γ into some group G is a mapping b : Γ → G such that b(γ1 γ2 ) = b(γ2 )b(γ1 ) ∀γ1 , γ2 ∈ Γ. If b is also injective then b is called an antimonomorphism. D EFINITION 1.1. Let F be a smooth manifold and H be a Lie group. Following [8] we say that a fiber bundle with the standard fiber F and the structure group H is defined over some orbifold (N , A) if: (i) for each chart (Ωi , Γi , pi ) ∈ A there are given: (a) a fiber bundle Pi with projection πi : Pi → Ωi , standard fiber F and structure group H; (b) an antimonomorphism bi : Γi → AutPi of Γi into the automorphism group AutPi of the fiber bundle such that γ −1 ◦ πi = πi ◦ bi (γ) ∀γ ∈ Γi ; (ii)

for each embedding of charts φij : Ωi → Ωj , i, j ∈ J, an isomorphism φ¯ij : Pj |φij (Ωi ) → Pi of fiber bundles is defined, where Pj |φij (Ωi ) is the restriction of the bundle Pj to φij (Ωi ), satisfying the following conditions: (a) bi (γ)◦ φ¯ij = φ¯ij ◦bj (ψij (γ)) ∀γ ∈ Γi where ψij : Γi → Γj is a monomorphism of groups induced by the embedding φij ; (b) if Ui ⊂ Uj ⊂ Uk with the corresponding embeddings of charts φij and φjk then φjk ◦ φij = φ¯ij ◦ φ¯jk .

Denote by ξ = {Pi , bi , φ¯ij }i,j∈J the fiber bundle over N described above. A fiber bundle over an orbifold can be defined starting from an arbitrary atlas; see [30]. For each orbifold N there exists some atlas B = {(Ωβ , Γβ , pβ ) | β ∈ B} with contractible coordinate neighborhoods of all charts. For such an atlas the fiber bundles Pβ are trivial; i.e, Pβ = Ωβ × F and πβ : Pβ → Ωβ is the canonical projection onto the first factor. Let ξ = {Pi , bi , φ¯ij }i,j∈J be a fiber bundle with standard fiber F and structure group H over some orbifold N . For each chart (Ωi , Γi , pi ) ∈ A the antimonomorphism bi determines the smooth left action Φi : Γi × Pi → Pi : (γ, z) 7→ bi (γ −1 )(z) of Γi on the manifold Pi . Since Γi is a finite group, the quotient space P¯i := Γi \Pi is a smooth orbifold of dimension dim N + dim F, and the diagram p¯i Pi −−−−→ P¯i = Γi \Pi   π π¯ y i y i pi

Ωi −−−−→

Ui

is commutative, where p¯i : Pi → Γi \Pi is the quotient mapping and π ¯i : P¯i → Ui F takes the ¯ orbit of z ∈ Pi into pi (πi (z)) ∈ Ui = pi (Ωi ). Denote by P the disjoint union i∈J P¯i . Define on P¯ the equivalence relation ρ : say that two points z¯i ∈ P¯i and z¯j ∈ P¯j are ρequivalent if: (a) π ¯i (¯ zi ) = π ¯j (¯ zj ) = x ∈ Ui ∩Uj ; (b) there exist two points zi ∈ (¯ pi )−1 (¯ zi ), −1 zj ∈ (¯ pj ) (¯ zj ) and a chart (Ωk , Γk , pk ) ∈ A with coordinate neighborhood Uk , such that x ∈ Uk ⊂ Ui ∩ Uj and zj = (φ¯kj )−1 ◦ φ¯ki (zi ). It is easy to check [6] that ρ is indeed an

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equivalence relation, the quotient space P = P¯ /ρ is naturally equipped with the structure of a smooth orbifold, and the submersions πi : Pi → Ωi define a submersion π : P → N of orbifolds. Thus we have proved P ROPOSITION 1.1. If a fiber bundle with the standard fiber F and the structure group H over an orbifold N is given then the smooth orbifold P of dimension dim N + dim F is naturally defined together with the submersion π : P → N of orbifolds. D EFINITION 1.2. The orbifold P is called the total space; and the mapping π : P → N is called the natural projection of the fibre bundle. N.

Remark that the standard fiber F is diffeomorphic to π −1 (x) for any regular point x ∈

D EFINITION 1.3. A smooth section of a fiber bundle ξ = {Pi , bi , φ¯ij }i,j∈J with the standard fiber F and the structure group H over an orbifold (N , A) is defined [8] as a family {si }i∈J of smooth sections si : Ωi → Pi of the fiber bundles Pi if the following conditions are satisfied: (a) bi (γ) ◦ si ◦ γ = si ∀γ ∈ Γi , i ∈ J; (b) φ¯ij ◦ sj ◦ φij = si for each embedding of charts φij : Ωi → Ωj , i, j ∈ J. Note that a family {si }i∈J determines a smooth mapping s : N → P of orbifolds satisfying the equality π ◦ s = idN .

1.4.

Tensors on Orbifolds

We define tensors on orbifolds as sections of a tensor bundles. Let (N , A) be a n-dimensional orbifold. Denote by πi : T Ωi → Ωi the tangent bundle of Ωi . For each γ ∈ Γi define a mapping bi (γ) : T Ωi → T Ωi by the equality bi (γ)(Xx ) := (γ −1 )∗x (Xx ), where Xx ∈ Tx Ωi is a tangent vector at some point x ∈ Ωi . For each embedding of charts φij : Ωi → Ωj , i, j ∈ J, define a mapping φ¯ij : T Ωj |φij (Ωi ) → T Ωi by the formula φ¯ij (Xφij (x) ) := (φij )−1 ∗x (Xφij (x) ), Xφij (x) ∈ Tφij (x) Ωj , x ∈ Ωi . Therefore, we have defined the fiber bundle with standard fiber a vector space isomorphic to Rn and structure group H = GL(n, R), which is called the tangent bundle to the orbifold N . The total space T N of this bundle is a smooth 2n-dimensional orbifold. Similarly, the cotangent bundle and the tensor bundle of type (p, q) over an orbifold are defined in [30, 8]. A smooth section of the tensor bundle of type (p, q) is called a tensor field of type (p, q) on the orbifold. In particular, a smooth vector field on an orbifold (N , A) is a smooth section of the tangent bundle of N ; i.e., a family {Xi }i∈J of Γi -invariant vector fields Xi on Ωi such that for each embedding of charts φij : Ωi → Ωj , i, j ∈ J, the equality (φij )∗ (Xi ) = Xj holds.

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2. G-structures on Orbifolds 2.1.

Proper Actions of Lie Groups on Orbifolds

D EFINITION 2.1. Let N be a smooth orbifold, G be a Lie group. A mapping Φ : N × G → N of the product orbifold N × G into N is called a smooth action of the Lie group G on orbifold N if Φ is a such action of G on N that for each g ∈ G the restriction Φg := Φ|N ×{g} is an automorphism of the orbifold category Orb. Recall that a continuous mapping f : X → Y of topological spaces is called a proper if the preimage of any compact subset of Y is a compact set of X. P ROPOSITION 2.1. Let Φ : N ×G → N be a smooth action of a Lie group G on an orbifold N . Then the following three conditions are equivalent: (i) the inducted mapping (idN , Φ) : N × G → N × N : (x, g) 7→ (x, x · g)

∀(x, g) ∈ N × G

is proper; (ii) if there exist such consequences {gn } ⊂ G and {xn } ⊂ N that xn → x and xn · gn → y, x, y ∈ N , then {gn } has a convergence subsequence in G; (iii) for all compact sets K and L in N the set {g ∈ G | K ∩ L · g 6= ∅} is compact in G. P ROOF is analogously to the proof of the respectively assertion for manifold ([23, p. 41]). D EFINITION 2.2. A smooth action of a Lie group G on an orbifold N is called proper if it satisfies at least one of the three conditions of Proposition 2.1. D EFINITION 2.3. A smooth action of a Lie group G on an orbifold N is called locally free if each isotropy group Gx , x ∈ N , is discrete in G. From item (iii) of Proposition 2.1 follows C OROLLARY 2.1. The isotropy group of a proper action of a Lie group on orbifold is compact. C OROLLARY 2.2. If a Lie group locally free proper acts on orbifold then all isotropy group is finite. E XAMPLE 2.1. Any smooth action of a compact Lie group G on an orbifold N is proper. Indeed, each sequence {gn } in compact group G has a convergent subsequence, therefore, item (iii) of Proposition 2.1 is satisfied. P ROPOSITION 2.2. The orbits of a proper action Φ : N × G → N of a Lie group G on an orbifold N are closed embedded submanifolds of N . P ROOF. As each g ∈ G is an automorphism of orbifold N , so any orbit x · G, x ∈ N , consists of points of the same orbifold type. Hence x · G belongs to the same stratum ∆k , with the restriction Φ|∆k ×G is the proper action of G on manifold ∆k . Therefore, by

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proposition 5.4 [23] the orbit x·G is an embedded submanifold of ∆k , consequently, x·G is an embedded submanifold in N . Since (idN , Φ) is a closed mapping as a continuous, proper mapping between topological spaces N ×G and N ×N , then (idN , Φ)(x, G) = {x}×x·G is closed in the fiber {x} × N of product N × N . Hence, the orbit x · G is closed in N .

2.2.

Principal Orbifold Bundles: Equivalent Approaches

We consider two equivalent approaches to the notion of a principal orbifold bundle. D EFINITION 2.4. A bundle ξ = {Pi , bi , φ¯ij }i,j∈J with the standard fiber F and the structure group H over an orbifold N is called principal bundle with structure group H if F = H and the group H acts by left translations on F. P ROPOSITION 2.3. Let ξ = {Pi , bi , φ¯ij }i,j∈J be a principal bundle over a n-dimensional orbifold N with structure group H. Then the following assertions are hold: (i) the total space P of ξ is a smooth orbifold of dimension n + dim H; (ii) a locally free proper action of the Lie group H on P is defined, with N = P/H and the quotient mapping π : P → P/H = N is submersion of orbifolds. P ROOF. The first statement follows from Proposition 1.1. Define a smooth action of the Lie group H on the total space P of the principal bundle ξ = {Pi , bi , φ¯ij }i,j∈J . For each i ∈ J the smooth right action Υi : Pi × H → Pi with (z, h) 7→ z · h where z ∈ Pi , h ∈ H, of H on the total space Pi is defined. Since bi (γ), γ ∈ Γi , is an automorphism of the principal bundle Pi , it follows that bi (γ)(z · h) = (bi (γ)(z)) · h; consequently, the ¯ i : P¯i × H → P¯i : (¯ mapping Υ z , h) 7→ p¯i (z · h), where z¯ ∈ P¯i , z F ∈ p¯−1 z ), h ∈ H, i (¯ ¯ ¯ defines a smooth right action of H on Pi = Γi \Pi . As above, P = i∈J P¯i . Denote by q : P¯ → P¯ /ρ = P the natural projection. The composition qi := q ◦ j : P¯i → P of the inclusion j : P¯i ֒→ P¯ with the projection q is a homeomorphism onto the image. Take u ∈ P, x = π(u), and some chart (Ωi , Γi , pi ) ∈ A with coordinate neighborhood Ui ∋ x. The formula Υ(u, h) := qi ◦ p¯i (z · h) where z ∈ (qi ◦ p¯i )−1 (u), h ∈ H, defines a smooth right action Υ : P × H → P of the Lie group H on orbifold P. The orbit space P/H of the action Υ is the orbifold N . The following diagram is commutative: (¯ pi , idH ) (qi , idH ) Pi × H −−−−−→ P¯i × H −−−−−→ P × H    ¯ Υ  yΥi y i yΥ

Pi  π y i

Ωi

p¯i

−−−−→ pi

−−−−→

P¯i  π¯ y i

Ui

qi

−−−−→ ֒→

P  π y

N.

Since each isotropy group Hu , u ∈ P, is isomorphic to some subgroup of the finite orbifold group Γ of the point x = π(u), so the action of H on P is locally free. Show that Υ is a proper action. Denote by [u] the orbit u·H and it will be considered as a point of orbit space N = P/H. Let {un } and {hn } be sequences in P and H, respectively, and un → u, un · hn → y, where u, y ∈ P. As the projection π : P → N is continuous

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mapping, so un → u and un · hn → y yield [un ] → [u] and [un ] = [un · hn ] → [y]. The uniqueness of the limit of a sequence in Hausdorff topological space N implies [u] = [y], i.e. u · H = y · H, consequently, there exists h0 ∈ H such that y = u · h0 . Since the isotropy group Hu is isomorphic to same subgroup of the orbifold group Γ of point x = π(u), then {h ∈ H | u · h = y} = h0 · Hu is a finite subset of H. The continuousness of the action Υ of H on P yields the convergence of the sequence u · hn to y. As it is known [3, Theorem 2], the topology of each group of homeomorphisms of a locally compact Hausdorff topological space acting continuously on this space contains all subsets open in the compact-open topology. Therefore, there exist such n0 ∈ N that the consequence {hn } belongs to some neighborhood V of h0 · Hu with compact closure V , hence, {hn } has a convergent subsequence. According to Proposition 2.1, the action Υ is proper. Using the fact that an action of H keeps the stratification of orbifold N and applying the theorem about existence of slices for a proper action of a Lie group on a manifold [23, Theorem 5.7, p. 44–45], we get the following statement. P ROPOSITION 2.4. Let P be a smooth m-dimensional orbifold and H be a Lie group, with H effective locally free proper acts on P. Then: (i) the orbit space N := P/H becames a n-dimensional orbifold by the natural a way, where n = m − dim H; (ii) the canonical projection π : P → N = P/H forms a principal bundle over N with structure group H; (iii) if P is a smooth manifold then: (a) the connected components of the fibers of the bundle π : P → N form a smooth foliation F of codimension n;

(b) if the Lie group H is connected then the holonomy group of a leaf L = π −1 (x) ∀x ∈ N is isomorphic to the orbifold group Γ of the point x ∈ N .

R EMARK 2.1. Thus, the family ξ = {Pi , bi , φ¯ij }i,j∈J forms a principal bundle over an orbifold N with structure group H and total space P if and only if an effective locally free proper action of the Lie group H on P is given, with the orbit space P/H is equal to N . R EMARK 2.2. Let P be a smooth manifold with effective proper locally free action of the Lie group H. According to Proposition 2.4, the orbit space N := P/H is a smooth orbifold and the quotient mapping π : P → P/H = N is a principal bundles over N . Let ∆ck be a connected component of a stratum ∆k of N . According to Theorem 1.1, ∆ck consists from orbifold points of the same orbifold type. Denote the orbifold group of points of ∆ck by Γ. Put Rck := π −1 (∆ck ) and πkc := π|Rck . Then πkc : Rck → ∆ck is a fibre bundle with the structure group G and the standard fibre G/Γ over manifold ∆ck .

2.3. G-structures on Orbifolds Let G be a Lie subgroup of the Lie group H and let ξ = {Pi , bi , φ¯ij }i,j∈J be a principal bundle over (N , A) with the structure group H. If, for every chart (Ωi , Γi , pi ) ∈ A, the

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structure group H of the bundle Pi over Ωi is reduced to the subgroup G, Ri is the reduced bundle, and moreover the conditions (a) bi (γ)(Ri ) = Ri ∀γ ∈ Γi ; (b) φ¯ij (Rj |φij (Ωi ) ) = Ri , i, j ∈ J, are satisfied, then the principal bundle ξ ′ = {Ri , bi , φ¯ij }i,j∈J over N with the structure group G is defined, where φ¯ij means φ¯ij |Rj |φ (Ω ) : Rj |φij (Ωi ) → Ri and bi : Γi → AutRi . ij

i

The principal bundle ξ ′ with structure group G over N is referred to as the reduced bundle. In this case we say that the structure group H of the bundle ξ is reduced to G. Let (Ωi , Γi , pi ) be a chart of a n-dimensional orbifold (N , A). We will regard a frame at a point x ∈ Ωi as a vector space isomorphism z : Rn → Tx Ωi , where Tx Ωi is the tangent space of Ωi at x. Denote by πi : Pi → Ωi the principal GL(n, R)-bundle of frames over Ωi . Define an anti-homomorphism bi of Γi into the automorphism group of bundles Pi as bi (γ)(z) := (γ −1 )∗x ◦ z, where γ ∈ Γi and z is a frame at x ∈ Ωi . For embedding φij : Ωi → Ωj , i, j ∈ J, define the mapping φ¯ij : Pj |φij (Ωi ) → Pi by the equality φ¯ij (z) := (φ−1 ij )∗φij (x) ◦ z where z is a frame at φij (x) ∈ φij (Ωi ) ⊂ Ωj . The so-constructed principal bundle with structure group GL(n, R) is called the frame bundle over the orbifold N . If the structure group GL(n, R) of the frame bundle over N is reduced to a Lie subgroup G ⊂ GL(n, R) then the reduced principal bundle is called a G-structure on the orbifold N . Let ξ = {Ri , bi , φ¯ij }i,j∈J be a G-structure on an orbifold N , let (Ωi , Γi , pi ) be an arbitrary chart of N . Fix x ∈ FixΓi , z ∈ πi−1 (x). As bi (γ) = (γ −1 )∗x is an automorphism of Ri for any γ ∈ Γi , so γ∗x (z) ∈ Ri and, consequently, the linear isomorphism z −1 ◦ γ∗x ◦ z : Rn → Rn belongs to the Lie group G. Straightforward verification shows that the mapping χz : Γi → G : γ 7→ z −1 ◦ γ∗x ◦ z correctly defines a faithful representation of the finite group Γi in the Lie group G. The equality bi (γ)(z) = (γ −1 )∗x ◦ z = z where z ∈ Ri , x = πi (z), γ ∈ Γi , implies γ∗x = idTx Ωi . As the group Γi is finite, so γ = idΩi . Therefore, the group Γi free acts on Ri and the total space R of G-structure is a manifold. Thus, applying Proposition 2.3 we have P ROPOSITION 2.5. Let R be a G-structure on a smooth n-dimensional orbifold N . Then R is a smooth manifold of dimension n+dim G and the connected components of the fibers of the bundle π : R → N constitute a smooth foliation F of codimension n; moreover, if the Lie group G is connected then the holonomy group of a leaf L = π −1 (x) is isomorphic to the orbifold group Γ of the point x ∈ N . Let R be a G-structure on a smooth n-dimensional orbifold N , let g be the Lie algebra of the Lie group G. Denote by V the smooth distribution on R tangent to the fibres of π. Given X ∈ g, xt be the global one-parameter subgroup in G generated by X. Then X d defines to the vector field X ∗ on R by the formula Xu∗ := dt (u · xt )|t=0 , u ∈ R. The vector ∗ field X is called the fundamental vector field corresponding to X. The vector field X ∗ is tangent to the fibres of π. A connection in R is a smooth n-dimensional distribution H on R satisfying the equalities: Hu ⊕ Vu = Tu R, (Rg )∗ (Hu ) = Hu·g for g ∈ G, u ∈ R, where Rg : R → R : u 7→ u · g is a left translation on element g ∈ G. Each vector Xu ∈ Tu R can be uniquely written

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down as Xu = HXu + V Xu , where HXu ∈ Hu , V Xu ∈ Vu . We call HXu (V Xu ) the horizontal (vertical) component of Xu . Observe that the distribution H is an Ehresmann connection for the foliation (R, F) in the sense of Blumenthal and Hebda [10]. Define the 1-form ω on R with values in the Lie algebra g of the group G as follows. By above, each element X ∈ g defines the fundamental vector field X ∗ on R, moreover, the mapping g → Vu : X 7→ Xu∗ is a vector space isomorphism. For each vector Xu ∈ Tu R, define ω(Xu ) to be the only X ∈ g for which Xu∗ equals the vertical component of Xu . We see from the definition that ω(Xu ) = 0 if and only if Xu ∈ Hu . The form ω is called the connection form for H. The following assertions are proved by analogy to the case of bundles over manifolds (see [20]). P ROPOSITION 2.6. The connection form ω satisfies the conditions ω(X ∗ ) = X and (Rg )∗ ω = Ad(g −1 )ω for every X ∈ g, g ∈ G, where Ad is the adjoint representation of G in g. Conversely, if ω is some g-valued 1-form on R satisfying these conditions then there is a unique connection H on R whose connection form is ω. The canonical form θ on R is the Rn -valued 1-form defined as follows: For any Xu ∈ Tu R and every chart (Ωi , Γi , pi ) at π(u) ∈ N , let Yz ∈ Tz Ri be such that (qi ◦ p¯i )∗ (Yz ) = ¯ i is the quotient mapping and qi := q ◦ j : R ¯i → Xu , where as above p¯i : Ri → Γi \Ri = R F ¯ ¯ ¯ R is the composition of the inclusion j : Ri ֒→ R = i∈J Ri and the quotient mapping ¯ → R/ρ ¯ = R. Then we put by definition θ(Xu ) := z −1 (πi )∗ (Yz ). It is easy to check q: R that the value of θ is independent on the choice of the chart (Ωi , Γi , pi ) and z ∈ Ri . The torsion form Σ is defined to be the exterior covariant differential of the canonical form θ. P ROPOSITION 2.7. The canonical form θ possesses the following properties: (i) if Xu ∈ Vu , then θ(Xu ) = 0; (ii) (Rg )∗ θ = g −1 θ, g ∈ G; (iii) Σ = dθ + ω ∧ θ. Recall that a diffeomorphism f : Ωi → Ωj is called an isomorphism of G-structure Ri and Rj on manifolds Ωi and Ωj respectively if for any point x ∈ Ωi and for any frame z ∈ Ri at x the frame f∗x ◦ z belongs to Rj . D EFINITION 2.5. Let R be a G-structure on an orbifold (N , A). By automorphism of G-structure R we mean an diffeomorphism f : N → N of the orbifold N , satisfying the following condition: for each point x ∈ N and each pair of charts (Ωi , Γi , pi ), (Ωj , Γj , pj ) ∈ A with coordinate neighborhoods Ui ∋ x and Uj such that f (Ui ) = Uj , there exists a local lift fij : Ωi → Ωj which is an isomorphism of G-structure Ri and Rj on manifolds Ωi and Ωj respectively. The definition the G-structure on the orbifold N implies that this definition is correct; i.e., it is independent on the choice of charts at x and f (x) and of the choice of a local lift. We denote the hole group of automorphisms of a G-structure on an orbifold N by A(N ).

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2.4.

459

Examples of G-structures

P SEUDO -R IEMANNIAN (R IEMANNIAN , L ORETZIAN ) STRUCTURE . Define an inner product of signature (p, q) on Rn , 0 ≤ p, q ≤ n, p + q = n, by setting hx, yi = x1 y1 + . . . + xp yp − xp+1 yp+1 − . . . − xp+q yp+q . Let R(p,q) denote Rn with this particular inner product and O(p, q) be the associated orthogonal group of all linear transformations of Rn preserved h·, ·i. The Lie group O(p, q) is a closed Lie subgroup of GL(n, R). The O(p, q)-structure on a n-dimensional orbifold N is called a pseudo-Riemannian structure. If p = n − 1, q = 1, then O(p, q)-structure is called a Loretzian structure. If p = n, q = 0, then h·, ·i is a positive inner product in Rn . Put O(n, R) := O(n, 0). The O(n, R)-structure on orbifold N is called a Riemannian structure. D EFINITION 2.6. A pseudo-Riemannian (Riemannian, Loretzian) metric g on an orbifold N is a family {gi }i∈J of Γi -invariant pseudo-Riemannian (Riemannian, Loretzian) metrics gi on the manifolds Ωi such that each embedding of charts φij : Ωi → Ωj , i, j ∈ J, is an isometry of pseudo-Riemannian (Riemannian, Loretzian) manifolds (Ωi , gi ) and (Ωj , gj ). The pair (N , g) is called a pseudo-Riemannian (Riemannian, Loretzian) orbifold. If g is not Riemannian metric then (N , g) is called proper pseudo-Riemannian orbifold. By analogy with manifolds, prescription of a pseudo-Riemannian metric (Riemannian, Loretzian) g on an orbifold N is equivalent to prescription of an pseudo-Riemannian (Riemannian, Loretzian) on N . It is known that each smooth orbifold admits a Riemannian metric, but not each ndimensional orbifold can be endowed a pseudo-Riemannian metric of signature (p, q), 0 < p < n, p + q = n. So a n-dimensional manifold admits a Lorentzian metric if and only if it admits a 1-dimensional distribution, n ≥ 2. On the other hand, for n ≥ 2 a n-dimensional compact manifold admits a 1-dimensional distribution if and only if its Euler–Poincare characteristic is zero. Therefore, any (2n + 1)-dimensional compact orientable manifold admits a Lorentzian metric. C ONFORMAL STRUCTURE . Let N be a n-dimensional orbifold, n ≥ 3. Let CO(n, R) := {A ∈ GL(n, R) | At A = cE, c ∈ R+ } where E is unit matrix. A CO(n, R)-structure on N is referred to as a conformal structure on orbifold N . There is another equivalent approach to the notion of conformal structure on an orbifold. Two Riemannian metrics g and g ′ on an orbifold N are said to be conformally equivalent if there is a smooth positive function λ on N such that g ′ = λg. The class of conformally equivalent metrics [g] determines the conformal structure on N . Conversely, each CO(n, R)-structure on N forms class of conformally equivalent Riemannian metrics [g] on N . A LMOST where

COMPLEX STRUCTURE .

Put GL(m, C) := {A ∈ GL(2m, R) | AJ = JA} J=



 0 −E . E 0

A GL(m, R)-structure on a 2m-dimensional orbifold N is called an almost complex structure. An orbifold N can be endowed by GL(m, R)-structure if and only if there exists tensor J of type (1, 1) on N satisfying the equality J ◦ J = −Id.

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A LMOST SYMPLECTIC AND SYMPLECTIC STRUCTURES . Let Sp(m, R) = {A ∈ GL(2m, R) | At JA = J} be a symplectic group. Recall that the Lie group Sp(m, R) consists from all linear transformations of R2m which preserve a skew-form dx1 ∧dxm+1 +. . .+ dxm ∧ dx2m where x1 , . . . , x2m are standard coordinates in R2m . A Sp(m, R)-structure on a 2m-dimensional orbifold N is called a almost symplectic structure. An almost symplectic structure is given on a 2m-dimensional orbifold N if and only if a skew 2-form ω of maximal rang (ω m 6= 0) is defined on N . If the skew 2-form ω on N is closed, then an almost symplectic structure is called a symplectic one and the form ω is called symplectic. A LMOST H ERMITIAN STRUCTURE . Put U (m) = GL(m, C) ∩ O(2m, R). A U (m)structure on a 2m-dimensional orbifold N is called an almost Hermitian structure. A U (m)-structure can be regarded as the intersection almost complex structure and Riemannian structure. T ENSOR G- STRUCTURE . Let K0 be a tensor of type (r, s) on Rn , G be the group of all linear transformation of Rn preserved tensor K0 . Since G is an algebraic group, then G is a closed Lie subgroup of Lie group GL(n, R). A G-structure of such type on a n-dimensional orbifold N is called tensor one. If R is a tensor G-structure on N associated with tensor K0 , then a tensor field K of type (r, s) on N is well-defined. Such tensor field is called O-deformable (see [15]). We stress that there exist non O-deformable tensor fields on N . It is easy to show that automorphism f of an orbifold N is an automorphism of tensor G-structure on N if and only if f conserves the corresponding O-deformable tensor field K. A pseudo-Riemannian, an almost complex, an almost symplectic structure are tensor G-structures.

2.5.

Inducted G-structures on Strata of Orbifolds

Let ξ = {Ri , bi , φ¯ij }i,j∈J be a G-structure on a n-dimensional orbifold N . Consider a connection component ∆ck of stratum ∆k of N . We will regarded Rn as Rk × Rn−k . Let Gn,k be the Grassmann manifold of all k-dimensional linear subspaces in Rn . Let ζ0 = Rk × {0} be a linear subspace, generated by the first k vectors of standard basis in Rn . The Lie group GL(n, R) transitive acts on Gn,k and hence the Lie subgroup G ⊂ GL(n, R) also acts on Gn,k . Denote the orbit of ζ0 of the group G by ζ0 · G, and the isotropy subgroup of G at ζ0 by G0 . Remark that the group G0 is formed by matrixes ! Aba Aβa , (∗) A= 0 Aβα where det A = det(Aba ) det(Aβα ) 6= 0, a, b = 1, . . . , k, α, β = k + 1, . . . , n. Since the mapping ζ0 · G → G/G0 : ζ0 · g 7→ g · G0 , g ∈ G, is a bijection, so we can identity the orbit ζ0 · G with the homogeneous space G/G0 . The Lie group G acts on G/G0 by left translations. Consider the mapping α : G0 → GL(k, R) : A 7→ (Aba ) where matrix A has the form (∗). It is clear that α is a Lie group homomorphism. Put H := α(G0 ). Then we have the isomorphism group α ˆ : G0 / ker α → H.

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According to Theorem 1.1, the connected component ∆ck is formed by points of the same orbifold type. Therefore, the orbifold groups of all points of ∆ck are isomorphic to the same finite group Γ. Without loss of generality, we may assume that for each point x ∈ ∆ck there exists a linearized chart (Ωi = Rk × Rn−k , Γ, pi ), where FixΓ = Rk × {0}. Then Vi := pi (Rk ×{0}) is an open subset in ∆ck and the pair (Vi , ϕi ), where ϕi = (pi |Rk ×{0} )−1 , is a chart of k-dimensional manifold ∆ck at x. We will say that the chart (Vi , ϕi ) is associated with the chart (Ωi , Γ, pi ). Denote the family of all such charts (Vi , ϕi ) by A(∆ck ). Let Ri0 = Ωi × G0 be the reduced bundle with structure group G0 . We will regarded Qi := Vi × H as the principal bundle with structure group H over Vi . The mapping µi : Ri0 = Ωi × G0 → Qi = Vi × H : ((y1 , y2 ), g) 7→ (pi (y1 , 0), α(g)), where (y1 , y2 ) ∈ Rk × Rn−k , g ∈ G0 , is a homomorphism of the principal bundles ¯ := F Qi , where over manifolds. Enter an equivalence relation on the disconnect sum Q ˆ ∈ Qi , (x′ , Aˆ′ ) ∈ Qj , with (Vi , ϕi ) (Vi , ϕi ) ∈ A(∆ck ), by the following way. Let (x, A) and (Vj , ϕj ) be charts from A(∆ck ) associated with (Ωi , Γ, pi ) and (Ωj , Γ, pj ) respectively. ˆ and (x′ , Aˆ′ ) are ρ′ -equivalent, iff: there exist A, A′ ∈ G0 such that The points (x, A) Aˆ = α(A), Aˆ′ = α(A′ ), with the points p¯i (x, A) and p¯j (x′ , A′ ) are ρ-equivalent in the above sense (see proof of Proposition 1.1). Here, as above p¯i and p¯j designate the quotient mappings p¯i : Ri0 → Γ\Ri0 and p¯j : Rj0 → Γ\Rj0 . Using the inclusion ker α ⊃ χ(Γ) where χ : Γ → G is a representation of Γ in G, we check that ρ′ is really an equivalence relation ¯ The quotient space Q := Q/ρ ¯ ′ is a smooth manifold of dimension n + dim H. Since on Q. ¯ induces the inclusion the any points from Qi are not ρ′ -equivalent, so the inclusion Qi → Q Qi → Q. The free smooth action of the Lie group H on Qi defines a free smooth action of H on Q. Thus, Q is a principal bundle with the structure group H over the manifold ∆ck . As the Lie group H is a closed Lie subgroup of GL(k, R), then Q can be regarded as a reduced bundle of the frame bundle over ∆ck to the group H, i. e., as G-structure on ∆ck . D EFINITION 2.7. The constructed G-structure Q on ∆ck is called an inducted G-structure on the connected component ∆ck of stratum ∆k of the orbifold N .

R EMARK 2.3. Since the closure ∆ck of a connected component ∆ck of a stratum ∆k of an orbifold N consists from connected components of strata of N , then a G-structure on N defines G-structure on ∆ck . We also call it the inducted G-structure on the closure ∆ck . P ROPOSITION 2.8. Let K be a nondegenerate tensor field of the type (0, 2) on an orbifold N , let ∆ck be a connection component of a stratum ∆k of N . Then K induces the nondegenerate tensor field of the type (0, 2) on ∆ck . P ROOF. Let A(∆ck ) be an atlas of the manifold ∆ck consisting of the charts (Vi , ϕi ) associated with the linearized charts (Ωi , Γi , pi ), i ∈ J. By definition K for each i ∈ J nondegenerate Γi -invariant tensor field Ki of the type (0, 2) on the manifold Ωi is defined. Determine a tensor field Si of the type (0, 2) on Vi by the formula Si := ϕ∗i Ki . Demonstrate that Si is nondegenerate. Let x ∈ Vi be an arbitrary point, Xx ∈ Tx ∆ck , y = ϕ(x) ∈ FixΓi = Rk × {0}. Put for short V = Ty Rn , V ′ = Ty (Rk × {0}) and Γ := {γ∗x : V → V | γ ∈ Γi }. Since ϕ∗x : Tx ∆ck → V ′ ⊂ V is a linear isomorphism, the equality Si (Xx , Yx ) = 0 ∀Yx ∈ Tx ∆ck is equivalent to the equality Ki (v, w) = 0 ∀w ∈ V ′ where v = ϕ∗x (Xx ), w = ϕ∗x (Yx ). Take any u ∈ V.

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A.V. Bagaev and N.I. Zhukova P 1 ′ Then the element w := |Γ| γ∈Γ γ(u) belongs to V . Indeed, for any γ0 ∈ Γ we have P P 1 1 1 P γ0 (w) = γ0 ( |Γ| γ∈Γ γ(u)) = |Γ| γ∈Γ γ0 ◦ γ(u) = |Γ| γ∈Γ γ(u) = w and, hence, ′ w ∈ V . UsingP the fact that Ki is Γ-invariant, we obtain the sequence of the equalities 1 P 1 P 1 Ki (v, u) = |Γ| γ∈Γ Ki (v, u) = |Γ| γ∈Γ Ki (γ(v), γ(u)) = |Γ| γ∈Γ Ki (v, γ(u)) = 1 P Ki (v, |Γ| γ∈Γ γ(u)) = Ki (v, w) = 0. Since Ki is a nondegenerate tensor field at y, so Ki (v, u) = 0 ∀u ∈ V implies v = 0. It means that the equality Si (Xx , Yx ) = 0 ∀Yx ∈ Tx ∆ck implies Xx = 0. Thus the tensor field Si is nondegenerate at x. As x is an arbitrary point of Vi , so Si is nondegenerate in Vi . The compatibility of the tensor fields {Ki } yields the compatibility of tensor fields {Si }. Hence the nondegenerate tensor field S of the type (0, 2) on ∆ck is well-defined. C OROLLARY 2.3. Let ∆ck be a connected component of a stratum ∆k of an orbifold N . A nondegenerate tensor field K of the type (0, 2) on N induces nondegenerate tensor field of the type (0, 2) on the closure ∆ck . P ROPOSITION 2.9. Let g be a pseudo-Riemannian (Riemannian) metric on an orbifold N , let ∆ck be a connection component of a stratum ∆k of N . Then g indices pseudoRiemannian (Riemannian) metrics on the manifold ∆ck and on the orbifold ∆ck . Applying Proposition 2.8 we present a short proof of the following famous assertion. This statement is proved in [28] with using a momentum map for the symplectic action and Sjamaar–Lermann theorem [31]. P ROPOSITION 2.10. Any symplectic structure on an orbifold N induces a symplectic structure on each connected component ∆ck of a stratum of N . P ROOF. A symplectic structure on an orbifold N defines a closed skew 2-form ω on N . Conversely, if such form ω gives a symplectic structure on N . Let ∆ck be a connection component of a stratum ∆k of N . According to Proposition 2.8, form ω induces skew 2form ω e on ∆ck . Consequently, ∆ck has even dimension. Using Γi -invariance of form ωi on Ωi and applying Darboux’s Theorem, we obtain that ω e is a closed form. Thus, ω e is a closed skew 2-form on ∆ck .

3. 3.1.

Automorphisms of Finite Type G-structures on Orbifolds Prolongation of G-structures

Denote by V the n-dimensional vector space Rn . Let g be an arbitrary Lie subalgebra of the Lie algebra gl(n, R). Given k = 0, 1, . . . denote by gk the set of symmetric polylinear mappings t: V . . × V} → V, | × .{z (k+1) times

such that for arbitrary given vectors v1 , . . . , vk in V the mapping t(·, v1 , . . . , vk ) : V → V : v 7→ t(v, v1 , . . . , vk ) belongs to g or it is equivalent that for any v ∈ V the mapping t(v, ·, . . . , ·) : V . . × V} → V : (v1 , . . . , vk ) 7→ t(v, v1 , . . . , vk ) | × .{z k times

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belongs to gk−1 . The set gk with pointwise addition and multiplication by a number is a vector space. The Lie bracket in gk is defined by the equality [t, s]k := t ◦ s − s ◦ t. The Lie algebra gk is called the kth prolongation of the Lie algebra g. The order of g is defined as the least k for which gk = 0. In this case, gk+l = 0 for every l ∈ N. If gk 6= 0 for every k = 0, 1, . . ., then g is called an algebra of infinite type. Let G be an arbitrary Lie subgroup of the group GL(n, R), g be the Lie algebra of G and gk be the k th prolongation of g. The first prolongation G1 of G is the set of transformations t¯ of the vector space V + g which are defined by the elements t ∈ g1 in accordance with the following formulas: t¯(v) := v + t(·, v), t¯(x) := x for v ∈ V and x ∈ g. The kth prolongation Gk of G is defined as the group of all linear transformations t¯ of the vector space V + g + g1 + . . . + gk−1 , induced by elements t ∈ gk by the following formulas: t¯(v) := v + t(·, . . . , ·, v), t¯(x) := x, for v ∈ V, x ∈ g + . . . + gk−1 , t ∈ gk . V Let V ∗ be the dual linear space to V . Then V ⊗ 2 V ∗ can be treated as the space of all skew-symmetric bilinear mappings from V × V into V and g ⊗ V ∗ can be regarded as the of linear mappings from V into g. Define the linear mapping ν : g ⊗ V ∗ → V2space ∗ V⊗ V by the formula (νf )(v1 , v2 ) := f (v1 )v2 −f (v2 )v1 for f ∈ g⊗V ∗ , v1 , v2 ∈ V . The definition of ν implies that f belongs to the kernel ker ν if and only if the mapping V × V → V : (v1 , v2 ) 7→ f (v1 )v2 belongs to the first prolongation g1 . V2 ∗ Fix an arbitrary vector subspace C in V ⊗ V , complementary to ν(g ⊗ V ∗ ), i.e., V2 ∗ satisfying the equality V ⊗ V = ν(g ⊗ V ∗ ) ⊕ C. Let R be a G-structure on a smooth n-dimensional orbifold N . Suppose that u ∈ R and Hu is an arbitrary n-dimensional subspace of Tu R, complementary to Vu . We call Hu a horizontal subspace at u ∈ R. Then the restriction θ|Hu of the canonical form θ on Hu is a vector space isomorphism Hu → V . Therefore, the restriction of the exterior differential dθ on Hu × Hu defines some V skew-symmetric bilinear mapping V × V → V , i.e., an element c(u, Hu ) ∈ V ⊗ 2 V ∗ . If Hu′ is another horizontal subspace at u then, using relation (iii) of Proposition 2.7, we can easily verify that the difference c(u, Hu′ ) − c(u, Hu ) belongs to ν(g ⊗ V ∗ ). Each horizontal subspace Hu at Tu R defines some frame at u ∈ R. Indeed, as mentioned, the connection form ω gives rise to an isomorphism of g onto the vertical space Vu and the canonical form θ yields an isomorphism of V on Hu . We thus obtain some isomorphism of the vector space V + g onto the tangent space Tu R = Hu ⊕ Vu . We call it the frame at u corresponding to the subspace Hu . We denote by R1 the set of frames on R, corresponding to the horizontal subspaces Hu , u ∈ R, such that c(u, Hu ) ∈ C. The following holds: P ROPOSITION 3.1. The set R1 is a G1 -structure on R, with G1 the first prolongation of the group G. By induction we define the kth prolongation Rk of a G-structure R as the first prolongation of the Gk−1 -structure Rk−1 , i.e., Rk = (Rk−1 )1 .

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Recall that an absolute parallelism of an n-dimensional manifold M is a tuple of n smooth vector fields on M that are linearly independent at each point of M. A G-structure R on an orbifold N is said to be of finite type and order k, if the Lie algebra g of the group G has order k. In this case Gk = {e} and the Gk -structure Rk is an e-structure which defines an absolute parallelism on the manifold Rk−1 . In the opposite case, the G-structure R is called a G-structure of infinite order. It is known [19] that a pseudo-Riemannian structure is a G-structure of first order, a conformal structure has second order. A tensor G-structure, in general, is not G-structure of finite type. So, an almost complex and an almost symplectic structure are tensor Gstructures of infinite type.

3.2.

The Automorphism Groups of Finite Type G-structures on Orbifolds

Let R be a G-structure on a n-dimensional orbifold N . According to Proposition 2.5, the connected components of action of the Lie group G on R forms a smooth foliation F of codimension n. Take f ∈ A(N ). Then f induces an automorphism fˆ of foliation (R, F). By an automorphism of a foliation we mean a diffeomorphism of a foliated manifold which carries leaves into leaves. Let z¯ be an arbitrary point R. As above, ξ = {Ri , bi , φ¯ij }i,j∈J is denoted a reduced bundle, p¯i : Ri → Γi \Ri is the quotient mapping. By the definition of f , for x = π(u) there exist charts (Ωi , Γi , pi ), (Ωj , Γj , pj ) ∈ A with coordinate neighborhoods Ui ∋ x and Uj such that f (Ui ) = Uj , and there exists a local lift fij : Ωi → Ωj which is an isomorphism of G-structure Ri and Rj on manifolds Ωi and Ωj respectively. It follows that the mapping (fij )∗ : Ri → Rj : z 7→ (fij )∗ ◦ z takes Ri onto Rj . Then there exists z ∈ Ri such that qi ◦ p¯i (z) = u where as above qi : Γi \Ri ֒→ R is an embedding into R. The direct check shows that the mapping fˆ: R → R : u 7→ qj ◦ p¯j ◦ (fij )∗ (z) is correctly defined diffeomorphism of manifold R, with fˆ keeps the leaves of foliation F and keeps the canonical form θ invariant. Conversely, every automorphism h of the foliation (R, F) which keeps the canonical form θ invariant is induced by some automorphism of the G-structure R. Moreover, the diffeomorphism fˆ: R → R is an automorphism of the G1 -structure R1 on R (see [5]). Thus, each automorphism f of a G-structure R on an orbifold N defines the sequence (f, fˆ, fˆ1 , . . .) of automorphisms fˆi : Ri → Ri which in the case of a G-structure R of finite type and order k terminates at step k and acquires the shape (f, fˆ, fˆ1 , . . . , fˆk ). This sequence is said to be the tower of the automorphism f. T HEOREM 3.1. Let A(N ) be the hole group of automorphisms of a G-structure of finite type and order m on a smooth n-dimensional orbifold N . Then (i) the group A(N ) admits a unique topology and a unique smooth structure that makes it into a Lie group; (ii) the dimension of A(N ) satisfies to the inequality dim A(N ) ≤ d(n, g) := n + dim g + dim g1 + . . . + dim gm−1 , where gi is the ith prolongation of the Lie algebra g of the group G; the equality dim A(N ) = d(n, g) is possible only in the case when N is a smooth homogeneous space with transitive action of A(N );

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(iii) the Lie group acts on N smoothly and nonproperly, in general. P ROOF. As we have said, the each automorphism f ∈ A(N ) of the G-structure R on the n-dimensional orbifold N induces some automorphism fˆm−1 : Rm−1 → Rm−1 of the Gm -structure Rm which is an e-structure on Rm−1 . We thus have the group isomorphism σ : f 7→ fˆm−1 from the group A(N ) onto some group K of automorphisms of an e-structure on Rm−1 . The manifold Rm−1 is endowed by e-structure a Riemannian metric g. Indeed, let h·, ·i be the standard Euclidean metric in Rs , where s = dim Rm−1 . The formula gx (Xx , Yx ) := hz −1 (Xx ), z −1 (Yx )i ∀Xx , Yx ∈ Tx Rm−1 , where z ∈ Rm and x = πm (z), defines a Riemannian metric on Rm−1 . Denote the hole isometry group of the Riemannian manifold (Rm−1 , g) by I(Rm−1 ). It is known [19], the group I(Rm−1 ) endowed with the compact-open topology is a Lie group of transformations of Rm−1 . Remark that the isometry h ∈ I(Rm−1 ) is induced by the automorphism f ∈ A(N ) if and only if h satisfies the equality f ◦ π e=π e ◦ h where π e : Rm−1 → N is the composition of the π π π π m−1 2 1 bundle projections Rm−1 −→ . . . −→ R1 −→ R −→ N . Then the group K is a closed subgroup of I(Rm−1 ) and hence it is a closed Lie subgroup of the Lie group I(Rm−1 ). The bijection σ induces on the set A(N ) the structure of a smooth manifold. Since σ is a group isomorphism, with respect to the induced smooth structure A(N ) is a Lie group. According to Proposition 1 [7], if an arbitrary group H is isomorphic to the some closed subgroup of the isometry group of some Riemannian manifold, then the group H admits a unique smooth structure that makes it into a Lie group. Thus, the group A(N ) admits a unique topology and a unique smooth structure that makes it into a Lie group and the item (i) is proved. ˆ : K×Rm−1 → Rm−1 : (h, u) 7→ h(u) of the Lie group K on the manifold The action Ψ m−1 R is smooth because it is a restriction of a smooth action of the Lie group I(Rm−1 ) on Rm−1 . Define a map Ψ : A(N ) × N → N by the rule Ψ(f, x) := f (x) for all f ∈ I(N ) ˆ and the equality π ◦ Ψ ˆ = and x ∈ N . Then the smoothness of the maps π, σ and Ψ Ψ ◦ (σ × π) imply the smoothness of the map Ψ. In Example 4.3 (Section 4) we construct a pseudo-Riemannian orbifold N whose the hole isometry group acts on N nonproperly. It shows that the automorphism group A(N ) acts on the orbifold N nonproperly in general. The item (iii) is proved. Note that the automorphism group A(Rm−1 ) of the absolute parallelism of Rm−1 is a closed Lie subgroup of the Lie group I(Rm−1 ) of isometries of the Riemannian manifold (Rm−1 , g). It is known that the group A(Rm−1 ) acts on Rm−1 freely and that dim A(Rm−1 ) ≤ dim Rm−1 . Thus, the dimension of the closed Lie subgroup K of the Lie group A(Rm−1 ) satisfies the inequality dim K ≤ dim Rm−1 , which implies that dim A(N ) = dim K ≤ dim Rm−1 = n + dim g + dim g1 + . . . + dim gm−1 =: d(n, g). Since K is a Lie subroup of the isometry group I(Rm−1 ), the action of the Lie group ˆ K on Rm−1 is proper and free; consequently, each orbit Ψ(K, u), u ∈ Rm−1 , is a closed ˆ embedded submanifold of Rm−1 diffeomorphic to K, and the orbit Ψ(K, u) is not connected m−1 in general. We can propose that R is connected. Let the equality dim A(N ) = d(n, g) ˆ holds. Then dim A(N ) = dim Rm−1 and hence each orbit Ψ(K, u), u ∈ Rm−1 , of K is ˆ open in Rm−1 . As Rm−1 is connected, so the orbit Ψ(K, u) coincides with Rm−1 ; i.e., m−1 the group K acts on R transitively. Consequently, the group A(N ) acts transitively on N , which is only possible in the case that N is a manifold. It means that N is a smooth

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homogeneous space.

3.3. G-structures on Good Orbifolds Let (N , A) be a smooth orbifold. A covering orbifold of the orbifold N is an orbifold ˜ , A) ˜ with a projection f : N ˜ → N such that for each point x ∈ N and every y ∈ (N −1 f (x) there exist charts (Ω, Γ, p) ∈ A and (Ω′ , Γ′ , p′ ) ∈ A˜ with the following properties: (a) Ω = Ω′ and Γ′ is a subgroup of the group Γ; (b) x ∈ U = p(Ω) and y ∈ U ′ = p′ (Ω′ ); ˜ → N is called a covering mapping for N . (c) f |U ′ ◦ p′ = p. The projection f : N ˜ → N is an automorphism h : N ˜ → N ˜ in the A covering transformation of f : N category Orb, such that f ◦ h = f. The set G(f ) of all covering transformations of f forms ˜ → N is called regular if N = N ˜ /G(f ). the group. A covering mapping f : N Recall that a discrete group G of diffeomorphisms acts properly discontinuously on a manifold M if for x, y ∈ M where y does not belong to the orbit of x under G, there exist two neighborhoods V and W of x and y respectively such that, for any g ∈ G, g 6= idM , g(V ) ∩ W = ∅ holds, and each isotropy group Gx is finite. ˜ → N where N ˜ is a manWhen an orbifold N has a regular covering mapping f : N ifold, N is called good. Remark that G(f ) is a discrete group of diffeomorphisms which ˜ . In this case if the group G(f ) is finite, acts properly discontinuously on the manifold N then N is called very good orbifold. ˜ → N is called universal if for any other covering mapping A covering mapping f : N ′ ′ ˜ → N ′ that f ′ ◦ f˜ = f. If f : N ˜ → N is f : N → N there exists such covering f˜: N ˜ is called the universal covering orbifold for N . In [33] universal covering mapping, then N it is shown that for any orbifold N there exists the universal covering orbifold defined up to isomorphisms in the category Orb. The fundamenthal group π1orb (N ) of an orbifold N is defined by Thurston [33] as a ˜ → N. group of the all covering transformations of the universal covering mapping f : N If an orbifold N is not good, it is called bad. Any proper orbifold N with the trivial fundamenthal group π1orb (N ) is bad. Each bad orbifold has such N as the universal covering orbifold. The simplest example of a bad orbifold is a drop N with only one orbifold point having linearized chart (R2 , Γ, p) where Γ ∼ = Zk = hγ | γ k i, k 6= 1, is a group of the order k of 2 rotations of the plane R with the fix point 0 ∈ R2 . As it is known [16] there exists a countable family of pair-wise nonisomorphic bad 3-dimensional simply connected orbifolds with underlying space S3 . T HEOREM 3.2. Let R be a G-structure of finite type on an orbifold N and f : N ′ → N be a regular covering of N by an orbifold N ′ with the deck transformations group Γ. Then: (i) the G-structure R forms some G-structure R′ on N ′ ; (ii) the mapping fˆ: R′ → R is defined satisfying the equality π ◦ fˆ = f ◦ π ′ where π : R → N and π ′ : R′ → N ′ are natural projections of bundles; (iii) the hole group A(N ′ ) of automorphisms of the G-structure R′ is isomorphic to the quotient group N(Γ)/Γ of the normalizer N(Γ) of Γ in the hole group A(N ) of automorphisms of the G-structure R.

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P ROOF. Let A = {(Ωi , Γi , pi ) | i ∈ J} and A′ = {(Ω′α , Γ′α , p′α ) | α ∈ J ′ } be the maximal atlases of orbifolds N and N ′ respectively. Let ξ = {Ri , bi , φ¯ij }i,j∈J be a G-structure on N . Take y ∈ N ′ . Put x = f (y) ∈ N . According to definition of the covering mapping f, there exist charts (Ωi , Γi , pi ) ∈ A and (Ω′α , Γ′α , p′α ) ∈ A′ such that: (a) Ωi = Ω′α and Γ′α is a subgroup of the group Γi ; (b) x ∈ Ui = pi (Ωi ) and y ∈ Uα′ = p′α (Ω′α ); (c) f |Uα′ ◦ p′α = pi . Since Γ′α ⊂ Γi , then a G-structure Rα′ := Ri on Ω′α = Ωi is defined. So we can form the family ξ ′ := {Rα′ , α ∈ J ′ } of G-structures over Ω′α , α ∈ J ′ . The compatibility of G-structures from ξ implies the compatibility of G-structures from ξ ′ . Denote the so-constructed G-structure on N ′ by R′ . Let π : R → N and π ′ : R′ → N ′ be natural projections of bundles. Take u′ ∈ R′ . Put y = π ′ (u′ ) ∈ N ′ , x = f (y) ∈ N . Then there exist charts (Ωi , Γi , pi ) ∈ A and (Ω′α , Γ′α , p′α ) ∈ A′ such that: (a) Ωi = Ω′α and Γ′α ⊂ Γi ; (b) x ∈ Ui = pi (Ωi ) and y ∈ Uα′ = p′α (Ω′α ); (c) f |Uα′ ◦ p′α = pi . As above, let p¯i : Ri → Γi \Ri and p¯′α : Rα′ → Γ′α \Rα′ be the quotient mapping, let qi : Γi \Ri → R and qα′ : Γ′α \Rα′ → R′ be embeddings. For u′ ∈ R′ there exists z ∈ Rα′ = Ri such that qα′ ◦ p¯′α (z) = u. Define a mapping fˆ: R′ → R by the formula fˆ(u′ ) := qi ◦ p¯i (z). The straightforward check shows that fˆ is correctly defined smooth mapping, with the equality π ◦ fˆ = f ◦ π ′ is satisfied. Determine a group homomorphism χ : N(Γ) → A(N ) : h′ 7→ h by the equality h(x) := f ◦ h′ (y) for all x ∈ N where y ∈ f −1 (x). The definitions of the deck transformations group Γ and the homomorphism χ imply that ker χ coincides with Γ. Note that for each automorphism h ∈ A(N ) there exists some automorphism h′ ∈ N(Γ) covering h, i.e., f ◦ h′ = h ◦ f. The covering automorphism h′ takes each orbit of the action of Γ into another orbit, h′ (Γ(x)) = Γ(h′ (x)), x ∈ N ′ . This implies that h′ Γh′−1 = Γ, i.e., h′ ∈ N(Γ). Hence, χ is surjective. Since Γ is a closed discrete subgroup of A(N ′ ), the normalizer N(Γ) is a closed subgroup of A(N ′ ). Consequently, N(Γ) is a closed Lie subgroup of the Lie group A(N ′ ). Therefore, A(N ) is isomorphic to the quotient Lie group N(Γ)/Γ.

3.4.

Influence Stratification of Orbifold on Dimension of the Automorphism Group

Inducted automorphism group of a connected component of a stratum Let N be an orbifold, H be a Lie group of automorphisms of N . Let ∆ck be a connected component of a stratum ∆k of N . Consider a subgroup H(N , ∆ck ) := {f ∈ H | f (∆ck ) = ∆ck }. Demonstrate that H(N , ∆ck ) is an open-closed Lie subgroup of the Lie group H. Suppose that f ∈ H(N , ∆ck ) and h : [0, 1] → H is a continuous path in H, with f = h(0). Take x ∈ ∆ck . Denote the action of the Lie group H on N by Φ. Since the path h is continuous ˜ : [0, 1] → N , defined by the equality and the group H smoothly acts on N , then the path h c ˜ h(t) := Φ(h(t), x) is continuous. As f preserves ∆k , i.e., f (∆ck ) = ∆ck , and g(∆k ) = ∆k ˜ for all g ∈ H, so h(t) ∈ ∆ck , ∀t ∈ [0, 1]. Therefore, for each t ∈ [0, 1] the automorphism h(t) keeps invariant each connected component ∆ck of stratum ∆k . Consequently,

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h(t) ∈ H(N , ∆ck ), ∀t ∈ [0, 1], and the group H(N , ∆ck ) contains whole connected component of element f. Remark that the group H(N , ∆ck ) also contains connected component of identity idN of H. Thus, the group H(N , ∆ck ) consists from connected components of the Lie group H and, hence, it is an open-closed subgroup Lie of the Lie group H, with dim H(N , ∆ck ) = dim H. So we obtain P ROPOSITION 3.1. Let H be an arbitrary Lie group of automorphisms of an orbifold N , ∆ck be a connected component of a stratum ∆k of N . Then the subgroup H(N , ∆ck ) in H, consisted from automorphisms of H which preserved ∆ck , is an open-closed Lie subgroup of the Lie group H. D EFINITION 3.1. The Lie group HN (∆ck ) = {f |∆ck | f ∈ H(N , ∆ck )} is called an inducted automorphism group of a connected component ∆ck . Let R be a G-structure of finite type and order m on a n-dimensional orbifold N , ∆ck be a connected component of a stratum ∆k of N . According to subsection 2.5, a G-structure Q on ∆ck is well-defined. Remark that the G-structure Q is a finite type G-structure and has order ≤ m. Therefore, the group A(∆ck ) of all automorphisms of G-structure Q on ∆ck is a Lie group and the group homomorphism χ : A(N , ∆ck ) → A(∆ck ) : f 7→ f |∆ck is correctly defined. Let ∆ck be the closure a connected component ∆ck of a stratum ∆k . Suppose that ∆ck 6= ∆ck . By Theorem 1.1, ∆ck is a k-dimensional orbifold. According to Remark 2.3, G-structure R forms a G-structure on the closure ∆ck . Denote the group of automorphisms of G-structure on ∆ck by A(∆ck ). By Theorem 3.1, the group A(∆ck ) is a Lie group. Since each automorphism f ∈ A(N , ∆ck ) is continuous and it keeps invariant ∆ck , so the equality f (∆ck ) = ∆ck is satisfied. Therefore, we can consider the homomorphism ¯ of χ ¯ by AN (∆ck ) and call χ ¯ : A(N , ∆ck ) → A(∆ck ) : f 7→ f |∆c . Denote the image imχ k

it an inducted automorphism group of the closure ∆ck of a connected component ∆ck . Remark that the mapping υ : AN (∆ck ) → AN (∆ck ) defined by the formula υ(f |∆c ) := f |∆ck , k f ∈ A(N , ∆ck ), is a group isomorphism. Using the definition of the topologies in the automorphism groups A(N , ∆ck ), A(∆ck ) and A(∆ck ) we obtain the following assertion.

T HEOREM 3.3. Let A(N ) be the hole automorphism group of a G-structure on a ndimensional orbifold N . If N admits a k-dimensional stratum ∆k , k < n, and ∆ck is a connected component of ∆k , then (i) the homomorphisms χ : A(N , ∆ck ) → A(∆ck ) : f 7→ f |∆ck , χ ¯ : A(N , ∆ck ) → A(∆ck ) : f 7→ f |∆c , k

υ:

AN (∆ck )

→

AN (∆ck ) :

are Lie group homomorphisms, with υ ◦ χ ¯ = χ;

f |∆c 7→ f |∆ck k

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(ii) the images imχ = AN (∆ck ) and imχ ¯ = AN (∆ck ) are closed Lie subgroups of the Lie groups A(∆ck ) and A(∆ck ) respectively; (iii) the kernels ker χ and ker χ ¯ are isomorphic; (iv) the following inequality is satisfied dim A(N ) ≤ dim AN (∆ck ) + dim ker χ.

(3.1)

T HEOREM 3.4. Let R be a G-structure of finite type and order m on a n-dimension orbifold N . If N has a k-dimensional stratum, then dim A(N ) ≤ k + dim g + dim g1 + . . . + dim gm−1 ;

(3.2)

moreover, if this stratum is not closed in the topology of N , then dim A(N ) < k + dim g + dim g1 + . . . + dim gm−1 .

(3.3)

P ROOF. Let ∆ck be a connected component of a stratum ∆k . Let π e : Rm−1 → N be π π π1 π m−1 2 the composition of the bundle projections Rm−1 −→ . . . −→ R1 −→ R −→ N . Put c Rcm−1 := π e−1 (∆ck ) and πm−1 := π e|Rcm−1 . Denote the component of idN of the Lie group A(N ) by Ae (N ). Each automorphism f ∈ Ae (N ) keeps invariant ∆ck . Therefore, the equality π e ◦ fˆm−1 = f ◦ π e implies fˆm−1 (Rcm−1 ) = Rcm−1 . It means that automorphisms of the Lie group H := σ(Ae (N )) keep Rcm−1 , i. e. any orbit H · u ∀u ∈ Rcm−1 belongs to Rcm−1 . Here σ : A(N ) → K is the Lie group isomorphism indicated in the proof of Theorem 3.1. As the orbit H · u is a closed embedded manifold in Rcm−1 , so dim H · u ≤ dim Rcm−1 = k + dim g + dim g1 + . . . + dim gm−1 .

(3.4)

Since each automorphism of H preserves e-structure on Rm−1 , then the Lie group H freely acts on Rm−1 and, consequently, on Rcm−1 . It means that each orbit H · u, u ∈ Rcm−1 , is diffeomorphic to the Lie group H. Then dim A(N ) = dim Ae (N ) = dim H = dim H · u.

(3.5)

The formulas (3.4) and (3.5) implies the estimate (3.2). Note the equality in (3.2) is possible only in the case when Rcm−1 = Rcm−1 . In this case Rm−1 \Rcm−1 is an open subset in Rm−1 . As π e is a submersion, so π e(Rm−1 \Rcm−1 ) = c c N \∆k is an open subset in N . Hence, ∆k is a closed subset in N . Consequently, ∆ck 6= ∆ck yields the estimate (3.3). C OROLLARY 3.1.[5, Theorem 3] If orbifold N admits an isolated singular point then dim A(N ) ≤ dim g + dim g1 + . . . + dim gm−1 .

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4.

Classes of Affinely Connected Orbifolds

4.1.

The Automorphism Groups of Affinely Connected Orbifolds

Let M be a smooth manifold. Denote the algebra of all smooth vector fields on M by X(M) and the algebra of all smooth function on M by F(M). Recall that an affine connection on manifold M is called a mapping ∇ : X(M) × X(M) → X(M) satisfying to the following conditions: (a) ∇X (Y + Z) = ∇X Y + ∇X Z, (b) ∇X+Y Z = ∇X Z + ∇Y Z, (c) ∇X f Y = (Xf )Y + f ∇X Y, (d) ∇f X Y = f ∇X Y, where X, Y, Z ∈ X(M), f ∈ F(M). A diffeomorphism of M is said to keep the affine connection ∇ if ∇f∗ X f∗ Y = f∗ (∇X Y ) for all X, Y ∈ X(M). D EFINITION 4.1. Let N be a smooth orbifold with the maximal atlas A = {(Ωi , Γi , pi ) | i ∈ J}. It is said that an affine connection is given on N if a family ∇ = {∇i }i∈J is defined where ∇i is an affine connection on a manifold Ωi satisfying to the following conditions: (a) each γ ∈ Γi keeps the affine connection ∇i ; (b) an embedding φij a chart (Ωi , Γi , pi ) into chart (Ωj , Γj , pj ) with coordinate neighborhood Ui and Uj , Ui ⊂ Uj , satisfies to the equality (φij )∗ (∇iX Y ) = ∇j(φij )∗ X (φij )∗ Y for all vector fields X, Y on Ωi . The pair (N, ∇) is called an affinely connected orbifold. Remark that an affine connection ∇ is given on an orbifold N if and only if a connection form ω is given on the frame bundle over N . D EFINITION 4.2. Let (N , ∇) be a affinely connected orbifold. By an automorphism of (N , ∇) we mean an diffeomorphism f : N → N of the orbifold (N , A) satisfying the condition: for each point x ∈ N and each pair charts (Ωi , Γi , pi ), (Ωj , Γj , pj ) ∈ A with coordinate neighborhoods Ui ∋ x and Uj such that f (Ui ) = Uj there exists a local lift fˆ: Ωi → Ωj fulfilled the equality ∇jˆ fˆ∗ Y = fˆ∗ (∇iX Y ) for all X, Y ∈ X(Ωi ). f∗ X

Denote the hole automorphism group of (N , ∇) by A(N , ∇). R EMARK 4.1. Let (Ωi , Γi , pi ) ∈ A be a chart, X and Y be Γi -invariant vector fields on Ωi , i.e. γ∗ X = X, γ∗ Y = Y ∀γ ∈ Γi . The consequence of the equalities γ∗ (∇iX Y ) = ∇iγ∗ X γ∗ Y = ∇iX Y implies that ∇iX Y is Γi -invariant vector field on Ωi . The equality ′ ∇i Y := ∇i Y where X, Y is vector fields on FixΓ defines a connection on FixΓ . This i i X X means that the connection ∇ induces the connection ′ ∇ on each connected component ∆ck of stratum ∆k . By analogy the connection ∇ induces the connection ′′ ∇ on the closure ∆ck which is a k-dimensional orbifold. T HEOREM 4.1. Let A(N , ∇) be the hole group of a n-dimensional affinely connected orbifold (N , ∇) and let ∆(N ) = {∆k }k∈{0,...,n} be the stratification of N . Then: (i) the group A(N , ∇) is a Lie group of dimension at most n2 + n, with A(N , ∇) acts on N smoothly; (ii) the group A(N , ∇) admits a unique topology and smooth structure which make it into a Lie group;

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(iii) the equality dim A(N , ∇) = n2 + n is satisfied if and only if (N , ∇) is the ordinary affine space with the flat affine connection; (iv) if N is a proper orbifold then dim A(N , ∇) ≤ n2 ; (v) if ∆ck 6= ∅, k < n, then dim A(N , ∇) ≤ n2 + n − (n − k)(2k + 1) ≤ n2 ,

(4.1)

moreover, if ∆ck 6= ∆ck , k < n, then dim A(N , ∇) ≤ n2 + n − (n − k)(2k + 1) − k;

(4.2)

(vi) the equality dim A(N , ∇) = n2 implies ∆k = ∅ for all k ∈ {1, . . . , n − 1}; if dim A(N , ∇) > n2 , then N is an affinely connected n-dimensional manifold with zero torsion; if dim A(N , ∇) > n2 and n ≥ 4, then N is the ordinary affine space with affine connection. P ROOF. The items (i), (iii), (iv) and (vi) are proved by authors in [6]. For proof of items (ii) and (v) we repeat some reasoning of proof of Theorem 3 [6]. In [6] some estimates of dimension of the Lie group A(N , ∇) were obtained. Here we improve these estimates. Let P be the linear frame bundle over a n-dimensional orbifold N with the natural projection π : P → N . The affine connection ∇ on N determines a connection form ω on P. Denote the Lie algebra of the Lie group G = GL(n, R) by g. Fix some Euclidean scalar products d0 and d1 on the vector spaces Rn and g respectively. Recall that the canonical form θ and the connection form ω receive the values in Rn and g respectively. Then the formula d(X, Y ) := d0 (θ(X), θ(Y ))+d1 (ω(X), ω(Y )), where X and Y are smooth vector fields on the manifold P, defines a Riemannian metric on P. According to Lemma 3 [6], the diffeomorphism fˆ of P induced by an automorphism f ∈ A(N , ∇) keeps invariant the connection form ω and the canonical form θ : fˆ∗ ω = ω,

fˆ∗ θ = θ,

(4.3)

and it fulfils the equality π ◦ fˆ = f ◦ π.

(4.4)

Conversely, if an diffeomorphism h of P satisfies the conditions (4.3) and (4.4) then h is induced by an automorphism f ∈ A(N , ∇). Remark that the equalities (4.3) yield f ∗ d = d, i. e. an induced diffeomorphism fˆ is isometry of the Riemannian manifold (P, d). Denote the group of all isometries of the Riemannian manifold (P, d) by I(P, d). Recall that the group I(P, d) endowed with the compact-open topology is a Lie group of transformations. Thus the mapping σ : f 7→ fˆ defines an isomorphism of the group A(N , ∇) onto some subgroup of the Lie group I(P, d). It is easy to see that the image imσ is a closed subgroup of the Lie group I(P, d). Therefore, we have the group isomorphism of the group A(N , ∇) onto the closed Lie subgroup of the isometry group I(P, d). Hence, by Proposition 1 [7], the group A(N , ∇) admits a unique topology and smooth structure which make it into a Lie group. The item (ii) is proved.

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The bijection σ forms on the set A(N , ∇) the structure of a smooth manifold. Since σ is a group isomorphism, with respect to the induced smooth structure A(N , ∇) is a Lie group. Receive the estimates (4.1) and (4.2). Let ∆ck 6= ∅, k < n. By Remark 4.1, the affine connection ∇ induces the affine connections on the connected component ∆ck of the stratum ∆k and on the closure ∆ck of ∆ck . So we can consider the inducted groups AN (∆ck ) and AN (∆ck ). According to Proposition 3.1, the subgroup A(N , ∆ck ) consisted from automorphisms of A(N , ∇) which preserved the connected component ∆ck is an open-closed Lie subgroup of the Lie group A(N , ∇). Using the topology of the Lie groups A(N , ∆ck ) and A(∆ck ), it is easy to check that the mapping χ : A(N , ∆ck ) → A(∆ck ) : f 7→ f |∆ck is a homomorphism of the Lie groups, with the image imχ = AN (∆ck ) is a closed Lie subgroup of the Lie group A(∆ck ). Then dim A(N , ∇) = dim A(N , ∆ck ) ≤ dim AN (∆ck ) + dim ker χ. Estimate the dimension of the kernel ker χ := {f ∈ A(N , ∆ck ) | f |∆ck = id∆ck }. Let x be an arbitrary point of ∆ck , let (Ω, Γ, p) be an chart with coordinate neighborhood U ∋ x such that y = p−1 (x) ∈ FixΓ. Since Γ is finite group of diffeomorphisms of the manifold Ω, so there exists Riemannian metric g on Ω for which Γ is an isometry group. In the point y choice a normal coordinate system (y 1 , . . . , y n ) such that (y 1 , . . . , y k ) is a coordinate system of k-dimensional manifold FixΓ. Then with relation to the selected coordinate system Jacobi matrix of transformation γ ∈ Γ at the point y has the form   E 0 , 0 C where E is the unit of the orthogonal group O(k, R) and C ∈ O(n − k, R). Let f ∈ ker χ. As f is an automorphism of the orbifold N , so the triple (Ω, Γ, p′ := f |U ◦ p) is a chart of N with coordinate neighborhood f (U ). By the definition of the automorphism f of the affinely connected orbifold (N , ∇) a representative f¯: Ω → Ω of f in the charts (Ω, Γ, p) and (Ω, Γ, p′ ) is automorphism of the affinely connected manifold Ω. The differential f¯∗y of f¯ we will consider as a linear transformation of Rn , with f¯∗y (0) = 0, 0 ∈ Rn . Since f ∈ ker χ, so f¯|FixΓ = idFixΓ and with relation to the selected normal coordinates at the point y Jacobi matrix of the mapping f¯ at y has the form   E A 0 B where B ∈ GL(n − k, R), A is a matrix of size k × (n − k) and E is the unit in the group GL(k, R). ′ ◦ f¯ . For any γ ∈ Γ there exists γ ′ ∈ Γ such that f¯ ◦ γ = γ ′ ◦ f¯; then f¯∗y ◦ γ∗y = γ∗y ∗y Consequently,       E A E 0 E 0 E A , = 0 B 0 C′ 0 C 0 B ˜ ′ := {C ∈ where C, C ′ ∈ O(n − k, R). Therefore, AC = A or C t At = At for all C ∈ Γ
O(n − k, R) | E0 C0 is Jacobi matrix of γ at y, γ ∈ Γ}, where At , C t is corresponding transposed matrixes. Hence, the lines of the matrix A = (aij ) defines the vecors ai = ˜ Suppose that there exists a (ai1 , . . . , ain−k ), fixed by any transformations of the group Γ.

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˜ Then γ∗y (X) = X, ∀γ ∈ Γ. vector X ∈ {0} × Rn−k , X 6= 0 such that CX = X ∀C ∈ Γ. Since the vectors from the tangent vector space Ty Ω fixed by the all transformations γ∗y , γ ∈ Γ, belong to Rk × {0}, hence the vector X is equal to the null vector. Thus, A = 0 and Jacobi matrix of the mapping f¯ at y ∈ Ω has the form   E 0 , (4.5) 0 B where B ∈ GL(n − k, R). Denote the subgroup of the matrixes having the form (4.5) by G. Since f¯ is an automorphism of the affinely connected manifold Ω, so the equality f¯∗y = idRn implies that there is an open subset W ∋ y of Ω such that f¯|W = idW . Then f |p(W ) = idp(W ) . As p : Ω → U is an open mapping, so f and idN are equal on the open subset p(W ). According to Lemma 4 [6], if automorphisms f and idN of the affinely connected orbifold (N , ∇) coincide on the open subset p(W ) ⊂ N , then they coincide on the whole orbifold N , i. e. f = idN . Consequently, the mapping µ : ker χ → G : f 7→ f¯∗y is a group monomorphism from the Lie group ker χ onto some subgroup of the Lie group G. So dim ker χ ≤ dim G = dim GL(n − k, R) = (n − k)2 . Thus, we have dim A(N , ∇) ≤ AN (∆ck ) + dim ker χ ≤ k 2 + k + (n − k)2 = n2 + n − (n − k)(2k + 1). Let ∆ck 6= ∆ck , k < n. The straightforward check to show that the mapping υ : AN (∆ck ) → AN (∆ck ) : f |∆c 7→ f |∆ck is a Lie group isomorphism. Hence, AN (∆ck ) k

and AN (∆ck ) have the same dimension. Since AN (∆ck ) is a closed Lie subgroup of the Lie group A(∆ck ) of all automorphisms of proper affinely connected orbifold ∆ck , so applying the item (iv) to the orbifold ∆ck we obtain dim AN (∆ck ) ≤ dim A(∆ck ) ≤ k 2 . So we have the sequence of the inequalities dim A(N , ∇) ≤ dim AN (∆ck ) + dim ker χ ≤ dim AN (∆ck ) + dim ker χ ≤ k 2 + (n − k)2 = n2 + n − (n − k)(2k + 1) − k. The following proposition proves the precision of the estimates (4.1) and (4.2).

P ROPOSITION 4.1. 1. For each pair of integer numbers (n, k), where 0 ≤ k < n, there exists a n-dimensional affinely connected orbifold (N1 , ∇1 ) having the k-dimensional stratum ∆k , with dim A(N1 , ∇1 ) = n2 + n − (n − k)(2k + 1). 2. For each pair of integer numbers (n, k), where 0 < k < n, there exists a ndimensional affinely connected orbifold (N2 , ∇2 ), with the k-dimensional stratum ∆k is not closed in the topology of N2 and dim A(N2 , ∇2 ) = n2 + n − (n − k)(2k + 1) − k. P ROOF. Let γ1 , γ2 : Rn → Rn be transformations of Rn given the matrixes     −Ek 0 Ek 0 , , A2 = A1 = 0 En−k 0 −En−k

respectively, where Ek is the unit in the group GL(k, R), En−k is the unit in the group GL(n − k, R). Put Γ1 = hγ1 | γ12 i ∼ = Z2 , Γ2 = Γ1 × hγ2 | γ22 i ∼ = n Z2 × Z2 . The quotient spaces N1 = R /Γ1 and N2 = Rn /Γ2 are smooth n-dimensional orbifolds with stratifications ∆(N1 ) = {∆n , ∆k } and ∆(N2 ) = {∆n , ∆n−k , ∆k , ∆0 }, respectively. Note that if k > 0 then the stratum ∆k of N2 is not closed in the topology of N2 and ∆k = ∆k ⊔ ∆0 . Further, we assume that k > 0 for the orbifold N2 .

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The transformations of the groups Γ1 and Γ2 keep the ordinary flat affine connection of affine space An . Hence, N1 and N2 are affinely connected orbifolds. According to Proposition 7 [6], the Lie group A(Ni , ∇i ), i = 1, 2, is isomorphic to the factor-group N(Γi )/Γi , where N(Γi ) is the normalizer of Γi in the hole group A(An ) of all affine transformations of the affine space An . The group A(An ) is isomorphic to the semi-direct product of the linear group GL(n, R) and the translation group Rn , which is a normal subgroup in this product. Therefore, any transformation of A(An ) can be consider as a pair hA, ai, where A ∈ GL(n, R), a ∈ Rn , with the multiply in A(An ) is given by the equality hA, ai · hB, bi := hAB, Ab + ai,

hA, ai, hB, bi ∈ A(An ).

Then γi has a form hAi , 0i, where 0 = (0, . . . , 0) ∈ Rn . Since N(Γ1 ) = {hA, ai ∈ A(An ) | hA, ai · hA1 , 0i = hA1 , 0i · hA, ai}, so hA, ai ∈ N(Γ1 ) iff   ′ A 0 , A′ ∈ GL(k, R), A′′ ∈ GL(n − k, R), A= 0 A′′ a = (a1 , . . . , ak , 0, . . . , 0) ∈ Rn . Thus the group A(N 1 , ∇1 ) is isomorphic to the semi-direct product of the groups ′ 0
A G1 := { 0 A′′ | A′ ∈ GL(k, R), A′′ ∈ GL+ (n − k, R)} and n G2 := {a = (ai ) ∈ R | ai = 0, i = k + 1, . . . , n}, where GL+ (n − k, R) is the group of nondegenerate matrixes with positive determinates. Consequently, dim A(N1 , ∇1 ) = k 2 + (n − k)2 + k = n2 + n − (n − k)(2k + 1). As N(Γ2 ) = {hA, ai ∈ A(An ) | hA, ai · hAi , 0i = hAi , 0i · hA, ai, i = 1, 2}, so the group N(Γ2 ) consists from transformations having the form A=



A′ 0 0 A′′



, A′ ∈ GL(k, R), A′′ ∈ GL(n − k, R).

Therefore the group A(N2 , ∇2 ) is isomorphic to the product GL+ (k, R) × GL+ (n − k, R), where GL+ (k, R) is the group of nondegenerate matrixes with positive determinates. Thus dim A(N2 , ∇2 ) = k 2 + (n − k)2 = n2 + n − (n − k)(2k + 1) − k.

4.2.

The Isometry Groups of Pseudo-Riemannian Orbifolds

T HEOREM 4.2. Let I(N ) be the hole isometry group of a n-dimensional pseudoRiemannian orbifold N and let ∆(N ) = {∆k }k∈{0,...,n} be the stratification of N . Then: (i) the group I(N ) is a Lie group of dimension at most n(n + 1)/2, with the action of the Lie group I(N ) on orbifold N is smooth and nonproper in general; (ii) the group I(N ) admits a unique topology and smooth structure which make it into a Lie group; (iii) the equality dim I(N ) = n(n + 1)/2 implies that pseudo-Riemannian orbifold N is a homogeneous n-dimensional pseudo-Riemannian manifold;

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(iv) if ∆ck 6= ∅, k < n, then dim I(N ) ≤

n(n + 1) − (n − k)(k + 1), 2

(4.6)

moreover, if ∆ck 6= ∆ck , k < n, then dim I(N ) ≤

n(n + 1) − (n − k)(k + 1) − k; 2

(4.7)

(v) if N is the proper orbifold, then dim I(N ) ≤ n(n − 1)/2; moreover, the equality dim I(N ) = n(n − 1)/2 yields ∆k = ∅ for all k ∈ {1, . . . , n − 2}. P ROOF. Let N be a n-dimensional pseudo-Riemannian orbifold of signature (p, q). Since the pseudo-Riemannian structure on N is a G-structure of first order, so using the equality dim O(p, q) = n(n−1) and applying Theorem 3.1, we obtain items (i), (ii) and (iii). 2 Let ∆ck be a connected component of the stratum ∆k of N . According to Proposition 2.9, g induces a pseudo-Riemannian metric on ∆ck . Denote the hole isometry group of the Riemannian manifold ∆ck by I(∆ck ). By Theorem 3.3 a Lie group homomorphism χ : I(N , ∆ck ) → I(∆ck ) : f 7→ f |∆ck is defined and the image imχ = IN (∆ck ) is a closed Lie subgroup of the Lie group I(∆ck ). Use the inequality (3.1) of Theorem 3.3: dim I(N ) ≤ dim IN (∆ck ) + dim ker χ. By analogy with affinely connected orbifold (Theorem 4.1) we obtain the following estimate dim ker χ ≤ dim O(p1 , q1 ) = (n−k)(n−k−1) where (p1 , q1 ) is 2 the signature of the k-dimensional pseudo-Riemannian manifold ∆ck , k = p1 + q1 . Apply. Thus, ing item (i) of Theorem 4.2 to I(∆ck ), we have dim IN (∆ck ) ≤ dim I(∆ck ) ≤ k(k+1) 2 n(n+1) k(k+1) (n−k)(n−k−1) c = 2 −(n−k)(k+1). dim I(N ) ≤ dim IN (∆k )+dim ker χ ≤ 2 + 2 The estimate (4.7) is received by analogy with the estimate (4.2). The estimate (4.6) implies item (v). According to Proposition 2.9, a pseudo-Riemannian metric of signature (p, q) on a ndimensional orbifold N induces the pseudo-Riemannian metrics on each connected component ∆cs . The following examples shows that the induced pseudo-Riemannian metric on ∆cs can have arbitrary signature (k, l) where 0 ≤ k ≤ p, 0 ≤ l ≤ q, s = k + l < n, in general. E XAMPLE 4.1. Let R(p,q) be the pseudo-Euclidean space of dimension n = p + q of signature (p, q) (see subsection 2.4). Let (k, l) be an arbitrary pair of the integer number such that 0 ≤ k ≤ p, 0 ≤ l ≤ q, k + l < n. (4.8) The mapping γk,l : R(p,q) → R(p,q) given by the matrix  Ek 0 0 0   0 −Ep−k 0 0  A=   0 0 El 0 0 0 0 −Eq−l 

(4.9)

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is an isometry of R(p,q) . Since A2 = E, then the group Γk,l generated by γk,l is isomorphic to Z2 . The quotient space N = R(p,q) /Γk,l is a pseudo-Euclidean orbifold with the stratification ∆(N ) = {∆n , ∆k+l }. The pseudo-Euclidean metric induced on the stratum ∆k+l has the signature (k, l). E XAMPLE 4.2. Let the group Γ of isometries of the n-dimensional pseudo-Euclidean space R(p,q) , p + q = n, generated by the isometries αi (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) := (x1 , . . . , xi−1 , −xi , xi+1 , . . . , xn ), ∼ (Z2 )n and where (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) ∈ R(p,q) , i = 1, . . . , n. Then Γ = the quotient space N = R(p,q) /Γ is a pseudo-Euclidean orbifold with the stratification ∆(N ) = {∆n , ∆n−1 , . . . , ∆1 , ∆0 }, and ∆i 6= ∅ ∀i ∈ {0, . . . , n} (see Example 1.7). Let (k, l) be an arbitrary pair satisfying (4.8). The isometry αk+1 ◦ . . . ◦ αp ◦ αp+l+1 ◦ . . . ◦ αn coincides with the isometry γk,l ∈ Γk,l given by matrix of form (4.9) (see Example 4.1). Consequently, the group Γ contains a subgroup Γk,l . Hence the orbifold N has the (k + l)dimensional stratum ∆k+l on which the pseudo-Euclidean metric of signature (k, l) is inducted according with Example 4.1.

4.3.

The Warped Product of Pseudo-Riemannian Orbifolds

Let (L, h) and (N , g) be two pseudo-Riemannian orbifolds of dimensions n and m respectively, given by maximal atlases B = {(Ω′α , Γ′α , p′α ) | α ∈ A} and A = {(Ωi , Γi , pi ) | i ∈ J}. Let f : L → R be a positive function. We will say that the product L × N of the orbifolds is endowed by metric of the warped product h⊕f g if for any charts (Ω′α , Γ′α , p′α ) ∈ B and (Ωi , Γi , pi ) ∈ A in the chart (Ω′α × Ωi , Γ′α × Γi , p′α × pi ) of L × N the pseudoRiemannian metric hα ⊕ f¯α gi is given where hα and gi are pseudo-Riemannain metrics on Ω′α and Ωi respectively, f¯α : Ω′α → R is a representative of f in the chart (Ω′α , Γ′α , p′α ). Remark that the function f¯α is defined up to the composition with the elements from group Γ′α . Since the transformations from Γ′α are isometries pseudo-Riemannian manifold (Ω′α , hα ), so the following definition is correct. D EFINITION 4.3. The family {hα ⊕ f¯α gi }i∈J,α∈A defines a pseudo-Riemannian metric on the product L × N of the orbifolds which is called a metric of the warped product and is denoted by h ⊕ f g. The pair (L × N , h ⊕ f g) is called a warped product of pseudoRiemannian orbifolds (L, h) and (N , g), it is denoted by L ×f N . If (L, h) and (N , g) are two pseudo-Riemannian manifolds, then the warped product L ×f N is a pseudo-Riemannian manifold. This construction is well known and it is widely used in geometry of pseudo-Riemannian geometry (see, for example [9, 4]). In the case when (L, h) and (N , g) are two Riemannian manifolds, the metric h ⊕ f g is also called semi-reducible, and L ×f N is called a semi-reducible Riemannian manifold. A semi-reducible structure on a complete and simply connected Riemannian space was investigated in [32]. P ROPOSITION 4.3. Let I(L ×f N ) be the hole isometry group of the warped product (L × N , h ⊕ f g) of pseudo-Riemannian orbifolds (L, h) and (N , g). Then the mapping ν : I(N , g) → I(L ×f N ) : ψ 7→ (id, ψ) is a monomorphism of the Lie groups.

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P ROOF. Let ψ be an arbitrary isometry of I(N , g). Then for any point x ∈ N there exist charts (Ωi , Γi , pi ), (Ωj , Γj , pj ) ∈ A with coordinate neighborhoods Ui = pi (Ωi ) ∋ x, Uj = pj (Ωj ) ∋ ψ(x) such that ψ(Ui ) ⊂ Uj and an isometry ψij : Ωi → Ωj of pseudoRiemannian manifolds (Ωi , gi ) and (Ωj , gj ) satisfying the equality pj ◦ ψij = ψ ◦ pi . Since the mapping f depends only on the coordinates of the orbifold L, then the mapping id × ψij : Ω′α ×Ωi → Ω′α ×Ωj given by the formula (id×ψ)(y, z) := (y, ψij (z)) ∀(y, z) ∈ Ω′α × Ωi is an isometry of the warped product Ω′α ×f¯α Ωi where f¯α : Ω′α → R is a representative of f in the chart (Ω′α , Γ′α , p′α ) ∈ B. By the definition, it follows that id × ψ is an isometry of the warped product L ×f N . Thus the mapping ν is correctly defined. It is clear that ν is an injective group homomorphism. The warped product L ×f N has the product topology. Using this fact and the way of introduction of topology in the isometry group of pseudo-Riemannian orbifold L ×f N , indicated in the proof of Theorem 3.1, we get that the convergence of a sequence of isometries {ψn } ⊂ I(N , g) to ψ implies the convergence of the sequence {id × ψn } to {id × ψ}. It means that ν is continuously mapping and consequently ν is a homomorphism of the Lie groups.

4.4.

Examples of Lorentzian Orbifolds with Noncompact Isometry Groups

Let M be a n-dimensional manifold admitted a Lorentzian metric, n ≥ 3. Let L(M) be the set of all Lorentzian metrics on M with C ∞ -topology. P. Mounout [25] showed that contrarily to Riemannian case, the subspace of Loretzian metrics without isometries is not always open in L(M). As well known [9, 1, 2, 4], the isometry groups of compact Loretzian manifolds can be noncompact unlike Riemannian manifolds. Examples show that the same is true for Lorentzian orbifolds. For any n ≥ 2 we constructed examples of compact Lorentzian ndimensional orbifolds with noncompact isometry group (see Example 4.3). A NOSOV DIFFEOMORPHISMS A diffeomorphism f of a manifold M is called an Anosov diffeomorphism if the following conditions are satisfied: (a) there is a splitting Tx M = Exs ⊕ Exu of the tangent space Tx M for each x ∈ M, which depends continuously of x ∈ M; (b) f∗x (Exs ) = Efs (x) and f∗x (Exu ) = Efu(x) for all x ∈ M; (c) there is a Riemannian metric g on M and for norm k · k induced by g there exist constants c > 0, λ > 0 such that for any integer m > 0 and x ∈ M, k(f m )∗ vk ≤ cλ−m kvk when v ∈ Exs and k(f m )∗ vk ≥ cλm kvk when v ∈ Exu . If a manifold M is compact then this definition is independent of the choice of a Riemannian metric g. For other Riemannian metric the numbers c and λ can be changed. Moreover, there exists a Riemannian metric for which c = 1. A diffeomorphism f of an orbifold N we call Anosov one, if f |∆n is an Anosov diffeomorphism of manifold ∆n . L EMMA 4.1. Let G be a Lie group of diffeomorphims of an orbifold N , which continuously acts on N . If there exist a Riemannian metric g on N , a vector X ∈ Tx N , X 6= 0, and a diffeomorphism f ∈ G such that the subset {k(f n )∗x Xk | n ∈ Z} in R is unbounded, then the Lie group G is noncompact.

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P ROOF. Suppose opposite, let G be the compact Lie group. Since the action of the group G on N is continuous, so the mapping α : G → R : h 7→ kh∗x Xk is continuous. Therefore, the image α(G) is compact in R and hence it is bounded. But according to the condition of Lemma the subset {k(f n )∗x Xk | n ∈ Z} of α(G) is unbounded. The contradiction shows that the Lie group G is a noncompact. C OROLLARY 4.1. Let G be a Lie group of diffeomorphims of an orbifold product L × N acting on L × N continuously. If the group G contains a diffeomorphism (id, f ), where f is an Anosov diffeomorphism of N , then G is noncompact. P ROOF. Let g be a Riemannian metric on N satisfying to the definition of Anosov diffeomorphism f. Consider a Riemannian metric h on L. Let h ⊕ g be a Riemannian metric on L × N . According to definition of f there exist constants c > 0, λ > 0 such that for any integer m > 0 and x ∈ N , k(f m )∗ Xk ≥ cλm kXk when X ∈ Exu where Tx N = Exs ⊕ Exu is a corresponding splitting of the tangent space Tx N . Remark that Y = 0 ⊕ X ∈ T(z,x) (L × N ) = Tz L ⊕ Tx N , (z, x) ∈ L × N , with k(id, f )n∗(z,x) Y k = k(f n )∗x Xk. Consequently, the subset {k(id, f )n∗(z,x) Y k | n ∈ Z} is unbounded in R. Thus, the all conditions of Lemma 4.1 are satisfied, hence G is noncompact Lie group. 2 0 E XAMPLE plane with the metric defined the matrix
4.3. Let (R , g ) be a pseudo-Euclidean 1 −1 2 → R2 of the plane R2 given by the matrix . The affine transformation f : R 0 −1 −1 A = ( 52 21 ) is an isometry of (R2 , g 0 ). As f0 ◦Z2 = Z2 ◦f0 , where Z2 is the translation group of R2 on arbitrary vectors with integer coordinates, so f0 projects to some diffeomorphism √ fA of the torus T 2 = R2 /Z2 . The matrix A has two proper numbers λ1,2 = 3 ± 2 2, and λ1 > 1, 0 < λ2 < 1. Thus fA is an Anosov diffeomorphism of the torus T 2 . The invariance of the Lorentzian metric g 0 relatively Z2 admits to define a Lorentzian metric g on the torus T 2 such that the quotient mapping π : R2 → R2 /Z2 is a local isometry, with fA is an isometry of (T 2 , g). Let (L, h) be a compact m-dimensional Riemannian orbifold, m ≥ 2. If m = 1, we take as (L, h) the segment [0, 1] considered as 1-dimensional Euclidean orbifold with two orbifold points {0, 1} and with orbifold groups isomorphic to Z2 . Let f : L → R be a smooth positive function on the manifold L. Then for each m ≥ 1 the warped product (L × T 2 , h ⊕ f g) is the (m + 2)-dimensional Lorentzian orbifold, with its hole isometry group I(L × T 2 , h ⊕ f g) contains the isometry {(id, fA )}. According to Corollary 4.1 the isotropy group of I(L × T 2 , h ⊕ f g) at the point (x, y), where x = π(0, 0), is noncompact, hence by Corollary 2.1 the action of I(L×T 2 , h⊕f g) on the orbifold L×f T 2 is nonproper.

4.5.

The Isometry Groups of Riemannian Orbifolds

Combining the obtained results (Theorem 4.2) and our results from [5, 7], present the following theorem. T HEOREM 4.3. Let I(N ) be the hole isometry group of a n-dimensional Riemannian orbifold N and let ∆(N ) = {∆k }k∈{0,...,n} be the stratification of N . Then:

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(i) the group I(N ) endowed with compact-open topology is a Lie group of dimension at most n(n + 1)/2, with the action of the Lie group I(N ) on orbifold N is smooth and proper; (ii) the group I(N ) admits a unique topology and smooth structure which make it into a Lie group; (iii) if N is compact then I(N ) is compact; (iv) the equality dim I(N ) = n(n + 1)/2 holds if and only if Riemannian orbifold N is isometric to one of the following n-dimensional Riemannian manifolds of constant curvature: (a) the Euclidean space En ; (b) the sphere S n ; (c) the projective space RP n ; (d) the simply connected hyperbolic space Hn ; (v) if ∆ck 6= ∅, k < n, then dim I(N ) ≤

n(n + 1) − (n − k)(k + 1), 2

(4.10)

moreover, if ∆ck 6= ∆ck , k < n, then dim I(N ) ≤

n(n + 1) − (n − k)(k + 1) − k; 2

(4.11)

(vi) if N is the proper orbifold, then dim I(N ) ≤ n(n − 1)/2; the equality dim I(N ) = n(n − 1)/2 implies ∆k = ∅ for all k ∈ {1, . . . , n − 2}; moreover, if in this case ∆n−1 6= ∅, then each connected component ∆cn−1 of the stratum ∆n−1 is the one of the following (n − 1)-dimensional Riemannian manifold of constant curvature: (a) the Euclidean space En−1 ; (b) the sphere S n−1 ; (c) the projective space RPn−1 ; (d) the simply connected hyperbolic space Hn−1 . The following proposition proves the precision of the estimates (4.10) and (4.11). P ROPOSITION 4.4. 1. For each pair of integer numbers (n, k), where 0 ≤ k < n, there exists a n-dimensional Riemannian orbifold N1 having the k-dimensional stratum ∆k which the hole isometry group I(N1 ) has the dimension n(n+1) − (n − k)(k + 1). 2 2. For each pair of integer numbers (n, k), where 0 < k < n, there exists a ndimensional Riemannian orbifold N2 having the nonclosed k-dimensional stratum ∆k , with dim I(N2 ) = n(n+1) − (n − k)(k + 1) − k. 2

P ROOF. Let N1 = Rn /Γ1 and N2 = Rn /Γ2 be smooth n-dimensional orbifolds constructed in the proof of Proposition 4.1. The orbifolds N1 and N2 have the stratifications ∆(N1 ) = {∆n , ∆k } and ∆(N2 ) = {∆n , ∆n−k , ∆k , ∆0 } respectively. Remark that the stratum ∆k of the orbifold N2 for k 6= 0 does not closed in the topology of N2 and ∆k = ∆k ⊔ ∆0 . Since the group Γ1 and Γ2 are isometry group of the Euclidean space En , than N1 and N2 are flat Riemannian orbifolds. Calculate the isometry groups I(N1 ) and I(N2 ). According to Theorem 3.2, the group I(Ni ), i = 1, 2, is isomorphic to the quotient group N(Γi )/Γi of the normalizer N(Γi ) of Γi in the group I(En ) of all isometries of

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the Euclidean space En . The group I(En ) is isomorphic to the semi-direct product of the orthogonal group O(n, R) and the translation group Rn , with En is a normal subgroup of this product. Each transformation of I(En ) has the form hA, ai where A ∈ O(n, R), a ∈ Rn . As hA, ai ∈ N(Γ1 ) if and only if   ′ A 0 , A′ ∈ O(k, R), A′′ ∈ O(n − k, R), A= 0 A′′ a = (a1 , . . . , ak , 0, . . . , 0) ∈ Rn ,


′ so the group I(N1 ) is isomorphic to the semi-direct product of the groups G1 := { A0 A0′′ | A′ ∈ O(k, R), A′′ ∈ SO(n − k, R)} and G2 := {a = (ai ) ∈ Rn | ai = 0, i = k + 1, . . . , n}. Therefore, dim I(N1 ) = k(k−1) + (n−k)(n−k−1) + k = n(n+1) − (n − k)(k + 1). 2 2 2 The group N(Γ2 ) consists from the transformations having the form   ′ A 0 , A′ ∈ O(k, R), A′′ ∈ O(n − k, R). A= 0 A′′ Then the group I(N2 ) is isomorphic to SO(k, R) × SO(n − k, R) where SO(k, R), + SO(n − k, R) are the special orthogonal groups. Thus we have dim I(N2 ) = k(k−1) 2 n(n+1) (n−k)(n−k−1) = 2 − (n − k)(k + 1) − k. 2 Let (N , g) be a Riemannian orbifold. Let Si be Ricci tensor of the Riemannian manifold (Ωi , gi ), (Ωi , Γi , pi ) ∈ A. The definition of the Riemannian metric g = {gi }i∈J implies that the family S = {Si }i∈J of tensors is a tensor of type (0, 2) on the orbifold N . The tensor S is called the Ricci tensor of (N , g). We say that a symmetric bilinear form t = {ti }i∈J on an orbifold N is negative (or nonpositive) definite at x ∈ N if there is some chart (Ωi , Γi , pi ) ∈ A with coordinate neighborhood Ui ∋ x such that the form ti is negative (respectively nonpositive) definite at x0 ∈ p−1 i (x). Conditions (a) and (b) in the definition of a section imply that this definition is independent of a choice of a chart (Ωi , Γi , pi ) with the coordinate neighborhood Ui ∋ x and a point x0 ∈ p−1 i (x). We say also that a symmetric bilinear form t is negative (respectively nonpositive) definite on N if t possesses this property at each x ∈ N . Using the integration on orbifolds introduced by Satake [30] and some obtained integral formulas, we receive [7] the following theorem which can be regarded as an analogy of the well-known Bochner’s theorem [11]. T HEOREM 4.4. If N is a compact Riemannian orbifold with nonpositive definite Ricci tensor and at some point of N the Ricci tensor is negative definite, then the isometry group of N is finite. A Riemannian orbifold (N , g) is said to be a Riemannian orbifold of constant curvature k ∈ R if the Riemannian manifold (Ωi , gi ) ∀(Ωi , Γi , pi ) ∈ A has constant curvature k. A Riemannian orbifold of constant curvature k is called hyperbolic (flat, elliptic) if k < 0 (respectively k = 0, k > 0). C OROLLARY 4.2. The isometry group of every compact hyperbolic orbifold is finite. In [7] we calculated the isometry groups of some hyperbolic orbifolds.

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The following theorem proved by the second author [37]. T HEOREM 4.5. Let N be a n-dimensional flat Riemannian orbifold. Then N is good and has the Euclidean space En as a covering manifold. If the fundamenthal group of N is finitely generated, then N is very good. In particular, for any n-dimensional compact flat Riemannian orbifold N there exists a regular covering mapping π : Tn → N where Tn is the flat torus, with the group G(π) of covering transformations is finite. Applying Theorem 3.2 to Theorem 4.5 and using the fact that the isometry group of the flat Riemannian torus has the dimension at most n we have C OROLLARY 4.3. The isometry group I(N ) of a n-dimensional compact flat Riemannian orbifold has the dimension at most n, with dim I(N ) = n iff N is a flat Riemannian torus T n.

References [1] Adams, S.; Stuck, G. The isometry group of a compact Lorentz manifold. Invent. Math. 1997, 129, 239-261. [2] Adams, S.; Stuck, G. The isometry group of a compact Lorentz manifold, II. Invent. Math. 1997, 129, 263-287. [3] Arens, R. A topology for spaces of transformations. Ann. of Math. 1946, 47, 3, 480– 495. [4] Arouche, A.; Deffaf, M.; Zeghib A. On Loretzian dynamics: from group actions to warped products via homogeneous spaces. Trans. Amer. Math. Soc. 2007, 359, 3, 12531263 [5] Bagaev, A. V.; Zhukova, N. I. The automorphism groups of finite type G-structures on orbifolds. Siberian Math. J. 2003. 44, 2, 213224. [6] Bagaev, A. V.; Zhukova, N. I. Affinely connected orbifolds and their automorphisms. In Non-Euclidean Geometry in Modern Physics and Mathematics: Proceedings of the International Conference BGL-4 (Bolyai-Gauss-Lobachevsky) (Nizhny Novgorod, Sept. 7–11, 2004); Ed.: Jenkovszky L., Polotovkiy G.; Kiev: Bogolubov Institute for heoretical Physics, 2004. 31–48. [7] Bagaev, A. V.; Zhukova, N. I. The isometry groups of Riemannain orbifolds. Siberian Math. J. 2007, 48, 4, 579592. [8] Baily, W. L. Jr. The decomposition theorem for V -manifolds. Amer. J. of Math. 1956, 78, 4, 862–888. [9] Beem, J. K.; Ehrlich, P. E. Global Lorentzian Geometry, Dekker, New York, 1981. [10] Blumenthal, R. A.; Hebda, J. J. Ehresmann connections for foliations. Indiana Univ. Math. J. 1984, 33, 4, 597–611.

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[11] Bochner, S. Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 1946, 52, 776– 797. [12] D’Ambra, G.; Gromov, M. Lectures on transformation groups: geometry and dynamics. In Surveys in Differential Geometry (supplement to the Journal of Differential Geometry), 1, 1991, pp. 19-111. [13] Dixon, L.; Harwey, J. A.; Vafa, C.; Witten, E. Strings on orbifolds (1). Nucl. Phys. B. 1985, 261, 4, 678–686. [14] Ehresmann, C. Sur les pseudo-groupes de Lie de type fini. C. R. Acad. Sci. Paris. 1958, 246, 360–362. [15] Ermolitski, A. A. Riemannian manifolds with geometric structures (Monograph), Minsk, BSPU, 1998. [16] Haefliger, A.; Salem, E. Action of tori on orbifolds. Ann. Global Anal. and Geom. 1991, 9, 1, 37–59. [17] Hano, J.; Morimoto, A. Note on the group of affine transformations of an affinely connected manifold. Nagoya Math. J. 1955, 88, 71–81. [18] Kobayashi, S. Groupe de transformations qui laissent invariante une connexion infinitesimale. C. R. Acad. Sci. Paris A. 1954, 238, 644-645. [19] Kobayashi, S. Transformation groups in diffrential geometry, Springer-Verlag, 1972. [20] Kobayashi, S.; Nomizu, K. Foundations of differential geometry, N. Y., John Wiley and Sons, 1963; Vol. 1. [21] Kowalsky, N. Actions of non-compact simple groups on Lorentz manifolds and other geometric manifolds. Ann. of Math. 1996, 144, 2, 611–640. [22] Lermann, E.; Tolman, S. Torus actions on symplectic orbifolds and toric varieties. Trans. Amer.Math. Soc. 1997, 349, 4201–4230. [23] Michor, P.W. Isometric actions of Lie groups and invariants; Lecture course at the University of Vienna, 1996/97. Vienna: University of Vienna, 1997. [24] Moerdijk, I.; Pronk, D. Orbifolds, sheaves and groupoids. K-theory. 1997, 12, 3–21. [25] Mounoud, P. Dynamical properties of the space of Lorentzian metrics. Comment.Math. Helv. 2003, 78, 463–485. [26] Myers, S.B.; Steenrod, N. The group of isometries of a riemannian manifold. Ann. Math. 1939, 40, 400–416. [27] Nomizu, K. On the group of affine transformations of an affinely connected manifold. Proc. Amer. Math. Soc. 1953, 4, 816–823. [28] Pflaum, M.J. On deformation quantization of symplectic orbispaces. Diff. geom. and its Appl. 2003, 19, 3, 343–368.

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[29] Satake, I. On a generalization of the notion of manifold. Proc. of the Nat. Ac. of Sciences. 1956, 42, 6, 359–363. [30] Satake, I. The Gauss-Bonnet theorem for V -manifolds. J. Math. Soc. Japan. 1957, 9, 464–492. [31] Sjamaar, R.; Lerman, E. Stratified symplectic spaces and reduction. Ann. of Math. 1991, 134, 375-422. [32] Solodovnikov, A. S. Global structure of semireducible Riemannian spaces of class C ∞ . (English. Russian original) Russ. Acad. Sci. Sb. Math. 1994, 79, 1, 1-14. [33] Thurston, W. P. The geometry and topology of 3-manifolds. Princeton: Princeton Univ., 1978. [34] Zeghib, A. The identity component of the isometry group of a compact Lorentz manifold. Duke Math. J. 1998, 92, 321–333. [35] Zeghib, A. Sur les espaces-temps homog`enes. In The Epstein birthday schrift, Geom. Topol. Monogr., Geom. Topol. Publ., Coventry, 1998; Vol. 1, pp. 551576. [36] Zhukova, N. On the stability of leaves of Riemannian foliations. Ann. Global Anal. and Geom. 1987, 5, 3, 261–271. [37] Zhukova, N.I. Cartan geometries on orbifolds. In Non-Euclidean Geometry in Modern Physics: Proceedings of the Fifth International Conference Bolyai-GaussLobachevsky (Belarus, Minsk, October 10-13, 2006); B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, 2006, 228–238. [38] Zimmer, R.J. On the automorphism group of a compact Lorentz manifold and other geometric manifolds. Invent. Math. 1986, 83, 411-426.

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 485-561

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 17

W RAP G ROUPS OF C ONNECTED F IBER B UNDLES , T HEIR S TRUCTURE AND C OHOMOLOGIES S.V. Ludkovsky Dept. of Applied Mathematics, Moscow State Technical Univ., Moscow, Russia

Abstract This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real R, complex C numbers, the quaternion skew field H and the octonion algebra O. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition (f, g) 7→ f −1 g is continuous or differentiable depending on a class of smoothness of groups. Moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. Nevertheless, it is proved that these groups does not necessarily satisfy the Campbell-Hausdorff formula even locally. Iterated wrap groups are studied as well. Their smashed products are constructed. Cohomologies of wrap groups and their structure are investigated. Sheaves of wrap groups are constructed and studied. Moreover, twisted cohomologies and sheaves over quaternions and octonions are investigated as well. CW-groups associated with wrap groups are studied.

1.

Introduction

Wrap groups of fiber bundles considered in this paper are constructed with the help of families of mappings from a fiber bundle with a marked point into another fiber bundle with a marked point over the fields R, C, H and the octonion algebra O. Conditions on fibers supplied with parallel transport structures are rather mild here. Therefore, they generalize geometric loop groups of circles, spheres and fibers with parallel transport structures over them. A loop interpretation is lost in their generalizations, so they are called here wrap groups. This paper continues previous works of the author on this theme, where generalized

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loop groups of manifolds over R, C and H were investigated, but neither for fibers nor over octonions [24, 32, 30, 31]. Loop groups of circles were first introduced by Lefshetz in 1930-th and then their construction was reconsidered by Milnor in 1950-th. Lefshetz has used the C 0 -uniformity on families of continuous mappings, which led to the necessity of combining his construction with the structure of a free group with the help of words. Later on Milnor has used the Sobolev’s H 1 -uniformity, that permitted to introduce group structure more naturally [37]. Iterations of these constructions produce iterated loop groups of spheres. Then their constructions were generalized for fibers over circles and spheres with parallel transport structures over R or C [14]. Wrap groups of quaternion and octonion fibers as well as for wider classes of fibers over R or C are defined and investigated here for the first time. Holomorphic functions of quaternion and octonion variables were investigated in [28, 29, 26]. There specific definition of super-differentiability was considered, because the quaternion skew field has the graded algebra structure. This definition of superdifferentiability does not impose the condition of right or left super-linearity of a superdifferential, since it leads to narrow class of functions. There are some articles on quaternion manifolds, but practically they undermine a complex manifold with additional quaternion structure of its tangent space (see, for example, [39, 52] and references therein). Therefore, quaternion manifolds as they are defined below were not considered earlier by others authors (see also [26]). Applications of quaternions in mathematics and physics can be found in [11, 16, 17, 23]. Fiber bundles and sheaves and cohomologies over quaternions and octonions are interesting in such a respect, that they take into account spin and isospin structures on manifolds, because there is the embedding of the Lie group U (2) into the quaternion skew field H. In this article wrap groups of different classes of smoothness are considered. Henceforth, we consider not only orientable manifolds M and N , but also nonorientable manifolds. In particular, geometric loop groups have important applications in modern physical theories (see [20, 34] and references therein). Groups of loops are also intensively used in gauge theory. Wrap groups defined below with the help of families of mappings from a manifold M into another manifold N with a dimension dim(M ) > 1 can be used in the membrane theory which is the generalization of the string (superstring) theory. Section 2 is devoted to the definitions of topological and manifold structures of wrap groups. The existence of these groups is proved and that they are infinite dimensional Lie groups not satisfying even locally the Campbell-Hausdorff formula (see Theorems 3, 6, 12, Corollaries 5, 8, 9 and Examples 10). In the cases of complex, quaternion and octonion manifolds it is proved that they have structures of complex, quaternion and octonion manifolds respectively. In Section 3 smashed products of wrap groups are constructed. Iterated wrap groups are studied as well. Their structure is investigated in more details. The main results of Section 3 are Theorems 2, 6, 9, 10, 20, 21, Propositions 3, 7, 8, 12, 13, 17 and Corollary 11. Section 4 is devoted to constructions and investigations of cohomologies and sheaves of wrap groups. Moreover, over quaternions and octonions twisted cohomologies and sheaves are studied. Twisted analogs of bar resolutions of sheaves and smooth Deligne cohomology

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are investigated as well. This is done over twisted multiplicative groups. Previously the complex case and with loop groups of fiber bundles on spheres was only studied. The main results of Section 4 are given in Theorems 34, 36, 44, 48.1, 55, 58, 60, Propositions 6, 14, 15, 19, 26, 27, 29, 32, Corollaries 7, 8, 33, 45 and 47. In Section 5 a structure of wrap groups as CW-groups is studied. All main results of this paper are obtained for the first time.

2.

Wrap Groups of Fibers

To avoid misunderstandings we first give our definitions and notations. 1.1. Note. Denote by Ar the Cayley-Dickson algebra such that A0 = R, A1 = C, A2 = H is the quaternion skew field, A3 = O is the octonion algebra. Henceforth we consider only 0 ≤ r ≤ 3. 1.2. Definition. A canonical closed subset Q of the Euclidean space X = Rn or of the standard separable Hilbert space X = l2 (R) over R is called a quadrant if it can be given by the condition Q := {x ∈ X : qj (x) ≥ 0}, where (qj : j ∈ ΛQ ) are linearly independent elements of the topologically adjoint space X ∗ . Here ΛQ ⊂ N (with card(ΛQ ) = k ≤ n when X = Rn ) and k is called the index of Q. If x ∈ Q and exactly j of the qi ’s satisfy qi (x) = 0 then x is called a corner of index j. If X is an additive group and also left and right module over H or O with the corresponding associativity or alternativity respectively and distributivity laws then it is called the vector space over H or O correspondingly. In particular l2 (Ar ) consisting of all sequences x = {xn ∈ Ar : n ∈ N} with the P ∗ 1/2 finite norm kxk < ∞ and scalar product (x, y) := ∞ n=1 xn yn with kxk := (x, x) ∗ is called the Hilbert space (of separable type) over Ar , where z denotes the conjugated Cayley-Dickson number, zz ∗ =: |z|2 , z ∈ Ar . Since the unitary space X = Anr or the separable Hilbert space l2 (Ar ) over Ar while considered over the field R (real shadow) is r isomorphic with XR := R2 n or l2 (R), then the above definition also describes quadrants in Anr and l2 (Ar ). In the latter case we also consider generalized quadrants as canonical closed subsets which can be given by Q := {x ∈ XR : qj (x + aj ) ≥ 0, aj ∈ XR , j ∈ ΛQ }, where ΛQ ⊂ N (card(ΛQ ) = k ∈ N when dimR XR < ∞). 1.2.2. Definition. A differentiable mapping f : U → U ′ is called a diffeomorphism if (i) f is bijective and there exist continuous mappings f ′ and (f −1 )′ , where U and U ′ are interiors of quadrants Q and Q′ in X. In the Ar case with 1 ≤ r ≤ 3 we consider bounded generalized quadrants Q and Q′ in Anr or l2 (Ar ) such that they are domains with piecewise C ∞ -boundaries. We impose additional conditions on the diffeomorphism f in the 1 ≤ r ≤ 3 case: ¯ = 0 on U , (ii) ∂f (iii) f and all its strong (Frech´et) differentials (as multi-linear operators) are bounded ¯ are differential (1, 0) and (0, 1) forms respectively, d = ∂ + ∂¯ is on U , where ∂f and ∂f an exterior derivative, for 2 ≤ r ≤ 3 ∂ corresponds to super-differentiation by z and ∂˜ = ∂¯ corresponds to super-differentiation by z˜ := z ∗ , z ∈ U (see [28, 29]). The Cauchy-Riemann Condition (ii) means that f on U is the Ar -holomorphic mapping.

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1.2.3. Definition and notation. An Ar -manifold M with corners is defined in the usual way: it is a metric separable space modelled on X = Anr or X = l2 (Ar ) respectively and is supposed to be of class C ∞ , 0 ≤ r ≤ 3. Charts on M are denoted (Ul , ul , Ql ), that is, ul : Ul → ul (Ul ) ⊂ Ql is a C ∞ -diffeomorphism for each l, Ul is open in M , ul ◦ uj −1 is biholomorphic for 1 ≤ r ≤ 3 from the domain uj (Ul ∩ Uj ) 6= ∅ onto ul (Ul ∩ Uj ) (that is, uj ◦ u−1 and ul ◦ u−1 are holomorphic and bijective) and ul ◦ u−1 j satisfy conditions l Sj (i − iii) from §1.2.2, j Uj = M . A point x ∈ M is called a corner of index j if there exists a chart (U, u, Q) of M with x ∈ U and u(x) is of index indM (x) = j in u(U ) ⊂ Q. A set of all corners of index j ≥ 1 is called a border ∂M of M , x is called an inner point of M if indM (x) = 0, so S ∂M = j≥1 ∂ j M , where ∂ j M := {x ∈ M : indM (x) = j}. For a real manifold with corners on the connecting mappings ul ◦ u−1 ∈ C ∞ of real j charts only Condition 1.2.2(i) is imposed. 1.2.4. Terminology. In an Ar -manifold N there exists an Hermitian metric, which in P each analytic system of coordinates is the following nj,k=1 hj,k dzj d¯ zk , where (hj,k ) is a ∞ positive definite Hermitian matrix with coefficients of the class C , hj,k = hj,k (z) ∈ Ar , z are local coordinates in N . As real manifolds we shall consider Riemann manifolds. In accordance with the definition above for internal points of N it is supposed that they can belong only to interiors of charts, but for boundary points ∂N it may happen that x ∈ ∂N belongs to boundaries of several charts. It is convenient to choose an atlas such that ind(x) is the same for all charts containing this x. 1.3.1. Remark. If M is a metrizable space and K = KM is a closed subset in M of codimension codimR N ≥ 2 such that M \ K = M1 is a manifold with corners over Ar , then we call M a pseudo-manifold over Ar , where KM is a critical subset. Two pseudo-manifolds B and C are called diffeomorphic, if B \ KB is diffeomorphic with C \ KC as for manifolds with corners (see also [14, 36]). Take on M a Borel σ-additive measure ν such that ν on M \ K coincides with the Riemann volume element and ν(K) = 0, since the real shadow of M1 has it. The uniform space Hpt (M1 , N ) of all continuous piecewise H t Sobolev mappings from M1 into N is introduced in the standard way [30, 31], which induces Hpt (M, N ) the uniform space of continuous piecewise H t Sobolev mappings on M , since ν(K) = 0, where R ∋ t ≥ [m/2]+1, m denotes the dimension of M over R, [k] denotes the integer part of k ∈ R, T [k] ≤ k. Then put Hp∞ (M, N ) = t>m Hpt (M, N ) with the corresponding uniformity. For manifolds over Ar with 1 ≤ r ≤ 3 take as Hpt (M, N ) the completion of the family of all continuous piecewise Ar -holomorphic mappings from M into N relative to the Hpt uniformity, where [m/2] + 1 ≤ t ≤ ∞. Henceforth we consider pseudo-manifolds with ′ connecting mappings of charts continuous in M and Hpt in M \ KM for 0 ≤ r ≤ 3, where t′ ≥ t. 1.3.2. Note. Since the octonion algebra O is non-associative, we consider a nonassociative subgroup G of the family M atq (O) of all square q × q matrices with entries in O. More generally G is a group which has a Hpt manifold structure over Ar and group’s operations are Hpt mappings. The G may be non-associative for r = 3, but G is supposed to be alternative, that is, (aa)b = a(ab) and a(a−1 b) = b for each a, b ∈ G. As a generalization of pseudo-manifolds there is used the following (over R and C

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see [14, 45]). Suppose that M is a Hausdorff topological space of covering dimension dim M = m supplied with a family {h : U → M } of the so called plots h which are continuous maps satisfying conditions (D1 − D4): (D1) each plot has as a domain a convex subset U in Anr , n ∈ N; (D2) if h : U → M is a plot, V is a convex subset in Alr and g : V → U is an Hpt mapping, then h ◦ g is also a plot, where t ≥ [m/2] + 1; (D3) every constant map from a convex set U in Anr into M is a plot; (D4) if U is a convex set in Anr and {Uj : j ∈ J} is a covering of U by convex sets in n Ar , each Uj is open in U , h : U → M is such that each its restriction h|Uj is a plot, then h is a plot. Then M is called an Hpt -differentiable space. A mapping f : M → N between two Hpt -differentiable spaces is called differentiable if it continuous and for each plot h : U → M the composition f ◦ h : U → N is a plot of N . A topological group G is called an Hpt -differentiable group if its group operations are Hpt -differentiable mappings. ′ ′ Let E, N , F be Hpt -pseudo-manifolds or Hpt -differentiable spaces over Ar , let also ′ G be an Hpt group over Ar , t ≤ t′ ≤ ∞. A fiber bundle E(N, F, G, π, Ψ) with a fiber space E, a base space N , a typical fiber F and a structural group G over Ar , a projection π : E → N and an atlas Ψ is defined in the standard way in §II.1 [47] (see also [14, 36]) ′ with the condition, that transition functions are of Hpt class such that for r = 3 a structure group may be non-associative, but alternative. t′ Local trivializations φj ◦ π ◦ Ψ−1 k : Vk (E) → Vj (N ) induce the Hp -uniformity in the ′ family W of all principal Hpt -fiber bundles E(N, G, π, Ψ), where Vk (E) = Ψk (Uk (E)) ⊂ X 2 (G), Vj (N ) = φj (Uj (N )) ⊂ X(N ), where X(G) and X(N ) are Ar -vector spaces on which G and N are modelled, (Uk (E), Ψk ) and (Uj (N ), φj ) are charts of atlases of E and N N , Ψk = ΨE k , φj = φj . If G = F and G acts on itself by left shifts, then a fiber bundle is called the principal fiber bundle and is denoted by E(N, G, π, Ψ). As a particular case there may be G = A∗r , where A∗r denotes the multiplicative group Ar \ {0}. If G = F = {e}, then E reduces to N. 2. Definitions. Let M be a connected Hpt -pseudo-manifold over Ar , 0 ≤ r ≤ 3 satisfying the following conditions: (i) it is compact; (ii) M is a union of two closed subsets over Ar A1 and A2 , which are pseudo-manifolds and which are canonical closed subsets in M with A1 ∩ A2 = ∂A1 ∩ ∂A2 =: A3 and a codimension over R of A3 in M is codimR A3 = 1, also A3 is a pseudo-manifold; (iii) a finite set of marked points s0,1 , ..., s0,k is in ∂A1 ∩ ∂A2 , moreover, ∂Aj are arcwise connected j = 1, 2; (iv) A1 \ ∂A1 and A2 \ ∂A2 are Hpt -diffeomorphic with M \ [{s0,1 , ..., s0,k } ∪ (A3 \ Int(∂A1 ∩ ∂A2 ))] by mappings Fj (z), where j = 1 or j = 2, ∞ ≥ t ≥ [m/2] + 1, m = dimR M such that H t ⊂ C 0 due to the Sobolev embedding theorem [35], where the interior Int(∂A1 ∩ ∂A2 ) is taken in ∂A1 ∪ ∂A2 . Instead of (iv) we consider also the case (iv ′ ) M , A1 and A2 are such that (Aj \ ∂Aj ) ∪ {s0,1 , ..., s0,k } are C 0 ([0, 1], Hpt (Aj , Aj ))-retractable on X0,q ∩ Aj , where X0,q is a closed arcwise connected

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subset in M , j = 1 or j = 2, s0,q ∈ X0,q , X0,q ⊂ KM , q = 1, ..., k, codimR KM ≥ 2. ˆ be a compact connected Hpt -pseudo-manifold which is a canonical closed subset Let M ˆ and marked points {ˆ ˆ : q = 1, ..., 2k} and an Hpt in Alr with a boundary ∂ M s0,q ∈ ∂ M ˆ → M such that mapping Ξ : M ˆ \ ∂M ˆ onto M \ Ξ(∂ M ˆ ) open in M , Ξ(ˆ (v) Ξ is surjective and bijective from M s0,q ) = ˆ Ξ(ˆ s0,k+q ) = s0,q for each q = 1, ..., k, also ∂M ⊂ Ξ(∂ M ). ′ A parallel transport structure on a Hpt -differentiable principal G-bundle E(N, G, π, Ψ) ˆ as above over the with arcwise connected E and G for Hpt -pseudo-manifolds M and M ′ t same Ar with t ≥ t + 1 assigns to each Hp mapping γ from M into N and points u1 , ..., uk ∈ Ey0 , where y0 is a marked point in N , y0 = γ(s0,q ), q = 1, ..., k, a unique Hpt ˆ → E satisfying conditions (P 1 − P 5): mapping Pγˆ,u : M ˆ → N such that γˆ = γ ◦ Ξ, then Pγˆ,u (ˆ (P 1) take γˆ : M s0,q ) = uq for each q = 1, ..., k and π ◦ Pγˆ,u = γˆ (P 2) Pγˆ,u is the Hpt -mapping by γ and u; ˆ and every φ ∈ Dif Hpt (M ˆ , {ˆ (P 3) for each x ∈ M s0,1 , ..., sˆ0,2k }) there is the equality t ˆ Pγˆ,u (φ(x)) = Pγˆ◦φ,u (x), where Dif Hp (M , {ˆ s0,1 , ..., sˆ0,2k }) denotes the group of all Hpt ˆ preserving marked points φ(ˆ homeomorphisms of M s0,q ) = sˆ0,q for each q = 1, ..., 2k; ˆ (P 4) Pγˆ,u is G-equivariant, which means that Pγˆ,uz (x) = Pγˆ,u (x)z for every x ∈ M and each z ∈ G; ˆ and γˆ0 , γˆ1 : U → N are Hpt′ -mappings (P 5) if U is an open neighborhood of sˆ0,q in M such that γˆ0 (ˆ s0,q ) = γˆ1 (ˆ s0,q ) = vq and tangent spaces, which are vector manifolds over Ar , for γ0 and γ1 at vq are the same, then the tangent spaces of Pγˆ0 ,u and Pγˆ1 ,u at uq are the same, where q = 1, ..., k, u = (u1 , ..., uk ). ′ Two Hpt -differentiable principal G-bundles E1 and E2 with parallel transport structures (E1 , P1 ) and (E2 , P2 ) are called isomorphic, if there exists an isomorphism h : E1 → E2 such that P2,ˆγ ,u (x) = h(P1,ˆγ ,h−1 (u) (x)) for each Hpt -mapping γ : M → N and uq ∈ (E2 )y0 , where q = 1, ..., k, h−1 (u) = (h−1 (u1 ), ..., h−1 (uk )). Let (S M E)t,H := (S M,{s0,q :q=1,...,k} E; N, G, P)t,H be a set of Hpt -closures of isomorphism classes of Hpt principal G fiber bundles with parallel transport structure. 3. Theorems. 1. The uniform space (S M E)t,H from §2 has the structure of a topological alternative monoid with a unit and with a cancelation property and the multiplication operation of Hpl class with l = t′ − t (l = ∞ for t′ = ∞). If N and G are separable, then (S M E)t,H is separable. If N and G are complete, then (S M E)t,H is complete. 2. If G is associative, then (S M E)t,H is associative. If G is commutative, then (S M E)t,H is commutative. If G is a Lie group, then (S M E)t,H is a Lie monoid. 3. The (S M E)t,H is non-discrete, locally connected and infinite dimensional for dimR (N × G) > 1. ′ Proof. If there is a homomorphism θ : G → F of Hpt -differentiable groups, then there exists an induced principal F fiber bundle (E ×θ F )(N, F, π θ , Ψθ ) with the total space (E ×θ F ) = (E × F )/Y, where Y is the equivalence relation such that (vg, f )Y(v, θ(g)f ) for each v ∈ E, g ∈ G, f ∈ F . Then the projection π θ : (E ×θ F ) → N is defined by π θ ([v, f ]) = π(v), where [v, f ] := {(w, b) : (w, b)Y(v, f ), w ∈ E, b ∈ F } denotes the equivalence class of (v, f ).

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Therefore, each parallel transport structure P on the principal G fiber bundle E(N, G, π, Ψ) induces a parallel transport structure Pθ on the induced bundle by the formula Pθγˆ,[u,f ] (x) = [Pγˆ,u (x), f ]. Define multiplication with the help of certain embeddings and isomorphisms of spaces of functions. Mention that for each two compact canonical closed subsets A and B in Alr Hilbert spaces H t (A, Rm ) and H t (B, Rm ) are linearly topologically isomorphic, where l, m ∈ N, hence Hpt (A, N ) and Hpt (B, N ) are isomorphic as uniform spaces. Let Hpt (M, {s0,1 , ..., s0,k }; W, y0 ) := {(E, f ) : E = E(N, G, π, Ψ) ∈ W, f = Pγˆ,y0 ∈ Hpt : ′ π ◦ f (s0,q ) = y0 ∀q = 1, ..., k; π ◦ f = γˆ , γ ∈ Hpt (M, N )} be the space of all Hpt principal G fiber bundles E with their parallel transport Hpt -mappings f = Pγˆ,y0 , where W is as in §1.3.2. Put ω0 = (E0 , P0 ) be its element such that γ0 (M ) = {y0 }, where e ∈ G denotes the unit element, E0 = N × G, π0 (y, g) = y for each y ∈ N , g ∈ G, Pγˆ0 ,u = P0 . ˆ → M from §2 induces the embedding The mapping Ξ : M ∗ t ˆ , {ˆ Ξ : Hp (M, {s0,1 , ..., s0,k }; W, y0 ) ֒→ Hpt (M s0,1 , ..., sˆ0,2k }; W, y0 ), ˆ ˆ ˆ where M and A1 and A2 are retractable into points. Let as usually A ∨ B := ρ(Z) be the wedge sum of pointed spaces (A, {a0,q : q = 1, ..., k}) and (B, {b0,q : q = 1, ..., k}), where Z := [A × {b0,q : q = 1, ..., k} ∪ {a0,q : q = 1, ..., k} × B] ⊂ A × B, ρ is a continuous quotient mapping such that ρ(x) = x for each x ∈ Z \ {a0,q × b0,j ; q, j = 1, ..., k} and ρ(a0,q ) = ρ(b0,q ) for each q = 1, ..., k, where A and B are topological spaces with marked points a0,q ∈ A and b0,q ∈ B, q = 1, ..., k. Then the wedge product g ∨ f of two elements f, g ∈ Hpt (M, {s0,1 , ..., s0,k }; N, y0 ) is defined on the domain M ∨ M such that (f ∨ g)(x × b0,q ) = f (x) and (f ∨ g)(a0,q × x) = g(x) for ˆ , {ˆ each x ∈ M , where to f, g there correspond f1 , g1 ∈ Hpt (M s0,1 , ..., sˆ0,2k }; N, y0 ) such that f1 = f ◦ Ξ and g1 = g ◦ Ξ. Let (Ej , Pγˆj ,uj ) ∈ Hpt (M, {s0,1 , ..., s0,k }; W, y0 ), j = 1, 2, then take their wedge prod−1 uct Pγˆ,u1 := Pγˆ1 ,u1 ∨ Pγˆ2 ,v on M ∨ M with vq = uq g2,q g1,q+k = y0 × g1,q+k for each q = 1, ..., k due to the alternativity of G, γ = γ1 ∨γ2 , where Pγˆj ,uj (ˆ sj,0,q ) = y0 ×gj,q ∈ Ey0 for every j and q. For each γj : M → N there exists γ˜j : M → Ej such that π ◦ γ˜j = γj . Denote by m : G × G → G the multiplication operation. The wedge product (E1 , Pγˆ1 ,u1 ) ∨ (E2 , Pγˆ2 ,u2 ) is the principal G fiber bundle (E1 × E2 ) ×m G with the parallel transport structure Pγˆ1 ,u1 ∨ Pγˆ2 ,v . The uniform space Hpt (J, A3 ; W, y0 ) := {(E, f ) ∈ Hpt (J, W ) : π ◦ f (A3 ) = {y0 }} has the Hpt -manifold structure and has an embedding into Hpt (M, {s0,1 , ..., s0,k }; W, y0 ) due to Conditions 2(i − iii), where either J = A1 or J = A2 . This induces the following embedding χ∗ : Hpt (M ∨ M, {s0,q × s0,q : q = 1, ..., k}; W, y0 ) ֒→ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ). Analogously considering Hpt (M, {X0,q : q = 1, ..., k}; W, y0 ) = {f ∈ H t (M, W ) : f (X0,q ) = {y0 }, q = 1, ..., k} and Hpt (J, A3 ∪ {X0,q : q = 1, ..., k}; W, y0 ) in the case (iv ′ ) instead of (iv) we get the embedding χ∗ : Hpt (M ∨ M, {X0,q × X0,q : q = 1, ..., k}; W, y0 ) ֒→ Hpt (M, {X0,q : q = 1, ..., k}; W, y0 ). Therefore, g ◦ f := χ∗ (f ∨ g) is the composition in Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ). There exists the following equivalence relation Rt,H in Hpt (M, {X0,q : q = 1, ..., k}; W, y0 ): f Rt,H h if and only if there exist nets ηn ∈ Dif Hpt (M, {X0,q : q = 1, ..., k}), also fn and hn ∈ Hpt (M, {X0,q : q = 1, ..., k}; W, y0 ) with limn fn = f and

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limn hn = h such that fn (x) = hn (ηn (x)) for each x ∈ M and n ∈ ω, where ω is a directed set and convergence is considered in Hpt (M, {X0,q : q = 1, ..., k}; W, y0 ). Henceforward in the case 2(iv) we get s0,q instead of X0,q in the case 2(iv ′ ). Thus there exists the quotient uniform space Hpt (M, {X0,q : q = 1, ..., k}; W, y0 )/Rt,H =: (S M E)t,H . In view of [41, 42] Dif Hpt (M ) is the group of diffeomorphisms for t ≥ [m/2] + 1. The Lebesgue measure λ in the real ˆ by the mapping Ξ induces the measure λΞ on M which is equivalent to ν, shadow of M ˆ) = 0 since Ξ is the Hpt -mapping from the compact space onto the compact space, λ(∂ M ˆ \ ∂M ˆ → M is bijective. and Ξ : M Due to Conditions (P 1 − P 5) each element f = Pγˆ,u up to a set QM of measure zero, ν(QM ) = 0, is given as f ◦ Ξ−1 on M \ QM , where π ◦ f = γˆ , γˆ = γ ◦ Ξ. Denote f ◦ Ξ−1 also by f . Thus, for each (E, f ) ∈ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ) the image f (M ) is compact and connected in E. Therefore, for each partition Z there exists δ > 0 such that for each partition Z ∗ with supi inf j dist(Mi , M ∗ j ) < δ and (E, f ) ∈ H t (M, W ; Z), f (s0,q ) = uq , there exists (E, f1 ) ∈ H t (M, W ; Z ∗ ) with f1 (s0,q ) = uq for each q = 1, ..., k such that f Rt,H f1 , where Mi and Mj∗ are canonical closed pseudo-submanifolds in M corresponding to partitions Z and Z ∗ , H t (M, W ; Z) denotes the space of all continuous piecewise H t -mappings from M into W subordinated to the partition Z such that Z and Z ∗ respect Hpt structure of M. Hence there exists a countable subfamily {Zj : j ∈ N} in the family of all partitions Υ ˜ such that Zj ⊂ Zj+1 for each j and limj diamZ j = 0. Then t M i (i) str − ind{H (M, {s0,q : q = 1, ..., k}; W, y0 ; Zj ); hZ Zj ; N}/Rt,H = (S E)t,H is separable if N and G are separable, since each space Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ; Zj ) is separable. i The space str − ind{H t (M, {s0,q : q = 1, ..., k}; W, y0 ; Zj ); hZ Zj ; N} is complete due to Theorem 12.1.4 [40], when N and G are complete. Each class of Rt,H -equivalent elements is closed in it. Then to each Cauchy net in (S M E)t,H there corresponds a Cauchy i ×Yi net in str − ind{H t (M × [0, 1], {s0,q × e × 0; W, y0 ; Zj × Yj ); hZ Zj ×Yj ; N} due to theorems about extensions of functions [35, 44, 50], where Yj are partitions of [0, 1] with ˜ limj diam(Y j ) = 0, Zj × Yj are the corresponding partitions of M × [0, 1]. Hence M (S E)t,H is complete, if N and G are complete. If f, g ∈ H t (M, X) and f (M ) 6= g(M ), then (ii) inf ψ∈Dif Hpt (M,{s0,q :q=1,...,k}) kf ◦ ψ − gkH t (M,X) > 0. Thus equivalence classes ˆ is arcwise connected. Take < f >t,H and < g >t,H are different. The pseudo-manifold M ˆ an Hpt -mapping with η(0) = sˆ0,q and η(1) = sˆ0,k+q , where 1 ≤ q ≤ k. η : [0, 1] → M ˆ H t -coordinates one of which is a parameter along η. Therefore, for each Choose in M p gq , gk+q ∈ G there exists Pγˆ,u with Pγˆ,u (s0,q ) = y0 × gq and Pγˆ,u (s0,k+q ) = y0 × gk+q for each q = 1, ..., k. Since E and G are arcwise connected, then N is arcwise connected and (S M E)t,H is locally connected for dimR N > 1. Thus, the uniform space (S M E)t,H is non-discrete. The tangent bundle T Hpt (M, E) is isomorphic with Hpt (M, T E), where T E is the ′ Hpt −1 fiber bundle, t′ ≥ t+1. There is an infinite family of fα ∈ Hpt (M, T E) with pairwise S distinct images in T E for different α such that fα (M ) is not contained in β<α fβ (M ),

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α ∈ Λ, where Λ is an infinite ordinal. Therefore, T (S M E)t,H is an infinite dimensional fiber bundle due to (ii) and inevitably (S M E)t,H is infinite dimensional. Evidently, if f ∨ g = h ∨ g or g ∨ f = g ∨ h for {f, g, h} ⊂ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ), then f = h. Thus χ∗ (f ∨ g) = χ∗ (h ∨ g) or χ∗ (g ∨ f ) = χ∗ (g ∨ h) is equivalent to f = h due to the definition of f ∨ g and the definition of equal functions, since χ∗ is the embedding. Using the equivalence relation Rt,H gives < f >t,H ◦ < g >t,H =< h >t,H ◦ < g >t,H or < g >t,H ◦ < f >t,H =< g >t,H ◦ < h >t,H is equivalent to < h >t,H =< f >t,H . Therefore, (S M E)t,H has the cancelation property. −1 −1 Since G is alternative, then a2,q [a−1 2,q (a2,q+k (a2,q a1,q+k ))] = a2,q+k (a2,q a1,q+k ), hence −1 −1 P1 ∨(P2 ∨P2 ) = (P1 ∨P2 )∨P2 ; also a2,q [a−1 2,q (a1,q+k (a1,q a1,q+k ))] = a1,q+k (a1,q a1,q+k ), consequently, P1 ∨ (P1 ∨ P2 ) = (P1 ∨ P1 ) ∨ P2 and inevitably for equivalence classes (aa)b = a(ab) and b(aa) = (ba)a for each a, b ∈ (S M E)t,H . Thus (S M E)t,H is alternative. If G is associative, then the parallel transport structure gives (f ∨ g) ∨ h = f ∨ (g ∨ h) on M ∨ M ∨ M for each {f, g, h} ⊂ Hpt (M, {s0,q : q = 1, ..., k; W, y0 ). Applying the embedding χ∗ and the equivalence relation Rt,H we get, that (S M E)t,H is associative < f >ξ ◦(< g >ξ ◦ < h >ξ ) = (< f >ξ ◦ < g >ξ )◦ < h >ξ . In view of Conditions 2(i − iv) there exists an Hpt -diffeomoprhism of (A1 \ A3 ) ∨ (A2 \ A3 ) with (A2 \ A3 ) ∨ (A1 \ A3 ) as pseudo-manifolds (see §1.3.1). For the measure ν on M naturally the equality ν(A3 ) = 0 is satisfied. If M ′ - is the submanifold may be with corners or pseudo-manifold, accomplishing the partition Z = Zf of the manifold M , then the codimension M ′ in M is equal to one and ν(M ′ ) = 0. For the point s0,q in (M \A3 )∪{s0,q } there exists an open neighborhood U having the Hpt -retraction F : [0, 1] × U → {s0,q }. Hence it is possible to take a sequence of diffeomorphisms ψn ∈ Dif Hpt (M, {s0,q : q = 1, ..., k}) such that limn→∞ diam(ψn (U )) = 0. Let w0 be a mapping w0 : M → W such that w0 (M ) = {y0 ×e}. Consider w0 ∨(E, f ) for some (E, f ) ∈ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ). If (E, f ) ∈ Hpt (M, {s0,q : q = 1, .., k}; W, y0 ) with the natural positive t ∈ N, then f is bounded relative to the uniformity of the uniform space Hpt (M ; E). If Un is a sequence of bounded open or canonical closed subsets in M such that limn diam(Un ) = 0, then limn→∞ ν(Vn ) = 0 for the sequence of ν-measurable subsets Vn such that Vn ⊂ Un . Therefore, for each bounded sequence {gn : gn ∈ Hpt (M ; E); n ∈ N} there exists the limit limn→∞ gn |Un = 0 relative to the Hpt uniformity, where Un is subordinated to the partition of M into H t submanifolds. Then if {gn : gn ∈ Hpt (M, {s0,q : q = 1, ..., k}; E, y0 ); n ∈ N} is a bounded sequence such that gn converges to g ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ) on M \ Wk for each k relative to the Hpt -uniformity, the given open Wk in M , where k, n ∈ N and limn→∞ ν(Wn △ Un ) = 0, then gn converges to g in the uniform space Hpt (M, {s0,q : q = 1, ..., k}; E, y0 ). Mention that for each marked point s0,q in M there exists a neighborhood U of s0,q in M such that for each γ1 ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ) there exists γ2 ∈ Hpt such that they are Rt,H equivalent and γ2 |U = y0 . Therefore, if C is an arcwise connected compact subset in M of codimension codimR C ≥ 1 such that s0,q ∈ C, then the standard proceeding shows that for each γ1 ∈ Hpt there exists γ2 ∈ Hpt such that γ1 Rt,H γ2 and γ2 |C = y0 . Since C is compact, then each its open covering has a finite subcovering and hence (Y0 ) there exists an open neighborhood U of C in M such that for each γ1 there exists

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γ2 such that γ1 Rt,H γ2 and γ2 |U = y0 . There exists a sequence ηn ∈ Dif Hpt (M, {s0,q : q = 1, ..., k}) such that limn→∞ diam(ηn (A2 \ ∂A2 )) = 0 and wn , fn ∈ Hpt (M, {s0,q : q = 1, ..., k}; E, y0 ) with (iii) limn→∞ fn = f , limn→∞ wn = w0 and limn→∞ χ∗ (fn ∨ wn )(ηn−1 ) = f due to π ◦ f (s0,q ) = s0,q in the formula of differentiation of compositions of functions (over H and O see it in [28, 29, 26]). In more details, the sequence ηn as a limit of ηn (A2 ) produces a pseudo-submanifold B in M of codimension not less than one such that B can be presented with the help of the wedge product of spheres and compact quadrants up to Hpt -diffeomorphism with marked points {s0,q : q = 1, ..., k}, but as well B may be a finite discrete set also. Then by induction the procedure can be continued lowering the dimension of B. Particularly there may be circles and curves in the case of the unit dimension. Two quadrants up to an Hpt quotient mapping gluing boundaries produce a sphere. Thus the consideration reduces to the case of the wedge product of spheres. The case of spheres reduces to the iterated construction with circles, since the reduced product S 1 ∧ S n is Hpt homeomorphic with S n+1 (see Lemma 2.27 [49] and [14]). For the particular case of the n-dimensional sphere ˆ n = Dn , where Dn is the unit ball (disk) in Rn or in a n dimensional Mn = S n take M over R subspace in Alr , D1 = [0, 1] for n = 1. But S n \ s0 has the retraction into the point in S n , where s0 ∈ S n , n ∈ N. Therefore, w0 ∨ (E, f ) and (E, f ) belong to the equivalence class < (E, f ) >t,H := {g ∈ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ) : (E, f )Rt,H g} due to (iii) and (Y0 ). Thus, < w0 >t,H ◦ < g >t,H =< g >t,H . The pseudo-manifold M ∨M \{s0,q ×s0,j : q, j = 1, ..., k} has the Hpt -diffeomorphism ψ (see definition in §1.3.1) such that ψ(x, y) = (y, x) for each (x, y) ∈ (M × M \ {s0,q × s0,j : q, j = 1, ..., k}). Suppose now, that G is commutative. Then (f ∨ g) ◦ ψ|(M ×M \{s0,q ×s0,j :q,j=1,...,k}) = g ∨ f |(M ×M \{s0,q ×s0,j :q,j=1,...,k}) . On the other hand, < f ∨ w0 >t,H =< f >t,H =< f >t,H ◦ < w0 >t,H =< w0 >t,H ◦ < f >t,H , hence, < f ∨ g >t,H =< f >t,H ◦ < g >t,H =< f ∨ w0 >t,H ◦ < w0 ∨ g >t,H =< (f ∨w0 )∨(w0 ∨g) >t,H =< (w0 ∨g)∨(f ∨w0 ) >t,H due to the existence of the unit element < w0 >t,H and due to the properties of ψ. Indeed, take a sequence ψn as above. Therefore, the parallel transport structure gives (g ∨ f )(ψ(x, y)) = (g ◦ f )(y, x) for each x, y ∈ M , consequently, (f ◦ g)Rt,H (g ◦ f ) for each f, g ∈ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ). The using of the embedding χ∗ gives that (S M E)t,H is commutative, when G is commutative. The mapping (f, g) 7→ f ∨ g from Hpt (M, {s0,q : q = 1, ..., k}; W, y0 )2 into Hpt (M ∨ M \{s0,q ×s0,j : q, j = 1, ..., k}; W, y0 ) is of class Hpt . Since the mapping χ∗ is of class Hpt , then (f, g) 7→ χ∗ (f ∨ g) is the Hpt -mapping. The quotient mapping from Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ) into (S M E)t,H is continuous and induces the quotient uniformity, T b (S M E)t,H has embedding into (S M T b E)t,H for each 1 ≤ b ≤ t′ − t, when t′ > t is ′ finite, for every 1 ≤ b < ∞ if t′ = ∞, since E is the Hpt fiber bundle, T b E is the fiber bundle with the base space N . Hence the multiplication (< f >t,H , < g >t,H >) 7→< f >t,H ◦ < g >t,H =< f ∨ g >t,H is continuous in (S M E)t,H and is of class Hpl with l = t′ − t for finite t′ and l = ∞ for t′ = ∞. 4. Definition. The (S M E)t,H from Theorem 3.1 we call the wrap monoid. 5. Corollary. Let φ : M1 → M2 be a surjective Hpt -mapping of Hpt -pseudo-

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manifolds over the same Ar such that φ(s1,0,q ) = s2,0,a(q) for each q = 1, ..., k1 , where {sj,0,q : q = 1, ..., kj } are marked points in Mj , j = 1, 2, 1 ≤ a ≤ k2 , l1 ≤ k2 , l1 := card φ({s1,0,q : q = 1, ..., k1 }). Then there exists an induced homomorphism of monoids φ∗ : (S M2 E)t,H → (S M1 E)t,H . If l1 = k2 , then φ∗ is the embedding. ˆ 1 → M1 with marked points {ˆ Proof. Take Ξ1 : M s1,0,q : q = 1, ..., 2k1 } as in §2, then ˆ ˆ take M2 the same M1 with additional 2(k2 −l1 ) marked points {ˆ s2,0,q : q = 1, ..., 2k3 } such ˆ 2 → M2 that sˆ1,0,q = sˆ2,0,q for each q = 1, .., k1 , k3 = k1 + k2 − l1 , then φ ◦ Ξ1 := Ξ2 : M is the desired mapping inducing the parallel transport structure from that of M1 . Therefore, ˆ 2 → N induces γˆ1 : M ˆ 1 → N and to Pγˆ ,u2 there corresponds Pγˆ ,u1 with each γˆ2 : M 2 1 additional conditions in extra marked points, where u1 ⊂ u2 . The equivalence class < (E2 , Pγˆ2 ,u2 ) >t,H ∈ (S M2 E)t,H gives the corresponding elements < (E1 , Pγˆ1 ,u1 ) >t,H ∈ ˆ 1 , {ˆ ˆ 1 , {ˆ (S M1 E)t,H , since Dif Hpt (M s0,q : q = 1, ..., 2k2 }) ⊂ Dif Hpt (M s0,q : q = ∗ ∗ 1, ..., 2k3 }). Then φ (< (E2 , Pγˆ2 ,u2 ) ∨ (E1 , Pηˆ2 ,v2 ) >t,H ) = φ (< (E2 , Pγˆ2 ,u2 ) >t,H ˆ 1 \ ∂M ˆ 1 ) coincides with )φ∗ (< (E1 , Pηˆ2 ,v2 ) >t,H ), since f2 ◦ φ(x) for each x ∈ Ξ1 (M f1 (x), where fj corresponds to Pγj ,y0 ×e (see also the beginning of §3). ˆ1 = M ˆ 2 and the group of diffeomorphisms Dif Hpt (M ˆ 1 , {ˆ If l1 = k2 , then M s0,q : ∗ ∗ q = 1, ..., 2k1 }) is the same for two cases, hence φ is bijective and inevitably φ is the embedding. 6. Theorems. 1. There exists an alternative topological group (W M E)t,H containing the monoid (S M E)t,H and the group operation of Hpl class with l = t′ − t (l = ∞ for t′ = ∞). If N and G are separable, then (W M E)t,H is separable. If N and G are complete, then (W M E)t,H is complete. 2. If G is associative, then (W M E)t,H is associative. If G is commutative, then M (W E)t,H is commutative. If G is a Lie group, then (W M E)t,H is a Lie group. 3. The (W M E)t,H is non-discrete, locally connected and infinite dimensional for dimR (N × G) > 1. Moreover, if there exist two different sets of marked points s0,q,j in A3 , q = 1, ..., k, j = 1, 2, then two groups (W M E)t,H,j , defined for {s0,q,j : q = 1, ..., k} as marked points, are isomorphic. 4. The (W M E)t,H has a structure of an Hpt -differentiable manifold over Ar . Proof. If γ ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ), then for u ∈ Ey0 there exists a unique hq ∈ G such that Pγˆ,u (ˆ s0,q+k ) = uq hq , where hq = gq−1 gq+k , y0 ×gq = Pγˆ,u (ˆ s0,q ), gq ∈ G. Due to the equivariance of the parallel transport structure h depends on γ only and we denote it by h(E,P) (γ) = h(γ) = h, h = (h1 , ..., hk ). The element h(γ) is called the holonomy of P along γ and h(E,P) (γ) depends only on the isomorphism class of (E, P) ˆ ; {ˆ due to the use of Dif Hpt (M s0,q : q = 1, ..., 2k}) and boundary conditions on γˆ at sˆ0,q for q = 1, ..., 2k. Therefore, h(E1 ,P1 )(E2 ,P2 ) (γ) = h(E1 ,P1 ) (γ)h(E2 ,P2 ) (γ) ∈ Gk , where Gk denotes the direct product of k copies of the group G. Hence for each such γ there exists the homomorphism h(γ) : (S M E)t,H → Gk , which induces the homomorphism h : (S M E)t,H → C 0 (Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ), Gk ), where C 0 (A, Gk ) is the space of continuous maps from a topological space A into Gk and the group structure (hb)(γ) = h(γ)b(γ) (see also [14] for S n ). Thus, it is sufficient to construct (W M N )t,H from (S M N )t,H . For the commutative monoid (S M N )t,H with the unit and the cancelation property there exists a commutative

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group (W M N )t,H . Algebraically it is the quotient group F/B, where F is the free commutative group generated by (S M N )t,H , while B is the minimal closed subgroup in F generated by all elements of the form [f + g] − [f ] − [g], f and g ∈ (S M N )t,H , [f ] denotes the element in F corresponding to f (see also about such abstract Grothendieck construction in [?, 48]). By the construction each point in (S M N )t,H is the closed subset, hence (S M N )t,H is the topological T1 -space. In view of Theorem 2.3.11 [12] the product of T1 -spaces is the T1 -space. On the other hand, for the topological group G from the separation axiom T1 it follows, that G is the Tychonoff space [12, 43]. The natural mapping η : (S M N )t,H → (W M N )t,H is injective. We supply F with the topology inherited from the topology of the P Tychonoff product (S M N )Z f nf,z [f ], t,H , where each element z in F has the form z = P nf,z ∈ Z for each f ∈ (S M N )t,H , f |nf,z | < ∞. By the construction F and F/B are T1 -spaces, consequently, F/B is the Tychonoff space. In particular, [nf ] − n[f ] ∈ B, hence (W M N )t,H is the complete topological group, if N and G are complete, while η is the topological embedding, since η(f + g) = η(f ) + η(g) for each f, g ∈ (S M N )t,H , η(e) = e, since (z + B) ∈ η(S M N )t,H , when nf,z ≥ 0 for each f , and inevitably in the general case z = z + − z − , where (z + + B) and (z − + B) ∈ η(S M N )t,H . ′ Using plots and Hpt transition mappings of charts of N and E(N, G, π, Ψ) and equivalence classes relative to Dif Hpt (M, {s0,q : q = 1, ..., k}) we get, that (W M E)t,H has the structure of the Hpt -differentiable manifold, since t′ ≥ t. The rest of the proof and the statements of Theorems 6(1-4) follows from this and Theorems 3(1-3) and [30, 31]. Since (S M E)t,H is infinite dimensional due to Theorem 3.3, then (W M E)t,H is infinite dimensional. 7. Definition. The (W M E)t,H = (W M,{s0,q :q=1,...,k} E; N, G, P)t,H from Theorem 6.1 we call the wrap group. 8. Corollary. There exists the group homomorphism h : (W M E)t,H → 0 C (Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ), Gk ). −1

Proof follows from §6 and putting hf (γ) = (hf (γ))−1 . 9. Corollary. If M1 and M2 and φ satisfy conditions of Corollary 5, then there exists a homomorphism φ∗ : (W M2 E)t,H → (W M1 E)t,H . If l1 = k2 , then φ∗ is the embedding. 10. Remarks and examples. Consider examples of M which satisfy sufficient condin , S n \ V with tions for the existence of wrap groups (W M E)t,H . Take M , for example, DR R n n n n s0 ∈ ∂V , DR \Int(Db ) with s0 ∈ ∂Db and 0 < b < R < ∞, where SR denotes the sphere of the dimension n > 1 over R and radius R, V is Hpt -diffeomorphic with the interior n ) of the n-dimensional ball D n := {x ∈ Rn : Pn 2 Int(DR n dimensional k=1 xk ≤ R} or in R P l n 2 over R subspace in Ar and is the proper subset in SR := {x ∈ Rn+1 : n+1 k=1 xk = R}. Instead of sphere it is possible to take an Hpt pseudo-manifold Qn homeomorphic with a sphere or a disk, particularly, Milnor’s sphere. Indeed, divide M by the equator {x1 = 0} into two parts A1 and A2 and take A3 = {x ∈ M : x1 = 0} ∪ P , where s0 ∈ ∂A1 ∩ ∂A2 , while P = ∅, P = ∂V , P = ∂Dbn correspondingly. Then take also V and Dbn such that n or D n respectively or their equators would be generated by the equator {x1 = 0} in SR R n more generally Q . S Take then M = Qn \ lk=1 Vk , where Vk are Hpt -diffeomorphic to interiors of bounded quadrants in Rn or in n dimensional subspace in Aar , where l > 1, l ∈ N, ∂Vk ∩ ∂Vj =

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{s0 } and Vk ∩ Vj = ∅ for each k 6= j, diam(Vk ) ≤ b < R/3. In more details it is possible make a specification such that if l is even, then [l/2] − 1 among Vk are displayed above the equator and the same amount below it, two of Vk have equators, generated by equators {x1 = 0} in Qn . If l odd, then [(l − 1)/2] among Vk are displayed above and the same amount below it, one of Vk has equator generated by that of {x1 = 0} in Qn , T s0 ∈ k ∂Vk ∩ {x ∈ M : x1 = 0}. Divide M by the equator {x1 = 0} into two parts A1 and A2 and let A3 = {x ∈ M : S x1 = 0}∪P , where P = lk=1 ∂Vk . Then either A1 \A3 and A2 \A3 are Hpt diffeomorphic as pseudo-manifolds or manifolds with corners and Hpt diffeomorphic with M \ [{s0 } ∪ (A3 \ Int(∂A1 ∩ ∂A2 ))] =: D or 2(iv ′ ) is satisfied, since the latter topological space D is obtained from Qn by cutting a non-void connected closed subset, n > 1, consequently, D is retractable into a point. In a case of a usual manifold M the point s0 ∈ ∂M (for ∂M 6= ∅) may be a critical point, but in the case of a manifold with corners this s0 is the corner point from ∂M , since for x ∈ ∂M there is not less than one chart (U, u, Q) such that u(x) ∈ ∂Q, M \ ∂M = S −1 S −1 k uk (Int(Qk )), ∂M ⊂ k uk (∂Qk ). Further, if M satisfies Conditions 2(i − v) or m = P also satisfies them for m ≥ 1, since D m is retractable (i − iii, iv ′ , v), then M × DR R m of P , where j = 1, 2, A (M ) into the point, taking as two parts Aj (K) = Aj (M ) × DR j m and are pseudo-submanifolds of M . Then A1 (P ) ∩ A2 (P ) = (A1 (M ) ∩ A2 (M )) × DR m , s (P ) ∈ s (M ) × {x ∈ D m : x = 0}. In it is possible to take A3 (P ) = A3 (M ) × DR 0 0 1 R particular, for M = S 1 and m = 1 this gives the filled torus. This construction can be naturally generalized for non-orientable manifolds, for examS ple, the M¨obius band L, also for M := L \ ( βj=1 Vj ) with the diameter bj of Vj less than the width of L, where each Vj is Hpt diffeomorphic with an interior of a bounded quadrant Ta1 +...+aq in R2 , s0,q ∈ ∂L ∩ ( j=a ∂Vj ), a0 := 0, a1 + ... + ak = β, q = 1, ..., k, 1 +...+aq−1 +1 1 since ∂L is diffeomorphic with S , also S 1 \ {s0,q } is retractable into a point, consequently, ˆ = I 2 , then take a connected curve A1 and A2 are retractable into a point. For L take M ηˆ consisting of the left side {0} × [0, 1] joined by a straight line segment joining points {0, 1} and {1, 0} and then joined by the right side {1} × [0, 1]. This gives the proper cutˆ which induces the proper cutting of L and of M with A3 ⊃ η ∪ ∂L up to an ting of M t Hp diffeomorphism, where η := Ξ(ˆ η ), hence the M¨obius band L and M satisfy Conditions 2(i − iii, iv ′ , v). Take a quotient mapping φ : I 2 → S 1 such that φ({s0,1 , s0,2 }) = s0 ∈ S 1 , s0,1 = (0, 0), s0,2 = (0, 1) ∈ I 2 , where I = [0, 1], hence there exists the embedding 1 2 φ∗ : (W S ,s0 E)t,H ֒→ (W I ,{s0,1 ,s0,2 } E)t,H . ˆ = I 2 with twisting equivalence relation on ∂I 2 so it satisThe Klein bottle K has M fies sufficient conditions. Moreover, K is the quotient φ : Z → K of the cylinder Z with twisted equivalence relation of its ends S 1 using reflection relative to a horizontal diameter. Thus A3 ⊃ φ(S 1 ). Therefore, there exists the embedding φ∗ : (W K,{s0 } E)t,H → (W Z,{s0,1 ,s0,2 } E)t,H , where s0,1 , s0,2 ∈ ∂Z, φ({s0,1 , s0,2 }) = s0 . Take a pseudo-manifold Qn Hpt -diffeomorphic with S n for n ≥ 2, cut from it β nonintersecting open domains V1 , ..., Vβ Hpt -diffeomorphic with interiors of bounded quadrants Ta1 +...+aq in Rn , s0,q ∈ j=a ∂Vj , a0 := 0, a1 + ... + ak = β, q = 1, ..., k. Then glue 1 +...+aq−1 +1 for V1 , ..., Vl , 1 ≤ l ≤ β, by boundaries of slits Hpt -diffeomorphic with S m−1 the reduced

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product L ∨ S n−2 , since ∂L = S 1 , S 1 ∧ S n−2 is Hpt -diffeomorphic with S n−1 [49]. We get the non-orientable Hpt -pseudo-manifold M , satisfying sufficient conditions. Since the projective space RP n is obtained from the sphere by identifying diametrically opposite points. Then take M Hpt -diffeomorphic with RP n for n > 1 also M with cut Ta1 +...+aq V1 , ..., Vβ Hpt -diffeomorphic with open subsets in RP n , s0,q ∈ ( j=a ∂Vj ) ∩ 1 +...+aq−1 +1 {x ∈ M : x1 = 0}, Vj ∩ Vl = ∅ for each j 6= l, a0 := 0, a1 + ... + ak = β, q = 1, ..., k. Then Conditions 2(i − v) or (i − iii, iv ′ , v) are also satisfied for RP n and M . In view of Proposition 2.14 [49] about H-groups [X, x0 ; K, k0 ] there is not any expectation or need on rigorous conditions on a class of acceptable M for constructions of wrap groups (W M E)t,H . If M1 is an analytic real manifold, then taking its graded product with generators {i0 , ..., i2r −1 } of the Cayley-Dickson algebra gives the Ar manifold (see [28, 26, 27]). Particularly this gives l2r dimensional torus in Alr for the l dimensional real torus T2 = (S 1 )l as M1 . Consider T2 . It can be slit along a closed curve (loop) C Hp∞ -diffeomorphic with S 1 1 × S1 and marked points s0,q ∈ C ⊂ T2 such that C rotates on the surface of T2 = SR b 1 1 on angle π around Sb while C rotates on 2π around SR , such that C rotates on 4π around 1 that return to the initial point on C, where 0 < b < R < ∞, q = 1, ..., k, k ∈ N. SR Therefore, the slit along C of T2 is the non-orientable band which inevitably is the M¨obius band with twice larger number of marked points {sL 0,j : j = 1, ..., 2k} ⊂ ∂L. ˆ take a quadrant in R2 with 2k pairwise opposite Therefore, for M = T2 as M ˆ , q = 1, ..., k, k ∈ N. Suitmarked points sˆ0,q and sˆ0,q+k on the boundary of M ˆ gives the mapping Ξ : M ˆ → T2 , Ξ(ˆ able gluing of boundary points in ∂ M s0,q ) = ˆ ˆ Ξ(ˆ s0,q+k ) = s0,q , q = 1, ..., k. Proper cutting of M into Aj , j = 1, 2, or of L induces that of T2 . Thus we get a pseudo-submanifold A3 (T2 ) =: A3 ⊃ C, while A1 and A2 are retractable into a marked point s0,q ∈ C for each q, hence T2 satisfies Conditions 2(i − iii, iv ′ , v). In view of Corollary 9 there exists the embedding L φ∗ : (W T2 ,{s0,q :q=1,...,k} E)t,H → (W L,{s0,q :q=1,...,2k} E)t,H , where φ : L → T2 is the L quotient mapping with φ({sL 0,q , s0,q+k }) = {s0,q }, q = 1, ..., k. For the n-dimensional torus Tn in Aar with n > 2 take a n − 1-dimensional surface B such that each its projection into T2 is Hpt -diffeomorphic with C for a loop C as above. Therefore, the slit along B up to a Hpt -diffeomorphism gives M0 := L × I n−2 for even n or M0 := S 1 × I n−1 for odd n, where I = [0, 1]. Since I m is retractable into a point, where m ≥ 1. Thus we lightly get for Tn a pseudo-submanifold A3 ⊃ B and two A1 and A2 retractable into points and satisfying sufficient Conditions 2(i − iii, iv ′ , v), where ˆ = I n up to a Hpt -diffeomorphism, s0,q ∈ B ⊂ A3 := A3 (Tn ), {sM0 , sM0 } ⊂ ∂M0 , M 0,q 0,q+k ˆ ˆ q = 1, ..., k, k ∈ N. Proper cutting of M into Aj , j = 1, 2, induces that of Tn . Thus there M0 0 exists an Hpt quotient mapping φ : M0 → Tn with φ({sM 0,q , s0,q+k }) = {s0,q } and the M0

embedding φ∗ : (W Tn ,{s0,q :q=1,...,k} E)t,H ֒→ (W M0 ,{s0,q :q=1,...,2k} E)t,H due to Corollary 9. More generally cut from Tn open subsets Vj which are Hpt diffeomorphic with interiors of bounded quadrants in Rn embedded into Alr , j = 1, ..., β, such that s0,q ∈ Ta1 +...+aq B ∩ ( j=a ∂Vj ), Vj ∩ Vi = ∅ for each j 6= i, Vj ∩ B = ∅ for each j, where 1 +...+aq−1 +1

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B is defined up to an Hpt diffeomorhism, a0 := 0, a1 + ... + ak = β, q = 1, ..., k, that gives the manifold M2 . Then from M0 cut analogously corresponding Vj,b , such that s0,q ∈ Ta1 +...+aq Ta1 +...+aq B ∩ ( j=a ∂Vj,1 ), s0,q+k ∈ B ∩ ( j=a ∂Vj,2 ), Vj,b1 ∩ Vi,b2 = ∅ for 1 +...+aq−1 +1 1 +...+aq−1 +1 each j 6= i or b1 6= b2 , a0 := 0, a1 + ... + ak = β, q = 1, ..., k, j = 1, ..., β, b = 1, 2, that produces the manifold M1 . We choose Vj,b such that for the restriction φ : M1 → M2 of the M1 1 mapping φ there is the equality φ(Vj,1 ∪ Vj,2 ) = Vj for each j, φ({sM 0,q , s0,q+k }) = {s0,q }. M1

This gives the embedding φ∗ : (W M2 ,{s0,q :q=1,...,k} E)t,H ֒→ (W M1 ,{s0,q :q=1,...,2k} E)t,H . Another example is M3 obtained from the previous M2 with 2k marked points and 2β cut out domains Vj , when s0,q is identified with s0,q+k and each ∂Vj is glued with ∂Vj+β for each j ∈ λq ⊂ {d : a1 + ... + aq−1 + 1 ≤ d ≤ a1 + ... + aq }, q = 1, ..., k, k ∈ N, by an equvalence relation υ. Such M3 is obtained from the torus Tn,m with m holes instead of one hole in the standard torus Tn,1 = Tn cutting from it Vj with S j ∈ {1, ..., 2β} \ ( q=1,...,k λq ), where m = m1 + ... + mk , mq := card(λq ). For Tn and M2 the surface B is Hpt diffeomorphic with (∂L) × I n−2 for even n or S 1 × I n−1 S for odd n. Take A3 ⊃ B ∪ ( j∈λq υ(∂Vj )), it is arcwise connected and contains all marked points. Therefore, M3 satisfies conditions of §2 and there exists the embedding M3

M2

υ ∗ : (W M3 ,{s0,q :q=1,...,k} E)t,H ֒→ (W M2 ,{s0,q :q=1,...,2k} E)t,H . This also induces the emTn,m

bedding (W Tn,m ,{s0,q

:q=1,...,k}

Tn,m

ement g ∈ (W Tn,m ,{s0,q

Tn

E)t,H ֒→ (W Tn ,{s0,q :q=1,...,2k−1} E)t,H such that each el-

:q=1,...,k}

E)t,H can be presented as a product g = (..(g1 g2 )...gm ) n :q=1,...,2k−1} Tn ,{sT 0,q (W E)t,H , gj =< fj >t,H , supp(π ◦ fj ) ⊂ Bj ,

of m elements gj ∈ B1 ∪ ... ∪ Bm = Tn , Bi ∩ Bj = ∂Bi ∩ ∂Bj for each i 6= j, each Bj is a canonical closed subset in Tn , s0,1 ∈ B1 , s0,2q , s2q+1 ∈ Bd for m1 + ... + m0 + 1 ≤ d ≤ m1 + ... + mq , q = 1, ..., k − 1, where m0 := 0. Evidently, in the general case for different manifolds M and N wrap groups may be non isomorphic. For example, as M1 take a sphere S n of the dimension n > 1, as M2 take M1 \ K, where K is up to an Hpt -diffeomorphism the union of non intersecting interiors Bj of quadrants of diameters d1 , ..., ds much less, than 1, K = B1 ∪ ... ∪ Bl , l ∈ N. Let N be a δ-enlargement for M2 in Rn+1 relative to the metric of the latter Euclidean space, where 0 < δ < min(d1 , ..., dl )/2. Then the groups (W M1 N )t,H and (W M2 N )t,H are not isomorphic. This lightly follows from the consideration of the element b :=< f >t,H ∈ (W M2 N )t,H , where f : M2 → N is the identity embedding induced by the structure of the δ-enlargement. Recall, that for orientable closed manifolds A and B of the same dimension m the degree of the continuous mapping f : A → B is defined as an integer number deg(f ) ∈ Z such that f∗ [A] = deg(f )[B], where [A] ∈ Hm (A) or [B] ∈ Hm (B) denotes a generator, defined by the orientation of A or B respectively [5]. Consider mappings fj : S n → N such that Vj ⊃ ∂Bj ∩ N , where Vj is a domain in Rn+1 bounded by the hyper-surface fj (Bj ), fj is w0 on each Bi with i 6= j, while the degree of the mapping fj from S n onto fj (S n ) is equal to one. If there would be an isomorphism θ : (W M2 N )t,H → (W M1 N )t,H , then θ(b) would have a non trivial decomposition into the sum of non canceling non zero additives, which is induced by mappings fj : S n → N . Nevertheless, an element b in (W M2 N )t,H has not such decomposition. If two groups G1 and G2 are not isomorphic, then certainly (W M E; N, G1 , P)t,H and

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(W M E; N, G2 , P)t,H are not isomorphic. The construction of wrap groups can be spread on locally compact non compact M ˆ is locally compact satisfying conditions 2(ii − iv) or (ii, iii, iv ′ ) changing (v) such that M t l ˆ non-compact Hp -domain in Ar , its boundary ∂ M may happen to be void. For this it is sufficient to restrict the family of functions to that of with compact supports f : M → W relative to w0 : M → W , that is suppw0 (f ) := clM {x ∈ M : f (x) 6= y0 × e} is compact, clM A denotes the closure of a subset A in M . Then classes of equivalent elements are given with the help of closures of orbits of the group of all Hpt diffeomorphisms g t (M, {s with compact supports preserving marked points Dif Hp,c 0,q : q = 1, ..., k}) that is suppid (g) := clM {x ∈ M : g(x) 6= x} are compact, where id(x) = x for each x ∈ M . Then wrap groups (W M E)t,H for manifolds M such as hyperboloid of one sheet, one sheet of two-sheeted hyperboloid, elliptic hyperboloid, hyperbolic paraboloid and so on in larger dimensional manifolds over Ar . For non compact locally compact manifolds it is possible also consider an infinite countable discrete set of marked points or of isolated singularities. These examples can be naturally generalized for certain knotted manifolds arising from the given above. Milnor and Lefshetz have used for M = S 1 and G = {e} the diffeomorphism group preserving an orientation and a marked point of S 1 . So their loop group L(S 1 , N ) may be non-commutative. The iterated loop group L(S 1 , L(S n−1 , N )) is isomorphic with L(S n , N ), where the latter group is supplied with the uniformity from the iterated loop group, so n times iterated loop group of S 1 gives loop group of S n [14]. For dimR M > 1 orientation preservation loss its significance. Here above it was used the diffeomorphism group without any demands on orientation preservation of M such that two copies of M in the wedge product already are not distinguished in equivalence classes and for commutative G it gives a commutative wrap group. Mention for comparison homotopy groups. The group πq (X) for a topological space X with a marked point x0 in view of Proposition 17.1 (b) [2] is commutative for q > 1. For q = 1 the fundamental group π1 (X) may be non-commutative, but it is always commutative in the particular case, when X = G is an arcwise connected topological group (see §49(G) in [43]). 11. Proposition. Let L(S 1 , N ) be an Hp1 loop group in the classical sense. Then the iterated loop group L(S 1 , L(S 1 , N )) is commutative. Proof. Consider two elements a, b ∈ L(S 1 , L(S 1 , N )) and two mappings f ∈ a, g ∈ b, (f (x))(y) = f (x, y) ∈ N , where x, y ∈ I = [0, 1] ⊂ R, e2πx ∈ S 1 . An inverse element d−1 of d ∈ L(S 1 , N ) is defined as the equivalence class d−1 =< h− >, where h ∈ d, h− (x) := h(1 − x). Then (1) f (x, 1 − y) = (f (x))(1 − y) ∈ a−1 and g(x, 1 − y) = (g(x))(1 − y) ∈ b−1 for L(S 1 , L(S 1 , N )) and symmetrically (2) (f (y))(1 − x) = f (1 − x, y) ∈ a−1 and (g(y))(1 − x) = g(1 − x, y) ∈ b−1 . On the other hand, f ∨ g corresponds to ab, and g ∨ f corresponds to ba, where the reduced product S 1 ∧S 1 is Hpt -diffeomorphic with S 2 in the sense of pseudo-manifolds up to critical subsets of codimension not less than two. Consider (S 1 ∨ S 1 ) ∧ (S 1 ∨ S 1 ) and (f ∨ w0 ) ∨ (w0 ∨ g) and (g ∨ w0 ) ∨ (w0 ∨ f ) and the ˆ = I 2 divided into four iterated equivalence relation R1,H . This situation corresponds to M quadrats by segments {1/2} × [0, 1] and [0, 1] × {1/2} with the corresponding domains for

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f , g and w0 in the considered wedge products, where < f ∨ w0 >=< w0 ∨ f >=< f > is the same class of equvalent elements. Since G = {e}, (ab)−1 = b−1 a−1 , then g(1 − x, y) ∨ f (1 − x, y) is in the same class of equivalent elements as g(x, 1 − y) ∨ f (x, 1 − y). But due to inclusions (1, 2) < g(1 − x, y) ∨ f (1 − x, y) >=< f (x, y) ∨ g(x, y) >−1 and < f (x, y) ∨ g(x, y) >=< g(x, 1 − y) ∨ f (x, 1 − y) >−1 and < h(x, y) >−1 =< h(x, 1 − y) >=< h(1 − x, y) > for h ∈ ab, consequently, < h(x, y) >=< h(1 − x, 1 − y) > and < (f ∨ g)(x, 1 − y) >=< f (x, 1 − y) ∨ g(x, 1 − y) >∈ (ab)−1 , since (x, y) 7→ (1 − x, 1 − y) interchange two spheres in the wedge product S 2 ∨ S 2 . Hence a−1 b−1 = b−1 a−1 and inevitably ab = ba. 12. Theorem. Let M and N be connected both either C ∞ Riemann or Ar holomorphic manifolds with corners, where M is compact and dimM ≥ 1 and dimN > 1. Then (W M N )t,H has no any nontrivial continuous local one parameter subgroup g b for b ∈ (−ǫ, ǫ) with ǫ > 0. Proof. Suppose the contrary, that {g b : b ∈ (−ǫ, ǫ)} with ǫ > 0 is a local nontrivial one parameter subgroup, that is, g b 6= e for b 6= 0. Then to g δ for a marked 0 < δ < ǫ there corresponds f = fδ ∈ Hp∞ such that < f >t,H = g δ , where f ∈ Hpt . If f (U ) = {y0 × e} for a sufficiently small connected open neighborhood U of s0,q in M , then there exists a sequence f ◦ ψn in the equivalence class < f >t,H with a family of diffeomorphisms ψn ∈ Dif Hpt (M ; {s0,q : q = 1, ..., k}) such that limn→∞ diamψn (U ) = 0 T and ∞ n=1 ψn (U ) = {s0,q }. If h(x) 6= y0 , then in view of the continuity of h there exists an open neighborhood P of x in M such that y0 ∈ / h(P ). Consider the covariant differentiation ∇ on the manifold M (see [22]). The set Sh of points, where ∇k h is discontinuous is a submanifold of codimension not less than one, hence of measure zero relative to the Riemann volume element in M . For others points x in M , x ∈ M \ Sh , all ∇k h are continuous. Take then open V = V (f ) in M such that V ⊃ U and ∇kν f |∂V 6= 0 for some k ∈ N, where ∇ν f (x) := limz→x,z∈M \V ∇ν f (z), ν is a normal (perpendicular) to ∂V in M at a point x in the boundary ∂V of V in M . Practically take a minimal k = k(x) with such property. Since M is compact and ∂V := cl(V ) ∩ cl(M \ V ) is closed in M , then ∂V is compact. The function x 7→ k(x) ∈ N is continuous, since f and ∇l f for each l are continuous. But N is discrete, hence each ∂q V := {x ∈ ∂V : k(x) = q} is open in V . Therefore, ∂V is a finite union of ∂q V , 1 ≤ q ≤ qm , where qm := maxx∈∂V k(x) < ∞ for f = fδ , since ∂V is compact. Thus, there exists a subset λ ⊂ {1, ..., qm } such that S ∂V = q∈λ ∂q V and ∂q V 6= ∅ for each q ∈ λ. If ∇l f (x) = 0 for l = 1, ..., k(x) − 1 and ∇k(x) f (x) 6= 0, then ∇k(x) f (ψ(y)) = ∇k(x) (ψ(y)).(∇ψ(y))⊗k(x) 6= 0 for y ∈ M such that ψ(y) = x, since ∇ψ(y) 6= 0, where ψ ∈ Dif Hp∞ (M ; {s0,q : q = 1, ..., k}). We can take ǫ > 0 such that {g b : b ∈ (−ǫ, ǫ)} ⊂ U , where U = −U is a connected symmetric open neighborhood of e in (W M N )t,H . Since g b1 + g b2 = g b1 +b2 for each b1 , b2 , b1 + b2 ∈ (−ǫ, ǫ), then limt→0 g b = e for the local one parameter subgroup and in particular limm→∞ g 1/m = e, where m ∈ N. Take δ = δm = 1/m and f = fm ∈ Hp∞ such that < fm >t,H = g 1/m . On the other hand, jg 1/m = g j/m for each j < mǫ, j ∈ N, hence fj/m (M ) = f1/m (M ) for each j < mǫ, since f ∨ h(M ∨ M ) = f (M ) ∨ h(M ) and using embedding η of (S M N )t,H into (W M N )t,H . k(x)

The function |∇ν

fδ (x)| for x ∈ ∂V is continuous by δ due to the Sobolev embedding

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theorem [35], 0 < δ < ǫ, consequently, inf x∈∂V |∇ν fδ (x)| > 0, since ∂V is compact. We can choose a family fδ such that z (l) (δ, x) := ∇l fδ (x) is continuous for each 0 ≤ l ≤ k0 by (δ, x) ∈ (−ǫ, ǫ) × M , since {g b : b ∈ (−ǫ, ǫ)} is the continuous by b one parameter subgroup, where k0 := qm (δ0 ). Therefore, for this family there exists a neighborhood [−ǫ+c, ǫ−c] such that δ0 ∈ [−ǫ+c, ǫ−c] ⊂ (−ǫ, ǫ) with 0 < c < ǫ/3 such that qm (δ) ≤ k0 for each δ ∈ [−ǫ+c, ǫ−c] with a suitable choice of V (fδ ), since N is discrete. On the other k(x) k(x) hand, supx∈∂V (fδ ),0<δ≤ǫ−c |∇ν fδ (x)| ≤ supx∈M,0<δ≤ǫ−c |∇ν fδ (x)| =: B < ∞, since M and [−ǫ + c, ǫ − c] are compact. Therefore, for this family there exists a neighborhood [−ǫ + c, ǫ − c] such that δ0 ∈ [−ǫ + c, ǫ − c] ⊂ (−ǫ, ǫ) with 0 < c < ǫ/3 such that qm (δ) ≤ k0 for each δ ∈ [−ǫ + c, ǫ − c] with a suitable choice of V (fδ ), since N is discrete. k(x) Then limδ→0,δ>0 |∇ν fδ (x)| =: b > 0 for x ∈ ∂V with a suitable choice of V = V (fδ ), since M is connected, dimM ≥ 1 and inf m∈N diamfj/m (M ) > 0 for a marked δ0 = j/m0 < ǫ with j, m > m0 ∈ N mutually prime, (j, m) = 1, (j, m0 ) = 1. To < fl/m >t,H there corresponds < f1/m >t,H ∨...∨ < f1/m >t,H =:< f1/m >∨l t,H which is the k(x)

l-fold wedge product. Thus there exists C = const > 0 for M such that |∇ν

k(y) Cl inf y∈∂V (f1/m ) |∇ν f1/m (y)|

fl/m (x)| ≥

≥ Clb, where C > 0 is fixed for a chosen atlas At(M ) with given transition mappings φi ◦ φ−1 j of charts. Consider δ0 ≤ l/m < ǫ−c and m and l tending to the infinity. Then this gives B ≥ Clb for each l ∈ N, that is the contradictory inequality, hence (W M N )t,H does not contain any non trivial local one parameter subgroup.

3.

Structure of Wrap Groups

1. Proposition. The Hpm uniformity in L(S m , N ) (see §2.10 in Section 2) for m > 1 is strictly stronger, than the m times iterated Hp1 uniformity. Proof. If f ∈ H m , then ∂ k f (x)/∂xk11 ...∂xkmm ∈ L2 for each 0 ≤ k ≤ m, k = k1 + ... + km , 0 ≤ kj , j = 1, ..., m. But g of m times iterated H 1 uniformity means that ∂ k g(x)/∂xk11 ...∂xkmm ∈ L2 for each 0 ≤ k ≤ m, k = k1 + ... + km , 0 ≤ kj ≤ 1, j = 1, ..., m. The latter conditions are weaker than that of H m . For m > 1 there may appear g for which such partial derivatives are not in L2 , when 1 < kj ≤ m. Using transition mappings of charts of atlases At(M ) and At(N ) and applying this locally we get the statement. 2. Theorem. For a wrap group W = (W M E)t,H (see Definition 2.7 in Section 2) there ˆ = W ⊗W ˜ which is an Hpl alternative Lie group and there exists a exists a skew product W ˆ , where l = t′ −t (l = ∞ for t′ = ∞), E = E(N, G, π, Ψ) is group embedding of W into W ′ a principal G-bundle of class Hpt with t′ ≥ t ≥ [dim(M )/2] + 1. If G is associative, then ˆ is associative. Moreover, the loop group L(S 1 , E) is Hpt isomorphic with (W ˆ S 1 E)t,H W in the particular case of S 1 . ˜ be a set of all elements (g1 a1 ⊗ g2 a2 ) ∈ (W ⊗ B)2 , where B is a free Proof. Let W non-commutative associative group with two generators a, b, ab 6= ba, g1 , g2 ∈ W . Take in ˜ the equivalence relation: g1 g2 a ⊗ g2 b= W ˜ g1 eB ⊗ eeB , for each g1 , g2 ∈ W , where e and eB denote the unit elements in W and in B.

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˜ the multiplication: Define in W ˜ 3 a3 ⊗ g4 a4 ) := ((g1 g3 )(a1 a3 ) ⊗ (g4 g2 )((a−1 (g1 a1 ⊗ g2 a2 )⊗(g 1 a4 a1 )a2 ) for each g1 , g2 , g3 , g4 ∈ W and every a1 , a2 , a3 , a4 ∈ B, hence ˜ ⊗ g2 a2 ) = e ⊗ (g2 g1 )(a2 a1 ), (e ⊗ g1 a1 )⊗(e ˜ 2 a2 ⊗ e) = (g1 g2 )(a1 a2 ) ⊗ e, (g1 a1 ⊗ e)⊗(g ˜ ⊗ g4 a4 ) = g1 a1 ⊗ g4 (a−1 (g1 a1 ⊗ e)⊗(e 1 a4 a1 ), ˜ (e ⊗ g4 a4 )⊗(g1 a1 ⊗ e) := g1 a1 ⊗ g4 a4 . ˜ of groups (W ⊗ B) ⊗s (W ⊗ B) is non-commutative, since Thus this semidirect product W b−1 aba−1 6= e, where e := e × eB , ⊗s denotes the semidirect product, ⊗ denotes the direct product. ˜ generated by Consider the minimal closed subgroup A in the semidirect product W −1 ˜ elements (g1 g2 a ⊗ g2 b)⊗(g1 eB ⊗ eeB ) , where B is supplied with the discrete topology ˜ is supplied with the product uniformity. Then put W ˆ := W ˜ /A =: W ⊗W ˜ and W and ˆ ˜ denote the multiplication in W as in W . ˆ and the multiplication Therefore, W has the group embedding θ : g 7→ (geB ⊗e) into W ˜ 2 eB ⊗ e)]. m[(g1 eB ⊗ e), (g2 eB ⊗ e)] = (g1 eB ⊗ e)⊗(g ˜ On the other hand, (ga1 ⊗e)⊗(e⊗ga1 a2 a−1 ˜, eˆ = e˜A = A 1 ) = ga1 ⊗ga2 = (e⊗e) =: e ˆ and (e ⊗ ga1 a2 a−1 ) = (ga1 ⊗ e)−1 is the inverse element of is the unit element in W 1 ˆ , a1 = ea1 , ˜ = (e ⊗ e)⊗A ˜ = A in W (ga1 ⊗ e), where a2 ∈ B is such that (a1 ⊗ a2 )⊗A ˜ that is a1 ⊗ a2 =e ˜ ⊗ e in W . ˆ is noncommutative and alternative. As From preceding formulas it follows, that W t ˆ the manifold W is the quotient of the Hp manifold W 2 by the Hpt equivalence relation, ˆ is the Hpt differentiable space, since Conditions (D1 − D4) of §2.1.3.2 in Section hence W ˆ combines the product in W 2 are satisfied. The group operation and the inversion in W and the inversion with the tensor product and the equivalence relation, hence they are Hpl differentiable with l = t′ − t, l = ∞ for t′ = ∞, (see §§1.11, 1.12, 1.15 in [45] and §2.1.3.1 in Section 2). ˜ 3 ⊗ g4 ))⊗(g ˜ 5 ⊗ g6 ) := ((g1 g3 )g5 ⊗ g6 (g4 g2 )) and Then ((g1 ⊗ g2 )⊗(g ˜ 3 ⊗ g4 ))⊗(g ˜ 5 ⊗ g6 )) := (g1 (g3 g5 ) ⊗ (g6 g4 )g2 ). (g1 ⊗ g2 )⊗((g ˆ is alternative, since W is alternative (see Theorem 2.6.1 in Section 2) and B Therefore, W ˆ is associative. is associative. If G is associative, then W is associative and W Consider the commutator ˜ 3 a3 ⊗ g4 a4 )]⊗[(g ˜ 1 a1 ⊗ g2 a2 )−1 ⊗ ˜ [(g1 a1 ⊗ g2 a2 )⊗(g −1 ˜ (g3 a3 ⊗ g4 a4 ) ] = {((g1 g3 )(a1 a3 ) ⊗ (g4 g2 )((a−1 a a 4 1 )a2 ))⊗ 1 −1 −1 −1 −1 −1 ˜ −1 −1 −1 −1 −1 [(g1 a1 ⊗ g2 (a1 a2 a1 ))⊗(g3 a3 ⊗ g4 (a3 a4 a3 ))] −1 −1 ˜ −1 −1 −1 −1 = ((g1 g3 )(a1 a3 ) ⊗ (g4 g2 )((a−1 1 a4 a1 )a2 )⊗((g1 g3 )(a1 a3 ) ⊗ (g4 g2 ) −1 −1 −1 −1 −1 −1 −1 −1 −1 (a1 (a3 a4 a3 )a1 )(a1 a2 a1 ))) = (((g1 g3 )(g1 g3 ))(a1 a3 a1 a3 )⊗ −1 −1 −1 −1 ((g4−1 g2−1 )(g4 g2 ))((a1 a3 )−1 [((a1 a3 )a−1 4 (a1 a3 ) )(a1 a2 a1 )](a1 a3 ))((a1 a4 a1 )a2 ). The minimal closed subgroup generated by products of such elements is the commutant ˜ ˜ . The group (W M N )t,H is commutative (see Theorem 6(2) in Section 2). We Wc of W have B/Bc = {e}, the quotient group G/Gc = Gab is the abelianization of G, particularly if G is commutative, then Gab = G, where Gc denotes the commutant subgroup of G. Therefore, (W M E; N, G, P)t,H /[(W M E; N, G, P)t,H ]c = (W M E; N, Gab , P)t,H ˜ /W ˜ c = (W M E; N, Gab , P)t,H . and inevitably W

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˜ we get W ˆ /W ˆ c = (W M E; N, Gab , P)t,H . Using the equivalence relation in W In the particular case of M = S 1 for g ∈ W take f ∈ g, that is < f >t,H = g. The equivalence class of f relative to the analogous closures of orbits of the right action of the subgroup Dif f+∞ (S 1 , s0 ) preserving a marked point and an orientation of S 1 induced by ˜, that of I = [0, 1] denote by [f ]t,H , then to [f ]t,H put into the correspondence ga ⊗ e in W − −1 − while to [f ]t,H counterpose e ⊗ gaba , where f (x) := f (1 − x) for each x ∈ [0, 1], the unit circle S 1 is parametrized as z = e2πix , z ∈ S 1 ⊂ C, x ∈ [0, 1]. Their equivalence ˜ give elements in W ˆ. ˜ and (e ⊗ gaba−1 )⊗A ˜ in W classes (ga ⊗ e)⊗A −1 − ˆ is isomorphic with Since [f ]t,H := [f ]t,H and [f1 ∨ f2 ]t,H = [f1 ]t,H [f2 ]t,H , then W L(S 1 , E)t,H . 2.1. Remark. Consider the group B 2 ⊗ B 2 /E, where an equivalence relation E is ˜ : (a ⊗ b) ≈ (e ⊗ e), the group B is the same as in §2 induced by that of in B 2 as in W with two generators a, b. Then this gives the equivalences: [(a ⊗ b) ⊗ (a ⊗ b)] E [(e ⊗ e) ⊗ (e ⊗ e)] E [(e ⊗ b) ⊗ (a ⊗ e)] ⊗ [(e ⊗ b) ⊗ (a ⊗ e)] E {(e ⊗ b) ⊗ [(a ⊗ e) ⊗ (e ⊗ b)]} ⊗ (a ⊗ e) E (e ⊗ a−1 ba) ⊗ (a ⊗ e) E [(e ⊗ ab) ⊗ (ba ⊗ e)] in B 2 ⊗ B 2 , since B 4 is the associative group. This implies the commutativity of the iterated skew product wrap group, ˆ M (W ˆ M E)t,H )t,H = (W M (W M E)t,H )t,H , G = Gab . when G is commutative, that is (W M M ˆ (W ˆ N )t,H )t,H = (W M (W M N )t,H )t,H , where G = {e}. Therefore, In particular, (W from this remark and Theorem 2 the new proof of Proposition 11 in Section 2 follows. ′ 3. Proposition. If there exists an Hpt -diffeomorphism η : N → N such that η(y0 ) = y0 ′ , where t ≤ t′ then wrap groups (W M E; y0 )t,H and (W M E; y0 ′ )t,H defined with marked points y0 and y0 ′ are Hpl -isomorphic as Hpl -differentiable groups, where l = t′ − t for finite t′ , l = ∞ for t′ = ∞. Proof. Let f ∈ Hpt (M, E), then η ◦ π ◦ f (s0,q ) = η(y0 ) = y0 ′ for each marked point s0,q in M , where π : E → N is the projection, π ◦ f = γ, γ is a wrap, that is an Hpt mapping from M into N with γ(s0,q ) = y0 for q = 1, ...., k. The manifold N is connected together with E and G in accordance with conditions imposed in Section 2. Consider the ′ Hpt -diffeomorphism η×e of the principal bundle E. Then Θ : Hpt (M, W ) → Hpt (M, W ) is the induced isomorphism such that π ◦ Θ(f ) := η ◦ π ◦ f : M → N and (η × e) ◦ f = Θ(f ) for f ∈ Hpt (M, E). The mapping Θ is Hpl differentiable by f , hence it gives the Hpl isomorphism of the considered Hpl -differentiable wrap groups (see Theorem 6(1) in Section 2). 4. Remark. As usually we suppose, that the principal bundle E, its structure group G and the base manifold N are arcwise connected. Let (P M E)t,H be a space of equivalence classes < f >t,H of f ∈ Hpt (M, W ) relative to the closures of orbits of the left action of Dif Hpt (M ; {s0,q : q = 1, ..., k}). This means, that (P M E)t,H is the quotient space of Hpt (M, W ) relative to the equivalence relation Rt,H . There is the embedding θ : Hpt (M, {s0,q : q = 1, ..., k}; W ) ֒→ Hpt (M ; W ) and the s0,q ) : q = k+1, .., 2k), evaluation mapping eˆv : Hpt (M ; W ) → N k such that eˆv(f ) := (fˆ(ˆ t ˆ ˆ ˆ ˆ ; W ) is such that f = f ◦ Ξ, Ξ : M ˆ → M eˆv sˆ0,q (f ) := f (ˆ s0,q ), where f ∈ Hp (M t is the quotient mapping. We get the diagram Hp (M, {s0,q : q = 1, ..., k}; W ) → Hpt (M ; W ) → N k with Hpt differentiable mappings, which induces the diagram Hpt,l+1 (M, {s0,q : q = 1, ..., k}; W, y0 ) → Hpt (M, Hpt,l (M, {s0,q : q = 1, ..., k}; W, y0 ) → Hpt,l (M, {s0,q : q = 1, ..., k}; W, y0 ) for each l ∈ N, where Hpt,l+1 (M, {s0,q : q =

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1, ..., k}; W, y0 ) := Hpt (M, {s0,q : q = 1, ..., k}; Hpt,l (M, {s0,q : q = 1, ..., k}; W, y0 )), Hpt,1 (M, {s0,q : q = 1, ..., k}; W, y0 ) := Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ). Therefore, there exist iterated wrap semigroups and groups (S M E)l+1;t,H := (S M (S M E)l;t,H )t,H and (W M E)l+1;t,H := (W M (W M E)l;t,H )t,H , where (S M E)1;t,H := (S M E)t,H and (W M E)1;t,H := (W M E)t,H . ′ Evidently, if there are Hpt and Hpt diffeomorphisms ρ : M → M1 and η : N → N1 mapping marked points into respective marked points, then Hpt (M, W ) is isomorphic with Hpt (M1 , W1 ) and hence (W M E)b;t,H is Hpt isomorphic as the Hpt -manifold and Hpl isomorphic as the Hpl -Lie group with (W M1 E1 )b;t,H for each b ∈ N, where l = t′ − t, l = ∞ for t′ = ∞, t′ ≥ t ≥ [dim(M )/2] + 1. If f : N → N1 is a surjective map and N is an Hpt -differentiable space, then N inherits a structure of an Hpt -differentiable space with plots having the local form f ◦ ρ : U → N1 , where ρ : U → N is a plot of N . ′ 5. Lemma. Let E be an Hpt principal bundle and let D be an everywhere dense subset in N such that for each y ∈ D there exists an open neighborhood V of y in N and a differentiable map p : V → Hpt (M, {s0,q : q = 1, ..., k}; V, y) := {f ∈ Hpt (M ; V ) : f (s0,q ) = y, q = 1, ..., k} such that eˆv sˆ0,q (ˆ p(y)) = y for each q = 1, ..., 2k and each t y ∈ N , where p ◦ Ξ = π ◦ pˆ. Then eˆv : Hp (M ; W ) → N k is an Hpt differentiable principal (S M E)t,H bundle. Proof. Let {(Vj , yj ) : j ∈ J} be a family such that yj ∈ Vj ∩ D for each j and there exists pj : Vj → Hpt (M, {s0,q : q = 1, ..., k}; Vj , yj ) so that pˆj (ˆ s0,q )(y) = y × e for each q = 1, ..., 2k and every j, where {Vj : j ∈ J} is an open covering of N , y is a ˆ into Vj with y(M ˆ ) = {y}, where pˆj (ˆ constant mapping from M s0,q ) is the restriction to M Vj of the projection pˆ(ˆ s0,q ) : (P E)t,H → E, while pj (Ξ(ˆ x))(y) = π ◦ pˆj (ˆ x)(y × e) for ˆ ˆ each y ∈ N and x = Ξ(ˆ x) in M , where x ˆ ∈ M , Ξ : M → M . Then (W M E)t,H and (P M E)t,H are supplied with the Hpt -differentiable spaces structure (see Remark 4 above and Theorem 6 in Section 2), where the embedding (S M E)t,H ֒→ (P M E)t,H and the projection eˆv sˆ0,q : (P M E)t,H → N are Hpt -maps. Let ψj ∈ Dif Hpt (N ) such that ψj (y) = yj . Specify a trivialization φj : s0,q )(Vj ) → Vj × (S M E)t,H of the restriction pˆj (ˆ s0,q )|Vj of the projection pˆj (ˆ s0,q ) : pˆ−1 j (ˆ M (P E)t,H → E by the formula φj (f ) = (f (ˆ s0,q ), ψj ◦ pˆj (ˆ s0,q )(f )) for each f ∈ (P M E)t,H with π ◦ f (ˆ s0,q ) = y, where ψj ◦ pˆj (f ) = ψj (ˆ pj (f )). Then φ−1 j (y, g) = −1 M g (ψj ◦ pˆj (y)) =: η, η ∈ (P E)t,H with π ◦ ψj ◦ f (ˆ s0,q ) = yj , since G is a group, where g = ψj ◦ pˆj (f ). Finally the combination of the family {ˆ ev sˆ0,q : q = k + 1, ..., 2k} induce the mapping eˆv : Hpt (M ; W ) → N k . By the construction a fiber of this bundle is the monoid (S M E)t,H . 6. Theorem. If N is a smooth manifold over Ar (holomorphic for 1 ≤ r ≤ 3 respectively), then there exists an Hpt -differentiable principal (S M E)t,H bundle eˆv : (P M E)t,H → N k . Proof. In view of Lemma 5 it is sufficient to prove that for each y ∈ N there exists a neighborhood U of y in N and an Hpt -map pq : U → Hpt (M, W ) such that evs0,q (pq (z)) = z for each q = 1, ..., k, z ∈ U , where evx (f ) = f (x). ˆ consider a rectifiable curve ζq : [0, 1] → M ˆ joining sˆ0,q with sˆ0,q+k , where In M ˆ such that x1 corresponds 1 ≤ q ≤ k. Then consider a coordinate system (x1 , ..., xm ) in M to a natural coordinate along ζq . This coordinate system is defined locally for each chart of

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ˆ and x1 is defined globally. M Consider a real shadow NR of N , then NR is the Riemann C ∞ manifold. Thus there exists a Riemannian metric g in N . For each y ∈ N there exists a geodesic ball U at y of radius less than the injectivity radius expN for g. Then there exists a map pq : U → (P M U )t,H with π ◦ [pq (ˆ s0,q+k )(z)] = z and π ◦ [pq (ˆ s0,q (z)] = y for each z ∈ U , where pq ◦ ζq =: γˆq,y,z is the shortest geodesic in U joining y with z, γˆq,y,z : [0, 1] → N , ˆ with values γˆq,y,z ◦ ζq−1 (x1 ) ∈ N for each x1 . Having initially γˆq,y,z extend it to pˆq on M in E such that pq ◦ Ξ = π ◦ pˆq . 7. Proposition. (1). The wrap group (W M E; N, G, P)t,H is the principal Gk bundle over (W M N )t,H . (2). The abelianization [(W M E; N, G, P)t,H ]ab of the wrap group (W M E; N, G, P)t,H is isomorphic with (W M E; N, Gab , P)t,H . n (3). For n ≥ 2 the iterated loop group (LS E)t,H is isomorphic with the wrap group n (W S E)t,H for the sphere S n and a principal fiber bundle E for dimR N ≥ 2 with k = 1. Proof. 1. The bundle structure π : E → N induces the bundle structure π ˆ : M M (W E; N, G, P)t,H → (W N )t,H , since π ◦ Pγˆ,u = γˆ . In view of Lemma 5 it is sufficient to show, that there exists a neighborhood UG of e in (W M E)t,H and a G-equivariant ˆ → N, mapping φ : UG → (W M N )t,H . Let < Pγˆ,u >t,H ∈ (W M E)t,H , where γˆ : M γˆ = γ ◦ Ξ, γ : M → N , γ(s0,q ) = y0 for each q = 1, ..., k. Then π ◦ Pγˆ,u = γˆ and Pγˆ,u is G-equivariant by the conditions defining the parallel transport structure, that ˆ and z ∈ G and every u ∈ Ey . We have that is Pγˆ,u (x)z = Pγˆ,uz (x) for each x ∈ M 0 −1 uG = π (y) for each u ∈ Ey and y ∈ N . Therefore, put φ = π∗ , where π∗ < Pγˆ,u >t,H =< γˆ , u >t,H and take UG = π∗−1 (U ), where U is a symmetric U −1 = U neighborhood of e in (W M N )t,H . The group G acts effectively on E. Since G is arcwise connected, then Gk acts effectively on (W M E)t,H . Indeed, for each ζq from §6 there is gq ∈ G corresponding to γˆ (ˆ s0,q+k ) with Ppˆq ,ˆs0,q ×e (ˆ s0,q+k ) = {y0 × gq } ∈ Ey0 , gq ∈ G for every −1 q = 1, ..., k. Moreover, π∗ (π∗ (< Pγˆ,u >t,H )) =< Pγˆ,u >t,H Gk . Then the fibre of π ˆ : (W M E; N, G, P)t,H → (W M N )t,H is Gk . Due to Conditions 2(P 1 − P 5) in Section 2 it is the principal Gk differentiable bundle of class Hpt . n 2, 3. In view of Proposition 1 the loop group (LS E)l,H is everywhere dense in the n 1 1 n times iterated loop group (LS (...(LS E)1,H ...)1,H , while the wrap group (W S E)l,H is 1 everywhere dense in the n times iterated wrap group (W S E)n;1,H for each l ≥ n. For each n > m there exists the natural projection πnm : S n → S m which induces the embeddings m n m n (W S E)t,H ֒→ (W S E)t,H and (LS E)t,H ֒→ (LS E)t,H in accordance with Corollary 9 in Section 2, since k = 1 and choosing a marked point s0 ∈ S 1 . Therefore, due to dimR N ≥ 2 the considered here wrap and loop groups are infinite dimensional. Therefore, statements (2, 3) follow from (1) and the proof of Theorem 2 above and Proposition 11 in 1 1 Section 2 in accordance with which the iterated loop group (LS (...(LS E)1,H ...)1,H is commutative. 8. Proposition. If E is contractible, then (P M E)t,H is contractible. Proof. Let g : [0, 1] × E → E be a contraction such that g is continuous and g(0, z) = z and g(1, z) = y0 × e for each z ∈ E. Then for each f ∈ Hpt (M, W ) we get g(0, f (x)) = f (x) and g(1, f (x)) = y0 × e for each x ∈ M . Moreover,

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g(s, < f >t,H ) ⊂< g(s, f ) >t,H for each s ∈ [0, 1], since f ∈ gs−1 (< g(s, f ) >t,H ) and g is continuous while < g(s, f ) >t,H by its definition is closed in Hpt (M, W ), where gs (z) := g(s, z). Therefore, id = g(0, ∗) : (P M E)t,H → (P M E)t,H and g(1, (P M E)t,H ) =< w0 >t,H . 8.1. Notation. Denote by Homtp ((W M E)t,H , G) or Homtp ((S M E)t,H , G) the group or the monoid of Hpt differentiable homomorphisms from (W M E)t,H or (S M E)t,H respectively into G. By A∗r is denoted the multiplicative group of Ar \ {0}, where 0 ≤ r ≤ 3. ′ 9. Theorem. Let Dif Hpt (N ) acts transitively on N , t ≤ t′ . For each H ∞ manifold N and an Hpt differentiable group G such that A∗r ⊂ G with 1 ≤ r ≤ 3 there exists a homomorphism of the Hpt differentiable space of all equivalence classes of (P M E)t,H ′ relative to Dif Hpt (N ) (see §§1.3.2 and 3 in Section 2) and Homtp ((S M E)t,H , Gk ). They are isomorphic, when G is commutative. Proof. Mention that due to Theorem 6 the Hpt -differentiable principal (S M E)t,H bunˆ γˆ,uz (x) = P ˆ γˆ,u (x)z for each dle eˆv : (P M E)t,H → N k has a parallel transport structure P −1 t ˆ x ∈ M and all γ ∈ Hp (M, N ) and u ∈ eˆv (γ(s0,k )) and every z ∈ G and the correˆ gives the ˆ → N such that γ ◦ Ξ = γˆ . If x = sˆ0,q with 1 ≤ q ≤ k, then P sponding γˆ : M M M M identity homomorphism from (S E)t,H into (S E)t,H . If θ : (S E)t,H → Gk is an Hpt ˆ θ on differentiable homomorphism, then the holonomy of the associated parallel transport P the bundle (P M E)t,H ×θ G → N k is the homomorphism θ : (S M E)t,H → Gk (see §2.3 in Section 2). At the same time the group G contains continuous one-parameter subgroups from A∗r , where 1 ≤ r ≤ 3. If g ∈ (W M N )t,H and g 6= e, then g is of infinite order, since w0 does not belong to g n for each n 6= 0 non-zero integer n, where w0 (M ) = {y0 }. This holonomy induces a map h : (P M E)t,H /Q → Homtp ((S M E)t,H , Gk ), where ′ Q is an equivalence relation caused by the transitive action of Dif Hpt (N ) such that (S M E)t,H with distinct marked points either {s0,q : q = 1, ..., k} in M and y0 or y˜0 in ′ N are isomorphic, since there exists ψ ∈ Dif Hpt (N ) such that ψ(y0 ) = y˜0 . If G is commutative, then this map is the homomorphism, since (S M E)t,H is the commutative monoid for a commutative group G (see Theorem 3.2 in Section 2) and ˆ and u, v1 , v2 ∈ uPγˆ1 ,v1 (x1 )Pγˆ2 ,v2 (x2 ) = uPγˆ2 ,v2 (x2 )Pγˆ1 ,v1 (x1 ) for each x1 , x2 ∈ M M M Ey0 . There is the embedding (S E)t,H ֒→ (W E)t,H , hence a homomorphism θ : (W M E)t,H → Gk has the restriction on (S M E)t,H which is also the homomorphism. For G ⊃ A∗r there exists a family of f ∈ Homtp ((S M E)t,H , Gk ) separating elements of the wrap monoid (S M E)t,H , hence there exists the embedding of (S M E)t,H into Homtp ((S M E)t,H , Gk ). The bundle (P M E)t,H ×θ G → N k has the induced parallel transport structure Pθ . The holonomy of the parallel transport structure on (P M N )t,H ×θ G → N k is θ. Therefore, the map Hpt ((S M E)t,H , Gk ) ∋ θ 7→ Pθ is inverse to h. ˆ 2 ֒→ M ˆ1 10. Theorems. Suppose that M2 ֒→ M1 and M = M1 \ (M2 \ ∂M2 ) and M t ˆ =M ˆ 1 \(M ˆ 2 \∂ M ˆ 2 ) and N2 ֒→ N1 are Hp -pseudo-manifolds with the same marked and M points {s0,q : q = 1, ..., k} for M1 and M2 and M and y0 ∈ N2 satisfying conditions of §2 in Section 2 and G2 is a closed subgroup in G1 with a topologically complete principal fiber bundle E with a structure group G1 . 1. Then (W M2 ,{s0,q :q=1,...,k} E; N2 , G2 , P)t,H has an embedding as a closed subgroup into (W M1 ,{s0,q :q=1,...,k} E; N1 , G1 , P)t,H .

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2. The wrap group (W M2 ,{s0,q :q=1,...,k} E; N, G2 , P)t,H is normal in (W M1 ,{s0,q :q=1,...,k} E; N, G1 , P)t,H if and only if G2 is a normal subgroup in G1 . 3. In the latter case (W M E; N, G, P)t,H is isomorphic with (W M1 E; N, G1 , P)t,H /(W M2 E; N, G2 , P)t,H , where G = G1 /G2 . ˆ 2 , N2 ), then it has an Hpt extension to γˆ1 ∈ Hpt (M ˆ 1 , N1 ) due Proof. 1. If γˆ2 ∈ Hpt (M ˆ 1 serves to Theorem III.4.1 [35]. Therefore, the parallel transport structure Pγˆ1 ,u over M t ˆ as an extension of Pγˆ2 ,u over M2 . The uniform spaces Hp (Mj , {s0,1 , ..., s0,k }; Wj , y0 ) are complete for j = 1, 2, since the principal fiber bundle E is topologically complete and the corresponding principal fiber sub-bundle E2 with the structure group G2 is also complete (see Theorem 8.3.6 [12]). Therefore, Hpt (M2 , {s0,1 , ..., s0,k }; W2 , y0 ) has embedding as the closed subspace into Hpt (M1 , {s0,1 , ..., s0,k }; W1 , y0 ). Each Hpt diffeomorphism of M2 has an Hpt extension to a diffeomorphism of M1 (see also §III.4 in [35] and [50]). Since G2 is a closed subgroup in G1 , then (S M2 ,{s0,q :q=1,...,k} E; N2 , G2 , P)t,H has an embedding as a closed sub-monoid into (S M1 ,{s0,q :q=1,...,k} E; N1 , G1 , P)t,H and inevitably (W M2 ,{s0,q :q=1,...,k} E; N2 , G2 , P)t,H has an embedding as a closed subgroup into (W M1 ,{s0,q :q=1,...,k} E; N1 , G1 , P)t,H due to Theorem 6.1 in Section 2. 2. The groups (W Mj ,{s0,q :q=1,...,k} N )t,H for j = 1, 2 are commutative and (W Mj ,{s0,q :q=1,...,k} E)t,H is the Gkj principal fiber bundle on (W Mj ,{s0,q :q=1,...,k} N )t,H (see Theorem 6.2 in Section 2 and Proposition 7.1 above). Therefore, (W M2 ,{s0,q :q=1,...,k} E)t,H is the normal subgroup in (W M1 ,{s0,q :q=1,...,k} E)t,H if and only if G2 is the normal subgroup in G1 . 3. Consider the principal fiber bundle E(N, G, π, Ψ) with the structure group G (see Note 1.3.2 in Section 2) and the parallel transport structure P for the Hpt pseudo-manifold ˆ , where G = G1 /G2 is the quotient group. If γˆ1 ∈ Hpt (M ˆ 1 , N ), then γˆ1 is the combinaM tion (i) γˆ1 = γˆ2 ∇ˆ γ, ˆ 2 and M ˆ correspondingly. On the other hand, where γˆ2 and γˆ are restrictions of γˆ1 on M t t ˆ , N ) has an extension γˆ1 ∈ Hp (M ˆ 1 , N ). The manifold M ˆ 1 is metrizable each γˆ ∈ Hp (M t ˆ by a metric ρ. For each ǫ > 0 there exists ψ ∈ Dif Hp (M1 ; {ˆ s0,q : q = 1, ..., 2k}) such Ss ˆ ˆ ˆ ˆ that (ψ(M ) ∩ M2 ) ⊂ l=1 B(M1 , xl , ǫ) for some xl ∈ M1 with l = 1, ..., s and s ∈ ˆ 1 and M ˆ 2 are compact pseudo-manifolds. N and ψ|Mˆ 1 \(Mˆ Ss B(Mˆ 1 ,xl ,ǫ)) = id, since M l=1 Therefore, using Lemma 2.1.3.16 [31] and charts of the manifolds gives < Pγˆ,u |M >t,H =< Pγˆ1 ,u |M1 >t,H / < Pγˆ2 ,u |M2 >t,H ′ due to decomposition (i), since Pγˆ,u |Mj ∈ Gj for j = 1, 2 and G = G1 /G2 is the Hpt quotient group with t′ ≥ t. Consequently, (W M E; N, G, P)t,H is isomorphic with (W M1 E; N, G1 , P)t,H /(W M2 E; N, G2 , P)t,H (see also §§3, 6 in Section 2). 11. Corollary. Let suppositions of Theorem 10 be satisfied. Then (W M N )t,H is isomorphic with (W M1 N )t,H /(W M2 N )t,H . Proof. For (W M N )t,H taking G = G1 = G2 = {e} we get the statement of this corollary from Theorem 10.3. 12. Proposition. Suppose that M = M1 ∨ M2 , where M1 and M2 are Hpt -pseudomanifolds satisfying Conditions 2.2(i − v) in Section 2 with the bunch taken by marked points {s0,q : q = 1, ..., k}, then (W M N )t,H is isomorphic with the internal direct product (W M1 N )t,H ⊗ (W M2 N )t,H .

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Proof. The manifold M has marked points {s0,q : q = 1, ..., k} such that s0,q corresponds to s0,q,1 glued with s0,q,2 in the bunch M1 ∨ M2 for each q = 1, ..., k, where s0,q,j ∈ Mj are marked points j = 1, 2. Since each Mj satisfies Conditions 2.2(i − v) in Section 2, then M satisfies them also. In view of Theorem 10.1 (W Mj ,{s0,q :q=1,...,k} N )t,H has an embedding as a closed subgroup into (W M,{s0,q :q=1,...,k} N )t,H for j = 1, 2. If γj ∈ Hpt (Mj , {s0,q : q = 1, ..., k}; N, y0 ) for j = 1, 2, then γ1 ∨ γ2 ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ). On the other hand, each γ ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ) has the decomposition γ = γ1 ∨ γ2 , where γj = γ|Mj for j = 1, 2. Therefore, < γ >t,H =< γ1 ∨ w0,2 >t,H ∨ < w0,1 ∨ γ2 >t,H , where w0 (M ) = {y0 }, w0,j = w0 |Mj for j = 1, 2, hence (W M N )t,H is isomorphic with (W M1 N )t,H ⊗ (W M2 N )t,H . 13. Propositions. 1. Let θ : N1 → N be an embedding with θ(y1 ) = y0 , or F : E1 → E be an embedding of principal fiber bundles over Ar such that π ◦ F |N1 ×e = θ ◦ π1 , then there exist embeddings θ∗ : (W M N1 )t,H → (W M N )t,H and F∗ : (W M E1 )t,H → (W M E)t,H . 2. If θ : N1 → N and F : E1 → E are a quotient mapping and a quotient homomorphism such that N1 is a covering pseudo-manifold of a pseudo-manifold N , then (W M N )t,H is the quotient group of some closed subgroup in (W M N1 )t,H and (W M E)t,H is the quotient group of some closed subgroup in (W M E1 )t,H . ′ 3. If there are an Hpt diffeomorphism f1 : M → M1 and an Hpt -isomorphism f2 : E → E1 , then wrap groups (W M1 E1 )t,H and (W M E)t,H are isomorphic. Proof. 1. If γ1 ∈ Hpt (M, {s0,q : q = 1, ..., k}; N1 , y1 ), then θ◦γ1 = γ ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ), < γ >t,H = θ∗ < γ1 >t,H , where θ∗ < γ1 >t,H := {θ◦f : f Rt,H γ1 }. In addition F |E1,v gives an embedding F : G1 → G, where G1 and G are structural groups of E1 and E. Therefore, for the parallel transport structures we get (1) F ◦ P1γˆ1 ,v (x) = Pγˆ,u (x) ˆ , where F (v) = u, π ◦ F = θ ◦ π1 , where P1 is for E1 and P for E. for each x ∈ M Define F∗ < P1γˆ1 ,v >t,H := {F ◦ g : gRt,H P1γˆ1 ,v }. Since θ and F are Hpt differentiable mappings, then θ∗ and F∗ are embeddings of Hpt manifolds and group homomorphisms of Hpl differentiable groups (see also Theorems 6 in Section 2). 2. If γ ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ), then there exists γ1 ∈ Hpt (M, {s0,q : q = 1, ..., k}; N1 , y1 ) such that θ ◦ γ1 = γ, since N1 is a covering of N , that is each y ∈ N has a neighborhood Vy for which θ−1 (Vy ) is a disjoint union of open subsets in N1 for each y ∈ N . This γ1 exists due to connectedness of M and γ(M ), where γ(M ) ⊂ N . To each parallel transport in E1 there corresponds a parallel transport in E so that Equation (1) above is satisfied. Put θ∗−1 < γ >t,H = {< γ1 >t,H : θ ◦ γ1 = γ} and F∗−1 < Pγˆ,u >t,H := {< P1γˆ1 ,v >t,H : F ◦ P1γˆ1 ,v = Pγˆ,u }, where F (v) = u. This gives quotient mappings θ∗ and F∗ from closed subgroups θ∗−1 (W M N )t,H and F∗−1 (W M E)t,H in (W M N1 )t,H and (W M E1 )t,H respectively onto (W M N )t,H and (W M E)t,H by closed subgroups θ∗−1 (e) and F∗−1 (e) correspondingly. 3. We have that g ∈ Hpt (M, {s0,q : q = 1, ..., k}; W, y0 ) if and only if f2 ◦ g ◦ f1−1 ∈ t Hp (M1 , {s0,q,1 : q = 1, ..., k; W1 , y1 ), where f1 (s0,q ) = s0,q,1 for each q = 1, ..., k, f2 (y0 × e) = y1 × e. At the same time ψ ∈ Dif Hpt (M ) if and only if f1 ◦ ψ ◦ f1−1 ∈ Dif Hpt (M1 ). Hence (S M E)t,H is isomorphic with (S M1 E1 )t,H and inevitably wrap

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groups (W M E)t,H and (W M1 E1 )t,H are Hpt diffeomorphic as manifolds and isomorphic as Hpl groups. 14. Note. If N is a manifold not necessarily orientable, then it contains up to equivalence of atlases a connected chart V open in N such that y ∈ V and V is orientable. Since (W M E|V )t,H is the infinite dimensional group, then (W M E)t,H is also infinite dimensional even if N is not orientable due to Proposition 13.1. If N is not orientable, then there exists an orientable covering manifold N1 and a quotient mapping θ : N1 → N as in Proposition 13(2) (see also about coverings and orientable coverings in §§50, 51 [43], §§II.4.18,19 [7]). It is necessary to mention that some circumstances of wrap groups are related also with their infinite dimensionality. 15. Note. Let G be a topological group not necessarily associative, but alternative: (A1) g(gf ) = (gg)f and (f g)g = f (gg) and g −1 (gf ) = f and (f g)g −1 = f for each f, g ∈ G and having a conjugation operation which is a continuous automorphism of G such that (C1) conj(gf ) = conj(f )conj(g) for each g, f ∈ G, (C2) conj(e) = e for the unit element e in G. If G is of definite class of smoothness, for example, Hpt differentiable, then conj is supposed to be of the same class. For commutative group in particular it can be taken the identity mapping as the conjugation. For G = A∗r it can be taken conj(z) = z˜ the usual conjugation for each z ∈ A∗r , where 1 ≤ r ≤ 3. Suppose that ˆ = G ˆ 0 i0 ⊕ G ˆ 1 i1 ⊕ ... ⊕ G ˆ 2r −1 i2r −1 such that G is a multiplicative group of (A2) G ˆ ˆ 0 , ..., G ˆ 2r −1 are pairwise isomora ring G with the multiplicative group structure, where G phic commutative associative rings and {i0 , ..., i2r −1 } are generators of the Cayley-Dickson algebra Ar , 1 ≤ r ≤ 3 and (yl il )(ys is ) = (yl ys )(il is ) is the natural multiplication of any ˆ = Anr . pure states in G for yl ∈ Gl . For example, G = (A∗r )n and G 16. Lemma. If G and K are two topological or differentiable groups twisted over {i0 , ..., i2r −1 } satisfying conditions 15(A1, A2, C1, C2) and K is a closed normal subgroup in G, where 2 ≤ r ≤ 3, then the quotient group is topological or differentiable and twisted over {i0 , ..., i2r −1 }. ˆ = G ˆ 0 i0 ⊕ G ˆ 1 i1 ⊕ ... ⊕ G ˆ 2r −1 i2r −1 , where G ˆ 0 , ..., G ˆ 2r −1 are Proof. Since G ˆ K ˆ = (G ˆ 0 /K ˆ 0 )i0 ⊕ ... ⊕ (G ˆ 2r −1 /K ˆ 2r −1 )i2r −1 is also pairwise isomorphic, then G/ ˆ twisted. Each Gj is associative, hence G/K is alternative, since 2 ≤ r ≤ 3 and using multiplicative properties of generators of the Cayley-Dickson algebra Ar . On the other hand, conj(K) = K, hence conj(gK) = Kconj(g) = conj(g)K ∈ G/K and conj(ghK) = conj(gh)K = (conj(h)conj(g))K = (conj(h)K)(conj(g)K) = conj(hK)conj(gK) = conj(gKhK). The subgroup K is closed in G, hence by the definition of the quotient differentiable structure G/K is the differentiable group (see also §§1.11, 1.12, 1.15 in [45]). ′ ′ 17. Proposition. Let η : N1 → N2 be an Hpt -retraction of Hpt manifolds, N2 ⊂ N1 , η|N2 = id, y0 ∈ N2 , where t′ ≥ t, M is an Hpt manifold, E(N1 , G, π, Ψ) and ′ E(N2 , G, π, Ψ) are principal Hpt bundles with a structure group G satisfying conditions of §2 in Section 2. Then η induces the group homomorphism η∗ from (W M E; N1 , G, P)t,H onto (W M E; N2 , G, P)t,H .

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Proof. In view of Proposition 7(1) the wrap group (W M E; N1 , G, P)t,H is the principal Gk bundle over (W M N1 )t,H . Extend η to ϑ : E(N1 , G, π, Ψ) → E(N2 , G, π, Ψ) such that π2 ◦ ϑ = η ◦ π1 and pr2 ◦ ϑ = id : G → G, where pr2 : Ey → G is the projection, y ∈ N1 . If f ∈ Hpt (M, N1 ), then η ◦ f := η(f (∗)) ∈ Hpt (M, N2 ). If f (s0,q ) = y0 , then η(f (s0,q )) = y0 , since y0 ∈ N2 . Since N2 ⊂ N1 , then Hpt (M, N2 ) ⊂ Hpt (M, N1 ). The parallel transport structure P is over the same manifold M . ˆ → N1 . In view of Theorems Put η∗ (< Pγˆ,u >t,H ) =< Pη◦ˆγ ,u >t,H , where γˆ : M ∗ 2.3 and 2.6 in Section 2 η∗ (< Pγˆ1 ,u ∨ Pγˆ2 ,u >t,H = η (< Pγˆ1 ,u >t,H )η∗ (< Pγˆ2 ,u >t,H ), and we can put η∗ (q −1 ) = [η∗ (q)]−1 , consequently, η∗ is the group homomorphism. Moreover, for each g ∈ (W M E; N2 , G, P)t,H there exists q ∈ (W M E; N1 , G, P)t,H such that η∗ (q) = g, since γ : M → N2 and N2 ⊂ N1 imply γ : M → N1 , while the structure group G is the same, hence η∗ is the epimorphism. 18. Definition. Let G be a topological group satisfying Conditions 15(A1, A2, C1, C2) ˆ where 1 ≤ r ≤ 2. Then define the such that G is a multiplicative group of the ring G, s ˆ s := G ˆ ⊗l G, ˆ where smashed product G such that it is a multiplicative group of the ring G ˆ ⊗l G ˆ is l = i2r denotes the doubling generator, the multiplication in G ∗ ∗ ˆ where v ∗ = (1) (a + bl)(c + vl) = (ac − v b) + (va + bc )l for each a, b, c, v ∈ G, conj(v). A smashed product M1 ⊗l M2 of manifolds M1 , M2 over Ar with dim(M1 ) = dim(M2 ) is defined to be an Ar+1 manifold with local coordinates z = (x, yl), where x in M1 and y in M2 are local coordinates. Its existence and detailed description are demonstrated below. ˆ s has a multiplicative group Gs containing all a + bl 6= 0 19. Proposition. The ring G ˆ If G ˆ is a topological or Hpt differentiable ring over Ar for t ≥ dim(G) + 1, with a, b ∈ G. ˆ s is a topological or Hpt differentiable over Ar+1 ring. then G Proof. For each 1 ≤ r ≤ 2 the group G is associative, since the generators ˆ s is non{i0 , ..., i2r −1 } form the associative group, when r ≤ 2. An element a + bl ∈ G zero if and only if (a + bl)(a + bl)∗ = aa∗ + bb∗ 6= 0 due to 15(A1, A2, C1, C2) and 18(1). For a + bl 6= 0 put u = (a∗ − lb∗ )/(aa∗ + bb∗ ), where aa∗ + bb∗ ∈ G0 , hence u(a + bl) = (a + bl)u = 1 ∈ G0 , since Gj is commutative for each j = 0, ..., 2r − 1, ˆ j . For r ≤ 2 the family of genwhere Gj denotes the multiplicative group of the ring G ˆs = G ˆ 0 i0 ⊕ ... ⊕ G ˆ 2r+1 −1 is erators {i0 , ..., i2r+1 −1 } forms the alternative group, hence G ˆ j are isomorphic with G ˆ 0 for each j. alternative, where G ˆ If an addition in G is continuous, then evidently (a + bl) + (c + ql) = (a + c) + (b + q)l ˆ is continuous, then Formula 18(1) shows that the is continuous. If the multiplication in G s ˆ multiplication in G is continuous as well. ˆ is Hpt differentiable, then from We have the decomposition Ar+1 = Ar ⊕ Ar l. If G ˆ s is Hpt differentiable over Ar+1 (see also in details the definition of plots it follows, that G 20(1 − 5)). 20. Theorem. Let M1 , M2 and N1 , N2 be Hpt manifolds over Ar with 1 ≤ r ≤ 2, and let G be a group satisfying Conditions 15(A1, A2, C1, C2), let also M1 ⊗l M2 , N1 ⊗l N2 be smashed products of manifolds and Gs be a smashed product group (see Proposition 19), where dim(M1 ) = dim(M2 ), dim(N1 ) = dim(N2 ), t ≥ max(dim(M1 ), dim(N1 ), dim(G)) + 1. Then the wrap group (W M1 ⊗l M2 ;{s0,j,1 ⊗l s0,v,2 :j=1,...,k1 ;v=1,...,k2 } E; N1 ⊗l N2 , Gs , Ps )t,H is twisted over

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{i0 , ..., i2r+1 −1 } and is isomorphic with the smashed product W M2 ;{s0,v,2 :v=1,...,k2 } E; N1 , (W M1 ;{s0,j,1 :j=1,...,k1 } E; N1 , G, P1 )t,H , P2 )t,H ⊗l W M2 ;{s0,v,2 :v=1,...,k2 } E; N2 , (W M1 ;{s0,j,1 :j=1,...,k1 } E; N2 , G, P1 )t,H , P2 )t,H of twice iterated wrap groups twisted over {i0 , ..., i2r −1 }. Proof. Let Mb and Nb be Hpt manifolds over Ar with 1 ≤ r ≤ 2, b = 1, 2 and let G be a group satisfying Conditions 15(A1, A2, C1, C2) such that E(Nb , G, π, Ψ) is a principal Gbundle. Consider the smashed products M1 ⊗l M2 , N1 ⊗l N2 of manifolds and the smashed product group Gs (see Proposition 19), where t ≥ max(dim(M1 ), dim(N1 ), dim(G)) + 1, where dim(Mb ) is a covering dimension of Mb (see [12]), dim(M1 ) = dim(M2 ), dim(N1 ) = dim(N2 ). For At(Mb ) = {(Uj,b , φj,b ) : j} an atlas of Mb its connecting t mappings φj,b ◦ φ−1 k,b are Hp functions over Ar for Uj,b ∩ Uk,b 6= ∅, where φj,b : Uj,b → Ar are homeomorphisms of Uj,b onto φj,b (Uj,b ). Then M1 ⊗l M2 consists of all points (x, yl) with x ∈ M1 and y ∈ M2 , with the atlas At(M1 ⊗l M2 ) = {(Uj,1 ⊗l Uq,2 , φj,1 ⊗l φq,2 ) : j, q} such that φj,1 ⊗l φq,2 : Uj,1 ⊗l Uq,2 → Am r+1 , where m is a dimension of M1 over Ar . Express for z = x + yl ∈ Ar with x, y ∈ Ar numbers x, y in the z representation, then denote by θj,q mappings corresponding to φj,1 ⊗l φq,2 in the z representation, hence the −1 transition mappings θj,q ◦ θk,n are Hpt over Ar+1 , when (Uj,1 ⊗l Uq,2 ) ∩ (Uk,1 ⊗l Un,2 ) 6= ∅. Therefore, M1 ⊗l M2 and N1 ⊗l N2 are Hpt manifolds over Ar+1 . In view of the Sobolev embedding theorem each H t mapping on M1 ⊗l M2 or N1 ⊗l N2 or Gs is continuous for t satisfying the inequality t ≥ max(dim(M1 ), dim(N1 ), dim(G)) + 1, where dim(M1 ) = dim(M2 ), dim(N1 ) = dim(N2 ). Each locally analytic function f (x, y) = f1 (x, y) + f2 (x, y)l by x ∈ U and y ∈ V can be written as the locally analytic function by z = x + yl with values in Ar+1 , where U w and V are open in Am r , fb (x, y) is a locally analytic function with values in Ar , b = 1, 2, m, w ∈ N. Indeed, write each variable xj and yj through zj with the help of generators of m Ar+1 , where xj , yj ∈ Ar , zj ∈ Ar+1 , x = (x1 , ..., xm ) ∈ Am r , z = (z1 , ..., zm ) ∈ Ar+1 (see Formulas 2.8(2) and Theorem 2.16 [29]). If z ∈ Ar+1 , then (1) z = v0 i0 + ... + v2r+1 −1 i2r+1 −1 , where vj ∈ R for each j = 0, ..., 2r+1 − 1, P2r+1 −1 ij (zi∗j )})/2, (2) v0 = (z + (2r+1 − 2)−1 {−z + j=1 P

r+1

2 −1 ij (zi∗j )}−zij )/2 for each s = 1, ..., 2r+1 −1, (3) vs = (is (2r+1 −2)−1 {−z + j=1 ∗ where z = z˜ denotes the conjugated Cayley-Dickson number z. At the same time we have for z = x + yl with x, y ∈ Ar , that (4) x = v0 i0 + ... + v2r −1 i2r −1 and (5) y = (v2r i2r + ... + v2r+1 −1 i2r+1 −1 )l∗ , where l = i2r denotes the doubling generator. Therefore, f (x, y) becomes Ar+1 holomorphic using the corresponding phrases arising canonically from expressions of xj , yj through zj by Formulas (1 − 5). The set of holomorphic functions is dense in Hpt in accordance with the definition of this space, hence using a Cauchy net we can consider for each f1 , f2 ∈ Hpt over Ar a representation of a function f = f1 + f2 l belonging to Hpt over Ar+1 (see also [29, 26]). Then E(N1 ⊗l N2 , Gs , π s , Ψs ) is naturally isomorphic with E(N1 , G, π1 , Ψ1 ) ⊗l E(N2 , G, π2 , Ψ2 ), where π s = π1 ⊗ π2 l : E(N1 ⊗l N2 , Gs , π s , Ψs ) → N1 ⊗l N2 is the natural projection.

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If γ : M1 ⊗l M2 → N1 ⊗l N2 is an Hpt mapping, then γ(z) = γ1 (x, y) × γ2 (x, y)l, where x ∈ M1 and y ∈ M2 , z = (x, yl) ∈ M1 ⊗l M2 , γb : M1 ⊗l M2 → Nb . We can write γb (x, y) as (γb,1 (x))(y) a family of functions by x and a parameter y or as (γb,2 (y))(x) a family of functions by y with a parameter x. If ηb,a : Ma → Nb , then Pηˆb,a ,ub ,a denotes the parallel transport structure on Ma over E(Nb , G, πb , Ψb ). Then Psγˆ,u (z) = [Pγˆ1,1 ,u1 ;1 (x)][Pγˆ1,2 ,u1 ;2 (y)] ⊗l [Pγˆ2,1 ,u2 ;2 (x)][Pγˆ2,2 ,u2 ;2 (y)] ∈ Ey0 (N1 ⊗l N2 , Gs , π s , Ψs ) is the parallel transport structure in M1 ⊗l M2 induced by that of in M1 and M2 , where u ∈ Ey0 (N1 ⊗l N2 , Gs , π s , Ψs ), u = u1 ⊗l u2 , ub ∈ Ey0,b (Nb , G, πb , Ψb ), y0,b ∈ Nb is a marked point, b = 1, 2, y0 = y0,1 ⊗l y0,2 . Then Ps is Gs equivariant. Therefore, < Psγˆ,u >t,H =< Pγˆ1 ,u1 >t,H ⊗l < Pγˆ2 ,u2 >t,H =< [Pγˆ1,1 ,u1 ;1 (x)][Pγˆ1,2 ,u1 ;2 (y)] >t,H ⊗l < [Pγˆ2,1 ,u2 ;2 (x)][Pγˆ2,2 ,u2 ;2 (y)] >t,H , where Pγˆb ,ub is the parallel transport structure on M1 ⊗l M2 over E(Nb , G, πb , Ψb ), b = 1, 2. Hence (W M1 ⊗l M2 ;{s0,j,1 ⊗l s0,v,2 :j=1,...,k1 ;v=1,...,k2 } E; N1 ⊗l N2 , Gs , Ps )t,H is isomorphic with the smashed product W M2 ;{s0,v,2 :v=1,...,k2 } E; N1 , (W M1 ;{s0,j,1 :j=1,...,k1 } E; N1 , G, P1 )t,H , P2 )t,H ⊗l W M2 ;{s0,v,2 :v=1,...,k2 } E; N2 , (W M1 ;{s0,j,1 :j=1,...,k1 } E; N2 , G, P1 )t,H , P2 )t,H of iterated wrap groups. 21. Theorem. There exists a homomorphism of iterated wrap groups θ : (W M E)a;∞,H ⊗ (W M E)b;∞,H → (W M E)a+b;∞,H for each a, b ∈ N, where G is an Hp∞ group, E(N, G, π, Ψ) is the principal Hp∞ bundle with the structure group G. Moreover, if G is either associative or alternative, then θ is either associative or alternative. Proof. Consider iterated wrap groups (W M E)a;∞,H as in §4, a ∈ N. If γa : M a → N , γb : M b → N are Hp∞ mappings such that γb (s0,j1 × ... × s0,jb ) = y0 for each jl = 1, ..., k and l = 1, ..., b, then γ := γa × γb : M a × M b → N × N = N 2 , where M a × M b = M a+b , s0,j are marked points in M with j = 1, ..., k and y0 is a marked point T in N , Hp∞ = t∈N Hpt . This gives the iterated parallel transport structure Pγˆ,u;a+b (x) := Pγˆa ,ua ;a (xa ) ⊗ Pγˆb ,u;b (xb ) on M a+b over E(N 2 , G2 , π, Ψ), where ub ∈ Ey0 (N, G, π, Ψ), u = ua × ub ∈ Ey0 ×y0 (N 2 , G2 , π, Ψ). The bunch M b ∨M b is taken by points sj1 ,...,jb in M b , where sj1 ,...,jb := s0,j1 ×...×s0,jb with j1 , ..., jb ∈ {1, ..., k}; s0,j are marked points in M with j = 1, ..., k. Then (M a ∨ M a ) × (M b ∨ M b ) \ {sj1 ,...,ja+b : jl = 1, ..., k; l = 1, ..., a + b} is Hpt homeomorphic with M a+b ∨ M a+b \ {sj1 ,...,ja+b : jl = 1, ..., k; l = 1, ..., a + b}, since sj1 ,...,ja × sja+1 ,...,ja+b = sj1 ,...,ja+b for each j1 , ..., ja+b . There is the embedding Dif Hp∞ (M a ) × Dif Hp∞ (M b ) ֒→ Dif Hp∞ (M a+b ) for each a, b ∈ N. If fa ∈ Dif Hp∞ (M a ) having a restriction fa |Ka = id, then fa × fb ∈ Dif Hp∞ (M a+b ) and fa × fb |Ka ×Kb = id for Ka ⊂ M a . Put θ(< Pγˆa ,ua ;a >∞,H;a , < Pγˆb ,ub ;b >∞,H;b ) =<< Pγˆa ,ua ;a >∞,H;a ⊗ < Pγˆb ,ub ;b >∞,H;b >∞,H;a+b is the group homomorphism, where the detailed notation < ∗ >t,H;a denotes the equivalence class over the manifold M a instead of M , a ∈ N. Therefore, < Pγˆ∨ˆη,u;a+b >∞,H;a+b :=<< Pγˆa ∨ˆηa ,ua ;a >∞,H;a ⊗ < Pγˆb ∨ˆηb ,ub ;b >∞,H;b >∞,H;a+b =< (< Pγˆa ,ua ;a >∞,H;a < Pηˆa ,ua ;a >∞,H;a ) ⊗ (< Pγˆb ,ub ;b >∞,H;b < Pηˆb ,ub ;b >∞,H;b ) >∞,H;a+b =< (< Pγˆa ,ua ;a >∞,H;a ⊗ < Pγˆb ,ub ;b >∞,H;b )(< Pηˆa ,ua ;a >∞,H;a ⊗ < Pηˆb ,ub ;b >∞,H;b ) >∞,H;a+b

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=<< Pγˆa ,ua ;a >∞,H;a ⊗ < Pγˆb ,ub ;b >∞,H;b >∞,H;a+b << Pηˆa ,ua ;a >∞,H;a ⊗ < Pηˆb ,ub ;b >∞,H;b >∞,H;a+b = θ(< Pγˆa ,ua ;a >∞,H;a , < Pγˆb ,ub ;b >∞,H;b )θ(< Pηˆa ,ua ;a >∞,H;a , < Pηˆb ,ub ;b >∞,H;b ). Thus θ is the group homomorphism. The mapping Hp∞ (M a , N ) × Hp∞ (M b , N ) ∋ (γa × γb ) 7→ (γa , γb ) ∈ Hp∞ (M a+b , N 2 ) is of Hp∞ class. The multiplication in Gv is Hp∞ for each v ∈ N, since it is such in G, since the multiplication in Gv is (a1 , ..., av ) × (b1 , ..., bv ) = (a1 b1 , ..., av bv ), where Gv is the v times direct product of G, a1 , ..., av , b1 , ..., bv ∈ G. The iterated wrap group (W M E)l;t,H for the bundle E is the principal Gkl bundle over the iterated commutative wrap group (W M N )l;t,H for the manifold N , since the number of marked points in M l is kl, where E is the principal G bundle on the manifold N , l ∈ N. Thus the iterated wrap group is associative or alternative if such is G. In view of Proposition 7 and Remark 4 the homomorphism θ is of Hp∞ class. From the wrap monoids it has the natural Hp∞ extension on wrap groups. If G is associative, then < Pγˆ,u;a+b+v >∞,H;a+b+v =<< (< Pγˆa ,ua ;a >∞,H;a ⊗ < Pγˆb ,ub ;b >∞,H;b ) >∞,H;a+b ⊗ < Pγˆv ,uv ;v >∞,H;v >∞,H;a+b+v =<< Pγˆa ,ua ;a >∞,H;a ⊗(< Pγˆb ,ub ;b >∞,H;b ⊗ < Pγˆv ,uv ;v >∞,H;v ) >∞,H;a+b+v = θ(θ(< Pγˆa ,ua ;a >t,H;a , < Pγˆb ,ub ;b >t,H;b ), < Pγˆv ,uv ;v >t,H;v ) θ(< Pγˆa ,ua ;a >t,H;a , θ(< Pγˆb ,ub ;b >t,H;b ), < Pγˆv ,uv ;v >t,H;v )), consequently, θ is the associative homomorphism. If G is alternative, then < Pγˆ,u;a+a+b >∞,H;a+a+b =<< (< Pγˆa ,ua ;a >∞,H;a ⊗ < Pγˆa ,ua ;a >∞,H;a ) >∞,H;a+a ⊗ < Pγˆb ,ub ;b >∞,H;v >∞,H;a+a+b =<< Pγˆa ,ua ;a >∞,H;a ⊗(< Pγˆa ,ua ;a >∞,H;a ⊗ < Pγˆb ,ub ;b >∞,H;b ) >∞,H;a+a+b = θ(θ(< Pγˆa ,ua ;a >t,H;a , < Pγˆa ,ua ;a >t,H;a ), < Pγˆb ,ub ;b >t,H;b ) θ(< Pγˆa ,ua ;a >t,H;a , θ(< Pγˆa ,ua ;a >t,H;a ), < Pγˆb ,ub ;b >t,H;b )), consequently, the homomorphism θ is alternative from the left, analogously it is alternative from the right.

4.

Cohomologies of Wrap Groups

1. Remarks and Definitions. Consider a triangulated compact polyhedron M may be embedded into Anr and its sub-polyhedron SM of codimension not less than two, codim(SM ) ≥ 2, where M \ SM is a C ∞ smooth manifold such that M \ SM is dense in M . If the covering dimension (see Chapter 7 [12]) of M \ SM is dim(M \ SM ) = b, then by the definition M is of dimension b. Then SM is called the singularity of M . A pseudo-manifold M is oriented, if M \ SM is oriented (see also §1.3.1 in Section 2). If M \ SM is without boundary, then the triangulated pseudo-manifold M is called a pseudo-manifold cycle. If (Y, ∂Y ) is the pair consisting of a triangulated pseudo-manifold Y and a boundary ∂Y , such that Y \ SY is a manifold with boundary ∂Y \ SY , ∂Y is a pseudo-manifold cycle with singularity SY ∩ ∂Y , then (Y, ∂Y ) is called the triangulated pseudo-manifold with boundary. A pre-sheaf F on a topological space X is a contra-variant functor F from the category of open subsets in X and their inclusions into a category of groups or rings (all either

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alternative or associative) such that F (U ) is a group or a ring for each U open in X and for each U ⊂ V with U and V open in X there exists a homomorphism sU,V : F (V ) → F (U ) such that sU,U = 1 and sU,V sV,Y = sU,Y for each U ⊂ V ⊂ Y with U, V, Y open in X. Let Fx denotes the family of all elements f ∈ F (U ) for all U open in X with x ∈ U . Elements f ∈ F (U ) and g ∈ F (V ) are called equivalent if there exists an open neighborhood Y of x such that sY,U (f ) = sY,V (g). This generates an equivalence relation and a class of all equivalent elements with f is called a germ fx of f at x. A set Fx of all germs of the pre-sheaf F at a point x ∈ X is the inductive limit Fx = ind − lim F (U ) taken by all open neighborhoods U of x in X. In the set F of all germs Fx take a base of topology consisting of all sets {fx ∈ Fx : x ∈ U }, where f ∈ F (U ). This induces a sheaf S generated by a pre-sheaf F . A sheaf of groups or rings (all either alternative or associative) on X is a pair (S, h) satisfying Conditions (S1 − S4): (S1) S is a topological space; (S2) h : S → X is a local homeomorphism; (S3) for each x ∈ X the set Fx = h−1 (x) is a group called a fiber of the sheaf S at a point x; (S4) the group or the ring operations are continuous, that is, S∆S ∋ (a, b) 7→ ab−1 ∈ S or S∆S ∋ (a, b) 7→ ab ∈ S and S∆S ∋ (a, b) 7→ a + b ∈ S are continuous respectively, where S∆S := {(a, b) : a, b ∈ S, h(a) = h(b)}. We can consider pre-sheafs and sheafs of different classes of smoothness, for example, t H or Hpt , when the corresponding defining sheaf and pre-sheaf mappings sU,V , h and group operations are such and S and F are H t or Hpt differentiable spaces respectively (see also §1.3.2 in Section 2). Consider a sheaf SN,G generated by a pre-sheaf U 7→ {f ∈ Homtp ((W M E)t,H , G) : supp(f ) ⊂ U }, where U is open in N and supp(f ) ⊂ U means that there exists y ∈ N ˆ , N ) with ηˆ(ˆ and ηˆ ∈ Hpt (M s0,q ) = x for each q = 1, ..., k and ηˆ(ˆ s0,q ) = y for each ˆ , {ˆ q = k + 1, ..., 2k and γˆ ∈ Hpt (M s0,q : q = 1, ..., 2k}; N, y) such that γˆ = γ ◦ Ξ and f =< ηˆ ∨ γˆ >t,H , where the wrap group (W M E)t,H is taken for a marked point y ∈ N , ˆ → M is the quotient mapping as in Section 2. Ξ:M In particular, we can take G = A∗r , and call SN,A∗r the sheaf of infinitesimal holonomies, where 1 ≤ r ≤ 3. In view of Property (P 4) in Section 2 for each non-singular points y ∈ N and u ∈ Ey in the fiber Ey of E over y there exists an Ar vector subspace Hu of the tangent bundle Tu E at u called a horizontal subspace of Tu E such that π∗ |Hu : Hu → Ty N is an isomorphism, where π(u) = y, t′ ≥ [dim(E)/2] + 2 or t′ = ∞, since there exist generalized derivatives ′ in the Sobolev space H t (see §III.3 [35]). This is the case for all y ∈ N and u ∈ Ey when ′ ′ N and E are of class H t instead of Hpt . Due to (P 1) the family {Hu } of horizontal subspaces of T E depends smoothly on u. Suppose that Y is a vector field in T E corresponding to a vector field X in T N such that π∗ (Y ) = X, then (CD1) Tu E = Hu ⊕ Vu , where Vu = π∗−1 (0) ⊂ Tu E is the space of vectors tangent to Eu at u. In accordance with (P 3) the horizontal spaces are G-equivariant, that is, (CD2) Huz = (Rz )∗ Hu , where Rz is the diffeomorphism of E given by the multiplication on z from the right and (Rz )∗ corresponds to the tangent mapping T Rz for the

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tangent fiber bundle T E. A family H = {Hu ⊂ Tu E : u ∈ E, π(u) = y ∈ N } is called the connection distribution of the principal fiber bundle E(N, G, π, Ψ), if Hu depends smoothly on u and the Conditions (CD1, CD2) are satisfied. 2. Definitions and Notes. Two smooth principal G fiber bundles E and E ′ with connection distributions (E, H) and (E ′ , H ′ ) are called isomorphic if there exists an isomorphism f : E → E ′ of smooth principal G fiber bundles f : E → E ′ such that f∗ (H) = H ′ . A connection distribution H on E determines a parallel transport structure PH on E t ˆ H (ˆ H = γ posing PH ˆ γ ˆ ,u ∈ Hp (M , E) with Pγ ˆ ,u s0,q ) = u for each q = 1, ..., k and π ◦ Pγ ˆ ,u H H H t ˆ ˆ such that Tx Pγˆ,u =: (Pγˆ,u (x), DPγˆ,u (x)) for each x ∈ M , where γˆ ∈ Hp (M , {ˆ s0,q : q = (x) ∈ H , T P is the tangent mapping of P (see [22]). 1, ..., 2k}; N, y0 ), DPH γ ˆ (x) γ ˆ ,u Thus there exists a bijective correspondence between parallel transport structures and connection distributions on E. Therefore, the mapping H 7→ PH induces a bijective correspondence between isomorphism classes of parallel transport structures and connection distributions. Using the exponential function on Ar gives exp(Ar ) = A∗r for 1 ≤ r ≤ 3 (see §3 [28, 29]). If E is a principal A∗r fiber bundle with 1 ≤ r ≤ 3, then for each v ∈ Vu there exists a unique z(v) ∈ Ar such that v = [d(y exp(b z(v))/db]|b=0 , where b ∈ R. Therefore, for each connection distribution {Hu : u ∈ E} on E a differential 1-form w over Ar exists such that w(Xh + Xv ) = z(Xv ) for each X = Xh + Xv ∈ Hu ⊕ Vu = Tu E and w is G-equivariant: (Rz )∗ w = w due to the G-equivariance of {Hu : u ∈ E}, here G = A∗r . A differential 1-form w on E so that it is G-equvariant and w(Xv ) = z(Xv ) for each Xv ∈ Vu is called a connection 1-form. Two smooth principal G fiber bundles with connections (E, w) and (E ′ , w′ ) are called isomorphic, if there exists an isomorphism f : E → E ′ of smooth principal G fiber bundles such that f ∗ (w′ ) = w. For w there exists a connection distribution H w on E for which Huw = ker(wu ) ⊂ Tu E, that induces a bijective correspondence between differential 1-forms and connection distributions on E. Hence w 7→ H w produces a bijective correspondence between isomorphism classes of connections and connection distributions. Then there exists a wrap group (W M E; N, A∗r , ∇)t,H , where a parallel transport structure P is associated with the covariant differentiation ∇ of the connection w. The curvature 2-form Ω over Ar , 1 ≤ r ≤ 3, of a connection 1-form w on a smooth principal fiber bundle E(N, G, π, Ψ) over Ar is given by Ω(X, Y ) = dw(hX, hY ), where hX and hY are horizontal components of the vectors X and Y . 3. Remark. If η ∈ Hpt (K, E), t ≥ 1, and ν is a differential form on E, then there exists its pull-back η ∗ ν which is a differential form on K, where K is an Hpt -pseudo-manifold. For orientable K and E and an Hpt diffeomorphism η of K onto E and ν with compact R R support K η ∗ ν = ǫ E ν, where ǫ = 1 if η preserves an orientation, ǫ = −1 if η changes an orientation (see [7, 53, 26]). In particular, K = E(M, G, πM , ΨM ) can be considered, η = (η0 , η1 ), η0 : M → N , η : E(M ) → E(N ), πN ◦ η = η0 ◦ πM , η1 ◦ pr2 = pr2 ◦ η, pr2 is a projection in charts of E from E into G, η1 = id may be as well. Suppose that M and E are an Ar holomorphic manifold and principal fiber bundle, such that E is orientable and 2r − 1-connected, which is not very restrictive due to Propositions

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ˆ , {ˆ 13 and Note 14 Section 3. If γˆ ∈ Hpt (M s0,q : q = 1, ..., k}; N, y0 ), then consider a ˆ . Therefore, path lq joining the point sˆ0,q with sˆ0,q+k , where 1 ≤ q ≤ k, ˆlq : [0, 1] → M ∗ ∗ ˆ pˆq := γˆ ◦ lq : [0, 1] → N and pˆq w := (ˆ pq , id) w is a differential form on [0, 1], where w is an Ar holomorphic connection one-form on E. We get that γˆ ∗ w is a differential one-form ˆ and there exists its restriction νγ,q := γ ∗ w|ˆ on M lq [0,1] . Then we have also γ ∈ Hpt (M, {s0,q : q = 1, ..., k}; N, y0 ) and lq : S 1 → M and pq : S 1 → N respectively, where γˆ = γ ◦ Ξ (see Section 2), S 1 is the unit circle in C with the center at zero, while C = CM is embedded into Ar as R ⊕ M R with M ∈ Ar , Re(M ) = 0, |M | = 1, when 2 ≤ r ≤ 3. R Since w is Ar holomorphic, then φ w does not depend on a rectifiable curve φ but only on the initial and final points φ(0) and φ(1), φ : [0, 1] → E (see Theorems 2.15 and 3.10 in [28, 29] and [31]). Consider now the principal fiber bundle E with the structure group A∗r , where 1 ≤ r ≤ 3. Then the pull-back p∗q E of the bundle E is a trivial A∗r -bundle over S 1 . The latter bundle carries a pull-back connection differential one-form p∗q w. Take the pull-back ρ∗ (p∗q w) one form, where ρ : S 1 → p∗q E is a trivialization of the fiber bundle p∗q E → S1 . ˆ induces the parallel The parallel transport structure Pγˆ,u (x) for (M, E) with x ∈ M 1 ∗ transport structures Ppˆq ,u∗ (s) for (S , pq E) with s ∈ [0, 1] for each q = 1, ..., k, where pq (u∗ ) = u. Then the holonomy along γ is given by R (H) h(γ) = (h1 , ..., hk ) ∈ Gk with hq = hq (γ) = exp[− S 1 ρ∗ (p∗q w)] for each q = 1, ..., k. If ζ : S 1 → p∗q E is another trivialization and f : S 1 → C∗M satisfies ζ = fR ρ, so that f (v) = exp(M 2πθ(v)), where θ(v) ∈ R, M 2πdθ(v) = dLn(f (v)), v ∈ S 1 , S 1 dθ is an integer number, since R is the center of the algebra Ar , where Ln is the natural logarithmic function over Ar (see §3.7 and Theorem 3.8.3 [29] and [28, 33]). Therefore, Formula (H) is independent of a trivialization ρ, since ζ ∗ (p∗q w) = ρ∗ (p∗q w) + dLn(f ), but R exp[ S 1 dLn(f )] = 1. 4. Non-associative bar construction. Let G be a topological group not necessarily associative, but alternative: (A1) g(gf ) = (gg)f and (f g)g = f (gg) and g −1 (gf ) = f and (f g)g −1 = f for each f, g ∈ G and having a conjugation operation which is a continuous automorphism of G such that (C1) conj(gf ) = conj(f )conj(g) for each g, f ∈ G, (C2) conj(e) = e for the unit element e in G. If G is of definite class of smoothness, for example, Hpt differentiable, then conj is supposed to be of the same class. For commutative group in particular it can be taken the identity mapping as the conjugation. For G = A∗r it can be taken conj(z) = z˜ the usual conjugation for each z ∈ A∗r , where 1 ≤ r ≤ 3. Denote by ∆n := {(x0 , ..., xn ) ∈ Rn+1 : xj ≥ 0, x0 + x1 + ... + xn = 1} the standard S simplex in Rn+1 . Consider (AG)n as the quotient of the disjoint union nk=0 (∆k × Gk+1 ) by the equivalence relations (1) (x0 , ..., xk , g0 , ..., gk ) ∼ (x0 , ..., xj + xj+1 , ..., xk , g0 , ..., gˆj , ..., gk ) for gj = gj+1 or xj = 0 with 0 ≤ j < k; (x0 , ..., xk , g0 , ..., gk ) ∼ (x0 , ..., xk−1 + xk , g0 , ..., gk−1 ) for gk−1 = gk or xk = 0.

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Consider non-homogeneous coordinates 0 ≤ t1 ≤ t2 ≤ ... ≤ tk ≤ 1 on the simplex related with the barycentric coordinates by the formula tj = x0 + x1 + ... + xj−1 and −1 h0 := g0 , hj = gj−1 gj for j > 0 on Gk+1 . Hence h0 h1 = g0 (g0−1 g1 ) = g1 , (h0 h1 )h2 = −1 g1 (g1−1 g2 ) = g2 and by induction ((...(h0 h1 )...)hk−1 )hk = gk−1 (gk−1 gk ) = gk . Then equivalence relations (1) take the form: (2) (t1 , ..., tk , h0 [h1 |...|hk ]) ∼ (t2 , ..., tk , (h0 h1 )[h2 |...|hk ]) for t1 = 0 or h0 = e; (t1 , ..., tk , h0 [h1 |...|hk ]) ∼ (t1 , ..., tˆj , ..., tk , h0 [h1 |...|hj hj+1 |...|hk ]) for tj = tj+1 or hj = e; (t1 , ..., tk , h0 [h1 |...|hk ]) ∼ (t1 , ..., tk−1 , h0 [h1 |...|hk−1 ]) for tk = 1 or hk = e. Denote by |x0 , ..., xk , g0 , ..., gk | the equivalence class of the sequence (x0 , ..., xk , g0 , ..., gk ); by |t1 , ..., tk , h0 [h1 |...|hk ]| denote the equivalence class of the sequence (t1 , ..., tk , h0 [h1 |...|hk ]). S k k+1 by the above equivalence relaThen the space AG is the quotient of ∞ k=0 ∆ × G k k+1 m m+1 tions (1), where (∆ × G ) ∩ (∆ × G ) is empty for k 6= m. n+1 Introduce in G the equivalence relation Y: (3) (g0 , ..., gn )Y(q0 , ..., qn ) if and only if there exist p1 , ..., pk ∈ G with k ∈ N such that gj = pk (pk−1 ...(p2 (p1 qj ))....) for each j = 0, ..., n. Evidently this relation is reflexive: (g0 , ..., gn )Y(g0 , ..., gn ) with p1 = e and k = 1. It is symmetric due to the alternativity of G, since gj = pk (pk−1 ...(p2 (p1 qj ))....) is equivalent −1 −1 −1 with qj = p−1 1 (p2 ...(pk−1 (pk gj ))...) for each j = 0, ..., n. This relation is transitive: (g0 , ..., gn )Y(q0 , ..., qn ) and (q0 , ..., qn )Y(f0 , ..., fn ) implies (g0 , ..., gn )Y(f0 , ..., fn ), since from gj = pk (pk−1 ...(p2 (p1 qj ))....) and qj = sl (sl−1 ...(s2 (s1 fj ))....) it follows gj = pk (pk−1 ...(p2 (p1 (sl (sl−1 ...(s2 (s1 fj ))....)))....) for each j = 0, ..., n, where k, l ∈ N, p1 , ..., pk , s1 , ..., sl ∈ G. In a particular case of an associative group G parameters k = 1 and l = 1 can be taken. S Consider in nk=0 ∆k × Gk the equivalence relations: (4) (x0 , ..., xk , [g0 : ... : gk ]) ∼ (x0 , ..., xj + xj+1 , ..., xk , [g0 : ... : gˆj : ... : gk ]) for gj = gj+1 or xj = 0 with 0 ≤ j < k; (x0 , ..., xk , g0 , ..., gk ) ∼ (x0 , ..., xk−1 + xk , [g0 : ... : gk−1 ]) for gk−1 = gk or xk = 0, where [g0 : ... : gk ] := {(q0 , ..., qk ) ∈ Gk+1 : (q0 , ..., qk )Y(g0 , ..., gk )} denotes the equivalence class of (g0 , ..., gk ) by the equivalence S relation Y. Put (BG)n to be the quotient of nk=0 ∆k × Gk by equivalence relations (4). Using the inhomogeneous coordinates on (BG)n rewrite the equivalence relation (4) in the form: (5) (t1 , ..., tk , [h1 |...|hk ]) ∼ (t2 , ..., tk , [h2 |...|hk ]) for t1 = 0 or h0 = e; (t1 , ..., tk , h0 [h1 |...|hk ]) ∼ (t1 , ..., tˆj , ..., tk , [h1 |...|hj hj+1 |...|hk ]) for tj = tj+1 or hj = e; (t1 , ..., tk , [h1 |...|hk ]) ∼ (t1 , ..., tk−1 , [h1 |...|hk−1 ]) for tk = 1 or hk = e. Denote by |x0 , ..., xk , [g0 : ... : gk ]| the equivalence class of the sequence (x0 , ..., xk , [g0 : ... : gk ]); by |t1 , ..., tk , [h1 |...|hk ]| denote the equivalence class of the seS k k quence (t1 , ..., tk , [h1 |...|hk ]). Then BG is the quotient of the disjoint union ∞ k=0 ∆ × G by the equivalence relations (4). A : AG → BG by the formula: Then there exists the projection πB A (6) πB : |x0 , ..., xk , g0 , ..., gk | 7→ |x0 , ..., xk , [g0 : ... : gk ]| or in the non-homogeneous A : |t , ..., t , h [h |...|h ]| 7→ |t , ..., t , [h |...|h ]|. coordinates by πB 1 1 1 k 0 1 k k k ∆k

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The conjugation in G induces that of in AG and BG such that: conj(t1 , ..., tk , h0 [h1 |...|hk ]) := (t1 , ..., tk , conj(h0 )[conj(h1 )|...|conj(hk )]) and conj(t1 , ..., tk , [h1 |...|hk ]) := (t1 , ..., tk , [conj(h1 )|...|conj(hk )]). Suppose that ˆ = G ˆ 0 i0 ⊕ G ˆ 1 i1 ⊕ ... ⊕ G ˆ 2r −1 i2r −1 such that G is a multiplicative group of (A2) G ˆ ˆ j \ {0}, G ˆ 0 , ..., G ˆ 2r −1 a ring G with the multiplicative group structure, where Gj = G are pairwise isomorphic commutative associative rings and {i0 , ..., i2r −1 } are generators of the Cayley-Dickson algebra Ar , 1 ≤ r ≤ 3 and (yl il )(ys is ) = (yl ys )(il is ) is the natural ˆ for yl ∈ G ˆ l . If a group G and a ring G ˆ satisfy multiplication of any pure states in G Conditions (A1, A2, C1, C2), then we call it a twisted group and a twisted ring over the set of generators {i0 , ..., i2r −1 }, where 1 ≤ r ≤ 3. The unit element of G denote by e. For ˆ = Anr . example, G = (A∗r )n and G ′ 5. Definitions. Let N. be a family {Nn : n ∈ N} of either C ∞ smooth or Hpt manifolds ′ together with either C ∞ or Hpt mappings ∂j : Nn → Nn−1 and sj : Nn → Nn+1 for each j = 0, 1, ..., n satisfying the identities: (1) ∂k ∂j = ∂j−1 ∂k for each k < j, (2) sk sj = sj+1 sk for each k ≤ j, (3) ∂k sj = sj−1 ∂k for k < j, ∂k sj = id|Nn for k = j, j + 1, ∂k sj = sj ∂k−1 for ′ k > j + 1, then N. is called a simplicial either C ∞ smooth or Hpt manifold. ` The geometric realization |N. | of N. consists of n≥0 ∆n ×Nn /E, where E is the equivalence relation generated by (∂ j x, y)E(x, ∂j y) for (x, y) ∈ ∆n−1 × Nn , (sj x, y)E(x, sj y) ` for (x, y) ∈ ∆n+1 ×Nn , where denotes the disjoint union of sets, the maps ∂ j : ∆n−1 → ∆n and sj : ∆n+1 → ∆n are such that ∂ j (x0 , ..., xn−1 ) = (x0 , ..., xj−1 , 0, xj , ..., xn−1 ) and sj (x0 , ..., xn+1 ) = (x0 , ..., xj−1 , xj + xj+1 , xj+2 , ..., xn+1 ) in barycentric coordinates. ′ A C ∞ or Hpt space structure on the geometric realization |N. | of N. consists of all ′ continuous C ∞ R-valued or Hpt Ar valued functions f on |N. | respectively, that is the `

`

q

f

composition n≥0 (∆n − ∂∆n ) × Nn ֒→ n≥0 ∆n × Nn −→ |N. | −→ Ar is either C ∞ or ′ Hpt , where q denotes the quotient mapping, r = 0 or 1 ≤ r ≤ 3 correspondingly, A0 = R, A1 = C, A2 = H, A3 = O. 6. Proposition. If a group G satisfies Conditions 4(A1, A2, C1, C2), then sets AG and BG can be supplied with group structures and they are twisted for 2 ≤ r ≤ 3. If G is a topological Hausdorff or Hpt differentiable alternative for r = 3 or associative for 0 ≤ r ≤ 2 group, then AG and BG are topological Hausdorff or C ∞ or Hpt differentiable alternative for r = 3 or associative for 0 ≤ r ≤ 2 groups respectively. Proof. Define on AG and BG group structures. Introduce a homeomorphism pairing: n ∆ × ∆k → ∆n+k , where σ is a permutation of the set {1, 2, ..., n + m + 1} such that tσ(1) ≤ tσ(2) ≤ ... ≤ tσ(n+k+1) , σ ∈ Sn+k+1 , Sm denotes the symmetric group of all permutations of the set {1, ..., m}. Define the multiplication for pure states in AG: (1) |t1 , ..., tn , h0 [h1 |...|hn ]| ∗ |tn+1 , ..., tn+k+1 , hn+k+2 [hn+1 |...|hn+k+1 ]| := |tσ(1) , ..., tσ(n+k+1) , (−1)q(σ) (h0 hn+k+2 )[hσ(1) |...|hσ(n+k+1) ]|, where hl = yl ij(l) , yl ∈ Gj(l) for each l = 0, ..., 2r − 1, q(σ) ∈ Z is such that (−1)q(σ) ij(0) (ij(1) ...(ij(n+k+1) ij(n+k+2) )...) = (ij(σ(0)) ij(σ(n+k+2)) )(ij(σ(1)) ... (ij(σ(n+k))) ij(σ(n+k+1)) )...) in Ar ; while in BG: (2) |t1 , ..., tn , [h1 |...|hn ]| ∗ |tn+1 , ..., tn+k+1 , [hn+1 |...|hn+k+1 ]| :=

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|tσ(1) , ..., tσ(n+k+1) , (−1)p(σ) [hσ(1) |...|hσ(n+k+1) ]|, where hl = yl ij(l) , yl ∈ Gj(l) , p(σ) ∈ Z is such that (−1)p(σ) ij(1) (ij(2) ...(ij(n+k) ij(n+k+1) )...) = ij(σ(1)) (ij(σ(2)) ...(ij(σ(n+k)) ij(σ(n+k+1)) )...) in Ar . ˆ Define also an addition in AG: (1′ ) |t1 , ..., tn , h0 [h1 |...|hn ]| + |tn+1 , ..., tn+k+1 , hn+k+2 [hn+1 |...|hn+k+1 ]| := |tσ(1) , ..., tσ(n+k+1) , (−1)q(σ) (h0 + hn+k+2 )[hσ(1) |...|hσ(n+k+1) ]| ˆ with j(0) = j(n + k + 2); as well as an addition in B G: (2′ ) |t1 , ..., tn , [h1 |...|hn ]| + |tn+1 , ..., tn+k+1 , [hn+1 , ..., hn+k+1 ]| = |tσ(1) , ..., tσ(n+k+1) , (−1)p(σ) [hσ(1) |...|hσ(n+k+1) ]| ˆ l for each l = 0, ..., 2r − 1. The multiplications (1, 2) for pure states hl = yl ij(l) , yl ∈ G extend to that of rings in the natural way, when some pure states are zero, hence due to the distributivity on the entire ring as well. ˆ is the ring, then these multiplications have unique extensions on AG and BG. Since G Verify, that AG and BG become groups with multiplications (1) and (2) respectively. Due to (1, 2) we get (3) v ∗ conj(v) = |t1 , ..., tk , (h0 conj(h0 ))[(h1 conj(h1 ))|...|(hk conj(hk ))]| for each v = |t1 , ..., tk , h0 [h1 |....|hk ]| in AG, while w ∗ conj(w) = |t1 , ..., tk , [(h1 conj(h1 ))|...|(hk conj(hk ))]| ˆ 0 for each h ∈ G, but G ˆ0 for each w = |t1 , ..., tk , [h1 |....|hk ]| in BG, where hconj(h) ∈ G ˆ ab = ba for each a ∈ G ˆ 0 and b ∈ G. ˆ The Moufang identities in is the center of the ring G: Ar for r = 3 (see [18]) induces that of in G such that (4) (xyx)z = x(y(xz)) and (x−1 yx)z = x−1 (y(xz)); (5) z(xyx) = ((zx)y)x and z(x−1 yx) = ((zx−1 )y)x; (6) (xy)(zx) = x(yz)x and (x−1 y)(zx) = x−1 (zy)x, since (7) x−1 = conj(x)(x conj(x))−1 , where (xconj(x)) ∈ G0 . The unit element in AG is e := {|t1 , ..., tk , e[e|...|e]| ∈ (AG)k : k = 0, 1, ...}, where i0 = 1, since |t1 , ..., tn , h0 [h1 |...|hn ]| ∗ |tn+1 , ..., tn+k+1 , e[e|...|e]| = |tn+1 , ..., tn+k+1 , e[e|...|e]| ∗ |t1 , ..., tn , h0 [h1 |...|hn ]| = |t1 , ..., tn , h0 [h1 |...|hn ]| due to equivalence relations 4(2), (1, ..., 1, e[e|...|e]) ∈ |t1 , ..., tk , e[e|...|e]|. ˆ is Z2 graded in the sense that elements yl jl ∈ G ˆ l jl are even for l = 0 The ring G r ˆ l il for each and odd for l = 1, ..., 2 − 1: (y0 i0 )(yl yl ) = (yl il )(y0 i0 ) = (y0 yl )il ∈ G r 2 2 ˆ ˆ s is 0 ≤ l ≤ 2 − 1, (yl il ) = −yl i0 ∈ G0 i0 , (yl il )(yk ik ) = −(yk ik )(yl il ) = (yl yk )is ∈ G r ˆ for 1 ≤ l 6= k ≤ 2 − 1, where is = il ik . For each pure states g0 , ..., gk ∈ G their product (...(g0 g1 )g2 ...)gk is a pure state, consequently, sets AG and BG are Z2 graded analogously ˆ having even and odd elements such that to G ˆ = (AG ˆ 0 )i0 ⊕ (AG ˆ 1 )i1 ⊕ ... ⊕ (AG ˆ 2r −1 )i2r −1 and (8) AG ˆ ˆ ˆ ˆ 2r −1 )i2r −1 . Each AGj and BGj is an (9) B G = (B G0 )i0 ⊕ (B G1 )i1 ⊕ ... ⊕ (B G t associative topological Hausdorff or Hp differentiable group isomorphic with AG0 or BG0 correspondingly for each j, since Gj are commutative and associative (see also Appendix

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ˆ 0 . Therefore, AG and B4 in [15]), where G0 denotes the multiplicative group of the ring G ˆ ˆ BG are the multiplicative groups of the rings AG and B G. If a ∈ AG0 or a ∈ BG0 , then ab = ba for each b ∈ AG or BG respectively. From Definition 6 it follows, that they are C ∞ or Hpt groups, when such is G. The inverse element is −1 −1 (10) {|t1 , ..., tk , h0 [h1 |...|hk ]| : k}−1 = {|t1 , ..., tk , h−1 0 [h1 |...|hk ]| : k} due to (2, 6), since −1 −1 −1 (h0 (h1 ...(hk−1 hk )...))((...(h−1 k hk−1 )...h1 )h0 ) = −1 −1 −1 −1 (...(((h0 h0 )(h1 h1 ))(h2 h2 )...)(hk hk ) = e for pure states for each k in view of the Moufang identities (4 − 6). In general it follows from (3, 8, 9), since v ∗conj(v) ∈ AG0 or BG0 for v ∈ AG or v ∈ BG respectively, hence v −1 = conj(v)(v ∗ conj(v))−1 = {|t1 , ..., tk , h0 [h1 |...|hk ]| : k}−1 . In view of (8, 9) and the existence of an inverse element we get, that AG is alternative, since 1 ≤ r ≤ 3. Putting h0 = 1 and applying the equivalence relation Y we get, that BG is also an alternative group, since the multiplicative group {i0 , ..., i7 } is alternative. If G is associative, for example, when 1 ≤ r ≤ 2, then AG and BG are associative, since the multiplicative group {i0 , i1 , i2 , i3 } is associative. Thus, groups AG and BG are Z2 graded, hence they are twisted over {i0 , ..., i2r −1 }. ˆ multiplication and addition operations, then they induce them for AG and Consider for G ˆ and B G ˆ are twisted rings. BG as above. It follows, that E G 7. Corollary. Let suppositions of Proposition 6 be satisfied, then AB m G and B m G are topological or C ∞ or Hpt differentiable groups respectively for each m ≥ 1. Moreover, all maps in the short exact sequence e → B a G → AB a G → B a+1 G → e are continuous or C ∞ or Hpt correspondingly. Proof. Define differentiable space structure by induction. Suppose that it is defined on a B G and ∆k ×(B a G)m for k, m ≥ 0, where a ≥ 1. Then f : AB a G → Ar is C ∞ or Hpt if `

q

f

′

A the composition n≥0 (∆n − ∂∆n ) × (B a G)n+1 −→ AB a G −→ Ar is either C ∞ or Hpt , ` qB while f : B a+1 G → Ar is C ∞ or Hpt if the composition n≥0 (∆n − ∂∆n ) × (B a G)n −→

f

′

B a+1 G −→ Ar is either C ∞ or Hpt , where 0 ≤ r ≤ 3. A function f : ∆k × (B a+1 G)m → Ar is C ∞ or Hpt if the composition ∆k × `

id×(qB )m

f

( n≥0 (∆n − ∂∆n ) × (B a G)n )m −→ ∆k × (B a+1 G)m −→ Ar is either C ∞ or ′ Hpt . From this it follows that all maps in the short exact sequences are of the same class of smoothness. Then the mappings B a G × B a G → B a G and AB a G × AB a G → AB a G of the form (f, g) 7→ f g −1 are C 0 or C ∞ or Hpt in respective cases due to Formulas 6(1 − 3, 8 − 10) (see also §1.3.2 in Section 2 and §1 and Appendix B in [15]). 8. Corollary. If a group G satisfies Conditions 4(A1, A2, C1, C2), then there exist Hpt groups AB a (W M,{s0,q :q=1,...,k} E)t,H and B a (W M,{s0,q :q=1,...,k} E)t,H for each a ∈ N. Proof. The wrap group (W M,{s0,q :q=1,...,k} E; N, G, P)t,H is a principal Gk bundle over M,{s M,{s0,q :q=1,...,k} N ) 0,q :q=1,..,k} N ) (W t,H , where (W t,H is commutative and associative (see Proposition 7(1) in Section 3). ˆ then g has the decomposition g = g0 i0 + ... + g2r −1 i2r −1 with gj ∈ G ˆ j for If g ∈ G, r each j = 0, 1, ..., 2 − 1 and

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(1) g0 = (g + (2r − 2)−1 {−g + s=1 is (gi∗s )})/2 and P r −1 (2) gj = (ij (2r − 2)−1 {−g + 2s=1 is (gi∗s )} − gij )/2 r for each j = 1, ..., 2 − 1. Therefore, each g0 , ..., g2r −1 has analytic expressions through g due to Formulas (1, 2). Fix this representations. Then the Hpt differentiable parallel transport structure P with the groups G induces the Hpt differentiable parallel transport structures j P with groups Gj . ˆ k is isomorphic with L ˆ ˆ Since G 0≤j(1),...,j(k)≤2r −1 (Gj(1) ij(1) , ..., Gj(k) ij(k) ) which is L k ˆ (ij(1) , ..., ij(k) ), then isomorphic with 0≤j(1),...,j(k)≤2r −1 G 0 M,{s :q=1,...,k} 0,q (W E; N, G, P)t,H is isomorphic with a group L {f = (f1 , ..., fk ) ∈ 0≤j(1),...,j(k)≤2r −1 [(W M,{s0,q :q=1,...,k} E; N, G0 , P)t,H ∪ {0}] (ij(1) , ..., ij(k) ) : f1 6= 0, ..., fk 6= 0}, where (ij(1) , ..., ij(k) ) ∈ (A∗r )k and (A∗r )k has the embedding into the family of all k × k matrices with entries in Ar as diagonal matrices, (W M,{s0,q :q=1,...,k} E; N, Gj , P)t,H is commutative for each j = 0, ..., 2r − 1 due to Theorem 2.6. The construction of Proposition 5 above has the natural generalization for Gk instead of G such that ˆk = L ˆ ˆ AG 0≤j(1),...,j(k)≤2r −1 (AGj(1) ij(1) , ..., AGj(k) ij(k) ) L k ˆ (ij(1) , ..., ij(k) ), also which is isomorphic with 0≤j(1),...,j(k)≤2r −1 AG 0 L k ˆ is isomorphic with ˆ k (ij(1) , ..., ij(k) ), consequently, BG B G r 0≤j(1),...,j(k)≤2 −1 0 A(W M,{s0,q :q=1,...,k} E; N, G, P)t,H is isomorphic with a group L M,{s0,q :q=1,...,k} E; N, G , P)k {v ∈ 0 0≤j(1),...,j(k)≤2r −1 [A(W t,H ∪ {0}](ij(1) , ..., ij(k) ) : vn = |t1 , ..., tn , h0 [h1 |...|hn ]|, hj 6= 0∀j, ∀n} and B(W M,{s0,q :q=1,...,k} E; N, G, P)t,H is isomorphic with L M,{s0,q :q=1,...,k} E; N, G , P)k {v ∈ 0 0≤j(1),...,j(k)≤2r −1 [B(W t,H ∪ {0}](ij(1) , ..., ij(k) ) : vn = |t1 , ..., tn , [h1 |...|hn ]|, hj 6= 0∀j, ∀n}. Continuing this by induction on a and using Corollary 7 we get the statement of this corollary for each a ∈ N. 9. Lemma. Let N be a C ∞ or Hpt manifold over Ar with 0 ≤ r ≤ 3 and G a C ∞ or Hpt differentiable group. If f : N → BG is a mapping such that for each y ∈ N there exists an open neighborhood V of y in N such that f |V = |f0 , f1 , ..., fn , [g1 |...|gn ]| with f0 , ..., fn being C ∞ or Hpt differentiable mappings, then f is either C ∞ or Hpt differentiable mapping correspondingly. Proof. If h : BG → Ar is a C ∞ or Hpt mapping, then for each n ≥ 1 the composition q

h

B ∆n × Gn −→ BG −→ Ar is of the corresponding class. For the commutative diagram f f¯ qB h consisting of N −→ BG −→ Ar and N −→ ∆n × Gn −→ BG and f = qb ◦ f¯, ¯ ¯ where f := (f0 , ..., fn , h1 , ..., hn ) both f and h ◦ qB are continuous C ∞ or Hpt . Then the composition h ◦ f = h ◦ qB ◦ f¯ is continuous and either C ∞ or H t , where as usually

p

h ◦ f (y) := h(f (y)). Thus f : N → BG is continuous either C ∞ or Hpt respectively. 10. Twisted bar resolution and hypercohomologies. For a twisted group G satisfying Conditions 4(A1, A2, C1, C2) the composition of the short exact sequences (1) e → B a G → AB a G → B a+1 G → e induces the long exact sequence σ σ σ σ σ (2) e → G → AG −→ ABG −→ AB 2 G −→ ... −→ AB a G −→ ..., where for each a ≥ 0 the homomoprhism σ : AB a G → AB a+1 G is the composition AB a G → B a+1 G → AB a+1 G of the surjection AB a G → B a+1 G and the monomor-

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phism B a+1 G → AB a+1 G. In view of Corollary 7 the short exact sequence (2) is a C ∞ or Hpt B a G-extension of a+1 B G. Hence the long exact sequence (2) induces the long exact sequence of twisted sheaves σ σ σ σ σ (3) e → GN → AGN −→ ABGN −→ AB 2 GN −→ ... −→ AB a GN −→ ..., which we will call the (twisted) bar resolution of the sheaf GN , where GN denotes the sheaf of C ∞ or Hpt functions on N with values in G. Suppose that S ∗ and F ∗ are complexes of sheaves of G-modules, where G is a sheaf of r rings, where S ∗ and F ∗ and G may be simultaneously twisted over {i0 , ..., i2 −1 }. Then a homomorphism mapping σ : S ∗ → F ∗ of such complexes induces a mapping of cohomology sheaves Hj (σ) : Hj (S ∗ ) → Hj (F ∗ ), where Hj (S ∗ ) is the sheaf associated with the pre-sheaf U 7→ [ker(Γ(U, F j )) → Γ(U, F j+1 ))]/im[(Γ(U, F j−1 )) → Γ(U, F j ))], where Γ(U, S j ) denotes the group of sections of the sheaf S j for a subset U open in X (see also §1). Then σ is called a quasi-isomorphism, if Hj (σ) is an isomorphism for each j. We consider complexes bounded below, that is there exists j0 such that S j = 0 for each j < j0 . A mapping σ : S ∗ → T ∗ is called an injective resolution of S ∗ if T ∗ is a complex of G-modules bounded below, σ is a quasi-isomorphism and the sheaves T b are injective, which means that Hom(B, T b ) → Hom(K, T b ) is surjective for each injective mapping K → B of sheaves of G-modules. Let G be a constant sheaf of rings, may be twisted over {i0 , ..., i2r −1 }. Suppose that S ∗ is a complex of G-modules bounded below. The hypercohomology group h Hb (X, S ∗ ) is defined to be the G-module such that b ∗ b b+1 )]/[im(Γ(X, T b−1 ) → Γ(X, T b )]. h H (X, S ) := [ker(Γ(X, T ) → Γ(X, T ∗ ∗ If σ : S → F is a quasi-isomorphism, then σ induces an isomorphism of the hypercohomology groups: σ : h Hb (X, S ∗ ) ∼ = h Hb (X, F ∗ ) (see also [15] and the reference [EV ] in it). In view of Lemma 16 in Section 3 the hypercohomology groups h Hb (X, S ∗ ) are twisted over {i0 , ..., i2r −1 }, when S ∗ and G are twisted over {i0 , ..., i2r −1 }. 11. Proposition. The sequence 10(3) is an acyclic resolution of the sheaf GN . Proof. Each standard simplex ∆n with n ≥ 1 has a C ∞ retraction zˆ : ∆n ×[0, 1] → {y} into a point y belonging to it. There exists a C ∞ deformation retraction (1) fˆ : AG × [0, 1] → AG supplied by the family of mappings (2) fˆn : (AG)n × [0, 1] → (AG)n+1 , where ` (3) (AG)n := qA ( j≤n ∆j × Gj+1 ) ⊂ AG and (4) fˆn (|t1 , ..., tn , h0 [h1 |...|hn ]|, t) := |Φ(0, t), Φ(t1 , t), ..., Φ(tn , t), h0 [h1 |...|hn ]|, where Φ : [0, 1]2 → [0, 1] is defined as the composition Φ(x, t) := φ(min(1, x + t)) taking φ a smooth nondecreasing function φ : [0, 1] → [0, 1] such that φ(0) = 0 and φ(1) = 1. Then for each C ∞ or Hpt differentiable mapping v : AG → Ar we get v◦ fˆ◦(qn ×id) = ˆ n and v ◦ qA = v n+1 , where qn × id : (∆n × Gn+1 ) × [0, 1] → AG × [0, 1], v n+1 ◦ h ˆ n : (∆n × Gn+1 ) × [0, 1] → ∆n+1 × Gn+2 is the smooth mapping given by the formula h ˆ hn (t1 , ..., tn , g0 , g1 , ..., gn , t) = (Φ(0, t), Φ(t1 , t), ..., Φ(tn , t), e, g0 , g1 , ..., gn ). At the same time ∆n+1 has a C ∞ retraction onto ∆n for each n ≥ 0 while the group G

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is C ∞ or Hpt differentiable and arcwise connected. Therefore, for each b > 0 the cohomology group Hb (N, AGN ) is trivial (see also §2.2 [3] and §54 below). 12. A sheaf S on X is called soft if for each closed subset Y of X the restriction map S(X) → S(Y ) is a surjection. 12.1. Lemma. For a C ∞ or Hpt differentiable group G satisfying conditions 4(A1, A2, C1, C2) the sheaf AGN is soft. Proof. Consider a closed subset Y of N and a section σY of AGN over Y . In accordance with the definition of a section over a closed subset there exists an open set U and an extension σU of σY from Y onto U such that Y ⊂ U ⊂ N . From the paracompactness of N there exists a neighborhood V of Y such that cl(V ) ⊂ U , where cl(V ) denotes the closure of V in N . Therefore, the extension σ of σY to a global section of AGN is provided by the formula σ(x) = fˆ(σU (x), ψ(x)), where fˆ : AG × [0, 1] → AG is a deformation retraction (see §11) and ψ : N → [0, 1] is either a C ∞ or Hpt differentiable function equal to 1 on V and equal to 0 on M \ U . 13. Remark. Let now N be a C ∞ or H ∞ manifold over Ar and T (N, G, π, Ψ) be a tangent bundle with T = T N and the projection π : T → N , where connecting mappings φj ◦ φ−1 k for Vj ∩ Vk 6= ∅ of the atlas At(N ) = {(Vj , φj ) : j} of the manifold N are Ar ∞ holomorphic for 1 ≤ r ≤ 3, φj ◦ φ−1 k ∈ H . Denote by T the sheaf of germs of smooth sections of T . Then BT denotes the sheaf associated with a the pre-sheaf assigning to each ` open subset V of N the group of sections of the natural projection y∈V B(π −1 (y)) → V , which are locally of the form (1) y 7→ |t1 (y), ..., tn (y), [σ1 (y)|...|σn (y)]|, where t1 , ..., tn are C ∞ for r = 1 or H ∞ for 1 ≤ r ≤ 3 functions and σ1 , ..., σn are C ∞ or H ∞ sections of the vector bundle T (N, G, π, Ψ). Using constructions above we define B a+1 T for each a ∈ N by induction. Then B a+1 T is the sheaf associated with the pre-sheaf assigning to an open subset V of N the group of sections of the natural projection ` a+1 (π −1 (y)) → V having the local form (1), where σ , ..., σ are sections of 1 n y∈V B B a T over V . Similarly we define AB a T . If now T = Λb T ∗ N is the b-th exterior power of the cotangent bundle of N , then b b the above construction produces the sheaves AB a SN,A and B a+1 SN,A of AB a Ar and r r B a+1 Ar valued respectively differential C ∞ forms on N , where the index Ar may be omitted, when the Cayley-Dickson algebra Ar is specified. In the equation P (2) w = J fJ (z)dxb1 ,j1 ∧ dxb2 ,j2 ∧ ... ∧ dxbk ,jk , where fJ : N → AB a Ar or fJ : N → B a+1 Ar , z = (z1 , z2 , ...) are local coordinates in N , zb = xb,0 i0 + xb,1 i1 + ... + xb,2r −1 i2r −1 , where zb ∈ Ar , xb,j ∈ R for each b and every j = 0, 1, ..., 2r − 1, J = (b1 , j1 ; b2 , j2 ; ...; bk , jk ). Since each topological vector space Z over Ar with 2 ≤ r ≤ 3 has the natural twisted structure Z = Z0 i0 ⊕ Z1 i1 ⊕ ... ⊕ Z2r −1 i2r −1 with pairwise isomorphic topological vector spaces Z0 , ..., Z2r −1 over R, then T N and T ∗ N and Λb T ∗ N have twisted structures, where X ∗ denotes the space of all continuous Ar additive and R homogeneous functionals on X with values in Ar , when 2 ≤ r ≤ 3, while X ∗ over C is the usual topologically dual space of continuous C-linear functionals on X. Therefore, due to Proposition 6 B a T and AB a T have the induced twisted structure for each a ∈ N. k can be written in the form: Each section σ of the sheaf AB a SN σ = a |h0 , .., hn , σ0 , ..., σn |, where σ0 , ..., σn are smooth differential B Ar valued differential k-

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forms on V and {hj : j = 0, 1, 2, ...} is a C ∞ smooth partition of unity on V . An addition of differential forms induces the additive group structure. As a multiplication there can be taken the external product of differential forms which is twisted over Ar for 2 ≤ r ≤ 3. k for an open subset V in N denote The group of sections of the sheaf AB b SN b k k ) → Γ(V, AS k ) → by Γ(V, AB SN ). The sequence of the groups 0 → Γ(V, SN n k Γ(V, BSN ) → 0 is exact for each open subset V in N , since the sequence of vector bundles 0 → Λk T ∗ N → AΛk T ∗ N → BΛk T ∗ N → 0 is exact. k induces the twisted structure of The twisted structure of the groups AB b SN k ). Therefore, the sequence of sheaves 0 → B b S k → AB b S k → B b+1 S k → Γ(V, AB b SN N N N 0 is exact as well as for each b ≥ 0. The composition of these sequences induces a long exact sequence σ σ σ σ k → AS k −→ k −→ k −→ (2) 0 → SN ABSN ... −→ AB b SN ..., N k → AB b+1 S k is the composition of mappings AB b S k → B b+1 S k → where σ : AB b SN N N N k . The sequence (2) will be called the bar resolution of the sheaf S k . AB b+1 SN N Let S be an arbitrary twisted sheaf on a topological space X. Denote by AS and BS sheaves associated with the pre-sheaves V 7→ A(Γ(V, S)) and V 7→ B(Γ(V, S)) correspondingly. The stalks of AS and BS are ASx and BSx at x, while the sequence (3) e → Sx → A(Sx ) → B(Sx ) → e is exact, consequently, the sequence of sheaves e → S → AS → BS → e is also exact. The composition of these sequences gives the bar resolution of S σ σ σ σ (4) e → S → AS −→ ABS −→ ... −→ AB b S −→ .... The complex of sheaves σ σ σ σ (5) B ∗ (S) : AS −→ ABS −→ ... −→ AB b S −→ ... is called the bar complex of S. The bar resolution of S is an acyclic resolution of S that is deduced analogously to the proofs of Proposition 11 and Lemma 12.1. Thus the cohomology of S is equal to the cohomology of the cochain complex σ σ σ σ (6) Γ(N, AS) −→ Γ(N, ABS) −→ ... −→ Γ(N, AB b S) −→ .... ∗ (S). The complex (6) will be called the bar cochain complex of S and will be denoted by CB Each short exact sequence of sheaves e → E → F → Y → e twisted over generators {i0 , i1 , ..., i2r −1 }, 2 ≤ r ≤ 3, induces a short exact sequence of complexes sheaves e → σ σ σ σ B ∗ (E) → B ∗ (F ) → B ∗ (Y ) → e, where B ∗ (F ) : AF −→ ABF −→ AB 2 F −→ ... −→ σ AB b F −→ ... is the bar complex of F . 14. Proposition. If a sequence of groups e → K → G → J → e is exact, where E and K, G, J are arcwise connected, then the sequence e → (W M E; N, K, P)t,H → (W M E; N, G, P)t,H → (W M E; N, J, P)t,H → e is exact. Proof. In view of Proposition 7.1 in Section 3 (W M E; N, K, P)t,H is the principal fiber bundle over (W M N )t,H with the structure group K k , πK,∗ : (W M E; N, K, P)t,H → (W M N )t,H , −1 πK,∗ < w0 >t,H =< w0 >t,H ×K k = e × K k , where e ∈ (W M N )t,H denotes the unit element. Since the sequence e → K k → Gk → J k → e is exact as well, then the corresponding sequence of wrap groups is exact. 15. Proposition. Let G be a C ∞ or Hpt differentiable twisted group over {i0 , i1 , ..., i2r −1 } satisfying Conditions 4(A1, A2, C1, C2). Then for each C ∞ or Hpt principal G -bundle E(N, G, π, Ψ) there exists a C ∞ or Hpt differentiable mapping φ : N →

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BG such that E → N is the pull-back of the universal principal G bundle by φ. Proof. Consider an open covering V = {Vj : j ∈ J} of N , where J is a set, such that for each j ∈ J there exists a trivialization ψj : π −1 (Vj ) → Vj × G. Define a mapping gj : E → G by the formula gj (x) = pr2 (ψj (x)) for x ∈ π −1 (Vj ), gj (x) = e for x ∈ / π −1 (Vj ), where e denotes the neutral element in G and pr2 : Vj × G → G is the projection on the second factor. ′ For a principal G bundle E(N, G, π, Ψ) consider a family of Hpt transition functions ′ {gi,j : i, j ∈ J} related with an open covering V := {Vj : j ∈ J} of an Hpt manifold N over Ar , where J is a set, gi,j : Vi ∩ Vj → A∗r , when the intersection Vi ∩ Vj 6= ∅ is non-void, 1 ≤ r ≤ 3, n ∈ N. Introduce the mapping (1) gE,N (x) := |fj(0) , fj(1) , ...fj(n) , [gj(0),j(1) |gj(1),j(2) |...|gj(n−1),j(n) ]| such that gE,N : N → BG, where {fj : j ∈ J} is an Hpt1 partition of unity subordinated ′ to U with t′ ≤ t1 ≤ ∞. Therefore, gE,N can be chosen of the smoothness class Hpt . A , ΨA ) by the Thus, E(N, G, π, Ψ) is the pull-back of the universal bundle AG(BG, G, πB A classifying mapping gE,N , where πB : AG → BG is as in §4. Show it in details. Take a partition of unity of class C ∞ or Hpt subordinated to to the covering V and Φ : A → AG be the following mapping Φ(y) := |fj0 (π(y)), fj1 (π(y)), ..., fjn (π(y)), gj0 (y), gj1 (y), ..., gjn (y)|, where j0 , ..., jn are indices such that fj (π(y)) 6= 0 for each j ∈ {j0 , ..., jn }. Then Φ is G equivariant, which means that Φ(yh) = Φ(y)h for all y and h ∈ G, since gj (yh) = pr2 (ψj (yh)) for yh ∈ π −1 (Vj ) and gj (yh) = e for yh ∈ / π −1 (Vj ). Indeed, y ∈ π −1 (y) is equivalent to yh ∈ π −1 (Vj ) for each h ∈ G, since π −1 (Vj ) = Vj × G, where y = (u, q) with u ∈ N and q ∈ G and (u, q)h = (u, qh) in local coordinates. Thus gj (yh) = gj (y)Rh , where Rh = h for y ∈ π −1 (Vj ) and Rh = e for y ∈ / π −1 (Vj ). Therefore, Φ induces a morphism of principal G-bundles Φ

A −→ AG ↓π ↓

N

φ

−→ BG

where the restriction of φ to Vj is φ(x)|Vj = |fj0 , fj1 (x), ..., fjn (x), [gj0 (σ(x)) : gj1 (σ(x)) : ... : gjn (σ(x))]|, σ : Vj → π −1 (Vj ) is a smooth section of the restriction π −1 (Vj ) → Vj for π : E → N . Consider equivalence classes qj ∼ gj if and only if there exist s1 , ..., sm ∈ G such that (sm (sm−1 ...(s1 (qj )...) = gj , hence qj h ∼ gj h, since qj h ∼ gj h if and only if h−1 qj ∼ h−1 gj , which is equivalent with h(sm (sm−1 ...(s1 (h−1 qj )...) = gj . Due to the alternativity of the group G we get −1 −1 −1 [(...(gj−1 s−1 1 )...sm−1 )sm ][(sm (sm−1 ...(s1 gl )...)] = gj gl . Therefore, in the non-homogeneous coordinates the mapping φ takes the form φ(x) = |fj0 (x), fj1 (x), ..., fjn (x), [gj0 ,j1 (x)|gj1 ,j2 (x)|...|gjn−1 ,jn (x)]|, where gj,l (x) = [gj (σ(x))]−1 gl (σ(x)) are transition functions associated with the open covering of N by the open sets {x ∈ N : fj (x) > 0}. Then the mapping φ(x) is independent from the choice of σ, since gj are G-equivariant. All functions gj are either C ∞ or Hpt , hence φ is either C ∞ or Hpt correspondingly. 16. Corollary. For each smooth C ∞ or Hpt principal B b A∗r bundle with 1 ≤ r ≤ 3 there exists a C ∞ or Hpt differentiable mapping φ : N → B b+1 A∗r such that E(N, G, π, Ψ)

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is the pull-back of the universal principal B b A∗r -bundle by φ, where G = B b A∗r . 17. Lemma. Let G be a differentiable (topological) group twisted over {i0 , ..., i2r −1 } for 1 ≤ r ≤ 3 satisfying Conditions 4(A1, A2, C1, C2). Then the group of isomorphism classes of C ∞ smooth (continuous) principal G-bundles over N is isomorphic with the group [N, BG]∞ of smooth (or [N, BG]0 of continuous respectively) homotopy classes of smooth (continuous) maps from N to BG. Proof. Each principal G-bundle over N has properties 4(A1, A2, C1, C2) induced by that of G, where locally π −1 (Vj ) = Vj × G, conj(y) = (u, conj(g)) for each y = (u, g) ∈ Vj × G, while {i0 , ..., i2r −1 } for 2 ≤ r ≤ 3 is the multiplicative group, which is associative for r = 2 and alternative for r = 3. Consider a short exact sequence e → GN → AGN → BGN → e. In view of Lemma 16 in Section 3 it induces the cohomology long exact sequence π∗ ... → C ∞ (N, AG) −→ C ∞ (N, BG) → H1 (N, GN ) → H1 (N, AGN ) → .... From H1 (N, AGN ) ∼ = e we get the isomorphism C ∞ (N, BG)/π∗ C ∞ (N, AG) ∼ = H1 (N, GN ). ∞ ∞ ∞ Then the image π∗ C (N, AG) of the group C (N, AG) in C (N, BG) consists of all smooth maps from N to BG for which there exist lift mappings from N to AG. On the other hand, f ∈ C ∞ (N, BG) has a lift F : N → AG if and only if f is smooth (or continuous) homotopic to a constant mapping, since [g0 : ... : gn ] in (BG)n is the equivalence class {(g0 , ..., gn ) ∼ (sm (...(s1 g0 )...), ..., (sm ...(s1 gn )...)) : s1 , ..., sm ∈ G, m ∈ N}, consequently, C ∞ (N, BG)/π∗ C ∞ (N, AG) ∼ = [N, BG]∞ . In the class of 0 0 continuous mappings we get analogously C (N, BG)/π∗ C (N, AG) ∼ = [N, BG]0 . 18. Notes. In view of §§4-6 there exists a short exact sequence e → G → AG → BG → e ′ of Hpt homomorphisms due to the twisted structures of G, AG and BG (see Equations 4(A2) and 6(8, 9)). To groups AG and BG are assigned simplicial topological groups AG. and BG. with face homomorphisms ∂j : AGn → AGn−1 given by: (1) ∂j (h0 [h1 |...|hn ]) = h0 h1 [h2 |...|hn ] for j = 0, ∂j (h0 [h1 |...|hn ]) = h0 [h1 |...|hj hj+1 |...|hn ] for 0 < j < n, ∂j (h0 [h1 |...|hn ]) = h0 [h1 |...|hn−1 ] for j = n. While ∂j : BGn → BGn−1 has the form: (2) ∂j ([h1 |...|hn ]) = [h2 |...|hn ] for j = 0, ∂j ([h1 |...|hn ]) = [h1 |...|hj hj+1 |...|hn ] for 0 < j < n, ∂j ([h1 |...|hn ]) = [h1 |...|hn−1 ] for j = n. The degeneracy homomorphisms sj : AGn → AGn+1 are prescribed by the formula: (3) sj (h0 [h1 |...|hn ]) = h0 [e|h1 |...|hn ] for j = 0, sj (h0 [h1 |...|hn ]) = h0 [h1 |...|hj |e|hj+1 |...|hn ] for 0 < j < n, sj (h0 [h1 |...|hn ]) = h0 [h1 |...|hn |e] for j = n. While sj : BGn → BGn+1 is given by: (4) sj ([h1 |...|hn ]) = [e|h1 |...|hn ] for j = 0, sj ([h1 |...|hn ]) = [h1 |...|hj |e|hj+1 |...|hn ] for 0 < j < n, sj ([h1 |...|hn ]) = [h1 |...|hn |e] for j = n. Analogous mappings are for simplices: (5) ∂ j (t0 , ..., tn+1 ) = (t0 , ..., tj , tj , tj+1 , ..., tn+1 ) and (6) sj (t0 , ...., tn+1 ) = (t0 , ..., tj , tˆj+1 , tj+2 , ..., tn+1 ), where tˆj+1 means that tj+1 is absent.

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The geometric realization |AG. | of the simplicial space AG. is defined to be the quotient n n+1 by the equivalence relations space of the disjoint union ⊔∞ n=0 ∆ × G j (7) (∂ x, g¯) ∼ (x, ∂j g¯) for each (x, g¯) ∈ ∆n−1 × Gn+1 , while (sj x, g¯) ∼ (x, sj g¯) for each (x, g¯) ∈ ∆n+1 × Gn+1 . At the same time the geometric realization |BG. | of the n n simplicial space BG. is the quotient of the disjoint union ⊔∞ n=0 ∆ × G by the equivalence relations (8) (∂ j x, g¯) ∼ (x, ∂j g¯) for each (x, g¯) ∈ ∆n−1 × Gn , while (sj x, g¯) ∼ (x, sj g¯) for each (x, g¯) ∈ ∆n+1 × Gn . Consider a non-commutative sphere Cr := {z ∈ Ir : |z| = 1} for r = 2, 3, where Ir := {z ∈ Ar : Re(z) = 0}. For r = 1 put Cr = {i, −i}, where i = (−1)1/2 . Let Z(Cr ) Q denotes the additive group ZCr /Z, where ZCr := b∈Cr Tb , Tb = Zb for each b ∈ Cr , Z is the additive group of integers, Z is the equivalence relation such that Tb × T−b /Z = Tb for each b ∈ Cr . For 2 ≤ r ≤ 3 the group Z(Cr ) is isomorphic with Zα , where card(α) = card(R) =: c. Particularly, Z(C1 ) = Zi for r = 1. Henceforth, we consider twisted sheaves and cohomologies over {i0 , ..., i2r −1 }, where 2 ≤ r ≤ 3. In particular, the complex case will also be included for r = 1, but the latter case is commutative over C. So we can consider simultaneously 1 ≤ r ≤ 3 and generally speak about twisting undermining that for r = 1 it is degenerate. 19. Proposition. Let G be the group either A∗r or Z(Cr ), where 1 ≤ r ≤ 3. Then for each H ∞ smooth manifold N over Ar and each b ≥ 2 the group Hb (N, Z(Cr )) is isomorphic with: (1) the group E(N, B b−2 G) of isomorphism classes of smooth principal B b−2 Gbundles over N ; (2) the group [N, B b−1 G]∞ of smooth homotopy classes of smooth mappings from N to B b−1 G. Proof. In view of Corollary 3.4 [28, 29] there exists the short exact sequence η (1) 0 → Z(Cr ) −→ Ar → A∗r → 1, since exp(M + 2πkM/|M |) = exp(M ) for each non-zero purely imaginary M ∈ Ir (with Re(M ) = 0) and every k ∈ Z, 1 ≤ r ≤ 3, where η(z) = 2πz for each z ∈ Ar . If f : Ar → A∗r is a differentiable function, then (dLnf ).h = w(h) is the differential oneform considering d as the external differentiation over R, where h ∈ Ar . In the particular case of G = A∗r with 1 ≤ r ≤ 3 there exist further short exact sequences (2) 1 → A∗r → AA∗r → BA∗r → 1 (3) 1 → BA∗r → ABA∗r → B 2 A∗r → 1 (4) 1 → B m A∗r → AB m A∗r → B m+1 A∗r → 1. Therefore, identifying the ends of these short exact sequences we get the long exact sequence (5) 0 → Z(Cr ) → Ar → AA∗r → ABA∗r → ... → AB m A∗r → ..., where σ : Ar → AA∗r ,..., σ : AB m−1 A∗r → AB m A∗r are homomorphisms, all terms Ar , AA∗r ,...,AB m A∗r ,... are contractible spaces. Suppose now that N and E are of class H ∞ . Let C∞ (N, AB m A∗r ) denotes the sheaf of germs of C ∞ functions from N into AB m A∗r . Thus, we get the functor C ∞ . Then the application of C ∞ functor to the long exact sequence (5) gives: (6) 0 → Z(Cr )N → C∞ (N, Ar ) → C∞ (N, AA∗r ) → C∞ (N, ABA∗r ) → ... → C∞ (N, AB m A∗r ) → ...,

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where σ∗ : C∞ (N, Ar ) → C∞ (N, AA∗r ),...,σ∗ : C∞ (N, AB m−1 A∗r ) → C∞ (N, AB m A∗r ) are induced homomorphisms. The latter exact sequence is called the bar resolution of Z(Cr )N . Sheaves C∞ (N, Ar ) and C∞ (N, AB m A∗r ) are contractible, since Ar and AB m A∗r are contractible. Therefore, the cohomology of the sheaf Z(Cr )N can be computed using the complex (7) C ∞ (N, Ar ) → C ∞ (N, AA∗r ) → ... → C ∞ (N, AB m A∗r ) → ... with homomorphisms σ∗ : C ∞ (N, Ar ) → C ∞ (N, AA∗r ),...,σ∗ : C ∞ (N, AB m−1 A∗r ) → C ∞ (N, AB m A∗r ). The long exact sequence (7) we call a bar cochain complex of Z(Cr )N . The cohomology of Z(Cr )N computed with the help of the bar complex is denoted by H∗b (N, Z(Ar )N ) and it is called the bar cohomology of Z(Cr )N . Then π0 C ∞ (N, BA∗r ) is the first bar cohomology H1b (N, Z(Ar )N ) of Z(Cr )N . For the generalized exponential sequence 0 → Z(Cr )N → AB b − 2, 1 b−2 consequently, the coboundary homomorphism H (N, B GN ) → Hb (N ; Z(Cr )) is an isomorphism (see also Chapter 2 §4 in [3] for abelian sheafs). The second statement of this proposition follows from Lemma 17. 20. Lemma. Let X be a topological vector space over Ar , 2 ≤ r ≤ 3. Then AX and BX with respect to the additive group structure of X and with respect to the multiplication on scalars from Ar in homogeneous coordinates are Ar vector spaces and the projection AX → BX is R-homogeoneous and Ar additive. Proof. Define the multiplications by: for Ar × AX → AX as s|t1 , ..., tn ; v0 [v1 |...|vn ]| = |t1 , ..., tn ; sv0 [sv1 |...|svn ]|, for AX × Ar → AX as |t1 , ..., tn ; v0 [v1 |...|vn ]|s = |t1 , ..., tn ; v0 s[v1 s|...|vn s]|, for Ar × BX → BX as s|t1 , ..., tn ; [v1 |...|vn ]| = |t1 , ..., tn ; [sv0 |...|sxn ]|, for BX × Ar → BX as |t1 , ..., tn ; [v1 |...|vn ]|s = |t1 , ..., tn ; [v1 s|...|vn s]|. Then if qj = sm (sm−1 ...(s1 vj )...) for each j, then for z 6= 0 we get zqj = z(sm ...(s1 (z −1 (zvj )))...) due to the alternativity of the octonion algebra O, while for z = 0 we trivially get 0 = (sm ...(s1 0)...). Thus such multiplication is compatible with the equiv-

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alence relations, since X = X0 i0 ⊕ ... ⊕ X2r −1 i2r −1 , where X0 , ..., X2r −1 are pairwise isomorphic topological vector spaces over R such that we put vx = xv for each v ∈ Xj and x ∈ R. Since R is the center of the algebra O, then the projection from AX to BX is R-linear. Evidently, it is additive as the additive group homomorphism. 21. Remark. Let N. be a simplicial smooth manifold over Ar , where 0 ≤ r ≤ 3. A smooth m-form w on the geometric realization |N. | of N. is defined to be as a family {wk : k} of smooth differential m-forms wk on ∆k × Nk with values in Ar being applied to vectors, satisfying for each 0 ≤ j ≤ n the compatibility conditions: (1) (∂ j × id)∗ wn = (id × ∂j )∗ wn−1 (2) (sj × id)∗ wn = (id × sj )∗ wn+1 , where ∂ j × id, id × ∂j , sj × id and id × sj are the maps as follows: (3) id × ∂j : ∆n−1 × Nn → ∆n−1 × Nn−1 , ∂ j × id : ∆n−1 × Nn → ∆n × Nn , id × sj : ∆n+1 × Nn → ∆n+1 × Nn+1 , sj × id : ∆n+1 × Nn → ∆n × Nn such that ∂ j and sj are coface and the codegeneracy maps on ∆n and ∂j , sj are the face and the degeneracy maps on Nn . We consider w taking values in a vector space or an algebra over Ar as is specified below. For a Lie group G either over R or may be twisted over Ar and its Lie algebra g put g −1 dg as the canonical g-valued connection 1-form on G (see also Lemma 20). Under the mapping g 7→ hg and dg 7→ hdg we have (g −1 h−1 )(hdg) = g −1 dg due to the alternativity of G and the Maufang identity (xy)(zx) = x(yz)x for each x, y, z ∈ O and de = d(g −1 g) = 0 = (dg −1 )g + g −1 dg = [(dg −1 )h−1 ](hg) + (g −1 h−1 )(hdg). Iterating this relation due to the alternativity of O and the Moufang identities in it we get the equivariance condition in homogeneous coordinates over O as well: −1 −1 −1 [(...(g −1 s−1 1 )...sm−1 )sm ][sm (sm−1 ...(s1 dg1 )...)] = g1 dg1 . The total space AG of the universal principal g-bundle AG → BG carries a smooth gvalued form w. The evaluation of w is w|x0 , ..., xn , g0 , ..., gn | = x0 g0−1 dg0 + x1 g1−1 dg1 + ... + xn gn−1 dgn , where x0 , ..., xn are barycentric coordinates in ∆n . Each term xn gn−1 dgn .s is in g for each s ∈ g such that gj−1 dgj = πj∗ (g −1 dg|Tgj G ), where πj : Gn+1 → G is the projection on the j-th factor and g −1 dg|Tgj G is the restriction of g −1 dg to the tangent space Tgj G of G at gj . For AA∗r with 2 ≤ r ≤ 3 define the canonical connection 1-form A(z −1 dz) by the family of AAr -valued 1-forms A(z −1 dz)n on ∆n × (A∗r )n+1 such that A(z −1 dz)n evaluated on a vector (v0 , ..., vn ) at a point |t1 , ..., tn , z0 [z1 |...|zn ]| is given by the formula (4) (A(z −1 dz)n ||t1 ,...,tn ,z0 [z1 |...|zn ]| .(v0 , ..., vn ) = |t1 , ..., tn , z0−1 v0 [z1−1 v1 |...|zn−1 vn ]| and formally denote it by (5) (A(z −1 dz)n ||t1 ,...,tn ,z0 [z1 |...|zn ]| = |t1 , ..., tn , z0−1 dz0 [z1−1 dz1 |...|zn−1 dzn ]|. For BA∗r with 2 ≤ r ≤ 3 the canonical connection 1-form B(z −1 dz) on BA∗r is defined by the family of BAr -valued 1-forms B(z −1 dz)n on ∆n × (A∗r )n , where (6) B(z −1 dz)n ||t1 ,...,tn ,[z1 |...|zn ]| = |t1 , ..., tn , [z1−1 dz1 |...|zn−1 dzn ]|. We have that (∂ j × id)∗ A(z −1 dz)n ||t1 ,...,tn−1 ;z0 [z1 |...|zn ]| = |t1 , ..., tj , tj , tj+1 , ..., tn−1 ; z0−1 dz0 [z1−1 dz1 |...|zn−1 dzn ]| and (∂ j × id)∗ B(z −1 dz)n ||t1 ,...,tn−1 ,[z1 |...|zn ]| =

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|t1 , ..., tj , tj , tj+1 , ..., tn−1 , [z1−1 dz1 |...|zn−1 dzn ]| and (id × ∂j )∗ A(z −1 dz)n−1 ||t1 ,...,tn−1 ;z0 [z1 |...|zn ]| = |t1 , ..., tn−1 ; (z0−1 dz0 ) + (z1−1 dz1 )[z2−1 dz2 |...|zn−1 dzn ]| for j = 0, (id × ∂j )∗ A(z −1 dz)n−1 ||t1 ,...,tn−1 ;z0 [z1 |...|zn ]| = −1 −1 −1 |t1 , ..., tn−1 ; z0−1 dz0 [z1−1 dz1 |...|zj−1 dzj−1 |(zj−1 dzj )+(zj+1 dzj+1 )|zj+2 dzj+2 |...|zn−1 dzn ]| for 0 < j < n, (id × ∂j )∗ A(z −1 dz)n−1 ||t1 ,...,tn−1 ;z0 [z1 |...|zn ]| = −1 |t1 , ..., tn−1 ; z0−1 dz0 [z1−1 dz1 |...|zn−1 dzn−1 ]| for j = n, while ∗ −1 n−1 (id × ∂j ) B(z dz) ||t1 ,...,tn−1 ;[z1 |...|zn ]| = |t1 , ..., tn−1 ; [z2−1 dz2 |...|zn−1 dzn ]| for j = 0, and (id × ∂j )∗ B(z −1 dz)n−1 ||t1 ,...,tn−1 ;[z1 |...|zn ]| = −1 −1 −1 |t1 , ..., tn−1 ; [z1−1 dz1 |...|zj−1 dzj−1 |(zj−1 dzj ) + (zj+1 dzj+1 )|zj+2 dzj+2 |...|zn−1 dzn ]| for 0<j
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the compatibility conditions: (7) id∆k × (∂ j × id)m )∗ wn = (id∆k × (id × ∂j )m )∗ wn−1 , (8) (id∆k × (sj × id)m )∗ wn = (id∆k × (id × sj )m )∗ wn+1 . It was shown above that the groups AB b Ar and B b+1 Ar also have the structures of Ar vector spaces. Therefore, the canonical connection 1-form AB b (z −1 dz) on AB b A∗r is a 1-form on AB b A∗r such that it satisfies the inductive formula: (9) AB b (z −1 dz)||t1 ,...,tn ,g0 [g1 |...|gn ]| = |t1 , ..., tn , B b (g0−1 g0 )[B b (g1−1 dg1 )|...|B b (gn−1 dgn )]|. Then the canonical connection 1-form B b+1 (z −1 dz) on B b+1 A∗r is a 1-form on B b+1 A∗r such that (10) B b+1 (z −1 dz)||t1 ,...,tn ,[g1 |...|gn ]| = |t1 , ..., tn , [B b (g1−1 dg1 )|...|B b (gn−1 dgn )]|, where g0 , g1 , ..., gn ∈ B b A∗r and B b (gj−1 dgj ) is the canonical connection 1-form B b (z −1 dz) on B b A∗r evaluated at gj . 22. Gerbes over quaternions and octonions. Consider twisted groups C, K, G satisfying Conditions 4(A1, A2, C1, C2). If (CE1) e → C0 → K0 → G0 → e is a topological central extension, then we say, that (CE2) e → C → K → G → e is a topological twisted extension. A gerbe on a topological space X is a sheaf S of categories satisfying the conditions (G1 − G3): (G1) for each open subset V in X the category S(V ) is a groupoid, which means that every morphism is invertible; (G2) each point x ∈ X has a neighborhood Vx for which S(Vx ) is non-empty; (G3) any two objects P1 and P2 of S(V ) are locally isomorphic, that is, each x ∈ V has a neighborhood Y for which the restrictions P1 |Y and P2 |Y are isomorphic. A gerbe S is called bound by a sheaf G of twisted groups over Ar satisfying Conditions 4(A1, A2, C1, C2), if for each open subset V in X and every object P of S(V ) there exists an isomorphism of sheaves ν : Aut(P ) → G|V , where G|V denotes the restriction of the sheaf G onto V , while Aut(P ) is the sheaf of automorphisms of P so that for an open subset Y in V the group Aut(P )(Y ) is the group of automorphisms of the restriction sY (P ). It is supposed that such an isomorphism commutes with with morphisms of S and must be compatible with restrictions to smaller open subsets. Two gerbes S and E bounded by G on a manifold N are equivalent, if they satisfy (G4, G5): (G4) if V is an open subset in X, then there exists an equivalence of categories µV : S(V ) → E(V ) so that for each object P of S(V ) there is a commutative diagram: µV : AutS(V ) (P ) → AutE(V ) (P ), νS : AutS(V ) (P ) → Γ(V, G), νE : AutE(V ) (P ) → Γ(V, G) such that νS = νE (µV ); (G5) for each pair of open subsets V and Y in N with Y ⊂ V there exists an invertible natural transformation: β : RE (µV ) = µY (RS ), where RS : S(V ) → S(Y ) denotes the natural restriction transformation. It is also imposed the condition, that for a triple of open subsets Y ⊂ V ⊂ J in N the compatibility conditions are satisfied. If there is a principal G-bundle E(B, G, π, Ψ) and and an extension (CE1, CE2) of topological groups, then there exists a gerbe Gπ bound by CN on B. This gerbe is constructed from the sheaf of sections of the bundle E(B, G, π, Ψ) by posing for each open

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subset V of B objects and morphisms of Gπ (V ) as follows. Associate with each section s : V → π −1 (V ) of π : π −1 (V ) → V the G-equivariant map ts : π −1 (V ) → G such that ts (z)s(π(z)) = z for every z ∈ π −1 (V ). We have as well the pull-back of principal C-bundle K → G from G to π −1 (V ) due to the mapping ts : π −1 (V ) → G. The composition π ◦ πs : Es → V is a principal K-bundle having a lifting of the structure group of π −1 (V ) → V to K. Then pairs (E, f ) of principal K-bundles πV : E(V, K, π, Ψ) → V and principal C-bundles f : E → π −1 (V ) such that there exists the commutative diagram with π(f (∗)) = πV (∗). A morphism of principal K-bundles η : E → E1 from (E, f ) to (E1 , f1 ) is described with the help of the condition f1 (η(∗)) = f with the corresponding commutative diagram. Therefore, the group of automorphisms of every object (E, f ) of Gπ (V ) is the group of mappings from V to C being the section of the sheaf CN over V , consequently, Gπ is the gerbe bound by CN . The constructed above gerbe Gπ has a global section if and only if there exists a lifting of the structure group from C to K. For G = BA∗r the extension (CE1, CE2) takes the form 1 → A∗r → AA∗r → BA∗r → 1. Then each principal AA∗r -bundle is trivial, since AA∗r is contractible. Hence the gerbe Gπ has a global section if and only if E(N, BA∗r , π, Ψ) is a trivial BA∗r -bundle. Construct now another gerbe Lπ of local sections of the bundle E(B, BA∗r , π, Ψ). For each open subset V in B the objects of Lπ (V ) are sections of E over V so that each local section s : V → π −1 (V ) induces a BA∗r -equivariant mapping ts : π −1 (V ) → BA∗r that induces the mapping τs = ts (s(∗)) : V → BA∗r . If Es is a principal A∗r -bundle over V induced by the mapping τs , then a morphism between the objects s, s1 ∈ Lπ (V ) induces the morphism Es → Es1 of the corresponding principal A∗r -bundles. Then Lπ is a gerbe bounded by (A∗r )N , where 2 ≤ r ≤ 3. Therefore, the natural transformation Lπ (V ) → Gπ (V ) sending a section s to the pull-back Es of the universal principal A∗r -bundle by ts is an equivalence of categories, which extends to an equivalence of gerbes Lπ → Gπ . For a gerbe G on N bounded by (A∗r )N with 2 ≤ r ≤ 3, assigning to each object Q in 1 G(V ) an SN,A -torsor COQ on V induces a connective structure. This torsor COQ consists r 1 of a sheaf on which SN,r acts so that for each point x ∈ N there exists a neighborhood V having the property that for each open subset Y ⊂ V the group COQ (Y ) is a principal 1 homogeneous space under the group Γ(Y, SN,A ). This assignment Q 7→ COQ (V ) need to r be functorial in accordance with restrictions from V onto Y . Moreover, for each morphism φ : Q → J of objects of G(V ) there exists an isomorphism φ∗ : COQ (V ) → COJ (V ) 1 of SN,A -torsors. Since G is a gerbe, then φ is an isomorphism and φ∗ is compatible with r compositions of morphisms and with restrictions to smaller open subsets, Y ⊂ V . If φ is an automorphism of Q induced by an A∗r -valued function g we suppose that φ∗ is an 1 automorphism ∇ 7→ ∇ − dLn(g) of the SN,A -torsor COQ (V ). r Consider a connection ω on a smooth principal BA∗r -bundle E(N, BA∗r , π, Ψ) and let V be an open subset in N such that Gπ (V ) is non-void and let ωV be the restriction of ω to ω (V ) of connections π −1 (V ). To each element (E, f ) of Gπ (V ) it is possible assign a set COE ∗ ∗ on E compatible with ω. If ω(q(∗)) = f ω for principal Ar -bundles q : AAr → BAr and f : E → π −1 (V ), then ω generates an element ω ˆ ∈ COE (V ). Therefore, the assignment

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ω is the connective structure on G . ω 7→ CO π The equivalence of gerbes Lπ → Gπ implies an extension of a pull-back of the connective structure from Gπ to Lπ . 23. Corollary. A mapping posing to the isomorphism class of a principal BA∗r -bundle E(B, BA∗r , π, Ψ), π : E = AA∗r → BA∗r , the equivalence class of the gerbe of section Lπ of E(B, BA∗r , π, Ψ) induces an isomorphism between the group of isomorphism classes of principal BA∗r -bundles and the group of equivalence classes of gerbes bound by A∗r . 24. Corollary. A mapping sending to the isomorphism class of a principal BA∗r -bundle E(B, BA∗r , π, Ψ), π : E = AA∗r → BA∗r , with a connection ω the equivalence class of the gerbe of section Lπ of E(B, BA∗r , π, Ψ) with the connective structure on Lπ induced by ω induces an isomorphism between the group of isomorphism classes of principal BA∗r bundles with connection on the group of equivalence classes of gerbes bound by A∗r with connective structures. Proof. This follows from Proposition 19 and §22, since the case of 2 ≤ r ≤ 3 is obtained from the complex case (see Theorems A1, A2 [15]) by additional doubling procedure of groups with doubling generators: H from C and O from H, while the considered groups satisfy Conditions 4(A1, A2, C1, C2). 25. Sheaves, geometric bars and gerbes for wrap groups. If G satisfies Conditions 4(A1, A2, C1, C2), then wrap groups (W M E)t,H satisfy these Conditions 4(A1, A2, C1, C2) as well, since Gk satisfies them being a multiplicative subˆ k and (W M,{s0,q :q=1,...,k} E; N, G, P)t,H is the principal Gk -bundle over group of the ring G the commutative group (W M,{s0,q :q=1,...,k} N )t,H (see Propositions 7(1, 2) in Section 3). Thus, wrap groups can be taken as the particular cases of groups for the sheaves, geometric bar and gerbes constructions (see §§1, 4, 11-13, 22, Corollary 9, Lemmas 16 in Section 3, 17, etc.). More concretely this can be done as follows. For a pseudo-manifold X = X1 × X2 over Ar , where X1 and X2 are Hpt -pseudo-manifolds over Ar , suppose that for each points s0,1 , ..., s0,k in X1 and every neighborhood U of {s0,1 , ..., s0,k } in X1 and a point y0 ∈ X2 and every neighborhood V of y0 in X2 there exist manifolds M and N such that {s0,1 , ..., s0,k } ⊂ M ⊂ U and y0 ∈ N ⊂ V for which a principal G-bundle E(N, G, π, Ψ) exists with a marked group G satisfying conditions of §2 in Section 2. If Q (1) J(Λ) = α∈Λ Jα is the product of topological groups Jα , where Λ is a set, and Λ2 ⊂ Λ1 , then there exists the natural projection group homomorphism (2) sˆΛ2 ,Λ1 : J(Λ1 ) → J(Λ2 ). Then define a pre-sheaf F on X such that Q (3) F (U × V ) = s0,1 ,...,s0,k ∈M ⊂U ;y0 ∈N ⊂V (W M,{s0,q :q=1,...,k} E; N, G, P)t,H and sU2 ×V2 ,U1 ×V1 : F (U1 × V1 ) → F (U2 × V2 ), since (W M2 ,{s0,q :q=1,...,k} E; N2 , G, P)t,H ⊂ (W M1 ,{s0,q :q=1,...,k} E; N1 , G, P)t,H for {s0,q : q = 1, ..., k} ⊂ M2 ⊂ M1 and y0 ∈ N2 ⊂ N1 satisfying conditions of Theorem 3.10, where U2 ⊂ U1 and V2 ⊂ V1 , while open subsets of the form U × V contain the base of topology of X. If S is a sheaf on X and S(U ) satisfies Conditions 4(A1, A2, C1, C2) for each U open in X, then we call S the twisted sheaf over {i0 , ..., i2r −1 }. For k = 1 consider x = {s0 ; y0 } ∈ X, but generally, consider x = {s0,1 , ..., s0,k ; y0 } ∈

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X1k × X2 instead of X1 × X2 . Then a set Fx of all germs of the pre-sheaf F at a point x ∈ X1k × X2 is the inductive limit Fx = ind − lim F (U × V ) taken by all open neighborhoods U k × V of x in X1k × X2 . Then applying the general construction of §1 gives the sheaf SW,X1 ,X2 of wrap groups. It is twisted over {i0 , ..., i2r −1 } for the group G twisted over generators {i0 , ..., i2r −1 } for 2 ≤ r ≤ 3. This sheaf is commutative, if G is commutative. This sheaf is obtained from the given below generalization taking a constant sheaf of the group G = G(U ) for each U open in X1 . More generally, if there is a sheaf G = GX1 on X1 of groups such that for each U open in X1 a group G(U ) satisfies conditions of §2 in Section 2, then put Q (4) F (U × V ) = s0,1 ,...,s0,k ∈M ⊂U ;y0 ∈N ⊂V (W M,{s0,q :q=1,...,k} E; N, G(U ), P)t,H , where sU2 ,U1 : G(U1 ) → G(U2 ) is the restriction mapping for each U2 ⊂ U1 so that the parallel transport structure for M ⊂ U is defined, GX1 , may be twisted for 2 ≤ r ≤ 3. Therefore, due to Theorem 3.10 and (1, 2) above there exists a restriction mapping sU2 ×V2 ,U1 ×V1 : F (U1 × V1 ) → F (U2 × V2 ) for each open U2 ⊂ U1 and V2 ⊂ V1 . Then this presheaf induces a sheaf SW,X1 ,X2 ,G of wrap groups. If G is a twisted over {i0 , ..., i2r −1 } for 2 ≤ r ≤ 3 sheaf, then the sheaf SW,X1 ,X2 ,G is twisted over {i0 , ..., i2r −1 }. If the sheaf G is commutative, then the sheaf SW,X1 ,X2 ,G is commutative. 26. Proposition. If hj : Xj → Yj are Hpt differentiable mappings from Xj onto Yj , j = 1, 2, where X = X1 × X2 and Y = Y1 × Y2 , X, X1 , X2 , Y, Y1 , Y2 are Hpt -pseudomanifolds over Ar , 0 ≤ r ≤ 3, h3 : GY1 → GX1 is an Hpt sheaf homomorphism, t ≥ [max{dim(X1 ), dim(X2 ), dim(Y1 ), dim(Y2 )}]/2 + 2. Then they induce homomorphisms (h1 , h3 )∗ : SW,Y1 ,X2 ,GY1 → SW,X1 ,X2 ,GX1 and h2,∗ : SW,X1 ,X2 ,GX1 → SW,X1 ,Y2 ,GX1 of wrap sheaves. −1 Proof. If M2 ⊂ U2 ⊂ Y1 , then h−1 1 (M2 ) =: M1 ⊂ h1 (U2 ) =: U1 ⊂ X1 and −1 h1 (U2 ) =: U1 is open in X1 for each U2 open in Y1 . In view of Corollary 9 in Section 2 and Proposition 7.1 and Theorem 3.10 there exists a homomorphism (h1 , h3 )∗ : (W M2 ,{v0,q :q=1,...,k2 } E; N, G(U2 ), P)t,H → (W M1 ,{s0,q :q=1,...,k1 } E; N, G(U1 ), P)t,H , where h3 : G(U2 ) → G(U1 ) is the group homomorphism, h1 (s0,q ) = v0,a(q) for each q = 1, ..., k2 , 1 ≤ a = a(q) ≤ k2 . Choose in particular s0,q such that k1 = k2 = k. Therefore, there exists the presheaf homomorphism (h1 , h3 )∗ : FY1 ,X2 ,GY1 (U2 × V ) → FX1 ,X2 ,GX1 (U1 × V ) for each U2 open in Y1 and V open in X2 . This presheaf homomorphism induces the sheaf homomorphism. If f : M1 → N1 ⊂ X2 , then h2 ◦ f : M1 → N2 for Hpt pseudo-manifolds M1 in X1 , N1 in X2 , N2 in Y2 . If f and h2 are Hpt mappings, then due to the Sobolev embedding theorem [35] for t ≥ [max{dim(X1 ), dim(X2 ), dim(Y1 ), dim(Y2 )}]/2+2 we have that f ′ exists and is continuous almost everywhere on X1 and h2 (f (∗)) is the Hpt mapping (see also [8]). Then h2,∗ (Pγˆ,u (x)) := Ph2 ◦ˆγ ,u (x) implies h2,∗ < Pγˆ,u >t,H =< Ph2 ◦ˆγ ,u >t,H for classes of Rt,H equivalent elements, since the group G(U ) and the manifold M are specified, and the same for N1 and N2 . Therefore, there exists the induced homomorphism h2,∗ : (W M,{s0,q :q=1,...,k} E; N1 , G(U ), P)t,H → (W M,{s0,q :q=1,...,k} E; N2 , G(U ), P)t,H , where N1 ⊂ V1 ⊂ X2 , N1 = h−1 2 (N2 ), y0,1 ∈ N1 , h2 (y0,1 ) = y0,2 , y0,2 ∈ N2 ⊂ V2 ⊂ Y2 . Consequently, there exists the homomorphism of pre-sheaves h2,∗ : FX1 ,X2 ,GX1 (U ×V1 ) → FX1 ,Y2 ,GX1 (U × V2 ) (see §25), where V1 = h−1 2 (V2 ), V2 is open in Y2 . Thus h2,∗ induces the homomorphism of the wrap sheaves.

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27. Proposition. Let e → G1 → G2 → G3 → e be an exact sequence of sheaves on X1 . Then there exists an exact sequence e → SW,X1 ,X2 ,G1 → SW,X1 ,X2 ,G2 → SW,X1 ,X2 ,G3 → e of wrap sheaves, where e is the unit element (see §25). Proof. For each U open in X1 there exists a short exact sequence of groups e → G1 (U ) → G2 (U ) → G3 (U ) → e such that G3 (U ) is isomorphic with the quotient group G2 (U )/G1 (U ), where G1 (U ) is the normal closed subgroup in G2 (U ). In view of Theorem 3.10 there exists the short exact sequence e → (W M E; N, G1 (U ), P)t,H → (W M E; N, G2 (U ), P)t,H → (W M E; N, G3 (U ), P)t,H → e. Then this induces the short exact sequence of wrap presheaves e → FG1 (U ) (U ) → FG2 (U ) (U ) → FG3 (U ) → e and the latter in its turn gives the short exact sequence of wrap sheaves (see also in general [3]). 28. Wrap sub-sheaf. In the construction of §25 consider a sub-pre-sheaf corresponding to FN (U ), that is, for V = N with a fixed marked point y0 ∈ N , where Q (1) FN (U ) := s0,1 ,...,s0,k ∈M ⊂U (W M,{s0,q :q=1,...,k} E; N, G(U ), P)t,H , where sG U2 ,U1 : G(U1 ) → G(U2 ) is the restriction mapping for each U2 ⊂ U1 . In view of Theorem 3.10 there exists a restriction mapping sU2 ,U1 : FN (U1 ) → FN (U2 ) for each open U2 ⊂ U1 . Then this presheaf induces a sheaf SW,X1 ,G (N ) of wrap groups, which is the subsheaf of SW,X1 ,X2 ,G . ′ ′ 29. Proposition. Let η : N1 → N2 be an Hpt -retraction of Hpt manifolds, N2 ⊂ N1 , η|N2 = id, y0 ∈ N2 , where t′ ≥ t, M is an Hpt manifold, E(N1 , G, π, Ψ) and ′ E(N2 , G, π, Ψ) are principal Hpt bundles with a structure group G satisfying conditions of §2 in Section 2, then there exists a sheaf homomorphism η∗ from SW,X1 ,G (N1 ) onto SW,X1 ,G (N2 ). Proof. In view of Proposition 3.17 there exists a group homomorphism η∗ (U ) from FN1 (U ) onto FN2 (U ) for each U open in X1 such that {s0,q : q = 1, ..., k} ⊂ M ⊂ X1 . If B is a sheaf on X and ηB(U ) = B(η −1 (U )) for each U open in X, then there exists a sheaf ηB which is called the image of the sheaf B (see [3]). On the other hand, η∗ (U2 ) ◦ sU2 ,U1 = sU2 ,U1 ◦ η∗ (U1 ) for each open U2 ⊂ U1 due to Condition 25(2). Then SW,X1 ,G (N2 ) is the image of SW,X1 ,G (N1 ), that is η∗ SW,X1 ,G (N1 ) = SW,X1 ,G (N2 ), since there exists an Hpt mapping id × η from M × N1 onto M × N2 (see §28). This gives the sheaf homomorphism (see also §3 [3]). 30. Remark. For a continuous mapping f : X → Y and a sheaf B on Y a inverse image f ∗ B is a sheaf on X such that f ∗ B = {(x, q) ∈ X × B : f (x) = π(q)} (see [3]). Particularly, if f : X → Y is an Hpt mapping such that f = (f1 , f2 ), f1 : X1 → Y1 , f2 : X2 → Y2 , then there exists a sheaf inverse image f ∗ SW,Y1 ,Y2 ,G2 , where f1∗ G2 = G1 . 31. Corollary. Let suppositions of Proposition 26 be satisfied, where hj are diffeomorphisms for j = 1, 2 and an isomorphism for j = 3, then SW,X1 ,X2 ,GX1 and SW,Y1 ,Y2 ,GY1 are isomorphic sheaves. Proof. This follows from Proposition 26 and Remark 30. 32. Proposition. Let a sheaf G be an inductive limit ind − limα∈Λ Gα of sheaves Gα , where Λ is a directed set. Then the wrap sheaf SW,X1 ,X2 ,G is the inductive limit ind − limα∈Λ SW,X1 ,X2 ,Gα . Proof. For each U open in X1 and all α < β ∈ Λ there exists a homomorphism α πβ : Gα (U ) → Gβ (U ). Then the sheaf G is defined as the sheaf generated by a pre-sheaf U 7→ ind − limα∈Λ Gα (U ) (see Chapter 1 §5 [3]). Each homomorphism πβα generates the homomorphism of principal bundles from E(N, Gα , π, Ψ) into E(N, Gβ , π, Ψ). In view

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of Proposition 26 for each α < β ∈ Λ and every U open in X1 there exists the group α : S α homomorphism πβ,∗ W,X1 ,X2 ,Gα (U ) → SW,X1 ,X2 ,Gβ (U ) generated by πβ . Thus, there exists SW,X1 ,X2 ,G := ind − limα∈Λ SW,X1 ,X2 ,Gα . 33. Corollary. Let X1 = ind − limα∈Λ X1,α and G = ind − limα∈Λ Gα satisfy conditions of Proposition 26, where Gα = GX1,α and X1,α is an Hpt pseudo-manifold for each α in a directed set Λ. Then SW,X1 ,X2 ,G = ind − limα∈Λ SW,X1,α ,X2 ,Gα . Proof. For an Hpt pseudo-manifold X1 its base of topology consists of all those subsets T U in X1 such that U = m v=1 Uα(v) for some m ∈ N and α(1), ..., α(m) ∈ Λ, where −1 Vα = πα (Uα(v) ) is open in Xα , πα : X1,α → X1 is an embedding for each α in Λ. In view of Proposition 26 for each U open in X1 and α < β ∈ Λ there exists a group α : S homomorphism πβ,∗ W,X1,α ,X2 ,Gα (U ) → SW,X1,β ,X2 ,Gβ (U ). Due to Proposition 32 this generates SW,X1 ,X2 ,G as the inductive limit of sheaves SW,X1,α ,X2 ,Gα . ′ 34. Theorem. Let X2 = X2,1 × X2,2 , where X1 , X1,2 and X2,2 are Hpt and Hpt pseudo-manifolds respectively over Ar as in §25. Then the restriction of the complete ˆ W,X1 ,X2,2 ,G2 on ∆1 × X2 is isomorphic tensor product of wrap sheaves SW,X1 ,X2,1 ,G1 ⊗S with SW,X1 ,X2 ,G , where G is the tensor product G := G1 ⊗G2 of sheaves G1 and G2 on X1 , ∆1 := {(x, x) : x ∈ X1 } is the diagonal in X12 . Proof. If B1 and B2 are sheaves on a topological space X, then B1 ⊗B2 denotes the sheaf on X generated by the presheaf U 7→ B1 (U ) ⊗ B2 (U ), where (B1 ⊗ B2 )x ∼ = B1,x ⊗ B2,x is the natural isomorphism of fibers. The sheaf B1 ⊗B2 is called the tensor product of sheaves. Consider the natural projections φ1 : X2 → X2,1 and φ2 : X2 → X2,2 having extensions id × φ1 : X1 × X2 → X1 × X2,1 and id × φ2 : X1 × X2 → X1 × X2,2 . Therefore, ˆ W,X1 ,X2,2 ,G2 := [(id × φ1 )∗ SW,X1 ,X2,1 ,G1 ] ⊗ there exists the sheaf S := SW,X1 ,X2,1 ,G1 ⊗S ∗ [(id × φ2 ) SW,X1 ,X2,2 ,G2 ] which is the complete tensor product of sheaves (see in general Chapter 1 §5 [3]). If γ : M → X2 is an Hpt mapping preserving marked points, then γ = (γ1 , γ2 ), where γj : M → X2,j for j = 1, 2, γ(s0,q ) = y0 , γj (s0,q ) = yj,0 for each q = 1, ..., k and ˆ → X2 such that γ ◦ Ξ = γˆ (see §§2, j = 1, 2, y0 = y1,0 × y2,0 . Then we get a lifting γˆ : M 3 and 6 in Section 2). Therefore, Pγˆ,u (ˆ s0,k+q ) = Pγˆ1 ,u1 (ˆ s0,k+q ) ⊗ Pγˆ2 ,u2 (ˆ s0,k+q ) ∈ G for each q = 1, ..., k, with G = G1 ⊗ G2 being the direct product of groups for G1 = G1 (U1 ) and G2 = G2 (U2 ) for every Uj open in X1 , j = 1, 2, where u ∈ Ey0 , uj ∈ Ej,yj,0 , N = N1 × N2 , Nj ⊂ Vj ⊂ X2,j , E = E(N, G, π, Ψ), Ej = E(Nj , Gj , πj , Ψj ) are principal bundles, y0 = y0,1 × y0,2 , yj,0 ∈ Nj are marked points, Vj is open in X2,j for j = 1, 2 (see also §25). For classes of equivalent parallel transport structures we get < Pγˆ,u >t,H =< Pγˆ1 ,u1 >t,H ⊗ < Pγˆ2 ,u2 >t,H , hence F (U × (V1 × V2 )) is isomorphic with (φ1 )∗ F (U × V1 ) ⊗ (φ2 )∗ F (U × V2 ) for each U open in X1 and all Vj open in X2,j , j = 1, 2, since open sets of the form V = V1 × V2 form a base of topology in X2 , where F (U × Vj ) is given for the group Gj (U ). Here U = U1 = U2 and (φ1 )∗ F (U ×V1 )⊗(φ2 )∗ F (U ×V2 ) is isomorphic with the restriction of (id × φ1 )∗ F (U × V1 ) ⊗ (id × φ2 )∗ F (U × V2 ) from U 2 × V1 × V2 onto ∆(U ) × V1 × V2 , where ∆(U ) denotes the diagonal in U 2 . Thus, SW,X1 ,X2 ,G is isomorphic ˆ W,X1 ,X2,2 ,G2 with the restriction of the complete tensor product of sheaves SW,X1 ,X2,1 ,G1 ⊗S on ∆1 × X2 . 35. Twisted Alexander-Spanier cohomologies. Let G be a group satisfying Conditions 4(A1, A2), which may be in particular a wrap

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group for Ar pseudo-manifolds with 2 ≤ r ≤ 3. For an open U in X denote by Am (U ; G) a group of all functions f : U m+1 → G with a pointwise multiplication in G as the group operation. Therefore, the functor U 7→ Am (U ; G) is a presheaf on X satisfying the condition: S (S2) if {Uj : j} is a family of open subsets of X such that j Uj = U , then for a family of elements sj ∈ Am (Uj ; G) such that sj |Uj ∩Uk = sk |Uj ∩Uk for each j, k there exists s ∈ Am (U ; G) such that s|Uj = sj for each j. To satisfy this put sj = fj : Ujm+1 → G to be functions here and s = f is their combination such that f |U m+1 = sj , while f on S

j

X m+1 \ ( j Ujm+1 ) is arbitrary. The property S (S1) if U = j Uj , where Uj is open in X and f, g ∈ Am+1 (U ; G) coincide on Ujm+1 S for each j, then f = g on j Ujm+1 is evident, since f, g are functions. Recall that a family φ of closed subsets in X is called a family of supports, if it satisfies conditions (SP 1, SP 2): (SP 1) if B is a closed subset in C, where C ∈ φ, then B ∈ φ; S (SP 2) if B1 , ..., Bm ∈ φ, m ∈ N, then m j=1 Bj ∈ φ. The family φ of supports is called paracompactfying, if satisfies two additional conditions: (SP 3) each element in φ is a paracompact space; (SP 4) each set from φ has a closed neighborhood belonging to φ. S The union C∈φ C =: E(φ) is called a spread of φ. Put Γφ (S) := {s ∈ S(X) : |s| ∈ φ} for a sheaf S on X, where |s| := {x ∈ X : s(x) 6= e} denotes its support. Clearly Γφ (S) is a subgroup in S(X). For a presheaf A on X put Aφ (X) := {s ∈ A(X) : |s| ∈ φ}. For a presheaf A on X put Aφ (X) := {s ∈ A(X) : |s| ∈ φ}. Let now Am (X; G) be a sheaf generated by the presheaf Am (.; G). Define the differential d : Am (U ; G) → Am+1 (U ; G) by the formula: P j ˆj , ..., xm+1 ), where f : U m+1 → G is an df (x0 , ..., xm+1 ) = m+1 j=0 (−1) f (x0 , ..., x P2r −1 ˆ k , {i0 , .., i2r −1 } are generators arbitrary function. Then f = k=0 fk ik , where fk ∈ G of Ar , 2 ≤ r ≤ 3. Hence d is the homomorphism of presheaves and d2 = 0, since P r −1 (dfk )ik . df = 2k=0 Then twisted Alexander-Spanier cohomologies are defined as m m ∗ ∗ AS Hφ (X; G) = H (Aφ (X; G))/A0 (X; G)). 36. Theorem. Let A be a pre-sheaf on X satisfying Condition 35(S2) and S be a sheaf generated by A, where S and A are twisted over {i0 , ..., i2r −1 } with 1 ≤ r ≤ 3. Then for each paracompatifying family φ of supports in X there exists the exact sequence θ e → A0 (X) → Aφ (X) −→ Γφ (S) → e, where θ : A(X) → S(X) is the natural mapping of the presheaf into the generated by it sheaf. Proof. Consider s ∈ Γφ (S) and a neighborhood U of |s| such that cl(U ) ∈ φ, where cl(U ) denotes the closure of U in X. Since cl(U ) is paracompact find a locally finite covering {Uj : j} of cl(U ), where each Uj is open in X and for which there exists sj ∈ A(Uj ) such that θ(sj ) = s|Uj . Let {Vj : j} be a refinement of {Uj : j} such that U ∩ cl(Vj ) ⊂ Uj . For x ∈ X the set J(x) := {j : x ∈ cl(Vj )} is finite, hence for each x ∈ X there exists a neighborhood W (x) such that W (x) ⊂ Uj and for each j ∈ J(x) and every y ∈ W (x) there is the inclusion J(y) ⊂ J(x).

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For j ∈ J(x) we get θ(sj (x)) = s(x). Take W (x) sufficiently small such that sj |W (x) =: sx does not depend on j ∈ J(x), since J(x) is finite, consequently, sx ∈ A(W (x)). Let x, y ∈ U , z ∈ W (x) ∩ W (y) and j ∈ J(z), where J(z) ⊂ J(x) ∪ J(y). Then sx |W (x)∩W (y) = sy |W (x)∩W (y) . Due to Condition (S2) there exists β ∈ A(U ) such that β|W (x) = sx for each x ∈ U , clearly, θ(β) = s|U . Take now C ∈ φ such that |s| ⊂ Int(C) and C ⊂ U , where Int(C) denotes the interior of C. If x ∈ C \ Int(C), then θ(β)(x) = s(x) = 0. Therefore, there exists a covering {Qj } of C \ Int(C) with open in X sets Qj such that Qj ⊂ U and t|Qj = 0 for every j. Choose the open covering {Qj } ∪ {Int(C), X \ C} of X and elements e ∈ A(Qj ), β|Int(C) ∈ A(Int(C)) and e ∈ A(X \ C). Restrictions of each two elements on a common part of their domains of definition coincide. Hence due to Condition (S2) such elements have a common extension q ∈ A(X) and inevitably θ(q) = s and |q| = |θ(q)| = |s| ∈ φ. θ The sequence e → A0 (X) → Aφ (X) −→ Γφ (S) → e is exact, since each subsequence P r −1 ˆ θ e → A0,k (X) → Aφ,k (X) −→ Γφ,k (S) → e is exact, where Aˆφ = 2k=0 Aφ,k ik and P2r −1 ˆ ˆ ˆ Γφ = k=0 Γφ,k ik , where each Aφ,k is commutative and they are pairwise isomorphic for ˆ φ,k are commutative and pairwise isomorphic for different values of different k, as well as Γ k, since the sheaf S is twisted over the group of standard generators {i0 , ..., i2r −1 } of the Cayley-Dickson algebra Ar . Mention, that for a pre-sheaf A satisfying Condition (S1) we have A0 (X) = e. 37. Corollary. Let conditions of Theorem 36 be satisfied. Then for a paracompactifying family φ of supports there exists the natural isomorphism: ∗ ∼ m Hm φ (X; G) = H (Γφ (S (X; G)). Proof. This follows immediately from Theorem 36 and §35. 38. Twisted singular cohomologies. Let B be a locally finite twisted sheaf on X, that is a group B(U ) satisfies Conditions 4(A1, A2, C1, C2) for each U open in X. For U ⊂ X denote by S m (U ; B) the group of singular m-dimensional cochains of the space U with coefficients in B. Each element f ∈ S m (U ; B) is a function posing for each m-dimensional simplex σ : ∆m → U a section f (σ) ∈ Γ(σ ∗ (B)), where ∆m is a standard m-dimensional simplex. The pre-sheaf S m (.; B) satisfies Condition (S2). The sheaf B is locally constant, then the sheaf σ ∗ (B) is constant on ∆m , since the simplex ∆m is simply connected, where m ≥ 1. Therefore, there exists a usual coboundary operator d : S m (U ; B) → S m+1 (U ; B). Consider the sheaf S m (U ; B) generated by a pre-sheaf U 7→ S m (U ; B). Then the differential d in the pre-sheaf induces the differential in the sheaf. For a locally constant sheaf B singular cohomologies with coefficients in B and supports in the family φ are m ∗ defined as ∆ Hm φ (X; B) = H (Sφ (X; B)). Since B is the twisted sheaf over {i0 , ..., i2r −1 }, ∗ then Sφ (X; B) and inevitably ∆ Hm φ (X; B) are twisted over {i0 , ..., i2r −1 }. Let U := {Uj : j} be an open covering of X and let S ∗ (U; B) be a group of singular cochains defined on singular simplices subordinated to the covering U. With the help of the subdivision we get, that the homomorphism bU : S ∗ (X; B) → S ∗ (U; B) induces the isomorphism of cohomologies, consequently, the complex KU = kerbU is acyclic. On S the other hand, S0∗ (X; B) = KU∗ = ind − lim KU , hence H∗ (S0∗ (X; B)) = H∗ (ind − lim KU∗ ) = ind − lim H∗ (KU∗ ) = e.

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Thus for a paracompatifying family of supports from the exactness of the sequence e → S0∗ → Sφ∗ → Γφ (S ∗ ) → e ∗ ∼ m and Theorem 36 it follows the isomorphism ∆ Hm φ (X; B) = H (Γφ (S (X; B))). 39. Twisted differential sheaves. A graded sheaf is a sequence {S m : m ∈ Z} of sheaves, which is called a differential sheaf if there are homomorphisms (1) d : S m → S m+1 such that d2 = 0 for each m. This sheaf may be twisted over {i0 , ..., i2r −1 }, where 2 ≤ r ≤ 3. In this case we suppose that (2) up to an automorphisms θm : S m → S m we have θm+1 ◦ d(Skm ) ⊂ Skm+1 for each k = 0, ..., 2r − 1. A differential sheaf S ∗ having S m = 0 for each m < 0 and supplied with the augmentation homomorphism ε : B → S 0 is called the resolvent of the sheaf, if the sequence ε

d

d

e → B −→ S 0 −→ S 1 −→ S 2 → ... is exact. The notion of differential graded pre-sheaves is formulated analogously. If S m is twisted, that is Sˆm = Sˆ0m i0 + ... + Sˆ2mr −1 i2r −1 , where Sˆkm and Sˆjm are pairwise isomorphic and commutative for each k 6= j, then Ker(d : S m → S m+1 ) and Im(d : S m−1 → S m ) are twisted as well, since up to isomorphisms θm : S m → S m we have θm+1 ◦ d(Skm ) ⊂ Skm+1 for each k = 0, ..., 2r − 1. A sheaf of cohomologies (in another words a derivative sheaf) is defined as Hm (S ∗ ) = Ker(d : S m → S m+1 )/Im(d : S m−1 → S m ). If S ∗ is generated by a differential pre-sheaf S ∗ , then Hm (S ∗ ) is generated by the pre-sheaf U 7→ Hm (S ∗ (U )). For a sheaf B on topological space X and an open subset U ⊂ X denote by Y0 (U ; B) a set of all mappings (may be discontinuous) f : U → B such that π ◦ f = id is the identity Q mapping on U , where π : B → X is the canonical projection. Thus Y0 (U ; B) = x∈U Bx and it is the group with the pointwise group operation. Therefore, U 7→ Y0 (U ; B) is the presheaf satisfying Conditions (S1, S2), hence it is the sheaf which we denote by Y 0 (X; B). If B is twisted, then Y 0 (U ; B) is twisted as well. The inclusion of all continuous sections of B into the family of all sections not necessarily continuous induces the augmentation homomorphism ε : B → Y 0 (X; B). For a family φ of supports put Yφ0 (X; B) = Γφ (Y 0 (X; B). If e → B1 → B2 → B3 → e is a short exact sequence of sheaves (may be twisted), then the sequence of pre-sheaves e → Y0 (X; B1 ) → Y0 (X; B2 ) → Y0 (X; B3 ) → e is exact. If f ∈ Yφ0 (X; B), then its support is |f | := cl{x : f (x) 6= e}. Therefore, f is an image of a section g of the sheaf B such that g is not necessarily continuous and g(x) = e if f (x) = e for x ∈ X, hence |g| = |f | ∈ φ. Denote by Z 1 (X; B) the cokernel of the homomorphism ε such that the sequence e → ε ∂ B −→ Y 0 (X; B) −→ Z 1 (X; B) is exact. Define by induction the sheaves Y m (X; B) = Y 0 (X; Z m (X; B)), Z m+1 (X; B) = Z 1 (X; Z m (X; B)). If B is twisted over {i0 , .., i2r −1 }, then Z 1 (X; B) is twisted as well and by induction Z m (X; B) and Y m (X; B) are twisted for each m ∈ N. Therefore, the sequence e → ε ∂ Z m (X; B) −→ Y m (X; B) −→ Z m+1 (X; B) → e is exact. Consider the composition ∂ ε d = ε ◦ ∂ for Y m (X; B) −→ Z m+1 (X; B) −→ Y m+1 (X; B), then the sequence

Wrap Groups of Connected Fiber Bundles, Their Structure and Cohomologies ε

d

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d

e → B −→ Y 0 (X; B) −→ Y 1 (X; B) −→ Y 2 (X; B) → ... is exact. Thus, Y ∗ (X; B) is the resolvent of the sheaf B, which is called the canonical resolvent. 40. Proposition. The canonical resolvent of the twisted sheaf B is fiberwise homotopically trivial. Proof. Consider the homomorphism Y0 (U ; B) → Bx such that U ∋ x 7→ f (x) ∈ Bx for each f ∈ Y0 (U ; B) and x ∈ U . The direct limit by neighborhoods of a point x induces the homomorphism ηx : Y 0 (X; B)x → Bx , consequently, ηx ◦ ε : Bx → Bx is the identity isomorphism, where ηx ◦ ε(z) = ηx (ε(z)). Define the homomorphism νx : Z 1 (X; B) → Y 0 (X; B) by the formula νx ◦ ∂ = 1 − ε ◦ ηx which defines νx in a unique way. Therefore, there exists a fiber splitting ε

∂

Z m (X; B)x −→ Y m (X; B) −→ Z m+1 (X; B)x and ηx νx Z m (X; B)x ←− Y m (X; B) ←− Z m+1 (X; B)x . Put Dx := νx ◦ ηx : Y m (X; B)x → m−1 Y (X; B)x for m > 0. Therefore, d ◦ Dx + Dx ◦ d = ε ◦ ∂ ◦ νx ◦ ηx + νx ◦ ηx ◦ ε ◦ ∂ = ε ◦ ηx + νx ◦ ∂ = 1 on Y m (X; B) for m > 0. At the same time on Y 0 (X; B)x we have Dx ◦ d = νx ◦ ηx ◦ ε ◦ ∂ = νx ◦ ∂ = 1 − ε ◦ ηx . This means, that Y ∗ (X; B)x is homotopically fiberwise trivial resolvent. 41. Remark. The functor Y 0 (X; B) is exact by B, hence Z 1 (X; B) is also the exact functor by B. Using induction we get, that all functors Y m (X; B) and Z m (X; B) are exact by B. For an arbitrary family φ of supports on X put Yφm (X; B) := Γφ (Y m (X; B)) = Yφ0 (X; Z m (X; B)). Since the functor Y0 (X; ∗) is exact, then the functor Yφm (X; B) is exact. 42. Definition. Cohomologies in X with supports in φ with coefficients in B are defined m ∗ as Hm φ (X; B) := H (Yφ (X; B)). 42.1. Note. The sequence e → Γφ (B) → Γφ (Y 0 (X; B)) → Γφ (Y 1 (X; B)) is exact, consequently, Γφ (B) ∼ = H0φ (X; B). If there is a short exact sequence of twisted sheaves e → B1 → B2 → B3 → e on X, then it implies the exact sequence of cochain complexes e → Yφ∗ (X; B1 ) → Yφ∗ (X; B2 ) → Yφ∗ (X; B3 ) → e, that in its turn induces the long exact sequence δ

m+1 m m ... → Hm (X; B1 ) → .... φ (X; B1 ) → Hφ (X; B2 ) → Hφ (X; B3 ) −→ Hφ 43. Definition. Let G be a topological group satisfying conditions 4(A1, A2, C1, C2) ˆ where 1 ≤ r ≤ 2. Then define the such that G is a multiplicative group of the ring G, s ˆ s := G ˆ ⊗l G, ˆ where smashed product G such that it is a multiplicative group of the ring G ˆ ˆ l = i2r denotes the doubling generator, the multiplication in G ⊗l G is ˆ where v ∗ = (1) (a + bl)(c + vl) = (ac − v ∗ b) + (va + bc∗ )l for each a, b, c, v ∈ G, conj(v). A smashed product M1 ⊗l M2 of manifolds M1 , M2 over Ar with dim(M1 ) = dim(M2 ) is defined to be an Ar+1 manifold with local coordinates z = (x, yl), where x in M1 and y in M2 are local coordinates. 44. Theorem. There exists smashed products S s := S1 ⊗l S2 on X = X1 = X2 and s ˆ ˆ l S2 on X = X1 × X2 over {i0 , ..., i2r+1 −1 } of isomorphic twisted sheaves S1 on S := S1 ⊗ X1 and S2 on X2 over {i0 , ..., i2r −1 } with X1 = X2 , in particular of wrap sheaves, where 1 ≤ r ≤ 2, l = i2r . Proof. If Sj is a sheaf on a topological space Xj twisted over {i0 , ..., i2r −1 }, then ˆ Sj = Sˆ0,j i0 ⊕ ... ⊕ Sˆ2r −1,j i2r −1 , where Sˆk,j (U ) = Sk,j (U ) ∪ {0} are commutative rings

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for each U open in Xj , Sˆk,j are sheaves on Xj pairwise isomorphic for different values of k. Then for X = X1 = X2 take Sxs := (S1 )x ⊗l (S2 )x for each x ∈ X in accordance with Definition 43, that defines the twisted sheaf S on X over {i0 , ..., i2r+1 −1 } due to Proposition 3.19. This sheaf S is the smashed tensor product of sheaves. ˆ l S2 = (π1∗ S1 ) ⊗l (π2∗ S2 ), which is the smashed If X = X1 × X2 , then take Sˆs := S1 ⊗ complete tensor product of sheaves, where π1 : X → X1 and π2 : X → X2 are projections. 45. Corollary. Let X2 = X2,1 ⊗l X2,2 be the smashed product, where X1 and X2 are ′ Hpt and Hpt pseudo-manifolds respectively over Ar+1 , 1 ≤ r ≤ 2. Then the restricˆ l SW,X1 ,X2,2 ,G tion the smashed complete tensor product of wrap sheaves SW,X1 ,X2,1 ,G ⊗ s on ∆1 × X2 is isomorphic with SW,X1 ,X2 ,G s , where G is the smashed tensor product G s := G⊗l G twisted over {i0 , ..., i2r+1 −1 } of a sheaf G twisted over {i0 , ..., i2r −1 } on X1 , ∆1 := {(x, x) : x ∈ X1 } is the diagonal in X12 . Proof. The smashed product of manifolds was described in details in the proof of Theorem 3.20. Consider an Ar shadow of X1 that exists, since Ar+1 = Ar ⊕ Ar l, where l = i2r . For each U open in X1 there exists a group G(U ), hence G(U ) ⊗l G(U ) is defined due to Proposition 3.19, that gives the sheaf G s on X1 . Then wrap sheaves SW,X1 ,X2,b ,G over Ar are defined, where b = 1, 2. Thus the statement of this corollary follows from Proposition 3.19 and Theorem 44, modifying the proof of §34 for the smashed complete tensor product instead of complete tensor product so that Pγˆ,u (ˆ s0,k+q ) = Pγˆ1 ,u1 (ˆ s0,k+q )⊗l s s Pγˆ2 ,u2 (ˆ s0,k+q ) ∈ G with E = E(N, G , π, Ψ), where G = G(U ), U = U1 = U2 , consequently, < Pγˆ,u >t,h =< Pγˆ1 ,u1 >t,H ⊗l < Pγˆ2 ,u2 >t,H . 46. Corollary. Let X1 = X1,1 ⊗l X1,2 and X2 = X2,1 ⊗l X2,2 are smashed products, ′ where X1 , and X2 are Hpt and Hpt pseudo-manifolds respectively over Ar+1 , 1 ≤ r ≤ 2. Then the wrap sheaf SW,X1 ,X2 ,G s is twisted over {i0 , ..., i2r+1 −1 } and is isomorphic with the smashed complete tensor product of twice iterated wrap sheaves ˆ l SW,X1,2 ,X2,2 ,SW,X ,X ,G , SW,X1,2 ,X2,1 ,SW,X1,1 ,X2,1 ,G ⊗ 1,1 2,2 s where G is the smashed tensor product G s := G⊗l G of a twisted sheaf G over {i0 , ..., i2r −1 } on X1 . Proof. Consider projections πb,j : Xb → Xb,j , where j, b = 1, 2. Each Ar+1 manifold has the shadow which is the Ar manifold, since Ar+1 = Ar ⊕ Ar l. If U is open in X1,j , −1 −1 then π1,j (U ) is open in X1 and there exists a group G(π1,j (U )), where j = 1, 2. −1 Hence there exist the projection sheaves Gj = π1,j G on X1,j induced by G such that −1 Gj (U ) := G(π1,j (U )). Denote Gj on X1,j also by G, since Gj is obtained from G by taking the specific subfamily of open subsets. For U1 open in X1,1 and U2 open in X1,2 take U = U1 × U2 open in X1 . The family of all such subsets gives the base of the topology in X1 . ˆ ) ⊗l G(U ˆ ) =: Gˆs (U ), that induces In accordance with Definition 43 there exists G(U G s on X1 such that Gˆxs = Gˆx ⊗l Gˆx for each x ∈ X1 . Therefore, every element q + vl ˆ ). Thus the statement of this corollary follows from §25, is in Gˆs (U ) for each q, v ∈ G(U Theorems 3.20 and 44. 47. Consider now the iterated wrap sheaf SW,X1 ,X2 ,G;b of iterated wrap groups (W M E)b,∞,H with b ∈ N instead of wrap groups for b = 1 such that for its presheaf Q (1) Fb (U × V ) = s0,1 ,...,s0,k ∈M ⊂U ;y0 ∈N ⊂V (W M,{s0,q :q=1,...,k} E; N, G(U ), P)b;∞,H , where sU2 ,U1 : G(U1 ) → G(U2 ) is the restriction mapping for each U2 ⊂ U1 so that the parallel transport structure for M ⊂ U is defined, where G is the sheaf on X1 , G(U ) =

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G(U ), pseudo-manifolds X1 and X2 and the sheaf G are of class Hp∞ (see also §25). Corollary. There exists a homomorphism of of iterated wrap sheaves θ : SW,X1 ,X2 ,G;a ⊗ SW,X1 ,X2 ,G;b → SW,X1 ,X2 ,G;a+b for each a, b ∈ N. Moreover, if G is either associative or alternative, then θ is either associative or alternative. Proof. For pre-sheaves the mapping (2) θ : Fa (U × V ) ⊗ Fb (U × V ) → Fa+b (U × V ) is induced by Formula 47(1) and due to Theorem 3.21. Then θ has the extension on the sheaf of iterated wrap groups, since (SW,X1 ,X2 ,G;a )z = ind − lim Fa (U × V ), where the direct limit is taken by open subsets U × V for a point z = x × y ∈ X1k × X2 , x ∈ X1k , y ∈ X2 , such that x ⊂ U , y ∈ V , U is open in X1 , V is open in X2 . The inductive limit topology in (SW,X1 ,X2 ,G;a )z is the finest topology relative to which each embedding Fa (U × V ) ֒→ (SW,X1 ,X2 ,G;a )z is continuous. If f ∈ (SW,X1 ,X2 ,G;a )z and g ∈ (SW,X1 ,X2 ,G;b )z , then there exist open U1 × V1 and U2 × V2 such that f ∈ Fa (U1 × V1 ) and g ∈ Fb (U2 ×V2 ), consequently, f ∈ Fa (U ×V ) and g ∈ Fb (U ×V ), where U = U1 ∪U2 and V = V1 ∪ V2 , hence θ(f, g) ∈ Fa+b (U × V ). From (2) and the definition of the inductive limit topology it follows, that θ is continuous, since on iterated wrap groups θ is Hp∞ differentiable. Moreover, in accordance with Theorem 3.21 θ is either associative or alternative if G is associative or alternative. 48. Note. Let φ be a family of supports in X and B be a sheaf on X, where B may be twisted. A sheaf B is called φ-acyclic, if Hbφ (X; B) = 0 for each b > 0. Let L∗ be a resolvent of B. Put Z b := Ker(Lb → Lb+1 ) = Im(Lb−1 → Lb ), where 0 Z = B. An exact sequence (1) e → Z b−1 → Lb−1 → Z b → e induces an exact sequence (2) e → Γφ (Z b−1 ) → Γφ (Lb−1 ) → Γφ (Z b ) → H1φ (X; Z b−1 ). Therefore, there exists the monomorphism (3) Hb (Γφ (L∗ )) = Γφ (Z b )/Im(Γφ (Lb−1 → Γφ (Z b )) → H1φ (X; Z b−1 ). Moreover, the sequence e → Z b−v → Lb−v → Z b−v+1 → e induces the homomorphism: b−v+1 ) → Hv (X; Z b−v ). (4) Hb−1 φ φ (X; Z Define κ as the composition (5) Hb (Γφ (L∗ )) → H1φ (X; Z b−1 ) → H2φ (X; Z b−2 ) → ... → Hbφ (X; Z 0 ). If all sheaves Lb are φ-acyclic, then (3, 4) are isomorphisms. We call κ natural, if from the commutativity of the diagram: B −→ L∗ ↓f ↓g E −→ M∗ where g is a homomorphism of resolvents the commutativity of the diagram κ

Hb (Γφ (L∗ ) −→ Hbφ (X; B) ↓ g∗ ↓ f∗ κ Hb (Γφ (M∗ ) −→ Hbφ (X; E)

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follows. Thus we get the statement. 48.1. Theorem. If L∗ is the resolvent of the sheaf B, consisting of φ-acyclic sheaves, then for each b ∈ N the natural mapping κ : Hb (Γφ (L∗ ) → Hbφ (X; B) is the isomorphism. In view of the latter theorem if g : L∗ → M∗ is the homomorphism of two resolvents of the sheaf B consisting of φ-acyclic sheaves, then the induced mapping Hb (Γφ (L∗ )) → Hb (Γφ (M∗ )) is an isomorphism. 48.2. Corollary. If e → L0 → L1 → L2 → ... is an exact sequence of φ-acyclic sheaves, then the corresponding sequence e → Γφ (L0 ) → Γφ (L1 ) → Γφ (L2 ) → ... is exact. Proof. In view of Theorem 48.1 Hb (Γφ (L∗ )) = Hbφ (X; e). On the other hand, Yφn (X; e) = e, since Y 0 (X; e) = e and hence Y n (X; e) = e for all n, consequently, Hb (Γφ (L∗ )) = e for each b. 49. Differential forms and twisted cohomologies over octonions. A bar resolution exists for any sheaf or a complex of sheaves. Consider differential forms on N . In local coordinates write a differential k-form as P (1) w = J fJ (z)dxb1 ,j1 ∧ dxb2 ,j2 ∧ ... ∧ dxbk ,jk , where fJ : N → Ar , z = (z1 , z2 , ...) are local coordinates in N , zb = xb,0 i0 + xb,1 i1 + ... + xb,2r −1 i2r −1 , where zb ∈ Ar , xb,j ∈ R for each b and every j = 0, 1, ..., 2r − 1, k J = (b1 , j1 ; b2 , j2 ; ...; bk , jk ). For the sheaf SN,A of germs of Ar valued k-forms on N r has a bar resolution: σ σ σ k k k (2) 0 → SN,A −→ SN,AA −→ SN,ABA −→ ..., r r r k m where SN,AB m A denotes the sheaf of germs of AB Ar valued k-forms on N . r Denote by Z(q, Cr ) the group analogous to Z(Cr ) with u ∈ Cr replaced on uq , where uq is considered as equivalent with (−u)q , q ∈ N. Therefore, the exponential sequence η exp (3) 0 → Z(Cr )N −→ C∞ (N, Ar ) −→ C∞ (N, A∗r ) → 0 can be considered as a quasi-isomorphism: Z(Cr )N ↓ 0

η

−→ C∞ (N, Ar ) ↓ exp ∞ −→ C (N, A∗r )

∞ ∞ ∗ between the complex Z(Cr )∞ D : Z(Cr )N → C (N, Ar ) and the sheaf C (N, Ar ) of germs ∞ ∗ ∞ ∗ of C functions from N into Ar placed in degree one, that is C (N, Ar )[−1], where η(z) = 2πz for each z and exp(0) = 1 (see also §19), Ar is considered as the additive group (Ar , +), while A∗r is the multiplicative group (A∗r , ×). More generally this gives the quasi-isomorphism: d d d q−1 1 (4) Z(1, Cr )N −→C∞ (N, Ar )−→SN,A −→...−→SN,A and r r

0 −→ C∞ (N, A∗r )

dLn

−→

d

d

q−1 1 SN,A −→...−→SN,A r r

e

id

1 with vertical homomorphisms Z(1, Cr )N → 0, C∞ (N, Ar ) −→ C∞ (N, A∗r ), SN,A −→ r id

q−1 q−1 1 −→ SN,A for 2 ≤ q ∈ N, where e(f ) := exp(f ) between a degree q SN,A ,...,SN,A r r r smooth twisted complex d d d q−1 ∞ 1 (5) Z(Cr )∞ D : Z(Cr )N → C (N, Ar )−→SN,Ar −→...−→SN,Ar and the complex S
Wrap Groups of Connected Fiber Bundles, Their Structure and Cohomologies dLn

d

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d

q−1 1 (6) S
tb

N b ∗ N ∗ (8) h Hb (S
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so that wk and νk,j and ηk,j,l are H ∞ differential 1-forms. Then from (6 − 8) it follows, that (9) wk − wl + νk,l = wk − wj + νk,j + wj − wl + νj,l + ηk,j,k and hence (10) ηk,j,l = νk,l − νk,j − νj,l . Generally ηk,j,l may be non-zero because of non-commutativity or non-associativity. In view of the alternativity of the octonion algebra O the identities eM eN = eK , eM = K e e−N , eN = e−M eK and e−K = e−N e−M are equivalent, that leads to the identities: (11) M = K(K(M, N ), −N ), N = K(−M, K(M, N )), K(M, N ) = −K(−N, −M ), where M, N, K are purely imaginary octonions, moreover, K(M, 0) = M , K(0, N ) = N , since e0 = 1. Let E(N, A∗r , π, Ψ) be an H ∞ principal A∗r -bundle with transition functions {gk,j : Vk ∩ Vj → A∗r : k, j} and consider a family {wk , νk,j , ηk,j,l : k, j, l} of 1-forms related by 1 1 Equations (6 − 8) so that wj ∈ Γ(Vj , SN,A ), νk,j ∈ Γ(Vk ∩ Vj , SN,A ) for Vk ∩ Vj 6= ∅, r r 1 ηk,j,l ∈ Γ(Vk ∩ Vj ∩ Vl , SN,Ar ) for Vk ∩ Vj ∩ Vl 6= ∅, where k, j, l ∈ J. Consider a C ∞ partition of unity {fj : j ∈ J} subordinated to the covering V. Then (12) −w(x) = |fj0 , fj1 , ..., fjn , −wj0 (x), −wj1 (x), ..., −wjn (x)| and (13) −ν(x) = |fj0 fk0 , fj1 fk1 , ..., fjn fkn , −νj0 ,k0 (x), −νj1 ,k1 (x), ..., −νjn ,kn (x)| and (14) −η(x) = |fj0 fk0 fl0 , fj1 fk1 fl1 , ..., fjn fkn fln , −ηj0 ,k0 ,l0 (x), −νj1 ,k1 ,l1 (x), ..., −νjn ,kn ,ln (x)|, where wj (x) and νj,k (x) and ηj,k,l (x) denote the restriction of wj and νj,k and ηj,k,l to Tx N so that wj (x) and νj,k (x) and ηk,j,l (x) are AAr -valued 1-forms on N , (15) π∗ (−w(x)) = |fj0 , fj1 (x), ..., fjn (x); [wj0 (x) − wj1 (x) + νj0 ,j1 (x)|...|wjn−1 (x) − wjn (x) + νjn−1 ,jn ]|, where π : EAr → BAr is the standard projection. The principal G-bundle E(N, G, π, Ψ) is a pull-back of the universal bundle AG → BG by a classifying mapping gE(N,G,π,Ψ) : N → BG. In terms of transition functions (16) gE(N,A∗r ,π,Ψ) = |fj0 (x), fj1 (x), ..., fjn (x); [gj0 ,j1 (x)|gj1 ,j2 (x)|...|gjn−1 ,jn (x)]|. Therefore, (17) π∗ (w) + dLn(gE(N,A∗r ,π,Ψ) ) = 0, where for any differentiable function g : U → BA∗r we have g(x) = |f0 (x), f1 (x), ..., fn (x); [g1 (x)|...|gn (x)]|. While (18) dLn(g(x)) := |f0 (x), f1 (x), ..., fn (x); [dLn(g1 (x))|...|dLn(gn (x))]|. ∗,

σ(g) = 0 which means that g is a differentiable mapping from N into B b−1 Ar ; σ(w1 ) + dLn(g) = 0 means that w1 is a connection on the differentiable principal b−2 B A∗r -bundle over N induced by g; σ(wj+1 ) + (−1)j dwj = 0 serves as the definition of a (j + 1)-connection on a differentiable principal B b−2 A∗r -bundle E → B associated with the mapping g for 1 ≤ j ≤ b − 2. Then the sequence (g, w1 , ..., wj ) is called the j-connection bar cocycle. There exists an equivalence relation in the group of differentiable principal B b−2 A∗r bundles with (b − 1)-connections which is induced by the cohomology equivalence rela∗,
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a group E(N, B b−2 A∗r , ∇b−1 ) of equivalence classes of differentiable principal B b−2 A∗r bundles with (b − 1)-connections. An assignment (g, w1 , w2 , ..., wb−1 ) 7→ (−1)b−1 dwb−1 induces a homomorphism b (N ) called the curvature of the b-connection K : E(N, B b−2 A∗r , ∇b−1 ) → SA r (g, w1 , w2 , ..., wb−1 ). The kernel ker(K) is isomorphic to the group E(N, B b−2 A∗r , ∇f lat ) of isomorphism classes of differentiable principal B b−2 A∗r -bundles with flat connections. 51. Curvature of holonomy. If v, w ∈ T0 Rn , put (1) γv,w (u) = 4uv for 0 ≤ u ≤ 1/4, γv,w (u) = v + 4(u − 1/4)w for 1/4 ≤ u ≤ 1/2, γv,w (u) = w −4(u−3/4)v for 1/2 ≤ u ≤ 3/4, γv,w (u) = 4(1−u)w for 3/4 ≤ u ≤ 1 and s (u) := γ γv,w sv,sw (u), where 0 ≤ u, s ≤ 1. For a sequence of vectors w = (w0 , w1 , ..., wq ) in T0 Rn with q ∈ N define a (q + 1)-dimensional parallelepiped p[w0 , ..., wq ] in the Euclidean space Rn with q < n if w0 , ..., wq are linearly independent. Then define γw0 ,w1 ,w2 (u1 , u2 ) := γγw0 ,w1 (u1 ),w2 (u2 ) and by induction s (u , ..., u ) := (2) γw0 ,...,wq (u1 , ..., uq ) = γγw0 ,...,wq−1 (u1 ,...,uq−1 ),wq (uq ) and γw 1 q γsw (u1 , ..., uq ), where 0 ≤ u1 , ..., uq , s ≤ 1. This gives the natural parametrization of the parallelepiped p[w0 , ..., wq ] and the mapping γw : ∂I q+1 → Rn which is continuous and piecewise C ∞ . Denote by ej = (0, ..., 0, 1, 0, ..., 0) the standard orthonormal basis in Rn with 1 in the j-th place. Put Ln(diag(a1 , ..., ak )) := diag(Ln(a1 ), ..., Ln(ak )), where Ln is the principal branch of the logarithmic function with Ln(1) = 0 and diag(a1 , ..., an ) is the diagonal matrix with entries a1 , ..., ak ∈ A∗r . If h is an (A∗r )k -valued C n holonomy or an homomorphism for a wrap group ˆ being Hp∞ diffeomorphic with ∂I m+1 and ψ = (y1 , ..., yn ) is a (W M E)∞,H with M coordinate system centered at y, ψ : V → Rn , V is an open neighborhood of a point y in N , then a curvature of h at y is a q-form P (3) Ky := 1≤j1 <...<jq ≤n Kj1 ,j2 ,...,jq (y)dyj1 ∧ dyj2 ∧ ... ∧ dyjq ∈ Λq Ty∗ N , where (4) Kj1 ,....,jq (y) = (−1)q lims→0 Ln[h(ψ −1 (γesj ,...,ejq ))]s−q−1 , 1 where m ≥ q. ˆ being Consider the inversion (wj , wj+1 ) 7→ (wj+1 , wj ). In view of Theorem 3.2 for M ∞ m+1 Hp diffeomorphic with ∂I using the iterated loops and the mapping uj 7→ (1 − uj ) we get, that (5) Ky (wg(1) , ..., wg(q+1) ) = (−1)|g| Ky (w1 , ..., wq+1 ), where g ∈ Sq+1 , Sq denotes the symmetric group of the set {1, ..., q}, |g| = 1 for odd g, while |g| = 2 for an even transposition g. 52. Remark. Consider an H ∞ manifold N and a pseudo-manifold X. A mapping γ : X → N is called piecewise C ∞ or H ∞ smooth if it is continuous and the restriction of γ to each top dimensional simplex of X is a C ∞ or H ∞ mapping. A piecewise smooth mapping γ : X → N is called an oriented singular pseudo-manifold q-cycle, if X is an ψ oriented pseudo-manifold q-cycle. Denote by Zψ q (N ) := Zq (X, N ) the group of oriented singular pseudo-manifold q-cycles in N . If there exists an oriented pseudo-manifold with boundary (Y, ∂Y ) with a pseudodiffeomorphism η : ∂Y → X and a piecewise smooth mapping ζ : Y → N such that γ = ζ|∂Y ◦ η −1 , where γ is an oriented singular pseudo-manifold q-cycle, then γ is called ψ an oriented singular pseudo-manifold q-boundary in N . Denote by Bψ q (N ) := Bq (X, N ) the group of oriented singular pseudo-manifold q-boundaries in N .

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Two oriented singular pseudo-manifold q-cycles γj : Xj → N , j = 1, 2, are homologous, if there exists an oriented (q + 1)-dimensional pseudo-manifold with boundary (Y, ∂Y ) and a piecewise differentiable mapping ζ : Y → N such that ∂Y is isomorphic with X1 ∪ X2 and ζ|Xj = γj up to an isomorphism ∂Y ∼ = X1 ∪ X2 for j = 1, 2. ψ (N )/Bψ (N ) of homology classes of oriented Then there exists the group Hψ (N ) = Z q q q singular pseudo-manifold q-cycles in N , where the group structure is given by the disjoint union. Consider a twisted Ar analog of Cheeger-Simons differential group functor consisting q+1 ∗ of pairs (h, α) ∈ Hom(Zψ q (N ),R Ar ) × SAr (N )0 satisfying the condition q (CS) h(∂η) = exp((−1) η α) for each η ∈ Sq+1 (N ), where Sq (N ) is the group of smooth singular q-chains in N , Sq+1 Ar (N )0 denotes the group of closed differential Ar -valued q-forms on N with 2πZ(Cr )-integral periods belonging to ˆ q (N, Z(Cr )) of Ir = {z ∈ Ar : Re(z) = 0}, 1 ≤ r ≤ 3. The Cheeger-Simons group H ψ degree q differential characters on N consists of homomorphisms h described above. ˜ ψ˜ Suppose that X is an Hp∞ pseudo-manifold. Construct quotients Zψ q (N ) and Bq (N ) as ψ quotients of Zψ q (N ) and Bq (N ) by the equivalence relation: (E1) if γ : X → N is an oriented singular pseudo-manifold q-cycle and ξ is a homeomorphism of X such that its restrictions on all top dimensional simplices of a refinement of a triangulation T of X is an Hp∞ diffeomorphism, then γ ∼ γ ◦ ξ and as a class of equivalent elements take < γ >∞,H which is the closure relative to the Hp∞ -uniformity of the family of all such γ ◦ ξ. In view of the Morse and the Sard theorems (see §§II.2.10, ˜ ψ˜ ψ˜ 11 [7]) if δ ∈< γ >∞,p , then δ is homologous to γ. Put Hψ q (N ) := Zq (N )/Bq (N ), then ˜ Hψ (N ) ∼ = Hψ (N ) are isomorphic. q

q

53. Higher twisted holonomies. Suppose that E(N, BA∗r , π, Ψ) is a differentiable principal BA∗r -bundle with a classifying mapping g : N → B q A∗r and a q-connection (g, w1 , ..., wq ), where 2 ≤ r ≤ 3. Consider a q-dimensional orientable closed pseudomanifold X over Ar and γ : X → N an H ∞ mapping. We have that B q A∗r is q-connected and g ◦ γ : X → B q A∗r is homotopic to a constant mapping. This implies an existence of a differentiable mapping g ◦ γ : X → AB q−1 A∗r with π ◦ g ◦ γ = g ◦ γ, where π : AB q−1 A∗r → B q A∗r . On the other hand, π∗ (γ ∗ w1 + dLng ◦ γ) = π∗ γ ∗ w1 + dLn(g ◦ γ) = γ ∗ (π∗ w1 + dLn(g)) = 0, then (γ ∗ w1 + dLn(g ◦ γ) is a BAr -valued 1-form on X. The projection π : AAr → BAr induces the surjective homomorphism π∗ : j SAAr (X) → SjBAr (X) for each j = 1, 2, .... Therefore, there exists an AAr -valued 1form w ¯j ∈ SjAAr (X) satisfying the equation: π∗ w ¯1 = γ ∗ w1 + dLn(g ◦ γ). Since σ(γ ∗ w2 − dw ¯1 ) = σγ ∗ w2 − dγ ∗ w1 = γ ∗ (σw2 − dw1 ) = 0, then γ ∗ w2 − dw1 is an Ar -valued 2-form on X. By induction we get, that there exists a differential j-form w ¯j ∈ SjAAr (X) such that π∗ w ¯j = γ ∗ wj + (−1)j−1 dγ ∗ wj−1 for each j = 2, ..., q. We have that σ(γ ∗ wj + j−1 (−1) dw ¯j−1 ) = σγ ∗ wj + (−1)j−1 dγ ∗ wj−1 = γ ∗ (σwj + (−1)j−1 dwj−1 ) = 0, conse∗ quently, γ wj + (−1)j−1 dw ¯j−1 is an Ar -valued j-form on X. The holonomy of the q-connection (g, w1 , ..., wq ) along γ : X → N is given by R ¯q−1 )). h(γ) = exp( X (γ ∗ wq + (−1)q−1 dw If there is some other lift w ˆj−1 , then w ˆj−1 = w ¯j−1 + vj−1 , where vj−1 is a is an Ar -valued

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(j −R 1)-form on X. Therefore, R R ∗ j−1 dw ¯j−1 ) + (−1)j−1 X dvj−1 = ˆj−1 ) = X (γ ∗ wj + (−1)j−1 dw X (γ wj + (−1) R ∗ j−1 dw ¯j−1 ) in the considered here case of X with ∂X = 0. X (γ wj + (−1) This holonomy can be generalized in an abstract way for an equivalence class η of a q-connection (g, w1 , ..., wq ) along a singular oriented pseudo-manifold X of dimension q with an H ∞ mapping γ : X → N such that hη (γ) ∈ Ar . Define q+1 (X \ S , Z(C )(q + 1)∞ ), where S Hq+1 (X, Z(Cr )(q + 1)∞ r X X is a singularD ) := H D q (X, A∗ ). Since ∼ ity of X. If the dimension of X is q, then Hq+1 (X, Z(Cr )(q + 1)∞ ) H = r D codim(SX ) ≥ 2, then Hq (X, A∗r ) has a fundamental class that induces an integration along the fundamental class isomorphism and Hq (X, A∗r ) ∼ = A∗r . Thus we get the isomorphism q q+1 ∞ ∗ TX : H (X, Z(Cr )(q + 1)D ) → Ar . Therefore, hη (γ) = TXq (γ ∗ (η)) is the holonomy of a q-connection corresponding to an element η ∈ Hq+1 (N, Z(Cr )(q + 1)∞ D ) along γ : X → N for a singular oriented Ar pseudo-manifold φ : X → N of a real dimension q, where 2 ≤ r ≤ 3. 54. Twisted cohomology. Consider a twisted sheaf B over {i0 , ..., i2r −1 }. Then a ˇ twisted analog of an Alexander-Spanier (or of isomorphic Cech) cohomology with coeffi∗ ∗ cients in B and supports in the family φ is AS Hφ (X; B) = H (Γφ (S ∗ ⊗ B)).
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S.V. Ludkovsky

a locally constant A∗r -valued cochain. Then the mapping η 7→ g¯jq induces the isomorphism
d


ψ

N,Ar

h(η) := (hη , K η ), where hη is the holonomy of η and K η denotes the curvature of η. If η = (gjq , wjq−1 , ..., wj0 ), then K η = K(gjq , wjq−1 , ..., wj0 ) = dwj0 , consequently, the right hand side of the above diagram commutes. ∗ The universal coefficient theorem isomorphism u : Hq (N, A∗r ) → Hom(Hψ q (N ), Ar ) ∞ is induced by a pairing assigning to a Hp mapping γ : M → N and a cohomology class η ∈ Hq (N, A∗r ) an octonion or quaternion number tqM ◦ γ ∗ (η), where tqM denotes the P q restriction of TM to Hq (N, A∗r ). Therefore, for η ∈ Hq (N, A∗r ) and a q-cycle j nj γj , P Q where γj : M → N we get u(η)( j nj γj ) = tqM ( j γj∗ (η)nj ). q From the equalities hη (γ) = u(η)(γ) and TM = tqM ◦ eqM and eqM ◦ iqM = id for an arbiq q ∗ q q q ∗ trary Hp∞ mapping γ : M → N it follows that hiN (η) (γ) = TM γ iN (η) = TM iM γ (η) = q q q ∗ q ∗ q ˆ tM eM iM γ (η) = tM γ (η) = iN u(η)(γ). Since h is the homomorphism, then the left hand side square of the diagram is commutative as well. ˆ q (N, Z(Cr )) is a holon56. Remark. In view of Theorem 55 every element of H ψ omy homomorphism. The operator TXq in the definition of the holonomy uses the integration which is invariant under the equivalence relation 52(E1). Then the quotient ψ˜ ˆq ˆq mapping Zψ q (N ) → Zq (N ) induces an isomorphism Hψ˜ (N, Z(Cr )) → Hψ (N, Z(Cr )), ˆ q (N, Z(Cr )) consists of pairs (h, v) ∈ Hom(Zψ˜ (N ), A∗ ) × S q+1 (N )0 so that where H

q r Ar ˜ ψ h(∂ζ) = ζ v) for each ∂ζ ∈ Bq (N ). A set theoretic inclusion Hp∞ (M, N ) → Zψ q (M, N ), where q is a dimension of M , ˜ M induces a group homomorphism κ : (W N )t,H → Zψ q (M, N ). q Denote by LN,A∗r the sheaf associated with the pre-sheaf U 7→ {γ ∈ Hom∞ ((W M N )∞,H , A∗r ) : supp(γ) ⊂ U }. Section 53 and Theorem 55 imply that K h is an 2πZ(Cr )-integral closed (q + 1)-form on N . 56.1. Lemma. For each Hp∞ mapping ζ : Y → N , where (Y, ∂Y ) is a pseudoψ˜

exp((−1)q

R

ˆ : manifold with boundary ∂Y being a pseudo-manifold over Ar and for each extension h ψ˜ M kb ∞ Zqb (M, N ) → G of an Hp differentiable homomorphism h : (W E)b;∞,H → Gkb ˆ q (N, Z(Cr )) there is the identity: being an element of H ψ˜

R ˆ h(∂ζ) = exp((−1)q ζ K h ), where b ∈ N, E = E(N, G, π, Ψ), G is a commutative subgroup in A∗r , G is isomorphic with C∗ .

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(Y, ∂Y ) and an Hp∞ mapping ζ : Y → N take a partition T of Y into small cubes Qj . Q ˆ ˆ Q ), since From the cancellation property of holonomies we get that h(∂ζ) = Qj ∈T h(γ| j Q ˆ is an extension of h, hence ˆ Q ) = G is commutative. On the other hand, h h(γ|

Q

Qj ∈T h(γ|Qj ). in Am r × R such

Qj ∈T

j

Thus the proof reduces to the case of Y being a (q + 1) dimensional cube q+1 . that ∂Y is embedded into Am r and has the real shadow ∂[0, 1] If µ is a Borel measure on Y relative to which the Sobolev uniformity is given, then µ(YS ) = 0, since codim(YS ) ≥ 2, where SY is the singularity of Y . Moreover, a Lebesgue ′ measure on Rq+1 induces µ on Y using the fact that Y \ YS is an H t -manifold with t′ > [(q + 1)/2] + 1. For matrix-valued over Ar differential forms w = (wj,k : j, k = 1, ..., m) put R R ξ w = ( ξ wj,k : j, k = 1, ..., m), for diagonal matrices (a1 , ..., am ) put exp(a1 , ..., am ) := (ea1 , ..., eam ), if a1 6= 0,...,am 6= 0, then Ln(a1 , ..., am ) = (Ln(a1 ), ..., Ln(am )). Without loss of generality h is additive and R homogeneous on Zq (N ). For each n ∈ N divide [0, 1] into n small subintervals, that induces a subdivision of [0, 1]q+1 into nq+1 cubes with vertices denoted by vj1 ,...,jq+1 (n), where j1 , ..., jq+1 = 0, 1, ..., n. Consider the wrap γjn1 ,...,jq+1 := γe1 /n,...,eq+1 /n + vj1 ,...,jq+1 (n), where e1 , ..., eq+1 is the standard basis of Rq+1 . Take ξ ∈R Hp∞ (Y, E) such that π ◦ ξ = γ. Therefore, R P ∗ ∗ −q−1 j1 ,...,jq+1 ξ K(vj1 ,...,jq+1 (n))n ξ K = Y ξ K = limn→∞ P = (−1)q limn→∞ j1 ,...,jq+1 lims→0 [Ln h(ξ ◦ γjn1 ,...,jq+1 )]s−q−1 n−q−1 , where ξ ∗ Ky = ξ ∗ K(y)dx1 ∧ ... ∧ dxq+1 for each y ∈ N . Taking s = 1/n gives P limn→∞ lims→0 j1 ,...,jq+1 [Ln h(ξ ◦ γjn1 ,...,jq+1 )]s−q−1 n−q−1 P = limn→∞ j1 ,...,jq+1 Ln h(ξ ◦ γjn1 ,...,jq+1 ) Q P = limn→∞ Ln( j1 ,...,jq+1 h(ξ ◦ γjn1 ,...,jq+1 )) = limn→∞ Ln h( j1 ,...,jq+1 ξ ◦ γjn1 ,...,jq+1 ) n = limn→∞ Lnh(ξ ◦ γ0,...,0 ) = Ln h(γ), −1 since h(γ1 λλ γ2 ) = h(γ1 γ2 ) and G is commutative, where λ : Y → N is a path joining marked points y1 and y2 of wraps γ1 and γ2 , that is γj (ˆ s0,q ) = yj and λ(ˆ s0,q ) = y1 , λ(ˆ s0,q+k ) = y2 while λ−1 (ˆ s0,q ) = y2 and λ−1 (ˆ s0,q+k ) = y1 for each j = 1, 2 and q = 1, ..., k. 57. Lemma. Suppose that φ : A ⊂ X is a pointed inclusion of CW-complexes and θ : X → X/A is the quotient mapping. Let a group G be twisted over {i0 , ..., i2r −1 }. Then θ∗ : (W M E; X, G, P)t,H → (W M E; X/A, G, P)t,H is a principal (W M E; A, G, P)t,H bundle. Proof. Let G, E and B be topological groups so that G acts effectively on E. Consider U open in B with e ∈ U . Suppose that π : E → B is an open surjective mapping. Each G-equivariant mapping ξ : π −1 → G induces a local trivialization of π : E → B over U . A group structure in E induces a system of local trivializations of E/B. It is described as follows. For each v ∈ E take an open subset Uv = π(vπ −1 (U )) in B. Then the family {Uv : v ∈ E} forms an open covering of B. For each v ∈ E there exists a G-equivariant mapping ξv : π −1 (Uv ) = vπ −1 (U ) → G given by ξv (x) = ξ(v −1 x). Therefore, an open surjective mapping π : E → B is a principal G-bundle if and only if there exists a neighborhood U of the unit element e in B and a G-equivariant mapping

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ξ : π −1 (U ) → G. Since the group G is twisted, then due to Proposition 19 and Theorem 3.20 it is sufficient to prove this Lemma for the commutative group G0 . Consider a deformation retraction η : [0, 1] × V → A of V onto A, where V is an open neighborhood of A, put U = θ∗ [(W M E; V, G0 , P)t,H ]. A (W M E; A, G0 , P)t,H equivariant mapping ξ : (θ∗ )−1 (U ) = (W M E; W, G0 , P)t,H → (W M E; A, G0 , P)t,H is given by the formula ξ(< Pγˆ,v >t,H ) =< η(1, Pγˆ,v >t,H due to Propositions 7.1 and 13(2) in Section 3. 58. Theorem. For each connected smooth manifold N , the homomorphism κ induces an isomorphism ˜

κ∗ : π0 ((W M E; N, A∗r , P)b;∞,H ) → Hψ qb (N, Z(Cr )), where 1 ≤ b ∈ N, q is a dimension of M . Proof. The uniform space Hp∞ (M, E) is everywhere dense in the uniform space 0 C (M, E) of all continuous mappings from M into E, where M is an Hp∞ -pseudomanifold. Therefore, there exists an extension of N 7→ π0 ((W M E; N, A∗r , P)b;∞,H ) to a functor on the category of pointed CW-complexes and pointed continuous mappings, that does not change a homotopy type. Recall a reduced homology theory. It is a functor H∗ from the category of pointed CWcomplexes and pointed continuous mappings into the category of graded twisted groups satisfying the properties (H1 − H4). (H1). For each pointed continuous mapping of CW-complexes f : X → Y and a ∈ Z, the induced homomorphism f∗ : Ha (X) → Ha (Y ) depends only on the homotopy type of f. (H2). For each pointed CW-complex X and a ∈ Z there is a natural isomorphism ΣX : Ha (X) → Ha+1 (ΣX), where ΣX is a reduced suspension of X. (H3). For each pointed inclusion i : A ⊂ X of CW-complexes and a ∈ Z the sequence i

g∗

∗ Ha (A) → Ha (X) → Ha (X/A) is exact, where g : X → X/A is the quotient mapping. (H4). Ha (S 1 ) = e for a 6= 1 and H1 (S 1 ) = Z. These properties are standard and they are demonstrated in Lemma 4.5 [14] for commutative groups. Due to Conditions 4(A1, A2) on twisted groups we get the reduced twisted homology theory. In view of Lemma 57 πj ((W M E; A, G, P)t,H ) → πj ((W M E; X, G, P)t,H ) → πj ((W M E; X/A, G, P)t,H ) is a fragment of the long exact homotopy sequence of the fibration θ∗ , where G is the twisted group over {i0 , ..., i2r −1 }, j = 0, 1, 2, .... Moreover, Conditions (H2, H3) follow from Lemma 57. Therefore, Properties (H1 − H4) for twisted groups are direct consequences of the corresponding properties for commutative groups. Though for the proof of this theorem the case of commutative graded groups is sufficient. Since κ is a natural transformation of homology theory and in view of Proposition 19 and Theorem 20 in Section 3 this induces the isomorphism κ∗ . q+1,cl 59. Proposition. The curvature morphism K : LqN,A∗r → SN,A is an isomorphism. r Proof. The family CM := R ⊕ M R with M ∈ Ar , Re(M ) = 0 and |M | = 1 is S such that its union gives M CM = Ar . In view of Theorem 55 and Lemma 56.1 K is a q+1,cl monomorphism and an epimorphism from LqN,A∗r onto SN,A . This gives the statement of r

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this proposition. ˜

∗ 60. Theorem. The restriction homomorphism κ∗ : Hom(Zψ qb (M, N ), Ar ) → M ∗ Hom((W N )b;∞,H , Ar ) induces an isomorphism ˆ qb (N, Z(Cr )) → Hom∞ ((W M N )b;∞,H , Ar ), where 1 ≤ b ∈ N, q is a covering κ ˆ:H ψ˜ dimension of M , M and N are over Ar . ˜ ˆ qb (N, Z(Cr )), Proof. Each homomorphism h : Zψ (M, N ) → A∗ is a holonomy of H

r ψ˜
is the identity: R ˆ h(ξ) = h(ζ) + exp((−1)qb η K h ) ˆ This implies that due to Section 53 and Lemma 59. Therefore, h has a unique extension h. κ ˆ is an isomorphism. 61. Remark. Mention that Theorems 55, 58 and 60 can be proved in another way using the corresponding statements over C and the twisted structure of sheaves over {i0 , ..., i2r −1 }.

5.

CW-groups for Wrap Groups

To avoid misunderstandings we first give our definitions and notations. 1. Definitions. Suppose that K is a Hausdorff space, which is a union of disjoint open cells, denoted by e, en , enj , satisfying the following conditions. The closure ¯en of each n-cell, en ∈ K, is an image of n-simplex σ n , in a mapping f : σ n → ¯en such that (CW 1) f |σn \∂σn is a homeomorphism onto en ; (CW 2) ∂en ⊂ K n−1 , where ∂en = f (∂σ n ) = ¯en \en , K n−1 is the (n−1)-dimensional section of K consisting of all cells whose dimensions do not exceed (n − 1), in another words a (n − 1)-skeleton, K −1 := ∅. Then K is called a cell complex or a complex. Such mapping f : σ n → ¯en is called a characteristic mapping for en . A sub-complex L ⊂ K is the union of a subset of cells of K, which are cells of L, so that if e ⊂ L, then ¯e ⊂ L. If X is a subset of points in K, then K(X) denotes the

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intersection of all sub-complexes of K containing X. A complex K is called closure finite if and only if K(e) is a finite sub-complex for each cell e ∈ K. A weak topology in K is characterized by the condition: a subset X is closed (or open) in K if and only if X ∩ ¯e is closed (or relatively open correspondingly) for each cell e of K. By a CW-complex we mean one which is closure finite and has the weak topology. A mapping f : K → L for CW-complexes K and L is called cellular, if f (K n ) ⊂ Ln for each n = 0, 1, 2, .... A topological group is called a CW-group if it is a CW-complex such that the inversion and product mappings G ∋ g 7→ g −1 ∈ G and G × G ∋ (g, f ) 7→ f g ∈ G are both cellular, that is, they carry the k-skeleton into the k-skeleton. Then a CW-group G is called countable, if it is a countable CW-complex. A mapping f : X → Y is called a homotopy equivalence, if and only if it has a homotopy inverse meaning a mapping g : Y → X such that gf ≈ 1X and f g ≈ 1Y (see [49, 51]). Denote by (P M E; y0 , y1 )t,H the quotient uniform space of Rt,H equivalence classes of ˆ into E such that γˆ : M ˆ → N, Hpt mappings of a parallel transport structure Pγˆ,u from M ˆ → M E = E(N, G, π, Ψ) is a principal fiber bundle with a structure group G, Ξ : M is a quotient mapping, γˆ (ˆ s0,q ) = y0 , γˆ (ˆ s0,q+k ) = y1 for each q = 1, ..., k. Recall that the equivalence relation Rt,H is generated by: f ∼ g if and only if there exists sequences ˆ , W ) when n tends to the infinity fn and gn converging to f and g respectively in Hpt (M ˆ preserving marked points sˆ0,j , such that fn = gn ◦ ψn , ψn is an Hpt -diffeomorphism of M j = 1, ..., 2k (see §§2.1-3). We call (P M E; y0 , y1 )t,H the quotient path space. Particularly, may be G = e, that is E = N is a manifold for G = e. As usually consider arcwise connected E, N and G, where G is a Lie either alternative or associative group. ˆ are compact connected Riemannian C ∞ manifolds may be 2. Theorem. If N and M with corners such that the Ricci tensor Rk,l of N is everywhere positive definite, then the quotient path space (P M N ; y0 , y1 )t,H for marked points y0 and y1 in N has the homotopy type of a CW-complex having only finitely many cells in each dimension. Proof. Theorem A in [37] states if X is the homotopy direct limit of {Xj } and Y is the homotopy direct limit of {Yj }, if also f : X → Y is a continuous map that carries each Xj into Yj by a homotopy equivalence, then f itself is a homotopy equivalence. The corollary on page 153 from Theorem A [37] states that if X is the homotopy direct limit of {Xj } and each Xj has the homotopy type of a CW-complex, then X itself has the homotopy type of a CW-complex. In particular, the quotient space relative to a continuous quotient mapping of a CW-complex has the homotopy type of a CW-complex. Therefore, it is sufficient to prove ˆ ˆ , W ) : π ◦ f (ˆ this theorem for the path space (P M N ; y0 , y1 )t,H := {f ∈ Hpt (M s0,q ) = y0 , π ◦ f (ˆ s0,q+k ) = y1 ∀q = 1, ..., k}. ˆ and N are C ∞ manifolds, then C 0 ⊂ Hpt due to Since t ≥ [dim(M )/2] + 1, while M ˆ

the Sobolev embedding theorem and the homotopy type of (P M N ; y0 , y1 )t,H is the same ˆ as (P M N ; y0 , y1 )∞,H . The manifold N is compact, hence it is finite dimensional and the space consisting of all vectors v of the unit length on N is compact. The Ricci tensor is the bilinear pairing

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ˆ 1 , w)u2 from R : Ty N ×Ty N → R, which is the trace of the linear transformation w → R(v ˆ Ty N into Ty N , where R denotes the Riemann curvature tensor and R is its contraction. Therefore, there exists min{R(v, v) : v ∈ Ty N, y ∈ N, kvk = 1} =: (n − 1)ρ−2 , where n denotes the dimension of N . ˆ is compact, consequently, there exists a finite partition T of M ˆ conThe manifold M m sisting of Uj such that each Uj is homeomorphic with a cube [0, 1] , while Uj \ ∂Uj is S ˆ , m denotes the dimension of M ˆ, C ∞ diffeomorphic with [0, 1]m \ ∂[0, 1]m , j Uj = M Uj ∩ Ul = ∂Uj ∩ ∂Ul , j = 1, ..., a0 , a0 ∈ N. ˆ → N such that γˆ (ˆ Consider a path γˆ : M s0,q ) = y0 and γˆ (ˆ s0,q+k ) = y1 for each ∞ ˆ q = 1, ..., k, where M is the corresponding C Riemannian manifold satisfying Conditions ˆ → M is the quotient mapping as in §2.2, Ξ(ˆ §2 in Section 2 and Ξ : M s0,q ) = Ξ(ˆ s0,q+k ) = ˆ s0,q for each q = 1, ..., k, s0,q and sˆ0,q , sˆ0,q+k are marked points in M and M respectively for every q = 1, ..., k, k ∈ N. Therefore, the path γˆ can be presented as the combination of its restrictions γˆ |Uj . Without loss of generality we can take a partition T such that each marked point sˆ0,q in S 0 ˆ M belongs to aj=1 ∂Uj . If Uj has less, than two distinct marked points s0,q , then introduce in Uj additional marked points x0,a,j such that to have not less than two distinct marked points in Uj . The manifold N has the homotopy type of a CW-complex, hence N b has the homotopy type of a CW-complex for each b ∈ N (see also [1, 38] and below). In view of the Sard theorem II.2.10.2 [7] and §III.6 [35] the set of all Hpt difˆ is everywhere dense in the uniform space Hpt (M ˆ,M ˆ ). Then feomorphisms of M ˆ

S

0 (P M N ; y0 , y1 )t,H has the homotopy type of ( aj=1 (P Uj N ; y0,j , y1,j )t,H ) × N 2a0 −2 , where y0,j , y1,j are 2a0 distinct marked points in N containing y0 , y1 with the corresponding marked points in Uj . In accordance with Proposition (H) [51] if L is a locally finite complex and K is a CW-complex, then K × L is a CW-complex. The sum of CW-complexes is a CW-complex, the product of CW-complexes is a CWˆ and complex in accordance with Section 5 and Proposition (H) of [51]. The manifolds M N are connected, consequently, it is sufficient to prove this theorem in the special case of ˆ = [0, 1]m . M Therefore, consider γˆ : [0, 1]m → N , γˆ (x) ∈ N , x = (x1 , ..., xm ), xj ∈ [0, 1] for each j = 1, ..., m. Suppose that ηs (xs ) is a geodesic between points as and bs ∈ N , where ηs (xs ) := η(z1 , ..., zs−1 , xs , zs+1 , ..., zm ) with marked values of z1 , ..., zs−1 , zs+1 , ..., zm ∈ [0, 1] and η : [0, 1]m → N , as = ηs (0), bs = ηs (1). If ηs (xs ) has a length greater than πρ, then it has an index λ ≥ 1 (see also §§16, 17, 19 in [37]). Let E(ζ) denotes the energy functional of a geodesic in the Riemannian manifold and E∗∗ be its Hessian (see §12 in [37]). Generally consider a geodesic ζ of length greater than gπρ, consequently, ζ has an index λ ≥ g, where g ∈ N. For each j = 1, ..., g there exists a vector field Yj in N such that Yj along ζ vanishes outside the interval ((j − 1)/k, j/k), and so that E∗∗ (Yj , Yl ) < 0. Since E∗∗ (Yj , Yl ) = 0 for each j 6= l, then Y1 , ...Yg span a g-dimensional subspace of S y∈ζ([0,1]) Ty N on which E∗∗ is negative definite (see §19 in [37]). Suppose that points y0,j and y1,j are not conjugate along any geodesic from y0,j to y1,j , hence there exists only a finite number of geodesics like ηs from y0,j to y1,j in N by the

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variable xs of length not greater than gπρ. Hence there exists only finitely many geodesics with index less than g. In accordance with Theorem 17.3 [37] if N is a complete Riemannian manifold and y0 , y1 ∈ N are two points, which are not conjugate along any geodesic, then (P [0,1] N ; y0 , y1 )t,H has the homotopy type of a countable CW-complex containing one cell of dimension λ for each geodesic from y0 to y1 of index λ. Together with Theorem 17.3 [37] this completes the proof for dim(M ) = 1. For m > 1 proceed by induction: m m−1 (P [0,1] N ; y0 , y1 )∞,H = (P [0,1] (P [0,1] N ; y0 , y1 )∞,H ; y0 , y1 )∞,H , where yb in l [0,1] (P N ; y0 , y1 )∞,H denotes the constant mapping yb : [0, 1]l → N , yb ([0, 1]l ) = {yb }, {yb } denotes the singleton in N , b = 1, 2, l ∈ N, here the notation yb corresponds to yb,j for some j. This procedure lowers a number of variables on each step by one. In view of Theorem 19.6 [37] (P [0,1] N )t,H has the homotopy type of a CW complex B, which is σ-compact, that is a countable union of compact sets. Consider now (P [0,1] B)t,H , where B is a countable union of compact Riemannian manifolds may be with corners, since each polyhedron in Rn with n ∈ N is a manifold with S S corners. Put B = j∈Λ Bj , B k := kj=1 Bj , where Bj is a compact Riemannian manifold with corners being a j-skeleton of a CW complex, Λ ⊂ N. Up to a homotopy type or bending Bj a little in the corresponding Euclidean space Rn of dimension n ≥ 2 dim (Bj ), Bj ֒→ Rj ֒→ Rn , we can consider, that each Bj is homotopy equivalent to a compact Riemannian manifold Xj with positive definite Ricci tensor. Therefore, we have to consider S S now (P [0,1] X)t,H , where X = j Xj . Put X j = k≤j Xk , then X j ⊂ X j+1 for each j ∈ Λ, dim (Xj ) = j. Each path from the compact manifold M into a CW-complex B has a compact image, consequently, it has a finite covering by cells. Hence a continuous path from M into X up to a homotopy equivalence has a finite covering by X j . If N1 and N2 are homotopy equivalent Riemannian manifolds, then (P [0,1] N1 ; y0,1 , y1,1 )t,H and (P [0,1] N2 ; y0,2 , y1,2 )t,H are homotopy equivalent, when y0,1 6= y0,2 and y0,2 6= y1,2 simultaneously. On the other hand, (P [0,1] X; y0 , y1 )t,H is homotopy S equivalent with a CW-complex K = j∈Λ Kj , where each Kj is a CW-complex homotopy equivalent with (P [0,1] X j ; y0 , y1 )t,H , where y0 , y1 ∈ X1 , so that Kj ⊂ Kj+1 for each j, since X j ⊂ X j+1 . Denote by W the class of all spaces having the homotopy type of a CW-complex. By a CW-n-ad K = (K; K1 , ..., Kn−1 ) is undermined a CW-complex together with (n − 1) numbered sub-complexes K1 , ..., Kn−1 . Then W n denotes the class of all n-ads which have the homotopy type of a CW-n-ad. As usually AC denotes the subspace of the space AC of all continuous functions f from A into C such that f : C → A is a mapping of n-ads, that is the induced mappings are fj : Cj → Aj from the j-skeleton to the j-skeleton for each 1 ≤ j ≤ n. In accordance with Theorem 3 [38] if A belongs to the class W n and C is a compact n-ad, then the function space AC belongs to W. In fact the n-ad (AC ; (A, A1 )(C,C1 ) , ..., (A, An−1 )(C,Cn−1 ) ) belongs to the class W n . ˆ

Thus, (P M N ; y0 , y1 )t,H has the homotopy type of the CW-complex.

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ˆ and N are manifolds Hpt and Hpt diffeomorphic with C ∞ 2. Corollary. If M and M ˆ 1 and N correspondingly, t′ ≥ t, where M1 , M ˆ 1 and Riemannian manifolds M1 and M ˆ M N1 satisfy conditions of the preceding theorem, then the path space (P N ; y0 , y1 )t,H and the quotient path space (P M N ; y0 , y1 )t,H for marked points y0 and y1 in N are of the homotopy types of CW-complexes having only finitely many cells in each dimension. ˆ1 → M ˆ and θ : N1 → N be homeomorphisms, which are Hpt and Hpt′ Proof. Let φ : M ′

ˆ

ˆ

diffeomorphisms. Then the uniform spaces (P M N ; y0 , y1 )t,H and (P M1 N1 ; y0,1 , y1,1 )t,H are isomorphic, where the mapping f 7→ θ−1 ◦ f ◦ φ establishes the isomorphism, ˆ f ∈ (P M N ; y0 , y1 )t,H , θ(yb,1 ) = yb for b = 1, 2. Using this isomorphism and ˆ applying the preceding theorem to (P M1 N1 ; y0,1 , y1,1 )t,H and the quotient path space (P M1 N1 ; y0,1 , y1,1 )t,H we get the statement of this corollary. 3. Corollary. Let M and N be satisfying conditions of the preceding Corollary. Then the wrap monoid (S M N )t,H and the wrap group (W M N )t,H have homotopy types of CWcomplexes having only finitely many cells in each dimension. Proof. The wrap monoid has the homotopy type of (P M N ; y0 , y0 )t,H . On the other hand, the wrap group is the quotient of the free commutative group F generated by (S M N )t,H by the closed equivalence relation, which is obtained factorizing by the minimal closed normal subgroup B containing all elements of the form [a + b] − [a] − [b], where a, b ∈ (S M N )t,H , [a] and [b] are the corresponding elements of F . Topologically F is isomorphic with [(S M N )t,H ]Z supplied with the weak (Tychonoff) product topology. Applying Corollary on page 153 from Theorem A [37] and the preceding theorem we get the statement of this corollary. 4. Corollary. Let M and N be satisfying conditions of Corollary 2, while E be a principal fibre bundle with the structure group G, which is up to the homotopy a CW-group. Then a wrap monoid (S M E)t,H and a wrap group (W M E)t,H have homotopy types of a CW-monoid and a CW-group correspondingly. Proof. By Proposition (N ) any covering complex of a CW-complex is a CW-complex [51]. Therefore, if prove that (S M E)t,H is a CW-complex, then it would mean that (W M E)t,H is a CW-complex. This follows immediately from the preceding corollary and Proposition 7.1 in Section 3 and Proposition (H) [51], since (S M E)t,H and (W M E)t,H have structures of principal Gk -bundles over (S M N )t,H and (W M N )t,H . On the other, hand the mapping (S M N )t,H ∋ (f, g) → f g ∈ (S M N )t,H is cellular, since if a, b ∈ K n , then a ∨ b ∈ K n ∨ K n , where the bunch K n ∨ K n of K n by a finite number of marked points consists of cells of dimension at most n. Therefore, in (W M N )t,H the group multiplication is cellular as well (see also §3). In (W M N )t,H the mapping f 7→ f −1 is cellular due to the definition of the wrap group. Since G is the CW-group, then Gk is the CW-group, consequently, (S M E)t,H and (W M E)t,H are the CW-monoid and the CW-group respectively. 5. Remark. A topological space P is said to be dominating a topological space X if and only if there are continuous mappings f : X → P and g : P → X such that gf ≈ 1X . In accordance with Theorem 1 [38] A belongs to the class W0 if and only if A is dominated by a countable CW-complex. If G is a compact simply connected Lie group, then in accordance with Theorem 21.7 [37] (P [0,1] G; y0 , y1 )t,H has the homotopy type of a CW-complex with no odd-dimensional

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cells and with only finite number of n-cells for each even number n. These two theorems imply that G also is a CW-group, since (P [0,1] G)t,H dominates G and applying the homotopy equivalence. If G is not associative, but alternative, then the corresponding CW-group is alternative as well, since if a1 ≈ a2 , b1 ≈ b2 are homotopy equivalent elements of G, then (a1 a1 )b1 = −1 a1 (a1 b1 ) ≈ a1 (a2 b2 ) ≈ a2 (a2 b2 ) = (a2 a2 )b2 and b1 = a−1 1 (a1 b1 ) ≈ a1 (a2 b2 ) ≈ −1 a−1 2 (a2 b2 ) = (a2 a2 )b2 = b2 and analogously for identities with aj on the right from bj . In accordance with Corollary 1 [38] every separable finite dimensional manifold belongs to the class W0 , where W0 denotes the class of topological spaces having the homotopy type of countable CW-complexes. Due to Corollary 2 [38] if A belongs to W0 and C is a compact metric space, then the function space AC in the compact open topology belongs to W0 . Therefore, modifying Theorem 2 and Corollary 4 we get. 6. Proposition. If N is a finite dimensional separable manifold, G is a CW-group, then (P M E; y0 , y1 )t,H has the homotopy type of a CW-complex, (S M E)t,H and (W M E)t,H have homotopy types of a CW-monoid and a CW-group respectively.

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[12] R. Engelking. ”General topology” (Moscow: Mir, 1986). [13] H. Esnault. ”Algebraic theory of characteristic classes of bundles with connection”. In: Algebraic K-theory. Proc. Symp. in Pure Math. 67 (1999), 13-25. [14] P. Gajer. ”Higher holonomies, geometric loop groups and smooth Deligne cohomology”. In: ”Advances in geometry”. Progr. in Math. 172, 195-235 (Boston: Birkh¨auser, 1999). [15] P. Gajer. ”Geometry of Deligne cohomology”. Invent. Mathem. 127 (1997), 155-207. [16] F. G¨ursey, C.-H. Tze. ”On the role of division, Jordan and related algebras in particle physics” (Singapore: World Scientific Publ. Co., 1996). [17] W.R. Hamilton. ”Selected papers. Optics. Dynamics. Quaternions” (Moscow: Nauka, 1994). [18] F.R. Harvey. ”Spinors and calibrations”. Perspectives in Mathem. 9 (Boston: Academic Press, 1990). [19] P. Iglesias. ”Diff´eologie d’espace singulier et petits diviseurs”. C.R. Acad. Sci. Paris. Ser. I. 302: 14 (1986), 519-522. [20] C.J. Isham. ”Topological and global aspects of quantum theory”. In: ”Relativity, groups and topology.II” 1059-1290, (Les Hauches, 1983). Editors: R. Stora, B.S. De Witt (Amsterdam: Elsevier Sci. Publ., 1984). [21] I.L. Kantor, A.S. Solodovnikov. ”Hypercomplex numbers” (Berlin: Springer-Verlag, 1989). [22] W. Klingenberg. ”Riemannian geometry” (Berlin: Walter de Gruyter, 1982). [23] H.B. Lawson, M.-L. Michelson. ”Spin geometry” (Princeton: Princ. Univ. Press, 1989). [24] S.V. Ludkovsky. ”Quasi-invariant measures on loop groups of Riemann manifolds”. Dokl. Akad. Nauk 370: 3 (2000), 306-308. [25] S.V. Ludkovsky. ”Poisson measures for topological groups and their representations”. Southeast Asian Bulletin of Mathematics. 25 (2002), 653-680. (shortly in Russ. Math. Surv. 56: 1 (2001), 169-170; previous versions: IHES/M/98/88, 38 pages, also Los Alamos Nat. Lab. math.RT/9910110). [26] S.V. Ludkovsky. ”Functions of several Cayley-Dickson variables and manifolds over them”. J. Mathem. Sci. 141: 3 (2007), 1299-1330 (previous variant: Los Alamos Nat. Lab. math.CV/0302011). [27] S.V. Ludkovsky. ”Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables”. Sovrem. Mathem. Fundam. Napravl. 18 (2006), 101-164 (previous variant: Los Alam. Nat. Lab. math.DG/0603006).

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[28] S.V. Ludkovsky, F. van Oystaeyen. ”Differentiable functions of quaternion variables”. Bull. Sci. Math. (Paris). Ser. 2. 127 (2003), 755-796. [29] S.V. Ludkovsky. ”Differentiable functions of Cayley-Dickson numbers and line integration”. J. Mathem. Sci. 141: 3 (2007), 1231-1298 (previous version: Los Alam. Nat. Lab. math.NT/0406048; math.CV/0406306; math.CV/0405471). [30] S.V. Ludkovsky. ”Stochastic processes on geometric loop groups, diffeomorphism groups of connected manifolds, associated unitary representations”. J. Mathem. Sci. 141: 3 (2007), 1331-1384 (previous version: Los Alam. Nat. Lab. math.AG/0407439, July 2004). [31] S.V. Ludkovsky. ”Geometric loop groups and diffeomorphism groups of manifolds, stochastic processes on them, associated unitary representations”. In the book: ”Focus on Groups Theory Research” (Nova Science Publishers, Inc.: New York) 2006, pages 59-136. [32] S.V. Ludkovsky. ”Generalized geometric loop groups of complex manifolds, Gaussian quasi-invariant measures on them and their representations”. J. Mathem. Sci. 122: 1 (2004), 2984-3011 (earlier version: Los Alam. Nat. Lab. math.RT/9910086, October 1999). [33] S.V. Ludkovsky. ”Quasi-conformal functions of quaternion and octonion variables, their integral transformations”. Far East J. of Math. Sci. (FJMS) 28: 1 (2008), 37-88. [34] M.B. Mensky. ”The paths group. Measurement. Fields. Particles” (Moscow: Nauka, 1983). [35] V.P. Mihailov. ”Differential equations in partial derivatives” (Moscow: Nauka, 1976). [36] P.W. Michor. ”Manifolds of differentiable mappings” (Boston: Shiva, 1980). [37] J. Milnor. ”Morse theory” (Princeton, New Jersey: Princeton Univ. Press, 1963). [38] J. Milnor. ”On spaces having the homotopy type of a CW-complex”. Transactions of the A.M.S. 90 (1959), 272-280. [39] N. Murakoshi, K. Sekigawa, A. Yamada. ”Integrability of almost quaternionic manifolds”. Indian J. Mathem. 42: 3, 313-329 (2000). [40] L. Narici, E. Beckenstein. ”Topological vector spaces” (New York: Marcel-Dekker Inc., 1985). [41] H. Omori. ”Groups of diffeomorphisms and their subgroups”. Trans. Amer. Math. Soc. 179 (1973), 85-122. [42] H. Omori. ”Local structures of groups of diffeomorphisms”. J. Math. Soc. Japan 24: 1 (1972), 60-88. [43] L.S. Pontrjagin. ”Continuous groups” (Moscow: Nauka, 1984).

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[44] R.T. Seeley. ”Extensions of C ∞ functions defined in a half space”. Proceed. Amer. Math. Soc. 15 (1964), 625-626. [45] J.M. Souriau. ”Groupes differentiels” (Berlin: Springer Verlag, 1981). [46] N. Steenrod. ”The topology of fibre budles” (Princeton, New Jersey: Princeton Univ. Press, 1951). [47] R. Sulanke, P. Wintgen. ”Differentialgeometrie und Faserb¨undel” (Berlin: Veb deutscher Verlag der Wissenschaften, 1972). [48] R.C. Swan. ”The Grothendieck ring of a finite group”. Topology 2 (1963), 85-110. [49] R.M. Switzer. ”Algebraic topology - homotopy and homology” (Berlin: SpringerVerlag, 1975). [50] J.C. Tougeron. ”Ideaux de fonctions differentiables” (Berlin: Springer-Verlag, 1972). [51] J.H.C. Whitehead. ”Combinatorial homotopy.I”. Bull. Amer. Mathem. Soc. 55 (1949), 213-245. [52] K. Yano, M. Ako. ”An affine connection in almost quaternion manifolds”. J. Differ. Geom. 3 (1973), 341-347. [53] V.A. Zorich. ”Mathematical analysis”, V. 2 (Moscow: Nauka, 1984).

In: Lie Groups: New Research Editor: Altos B. Canterra, pp. 563-594

ISBN 978-1-60692-389-4 c 2009 Nova Science Publishers, Inc.

Chapter 18

G ROUPS OF D IFFEOMORPHISMS AND W RAPS OF M ANIFOLDS OVER N ON - ARCHIMEDEAN F IELDS S.V. Ludkovsky Dept. of Applied Mathematics, Moscow State Technical Univ., Moscow, Russia

Abstract The article is devoted to the investigation of groups of diffeomorphisms and wraps of manifolds over non-archimedean fields of zero and positive characteristics. Different types of topologies are considered on groups of wraps and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies pairwise incomparable are found as well. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. There are proved theorems about projective limit decompositions of these groups and their compactifications for compact manifolds. Moreover, an existence of one-parameter local subgroups of diffeomorphism groups is investigated.

1.

Introduction

Non-archimedean analysis has rather long history, but it is much less developed in comparison with the classical analysis over the fields R and C. Therefore, the theory of groups on manifolds over non-archimedean fields is not so well investigated as for Riemann or complex manifolds [35, 16, 12, 5]. As it is known fields with multiplicative ultra-norms such as the field of p-adic numbers were first introduced by K. Hensel [14]. Several years later on it was proved by A. Ostrowski [33] that on the field of rational numbers each multiplicative norm is either the usual norm as in R or is equivalent to a non archimedean norm |x| = p−k , where x = npk /m ∈ Q, n, m, k ∈ Z, p ≥ 2 is a prime number, n and m and p are mutually pairwise prime numbers. Each locally compact infinite field with a non trivial non archimedean valuation is either a finite algebraic extension of the field of p-adic numbers or is isomorphic to the field Fpk (θ) of power series of the variable θ with expansion coefficients in the finite field Fpk of pk elements, where p ≥ 2 is a prime number, k ∈ N is a natural number [38, 43]. The valuation group ΓK := {|x| : x ∈ K, x 6= 0} ⊂ (0, ∞) of a locally compact

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field K is discrete. Non locally compact fields are wide spread as well and among them there are fields with the valuation group ΓK = (0, ∞) [9, 38, 40]. The non-archimedean analog of the field of complex numbers is Cp which is complete algebraically and as the uniform space relative to its multiplicative norm and ΓCp = {x ∈ Q : x > 0}. The importance of transformation groups of manifolds in the non-Archimedean functional analysis, representation theory and mathematical physics is clear and also can be found in the references given below [2, 21, 22, 24, 25, 38, 39]. Here we use the terminology a wrap group instead of a (geometric) loop group, because in the non-archimedean case there are not any circles like as in classical case, moreover, manifolds are already totally disconnected and of arbitrary dimension over a non-archimedean field. It was shown earlier in preceding works of the author that such groups are smooth manifolds with differentiable group operations depending on a considered class of smoothness, that is they are Lie groups. This article is devoted to several aspect of such groups and to further investigation of their structure. One of them is on their structure from the point of view of the nonarchimedean compactficiation (see also about Banaschewski compactification in [38]). Though a new topology used for compactification may be different from the initial topology of a group or even may be non-comparable, because on the same group it is possible an existence of several different topologies making it a topological group. This is useful also for studying their representations as restrictions of representations of non-archimedean compactifications, which are constructed below such that they also are groups. Apart from previous works [21, 22, 23, 24, 25], where the characteristic char(K) = 0 was zero, in this paper groups on manifolds over fields with non-zero characteristics also are defined and investigated. Different types of topologies are considered on groups of wraps and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies pairwise incomparable are found as well. It is caused by the fact that repeated application of projective and inductive limits of topological spaces generate topologies and spaces in general dependent from an order of taking limits and their types, so that such topologies may appear incomparable on a subset contained in these topological spaces. It is proved, that relative to the C ∞ bounded-open topology groups of geometric wraps and groups of diffeomorphisms of manifolds over non-archimedean ultra-normed fields are the generalized Lie groups. Previously one-parameter subgroups over fields of non-zero characteristics were not studied. This article contains as well results on one-parameter subgroups over K with char(K) = p > 1 using its multiplicative subgroup K∗ := K \ {0}. It is proved below that the diffeomorphism group of a compact manifold is topologically simple relative to the C ∞ compact-open topology, that develops previous results [24], where topological simplicity and perfectness was proved over fields of zero characteristic. At first in Section 2 we remind basic facts and notations, which are given in detail in references [38, 40, 21, 22, 23]. A wrap group Lt (M, N ) is defined as a quotient space of a family of mappings f : M → N of class C t of one Banach manifold M into another ¯ m f )(z; h1, ..., hm; ζ1, ..., ζm) = 0 or N over the same local field K such that limz→s (Φ m [m] limz→s (Υ f )(z ) = 0 for each 0 ≤ m ≤ t, where M and N are embedded into the corresponding Banach spaces X and Y , cl(M ) = M ∪ {s}, cl(M ) and N are clopen ¯ m f )(z; h1, ..., hm; ζ1, .., ζm) and (Υm f )(z [m] ) are in X and Y respectively, 0 ∈ N , (Φ

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continuous extensions of difference quotients by variables corresponding to z or by all appearing inductively variables over a non-archimedean field K of zero characteristic or of char(K) = p > 0 respectively, z ∈ M , h1 , ..., hm are nonzero vectors in X, ζ1, ..., ζm ∈ K such that z + ζ1 h1 + ... + ζm hm ∈ M , z [m+1] := (z [m] , v [m], ζm+1 ), z 1 = (z, v [0], ζ1), z [m] , v [m] ∈ K[m] , K[m+1] = K[m] ⊕ K[m] ⊕ K, U [1] = U , z [m] + ζm+1 v [m] ∈ U [m] (see also §2.1). In Section 2 preliminary investigations on structures of Dif f t (M ) as the topological groups and Lie groups are studied. Non-archimedean completions of clopen subgroups W of wrap groups G and diffeomorphism groups G are considered in Sections 3 and 7. Completions are considered relative to uniformities associated with projective decompositions. They produce topologies incomparable with the initial one. Relative to them they remain topological groups. In the case of the wrap group the non-archimedean completion produces a new topological group V in which the initial group W is embedded as a dense subgroup such that V 6= W . Such topologies have purely non-archimedean origin related with non-archimedean uniformities or families of non-archimedean semi-norms on spaces of continuous or more narrow classes of functions between non-archimedean manifolds. In the classical case over R one might expect instead of this some repeated combination of an inductive and a projective limits, which is quite different thing. For the compact manifold M in the case of the diffeomorphism group the nonarchimedean completion of W produces profinite group. For the locally compact manifolds M and N in the case of the wrap group Lt (M, N ) one of the non-archimedean completion of W produces its embedding into ZpN and also there exists the completion isomorphic with (νZ)ℵ0 , where νZ is the one-point Alexandroff compactification of Z. When W is bounded relative to the corresponding metric in Lt (M, N ), then W is embedded into ZpN . Moreover, topologies of Dif f w (M ) and Dif f t (M ) or Lt(M, N ) and Lw (M, N ) are incomparable for compact manifolds M and N , where the groups Dif f w (M ) and Lw (M, N ) are supplied with the weak projective limit topologies τw . The group Dif f t(M ) is topologically simple, on the other hand, the group Lt (M, N ) is commutative. An existence of one parameter subgroups of Dif f t (M ) is investigated in Section 5. It is proved in Section 6, that Dif f t (M ) is topologically simple relative to its C t compactopen topology, as well as the theorem about continuous automorphisms of Dif f t (M ) is proved. The notation given below and the corresponding definitions are given in detail in [21, 23]. All results of this paper over the fields of positive characteristics are obtained for the first time.

2.

Groups of Diffeomorphisms

1. Definitions. Let K be an infinite field with a non trivial non archimedean valuation, let also X and Y be topological vector spaces over K and U be an open subset in X. For a function f : U → Y consider the associated function f [1](x, v, t) := [f (x + tv) − f (x)]/t on a set U [1] at first for t 6= 0 such that U [1] := {(x, v, t) ∈ X 2 × K, x ∈ U, x + tv ∈ U }. If f is continuous on U and f [1] has a continuous extension on U [1] , then we say, that

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f is continuously differentiable or belongs to the class C 1 . The K-linear space of all such continuously differentiable functions f on U is denoted C [1] (U, Y ). By induction we define functions f [n+1] := (f [n] )[1] and spaces C [n+1] (U, Y ) for n = 1, 2, 3, ..., where f [0] := f , f [n+1] ∈ C [n+1] (U, Y ) has as the domain U [n+1] := (U [n] )[1]. The differential df (x) : X → Y is defined as df (x)v := f [1](x, v, 0). Define also partial difference quotient operators Φn by variables corresponding to x only such that Φ1f (x; v; t) = f [1](x, v, t) at first for t 6= 0 and if Φ1f is continuous for t 6= 0 and has a continuous extension on ¯ 1 f (x; v; t). Define by induction U [1] =: U (1), then we denote it by Φ n+1 Φ f (x; v1, ..., vn+1; t1, ..., tn+1) := Φ1 (Φn f (x; v1, ..., vn; t1 , ..., tn))(x; vn+1; tn+1 ) at first for t1 6= 0, ..., tn+1 6= 0 on U (n+1) := {(x; v1, ..., vn+1; t1, ..., tn+1) : x ∈ U ; v1, ..., vn+1 ∈ X; t1, ..., tn+1 ∈ K; x + v1 t1 ∈ U, ..., x + v1t1 + ... + vn+1 tn+1 ∈ U }. If f is continuous on U and partial difference quotients Φ1 f ,...,Φn+1 f has continuous ¯ 1f ,..., Φ ¯ n+1 f on U (1),...,U (n+1) respectively, then we say that f extensions denoted by Φ n+1 . The K linear space of all C n+1 functions on U is denoted is of class of smoothness C by C n+1 (U, Y ), where Φ0 f := f , C 0 (U, Y ) is the space of all continuous functions f :U →Y. := Then the differential is given by the equation dn f (x).(v1, ..., vn) ¯ n!Φn f (x; v1, ..., vn; 0, ..., 0), where n ≥ 1, also denote Dn f = dn f . Shortly we ¯ n f as x(n) ∈ U (n) , where shall write the argument of f [n] as x[n] ∈ U [n] and of Φ x[0] = x(0) = x, x[1] = x(1) = (x, v, t), v [0] = v (0) = v, t1 = t, x[k] = (x[k−1], v [k−1], tk ) for each k ≥ 1, x(k) := (x; v1, ..., vk; t1 , ..., tk). ¯ kf Subspaces of uniformly C n or C [n] bounded continuous functions together with Φ k (k) [k] or U for k = 1, ..., n we denote by or Υ f on bounded open subsets of U and U [n] n Cb (U, Y ) or Cb (U, Y ) respectively. Consider partial difference quotients of products and compositions of functions and relations between partial difference quotients and differentiability of both types. Denote by L(X, Y ) the space of all continuous K-linear mappings A : X → Y. By Ln (X ⊗n , Y ) denote the space of all continuous K n-linear mappings A : X ⊗n → Y , particularly, L(X, Y ) = L1 (X ⊗1, Y ). If X and Y are normed spaces, then Ln (X ⊗n , Y ) is supplied with the operator norm: kAk := suph1 6=0,...,hn 6=0;h1 ,...,hn ∈X kA.(h1, ..., hn)kY /(kh1kX ...khnkX ). 2. Lemma. The spaces C [1] (U, Y ) and C 1 (U, Y ) are linearly topologically iso¯ n f (x; ∗; 0, ..., 0) : X ⊗n → Y is a K n-linear morphic. If f ∈ C n (U, Y ), then Φ 0 ⊗n C (U, Ln (X , Y )) symmetric map. ¯ 1 f (x; v; t) on U [1] = U (1) , Proof. From Definition 1 it follows, that f [1](x, v, t) = Φ so both K-linear spaces are linearly topologically isomorphic. On the other hand, due to ¯ n f (x; ∗, 0, ..., 0) is the K n-linear symmetric mapping for each x ∈ U and its definition Φ ¯ n f (x; v1, ..., vn; t1 , ..., tn) is continuous on U (n) it belongs to C 0 (U, Ln(X ⊗n , Y )), since Φ and for each x ∈ U and v1 , ..., vn ∈ X there exist neighborhoods Vi of vi in X and W of zero in K such that x + W V1 + ... + W Vn ⊂ U . ¯n : 3. Lemma. Operators Υn (f ) := f [n] from C [n] (U, Y ) into C 0 (U [n] , Y ) and Φ n 0 (n) C (U, Y ) → C (U , Y ) are K-linear and continuous. Proof. Since [(af + bg)(x + vt) − (af + bg)(x)]/t = a(f (x + vt) − f (x))/t + b(g(x +

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vt) − g(x))/t for each f, g ∈ C 1 (U, Y ) and each a, b ∈ K, then applying this formula by ¯ n we get their K-linearity. Indeed, induction and using definitions of operators Υn and Φ n [n] 1 n−1 Υ (af + bg)(x ) = Υ (Υ (af + bg)(x[n−1]))(x[n]) = Υ1 (af [n−1] + bg [n−1])(x[n]) = af [n] (x[n] ) + bg [n](x[n] ) and ¯ 1 (Φ ¯ n−1 (af + bg)(x(n−1)))(x(n)) = Φ ¯ 1(af (n−1) + ¯ n (af + bg)(x(n)) = Φ Φ bg (n−1))(x(n)) = af (n) (x(n) ) + bg (n)(x(n) ). ¯ n follows from definitions of spaces C [n] (U, Y ) and C n (U, Y ) The continuity of Υn and Φ respectively. 4. Definitions. Let M be a manifold modelled on a topological vector space X over K 0 such that its atlas At(M ) := {(Uj , M φj ) : j ∈ ΛM } is of class Cβα0 , that is the following four conditions are satisfied: (M 1) {Uj : j ∈ ΛM } is an open covering of M , Uj = M Uj , S (M 2) j∈ΛM Uj = M , (M 3) M φj := φj : Uj → φj (Uj ) is a homeomorphism for each j ∈ ΛM , φj (Uj ) ⊂ X, α0 (M 4) φj ◦ φ−1 i ∈ Cβ 0 on its domain for each Ui ∩ Uj 6= ∅, T

[∞]

T

l [l] 0 where ΛM is a set, C ∞ := ∞ := ∞ l=1 Cβ , Cβ l=1 C , α ∈ {n, [n] : 1 ≤ n ≤ ∞}, 0 0 β ∈ {∅, b}, C∅α := C α . Supply Cβα (U, Y ) with the bounded-open Cβα topology (denoted by τα,β generally or τα for β = ∅ or for compact U ) with the base W (P, V ) = {f ∈ Cβα (X, Y ) : S k f |P ∈ V, k = 0, ..., n} of neighborhoods of zero, where P is bounded and open in U ⊂ X, ¯ k or S k = Υk for α = n or α = [n] respectively, P ⊂ U , V is open in Y , 0 ∈ V , S k = Φ [k] v1 , ..., vn ∈ (P − y0 ), vl ∈ (P − y0 ) for each k, l for some marked y0 ∈ P and |tj | ≤ 1 for every j. 0 If M and N are Cβα manifolds on topological vector spaces X and Y over K, then consider the uniform space Cβα (M, N ) of all mappings f : M → N such that fj,i ∈ Cβα on its domain for each j ∈ ΛN , i ∈ ΛM , where fj,i := N φj ◦ f ◦ M φ−1 i is with values in Y , α ≤ α0 . The uniformity in Cβα (M, N ) is inherited from the uniformity in Cβα (X, Y ) with the help of charts of atlases of M and N . If M is compact, then Cbα (M, N ) and C α (M, N ) coincide. The family of all homeomorphisms f : M → M of class Cβα denote by Dif fβα(M ). Let γ be a set, then denote by c0 (γ, K) the normed space consisting of all vectors x = {xj ∈ K : j ∈ γ, for each  > 0 the set {j : |xj | > } is finite }, where kxk := supj∈γ |xj |. In view of the Kuratowski-Zorn lemma it is convenient to consider γ as an ordinal. Henceforth, suppose that X = c0(γX , K) and Y = c0 (γY , K). 5. Theorem. The uniform space Dif fbα (M ) (see §4) is the topological group relative to compositions of mappings. Proof. The group operation in Dif fβα(M ) is (f, g) 7→ f ◦g, where f ◦g(x) := f (g(x)) for each x ∈ M . Then f = id is the unit element in Dif fβα(M ), where id(x) = x for each x ∈ M . Since the composition of mappings is associative, then f ◦ (g ◦ h) = (f ◦ g) ◦ h is associative as the group operation. For each f ∈ Dif fβα there exists its inverse mapping f −1 such that f −1 (y) = x for each y = f (x), x ∈ M , since f : M → M is the homeomorphism. It remains to verify that f −1 ∈ Dif fβα (M ) for each f ∈ Dif fβα (M ) and the composition (Dif fβα)2 3 (f, g) 7→ f ◦ g ∈ Dif fβα(M ) and inversion f 7→ f −1 are continuous operations.

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In the normed space Y = c0(γY , K) a subspace spanK {ej : j ∈ γX } consisting of all finite K-linear combinations of vectors ej = (0, ..., 0, 1, 0, ...) with 1 on the j-th place is everywhere dense. Therefore, each f ∈ Cbα (U, Y ) is the uniform limit of mappings ¯ k f or Υk f on bounded subsets of U and U (k) (f1 , ..., fj, 0, ...) ∈ Cbα (U, Y ) together with Φ [k] or U for 1 ≤ k ≤ n, n ∈ N, n ≤ α. In particular, consider N φl ◦ f ◦ M φ−1 i for f ∈ Cbα (M, N ) taking U as a finite union of M φi ( M Uj ). Consider all possible embeddings of Kv into X, particularly, containing x(k) or x[k] for each 0 ≤ k ≤ n, where n ∈ N, n ≤ α. In view of Formulas 6(1) or 7(1) of the Appendix using restrictions on different embedded subspaces K[k] or K(k) into X [k] or X (k) and uniform continuity of Υk and ¯ k on bounded open subsets, k = 0, 1, 2, ..., we get, that f ◦ g ∈ C α (M, M ) for each Φ b f, g ∈ Dif fbα (M ), since fl,s ◦ gs,i ∈ Cbα (Ul,s,i, X) on corresponding domains Ul,s,i in X. From f, g ∈ Hom(M, M ) it follows, that f ◦g ∈ Hom(M, M ), hence f ◦g ∈ Dif fbα(M ). −1 Applying to idl,i = fl,s ◦ fs,i on corresponding domains Formulas A.6 (1) or A.7(1) and restricting on different embedded subspaces K[k] or K(k) in X [k] or X (k) and using uniform ¯ k , k = 0, 1, 2, ..., to both sides of this continuity on bounded open subsets for Υk or Φ equality gives that f −1 ∈ Dif fbα(M ) for each f ∈ Dif fbα(M ). The space Cbφ (U, Y ) is normed for U bounded in X for φ ∈ {n, [n]} with n ∈ N such that ¯ k f (z)kY or (1) kf kCbn (U,Y ) := sup0≤k≤n;z∈V (k) kΦ (2) kf kC [n] (U,Y ) := sup0≤k≤n;z∈V [k] kΥk f (z)kY , b

where V (k) := {z ∈ U (k) : z = (x; v1, v2, ...; t1, t2 , ...), kvj kX = 1∀j}, V [k] := [k−1] [q] [q] {z = x[k] ∈ U [k] : kv1 kX = 1, | l v2 tq+1 | ≤ 1, |v3 | ≤ 1 ∀l, q}. The uni[∞] formity of Cb∞ (U, Y ) or Cb (U, Y ) is defined by the family of such norms. Then the α uniformity in Cb (M, N ) is induced by the uniformity in Cbα (U, Y ) by all bounded subsets U in finite unions of M φi ( M Ui ), since to each f ∈ Cbα (M, N ) there corresponds in Cbα (Uj,i, Y ) with a corresponding domain Uj,i ⊂ X. Then fj,i = N φj ◦ f ◦ M φ−1 i application of Formulas A.6(1) or A.7(1) by induction on k and restricting on different embedded subspaces K[k] or K(k) in X [k] or X (k) and using uniform continuity on bounded open subsets gives that (f, g) 7→ f −1 ◦ g is Cbα uniformly continuous on bounded subsets of U and U (k) or U [k] , where U is a finite union of charts Uj of M . 6. Definition. A topological group G is called a Cβα Lie group if and only if G has 0 a structure of a Cβα manifold and the mapping G2 3 (f, g) 7→ f −1 g ∈ G is of class of smoothness Cβα , where α ≤ α0 . 7. Theorem. If M is a Cbα manifold on X = c0 (γX , K), where either α = ∞ or α = [∞], then Dif fbα (M ) is the Cbα Lie group. Proof. In view of Theorem 5 it remains to demonstrate that G can be supplied with a structure of Cbα manifold and that G2 3 (f, g) 7→ f −1 g ∈ G is of class of smoothness Cbα . It is possible to take an equivalent atlas of M consisting of clopen (closed and open simultaneously) charts Uj shrinking it a little in case of necessity. Take a base W of neighborhoods of id in Dif fbα(M ) from the proof of Theorem 5. Then consider a subgroup Ω in Dif fbα (M ) such that ΩW covers Dif fbα(M ) for each W ∈ W, where ΩW := {gW : g ∈ Ω}, gW := {gf : f ∈ W }. Therefore, the base ΩW generates a topology in Dif fbα (M ) equivalent with the initial one (see Chapter 8 in [11]). Moreover, ΩW generates a left uniformity in the group of diffeomorphisms.

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Consider a subset WU := {f |U ∈ W : f ∈ Dif fbα(M )} for a bounded open U in Uj for some j ∈ ΛM . Then φi ◦ (gWU ) ◦ φ−1 ⊂ Cbα (Vj , X) and φi ◦ (gWU ) ⊂ Cbα (U, X) j for each i, where Vj := φj (Uj ) ⊂ X. Therefore, gWU ∩ hWV = {f ∈ Dif fbα (M ) : g −1f |U ∈ W, h−1f |V ∈ W }, WU ∩ WV = {f ∈ Dif fbα (M ) : f |U ∪V ∈ W }. Put −1 −1 α ψi,g,j := φi ⊗ g −1 ⊗ φ−1 j such that ψi,g,j (gWU ) := φi ◦ g ((gWU ) ◦ φj ) ⊂ Cb (A, X), −1 −1 −1 where A = φj (U ), ψl,h,b := φ−1 l ⊗h⊗φb , hence ψl,h,b ◦ψi,g,j (gWU ∩hWV ) = (φl ◦φi )◦ −1 for U ⊂ Uj , V ⊂ Ub with Uj ∩ Ub 6= ∅ and Ul ∩ Ui 6= ∅. h(WU ∩ g −1 hWV ) ◦ (φ−1 b ◦ φj ) −1 Thus ψi,g,j ◦ ψl,h,b gives the transition mapping for Dif fbα (M ). On the other hand, each Cbα (A, X) has the natural embedding into Cbα (X, X), since X is totally disconnected and A can be taken clopen in X such that each f ∈ Cbα (A, X) has a Cbα (X, X) extension. For Uj ∩ Ub 6= ∅ the intersection WU ∩ WV is non void. Take At(Dif fbα (M )) := {(Wg,U , ψi,g,j ) : g ∈ Ω, i, j ∈ ΛM } with charts Wg,U := gWU , W ∈ W and g ∈ Ω, U bounded in some Uj and with transition mappings ψi,g,j ◦ −1 −1 ψl,h,b for Wh,U and Wg,V when U ∩ V 6= ∅. Since ψi,g,j ◦ ψl,h,b ∈ Cbα , then this is the α −1 α α Cb atlas. The mapping (f, g) 7→ f g is of class Cb (Dif fb (M ), Dif fbα(M )) due to −1 ◦ gi,l from Cbα (U, X) Formulas A.6(1) and A.7(1), since the mappings (fi,j , gi,l) 7→ fi,j α α into Cb (V, X) with U ⊂ φl (Ul) 6= ∅ and V ⊂ φj (Uj ) are of class Cb . 8. Theorem. If an ultrametric field K is complete relative to its multiplicative norm, then Dif fbα(M ) is complete as a left uniform space. Proof. Recall some facts about uniform spaces. A subset A of the product S × S of a set S is called a relation in S. The relation inverse to A is denoted −A such that −A = {(x, y) : (y, x) ∈ A} and the composition of relations is denoted A + B such that A + B = {(x, z) : there exists a y ∈ S such that (x, y) ∈ A and (y, z) ∈ B}. Denote by ∆ := {(x, x) : x ∈ S} the diagonal of the product S × S. Every subset in S × S containing ∆ is called an entourage of the diagonal ∆. The family of all entourages of the diagonal is denoted by DS . One writes |x − y| < V if (x, y) ∈ V and one says that x and y are at a distance less than V . If the condition |x − y| < V is not satisfied, then one writes |x − y| ≥ V . If A ⊂ S and |x − y| < V for each x, y ∈ A, then one says that the diameter δ(A) of A is less than V . Denote 1A := A, nA := (n − 1)A + A. A uniformity U in a set S is a non-empty subfamily in DS satisfying the following four conditions: (U 1) If V ∈ U and V ⊂ W ∈ DS , then W ∈ U ; (U 2) If V1, V2 ∈ U , then V1 ∩ V2 ∈ U ; (U 3) For each V ∈ U there exists W ∈ U such that 2W ⊂ V ; T (U 4) V ∈U V = ∆. A topological space S is called a T1 space if for each x 6= y ∈ S there exists an open set U in S such that x ∈ U and y ∈ / U . A topological space S is called a Tychonoff space and it is denoted by T3 1 if it is a T1 space and for each point x ∈ S and each closed set J in S 2 with x ∈ / J there exists a continuous function f : S → [0, 1] ⊂ R such that f (x) = 0 and f (y) = 1 for each y ∈ J. If S is a Tychonoff space then for every finite family of functions f1 , ..., fn ∈ C 0 (S, R) or Cb0 (S, R) the formula ρf1 ,...,fn (x, y) := maxni=1 |fi (x) − fi (y)| defines a pseudo-metric in S. The families of such pseudo-metrics denote by P and Pb respectively. They generate uniformities denoted by C and Cb correspondingly (see Chapter 8 in [11]).

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If U is a uniformity in S, then the family O := {U ⊂ S : ∀x ∈ U ∃V ∈ U such that B(x, V ) ⊂ U } is the topology in S and it is called the topology induced by the uniformity U . Every covering of S for which there exists V ∈ U such that C(V ) is a refinement of it is called a uniform covering relative to U , where C(V ) := {B(x, V ) : x ∈ S}, B(x, V ) := {y ∈ S : |x − y| < V }. If (S, U ) is a uniform space and F is a family of subsets of S, then one says that F contains arbitrary small sets if for every V ∈ U there exists F ∈ F such that δ(F ) < V . A uniformity U in S is called complete or (S, U ) is called complete, if for each family of closed subsets {Fu : u ∈ Ψ} of the topological space S with the topology induced by U which has the finite intersection property and contains arbitrary small sets the intersection T u∈Ψ Fu is non-empty. If C is an ordered set and {xu : u ∈ C} is a net in S such that for each V ∈ U there exists u0 ∈ C for which |xu − xv | < V for each C 3 u, v ≥ u0, then it is called the Cauchy net. In accordance with theorem of Chapter 8 [11] a uniform space (S, U ) is complete if and only if each Cauchy net in this space is convergent. On the other hand, each closed subset A of a complete uniform space (S, U ) is complete relative to the uniformity UA in A inherited from U in S. If G is a topological group and B(e) = B is a base of neighborhoods at the unit element e, then each F ∈ B determines three coverings of the topological space G: Cl (F ) = {xF : x ∈ G}, Cr (F ) := {F x : x ∈ G}, C(F ) := {xF y : x, y ∈ G}. By Cl , Cr and C are denoted the families of those coverings of G which have refinements of the form Cl (F ), Cr (F ) or C(F ) respectively. Each of these families generates a uniformity in G. The topology of each of these uniformities is the same as the initial topology in G. The uniform space Cbα (M, M ) is complete for complete M . The group of homeomorphisms Hom(M ) of the manifold M is contained in C 0 (M, M ) and it is characterized by the continuous condition: for each f ∈ Hom(M ) there exists f −1 ∈ Hom(M ) such that f −1 ◦ f = id. Therefore, Hom(M ) is closed in C 0 (M, M ) and hence complete [11]. If U is a bounded canonical closed subset of M contained in a finite number of charts of M , then Dif fbα (U ) for each α ≥ 1 is the neighborhood of id|U in Cbα (U, X), since from f ∈ Cbα (U, X) with k(idi,j − fi,j )|φj (U )kn,U,X ≤ |π| for each i, j and 1 ≤ n ≤ α, n ∈ N, it follows that f ∈ Dif fbα (U ), where π ∈ K, 0 < |π| < 1, k ∗ kn,U,X is either k ∗ kCbn (U,X) or k ∗ kC [n] (U,X) for α ∈ N ∪ {∞} or α ∈ {[1], [2], ...} ∪ {[∞]} respectively. b

Let {fw : w ∈ C} be a Cauchy net in Dif fbα (M ), where C is a directed set. This means that for each neighborhood W of id in Dif fbα (M ) there exists w0 ∈ C such that fw−1 fv ∈ W for each w0 ≤ w, v ∈ C, hence fv ∈ fw0 W for each v ≥ w0 . Since Dif fbα (M ) ⊂ Cbα (M, M ) and Cbα (M, M ) is complete, then fv converges in Cbα (M, M ) to a function f . We can take as W a canonical closed subset in Dif fbα(M ), id ∈ W . For each bounded U in M as above the restriction f |U is the diffeomorphism of u onto f (U ), f |U ∈ fw0 WU . Since K is complete, then X is complete, hence X is the Banach space. Thus Cbα (U, X) is complete for each U bounded clopen subset in X. Consider a neighborhood W from the base of neighborhoods of id, W = WU,n,,i,j = {f ∈ Dif fbα (M ) : kfi,j kn,U,X < }, where i, j ∈ ΛM , n ∈ N, n ≤ α, U is a clopen bounded subset in X such that

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U ⊂ φj (Uj ), 0 <  < ∞, k ∗ kn,U,X := k ∗ kCbn (U,X) or k ∗ kn,U,X := k ∗ kC [n] (U,X) b

respectively. Take without loss of generality 0 <  ≤ |π|. The manifold M is on the normed space X, hence M is paracompact and it has a locally finite refinement of its atlas (see [11]). Each ψi ◦ fw ◦ φ−1 j is converging to a function denoted fi,j and by transfinite induction this consistent family {fi,j } induces f ∈ Cbα (M, M ) such that fw converges to [k] ) is bounded for each V [k] corresponding to bounded U , f . Since Υk φi ◦ fw ◦ φ−1 j (V [k] ) is bounded and kf k ¯k then Υk φi ◦ f ◦ φ−1 i,j n,U,X < ∞, analogously for Φ f . Thus j (V f ∈ Cbα (M, M ), consequently, f ∈ Dif fbα (M ). 9. Theorem. If a manifold M has a finite atlas having clopen bounded φj (Uj ) ⊂ X, 0 ≤ α < ∞, then Dif fbα (M ) is metrizable by a left invariant metric. Proof. The metrization Theorem 8.3 [16] states that if G is a T0 topological group, then G is metrizable if and only if there is a countable open basis at e. In this case, the metric can be taken left-invariant. If At(M ) is finite such that each φj (Uj ) is a clopen bounded subset in X, then the base of neighborhoods of id in Dif fbα (M ) is countable and Dif fbα(M ) is metrizable by a left invariant metric in accordance with the general metrization theorem. Practically take as the metric ρ(f, g) := ρ(id, f −1g) := maxi,j∈ΛM kidi,j − (f −1 g)i,j kCbα (Vj ,X), where Vj := φj (Uj ). 10. Remark. As it is known left and right uniformities in a topological group may be different. Here a left uniformity was considered above. On the same group there may exist several topologies supplying it with structures of a topological group and these topologies need not be comparable.

3.

Projective Decomposition of Diffeomorphism Groups

1. Notations and Notes. Let M and N be compact manifolds over a locally compact field K. Suppose that M and N are embedded into B(Km , 0, 1) and B(Kn , 0, 1) as clopen (closed and open at the same time) subsets [6, 29], where m, n ∈ N, B(X, y, r) := {z : z ∈ X; dX (y, z) ≤ r} denotes a clopen ball in a space X with an ultra-metric dX . The unit ball B(Kn , 0, 1) has the ring structure with coordinate wise addition and multiplication, where char(K) = 0 or char(K) = p > 1 is a prime number. This ring is isomorphic to a subring of diagonal matrices in the ring Mn (K) of n × n square matrices over K. Then B(Kn , 0, |π|k) for k ≥ 1, π ∈ K, 0 < |π| < 1 is its two-sided ideal, since K is commutative and | ∗ | = | ∗ |K is the multiplicative norm in K. Thus there exists the quotient ring B(Kn , 0, 1)/B(Kn, 0, |π|k) [4]. The ring B(Kn , 0, 1) is algebraically isomorphic with the projective limit B(Kn , 0, 1) = pr − limk Spk n , Spk is a finite ring consisting of pkc elements such that Spk = Spk (K) is equal to the quotient ring B(K, 0, 1)/B(K, 0, p−k), Spk n = Spk ⊗n is an external product of n copies of Spk , c is a natural number. Though their structure depends on char(K) we denote these rings by the same symbol depending on K and omitting it, when a field is specified. k In particular B(Fpn (θ), 0, 1)/B(Fpn (θ), 0, p−k ) = (Fpn )⊗p = Spnk = Spnk (Fpn (θ)) and B(Qp , 0, 1)/B(Qp, 0, p−k ) = Zp/(pk Zp ) = Spk = Spk (Qp ) are finite rings consisting of pnk and pk elements respectively, Zp is the ring of p-adic integer numbers, aB := {x : x = ab, b ∈ B} for a multiplicative group B and its element a ∈ B, k ∈ N [38, 43]. The quotient mapping πk : K → K/B(K, 0, p−k ) is defined as well, where

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k ∈ N. Decompositions of continuous and differentiable functions on compact subsets of locally compact fields K of zero and non-zero characteristics with values in K into series of polynomials were studied in [1, 40, 7] and references therein. Each function f ∈ C t (M, N ) has a C t (B(Km , 0, 1), Kn)-extension by zero on B(Km , 0, 1). Therefore, it has the decomposition P l ¯ Qm el (1) f = l,m fm ¯ m , where el is the standard orthonormal in the non-archimedean in the Amice’s basis Q sense [38] basis in Kn such that el = (0, ..., 0, 1, 0, ...) with 1 in the l-th place, l ∈ K are expansion coefficients such that m ∈ Zn , ml ≥ 0, m = (m1, ..., mn), fm l | J(t, m) = 0, Q ¯ m are polynomials on B(Km , 0, 1) with values in K, liml+|m|→∞ |fm K ¯ J(t, m) := kQm kC t (B(Km,0,1),K). The space C t (M, N ) is supplied with the uniformity b inherited from the space Cbt (Km , Kn ). 2. Lemma. Each f ∈ C t (M, N ) is a projective limit f = pr − limk fk of polynomials P l ¯ m,k el on rings Smk = Smk (K) with values in Sn k = Sn k (K), where Q fk = l,m fm,k p p p p l ¯ m,k are polynomials on Smk with values in S k . fm,k ∈ Spk and Q p p Proof. For each m ≥ k consider the quotient mappings (ring homomorphisms): πm : B(K, 0, 1) → Spm and πkm : Spm → Spk (see §1). This induces the quotient mappings πm : N → Nm and πkm : Nm → Nk , where Nm ⊂ Spm , πkm ◦ πm = πk , πkk = idk : Spk → Spk . Let now M and N be two analytic compact manifolds embedded into B(Km , 0, 1) and B(Kn , 0, 1) respectively as clopen subsets and f ∈ C t (M, N ), where C t (M, N ) denotes the space of functions f : M → N of class C t , t ≥ 0. There exists s ∈ N such that if x ∈ M and y ∈ N , then B(Km , x, p−s) ⊂ M and B(Kn , y, p−s) ⊂ N . Therefore, consider the cofinal set Λs := {k : k ≥ s, k ∈ N} in N. For an integer t it is a space of t-times continuously differetiable functions in the sense of partial difference quotients (see Section 2 and [21, 23, 40]). Thus f = pr − limk fk , where fk := πk∗(f ), πk∗ is naturally induced by πk using the polynomial expansion of f (see in details below), where such decomposition exists for each continuous f : M → N due to Formula 1(1) (for the limit of an inverse sequence, see [4], §2.5 [11] and §§3.3, 12.202 [32]). Put C t (Mk , Nk ) := πk∗ ◦ C t (M, N ) = {fk : f ∈ C t (M, N )}, hence (1) C t (M, N ) ⊂ pr − limk C t (Mk , Nk ) algebraically without taking into account topologies. Thus write it in the form: (2) C t (M, N ) = T − pr − lim{C t (Mk , Nk ), πlk , Λs|C t (M, N )} algebraically, where (3) T −pr −lim{Pk , πlk , Λ|G} := {f : pr −lim{fk , πlk , Λ} = f, f ∈ G, fk ∈ Pk ∀k ∈ Λ} denotes the conditional projective limit with a condition G, since T − pr − lim{Pk , πlk , Λ|G} = G ∩ pr − lim{Pk , πlk , Λ}. Indeed, in accordance with §1 fk = πk∗(f ) and P l ¯ m (x))el, )(πk∗Q (4) πk∗(f (x)) = l,m (πk (fm since πk is the ring homomorphism and πk (el) = el . Then πk (axm ) = ak xm (k) for each mm 1 a ∈ K and x ∈ B(Km , 0, 1), where xm = xm 1 ...xm , ak = πk (a) with ak ∈ Spk and xm (k) := πk (xm ) with x(k) ∈ Sm , hence pk ∗ ¯ ¯ (5) π (Qm (x)) = Qm,k (x(k)). k

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P

q ¯ m (x) = If Q q,0≤qj ≤mj ∀j bq x , where q = (q1, ..., qm), m = (m1, ..., mm), mj ∈ {0, 1, 2, ...} =: N0 , j = 1, ..., m, bq ∈ K, xq := xq11 ...xqmm , then ¯ m,k (x(k)) = Pq,0≤q ≤m ∀j πk (bq )x(k)q , Q j j x(k) = (πk (x1), ..., πk(xm )), since πk : K → K/B(K, 0, p−k ) is defined as well. The series for fk is finite, since πk (a) = 0 for each a ∈ K with |a| < p−k and l | J(t, m) = 0. Therefore, (6) liml+|m|→∞ |fm K (7) f (x) = pr − lim{fk (x(k)), πlk, Λs} for each x ∈ M . As shows this proof the index t can be omitted from C t (Mk , Nk ), where k ∈ Λs . More precisely we have the following corollary. 3. Corolary. The space C t (Mk , Nk ) is independent from t and it is algebraically isomorphic with the space NkMk of all mappings from Mk into Nk for each k ∈ Λs . Moreover, (Smk )

(Snpk )

p

is a finite-dimensional space over the ring Spk .

Proof. In view of Lemma 2 in the module C t (Sm , Spk ) of the ring Spk there pk ¯ m,k (x(k)), since the is only a finite number of Spk -linearly independent polynomilas Q and Spk are finite, also z a = z b for each natural numbers a and b such that rings Sm pk a = b (mod (pk )) and each z ∈ Spk . The space C t (Mk , Nk ) is isomorphic with NkMk , since Mk and Nk are discrete. Therefore, denote C t (Mk , Nk ) by C(Mk , Nk ). 4. Corollary. There exists the group πk∗ ◦ Dif f α(M ) isomorphic with the symmetric group Σnk for every k ∈ Λs , where nk is the cardinality of Mk , α ∈ {n, [n] : 0 ≤ n ≤ ∞}. Proof. For each k ∈ Λs there exists the mapping πk : M → Mk (see the proof of lemma 2). Since πk is the quotient continuous mapping, then πk (M ) = Mk . If f : M → M is a continuous epimorphism, f (M ) = M , then πk (f (M )) = Mk . Let z ∈ Mk , k ∈ Λs , then there exists x ∈ M such that πk (x) = z, hence fk (z) = πk∗(f (πk−1(z))) due to Formulas 2(4, 5), since πk (πk−1(z)) = z and πk (x) = x(k) = z. Then πk−1 (Mk ) = M and S πk∗−1 (fk (Mk )) = z∈Mk {f (x) : x ∈ πk−1(z)}, consequently, πk∗−1 (fk (Mk )) = M , since for each x ∈ M there exists z ∈ Mk such that πk (x) = z. Therefore, if h ∈ Dif f α(M ), then hk (Mk ) = Mk , since h(M ) = M and Mk is finite. In accordance with Theorem 3.2.14 [11] if {φ, fσ0 } is a mapping of an inverse spectrum {Xσ , πρσ , Ψ} of compacts into 0 an inverse spectrum {Yσ0 , πρσ0 , Ψ0 } of T1-spaces and all fσ0 are epimorphisms on Yσ0 , then the limit mapping f = pr − lim{φ, fσ0 } is also the epimorphism. In view of Corollary 3 πk∗ ◦ Dif f α (M ) is algebraically isomorphic with the following discrete group Hom(Mk ) of all homeomorphisms hk of Mk , that is, bijective surjective mappings hk : Mk → Mk . Using an enumeration of elements of Mk we get an isomorphism of Hom(Mk ) with Σnk . 5. Suppose that Cw (M, N ) := pr − limk {NkMk , πlk , Λs} is an uniform space of continuous mappings f : M → N supplied with an uniformity and projective weak topolQ Mk (see also §8.2 [11] ogy as well inherited from products of uniform spaces ∞ k=1 Nk and §1 above). Denote the corresponding projective weak topology in Cw (M, N ) by τw . The spaces C α (M, N ) and Cw (M, N ) are subsets of K-linear spaces C α (M, Kn ) and C 0 (M, Kn ) respectively. Supply with algebraic structures subsets of the latter K-linear spaces as inherited from them. Corollary. The uniform space C α (M, N ) is not algebraically isomorphic with Cw (M, N ), when α > 0. The topological space Cw (M, N ) is compact.

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Proof. In accordance with appendix in [4] and §2.5 [11] and Formulas 2(1−6) above the spaces C 0 (M, N ) and Cw (M, N ) coincide algebraically, since the connecting mappings πnm are uniformly continuous for each m ≥ n. The space C 0 (M, Kn) is K-linear and its uniformity is completely determined by a neighborhood base of zero. The space Cw (M, N ) is uniformly homeomorphic with pr − limk (Spk )Mk , which is compact by the Tychonoff Theorem 3.2.4 [11]. Since C 0 (M, N ) 6= C α (M, N ) for α > 0, then Cw (M, N ) and C α (M, N ) are different algebraically see Formulas 2(1 − 3)). Remark. In general in two consequtive projective limits of topological spaces two limits may be non commuting. 6. Suppose that Dif fw (M ) := pr − limk Hom(Mk ) is supplied with the uniformity inherited from Cw (M, M ). The group Dif fw (M ) is called the non-archimedean compactification of Dif f α(M ). Theorem. The group Dif fw (M ) is the compact topological group and it is the compactification of Dif f α (M ) in the projective weak topology τw . If α > 0, then Dif f α (M ) does not coincide with Dif fw (M ). Proof. From Dif f α (M ) ⊂ C α (M, M ) it follows that Dif f α(M ) has the corresponding algebraic embedding into Cw (M, M ) as the set. Since Cw (M, M ) is compact and Hom(M ) is a closed subset in Cw (M, M ), then due to Corollary 5 Hom(M ) ∩ Cw (M, M ) = Dif fw (M ) is compact. The topological space C α (M, M ) is dense in C 0 (M, M ), consequently, Dif f α(M ) is dense in Dif fw (M ). If α > 0, then Dif f α (M ) 6= Hom(M ), hence Dif f α(M ) and Dif fw (M ) do not coincide algebraically. It remains to verify, that Dif fw (M ) is the topological group in its projective ¯ m (g(x))) = Q ¯ m,k (gk (x(k)) due to weak topology τw . If f, g ∈ C α (M, N ), then πk∗(Q P ∗ l ¯ Formula 2(4), consequently, πk (f ◦ g) = l,m πk (fm )Qm,k (gk (x(k))el, hence (1) (f ◦ g)k = fk ◦ gk . Since πk (x) = x(k), then πk∗(id(x)) = idk (x(k)), where id(x) = x for each x ∈ M . Therefore, for f = g −1 we have (f ◦ g)k = fk ◦ gk = idk , hence (2) πk∗(g −1) = gk−1. The associativity of the composition (fk ◦ gk ) ◦ hk = fk ◦ (gk ◦ hk ) of all functions fk , gk , hk ∈ Hom(Mk ) together with others properties given above means, that Dif fw (M ) is the algebraic group. Indeed, inverse limits of mappings f = pr − limk fk , g = pr − limk gk and h = pr − limk hk satisfy the associativity axiom as well, each f has the inverse element f −1 = pr − limk fk−1 such that f −1 (f (x)) = id and e = id is the unit element. By the definition of the weak topology in Dif fw (M ) for each neighborhood of e = id in Dif fw (M ) there exists k ∈ N and a subset Wk ⊂ Hom(Mk ) such that ek ∈ πk−1 (Wk ) ⊂ W . On the other hand, Hom(Mk ) is discrete, hence there are ek ∈ Vk ⊂ Hom(Mk ) and ek ∈ Uk ⊂ Hom(Mk ) such that Vk Uk ⊂ Wk , hence there are neighborhoods e ∈ V ⊂ Dif fw (M ) and e ∈ U ⊂ Dif fw (M ) such that V U ⊂ W , where V = πk−1 (Vk ), U = πk−1(Uk ) and V U = {h : h = f ◦ g, f ∈ V, g ∈ U }. Consider a neighborhood W 0 of f −1 , then V := W 0 f −1 is the neighborhood of e and there exists k ∈ N such that πk−1(ek ) =: U ⊂ V −1 , since e−1 k = ek and πk is the homomorphism. Thus, f U := W is the neighborhood of f such that W −1 ⊂ W 0 , hence the inversion operation f 7→ f −1 is continuous. 7. Theorem. The initial C α topology τα and the weak projective limit topology τw in Dif f α (M ) are incomparable, where M is compact.

Groups of Diffeomorphisms and Wraps of Manifolds...

575

Proof. Remind that a topology O1 in a topological space F is called weaker, than a topology O2 , if O1 ⊂ O2 or one says that O2 is stronger than O1 . Up to a diffeomorphism as above consider M clopen in B(Kn , 0, 1). Since Dif f β (M ) is contained in Dif f α (M ) for each β > α ≥ 0 and τβ is stronger than τα in Dif f β (M ), then it is sufficient to prove this theorem for α ≥ 1. Each projection πk∗ : C α (M, Kn) → (Knk )Mk induces the quotient metric ρk in the Kk -module (Knk )Mk such that ρk (fk , gk ) := inf z,πk (z)=0 kf − g + zkC α (M,Kn) , where Kk := K/B(K, 0, p−k ) is the quotient ring and πk is induced by such quotient mapping from K onto Kk . In view of the Kaplansky theorem which is the non-archimedean analog of the StoneWeierstrass theorem and true over fields of zero and non-zero characteristics in C α (B(Kn , 0, 1), K) the set of polynomials is dense [40, 7]. Thus after application of the quotient mapping we get that in the module C(Spk n , Spk ) over the finite ring Spk the set of polynomials is dense. Moreover, each continuous function has a decomposition into converging series of polynomials. The locally compact field K is commutative [43], hence the ring Spk is commutative. This means that the multiplicative group Spk ∗ := Spk \ {0} is k commutative and consists of pk − 1 elements. Thus, if 0 6= x ∈ Spk , then xp = 1. Therefore, over Spk the set of polynomials is finite dimensional, hence each fk ∈ C(Spk n , Spk ) is polynomial over Spk . Again consider C α (B(Kn , 0, 1), K), which is the algebra over K. If fi (x) = xi , where Q Q x = (x1 , ..., xn) ∈ B(Kn , 0, 1), then ni=1 fi (x)si = ni=1 xsi i ∈ C α (B(Kn , 0, 1), K), Q where xsi := sj=1 (xi)j with (xi )j = xi for each j, where s, si ∈ N. In particular, it contains a subalgebra Ak containing all constants from K and all polynomials of the form P (1) k1 =pk l(1),...,kn=pk l(n) ak1 ,...,kn xk11 ...xknn , where ak1 ,...,kn ∈ K, l(i) ∈ {0, 1, 2, ...} with P (2) k1 =pk l(1),...,kn=pk l(n) ak1 ,...,kn = 0, 1 ≤ k ∈ Z. This algebra Ak over K separates points in C α (B(Kn , 0, 1), K) and by the Kaplansky k theorem Ak is everywhere dense in C α (B(Kn , 0, 1), K)). But πk (xki i ) = (πk (xpi ))l(i) = 1 for each i and ki = pk l(i) with l(i) ∈ {0, 1, 2, ...}. Consider a polynomial with values in Kn of the form P P (3) f = id + ni=1 k1 =pk l(1),...,kn=pk l(n) ai,k1 ,...,kn xk11 ...xknn ei , where (f − id) ∈ Ank , |ai,k1 ,...,kn | ≤ |π| for each i, l(1), ..., l(n), where π ∈ B(K, 0, 1), |π| < 1, |π| = max{y ∈ ΓK : 0 < y < 1}. Then kf − idkC α (B(Kn,0,1),K) ≤ |π|, consequently, f is the isometry and f ∈ Dif f α(M ) and inevitably P P fk := πk∗ (f ) = πk∗(id) + ni=1 k1 =pk l(1),...,kn=pk l(n) πk (ai,k1 ,...,kn )ei = πk∗ (id) due to Condition (2). Therefore, (πk∗)−1 (ek ) is everywhere dense in a neighborhood of e = id in Dif f α(M ), where ek ∈ πk∗(Dif f α(M )) = Hom(Mk ) is the unit element, k > 1, card(Mk ) > 1, Hom(Mk ) is the symmetric group of Mk elements. On the other hand, there is c = card(R) elements f ∈ Dif f α(M ) with kf − idkC α (B(Kn,0,1),K) ≤ |π| such that P P f = id + ni=1 k1 =pk l(1),...,kn=pk l(n) ai,k1 ,...,kn xk11 ...xknn ei , but with P ∗ k1 =pk l(1),...,kn=pk l(n) ai,k1 ,...,kn 6= 0 for which πk (f ) 6= ek . / τα . At the same time, Thus the set πk∗−1 (ek ) is open in (Dif f α(M ), τw ), but πk∗ −1 (ek ) ∈ {f ∈ Dif f α (M ) : kf − idkC α (B(Kn,0,1),Kn) ≤ |π|} is open in (Dif f α(M ), τα), but it is not open in τw topology. This proves the assertion of this theorem, since neither τα nor τw

576

S.V. Ludkovsky

is weaker among two of them and they are different. 8. Theorem. Let M be a C α manifold finite dimensional over a non-archimedean infinite field K with a non trivial multiplicative norm complete relative to its uniformity, where K may be non locally compact. Suppose that Dif fbα(M ) is the group of all uniformly C α continuous diffeomorphisms of M , where M is embedded into Kn as the bounded clopen subset, α ∈ {t, [t]}, 1 ≤ t ≤ ∞. Then Gs := {g ∈ Dif fbα(M ) : kg − idkC α (M,Kn) < |π|s} has a non-archimedean completion, which is a group, where π ∈ K, |π| < 1, 1 ≤ s ∈ N. Proof. If g ∈ Gs , then g is the isometry: |g(x) − g(y)| = |x − y| for each x, y ∈ ¯ 1 g(x; x − y; 1), where 1 ≤ s. Without loss of M ⊂ Kn , since 1 ≤ t and g(x) − g(y) = Φ generality up to an affine diffeomorphism of Kn we can consider, that M ⊂ B(Kn , 0, 1). Consider the ring homomorphism πk : B(Kn , 0, 1) → B(Kn , 0, 1)/B(Kn, 0, |π|k), where the quotient ring is discrete and can be supplied with the quotient norm. Then πk (M ) =: Mk is the discrete topological space. Since M is clopen in Kn then each g ∈ Dif fbα (M ) has the extension as id on Kn \ M . Therefore, without loss of generality consider M such that there exists q ∈ N for which from x ∈ M it follows that B(Kn , x, |π|q) ⊂ M . Consider the cofinal set Λq := {k ∈ N : k ≥ q}. If g ∈ Gs , then (1) g(B(Kn , x, |π|k)) = B(Kn , g(x), |π|k) for each B(Kn , x, |π|k) ⊂ M , since g is the isometry. Consequently, (2) f ◦ g(B(Kn , x, |π|k)) = f (B(Kn , g(x), |π|k)) = B(Kn , f (g(x)), |π|k) and g −1(B(Kn , x, |π|k) = B(Kn , g −1(x), |π|k) for each f, g ∈ Gs . Therefore, g and πk generate the natural mapping k g : Mk → Mk such that it is bijective and epimorphic, since πk−1 (z) = B(Kn , x, |π|k) for each z ∈ πk (B(Kn , 0, 1)), and (3) B(Kn , x, |π|k) = x + B(Kn , 0, |π|k), where x ∈ B(Kn , 0, 1) is such that πk (x) = z, πk∗(g(x)) = k g(z) := πk ◦ g ◦ πk−1 (z) for each z ∈ Mk . Thus due to Equations (1 − 3) there exists the discrete group πk∗(Gs ) =: k Gs and πkl ( l Gs ) = k Gs for each l ≥ k, where πkl are mappings of the inverse system such that πkl ◦ πl = πk such that πkl and πl , πk are algebraic homomorphisms, πkl are written without star for simplicity of notation. As in Theorem 6 it gives the inverse sequence of discrete groups S = { l Gs , πkl , Λq }. In view of Lemma 2.5.9 [11] if {φ, fσ0 } is a mapping of an inverse system S = {Xσ , πρσ , Ψ} into 0 an inverse system S 0 = {Yσ0 , πρσ0 , Ψ0} and all mappings fσ0 are injective, then the limit mapping f = lim{φ, fσ0 } is also injective. If moreover, all fσ0 are surjective, then f is also surjective. Each discrete topological group is complete relative to its left uniformity generated by its topology. Thus, the limit lim S of the inverse system of discrete groups is the Tychonoff topological group relative to the projective weak topology inherited from the product TyQ chonoff topology τw , G ⊂ lim S ⊂ l∈Λq l Gs . Moreover, lim S =: Gw s is the complete uniform space with the left uniformity Tw generated by the left shifts and the neighborhood base of e in Gw s , since each k Gs is complete (see Theorems 2.5.13, 8.3.6 and 8.3.9 [11]). We have that algebraically Gs ⊂ Gw s , Gs is the topological group relative to the topology w inherited from Gs . In view of Theorem 7 the τw |Gs topology is incomparable with the Cbα uniformly bounded continuous topology. 9. Corollary. Let Dif fbα (M ) be the group as in Theorem 8. Then Dif fbα (M ) has the

Groups of Diffeomorphisms and Wraps of Manifolds...

577

non-archimedean completion which is the topological group. Proof. Let Gs be the subgroup as in Theorem 8, then Gs is clopen in Dif fbα (M ). There S exists a family gj ∈ Dif fbα (M ) such that j∈Ψ gj Gs = Dif fbα(M ), where Ψ is a set. We have that gGs is clopen in Dif fbα (M ) for each g ∈ Dif fbα(M ), since Lg : Dif fbα (M ) → Dif fbα (M ) is the homeomorphism, where Lg (h) = gh for each g, h ∈ Dif fbα(M ). Then gj Gs ∩ giGs = gj (Gs ∩ gj−1 gi Gs ). Let gj0 = id and gj 6= id for each j 6= j0. The group Dif fbα (M ) is metrizable, since Cbα (M, Kn) ⊂ Cbα (B(Kn , 0, 1), Kn) is metrizable, hence it is paracompact. Each two balls in Cbα (M, Kn ) either are disjoint or coincide, since it is the normed space. We have that Dif fbα(M ) is contained in Cbα (M, Kn ) and at the same time Dif fbα(M ) is the neighborhood of id in Cbα (M, Kn ), B(Cbα (M, Kn ), id, |π|s) = Gs for s ≥ 1 and 1 ≤ t, α ∈ {t, [t]}. Therefore choose Ψ such that gj Gs ∩ gi Gs = ∅ for each i 6= j ∈ Ψ, hence gi−1 gj ∈ / Gs for each i 6= j. S α The minimal group gr( j∈Ψ gj Gs ) = Dif fb (M ) and it is contained in the minimal S w algebraic group gr( j∈Ψ gj Gw s ) =: G . Supply the latter group with the uniformity Tw induced from the base of neighborhoods of e in Gw s with the help of left shifts. Then the w coincides with that of T in Theorem 8. Since Gw restriction of Tw on Gw w s s is clopen in G w w and Gs is complete, then for each Cauchy net {hq : q ∈ ν} in G , where ν is an ordered w set, there exists q0 such that for each q, l ≥ q0 there is the inclusion h−1 q hl ∈ Gs , hence −1 w w w {hq0 hl : l > q0 } converges in Gs to some gs ∈ Gs , since Gs is complete, consequently, {hq : q ∈ ν} converges in Gw to hq0 gs ∈ Gw and inevitably Gw is complete (see also Theorem 8.3.20 [11]). This Gw is the desired non-archimedean completion.

4.

Example of the Group of Diffeomorphisms

This section contains the example of the group of diffeomorphisms. It illustrates the general theory. For the group of diffeomorphisms of Zp of class C t , where 0 ≤ t ≤ ∞, formulas for expansion coefficients in the Mahler base of compositions g ◦ f and inverse elements f −1 of diffeomorphisms f and g are found. Let C t be a class of smoothness of functions f : M → K as in §1 (see also [1, 40, 21, 22]), where M is a Banach manifold over a complete (as an uniform space) non-Archimedean infinite field K with non-trivial valuation and of zero characteristic char(K) = 0. Suppose that Lb f (x) := f (x + b) is a shift operator, ∆b f (x) := (Lb − I)f (x) is a difference operator, L0 = I, ∆0 = I, L := L1 , ∆ := ∆1, where x, b ∈ K. For a product of two functions f, g : K → K there are formulas: (1) ∆[f (x)g(x)] = (∆f )(x)(Lg)(x) + f (x)(∆g)(x) and (2) L∆ = ∆L. Therefore,
P ∆k [f (x)g(x)] = kj=0 kj [∆j f (x)]Lj [∆k−j g(x)] for each k ∈ N, consequently, P

(3) ∆k f1 (x)...fn(x) = k1 +...+kn =k (k!/(k1!...kn!))[∆k1 f1 (x)]Lk1 [∆k2 f2 (x)]Lk2 ... [∆kn−1 fn−1 (x)]Lkn−1 [∆kn fn (x)], where Lkj acts on all functions situated
on the right from it. If f ∈ C t (Zp , Qp), then there is its expansion in the Mahler base nx as a series: P x
(4) f (x) = ∞ j=0 fj j , where fj ∈ Qp , (5) limj→∞ |fj |j t = 0 for 0 ≤ t < ∞ and (6) fj = [∆j f (x)]|x=0, since

578

S.V. Ludkovsky x











x x (7) ∆ j = j−1 for each j ∈ N, x0 = 1 and m := 0 for each 0 > m ∈ Z (see §52 [40], [36] and [22]). Therefore,P ∞ x
P (8) (g
◦ f )k = n gn ∆k m=0nfm (m) |x=0 . Since (9) nx = x(x − 1)...(x − n + 1)/n!, then f (x)


P = (k!/(k !...k !))[∆k1 f (x) ]Lk1 [∆k2 ( f (x) − (10) ∆k ( m ) n

1

k1 +...+kn =k







n

m

m

(x) (x) − n + 2)]Lkn−1 [∆kn ( fm − n + 1)]/n!, where 1)]Lk2 ...[∆kn−1 ( fm

P f (x) k f (x)
k k −1 l k−l (11) ∆ m = m l=0 l [∆ L m−1
− δl,0 (m − 1)]. Since P∞ P k−l
l k−l l x+k−l (12) ∆ L f (x)|x=0 = m=0 fm ∆ |x=0 = m fm m−l , then m

(13) ∆

[

k

∞ X

!

f (x)
m

n

fm

m=0

|x=0 = (n!)

X

−1

l1 +...+ln =k

k l1

!

!

k − l1 k − l1 − ... − ln−2 ... l2 ln−1

!

!

!

∞ X k − l1 k − l1 − l2 − δl1 ,0 (n − 1)][ fm − δl2 ,0 (n − 2)]... m − l1 m − l2 m=0 ∞ X

[

!

k − l1 − ... − ln−1 − δln−1 ,0 ]fln , m − ln−1

fm

m=0

so that it is necessary to evaluate coefficients (14)

Ωk,n m1 ,...,mn

X

:=

l1 +...+ln−1 =k−mn

k l1

!

!

k − l1 k − l1 − ... − ln−2 ... l2 ln−1

!

!

k − l1 m1 − l1

!

!

k − l1 − l2 k − l1 − ... − ln−1 .. . m2 − l2 mn−1 − ln−1 There are identities:

(15) ml = (−1)l l−m−1 , l P b
ac+e
d e (16) (1 + x) (1 + (1 + x)a)b = c d x , such that for e = b and a = −1 this gives ! ! ! ! X k − l1 k − l1 − l2 k k − l1 = (17) m1 − l1 m2 − l2 l2 l1 l ,l 1 2

X

k − l1 m1 − l1

l1

!

!

k X m1 m1 l l1 1

X

b−l m−l

l

!

!

k l1

!

!

k − l1 m 2 2 = m2 !

k − l1 m2 2 , since m2

!

X b = b!/[(m − l)!(b − m + l)!l!]. l l

General expressions are too complicated, but it is suffucient to construct a generating function for these coefficients: m

m

n−2 n−1 (18) y1m1 ...yn−2 yn−1

+mn

(x1 y2 ...yn + (y1 + z2 )(x2 y3 ...yn−1 + (y2 + z3 )(x3 y4 ...yn−1 +

Groups of Diffeomorphisms and Wraps of Manifolds...

579

(y3 + z4 )(x4 y5 ...yn−1 + ...))...)k =

m

m

n−2 n−1 y1m1 ...yn−2 yn−1

+mn

X k

(x1 y2 ...yn−1 )l1 (y1 + z2 )k−l1 (x2 y3 ...yn−1 + (y2 + z3 )(x3 y4 ...yn−1 +

l1

l1

(y3 + z4 )(x4 y5 ...yn−1 ... + (xn−1 + yn−1 )...)k−l1 =

X  k  k − l1 k − l1 

mn−2 mn−1 +mn y2m2 ...yn−2 yn−1

l1

l1 ,q1 ,l2

q 1 − l1

l2

(x1 y2 ...yn−1 )l1 y1k−q1 +m1 z2q1 −l1

l2

(x2 y3 ...yn−1 ) (y2 + z3 )k−l1 −l2 (x3 y3 ...yn−1 + (y3 + ...)...)k−l1−l2 = m

m

n−2 n−1 y3m3 ...yn−2 yn−1

 

X

+mn

k l1

l1 ,q1 ,l2 ,q2

k − l1 q1 − l 1



k − l1 l2

z2q1−l1 (x2 y3 ...yn−1 )l2 y2k+m2 −q2 z3q2 −l2 (x3 y4 ...yn−1





k − l1 − l2 (x1 y3 ...yn−1)l1 y1k+m1 −q1 q2 − l 2

+ (y3 + z4 (x4 y5 ...yn−1 + ...)...)k−l1−l2 = .... m

n−1 k k,n 1 m2 In this series coefficients in front of xm 1 x2 ...xn−1 (y1...yn−1 ) are equal to Ωm1 ,...,mn , where xj = zj+1 for each j = 1, ..., n−1, xj and yj are variables. Therefore, the generating function has the form:

m

m

n−2 n−1 (19) y1m1 ...yn−2 yn−1

+mn

(x1y2 ...yn + (x1 + y1 )(x2y3 ...yn−1 + (x2 + y2 )(x3y4 ...yn−1 +

(x3 + y3 )(x4y5 ...yn−1 ... + (xn−1 + yn−1 )...)k = X

m

m1 m2 n−1 k Ωk,n m1 ,...,mn x1 x2 ...xn−1 (y1 ...yn−1 ) +

m1 ,...,mn−1

X

q

q1 q2 n−1 Υk,n m1 ,q1 ,...,mn−1 ,qn−1 x1 x2 ...xn−1 ×

m1 ,q1 ,...,mn−1 ,qn−1 k+m

y1k+m1 −q1 ...yn−2 n−2

−qn−2 k+mn−1 −qn−1 +mn yn−1 ,

where coefficients Υk,n m1 ,q1 ,...,mn−1 ,qn−1 are given by Equation (18). In particular id = f −1 ◦ f and ∆k id(x)|x=0 = δk,1 , hence P −1 −1 ) Qk,n (f ), where coefficients Qk,n are given by Equa(21) δk,1 = ∞ n n=0 (n!) (f tions (8, 13, 14, 19), that is X

(22) Qk,n (f ) =

X

Ωk,n mi ,...,mi 1

m1 ,...,mn i1 <...
n−p

fmi1 ...fmin−p !

Y (js∈(1,...,n)\{i1,...,in−p};j1 <j2 <...;lj1 =0,...,ljp−1

k − l1 − ... − ls−1 (js − n) . ms =0)

For analytic functions there are equalities: k n

(23) ∆ x =

n−1 X l1 =0

n−1 1 −1 X lX l1 =0 l2 =0

lk−1 −1

...

X

lk =0

n l1

!

!

!

n ∆k−1 xl1 = l1 !

l1 lk−1 lk ... x , consequently, l2 lk

580

S.V. Ludkovsky k n

(24) (∆ x )x=0 =

n−1 1 −1 X lX l1 =0 l2 =0

lk−2−1

...

X

lk−1 =0

n l1

!

!

l1 lk−2 ... l2 lk−1

!

=: Tn,k .

On the other
hand, P x m−l (−1)l α (1, ..., m − 1) = (25) m = (m!)−1 m l l=0 x P l S x , where m,l l P (26) αl (z1, ..., zm) := i1
X

am (g)[m!/(l1!...lm!)]alk11 (f )...alkmm (f )]xn,

m,lj ,kj ,n

where in the last series k1 l1 + ... + km lm = n, l1 + ... + lm = m, 0 ≤ lj , kj ∈ Z. For estimations of Ωk,n m1 ,...,mn fm1 ...fmn it can be used k
−λ(k)+λ(q)+λ(k−q) , where (31) | l |p = p (32) λ(n) = (n − sn )/(p − 1), n = a0 + a1 p + ... + aj pj , sn := a0 + a1 + ... + aj , where ai ∈ {0, 1, ..., p − 1}, λ(q) + λ(k − q) − λ(k) = (sk − sq − sk−q )/(p − 1) (see also §25 in [40] and [36]).

5.

One-parameter Subgroups of Diffeomorphism Groups

1. Theorem. If M is a compact manifold over a locally compact field K of characteristic char(K) = p > 1, 1 ≤ t ∈ N, α ∈ {t, [t]}, then there exists a clopen subgroup W in Dif f α(M ) such that each element g ∈ W lies on a local one-parameter subgroup g x continuous by x relative to the multiplicative group (K∗, ×) with x ∈ K∗ . Proof. Since M is compact, then M is finite dimensional dimK M = n over K and the manifold M can be supplied with a finite disjoint analytic atlas, that is, with disjoint clopen charts a finite union of which covers M . Therefore, there exists a natural C α embedding of M into Kn as the clopen subset. As it was proved above Dif f α(M ) is mertizable with the left-invariant metric ρα. Choose k0 ∈ N and the clopen subgroup W := {g ∈ Dif f α (M ) : ρα(g, id) < |π|k0 } such that g ∈ W implies kg − idkCbα (M,Kn) < |π|k0 . Consider x ∈ B(K, 1, |π|s), where 1 ≤ s ∈ N, then |x| = 1, since |x − 1| ≤ |π|s and due to the ultrametric inequality. If g ∈ W , then for the proving of this theorem it is necessary to find a one-parameter local subgroup {gx : x ∈ B(K, 1, |π|s)} satisfying Conditions (1 − 3): (1) g 1 = id; (2) g x1 g x2 = g x1x2 for each x1, x2 ∈ B(K, 1, |π|s); (3) g x0 = g for some x0 ∈ B(K, 1, |π|s). Each g ∈ Dif f α (M ) has the form g = (g1, ..., gn), where gj : M → K for each j ∈ {1, ..., n}. Since M is embedded as the compact subset in Kn , then its covering by balls B(Kn , x, r) ⊂ M has a finite subcovering {B(Kn , xj , rj ) : j = 1, ..., m}, where x ∈ M , 0 < rj < ∞, m ∈ N, consequently, minj=1,...,m rj = r > 0. If g x1 and f x2 are two commuting local one-parameter subgroups for each x1, x2 ∈ B(K, 1, |π|s) in Dif f α(M ), then g xf x =: (gf )x is a local one-parameter subgroup in

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Dif f α (M ), since (gf )x1 (gf )x2 = (g x1 f x1 )(g x2 f x2 ) = g x1 g x f x1 f x2 = g x1x2 f x1 x2 = (gf )x1x2 for each x1, x2 ∈ B(K, 1, |π|s), also g 1 = f 1 = e = id. For example, if supp(g x) ⊂ A and supp(f x ) ⊂ C for each x ∈ B(K, 1, |π|s) such that A ∩ C = ∅, where A and C are closed subsets in M , supp(g) := clM {y ∈ M : g(y) 6= y}, clM (V ) denotes the closure of a subset V in M , then g x1 f x2 = f x2 g x1 for each x1 6= x2 ∈ B(K, 1, |π|s). On the other hand, each g ∈ Dif f α (M ) can be decomposed into the product g = h1 ...hm , where supp(hj ) ⊂ B(Kn , xj , rj ) for each j = 1, ..., m. Therefore, the proof of this theorem reduces to the case Dif f α (B), where B is a clopen compact ball in Kn , since W decomposes into the internal direct product of its subgroups Wj := {g ∈ W : supp(g) ⊂ B(Kn , xj , rj )} and Wj has the natural embedding into Dif f α(B(Kn , xj , rj )) for each j = 1, ..., m. For g = id put idx = id for each x ∈ B(K, 1, |π|s). Therefore, due to Equations 3.8(1, 2) it remains the case of g 6= id in W , consequently, x0 6= 1 for such g. In accordance with Lemma 7.6 [15] if K is a field and G is a finite subgroup of the multiplicative group of nonzero elements of K, then G is a cyclic group. Then by Theorem 7.b the multiplicative group of nonzero elements of a finite field is cyclic. In view of the Wedderburn Theorem 7.c a finite division ring is necessarily a commutative field [15]. The field K is isomorphic with Fpu (θ), where Fpu is the finite field consisting of pu elements, u u ∈ N. The finite field Fpu is the splitting field of the polynomial xp − x. Thus χ : K∗ → K∗ is a continuous multiplicative character, where K∗ = K \ {0}, if and only if it can be written in the form χ(x) = φ(x)ψ(x), where φ and ψ are continuous multiplicative characters such that φ : F∗pu → F∗pu is some homomorphism of the multiplicative group of Fpu and φ(θ) = θ, ψ(θ) = θk for some nonnegative integer k and the restriction of ψ on Fpu is the identity mapping, since there exists the natural embedding of Fpu into K. Therefore, |χ(θ)| = |θ|k . Denote by Ω the family of all continuous multiplicative characters χ : K∗ → K∗ . In particular, χ1 (x) = 1 and χid (x) = x for each in x ∈ K∗ are the trivial character and the identity character respectively. If g x is a local one-parameter subgroup in Dif f t (M ), then g χ(x) is also a local oneparameter subgroup in Dif f α (M ). If g x0 = g, then g z = g for each χ(z) = x0 with χ ∈ Ω, since χ(1) = 1. Therefore, if x0 ∈ B(K, 1, |π|s) is a marked point, then it is sufficient to satisfy Condition (3) for some x ∈ Vs := {χ−1 (x0 ) : χ ∈ Ω} ∩ B(K, 1, |π|s), where x0 6= 1 and x 6= 1 for g 6= 1. Thus, if each hj 6= id in Wj has x0 (hj ) ∈ Vs , then the desired local one-parameter subgroup g x will be found for id 6= g = h1 ...hm ∈ W . Take, for example, x0 = 1 + θs . We say that a family A of functions f : X → K separate points of X if for each x 6= y ∈ X there exists f ∈ A such that f (x) 6= f (y). If G is a cyclic group of order k and a ∈ G is such that {1, a, a2, ..., ak−1} = G and m is a natural number mutually prime with k, (m, k) = 1, then ζ : G → G such that ζ(al ) = alm for each l = 0, ..., k − 1 is the automorphism of G, in particular, for k = pu − 1. Since the multiplicative group G = F∗pu is cyclic, then the family of all φ separate points of G, hence the family Ω of all continuous multiplicative characters χ of K∗ separate points of K∗. In view of the Kaplansky theorem a subalgebra A of C 0 (X, K) containing all constant functions and separating points of a locally compact totally disconnected Hausdorff space X is dense in C 0 (X, K) [18, 3]. From [1] it follows, that C 0 (M, Kn ) has the polynomial basis {Qm ¯ ∈ N0 n , i = 1, ..., n} with N0 = {0, 1, 2, ...} and ¯ (y)ei : m ei = (0, ..., 0, 1, 0, ...) ∈ Kn with 1 on the i-th place.

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In view of the proof above up to the affine C ∞ diffeomorphism it is sufficient to consider W for Dif f t (B), where B = B(Kn , 0, 1). We seek a solution in the form P (4) gjx(y) = yj + ∞ k=1 bj,k (y, x) for each j = 1, ..., n with the converging series of functions bj,k (y, x) : M × B(K, 1, |π|s) → K of class C t by y ∈ M and continuous by x ∈ B(K, 1, |π|s) for each j = 1, ..., n such that kbj,k kC t (B,K) < |π|k0 and limk→∞ maxj=1,...,n kbj,k kC t (B,K) = 0, b where g = (g1, ..., gn). Each function bj,k (y, x) has the decomposition P n cj,k,m (5) bj,k (y, x) = χj,k (x) m∈N ¯ Qm ¯ (y), ¯ 0 where χj,k (x) ∈ Ω, y ∈ B, x ∈ B(K, 1, |π|s), cj,k,m ¯ ∈ K. Without loss of generality we can suppose that |χj,k (x)| ≤ 1 on B(K, 1, |π|s) for each j, k, m, since B is the unit ball and s ≥ 1. The convergence of Series (4) is equivalent to: maxj=1,...,n kcj,k,m (6) limk+|m|→∞ ¯ Qm ¯ (y)kCbα(M,K) kχj,k (x)kC 0 (B(K,1,|π|s),K) = 0, ¯ b where |m| ¯ := m1 + ... + mn , m = (m1 , ..., mn). Then Condition (1) is equivalent to P∞ P ¯ and j, while Condition (3) is equivalent to m ¯ = 0 for each m ¯ Qm ¯ (y) = k=1 cj,k,m ¯ cj,m P∞ ¯ ∈ N0 n } gj (y) − yj with cj,m ¯ := ¯ χj,k (x0) , since χj,k (1) = 1, {Qm ¯ (y) : m k=1 cj,k,m are linearly independent over K. Consider two monotone increasing sequences {sv ∈ N : v ∈ N} and {qv ∈ N : v ∈ N} such that qv ≥ sv for each v. For each q > l ∈ N there exists the quotient algebraic homomorphism of rings πlq : B(K, 0, 1)/B(K, 0, |π|q) → B(K, 0, 1)/B(K, 0, |π|l) and πq : B(K, 0, 1) → B(K, 0, 1)/B(K, 0, |π|q) such that πlq ◦ πq = πl . These algebraic homomorphisms induce natural homomorphisms of functions in the same notation such that πq∗(g) =: q g, where q g ∈ Hom(Mq ), Mq = πq (M ) is a finite set, g ∈ W ⊂ Dif f t (B), q g : Mq → Mq is a bijective surjective mapping of Mq onto Mq (see Formulas 3.2(4 − 7)). Then also πsv (B(K∗, 1, |π|s)) is the finite set for sv > s. For sv > 1 we have that πsv (K) is the discrete commutative ring (see §3.1). If x ∈ K and |x| = 1, then |x−1| = 1. The multiplicative norm in K induces the norm in πsv (K). If |x − 1| < P l 1, then x−1 = (1 + (x − 1))−1 = 1 + ∞ l=1 (1 − x) , thus πsv (x) has the inverse in s s πsv (B(K, 1, |π| )), when |x − 1| ≤ |π| , s ≥ 1, since πsv ((x − 1)l) = 0 for ls ≥ sv . Then 1 ∈ πsv (B(K, 1, |π|s) and πsv (x1x2 ) = πsv (x1 )πsv (x2 ) ∈ πsv (B(K, 1, |π|s)) for each x1 , x2 ∈ B(K, 1, |π|s), since |x1x2 −1| ≤ max(|x1 −1||x2|, |x2 −1|) ≤ |π|s and |x2 | = 1. Thus, B(Fpu , 1, |π|s) for a natural number s ≥ 1 is the multiplicative commutative group and B(Fpu , 1, |π|s1 )/B(Fpu , 1, |π|s2) for each natural numbers 1 ≤ s1 < s2 is the finite multiplicative commutative quotient group. If g ∈ W and k0 ≥ 1, then |g(y)−y| ≤ kg −idkC 1 |y| ≤ |π|k0 |y| and |g(y1)−g(y2 )| ≤ kgkC 1 |y1 − y2 | and |g −1 (y1) − g −1 (y2)| ≤ kg −1kC 1 |y1 − y2 |, since t ≥ 1, consequently, g is the isometry, that is, |g(y)| = |y| for each y ∈ M . For given s and s1 choose k0 and q1 ≥ s1 such that for each g ∈ W we have q1 g = id on Mq1 . Then for a given g ∈ W there exists the algebraic homomorphism η1 from πs1 (B(K, 1, |π|s)) into πq1 (W ) satisfying Conditions (1 − 3) with πs1 (x0) instead of x0 . Each g ∈ W and each x ∈ K is the projective limit of the inverse sequences g = pr − lim{ qv g, πqqlv , Λq1 } and x = pr − lim{ sv x, πsslv , N}, since sequences {qv } and {sv } are cofinal with N, where v ≥ l ∈ N (see also §2.5 [11]). For example, we can q take sv+1 = sv + 1 for each v ∈ N. Consider qv+1 g, then πqvv+1 ( qv+1 g) = qv g. Each qv g ∈ Hom(Mqv ) is the element of the symmetric group Sbv , where bv is the cardinality of the finite set πqv (M ) = Mqv (see also Theorem 3.6 above). Thus, qv g has a decomposition

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into a product of nonintersecting finite cycles. If yv ∈ Mqv , then the cardinality of each q (πqvv+1 )−1 (yv ) is the same for each yv ∈ Mqv . The homomorphism qv+1 g on each subset q (πqvv+1 )−1 (yv ) acts as the isometry. Then each cycle of qv g splits into the product of cycles or becomes a cycle of greater length for qv+1 g. Moreover, qv+1 g is the product of two hoq q momorphisms h and f , where f ((πqvv+1 )−1 (yv )) = (πqvv+1 )−1 (yv ) for each yv ∈ Mqv and qv+1 ∗ πqv (f ) = πqv (id) is the identity on πqv (M ), while h(zv+1 ) = h(yv+1,0 )+(zv+1 −yv+1,0 ) q q with a marked yv+1,0 ∈ (πqvv+1 )−1 (yv ) and for each zv+1 ∈ (πqvv+1 )−1 (yv ). If σ ∈ Σl and σ is a cycle of length u, then its algebraic order is u, that is, the cyclic group {σ a : a ∈ N} is of order u. On the step v there is the commutative local subgroup xv = η (x ) for each x ∈ π (B(K, 1, |π|s)). Therefore, on the v + 1-th step choose qv g v v v sv qv+1 sufficiently large such that a number of nonintersecting cycles σv,j or their length are sufficiently large in the sense that a commutative subgroup of πqv+1 (W ) generated by bj , bj ∈ N, is of sufficiently large order that to provide the algebraic finite products of σv,j homomorphism ηv+1 from πsv+1 (B(K, 1, |π|s)) into πqv+1 (W ) satisfying Conditions (1 − s 3) with πqv+1 (x0) instead of x0 and πsvv+1 (ηv+1) = ηv . In accordance with Lemma 2.5.10 [11] if {φ, fj 0 } is a mapping of an inverse system S = 0 {Xj , πij , Ψ} into an inverse system S 0 = {Yj 0 , πij0 , Ψ0} and all fj 0 are homeomorphisms, then the limit mapping f = pr−lim{φ, fj 0 } is also the homeomorphism of X = pr−lim S onto Y = pr−lim S 0, where φ : Ψ0 → Ψ is a nondecreasing function, Ψ0 and Ψ are directed sets. Using the same notation which can not cause a confusion we mention the following. Each this induction step gives the corresponding solution of (4 − 6) ∗ with ηv (πsv (x)) ∈ πqv (B(K, 1, |π|s)) and πqv (y) and πqv (cj,k,m ¯ ), πqv (Qm ¯ (y)) and s πsv (χj,k (x)) ∈ πqv (B(K, 1, |π| )) instead of x, y, cj,k,m ¯ , Qm ¯ (y) and χj,k (x) respectively. If f ∈ Cbα (M, Kn ) and kf − idkCbα (M,Kn) < 1, then f ∈ Dif f α (M ). We say that a subset A in M is an -net, if for each y ∈ M there exists z ∈ A such that |z − y| < , where 0 <  < ∞. We have limv→∞ |π|v = 0 and representatives of Mqv in M form a ρv net for suitable subsequence qv , which is denoted by the same notation, where 0 < ρ < 1 is a constant and ρ is a parameter characterizing a net Aqv corresponding to {Qm ¯ ∈ N0 n } ¯ :m for mj ≤ qv for each j = 1, ..., n, where each z ∈ Aqv is a zero of Qm ¯ as soon as mj ≥ mj,0 for some j = 1, ..., n, where m ¯ 0 := (m1,0, ..., mn,0) corresponds to z. On the other hand, |χj,k,m (x)| ≤ 1 on B(K, 1, |π|s) for each j, k, m, moreover, x0 = 1 + θs , |x0| = 1. Then the corresponding to it series converges, since qv g ηv (xv ) ∈ πq∗v (W ) for each xv ∈ πsv (B(K, 1, |π|s)) and W is complete relative to the Cbα uniformity. Thus, the inverse sequence of qv g ηv(πsv (x)) converges to a local multiplicative continuous oneparameter subgroup g x(y) ∈ W relative to the Cbα × Cb0 uniformity which is Cbα by y ∈ M and Cb0 by x ∈ B(K, 1, |π|s). 2. Theorem. If M is a manifold on the Banach space X := c0(γX , K) with a finite atlas and charts with bounded φj (Uj ) in X for each j ∈ ΛM , where char(K) = p > 1, then in Dif fbα (M ) in each neighborhood of id there are g 6= id which does not belong to any non-trivial one-parameter subgroup g y relative to the additive group (K, +). Moreover, if M is embedded into X and (i) (h − h ◦ g −1 ), (h − h ◦ g −1 ) ◦ g 2,...,(h− h ◦ g −1 ) ◦ g p−1 are K-linearly independent and g 6= id, where h := g − id, then g does not belong to any one-parameter subgroup

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{g y : y ∈ (K, +)}. Proof. From the conditions of this theorem it follows, that M has a C t embedding as an open bounded subset in X. In view of Theorem 2.9 the group Dif fbα (M ) is metrizable. Since Dif fbβ (M ) is everywhere dense in Dif fbα (M ) for each β > α, then it is sufficient to consider the case t ≥ 1, α ∈ {t, [t]}. If g ∈ Dif fbα(M ), then up to a diffeomorphism of manifolds g|U ∈ Dif fbα (U ) for any U open in M , since g : U → g(U ) ⊂ M . Let U ⊂ M be a clopen bounded subset such that U ⊂ Uj for some j ∈ ΛM , g ∈ Dif fbα (M ) and supp(g) ⊂ U , then g|U ∈ Dif fbα(U ). Take in particular U := φ−1 j (B(X, 0, |π|)), where π ∈ K, 0 < |π| < 1. Thus, if prove theorem for U = B(X, 0, |π|), then it will be also true for Dif fbα (M ). Consider in Dif fbα(U ) a left-invariant metric ρα (see Theorem 2.9). Take id 6= g ∈ W := {f ∈ Dif fbα(U ) : ρα (f, e) ≤ |π|}, then g = id + h, where 0 < khkCbα (U,X) ≤ |π|. If we consider pn as the element of K, where n ∈ N := {1, 2, ...}, then pn = 0 ∈ n K, since char(K) = p. So we need to have g p = g 0 = id
on U , where g 1 = g, P n n g k+1 = g k g 1,...,gp = g p −1 g 1. Consider (1 + a)k = ls=0 ks as , where a ∈ K, then n n n n (1 + a)p = 1 + ap and inevitably |(1 + a)p | = |1 + ap |, since each binomial coefficient
pn
n − 1 and hence pn is equal to zero in K. Then is divisible on p for each 1 ≤ l ≤ p l l l
l+(l−1)s s a and s=0 s xi p2 p+1 xi a for l = p and so

(xi + axli ) ◦ (xi + axli ) = xi + axli + a axpi )

axpi )

2axpi

Pl

so on. In particular,

◦ (xi + = xi + + on. This shows, that (xi + elements of the form g(x) = id(x) + axli ei with a 6= 0 and 1 < l ∈ N can not lie on any one-parameter subgroups, since g p 6= id. Demonstrate in general for M embedded into X, that each id 6= g ∈ W satisfying Condition (i) does not belong to any one-parameter subgroup. On the other hand, g 2 (x) = g◦g(x) = id◦(id+h)(x)+h◦(id+h)(x) = g(x)+h◦g(x), g n (x) = g n−1(x)+h◦g n−1(x), consequently, ρα(g n, g n−1) = ρα (g 2, g) ≤ |π| for each n ∈ N and kh ◦ g n−1 kCbα ≤ |π|. Then by induction g n (x) = g(x) + h ◦ g(x) + ... + h ◦ g n−1 (x) for each n ≥ 2, hence ρα(g n , g) ≤ |π|. Then |h(y) − h(x)| ≤ khkCbα (U,X)|x − y| for each x, y ∈ U , in particular, for y = g(x), where y − x = g(x) − x = h(x), hence |h ◦ g(x) − h(x)| ≤ khkCbα (U,X)|h(x)| ≤ |π|2. Therefore, |g n(x) − g(x) − (n − 1)h(x)| ≤ khkCbα (U,X)|h(x)| for each x ∈ U , where |h(x)| ≤ khkCbα (U,X)|x| ≤ |π||x|, consequently, |g n(x) − g(x) − k

(n − 1)h(x)| ≤ (khkCbα (U,X))2 |x| ≤ |π|2|x|. Thus |g p (x) − id(x)| ≤ (khkCbα (U,X))2|x|, k

since |pk h(x)| = 0. Suppose that g p = id for each k ∈ N, then 0 = g p(x) − id(x) = g p(x) − g(x) − (p − 1)h(x) = h(x) + h ◦ g(x) + ... + h ◦ g p−1(x) = (h ◦ g(x) − h(x)) + ... + (h ◦ g p−1(x) − h(x)) = h(x) + p(h ◦ g 1(x) − h(x)) + (p − 1)(h ◦ g 2(x) − h ◦ g(x)) + ... + 3(h ◦ g p−2(x) − h ◦ g p−3(x)) + 2(h ◦ g p−1(x) − h ◦ g p−2(x)) − h ◦ g p−1(x), since ph(x) = 0 and p(h ◦ g − h)(x) = 0 identically and g p−1 (x) = g −1 (x). Therefore, it would be 0 = |h(x) − h ◦ g −1(x)| on U , since Condition (i) is supposed to be satisfied, that leads to the contradiction, since h 6= 0, g 6= id, h ◦ g 6= h. Thus g p(x) 6= id(x) for each id 6= g ∈ W satisfying Condition (i), consequently, g does not belong to any one-parameter subgroup {g y : y ∈ (K, +)}. 2.1. Remark. On the other hand, if M ⊃ B(Kn , x0, r), then g y (x) := x + yz for x ∈ B(Kn , x0, r) and g y (x) := x for x ∈ M \ B(Kn , x0, r) is the non-trivial oneparameter subgroup for a marked z ∈ B(Kn , 0, r), where y ∈ B(K, 0, 1), 0 < r, x0 ∈ M . Therefore, for each neighborhood W of id there exists 0 6= z ∈ B(Kn , 0, r) such that

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id 6= g ∈ W .

6.

Topological Perfectness of Diffeomorphism Groups

1. Definition. A group G is called algebraically perfect, if its commutator group [G, G] coincides with G, where [G, G] is the minimal group generated by all commutators [f, g] := f −1 g −1 f g, f, g ∈ G. A group G is called algebraically simple, if it does not contain any normal subgroup other than {e} and G. A topological group G is called topologically perfect if clG [G, G] = G, G is called topologically simple if it does not contain any closed normal subgroup other than {e} or G, where clG V denotes the closure of a subset V in G. 2. Remark. It is well known that the symmetric group Sn is perfect for n 6= 2 and n 6= 6, but it is not simple for n ≥ 3, since the subgroup An consisting of even permutations is its normal subgroup different from {e} and Sn [19]. Below the topological perfectness and simplicity of the diffeomorphism group Dif f α (M ) is considered relative to its C α topology. 3. Theorem. Let M be a compact manifold, t ≥ 0, α ∈ {t, [t]}. Then Dif f α (M ) supplied with the C α compact-open topology is topologically perfect. Proof. In view of Theorem 3.6 above and Theorem 2.5 [24] the diffeomorphism group Dif f α (M ) algebraically is the projective limit of an inverse sequence S = {Gq , πsq , N} of finite groups Gq , where Gq is isomorphic with the symmetric group Sb with b = card(Mq ), Mq = πq (M ). There exists q0 ∈ N such that for each q > q0 the cardinality of Mq is greater than 6, consequently, Gq is perfect for each q > q0 , since Sb is perfect with b = card(Mq ) by the H¨older Theorem 5.3.1 [19]. Each element h ∈ Dif f t(M ) is decomposable into the thread {hq , πsq , N}. Every hq is decomposable as a finite product of commutators in Gq for q > q0 . Therefore, the commutator group [G, G] is dense in G (see also Theorem 3.7 above), since the projective limit of a thread of products of commutators is the product of commutators, where G = Dif f α(M ). 4. Corollary. Let M be a locally compact manifold, t ≥ 0, α ∈ {t, [t]}. Then the diffeomorphism group Dif f α(M ) and the group Dif fcα (M ) of all C α diffeomorphisms with compact supports supplied with the C α compact-open topology are topologically simple. Proof. The group Dif fcα (M ) is everywhere dense in Dif f α(M ). Therefore, it is sufficient to prove this corollary for Dif fcα (M ). The group G = Dif fcα(M ) satisfies the Epstein system of axioms. Let X be a paracompact Hausdorff topological space, G a group of homeomorphisms of X, and U a basis of open sets for the topology of X. The Epstein axioms are the following: (E1) if U ∈ U and g ∈ G, then gU ∈ U ; (E2) G acts transitively on U ; (E3) let g ∈ G, U ∈ U and B be an open cover of X, then there exist m ∈ N and g1, ..., gm ∈ G and V1 , ..., Vm ∈ B such that: (i) g = gm gm−1 ...g1; (ii) supp(gi) ⊂ Vi ; (iii) supp(gi ) ∪ (gi−1...g1cl(U )) 6= X for each i : 1 ≤ i ≤ n. The manifold M has an C α embedding θ as the clopen subset into either Kn or a direct topological sum of copies of Kn such that θ(M ) is a disjoint union of clopen balls, since

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M is totally disconnected and has a base of topology consisting of clopen compact subsets in M . As U take a system of all clopen proper subsets U in M for which there exists g ∈ G such that θ(gU ) is a ball of finite positive radius in Kn . Since M is totally disconnected and modelled on Kn as the manifold, then U forms the base of the topology of M . In particular, for the diffeomorphism group G = Dif fcα(M ) and the family U of M = X we have gU ∈ U for each g ∈ G, since g −1(gU ) = U and g : M → M is the continuous bijective epimorphism together with g −1. If U = M , then g(M ) = M . A disjoint clopen covering C of θ(M ) by balls form a C ∞ atlas (moreover, it is the analytic atlas) such that if f |U is of C α class for each U ∈ C, then f is C α on θ(M ). If U ∈ U and U 6= M , then M \ U is a clopen nonvoid subset in M . Therefore, if U1 and U2 are two clopen nonvoid subsets in U different from M , then there exists g ∈ G such that g(U1) = U2 , since each two clopen balls in Kn are analytically diffeomorphic. Thus axioms (E1, E2) are satisfied. If g ∈ G, then supp(g) := cl{x ∈ M : g(x) 6= x} is compact and for an open covering B of M there are V1, ..., Vm in B such that supp(g) ⊂ V1 ∪ ... ∪ Vm . Suppose that U ∈ U and cl(U ) 6= M , where cl(U ) denotes the closure of U in M . Since U is clopen in M , then cl(U ) = U . The topological Tychonoff space M is totally disconnected and locally compact paracompact, since it is modelled on Kn . In view of Theorem 6.2.9 [11] it is strongly zero dimensional. Therefore, V1, ..., Vm has a refinement P1 , ..., Pm consisting of clopen compact subsets in M such that supp(g) ⊂ P1 ∪ ... ∪ Pm . Take P0 := ∅ and Wi := Pi \j
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3.6 above or Theorem 2.5 [24] for each v ∈ N, v ≥ s, there exists the quotient mapping πv∗ : Dif f α (M ) → Σbv , where Σm is the symmetric group of the set {1, ..., m}, bv is the cardinality of the finite set Mv = πv (M ), πv∗ is induced by the quotient mapping πv : K → K/B(K, 0, |π|v ) with the help of polynomial expansions by Formulas 3.2(4, 5). Consider M embedded into Kn . The set of diffeomorphisms f such that f − id is a piecewise affine on balls of the covering of M is contained in Dif f α (M ), consequently, πv∗ (Dif f α (M )) = Sbv is the epimorphism. This also follows from Formulas 3.2(4−7). The automorphism ψ induces the automorphism ψv of πv∗ (Dif f α (M )) such that πv∗ (ψ(g)) =: ψv (πv∗ (g)), since then πv∗ (ψ(gh)) = πv∗ (ψ(g)ψ(h)) = πv∗ (ψ(g))πv∗ (ψ(h)) = ψv (πv∗ (g))ψv (πv∗ (h)) for each g, h ∈ Dif f α (M ). Consider v sufficiently large such that bv > 6. In view of the H¨older Theorem 5.3.1 [19] the automorphism ψv is internal: ∗ α ∗ α ψv (a) = hv ah−1 v for each a ∈ πv (Dif f (M )), where hv ∈ πv (Dif f (M )) is a marked v v element. This defines the inverse sequence {hv , πl , Λs } such that πl (hv ) = hl for each v ≥ l ∈ Λs , where πlv ◦ πv∗ = πl∗ and πlv : Gv → Gl are algebraic epimorphisms of discrete groups Gv := πv∗ (Dif f α (M )) for each v ≥ l ∈ Λs , πlv are written without star for simplicity of notation. Each hv : Mv → Mv is the homeomorphism and each ψv : Gv → Gv is the homeomorphism. A limit of an inverse mapping system of homeomorphic mappings is a homeomorphism by Proposition 2.5.10 [11]. Therefore, h = pr − lim{hv , πlv , Λs } is the element of Dif fw (M ) (see Theorems 3.6, 3.7 above and Theorem 2.5 [24]), moreover, v −1 for each g ∈ Dif f α (M ). On the other hand, ψ(g) = lim{hv πv∗ (g)h−1 v , πl , Λs } = hgh Dif fw (M ) is algebraically isomorphic with Hom(M ), hence h ∈ Hom(M ). 7. Remark. Theorem 6 is not true for a locally compact noncompact manifold M for the group Dif fcα (M ) of compactly supported diffeomorphisms of Dif f α (M ), since then Dif fcα (M ) has the external automorphisms φf (g) := f gf −1 for f ∈ Dif f α (M ) \ Dif fcα (M ), where g ∈ Dif fcα (M ). Moreover, Dif fcα (M ) is the proper normal subgroup in Dif f α (M ), but Dif fcα (M ) for a locally compact noncompact manifold M is the proper non-closed subgroup everywhere dense in Dif f α (M ).

7.

Projective Decomposition of Wrap Groups

¯ and N be two compact manifolds over a locally compact non1. Let as in §2.1 M archimedean infinite field K with a multiplicative non trivial norm relative to which K ¯ ) be a subgroup in Dif f α (M ¯ ) of all is complete as the uniform space and Dif f0α (M α ¯ ) such that ψ(s0 ) = s0 , where s0 is a marked point in M ¯, elements ψ ∈ Dif f (M α α ¯ α ∈ {t, [t]}. Denote by C0 (M, N ) a subspace in C (M , N ) of all elements f ∈ ¯ , N ) such that lim|ζ |+...+|ζ |→0 Φ ¯ v (f − w0 )(s0 ; h1 , ..., hn ; ζ1 , ..., ζn ) = 0 for α = t C α (M n 1 [n]

or lim|ζ1 |+...+|ζn |→0 Υv (f − w0 )(s0 ) = 0 for α = [t] for each v ∈ {0, 1, ..., t}, where ¯ \ s0 and w0 (M ¯ ) = {y0 }, x[k+1] = (x[k] , v [k] , ζk ) (see §2.6 [23] and Section 2 M = M above). Geometric wrap monoids Ωα (M, N ) and wrap groups Lα (M, N ) for C α classes of mappings were constructed in [23]. The same construction is for char(K) = p > 0. Theorem. Let Ωα (M, N ) be a commutative wrap monoid, then the quotient mappings πk induce the corresponding inverse sequence {Ω(Mk , Nk ) : k ∈ N} such that Ωw (M, N ) := pr − limk Ω(Mk , Nk ) is the commutative compact topological monoid, where πk∗ : Ωα (M, N ) → Ω(Mk , Nk ), πkl : Ω(Ml , Nl ) → Ω(Mk , Nk ) are surjective map-

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pings for each l ≥ k, Ω(Mk , Nk ) = {fk : fk ∈ NkMk , fk (s0,k ) = y0,k }/Kα,k , Kα,k is an equivalence relation induced by an equivalence relation Kα . Moreover, Ωw (M, N ) is a compactification of Ωα (M, N ) relative to the projective weak topology τw . Proof. In view of Corollary 3.3 πk (C0α (M, N )) is isomorphic with {fk : fk ∈ Mk Nk , fk (s0,k ) = y0,k }, where the quotient mapping is denoted by πk both for M and N , since it is induced by the same ring homomorphism πk : K → K/B(K, 0, 1), s0,k := πk (s0 ) and y0,k := πk (y0 ), where k ≥ s = max(s(M ), s(N )). Then πk∗ (Dif f0α (M )) is isomorphic with Hom0 (Mk ) := {ψk : ψk ∈ Hom(Mk ), ψk (s0,k ) = s0,k }. All of this is also applicable with the corresponding changes to classes of smoothness C α (or C(α) in the ¯ k on Υk in the latter case. If notation of [23]), where α = t or α = [t] with substitution of Φ f and g are two Kα -equivalent elements in C0α (M, N ), that is, there are sequences fn and gn in C0α (M, N ) converging to f and g respectively and also a sequence ψn ∈ Dif f0α (M ) such that fn (x) = gn (ψn (x)) for each x ∈ M , then πk∗ (fn ) =: fn,k and gn,k := πk∗ (gn ) converge to πk∗ (f ) and πk∗ (g) respectively and also ψn,k := πk∗ (ψn ) ∈ Hom0 (Mk ). From the equality fn,k (x(k)) = gn,k (ψn,k (x(k))) for each n ∈ N and x(k) ∈ Mk it follows, that the equivalence relation Kα induces the corresponding equivalence relation Kα,k in πk∗ (C0α (M, N )) such that classes < πk∗ (f ) >K,α,k of Kα,k -equivalent elements are closed. Each element fk ∈ πk∗ (C0α (M, N )) is characterized by the equality fk (s0,k ) = y0,k . This induces the quotient mapping πk∗ : Ωα (M, N ) → Ω(Mk , Nk ) and surjective mappings πkl : Ω(Ml , Nl ) → Ω(Mk , Nk ) for each l ≥ k. Each Ω(Mk , Nk ) is the finite discrete set, since each NkMk is the finite discrete set. This produces the inverse sequence of finite discrete spaces, hence the limit Ωw (M, N ) := pr − lim{Ω(Mk , Nk ), πlk , Λs } of the inverse sequence is compact and totally disconnected. It remains to verify that Ωw (M, N ) is the commutative topological monoid with the unit element and the cancelation property. ¯ \ {s0 }, it follows that Mk = M ¯ k , since for each k ∈ N From the equality M = M m −k there exists x ∈ M such that x + B(K , 0, p ) ∋ s0 . Moreover, Mk and Nk are finite discrete spaces. Then πk (M ∨ M ) = Mk ∨ Mk , where A ∨ B := A × {b0 } ∪ {a0 } × B ⊂ A × B is the wedge product of pointed spaces (A, a0 ) and (B, b0 ), A and B are sets with marked points a0 ∈ A and b0 ∈ B. The composition operation is defined on threads {< fk >K,α,k : k ∈ N} of the inverse sequence in the following way. There was fixed a C [∞] -diffeomorphism χ : M ∨ M → M [23]. Let x ∈ M , then πk (x) ∈ Mk and χ−1 (U ) ∈ M ∨ M , where U := πk−1 (x + B(K, 0, p−k )) ∩ M. On the other hand χ−1 (U ) is a disjoint union of balls of radius p−2k in B(K2m , 0, 1), hence there is defined a surjective mapping χk : M2k ∨M2k → Mk induced by χ, πk and π2k such that χk (χ−1 (U )) = πk (x). If f and g ∈ C α (M, N ), then f ∨ g ∈ C α ((M ∨ M ), N ) and χ(f ∨ g) ∈ C α (M, N ) as in §2.6 [23]. Hence χk (f2k ∨ g2k ) ∈ C α (Mk , Nk ) and inevitably χk (< f2k ∨ g2k >K,α,2k ) = χk (< f2k >K,α,2k ∨ < g2k >K,α,2k ) ∈ Ω(Mk , Nk ). ¯ , N ) and {fk : There exists a one to one correspondence between elements f ∈ Cw (M Mk k} ∈ {Nk : k ∈ Λs }. Therefore, pr − limk Ω(Mk , Nk ) algebraically this is the commutative monoid with the cancelation property. Let U be a neighborhood of e in Ωw (M, N ), then there exists Uk = πk−1 (Vk ) such that Vk is open in Ω(Mk , Nk ), e ∈ Uk and Uk ⊂ U . −1 On the other hand there exists U2k = π2k (V2k ) such that V2k is open in Ω(M2k , N2k ), e ∈ U2k and U2k + U2k ⊂ Uk . Therefore, (f + U2k ) + (g + U2k ) ⊂ f + g + Uk ⊂ f + g + U for each f, g ∈ Ωw (M, N ), consequently, the composition in Ωw (M, N ) is continuous. ¯ , N ), then Ωα (M, N ) is dense in Ωw (M, N ) relative Since C0α (M, N ) is dense in C0,w (M

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to the projective weak topology τw . 2. Corollary. The wrap group Lα (M, N ) has a non-archimedean compactification w L (M, N ) relative to the projective weak topology τw . Proof. Using the Grothendieck construction we get a compactification Lw (M, N ) = ¯ ¯ F /B of a wrap group Lα (M, N ), where F¯ is a closure in (Ωw (M, N ))Z of a free com¯ is a closure of a subgroup B generated mutative group F generated by Ωw (M, N ) and B by all elements [a + b] − [a] − [b], since the product of compact spaces is compact by the Tychonoff theorem. ¯ 3. Let now s0 = 0 and y0 = 0 be two marked points in the compact manifolds M m n [∞] and N embedded into K and K respectively. There is defined the following C diffeomorphism inv : (Km )′ → (Km )′ for (Km )′ := Km \ {x : there exists j with −1 ′ m ′ ′ xj = 0} such that inv(x1 , ..., xm ) = (x−1 1 , ..., xm ). Let M = M ∩ (K ) , then inv(M ) is locally compact and unbounded in Km , consequently, πk (inv(M ′ )) = (inv(M ′ ))k is ′ ′ a discrete infinite subset in Km k for each k ∈ N. Analogously πk (inv(M ∨ M )) = [∞] ′ ′ 2m (inv(M ∨ M ))k ⊂ Kk . There exists a C -diffeomorphism χ : M ∨ M → M such that inv ◦χ◦inv is the C [∞] -diffeomorphism of inv(M ′ ∨M ′ ) with inv(M ′ ) and it induces bijective mappings χk of inv((inv(M ′ ∨ M ′ ))k ) with inv((inv(M ′ ))k ) for each k ∈ N such that π ˆkl ◦ χl = χk for each l ≥ k, where π ˆkl := inv ◦ πkl ◦ inv. This produces ′ ˆ k , inv((inv(M ′ ∨ M ′ ))k ) = inverse sequences of discrete spaces inv((inv(M ))k ) =: M ˆk ∨ M ˆ k and their bijections χk such that pr − limk M ˆ k is homeomorphic with M ′ and M m pr − limk χk is equal to χ up to the homeomorphism, since pr − limk Km k = K (see also ¯ ), about admissible modifications and polyhedral expansions in [27, 28]). If ψ ∈ Dif f0α (M α ˆ ). Let Jf,k := {hk : hk = fk ◦ ψk , ψk ∈ Hom(M ˆ k ), ψk (s0,k ) = s0,k } then ψˆ ∈ Dif f (M ˆ

for fk ∈ NkMk with limx→0 fk (x) = 0, then Jf,k is closed and π ˆk∗ (< f >K,α ) ⊂ Jf,k . ˆ α,k -equivalent if and only if there exists ψk ∈ Hom(M ˆ k ) such Therefore, gk and fk are K ˆ ˆ that ψk (s0,k ) = s0,k and gk (x) = fk (ψk (x)) for each x ∈ Mk . Let Ω(Mk , Nk ) := π ˆk∗ (Ωα (M, N )). ˆ k , Nk ) forms an inverse sequence Theorem. The set of Ω(M l ˆ S = {Ω(Mk , Nk ); π ˆk ; k ∈ Λs } such that pr − lim S =: Ωi,w (M, N ) is an associative topological wrap monoid with the cancelation property and the unit element e. There exists an embedding of Ωα (M, N ) into Ωi,w (M, N ) such that Ωα (M, N ) is dense in Ωi,w (M, N ) relative to the projective weak topology τi,w . Proof. Let U ′ i be an analytic disjoint atlas of inv(M ′ ), f ∈ C α (inv(M ′ ), K), ψ ∈ Dif f α (inv(M ′ )), then each restriction f |U ′ i has the form f |U ′ i (x) = P ′ ′ ¯ ¯ m fi,m Qi,m (x) for each x ∈ U i , where Qi,m are basic Amice polynomials for U i , ∗ fi,m ∈ K. Therefore f is a combination f = ∇i f |U ′ i , hence π ˆk (f ◦ ψ(x)) = P ∗ (f ∗ ((f ◦ ψ)(x)) = f ◦ ¯ ′ [ˆ π )∇ Q (ψ (x(k)))] and inevitably π ˆ k k (i,ψk (x(k))∈ˆ πk (U k ) i,m,k m k i,m k ¯ i,m,k := π ¯ i,m ), x ∈ inv(M ′ ) and x(k) = π ψk (x(k)), where Q ˆk∗ (Q ˆk (x). As in §2.6.2 [23] we choose an infinite atlas At′ (M ) := {(U ′ j , φ′ j ) : j ∈ N} such that ′ φ j : U ′ j → B(X, y ′ j , r′ j ) are homeomorphisms, limk→∞ r′ j(k) = 0, limk→∞ y ′ j(k) = 0 S ′ for an infinite sequence {j(k) ∈ N : k ∈ N} such that clM¯ [ ∞ k=1 U j(k) ] is a clopen ¯ . We ¯ , where cl ¯ A denotes the closure of a subset A in M neighborhood of zero in M M −1 ′ −1 ′ ′ ′ ′ ′ ′ take |y j(k) | > r j(k) for each k, hence inv(B(X, y j , r j ) ∩ X ) = B(X, y j , r j ) ∩ X ′ S and k inv(U ′ j(k) ∩ X ′ ) is open in X ′ , where X = Km . For an atlas At′ (M ∨ M ) :=

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{(Wl , αl ) : l ∈ N} with homeomorphisms αl : Wl → B(X, zl , al ), limk→∞ al(k) = 0, limk→∞ zl(k) = 0 for an infinite sequence {l(k) ∈ N : k ∈ N} such that S ¯ ¯ clM¯ ∨M¯ [ ∞ k=1 Wl(k) ] is a clopen neighborhood of 0 × 0 in M ∨ M we also choose |zl | > al for each l, where card(N \ {l(k) : k ∈ N}) = card(N \ {j(k) : k ∈ N}). Then we take χ(Wl(k) ) = U ′ j(k) for each k ∈ N and χ(Wl ) = U ′ κ(l) for each l ∈ (N \ {l(k) : k ∈ N}), where κ : (N \ {l(k) : k ∈ N}) → (N \ {j(k) : k ∈ N}) is a bijective mapping such that p−1 ≤ r′ j(k) /al(k) ≤ p for each k and p−1 ≤ r′ κ(l) /al ≤ p for each l ∈ (N \ {l(k) : k ∈ N}). We can choose the locally affine mapping χ on ¯ \ {s0 } such that Φ ¯ n χ = 0 or Υn χ = 0 for each n ≥ 2 and B(X ′ , y ′ −1 , r′ −1 ) M = M l l are diffeomorphic with inv(U ′ l ∩ X ′ ) and B(X ′ ∨ X ′ , zl−1 , a−1 ) are diffeomorphic with l inv(Wl ∩ (X ′ ∨ X ′ )). ˆ ∨M ˆ → M ˆ and χ This induces the diffeomorphisms χ ˆ := inv ◦ χ ◦ inv : M ˆ∗ : ˆ ∨M ˆ , ∞ × ∞), (N, y0 )) → C α ((M ˆ , ∞), (N, y0 )), since each Φ ¯ n (f ∨ g)(χ C0α ((M ˆ−1 ) or 0 n −1 l j −1 l ¯ ¯ Υ (f ∨g)(χ ˆ ) has an expression through Φ (f ∨g) and Φ (χ ˆ ) or Υ (f ∨g) and Υj (χ ˆ−1 ) ′ ˆ := inv(M ) and conditions respectively with l, j ≤ q and q subordinated to α, where M α ˆ defining the subspace C0 ((M , ∞), (N, y0 )) differ from that of C0α ((M, s0 ), (N, y0 )) by substitution of limx→s0 on lim|x|→∞ . Then lim|x|→∞ |χ(x)| ˆ = ∞, consequently, there ˆ ˆ ˆ exists k0 ∈ N such that χ ˆk : Mk ∨ Mk → Mk are bijections for each k ≥ k0 , where ˆ ¯ ) and ψ(0) = 0, then lim|x|→∞ ψ(x) χ ˆk := π ˆk ◦ χ. ˆ If ψ ∈ Dif f α (M = ∞ and −1 lim|x|→∞ ψˆ (x) = ∞. Then considering ψˆk we get an equivalence relation Kα,k in ˆ ˆ k is supplied with the {fk : fk ∈ NkMk , lim|x|→∞ fk (x) = 0} induced by Kα , where M ˆ k. quotient norm induced from the space X, since X ′ ⊂ X, x ∈ M Let Jk denotes the quotient mapping corresponding to Kα,k . Therefore analogously to ˆ k , Nk ) are commutative monoids with the cancelation property §2.6 [23] we get, that Ω(M ˆ k , Nk ) = {fk : fk ∈ C 0 (M ˆ k , Nk ), lim|x|→∞ fk (x) = and the unit elements ek , since Ω(M l m ′ m ′ ˆ α,k and mappings π 0}/K ˆk : (K ) l → (K ) k and mappings πkl : Kn l → Kn k induce ˆ l , Nl ) → Ω(M ˆ k , Nk ) for each l ≥ k. Let the topology in {fk : fk ∈ mappings π ˆkl : Ω(M 0 ˆ k , Nk ), lim|x|→∞ fk (x) = 0} be induced from the Tychonoff product topology in C (M ˆ ˆ k , Nk ) be in the quotient topology. N Mk and Ω(M k

ˆ

The space NkMk is metrizable by the Baire metric ρ(x, y) := |π|−j , where j = min{i : ˆ k is enumerated xi 6= yi , x1 = y1 , ..., xi−1 = yi−1 }, x = (xl : xl ∈ Nk , l ∈ N), M as N, π ∈ K, 0 < |π| < 1 is the generator of the valuation group ΓK . Therefore, ˆ k , Nk ) is metrizable and the mapping (fk , gk ) → fk ∨ gk is continuous, hence the Ω(M mapping (Jk (fk ), Jk (gk )) → Jk (fk ) ◦ Jk (gk ) is also continuous. Then Jk (w0,k ) is the ˆ k ) = 0. Hence Ωi,w (M, N ) is the commutative monoid with unit element, where w0,k (M Q ˆ k , Nk ) is the topological the cancelation property and the unit element. Certainly k Ω(M monoid and pr − lim S is a closed in it topological totally disconnected monoid. For each ˆk∗ (f ), k ∈ Λs } such that f ∈ C0α (M, N ) there exists an inverse sequence {fk : fk = π f (x) = pr − limk fk (x(k)) for each x ∈ M ′ , but M ′ is dense in M . Therefore there exists an embedding Ωα (M, N ) ֒→ Ωi,w (M, N ), hence Ωα (M, N ) is dense in Ωi,w (M, N ) relative to the projective weak topology τi,w , since C α (M, N ) is dense in Cw (M, N ) relative to the τw topology. 4. Corollary. The inverse sequence of wrap monoids induces the inverse sequence of

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ˆ k , Nk ); π wrap groups SL := {L(M ˆkl ; Λs }. Its projective limit Li,w (M, N ) := pr − lim SL is a commutative topological totally disconnected group and Lα (M, N ) has an embedding in it as a dense subgroup. Proof. Due to the Grothendieck construction the inversion operation fk 7→ fk−1 is ˆ k , Nk ) and homomorphisms π continuous in L(M ˆkl and π ˆk have continuous extensions ˆ ˆ k , Nk ) is tofrom wrap submonoids onto wrap groups L(Mk , Nk ). Each monoid Ω(M ˆ ˆ k , Nk ) is supplied with tally disconnected, since NkMk is totally disconnected and Ω(M ˆ k , Nk ) is the quotient ultrametric, hence the free Abelian group Fk generated by Ω(M ˆ also totally disconnected and ultramertizable, consequently, L(Mk , Nk ) is ultrametrizable. Evidently their inverse limit is also ultrametrizable and the equivalent ultrametric ˜ K := {|z| : z ∈ K}, where Γ ˜ K ∩ (0, ∞) is discrete in can be chosen with values in Γ (0, ∞) := {x : 0 < x < ∞, x ∈ R}. Then the projective limit (that is, weak) topology of Li,w (M, N ) is induced by the projective weak topology of Cw (M, K). 5. Theorem. For each prime number p the wrap group Lα (M, N ) in its weak topology inherited from Li,w (M, N ) has the non-archimedean compactification isomorphic with Zp ℵ0 , moreover, Li,w (M, N ) has the compactification (νZ)ℵ0 , where νZ is the one-point Alexandroff compactification of Z. Proof. The projective ring homomorphism πk : K → Kk induces ¯ m (f (x; h1 , ..., hm ; ζ1 , ..., ζm )) = Φ ¯ m fk (x(k); h1 (k), ..., hm (k); ζ1 (k), ..., ζm (k)) π ˆk∗ (Φ m [m] [m] [m] ∗ and π ˆk (Υ (f (x )) = Υ fk (x(k) ), ¯ m fk and Υm fk are defined for the field of fractions generated by Kk , where m ∈ N, Φ since K is the commutative field (see also [4] and §§2.1-2.6 [23]). Then the condition ¯ m f (x; h1 , ..., hm ; ζ1 , ..., ζm ) = 0 or lim Φ

|x|→∞

lim Υm f (x[m] ) = 0

|x|→∞

implies the condition lim

|x(k)|→∞

¯ m fk (x(k); h1 (k), ..., hm (k); ζ1 (k), ..., ζm (k)) = 0 or Φ lim

|x(k)|→∞

Υm fk (x(k)[m] ) = 0

respectively, where x[1] = (x, v [0] , ζ1 ), x[m+1] := (x[m] , v [m] , ζm+1 ). Therefore, ˆ f := {x(k) : fk (x(k)) 6= 0} is a finite subset of the discrete space M ˆk supp(fk ) := M k ˆ

for each k ∈ N. Then evidently, π ˆk∗ (< g >K,α ) is a closed subset in NkMk for each ˆ , ∞), (N, 0)), since for each limit point fk of π ˆk∗ (< g >K,α ) its support is the g ∈ C0α ((M ˆ k . Let k0 be such that Nk 6= {0}, then this is also true for each k ≥ k0 . finite subset in M 0 ∗ If fk ∈ /π ˆk (< w0 >K,α ) and k ≥ k0 , then fk∨n ∈ /π ˆk∗ (< w0 >K,α ) for each n ∈ N, where ∨n fk := fk ∨ ... ∨ fk denotes the n-times wedge product, since kf ∨n kC α ≥ kf kC α > 0 n ∗ α m n and kfk∨n kC(Kmk ,Knk ) ≥ kf kC(Kmk ,Knk ) > 0, where C(Km k , Kk ) = πk (Cb (K , K )) is the ∗ quotient module over the ring Kk . Each π ˆk (< f >K,α ) can be presented as the following ˆ k , Nk ), where each bi corresponds composition z1 b1 + ... + zl bl in the additive group L(M

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ˆ k , Nk ) into L(M ˆ k , Nk ), zi ∈ {−1, 0, 1}, to π ˆk∗ (< gi >K,α ) and the embedding of Ω(M f gi ˆ ˆ l = card(Mk ), Mk are singletons for each i = 1, ..., l. ˆ k is the finite discrete set as well as Nk . For each x 6= y ∈ M ˆ k there exists ψ ∈ Each M Hom0 (Mk ) such that ψ(x) = y, where 0 corresponds to s0 for convenience of the notation. ˆ k , Nk ) is isomorphic with Znk , where nk = Using the group Hom0 (Nk ) we get that L(M card(Nk ) > 1. For each prime number p > 1 there exists the p-adic completion of Z which is Zp . In view of Corollary 4 Lα (M, N ) has the non-archimedean completion isomorphic with Zℵp 0 , since Z is dense in Zp and pr − limk Znk = Zℵ0 .

On the other hand, we can take the multiplicative subgroup {θl : l ∈ Z} of the locally compact field Fpu (θ) which gives the embedding φ of Z into Fpu (θ), where θ0 = 1. The completion of φ(Z) in Fpu (θ) is φ(Z) ∪ {0} which is the one-point Alexandroff compactification νZ of Z. This gives the non-archimedean completion (νZ)ℵ0 of Lα (M, N ). Moreover, Zℵp 0 and (νZ)ℵ0 are compact as products of compact spaces and Li,w (M, N ) has the aforementioned embeddings into them. 6. Note. Using quotient mappings ηp,s : Z → Z/ps Z we get that Lα (M, N )ℵ0 has Q the compactification equal to { p∈P Zp ℵ0 } × (νZ)ℵ0 relative to the product Tychonoff topology, where P denotes the set of all prime numbers p > 1, s ∈ N. These compactifications produce characters of Li,w (M, N ), since each compact Abelian group has only one-dimensional irreducible unitary representations [16]. On the other hand, there are irreducible continuous representations of compact groups in non-archimedean Banach spaces [39]. Among them there are infinite-dimensional [10, 37]. Moreover, in their initial C α topologies diffeomorphism and wrap groups also have infinite-dimensional irreducible unitary representations [22, 23]. At the same time topologies of Lα (M, N ) and Lw (M, N ) or Li,w (M, N ) are incomparable, since the topologies of C α (M, N ) and Cw (M, N ) are incomparable (see Theorem 3.7 above). Projective limits of groups obtained above have the non-archimedean origin related with non-archimedean families of semi-norms on spaces of continuous or more narrow classes of functions between manifolds over ultra-normed fields. Generally, if a topological space X has a projective limit decomposition X = pr − lim{Xα , πβα , Λ}, then if fβ : Xβ → Y is a continuous function into a topological space Y , then f := fβ ◦ πβ : X → Y is a continuous function, where the mapping πβα : Xα → Xβ is continuous for each α ≥ β ∈ Λ, Λ is a directed set, πβα ◦ πα = πβ , πα : X → Xα is continuous and epimorphic. Therefore, fα = π ˜αβ (fβ ) := fβ ◦πβα for each α ≥ β generate the inductive limit ind−lim{C(Xβ , Y ); π ˜αβ ; Λ}, where C(X, Y ) denotes the family of all continuous mappings from X into Y . On the other hand, if Y = pr − lim{Yγ , pγδ , Ψ}, then one gets pr − lim{C(X, Yγ ); pγδ , Ψ}. Then these two constructions can be combined with repeated application of projective and inductive limits, which may be dependent on the order of taking limits. If card(Ψ) ≥ ℵ0 , then a Q suitable box topology in γ∈Ψ C(X, Yγ ) is strictly stronger than a weak topology in it and in its projective limit subspace (see also [32]). ¯ and N are compact manifolds, α ∈ {∞, [∞]}, then Lα (M, N ) is 7. Theorem. If M α the C Lie group. Proof. The uniform space C α ((M, s0 ), (N, y0 )) has the structure of the C α manifold, since M and N are C α manifolds, where α ∈ {∞, [∞]}. Therefore, Ωα (M, N ) and Lα (M, N ) are C α manifolds. The wedge product (f, g) 7→ f ∨ g in C α ((M, s0 ), (N, y0 ))

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¯ \ {s0 }. Using the quotient mapping by closures of is the C α mapping, since M = M equivalence relation caused by the action of Dif f0α (M ) becomes the C α manifold and C α monoid with the C α smooth composition. Using the construction of Lα (M, N ) we get, that Lα (M, N ) is the C α Lie group (see also for more details [23]). 8. Remark. Theorem 7 can be generalized in the Cbα class for noncompact Cbα manifolds M and N .

8.

Appendix

1. Lemma. Let either f, g ∈ C [n] (U, Y ), where U is an open subset in X, Y is an algebra over K, or f ∈ C [n] (U, K) and g ∈ C [n] (U, Y ), where Y is a topological vector space over K, then (1) (f g)[n] (x[n] ) = (Υ ⊗ Pˆ + π ˆ ⊗ Υ)n .(f ⊗ g)(x[n] ) [n] 0 [n] and (f g) ∈ C (U , Y ), where (ˆ π k g)(x[k] ) := g ◦ π10 ◦ π21 ◦ ... ◦ πkk−1 (x[k] ), Pˆ n g := Pn Pn−1 ...P1 g, πkk−1 (x[k] ) := x[k−1] , (A ⊗ B).(f ⊗ g) := (Af )(Bg) for A, B ∈ L(C n (U, Y ), C m (U, Y )), m ≤ n, (A1 ⊗ B1 )...(Ak ⊗ Bk ).(f ⊗ g) := (A1 ...Ak ⊗ B1 ...Bk ).(f ⊗ g) := (A1 ...Ak f )(B1 ...Bk g) for corresponding operators, Υn f := f [n] , (Pk g)(x[k] ) := g(x[k−1] + v [k−1] tk ), Pˆ k π ˆ a1 Υb1 ...ˆ π al Υbl g = Pk+s ...Ps+1 π ˆ a1 Υb1 ...ˆ π al Υbl g with s = b1 +...+bl −a1 −...−al ≥ 0, a1 , ..., al , b1 , ..., bl ∈ {0, 1, 2, 3, ...}. Proof. Let at first n = 1, then (2) (f g)[1] (x[1] ) = [(f g)(x + vt) − (f g)(x)]/t = [(f (x + vt) − f (x))g(x + vt) + f (x)(g(x + vt) − g(x))]/t = (Υ1 f )(x[1] )(P1 g)(x[1] ) + (ˆ π10 f )(x[1] )Υ1 g(x[1] ), 0 [1] since π ˆ1 (x ) = x and P1 is the composition of the projection π ˆ10 and the shift operator on vt. Let now n = 2, then applying Formula (2) we get: (3) (f g)[2] (x[2] ) = ((f g)[1] (x[1] ))[1] (x[2] ) = (Υ1 (f [1] (x[1] )(x[2] ))g(x + (v [0] + [1] [1] [1] [1] [1] [1] v2 t2 )(t1 + v3 t2 ) + v1 t2 ) + f [1] (x[1] )g [1] (x + v [0] t1 , v1 + v2 (t1 + v3 t2 ), t2 ) + [1] [1] f [1] (x, v1 , t2 )g [1] (x[1] + v1 t2 ) + f (x)g [2] (x[2] ), [k] [k] [k] [0] where v [k] = (v1 , v2 , v3 ) for each k ≥ 1 and v [0] = v1 such that x[k] + v [k] tk+1 = [k] [k] [k] (x[k] + v1 tk+1 , v [k−1] + v2 tk+1 , tk + v3 tk+1 ) for each 1 ≤ k ∈ Z. For n = 3 we get (4) (f g)[3] (x[3] ) = [(Υ3 f )(Pˆ 3 g) + (ˆ π 1 Υ2 f )(Υ1 Pˆ 2 g) + (Υ1 (ˆ π 1 Υ1 f ))(Pˆ 1 Υ1 Pˆ 1 g) +(ˆ π 2 Υ1 f )(Υ2 Pˆ 1 g) + (Υ2 π ˆ 1 f )(Pˆ 2 Υ1 g) + (ˆ π 1 Υ1 π ˆ 1 f )(Υ1 Pˆ 1 Υ1 g) 1 2 1 2 3 3 [3] ˆ +(Υ (ˆ π f ))(P Υ g) + (ˆ π f )(Υ g)](x ), since by our definition Pˆ k π ˆ a1 Υb1 ...ˆ π al Υbl g = Pk+s ...Ps+1 π ˆ a1 Υb1 ...ˆ π al Υbl g with s = b1 + ... + bl − a1 − ... − al ≥ 0, a1 , ..., al , b1 , ..., bl ∈ {0, 1, 2, 3, ...}. Therefore, Formula (1) for n = 1 and n = 2 and n = 3 is demonstrated by Formulas (2 − 4). If f, g ∈ C 0 (U [k] , Y ), a, b ∈ K, then (Pk (af + bg))(x[k] ) := (af + bg)(x[k−1] + v [k−1] tk ) = af (x[k−1] + v [k−1] tk ) + bg(x[k−1] + v [k−1] tk ), moreover, π ˆ k (af + bg)(x[k] ) = k−1 [k] 1 0 π k f (x[k] ) + (af + bg) ◦ π1 ◦ π2 ◦ ... ◦ πk (x ) = (af + bg)(x) = af (x) + bg(x) = aˆ bˆ π k g(x[k] ) for each x[k] ∈ U [k] , hence π ˆ k and Pk and Pˆ k are K-linear operators for each k ∈ N. Suppose that Formula (1) is proved for n = 1, ..., m, then for n = m + 1 it follows by application of Formula (2) to both sides of Formula (1) for n = m: (f g)m+1 (x[m+1] ) = ((f g)[m] (x[m] ))[1] (x[m+1] ) = ((Υ ⊗ Pˆ + π ˆ ⊗ Υ)m .(f ⊗ g)(x[m] ))[1] (x[m+1] ) = (Υ ⊗ Pˆ + π ˆ ⊗ Υ)m+1 .(f ⊗ g)(x[m+1] ),

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since x[m+1] = (x[m] )[1] and more generally x[m+k] = (x[m] )[k] for each nonnegative integers m and k such that πkk−1 (x[m+k] ) = x[m+k−1] for k ≥ 1; Υk , Pˆ k and π ˆ are K-linear operators on corresponding spaces of functions (see above and Lemma 2.3) and (Υ ⊗ Pˆ + π ˆ ⊗ Υ)m+1 .(f ⊗ g)(x[m+1] ) = P a1 ˆ a1 a1 +...+am+1 +b1 +...+bm+1 =m+1 (Υ ⊗ P ) (ˆ π b1 ⊗ Υb1 )...(Υam+1 ⊗ Pˆ am+1 )(ˆ π bm+1 ⊗ Υbm+1 ).(f ⊗ g)(x[m+1] ), where aj and bj are nonnegative integers for each j = 1, ..., m + 1, (A1 ⊗ B1 )...(Ak ⊗ Bk ).(f ⊗ g) := (A1 ...Ak ⊗ B1 ...Bk ).(f ⊗ g) := (A1 ...Ak f )(B1 ...Bk g). 2. Note. Consider the projection (1) ψn : X m(n) × Ks(n) → X l(n) × Kn , where m(n) = 2m(n − 1), s(n) = 2s(n − 1) + 1, l(n) = n + 1 for each n ∈ N such that m(0) = 1, s(0) = 0, m(n) = 2n , s(n) = 1 + 2 + 22 + ... + 2n−1 = 2n − 1. Then m(n), s(n), l(n) and n correspond to number of variables in X, K for Υn , in X and K ¯ n respectively. Therefore, ψ(x[n] ) = x(n) and ψn (U [n] ) = U (n) for each n ∈ N for for Φ ¯ n f (x(n) ) = ψˆn Υn f (x[n] ) = f [n] (x[n] )| (n) , where suitable ordering of variables. Thus Φ W ψˆn g(y) := g(ψn (y)) for a function g on a subset V in X l(n) × Kn for each y ∈ ψn−1 (V ) ⊂ X m(n) × Ks(n) , W (n) = U (n) × 0, 0 ∈ X m(n)−l(n) × Ks(n)−n for the corresponding ordering of variables. 3. Corollary. Let either f, g ∈ C n (U, Y ), where U is an open subset in X, Y is an algebra over K, or f ∈ C n (U, K) and g ∈ C n (U, Y ), where Y is a topological vector space over K, then ¯ n (f g)(x(n) ) = (Φ ¯ ⊗ Pˆ + π ¯ n .(f ⊗ g)(x(n) ) (1) Φ ˆ ⊗ Φ) ¯ n (f g) ∈ C 0 (U (n) , Y ). In more details: and Φ P ¯ n (f g)(x(n) ) = P (2) Φ 0≤a,0≤b,a+b=n j1 <...<ja ;s1 <...<sb ;{j1 ,...,ja }∪{s1 ,...,sb }={1,...,n} ¯ a f (x; vj , ..., vja ; tj , ..., tja )Φ ¯ b g(x + vj tj + ... + vja tja ; vs , ..., vs ; ts , ..., ts ). Φ 1 1 1 1 1 1 b b Proof. The operator ψˆn is K-linear, since ψˆn (af + bg)(y) = (af + bg)(ψn (y)) = af (ψn (y)) + bg(ψn (y)) for each a, b ∈ K and functions f, g on a subset V in X l(n) × Kn and each y ∈ ψn−1 (V ) ⊂ X m(n) × Ks(n) . Mention that the restrictions of π ˆkk−1 and Pk k−1 (k) (k) (k−1) (k) (k−1) on W gives πk (x ) := x and (Pk g)(x ) := g(x + vk tk ) in the notation ˆ of §1.1. The application of the operator ψn to both sides of Equation 1(1) gives Equation ¯ n for each nonnegative integer n, where Υ0 = I and (1) of this corollary, since ψˆn Υn = Φ ¯ 0 = I and ψˆ0 = I are the unit operators. Φ 4. Lemma. Let f1 , ..., fk ∈ C [n] (U, Y ), where U is an open subset in X, either Y is an algebra over K, or f1 , ..., fk−1 ∈ C [n] (U, K) and fk ∈ C [n] (U, Y ), where Y is a topological vector space over K, then P ˆ ⊗α ⊗ Υ ⊗ Pˆ ⊗(k−α−1) ]n .(f1 ⊗ ... ⊗ fk )(x[n] ) (1) (f1 ...fk )[n] (x[n] ) = [ k−1 α=0 π [n] 0 [n] and (f1 ...fk ) ∈ C (U , Y ), where π ˆ ⊗α ⊗ Υ ⊗ Pˆ ⊗(k−α−1) .(f1 ⊗ ... ⊗ fk ) := (ˆ π (f1 ...fα ))(Υfα+1 )(Pˆ (fα+2 ...fk )), where 0 0 π ˆ := I, Pˆ = I is the unit operator, π ˆ f0 := 1, Pˆ fk+1 := 1 (see Lemma 1). Proof. Consider at first n = 1 and apply Formula 1(1) by induction to appearing products of functions, then (2) Υ1 (f1 ...fk )(x[1] ) = [(Υ1 (f1 ...fk−1 ))(P1 fk ) + (ˆ π 1 (f1 ...fk−1 ))(Υ1 fk )](x[1] ) = 1 1 1 [(Υ (f1 ...fk−2 ))(P1 fk−1 )(P1 fk ) + (ˆ π (f1 ...fk−2 ))(Υ fk−1 )(P1 fk ) 1 1 [1] +(ˆ π (f1 ...fk−1 ))(Υ fk )](x ) = ...

Groups of Diffeomorphisms and Wraps of Manifolds... P

595

⊗(k−α−1)

= ( k−1 π 1 )⊗α ⊗ Υ1 ⊗ P1 ).(f1 ⊗ ... ⊗ fk ), α=0 (ˆ ⊗α where A ⊗ B ⊗ C ⊗(k−α−1) .(f1 ⊗ ... ⊗ fk ) := (A(f1 ...fα ))(Bfα+1 )(C(fα+2 ...fk )) for operators A, B and C and each nonnegative integer α, where A0 := I, C 0 = I is the unit operator, Af0 := 1, Cfk+1 := 1, in particular, A = π ˆ 1 , B = Υ1 , C = P1 . Thus, acting by 1 induction on both sides by Υ from Formula (2) we get Formula (1) of this lemma, since the product of n terms Υ1 ...Υ1 is equal to Υn . 5. Corollary. Let f1 , ..., fk ∈ C n (U, Y ), where U is an open subset in X, either Y is an algebra over K, or f1 , ..., fk−1 ∈ C n (U, K) and fk ∈ C n (U, Y ), where Y is a topological vector space over K, then ⊗α ⊗ Φ ¯ n (f1 ...fk )(x(n) ) = [Pk−1 π ¯ ⊗ Pˆ ⊗(k−α−1) ]n .(f1 ⊗ ... ⊗ fk )(x(n) ) (1) Φ α=0 ˆ ¯ n (f1 ...fk ) ∈ C 0 (U (n) , Y ), where and Φ ¯ ⊗ Pˆ ⊗(k−α−1) .(f1 ⊗ ... ⊗ fk ) := (ˆ ¯ α+1 )(Pˆ (fα+2 ...fk )) (see π ˆ ⊗α ⊗ Φ π (f1 ...fα ))(Φf Lemma 3). Proof. Applying operator ψˆn from Note 2 to both sides of Equation 4(1) we get Formula (1) of this Corollary. 6. Lemma. Let u ∈ C [n] (Ks , Km ), u(Ks ) ⊂ U and f ∈ C [n] (U, Y ), where U is an open subset in Km , s, m ∈ N, Y is a K-linear space, then P Pm(n) 1 (1) (f ◦ u)[n] (x[n] ) = [ m j1 =1 ... jn =1 (Ajn ,v [n−1] ,tn ...Aj1 ,v [0] ,t1 f ◦ u)(Υ ◦ pjn Sˆjn−1 +1,v[n−2] tn−1 ...Sˆj1 +1,v[0] t1 un−1 )(Pn Υ1 ◦pjn−1 Sˆjn−2 +1,v[n−3] tn−2 ...Sˆj1 +1,v[0] t1 un−2 )...(Pn ...P2 Υ1 ◦pj1 u) P Pm(n−1) 1 P ˆ ⊗α ⊗ Υ ⊗ Pˆ ⊗(n−α−2) ] (ˆ π (A ...A [0] f ◦ u)[ n−2 π + m ... [n−2] j1 =1

jn−1 =1

jn−1 ,v

,tn−1

j1 ,v

,t1

α=0

((Υ1 ◦ pjn−1 Sˆjn−2 +1,v[n−3] tn−2 ...Sˆj1 +1,v[0] t1 un−2 ) ⊗ ... ⊗ (Pn−1 ...P2 Υ1 ◦ pj1 u)) Pm(n−2) 1 P P (ˆ π (A ...A ˆ ⊗α ⊗ Υ ⊗ Pˆ ⊗(n−α−2) ]( m ... +[ n−2 π [n−3]

j1 =1 jn−2 ,v ,tn−2 j1 ,v [0] ,t1 f ◦ jn−2 =1 ⊗(n−α−3) 1 ˆ ˆ ˆ ⊗Υ⊗P ]((Υ ◦ pjn−2 Sjn−3 +1,v[n−4] tn−3 ...Sj1 +1,v[0] t1 un−3 ) ⊗ u)) ⊗ [ α=0 ... ⊗ (Pn−2 ...P2 Υ1 ◦ pj1 u)) + ... Pm(2) 1 P P π Aj2 ,v[1] ,t2 Aj1 ,v[0] ,t1 f ◦ u)(Υ1 ⊗ Pˆ 1 + ˆ ⊗α ⊗ Υ ⊗ Pˆ ⊗(2−α) ]n−3 { m +[ 2α=0 π j1 =1 j2 =1 (ˆ π ˆ 1 ⊗ Υ1 )((Υ1 ◦ pj2 Sˆj1 +1,v[0] t1 u) ⊗ (P2 Υ1 ◦ pj1 u))} P π 1 Aj1 ,v[0] ,t1 f ◦ u) ⊗ (Υ2 ◦ pj1 u)}](x[n] ) +(Υ ⊗ Pˆ + π ˆ ⊗ Υ)n−2 { m j1 =1 (ˆ 0 s [n] and f ◦ u ∈ C ((K ) , Y ), where Sj,τ u(y) := (u1 (y), ..., uj−1 (y), uj (y + τ(s) ), uj+1 (y + τ(s) ), ..., um (y + τ(s) )), u = (u1 , ..., um ), uj ∈ K for each j = 1, ..., m, y ∈ Ks , τ = (τ1 , ..., τk ) ∈ Kk , k ≥ s, τ(s) := (τ1 , ..., τs ), pj (x) := xj , x = (x1 , ..., xm ), xj ∈ K for each j = 1, ..., m, Sˆj+1,τ g(u(y), β) := g(Sj+1,τ u(y), β), y ∈ Ks , β is some parameter, Aj,v,t := (Sˆj+1,vt ⊗ tΥ1 ◦ pj )∗ Υ1j , where Υ1 is taken for variables (x, v, t) or corresponding to them after actions of preceding operations as Υk , Υ1j f (x, vj , t) := [f (x + ej vj t) − f (x)]/t, (B ⊗ A)∗ Υ1 fi ◦ ui (x, v, t) := Υ1j fi (Bui , v, Aui ), B : Km(i) → Km(i) , A : Km(i) → K, ej = (0, ..., 0, 1, 0, ..., 0) ∈ Km(i) with 1 on j-th place; m(i) = m + i − 1, ji = 1, ..., m(i); u1 := u, u2 := (u1 , t1 Υ1 ◦ pj1 u1 ),...,un = (un−1 , tn−1 Υ1 ◦ pjn−1 un−1 ), Aj1 ,v[0] ,t1 f ◦ u =: f1 ◦ u1 , Ajn ,v[n−1] ,tn fn−1 ◦ un−1 =: fn ◦ un , Sˆ∗ Υ1 f (z) := Υ1 f (Sˆ∗ z). Proof. At first consider n = 1, then (f ◦ u)[1] (t0 , v, t) = [f (u(t0 + vt)) − f (u(t0 ))]/t, where t0 ∈ Ks , t ∈ K, v ∈ Ks . Though we consider here the general case mention, that in the particular case s = 1 one has t0 ∈ K, v ∈ K. Then α=0

Pn−3

π ˆ ⊗α

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(f ◦ u)[1] (t0 , v, t) = [f (u(t0 + vt)) − f (u1 (t0 ), u2 (t0 + vt), ..., um (t0 + vt))]/t + [f (u1 (t0 ), u2 (t0 + vt), u3 (t0 + vt), ..., um (t0 + vt)) − f (u1 (t0 ), u2 (t0 ), u3 (t0 + vt), ..., um (t0 + vt))]/t+ ... + [f (u1 (t0 ), ..., um−1 (t0 ), um (t0 + vt)) − f (u(t0 ))]/t, where u = (u1 , ..., um ), uj ∈ K for each j = 1, ..., m. Since uj (t0 + vt) − uj (t0 ) = [1] tuj (t0 , v, t), hence (f ◦ u)[1] (t0 , v, t) = Υ1 f ((u1 (t0 ), u2 (t0 + vt), ..., um (t0 + vt)), e1 , tΥ1 u1 (t0 , v, t)) 1 1 1 (t0 , v, t) + Υ f ((u1 (t0 ), u2 (t0 ), u3 (t0 + vt), ..., um (t0 + vt)), e2 , tΥ u2 (t0 , v, t)) Υ1 u2 (t0 , v, t) + ... + Υ1 f (u(t0 ), em , tΥ1 um (t0 , v, t))Υ1 um (t0 , v, t), since uj ∈ K for each j = 1, ..., m and K is the field, where ej = (0, ..., 0, 1, 0, ..., 0) ∈ Km with 1 on j-th place for each j = 1, ..., m. With the help of shift operators it is possible to write the latter formula shorter: Pm ˆ 1 1 1 (2) Υ1 (f ◦ u)(y, v, t) = j=1 Sj+1,vt Υ f (u(y), ej , tΥ ◦ pj u(y, v, t))(Υ ◦ pj u(y, v, t)), where pj (x) := xj , x = (x1 , ..., xm ), xj ∈ K for each j = 1, ..., m, Sˆj+1,τ g(u(y), β) := g(Sj+1,τ u(y), β), y ∈ Ks , τ ∈ Kk , k ≥ s, β is some parameter. Introduce operators Aj,v,t := (Sˆj+1,vt ⊗ tΥ1 ◦ pj )∗ Υ1j , where Υ1 is taken for variables (y, v, t) or corresponding to them after actions of preceding operators as Υk remembering that [k] [k] [k] [k] [k] [k] y [k] , v [k] ∈ (Ks )[k] , t ∈ K, v [k] = (v1 , v2 , v3 ) with v1 , v2 ∈ (Ks )[k−1] , v3 ∈ Kk for [0] each k ≥ 1, in particular, v [0] = v1 for k = 0, Υ1j f (x, v, t) := [f (x + ej vj t) − f (x)]/t, (B ⊗ A)∗ Υ1 fi ◦ ui (y, v, t) := Υ1j fi (Bui , v, Aui ), B : Km(i) → Km(i) , A : Km(i) → K. For example, in the particular case of s = 1 we have v [k] ∈ (K)[k] . Therefore, in the general case Formula (2) takes the form: P 1 (3) Υ1 f ◦ u(y, v, t) = m j=1 (Aj,v,t f ◦ u)(Υ ◦ pj u)(y, v, t). Take now n = 2, then P 1 [2] Υ2 f ◦ u(y [2] ) = Υ1 m j=1 [(Aj,v,t f ◦ u)(Υ ◦ pj u)(y, v, t)](y ). In the square brackets there is the product, hence from Formula 1(1) and Lemma 2.3 we get: P 1 1 π 1 Aj,v[0] ,t f ◦ u)(Υ2 ◦ (4) Υ2 f ◦ u(y [2] ) = m j=1 [(Υ Aj,v [0] ,t f ◦ u)(P2 Υ ◦ pj u) + (ˆ pj u)](y [2] ). Then from Formula (3) applied to terms Aj,v,t f ◦u it follows, that Υ1 Aj1 ,v[0] ,t1 f ◦u(y [2] ) = Υ1 u

Pm(2)

j2 =1 (Aj2 ,v [1] ,t2 Aj1 ,v [0] ,t1 f

◦ u)(Υ1 ◦ pj2 Sj1 +1,v[0] t1 u)(y [2] ), where v [0] = v, t1 = t (see

also Lemma 1). Therefore, Pm(2) P 1 ◦ (5) Υ2 f ◦ u(y [2] ) = [ m j1 =1 j2 =1 (Aj2 ,v [1] ,t2 Aj1 ,v [0] ,t1 f ◦ u)(Υ P m π 1 Aj1 ,v[0] ,t1 f ◦ u)(Υ2 ◦ pj1 u)](y [2] ). pj2 Sˆj1 +1,v[0] t1 u)(P2 Υ1 ◦ pj1 u) + j1 =1 (ˆ Then for n = 3 applying Formulas (3) and 4(1) to (5) we get: Pm(2) Pm(3) P (6) Υ3 f ◦ u(y [3] ) = [ m j1 =1 j3 =1 (Aj3 ,v [2] ,t3 Aj2 ,v [1] ,t2 Aj1 ,v [0] ,t1 f ◦ u) j2 =1 1 2 1 ˆ ˆ (Υ ◦ pj3 Sj2 +1,v[1] t2 Sj1 +1,v[0] t1 u )(P2 Υ ◦ pj2 Sˆj1 +1,v[0] t1 u)(P3 P2 Υ1 ◦ pj1 u)+ Pm Pm(2) 1 π (Aj2 ,v[1] ,t2 Aj1 ,v[0] ,t1 f ◦ u))(Υ2 ◦ pj2 Sˆj1 +1,v[0] t1 u)(P3 P2 Υ1 ◦ pj1 u)+ j1 =1 j2 =1 [(ˆ 1 1 (ˆ π 1 {(A [1] A [0] f ◦ u)(Υ1 ◦ pj Sˆ [0] u)}(Υ P2 Υ ◦ pj u)]+

2 j1 +1,v t1 1 1 1 ˆ ˆ Aj1 ,v[0] ,t1 f ◦ u)(Υ ◦ pj3 Sj1 +1,v[0] t1 u)(P3 Υ2 j1 =1 j3 =1 (Aj3 ,v [2] ,t3 π P m + j1 =1 (ˆ π 2 Aj1 ,v[0] ,t1 f ◦ u)(Υ3 ◦ pj1 u)](y [3] ).

Pm

j2 ,v

,t2

Pm(3)

j1 ,v

,t1

◦ pj1 u)

Groups of Diffeomorphisms and Wraps of Manifolds...

597

Thus Formula (1) is proved for n = 1, 2, 3. Suppose that it is true for k = 1, ..., n and prove it for k = n + 1. Applying Formula 4(1) to both sides of (1) we get: Pm

Pm(n+1) j1 =1 ... jn+1 =1 (Ajn+1 ,v [n] ,tn+1 ...Aj1 ,v [0] ,t1 f ◦ u) n 1 n−1 )... ˆ ˆ n+1 jn +1,v tn j1 +1,v [0] t1 u )(Pn+1 Υ ◦pjn Sjn−1 +1,v [n−2] tn−1 ...Sj1 +1,v [0] t1 u P P m(n) 1 (Pn+1 ...P2 Υ1 ◦ pj1 u) + m π (Ajn ,v[n−1] ,tn ...Aj1 ,v[0] ,t1 f ◦ u) j1 =1 ... jn =1 (ˆ 1 1 Υ ((Υ ◦ pjn Sˆjn−1 +1,v[n−2] tn−1 ...Sˆj1 +1,v[0] t1 un−1 )...(Pn ...P2 Υ1 ◦ pj1 u))+ P Pm(n−1) 1 Υ1 ( m π (Ajn−1 ,v[n−2] ,tn−1 ...Aj1 ,v[0] ,t1 f ◦ u))Υ1 ((Υ1 ◦ j1 =1 ... jn−1 =1 (ˆ

(7) Υn+1 f ◦ u(y [n+1] ) = [ (Υ1 ◦pj Sˆ ...Sˆ [n−1]

pjn−1 Sˆjn−2 +1,v[n−3] tn−2 Pm(2) P ...Sˆj1 +1,v[0] t1 un−2 )...(Pn−1 ...P2 Υ1 ◦ pj1 u)) + ... + Υn−2 { m j1 =1 j2 =1 1 (ˆ π 1 A [1] A [0] f ◦ u)Υ1 ((Υ1 ◦ pj Sˆ [0] u)(P2 Υ ◦ pj u))}+ j2 ,v

Pm,t2

Υn−1 { Pm

j1 ,v

,t1

2

j1 +1,v

t1

1

ˆ 1 Aj1 ,v[0] ,t1 f ◦ u)(Υ2 ◦ pj1 u)}](y [n+1] ) = j1 =1 π Pm(n+1)

1 ˆ jn+1 =1 (Ajn+1 ,v [n] ,tn+1 ...Aj1 ,v [0] ,t1 f ◦ u)(Υ ◦ pjn+1 Sjn +1,v [n−1] tn ...Sˆj1 +1,v[0] t1 un )(Pn+1 Υ1 ◦ pjn Sˆjn−1 +1,v[n−2] tn−1 ...Sˆj1 +1,v[0] t1 un−1 )...(Pn+1 ...P2 Υ1

[

j1 =1 ...

◦ pj1 u) P Pm(n) 1 Pn−1 ⊗α + m π (Ajn ,v[n−1] ,tn ...Aj1 ,v[0] ,t1 f ◦ u)[ α=0 π ˆ ⊗ Υ ⊗ Pˆ ⊗(n−α−1) ] j1 =1 ... jn =1 (ˆ ((Υ1 ◦ pjn Sˆjn−1 +1,v[n−2] tn−1 ...Sˆj1 +1,v[0] t1 un−1 ) ⊗ ... ⊗ (Pn ...P2 Υ1 ◦ pj1 u)) Pm(n−1) 1 Pn−1 ⊗α P π (Ajn−1 ,v[n−2] ,tn−1 ...Aj1 ,v[0] ,t1 f ◦ +[ α=0 π ˆ ⊗ Υ ⊗ Pˆ ⊗(n−α−1) ]( m j1 =1 ... jn−1 =1 (ˆ Pn−2 ⊗α ⊗(n−α−2) u)) ⊗ [ α=0 π ˆ ⊗ Υ ⊗ Pˆ ] 1 ˆ ˆ ((Υ ◦ pjn−1 Sjn−2 +1,v[n−3] tn−2 ...Sj1 +1,v[0] t1 un−2 ) ⊗ ... ⊗ (Pn−1 ...P2 Υ1 ◦ pj1 u)) Pm(2) 1 P P π Aj2 ,v[1] ,t2 Aj1 ,v[0] ,t1 f ◦ u)(Υ1 ⊗ Pˆ 1 + ˆ ⊗α ⊗ Υ ⊗ Pˆ ⊗(2−α) ]n−2 { m +[ 2α=0 π j1 =1 j2 =1 (ˆ π ˆ 1 ⊗ Υ1 )((Υ1 ◦ pj2 Sˆj1 +1,v[0] t1 u) ⊗ (P2 Υ1 ◦ pj1 u))} P π 1 A [0] f ◦ u) ⊗ (Υ2 ◦ pj u)}](y [n+1] ). +(Υ ⊗ Pˆ + π ˆ ⊗ Υ)n−1 { m (ˆ j1 =1

j1 ,v

,t1

1

Mention that in general (Υn+1 f ◦ u)(y [n+1] ) may depend nontrivially on all components of the vector y [n+1] through several terms in Formula (7). Thus Formula (1) of this Lemma is proved by induction. 7. Corollary. Let u ∈ C n (Ks , Km ), u(Ks ) ⊂ U and f ∈ C n (U, Y ), where U is an open subset in Km , s, m ∈ N, Y is a K-linear space, then ¯ n (f ◦ u)(x(n) ) = [Pm ... Pm(n) (B (n−1) ...B (0) f ◦ u) (1) Φ j1 =1 jn ,v ,tn j1 ,v ,t1 jn =1 1 ¯ ˆ ˆ ¯1 (Φ ◦ pjn Sjn−1 +1,v(n−2) tn−1 ...Sj1 +1,v(0) t1 un−1 )(Pn Φ pjn−1 Sˆj +1,v(n−3) ,t ...Sˆj1 +1,v(0) t1 un−2 ) n−2

n−2

0

P

¯1 ...(Pn ...P2 Φ

Pm(n−1)

◦

m π 1 (Bjn−1 ,v(n−2) ,tn−1 ...Bj1 ,v(0) ,t1 f ◦ ◦ pj1 u) + j1 =1 ... jn−1 =1 (ˆ Pn−2 ⊗α ¯ ⊗ Pˆ ⊗(n−α−2) ] ˆ ⊗Φ u)[ α=0 π 1 n−2 ) ⊗ ... ⊗ (P ¯ ◦ pj Sˆ ˆ ¯1 ((Φ n−1 ...P2 Φ ◦ pj1 u)) n−1 jn−2 +1,v (n−3) tn−2 ...Sj1 +1,v (0) t1 u Pn−2 ⊗α 1 ¯ ⊗ Pˆ ⊗(n−α−2) ](Pm ... Pm(n−2) (ˆ +[ α=0 π ˆ ⊗Φ j1 =1 jn−2 =1 π (Bjn−2 ,v (n−3) ,tn−2 ...Bj1 ,v (0) ,t1 f ◦ Pn−3 ⊗α ¯ n−3 ) ⊗ ¯ 1 ◦ pj Sˆ ˆ u)) ⊗ [ α=0 π ˆ ⊗ Φ ⊗ Pˆ ⊗(n−α−3) ]((Φ n−2 jn−3 +1,v (n−4) tn−3 ...Sj1 +1,v (0) t1 u ¯ 1 ◦ pj u)) + ... ... ⊗ (Pn−2 ...P2 Φ 1 Pm(2) 1 P2 P ⊗α ⊗(2−α) ¯ 1 ⊗ Pˆ 1 + ¯ ˆ π Bj2 ,v(1) ,t2 Bj1 ,v(0) ,t1 f ◦ u)(Φ +[ α=0 π ˆ ⊗Φ⊗P ]n−3 { m j1 =1 j2 =1 (ˆ ¯ 1 )((Φ ¯ 1 ◦ pj Sˆ ¯1 π ˆ1 ⊗ Φ (0) u) ⊗ (P2 Φ ◦ pj u))} 2

j1 +1,v

t1

1

598

S.V. Ludkovsky Pm

¯ ⊗ Pˆ + π ¯ n−2 { ¯ 2 ◦ pj u)}](x(n) ) +(Φ ˆ ⊗ Φ) π 1 Bj1 ,v(0) ,t1 f ◦ u) ⊗ (Φ 1 j1 =1 (ˆ and f ◦ u ∈ C 0 ((Ks )(n) , Y ) (see notation of Lemma 9), where Bj,v,t := ¯ 1 ◦ pj )∗ Φ ¯ 1 , where Φ ¯ 1 is taken for variables (x, v, t) or corresponding to them (Sˆj+1,vt ⊗ tΦ j ¯ k, Φ ¯ 1 f (x, v, t) := [f (x + ej vj t) − f (x)]/t, after actions of preceding operations as Φ j ¯ 1 fi ◦ ui (x, v, t) := Φ ¯ 1 fi (Bui , v, Aui ), B : Km(i) → Km(i) , A : Km(i) → K, (B ⊗ A)∗ Φ j ¯ 1 ◦ pj u1 ), m(i) = m + i − 1, ji = 1, ..., m(i), u1 = u, u2 := (u1 , t1 Φ 1 n n−1 1 n−1 1 1 ¯ ˆ ¯ ¯ ˆ u := (u , tn−1 Φ ◦ pjn−1 u ), S∗ Φ f (x) := Φ f (S∗ x). Proof. The restriction of operators of Lemma 6 on W (n) from Note 2 gives Formula (1) of this corollary, where v (k) ∈ (Ks )k × Kk .

References [1] Y. Amice. ”Interpolation p-Adique”. Bull. Soc. Math. France 92(1964), 117-180. [2] Aref’eva I.Ya., Dragovich B., Frampton P.H., Volovich I.V. ”Wave functions of the universe and p-adic gravity”. Int. J. Modern Phys. 6 (1991), 4341-4358. [3] J. Araujo, W.H. Schikhof. ”The Weierstrass-Stone approximation theorem for p-adic C n -functions”. Ann. Math. Blaise Pascal. 1 (1994), 61-74. [4] Yu.A. Bachturin. ”Basic structures of modern algebra” (Moscow: Nauka, 1990). [5] A. Banyaga. ”The structure of classical diffeomorphism groups”. Mathem. and its Applic. 400 (Dordrecht: Kluwer, 1997 [6] N. Bourbaki. ”Vari´et´es diff´erentielles et analytiques”. Fasc. XXXIII (Paris: Hermann, 1967). [7] P.-J. Cahen, J.-L. Chabert. ”On the ultrametric Stone-Weierstrass theorem on Mahler’s expansion”. J. de Th´eorie des Nombres de Bordeaux 14 (2002), 43-57. [8] S. De Smedt. ”Local invertibility of non-archimedean vector-valued functions”. Ann. Math. Blaise Pascal 5 (1998), 13-23. [9] B. Diarra. ”Ultraproduits ultrametriques de corps values”. Ann. Sci. Univ. Clermont II, S´er. Math. 22 (1984), 1-37. [10] B. Diarra. ”On reducibility of ultrametric almost periodic linear representations”. Glasgow Math. J. 37 (1995), 83-98. [11] R. Engelking. ”General topology”. 2-nd ed., Sigma Series in Pure Mathematics, V. 6 (Berlin: Heldermann Verlag, 1989). [12] J.M.G. Fell, R.S. Doran. ”Representations of ∗-algebras, locally compact groups, and Banach ∗-algebraic bundles”. V. 1 and V. 2 (Boston: Acad. Press, 1988). [13] S. Haller, J. Teichmann. ”Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones”. Annals of Global Analysis and Geometry 23 (2003), 53-63.

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[14] K. Hensel. Jahresber. Deutsch. Math. Ver. 6: 1 (1899), 83-88. [15] I.N. Herstein. ”Topics in algebra” (New-York: Blaisdell Publ. Co., 1964). [16] E. Hewitt, K.A. Ross. ”Abstract harmonic analysis” (Berlin: Springer, 1979). [17] Y. Jang. ”Non-Archimedean quantum mechanics”. Tohoku Math. Publ. 10 (1998). [18] I. Kaplansky. ”The Weiertsrass theorem in fields with valuations”. Proc. Amer. Math. Soc. 1 (1950), 356-357. [19] M.I. Kargapolov, Yu.I. Merzljakov. ”Principles of group theory” (Moscow: Nauka, 1977). [20] W. Ling. ”Factorizable groups of homeomorphisms”. Compositio Mathem. 51 (1984), 41-50. [21] S.V. Ludkovsky. ”Irreducible unitary representations of non-Archimedean groups of diffeomorphisms”. Southeast Asian Bull. of Mathem. 22: 3 (1998), 419-436. [22] S.V. Ludkovsky. ”Measures on groups of diffeomorphisms of non-Archimedean manifolds, representations of groups and their applications”. Theoret. i Math. Phys. 119: 3 (1999), 381-396. [23] S.V. Ludkovsky. ”Quasi-invaraint measures on non-Archimedean groups and semigroups of loops and paths, their representations. I, II”. Ann. Math. Blaise Pascal 7: 2 (2000), 19-53, 55-80. [24] S.V. Ludkovsky. ”A structure and representations of diffeomorphism groups of nonarchimedean manifolds”. Southeast Asian Bull. of Mathem. 26 (2003), 975-1004 (previous variant: Los Alamos Nat. Lab. math.GR/0004126, 32 pages, 19 April 2000). [25] S.V. Ludkovsky, B. Diarra. ”Profinite and finite groups associated with loop and diffeomorphism groups of non-archimedean manifolds”. Int. J. of Math. and Mathematical Sciences 2003: 42 (2003), 2673-2688. [26] S.V. Ludkovsky. ”Quasi-invariant and pseudo-differentiable measures on nonarchimedean Banach spaces with values in non-archimedean fields”. J. Math. Sci. 122: 1 (2004), 2949-2983 (previous variant: Los Alamos Nat. Lab. preprint math.GM/0106170). [27] S.V. Ludkovsky. ”Non-archimedean polyhedral expansions of ultrauniform spaces”. Russ. Math. Surveys. V. 54: 5 (1999), 163-164. (in detail: Los Alamos National Laboratory, USA. Preprint math.AT/0005205, 39 pages, 22 May 2000). [28] S.V. Ludkovsky. ”Non-archimedean polyhedral decompositions of ultrauniform spaces”. Fund. Prikl. Mathem. 6: 2 (2000), 455-475. [29] S.V. Ludkovsky. ”Embeddings of non-archimedean Banach manifolds into nonarchimedean Banach spaces”. Russ. Math. Surv. 53 (1998), 1097-1098.

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[30] S.V. Ludkovsky. ”Approximate and global differentiability of functions over nonarchimedean fields”. Sovrem. Mathem. i ee Pril. 52 (2008) (previous variant: Los Alamos Nat. Lab. math.CA/0608724). [31] P.W. Michor. ”Manifolds of differentiable mappings” (Orpington: Shiva Publishing Limited, 1980). [32] L. Narici, E. Beckenstein. ”Topological vector spaces” (New York: Marcel Dekker Inc., 1985). [33] A. Ostrowski. ”Untersuchungen zur arithmetischen Theorie der K¨orper”. Math. Zeit. 39 (1935), 269-404. [34] ”p-Adic Mathematical Physics.” 2nd International Conference (Belgrad, 2005). Editors: A.Yu. Khrennikov, Z. Raki´c, I.V. Volovich. AIP Conference Proceedings 826, New York, 2006. [35] L.S. Pontrjagin. ”Continuous groups” (Moscow: Nauka, 1984). [36] J. Riordan. ”Combinatorial identities” (New York: John Wiley and Sons Inc., 1968). [37] A. Robert. ”Representations p-adiques irr´eductibles de sous-groupes ouverts de SL2 (Zp )”. C.R. Acad. Sci. Paris S´er. I Math. 298: 11 (1984), 237-240. [38] A.C.M. van Rooij. ”Non-Archimedean functional analysis” (New York: Marcel Dekker Inc., 1978). [39] A.C.M. van Rooij, W.H. Schikhof. ”Groups representations in non-Archimedean Banach spaces”. Bull. Soc. Math. France. Memoire. 39-40 (1974), 329-340. [40] W.H. Schikhof. ”Ultrametric calculus” (Cambridge: Cambr. Univ. Press, 1984). [41] W.H. Schikhof. ”Non-Archimedean calculus”. Nijmegen: Math. Inst., Cath. Univ., Report 7812, 130 pages, 1978. [42] V.S. Vladimirov, I.V. Volovich, E.I. Zelenov. ”p-adic analysis and mathematical physics” (Moscow: Nauka, 1994). [43] A. Weil. ”Basic number theory” (Berlin: Springer, 1973).

INDEX A ABC, 248 Abelian, 17, 18, 99, 291, 402, 403, 406, 412, 419, 432, 591, 592 absorption, 2 absorption spectra, 2 academic, 27 access, 2 accounting, 91 accuracy, ix, 169, 180, 184 ad hoc, 201 Adams, 448, 481 adaptation, 92 additives, 499 affine group, 201, 210 age, 49 aid, 6, 9, 14, 62, 72, 76, 77, 212, 220, 222, 240, 243, 261 AIP, 600 Albert Einstein, 1 algorithm, 84, 124, 170, 172, 180 alternative, 54, 120, 209, 215, 216, 217, 284, 439, 488, 489, 490, 493, 495, 502, 503, 510, 511, 513, 514, 515, 517, 519, 521, 527, 543, 545, 554, 558 alternatives, 27 ambivalence, 5 amplitude, 46 AMS, 58, 187, 305, 348, 447 Amsterdam, 87, 559 analog, 262, 362, 545, 548, 549, 564, 575 angular momentum, vii, 3, 4, 5, 6, 7, 20, 21, 231, 246, 256, 294, 295 angular velocity, 62, 63, 74, 76, 77, 78, 82, 83, 294, 303, 340 annihilation, vii, 3, 17, 32, 42, 43 anomalous, 2, 4 antagonistic, 51 anthropology, 46 antimatter, 35, 247 appendix, 574

application, x, xi, xiii, 44, 62, 86, 126, 131, 134, 140, 142, 175, 203, 242, 283, 285, 294, 305, 326, 337, 376, 402, 528, 564, 568, 575, 592, 593, 594 Argentina, 325 argument, 35, 43, 46, 114, 211, 213, 227, 239, 275, 566 arithmetic, 122, 449 Arizona, 56 Arizona State University, 56 Asian, 559, 599 assignment, 352, 533, 547 assumptions, 79, 377, 392 astrophysics, 36 asymmetry, 37, 53 asymptotic, viii, 55, 89, 90, 91, 92, 99, 101, 103, 104, 106, 107, 108, 109, 110, 112, 114, 116, 268, 269, 271, 275, 281, 290, 360, 371, 383 asymptotically, 110, 114, 251, 360, 365 asymptotics, 91 Australia, 37, 186 Austria, 1 authority, 28

B Banach spaces, 564, 592, 599 baryon, vii, 3, 13, 23, 27, 28, 32, 40, 42, 43 baryonic matter, 36 baryons, 28, 44, 55 beams, 87 behavior, 104, 116, 118, 121, 214, 216, 267, 269, 371, 383, 402, 419 behaviours, viii, 89, 90, 92, 98 Beijing, 139, 169 Belarus, 483 Bessel, 118, 376, 377 Bianchi identity, 240, 241, 253 birth, 390 black hole, 247 blocks, 41 Bogolubov, 481 boson, 30, 34, 53 bosons, 35, 44, 52, 53, 201

602

Index

Boston, 56, 558, 559, 560, 598 boundary conditions, viii, 123, 124, 128, 129, 133, 135, 136, 137, 258, 495 boundary value problem, 124 bounded solution, 159 branching, 402, 411, 412 Brazil, 58 breakdown, 201, 202, 262 Brownian motion, 89, 90, 91, 102, 103, 104, 113, 117, 118, 120, 121, 122, 355, 356, 357, 365, 366, 376, 377, 380, 381, 382, 383 Buddhist, 27 building blocks, 39 bun, 507

C C*-algebra, 3, 7, 10, 11, 32, 57 Calabi-Yau, 280 calculus, 25, 57, 305, 600 candidates, 36, 268, 411 Carnot, 102, 323 Cartesian coordinates, 178 Casimir operators, 411, 412 cast, 229 category b, 450 catholic, 1 causality, 35, 247, 257, 258 causation, 247, 257 cell, 553, 554, 556 CERN, 1, 55 charge density, 52 charm, 34, 40 China, 139, 169, 184 chiral, 35, 201 chirality, 54 chromosome, 46 classes, viii, xii, 22, 25, 44, 53, 54, 89, 90, 124, 126, 214, 252, 358, 401, 409, 410, 413, 418, 420, 426, 430, 436, 439, 448, 486, 490, 492, 493, 496, 500, 504, 507, 515, 516, 526, 527, 528, 534, 535, 537, 547, 548, 554, 558, 559, 565, 587, 588, 592 classical, ix, 10, 46, 55, 123, 124, 128, 129, 169, 171, 183, 184, 186, 189, 264, 272, 284, 285, 286, 291, 293, 295, 297, 304, 305, 326, 327, 328, 329, 330, 337, 347, 348, 364, 372, 386, 409, 411, 500, 563, 564, 565, 598 classical mechanics, 305, 327, 330, 347 classification, xii, 23, 28, 30, 43, 50, 126, 296, 385, 396, 402, 412, 418, 440, 448 closure, 130, 188, 209, 366, 378, 409, 448, 451, 456, 461, 462, 468, 470, 472, 500, 524, 538, 548, 553, 554, 581, 585, 586, 589 clustering, 370, 371 cognition, 30 cognitive, 3, 55 cold dark matter, 36 collateral, 52

colors, 33 communication, 28, 57 community, viii, 61, 86 commutativity, 12, 293, 326, 504, 543, 545 compatibility, 462, 467, 530, 531, 532 competition, 27 complement, 18, 37, 62, 75, 76, 97, 114, 278, 286, 364, 368 complex numbers, 564 components, 5, 7, 20, 48, 52, 53, 85, 128, 201, 203, 207, 217, 218, 219, 220, 221, 222, 223, 226, 244, 246, 256, 257, 260, 261, 295, 338, 367, 402, 456, 457, 461, 464, 468, 516, 597 composition, xiii, 125, 204, 212, 259, 449, 450, 455, 465, 469, 476, 485, 489, 491, 519, 521, 522, 523, 525, 533, 540, 543, 545, 567, 569, 574, 588, 591, 593 comprehension, 27, 65 computation, 93, 94, 106, 111, 138, 207, 208, 218, 269, 297, 298, 325, 361, 402, 406, 428, 436 Computational Fluid Dynamics, 186 computing, 345, 422, 430 concentrates, 36 conception, 36, 50, 142 concrete, 28, 37, 44 conditioning, 115, 117 conductivity, 291 configuration, x, xi, 171, 197, 283, 284, 302 confinement, 10, 201 confusion, 69, 71, 394, 583 Congress, 281 congruence, 98, 99 conjecture, 347 conjugation, 7, 25, 26, 333, 352, 375, 377, 510, 517, 519 conservation, 137, 138, 175, 176, 186, 203, 227, 229, 230, 231, 246, 247, 248, 249, 253, 256, 257, 258 constraints, 24, 79, 178, 210, 406, 413, 433, 437, 438 construction, viii, 2, 3, 8, 9, 12, 13, 15, 27, 40, 42, 43, 75, 123, 124, 126, 203, 272, 273, 274, 279, 302, 308, 337, 344, 380, 476, 486, 494, 496, 497, 500, 505, 517, 522, 524, 535, 536, 550, 587, 589, 591, 593 continuity, 107, 113, 119, 372, 373, 379, 501, 567, 568 contractions, xii, xiii, 248, 401, 402, 408, 409, 410, 411, 412, 413, 417, 418, 419, 425, 428, 432, 434, 435, 436, 437, 438, 439 contracts, 410, 435 control, x, 283, 284, 285, 292, 293, 294, 296, 302, 304, 337, 402 convergence, viii, 89, 90, 175, 268, 274, 279, 280, 309, 317, 358, 359, 368, 379, 382, 454, 456, 477, 492, 582 convex, 366, 489 conviction, 48, 53 correlation, 3 correlations, 114 cosmological constant, 36, 227, 242

Index couples, 267 coupling, 225, 242, 248, 252, 253, 254 covering, 2, 100, 121, 267, 272, 449, 466, 467, 481, 489, 493, 505, 509, 510, 512, 514, 526, 538, 539, 545, 546, 551, 553, 556, 557, 567, 570, 580, 586, 587 CRC, 137, 138, 398 creativity, 2 critical density, 36 cycles, 583

D damping, 186, 285, 292 danger, 69, 71 dark matter, 35, 36, 37, 39 decay, 27, 114 decomposition, xii, 49, 50, 54, 97, 203, 204, 212, 216, 278, 336, 340, 343, 351, 364, 366, 367, 368, 370, 371, 376, 378, 382, 383, 402, 403, 406, 408, 410, 411, 412, 413, 418, 419, 420, 425, 432, 438, 443, 481, 499, 508, 509, 511, 521, 572, 575, 582, 592, 598 deconvolution, 50, 51 definition, viii, ix, xii, 14, 61, 69, 70, 73, 77, 86, 99, 110, 117, 120, 139, 140, 141, 145, 146, 147, 148, 157, 165, 192, 205, 208, 209, 219, 224, 237, 239, 248, 251, 279, 291, 330, 365, 371, 402, 408, 409, 449, 450, 458, 461, 463, 464, 467, 468, 472, 476, 477, 478, 480, 486, 487, 488, 493, 494, 507, 510, 511, 512, 514, 524, 529, 539, 543, 546, 550, 557, 566, 574, 593 deformation, viii, x, xii, 14, 32, 42, 84, 123, 127, 199, 222, 225, 256, 275, 276, 401, 402, 405, 409, 411, 412, 413, 414, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 447, 482, 523, 524, 552 degenerate, 14, 32, 102, 104, 124, 291, 305, 326, 327, 329, 335, 336, 342, 358, 431, 528 degrees of freedom, x, 24, 41, 47, 50, 199, 202, 256, 304 demand, 25 density, 79, 229, 233, 242, 256, 259, 358, 359, 360, 361, 365, 366 dependent variable, 129, 386 derivatives, x, 52, 62, 72, 74, 76, 81, 145, 153, 154, 157, 158, 164, 165, 170, 199, 225, 236, 239, 240, 241, 245, 253, 275, 285, 289, 300, 301, 352, 392, 502, 515, 560 deviation, 111, 309 differential equations, vii, viii, ix, xii, 63, 79, 121, 123, 127, 137, 138, 147, 169, 170, 175, 184, 185, 339, 385, 386, 396, 397, 398, 399, 411 differentiation, 53, 224, 494, 501, 516, 528 diffusion, 90, 121, 160, 292, 354, 355, 358, 359, 360, 362, 365, 377, 381, 382 diffusion process, 121, 354, 358, 362, 377

603

dilation, x, 199, 210 dimensionality, 510 Dirac equation, 2, 5, 54, 234, 246 Dirac spinor, 236, 256 Dirac spinors, 256 discipline, 1 discontinuity, 110, 127 discretization, 62, 113, 176, 284 dispersion, 157 displacement, 52, 53, 79, 80, 209 disputes, 2 distribution, 224, 284, 291, 299, 302, 353, 354, 355, 356, 358, 359, 360, 361, 362, 364, 365, 366, 367, 372, 373, 374, 375, 376, 378, 379, 380, 381, 457, 458, 459, 516 divergence, 242, 256, 297, 298 division, 559, 581 Doppler, 58 duality, 12, 100, 102, 226, 260, 354 Duality, 12 dynamical system, 280, 295, 368, 381 dynamical systems, 381

E earth, 59 electric field, 50, 52, 181 electricity, 175, 176 electromagnetic, 39, 52, 200, 227 electron, vii, 2, 3, 4, 5, 26, 285, 291, 293 electron gas, 285, 291, 293 electrons, 53 electroweak interaction, 11, 37, 44, 51 elementary particle, 57, 408 emission, 34 energy, 2, 13, 17, 28, 32, 34, 36, 42, 46, 52, 105, 175, 176, 246, 256, 281, 297, 393, 394, 555 energy-momentum, 229, 231, 245, 246, 256, 257 entanglement, 247, 257, 258 epistemological, 253 EPR, 257 equality, 119, 148, 153, 154, 157, 160, 162, 213, 288, 331, 334, 335, 360, 424, 437, 448, 450, 453, 457, 459, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 473, 474, 475, 477, 479, 490, 493, 499, 568, 588 equating, 245 equilibrium, 128, 160, 329, 330, 339 equilibrium state, 160 ergodic theory, 381 ESI, 1, 55, 446 Euclidean space, xi, 4, 65, 68, 351, 352, 357, 365, 451, 479, 480, 481, 487, 499, 547, 556 Euler equations, xi, 63, 79, 186, 283, 285, 286, 287, 289, 290, 291, 292, 295, 297, 300, 303, 305, 347 Eulerian, 62 Euler-Lagrange equations, 246, 249

604

Index

evolution, viii, 36, 47, 123, 128, 254, 258, 263, 325, 327, 408 excitation, 35 exclusion, 30 exercise, 394 expansions, 587, 589, 599 eyes, 3

F failure, 170 family, viii, 16, 33, 91, 104, 123, 125, 127, 128, 129, 132, 133, 136, 137, 144, 145, 148, 189, 191, 195, 196, 214, 216, 254, 280, 288, 326, 353, 369, 370, 371, 393, 408, 414, 416, 420, 421, 425, 427, 429, 430, 449, 451, 453, 456, 459, 461, 466, 467, 470, 476, 480, 488, 489, 492, 500, 501, 502, 505, 507, 511, 513, 515, 516, 519, 522, 523, 526, 530, 531, 538, 539, 540, 541, 542, 543, 545, 546, 548, 549, 551, 552, 564, 567, 568, 569, 570, 571, 577, 581, 586, 592 Far East, 560 Fermi, 28, 199, 385 fermions, 10, 28, 30, 34, 36, 38, 42, 44 Feynman, 28 fiber, xiii, 202, 204, 205, 206, 208, 209, 211, 212, 452, 453, 455, 485, 487, 489, 490, 491, 492, 493, 494, 505, 506, 507, 508, 509, 515, 516, 517, 525, 541, 554, 598 fiber bundles, xiii, 204, 208, 452, 453, 485, 487, 490, 491, 509, 516 fibers, xiii, 205, 208, 209, 212, 213, 214, 259, 456, 457, 485, 486, 537 field theory, 55 finite volume, 90, 121 Finland, 87 fixation, 2 flavor, 26, 30, 42, 45, 47 flow, x, xi, 89, 90, 91, 106, 107, 110, 119, 121, 177, 186, 274, 275, 279, 280, 281, 283, 284, 285, 286, 289, 290, 291, 293, 295, 296, 297, 298, 301, 302, 303, 332, 333, 341, 344, 368, 369, 370, 371, 382, 388 fluid, x, xi, 283, 284, 285, 286, 290, 291, 293, 297, 299, 301, 302, 303, 305, 558 Fourier, 358, 365, 382, 383, 446 Fourier analysis, 382, 383 fractals, 32 framing, 209 France, 89, 383, 397, 598, 600 freedom, x, 4, 24, 25, 28, 33, 41, 47, 50, 199, 202, 256, 304, 342 friction, 285, 292, 294 frog, 175 fulfillment, 131 functional analysis, 10, 600

G gas, 124, 138, 298 gauge, x, 5, 27, 35, 53, 54, 199, 200, 201, 202, 203, 204, 205, 208, 209, 212, 213, 214, 215, 216, 217, 219, 221, 222, 223, 224, 225, 227, 232, 233, 235, 239, 241, 243, 246, 247, 251, 255, 256, 257, 264, 486 gauge fields, x, 199, 200, 201, 202, 203, 219, 221, 232, 233, 256 gauge group, 5, 35, 200, 201, 202, 208, 209, 239 gauge theory, x, 53, 54, 199, 200, 201, 202, 203, 224, 227, 256, 486 Gauss-Bonnet, 483 Gaussian, 91, 103, 104, 108, 372, 373, 560 GBA, 260 gender, 140 gene, 287, 397, 419, 485 General Relativity, x, 55, 199, 200, 201, 202, 219, 242, 247, 249, 251, 253, 254, 255, 256, 257, 262 generalization, 210, 227, 256, 280, 290, 296, 303, 327, 339, 342, 483, 486, 488, 522, 535 generalizations, 287, 397, 419, 485 generation, 247, 250 generators, ix, xii, 13, 14, 22, 53, 99, 100, 125, 130, 139, 140, 141, 142, 148, 150, 152, 157, 160, 161, 206, 210, 211, 230, 231, 232, 385, 386, 387, 389, 394, 407, 409, 412, 414, 437, 498, 502, 504, 510, 511, 512, 519, 525, 534, 535, 538, 539 genes, 46 goals, 308 grades, 5, 18, 29, 30, 32 grading, 25, 32, 37, 47 gravitation, x, 56, 58, 199, 200, 201, 202, 208, 209, 230, 241, 256, 264 gravitational field, x, 1, 200, 226, 242, 246, 256, 306 gravitational force, 200 gravity, 200, 201, 202, 203, 224, 227, 242, 246, 247, 252, 253, 256, 263, 264, 598 groups, vii, ix, x, xi, xii, xiii, 2, 3, 5, 10, 11, 28, 35, 36, 40, 45, 48, 50, 51, 53, 54, 55, 61, 63, 70, 98, 121, 122, 124, 130, 138, 139, 140, 141, 148, 170, 172, 185, 187, 188, 200, 201, 202, 203, 208, 209, 210, 214, 260, 264, 283, 284, 285, 290, 291, 293, 296, 302, 305, 306, 323, 325, 337, 347, 348, 349, 351, 352, 353, 358, 366, 367, 381, 382, 383, 398, 401, 404, 405, 408, 411, 448, 449, 450, 452, 461, 468, 469, 472, 474, 476, 477, 478, 479, 480, 481, 482, 485, 486, 487, 490, 495, 496, 498, 499, 500, 503, 504, 505, 506, 508, 509, 510, 512, 513, 514, 515, 520, 521, 522, 523, 525, 527, 529, 532, 534, 535, 536, 537, 542, 543, 551, 552, 559, 560, 563, 564, 565, 576, 585, 586, 587, 591, 592, 598, 599, 600

Index

H hadrons, 27, 28 Hamiltonian, 3, 31, 32, 88, 169, 183, 185, 284, 287, 290, 291, 295, 296, 302, 306, 325, 326, 329, 330, 331, 332, 333, 335, 336, 338, 339, 340, 341, 342, 343, 344, 346, 347 handling, 202 Harmonic analysis, 347 Hebrew, 55 height, 97, 98, 106, 107, 112, 115, 129 Heisenberg, 186, 326, 327, 339, 340, 343, 344, 345, 348, 410 Heisenberg equations, 327 Higgs, 35, 36, 37, 39, 58 Higgs boson, 37, 39 Hilbert, 8, 9, 12, 13, 32, 56, 201, 487, 491 Hilbert space, 8, 9, 12, 13, 32, 56, 487, 491 homeomorphic, 205, 209, 447, 494, 496, 513, 555, 574, 587, 589 homogeneity, xi, 307, 308, 309 homogenous, 235 homology, 90, 97, 121, 548, 552, 561 homomorphism, 358, 460, 461, 467, 468, 475, 477, 490, 495, 496, 507, 509, 510, 511, 513, 514, 515, 523, 529, 530, 534, 535, 536, 537, 538, 539, 540, 541, 543, 544, 547, 548, 550, 551, 552, 553, 572, 574, 576, 581, 582, 583, 588, 591 homomorphisms, 11, 468, 507, 509, 527, 528, 529, 535, 540, 544, 548, 572, 576, 582, 591 hydrodynamic, x, xi, 283, 284, 285, 291, 302, 303 hydrodynamics, 284, 285, 286, 290, 291, 301, 302, 305 hydrostatic pressure, 129, 135 hyperbolic, viii, 89, 90, 91, 92, 93, 95, 96, 102, 105, 107, 108, 110, 113, 116, 121, 122, 123, 124, 128, 129, 137, 365, 479, 480, 500 hyperbolicity, 90 hypothesis, 188

I IAEA, 58 identification, 90, 91, 95, 99, 236, 253, 285, 340, 380, 404 identity, 2, 5, 12, 14, 48, 67, 72, 74, 76, 103, 117, 142, 176, 204, 205, 212, 214, 234, 240, 241, 253, 277, 291, 299, 328, 342, 352, 353, 361, 364, 395, 403, 405, 420, 421, 429, 451, 460, 468, 483, 499, 507, 510, 517, 530, 540, 541, 550, 553, 581, 583, 586 IMA, 185, 398 images, 63, 212, 469, 492 immersion, ix, 187, 190, 195, 196 implementation, viii, 35, 61, 173, 185, 203 imprisonment, 1 IMS, 381

605

inclusion, 17, 317, 404, 449, 455, 461, 538, 540, 550, 551, 552, 577 incompressible, 177, 186, 284, 290, 558 independence, viii, 50, 89, 90, 92, 103, 112, 113, 114, 450 independent variable, 52, 124, 126, 128, 129, 137, 388, 394, 395, 396 India, 58 Indian, 560 Indiana, 306, 348, 481 indicators, 3 indices, 37, 40, 44, 194, 231, 237, 238, 239, 240, 255, 259, 292, 301, 395, 462, 526 induction, 318, 321, 463, 494, 518, 521, 522, 524, 531, 540, 541, 547, 548, 556, 566, 567, 568, 571, 583, 584, 594, 595, 597 inequality, 143, 278, 280, 410, 411, 464, 465, 469, 475, 502, 512, 580 inertia, 286, 288, 292, 294, 295, 296, 297, 300, 303, 304 infinite, viii, xiii, 86, 123, 129, 130, 135, 137, 209, 272, 286, 287, 291, 353, 365, 386, 393, 463, 464, 485, 486, 490, 492, 493, 495, 496, 500, 506, 507, 510, 558, 563, 565, 576, 577, 587, 589, 590 inherited, 496, 567, 570, 572, 573, 574, 576, 591 injection, 190, 211 inspection, 250, 391, 411, 435 instruments, 3, 50 integration, viii, 61, 62, 63, 80, 81, 86, 88, 169, 174, 186, 245, 284, 301, 306, 347, 397, 480, 549, 560 intensity, 59, 181 interaction, 44, 53, 200, 202, 255 interactions, x, xii, 11, 55, 199, 200, 202, 222, 247, 253, 256, 401 interference, 53 interpretation, x, xi, xii, 5, 24, 35, 39, 64, 71, 133, 136, 198, 210, 250, 251, 283, 284, 285, 302, 402, 405, 411, 485 interval, ix, 113, 125, 170, 183, 187, 188, 285, 294, 337, 555 intrinsic, 201, 221 invariants, xi, 55, 224, 248, 284, 285, 286, 289, 291, 300, 301, 302, 303, 395, 396, 398, 405, 410, 411, 412, 413, 416, 420, 421, 431, 444, 482 inversion, 35, 503, 547, 554, 567, 574, 591 involution, 8, 326, 332, 337, 345, 346, 364, 366, 367 isomorphism, 37, 68, 100, 189, 205, 290, 329, 331, 336, 352, 412, 418, 420, 439, 449, 450, 452, 457, 458, 460, 461, 463, 464, 465, 468, 469, 471, 472, 473, 490, 495, 499, 504, 515, 516, 523, 527, 528, 529, 532, 533, 534, 536, 537, 539, 540, 541, 544, 545, 547, 548, 549, 550, 551, 552, 553, 557, 573 isospin, 17, 19, 20, 21, 22, 40, 41, 486 isotropic, 231, 381 isotropy, 291, 299, 328, 334, 353, 377, 378, 450, 454, 455, 456, 460, 466, 478 Israel, 55 Italy, 57, 199, 385, 399 ITEP, 59

606

Index

iteration, 82, 83, 172

J Jacobian, 124, 178, 228 January, 57, 59 Japan, 122, 483, 560 joints, 188 Jordan, 559 justification, 52, 77

K K+, 28 kernel, 100, 117, 189, 267, 268, 269, 276, 354, 378, 379, 389, 463, 472, 547 kinematics, 188, 198 Klein-Gordon, 162, 165, 249, 253 Kleinian, 121, 122 Kolmogorov, 302 Korteweg-de Vries, 166, 167

L Lagrangian, x, 27, 62, 80, 87, 104, 169, 174, 199, 201, 223, 224, 227, 229, 230, 232, 233, 235, 242, 243, 244, 248, 256, 283, 284, 285, 292, 302 Lagrangian density, 224, 227, 229, 230, 232, 233, 242, 244, 248 Lagrangian formulation, 62, 80 lambda, 17 language, xi, 204, 283, 387, 397 lattice, 53, 54, 59, 326, 327, 338, 348 lattices, 53 law, viii, 89, 90, 91, 92, 96, 103, 104, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 142, 203, 208, 212, 215, 216, 221, 237, 238, 257, 258 laws, 89, 119, 120, 137, 138, 170, 203, 204, 227, 229, 230, 231, 246, 247, 248, 249, 253, 255, 256, 257, 403, 487 lead, 2, 12, 36, 250, 269, 275, 396, 426, 429, 430, 436 Lebesgue measure, 492 Lie algebra, vii, ix, x, xii, 3, 4, 9, 10, 11, 13, 15, 16, 17, 18, 22, 23, 25, 26, 36, 41, 43, 44, 50, 67, 68, 93, 125, 126, 130, 137, 139, 140, 141, 142, 144, 151, 152, 166, 170, 171, 185, 187, 188, 191, 195, 205, 206, 207, 216, 217, 218, 219, 222, 226, 230, 232, 255, 259, 261, 268, 272, 277, 280, 283, 284, 285, 286, 287, 288, 289, 290, 292, 294, 297, 299, 300, 303, 304, 325, 326, 327, 328, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 347, 348, 352, 353, 354, 357, 358, 360, 362, 364, 366, 367, 369, 372, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414,

416, 418, 419, 420, 421, 422, 423, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 437, 438, 439, 441, 446, 448, 457, 458, 462, 463, 464, 471, 530 Lie group, vii, viii, ix, x, xi, xii, xiii, 2, 3, 5, 10, 11, 13, 18, 26, 36, 39, 40, 44, 50, 52, 53, 61, 63, 67, 70, 71, 121, 124, 125, 137, 138, 139, 140, 141, 144, 145, 146, 147, 148, 149, 157, 163, 164, 165, 169, 170, 171, 172, 175, 176, 184, 185, 186, 187, 188, 191, 198, 201, 205, 209, 210, 274, 283, 284, 285, 286, 287, 289, 290, 291, 292, 293, 294, 296, 297, 299, 300, 301, 302, 303, 304, 305, 306, 326, 327, 328, 333, 334, 335, 337, 340, 343, 344, 345, 347, 348, 349, 351, 352, 353, 355, 357, 358, 359, 360, 361, 363, 364, 365, 366, 367, 369, 371, 372, 373, 376, 377, 381, 382, 383, 398, 408, 447, 448, 452, 454, 455, 456, 457, 459, 460, 461, 464, 465, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 482, 485, 486, 490, 495, 502, 530, 557, 563, 564, 565, 568, 592, 593 limitation, 10, 39 linear, vii, xi, xii, 3, 8, 10, 11, 15, 21, 43, 46, 52, 65, 67, 69, 70, 71, 72, 73, 80, 84, 86, 100, 124, 127, 128, 130, 131, 140, 142, 143, 148, 149, 151, 153, 170, 172, 173, 174, 185, 186, 191, 201, 202, 205, 209, 210, 215, 217, 219, 222, 251, 255, 256, 291, 301, 305, 325, 326, 327, 329, 331, 332, 333, 336, 340, 342, 343, 344, 346, 347, 352, 357, 358, 364, 366, 367, 385, 386, 387, 388, 389, 390, 391, 392, 394, 402, 403, 404, 405, 408, 413, 414, 415, 417, 420, 421, 422, 423, 425, 426, 427, 428, 430, 431, 432, 457, 459, 460, 463, 471, 472, 474, 555, 566, 598 linear dependence, 301, 388 linear function, 8, 10, 11, 86, 366 linear systems, 305 location, 11 locus, 46, 193, 195, 198 London, 2, 52, 55, 59, 87, 121, 267, 306, 323, 347, 348 luminal, 250 Lyapunov, 369, 370 Lyapunov exponent, 369, 370

M machinery, 23 magnetic, 50, 52, 53, 58, 181 magnetic field, 52, 53, 58, 181 magnetization, 181 manifold, vii, viii, ix, x, xi, xii, xiii, 3, 8, 9, 20, 21, 22, 25, 36, 40, 41, 42, 44, 48, 52, 53, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 84, 86, 89, 90, 96, 98, 121, 122, 139, 140, 142, 143, 144, 165, 169, 170, 171, 175, 176, 186, 189, 190, 198, 199, 201, 202, 204, 205, 209, 224, 238, 239, 241, 246, 247, 248, 249, 251, 256, 257, 259, 267, 272, 274, 279, 280, 281, 283, 284, 285, 286, 290, 291, 293, 296, 297, 299, 300,

Index 302, 303, 306, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 338, 349, 351, 352, 353, 361, 368, 369, 370, 371, 376, 377, 381, 402, 403, 404, 405, 406, 447, 448, 449, 450, 451, 452, 454, 456, 457, 458, 459, 460, 461, 462, 464, 465, 466, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 485, 486, 488, 490, 493, 495, 496, 497, 498, 499, 500, 501, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 516, 519, 522, 524, 526, 528, 530, 532, 534, 535, 536, 541, 542, 545, 547, 548, 550, 552, 554, 555, 556, 557, 558, 559, 560, 561, 563, 564, 565, 567, 568, 570, 571, 572, 576, 577, 580, 583, 584, 585, 586, 587, 589, 592, 593, 599 manifolds, ix, x, xiii, 3, 20, 21, 22, 25, 40, 41, 48, 52, 53, 61, 62, 63, 121, 122, 169, 170, 171, 175, 176, 186, 189, 190, 204, 209, 280, 281, 283, 284, 285, 291, 293, 296, 302, 306, 328, 334, 335, 349, 352, 369, 370, 371, 381, 448, 449, 450, 451, 458, 459, 461, 464, 476, 477, 479, 482, 483, 485, 486, 488, 490, 495, 497, 499, 500, 501, 508, 509, 510, 511, 512, 519, 534, 536, 541, 542, 554, 555, 556, 557, 558, 559, 560, 561, 563, 564, 565, 567, 571, 572, 584, 587, 589, 592, 599 manners, 321 mapping, ix, xi, 63, 65, 67, 72, 75, 77, 86, 94, 95, 98, 100, 170, 172, 187, 188, 189, 190, 191, 195, 198, 205, 208, 298, 307, 312, 316, 323, 449, 450, 452, 453, 454, 455, 456, 457, 458, 460, 461, 462, 463, 464, 466, 467, 468, 470, 471, 472, 473, 475, 476, 477, 478, 481, 487, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 504, 505, 506, 509, 510, 512, 513, 514, 515, 516, 517, 519, 522, 523, 525, 526, 527, 530, 533, 534, 535, 536, 537, 538, 540, 542, 543, 544, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 566, 567, 568, 569, 571, 573, 575, 576, 581, 582, 583, 587, 588, 590, 592, 593 marches, 383 Markov, xii, 113, 114, 116, 351, 354, 361, 376, 377, 379, 380, 382 Markov chain, 114, 116 Markov process, xii, 351, 354, 361, 376, 377, 380, 382 martingale, 112, 113, 114, 118, 372, 373, 375, 381 Massachusetts, 166, 348 massive particles, 36, 250 mathematics, vii, 1, 9, 10, 46, 53, 140, 187, 447, 486 Matrices, 44 matrix, 2, 3, 4, 24, 34, 35, 37, 38, 44, 93, 94, 103, 149, 155, 156, 170, 171, 172, 173, 176, 178, 179, 182, 183, 185, 206, 234, 252, 259, 260, 285, 294, 295, 296, 303, 308, 311, 312, 328, 338, 352, 353, 355, 356, 357, 358, 359, 360, 366, 367, 372, 374, 377, 411, 420, 451, 459, 460, 472, 473, 475, 476, 478, 488, 547 matrix algebra, 3, 35 Maxwell equations, 51, 227 measurement, 58

607

measures, 11, 13, 32, 42, 89, 90, 91, 92, 107, 108, 116, 120, 281, 353, 358, 360, 361, 362, 363, 364, 367, 373, 374, 382, 559, 560, 599 mechanical properties, 188 media, 124, 138 memory, 54, 58 mesons, 28, 55 metric, viii, x, xi, 10, 36, 49, 50, 51, 54, 89, 92, 93, 94, 96, 102, 104, 199, 200, 203, 210, 211, 221, 222, 238, 241, 242, 243, 244, 245, 247, 249, 251, 252, 253, 255, 257, 259, 260, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 283, 284, 285, 286, 287, 288, 290, 291, 293, 297, 298, 302, 307, 309, 326, 335, 337, 338, 340, 342, 343, 344, 345, 346, 360, 361, 369, 371, 380, 420, 448, 459, 462, 465, 471, 472, 475, 476, 477, 478, 480, 488, 499, 506, 508, 558, 565, 571, 575, 580, 584, 590 metric spaces, xi Mexico, 2, 123 military, 1 Minkowski spacetime, 3, 10, 36, 44, 202, 210 MIT, 166 mixing, 92, 110 models, 12, 28, 37, 175 modules, ix, 37, 139, 141, 151, 404 modulus, ix, 169, 170, 175, 176, 177, 179, 180, 182, 183, 184, 201, 254 momentum, vii, 3, 4, 5, 6, 7, 20, 21, 45, 231, 246, 251, 256, 274, 285, 286, 287, 288, 289, 291, 292, 293, 294, 295, 296, 297, 298, 303, 304, 327, 462 monoids, 495, 514, 587, 590 monotone, 582 morning, 1 morphogenesis, 46 Moscow, 323, 485, 558, 559, 560, 561, 563, 598, 599, 600 motion, x, xi, 2, 5, 7, 12, 39, 45, 46, 47, 48, 50, 52, 53, 54, 55, 58, 59, 63, 65, 71, 79, 80, 86, 89, 90, 91, 92, 102, 103, 104, 113, 114, 116, 117, 118, 120, 121, 122, 124, 178, 200, 246, 249, 250, 256, 303, 325, 326, 327, 329, 330, 332, 337, 338, 339, 344, 347, 348, 355, 356, 357, 365, 366, 376, 377, 380, 381, 382, 383, 558 movement, 47 multiples, 62 multiplication, 7, 13, 21, 29, 170, 181, 259, 327, 337, 463, 490, 491, 494, 503, 510, 511, 514, 515, 519, 521, 525, 529, 538, 541, 557, 571 multiplicity, 369 multiplier, 251, 294, 301

N National Academy of Sciences, 55, 483 natural, vii, 3, 23, 30, 46, 48, 54, 90, 91, 93, 97, 106, 176, 247, 251, 253, 257, 258, 268, 270, 272, 273, 287, 326, 331, 340, 346, 353, 354, 357, 361, 370,

608

Index

373, 374, 375, 377, 378, 380, 449, 451, 453, 455, 456, 466, 467, 471, 493, 496, 505, 506, 510, 512, 514, 517, 519, 520, 522, 524, 532, 533, 534, 537, 538, 539, 543, 544, 545, 547, 552, 563, 569, 571, 573, 576, 580, 581, 582 Netherlands, 59 neutrinos, 36, 39 New Jersey, 560, 561 New York, 55, 58, 87, 137, 138, 166, 184, 262, 264, 347, 348, 349, 397, 399, 481, 558, 560, 600 Newton, 252 Newtonian, 201 non-Abelian, 200, 402 nonlinear, ix, x, xii, 7, 21, 82, 86, 127, 138, 139, 140, 142, 156, 160, 166, 167, 169, 170, 175, 176, 177, 184, 186, 199, 200, 201, 202, 203, 209, 210, 213, 215, 218, 219, 221, 222, 223, 246, 255, 256, 263, 385, 386, 398, 422 nonlinear optics, 175 nonlinear wave equations, ix, 142 non-random, 368, 370, 371 normal, ix, 23, 91, 95, 99, 178, 187, 195, 278, 312, 333, 367, 374, 472, 474, 480, 501, 508, 510, 536, 557, 585, 587 normalization, 271 norms, 39, 568 Norway, 58 numerical analysis, 169

O oat, 301 objectivity, 44, 61, 62, 83 observations, x, 200, 409 obstruction, xii, 281, 401, 421 omentum, 256 one dimension, 420, 423, 434 operator, xi, 2, 5, 6, 10, 19, 39, 40, 42, 43, 51, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 80, 83, 84, 86, 114, 125, 127, 130, 131, 142, 149, 154, 176, 178, 220, 223, 225, 249, 252, 253, 259, 260, 268, 274, 276, 277, 278, 286, 287, 288, 292, 295, 296, 297, 298, 300, 303, 304, 307, 309, 310, 315, 316, 318, 320, 322, 327, 342, 354, 355, 359, 360, 362, 363, 365, 374, 375, 377, 380, 392, 393, 394, 404, 407, 408, 410, 539, 550, 566, 577, 593, 594, 595 optics, 175 orbit, ix, xi, 9, 126, 157, 187, 209, 212, 213, 284, 285, 286, 289, 291, 295, 296, 301, 303, 327, 328, 334, 335, 336, 338, 340, 343, 344, 346, 348, 353, 376, 377, 380, 406, 409, 450, 451, 452, 454, 455, 456, 460, 465, 466, 467, 469 ordinary differential equations, 80, 126, 140, 141, 144, 145, 147, 172, 185, 289, 398 organization, 45 orientation, 35, 49, 56, 188, 450, 499, 500, 504, 516 orthogonality, 3, 11, 25, 29, 54, 103, 219, 230, 388 oscillation, 34, 46, 47, 49

oscillations, 329, 348 oscillator, 30, 32, 33, 35, 254, 327, 339, 391, 394

P PACS, 56 pairing, 281, 300, 519, 550, 554 paper, 10, 28, 37, 50, 52, 54, 55, 61, 62, 73, 81, 86, 170, 188, 202, 203, 210, 255, 267, 280, 309, 397, 399, 448, 485, 487, 564, 565 parabolic, 94, 95, 96, 98, 100, 406 paradox, 30 parallelism, 464, 465 parameter, viii, 63, 65, 67, 68, 69, 72, 75, 85, 91, 92, 93, 104, 108, 113, 114, 123, 125, 127, 128, 133, 134, 137, 146, 158, 159, 160, 175, 210, 212, 215, 230, 231, 254, 279, 288, 289, 330, 333, 405, 409, 425, 427, 428, 437, 492, 501, 502, 513, 565, 581, 583, 584, 595, 596 Paris, 55, 121, 482, 558, 559, 560, 598, 600 partial differential equations, viii, ix, xii, 123, 124, 125, 126, 127, 128, 137, 138, 385, 386, 394, 411 particle creation, 53 particle mass, 254 particle physics, 200 particle-like, 253 particles, 17, 35, 36, 37, 42, 57, 202, 203, 247, 249, 250, 253, 254, 255, 257, 298, 329, 338 partition, 41, 209, 210, 214, 372, 379, 492, 493, 525, 526, 546, 551, 555 passenger, 258 passive, 53, 205 pedestrians, 5, 50 periodic, 46, 47, 49, 106, 598 permit, 49 personal, 12, 58 perturbation, 258 perturbations, 258, 404 pH, 211, 212 phase space, 28, 145, 289, 293, 297, 326, 327, 338, 342, 344, 347 phenomenology, 2, 46 Philadelphia, 263 photon, 39 photons, 37, 39, 53 physical fields, 200 physical interaction, x, 199 physical sciences, 138 physics, vii, xii, 1, 3, 12, 13, 17, 28, 50, 53, 55, 137, 138, 175, 200, 247, 249, 251, 257, 347, 387, 393, 398, 399, 401, 408, 447, 486, 559, 564, 600 planar, 91, 113, 196 plane waves, 45, 46 plasma, 175, 387 plasma physics, 175, 387 plastic, 129, 134 plasticity, viii, 123, 124, 128, 129, 135, 137, 138 play, 188, 223, 247, 290, 296

Index Poincare group, 50 Poisson, 117, 169, 287, 295, 296, 299, 304, 306, 326, 331, 332, 333, 337, 339, 345, 346, 355, 356, 357, 372, 373, 381, 559 Poland, 1 polar coordinates, 129, 135, 136, 376, 383 polarization, 45, 267, 272 polynomial, 16, 21, 174, 223, 275, 365, 393, 572, 575, 581, 587 polynomials, 326, 332, 394, 572, 575, 589 positive linear functionals, 11 positron, 2, 45 power, 524, 563 Prandtl, 129, 131, 132 predicate, 25 pressure, x, 129, 135, 283, 290, 291 private, 28, 55, 57 probability, 11, 104, 107, 108, 110, 111, 112, 113, 114, 120, 353, 354, 358, 362, 363, 364, 368, 369, 372, 374, 378, 379, 381, 382 probe, 55 production, viii, 123 prognosis, 2 program, 40, 57, 211, 279 projector, 54 propagation, 127 property, vii, viii, ix, 26, 85, 113, 114, 123, 140, 169, 175, 176, 184, 187, 190, 216, 222, 233, 272, 326, 354, 358, 368, 371, 379, 388, 392, 393, 394, 402, 406, 411, 480, 490, 493, 495, 501, 533, 538, 551, 570, 588, 589, 590 proposition, 101, 198, 421, 437, 455, 473, 479, 529, 553 prototype, x, 199, 203 pseudo, xiii, 5, 39, 335, 447, 462, 474, 476, 477, 548, 550

Q QCD, 57 QED, 196 quadrupole, 57 qualitative research, 157 quanta, 32, 33, 42, 52, 53 quantization, 5, 12, 24, 30, 52, 55, 200, 274, 349, 447, 482, 558 quantum, vii, viii, 2, 3, 4, 5, 10, 11, 12, 13, 20, 24, 25, 30, 34, 40, 42, 44, 49, 55, 59, 200, 201, 202, 247, 255, 326, 327, 348, 391, 394, 398, 559, 599 quantum chromodynamics, 200, 201 quantum field theory, 200 quantum gravity, 202 quantum groups, 10 quantum mechanics, 5, 20, 55, 59, 327, 398, 599 quantum state, 10, 247 quantum theory, 2, 4, 10, 559 quark, 3, 28, 35, 38 quarks, 27, 32, 33, 34, 55

609

quasilinear, 124, 127, 128, 129, 137, 138, 386 quasi-periodic, 296

R radiation, 2, 59 radical, 2, 402, 403, 406, 407, 408, 410, 412, 413, 423, 425, 432 radius, 92, 106, 110, 111, 129, 135, 136, 255, 269, 496, 506, 586, 588 random, 92, 113, 114, 355, 356, 357, 358, 366, 369, 371, 372, 373, 374, 375, 378, 379, 381, 382, 383 random matrices, 381, 383 random walk, 358, 366, 383 real forms, 19, 403 real numbers, 402 real time, 46 reality, 81 reasoning, 471 recall, 5, 6, 20, 47, 98, 99, 109, 112, 113, 114, 188, 189, 208, 388, 392, 393, 394, 411, 439 recalling, 33, 99 reciprocity, 50, 53 recombination, 258 recursion, xii, 16, 385, 386, 392, 393, 394 reduction, x, xi, xii, 185, 202, 210, 247, 250, 251, 253, 254, 255, 257, 281, 283, 284, 285, 286, 287, 294, 297, 302, 303, 385, 386, 390, 394, 397, 406, 410, 412, 432, 447, 483 reference system, 50, 58 reflection, 293, 497 regular, 97, 98, 109, 110, 127, 144, 287, 367, 378, 379, 450, 451, 453, 466, 481 relatives, 36 relativity, 2, 50, 52, 58 relevance, 27, 263 reproduction, viii, x, 123, 127, 138, 200, 203 research, vii, xii, 2, 49, 55, 87, 140, 157, 166, 401, 439 residues, 90, 91, 97, 98 resolution, 124, 137, 188, 522, 523, 525, 529, 544, 545 rings, 510, 514, 515, 519, 520, 521, 523, 541, 571, 572, 573, 582 robotics, ix, 187 rolling, 303 rotation axis, 65, 76 rotation transformation, 134 rotations, viii, 5, 26, 33, 61, 62, 63, 67, 70, 71, 81, 83, 86, 87, 88, 239, 295, 365, 383, 466 Royal Society, 55 Russia, 447, 485, 563 Russian, 137, 138, 306, 307, 323, 483

S SAE, 58

610

Index

sample, 369 sampling, 118 scalar, x, 4, 5, 9, 12, 18, 29, 34, 47, 50, 51, 52, 53, 59, 201, 221, 229, 232, 233, 242, 248, 251, 252, 253, 254, 256, 257, 267, 264, 280, 281, 286, 297, 387, 471, 487 scalar field, 50, 52, 248, 251, 252, 253, 254, 257 scaling, 103, 113, 250, 269, 386, 387, 409, 417 scattering, 55 school, 28 Schrodinger equation, ix, xii, 169 scientists, 55 s-compact, 556 S-curves, 133 search, 27, 337 searching, 1, 6 self-interactions, 249 semigroup, 353, 354, 361, 362, 363, 364, 367, 368, 369, 371, 372, 374, 376, 377, 379, 380 separation, 258, 496 series, 17, 50, 98, 143, 185, 202, 255, 284, 308, 309, 358, 359, 360, 563, 572, 573, 575, 577, 579, 580, 582, 583 Shanghai, 167 shape, 210, 464 shear, x, 129, 199, 210, 215, 219, 225, 255 Shell, 32, 58 shock, 127 shock waves, 127 shy, 1 SIGMA, 397, 399 sign, 34, 107, 250 similarity, 243 simulation, 186 sine, 34 Singapore, 559 singular, 90, 92, 97, 98, 109, 110, 112, 121, 128, 157, 197, 279, 280, 333, 341, 342, 390, 430, 450, 469, 539, 547, 548, 549 singularities, ix, 53, 127, 187, 188, 189, 193, 195, 198, 247, 447, 500 smoothness, xiii, 171, 352, 449, 465, 485, 486, 510, 515, 517, 521, 526, 564, 566, 568, 577, 588 Sobolev space, 515 social competence, 1 sociologists, 1 sociology, 1 software, 2 solitons, 175 solutions, viii, ix, 2, 6, 19, 21, 45, 55, 123, 124, 125, 126, 127, 128, 129, 130, 135, 137, 138, 139, 140, 142, 156, 158, 161, 162, 164, 167, 169, 170, 175, 180, 203, 252, 253, 254, 255, 257, 268, 305, 308, 312, 330, 337, 386, 393, 394, 396, 397, 411, 424, 433 sorting, 73 Southeast Asia, 559, 599 spacetime, x, 3, 10, 11, 12, 18, 20, 23, 25, 27, 28, 32, 33, 34, 35, 36, 39, 42, 43, 44, 45, 46, 47, 48, 49,

50, 51, 56, 199, 200, 201, 202, 203, 210, 217, 221, 222, 226, 227, 230, 231, 233, 235, 236, 241, 242, 244, 246, 247, 249, 250, 253, 255, 256, 257, 258, 408 spatial, 13, 46, 48, 63, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 81, 86 spatial representations, 63, 79 spectrum, 49, 57, 255, 420, 422, 428, 430, 431, 573 speed, 50, 58, 91, 105, 106, 250 speed of light, 50, 58 spheres, viii, 89, 90, 92, 485, 486, 487, 494, 501 spin, vii, 2, 3, 4, 5, 7, 8, 19, 22, 28, 31, 39, 45, 54, 78, 181, 186, 201, 231, 237, 239, 242, 246, 251, 256, 262, 486 spin-1, 35 spinning particle, 202 spinor fields, 202, 243 SRT, 59 stability, ix, 169, 175, 180, 182, 184, 187, 190, 193, 195, 196, 198, 281, 284, 301, 304, 382, 402, 406, 407, 412, 413, 418, 419, 483 stabilizers, 274, 277, 279 standard model, 4, 5, 11, 27, 35, 37, 39, 44, 48, 50, 200 statistics, 28, 381 stochastic, 102, 104, 121, 354, 355, 356, 357, 366, 368, 369, 370, 371, 372, 373, 375, 376, 381, 382, 560 stochastic processes, 560 stratification, 451, 456, 470, 474, 476, 478 streams, 50 strength, x, 39, 52, 200, 222, 223, 240, 256 stress, viii, 89, 92, 128, 129, 207, 215, 241, 248, 249, 253, 256, 261, 396, 460 string theory, 447 stroke, 44 strong force, 47, 51, 200 strong interaction, 1, 27, 48, 49, 55, 56 STRUCTURE, 139, 199, 459, 460, 485 subgroups, xiii, 17, 35, 43, 93, 188, 209, 279, 291, 348, 366, 367, 450, 469, 507, 509, 560, 563, 564, 565, 580, 581, 584 substitution, 117, 213, 216, 227, 233, 588, 590 summer, 28 Sun, 170, 172, 174, 176, 178, 180, 182, 184, 186 superconductivity, 305 superposition, 254 supersymmetry, 36, 39 supply, 496 suppression, 27, 28 switching, xii, 401 symbols, 13, 242, 249, 309, 312 symmetry, vii, viii, x, xi, xii, 1, 21, 27, 32, 33, 34, 36, 37, 38, 39, 42, 50, 51, 55, 58, 123, 124, 125, 127, 128, 140, 166, 172, 176, 178, 182, 184, 194, 199, 200, 201, 202, 208, 210, 224, 226, 231, 239, 248, 256, 283, 284, 285, 288, 289, 290, 291, 298, 299, 300, 302, 303, 306, 385, 386, 387, 388, 389,

Index 390, 391, 392, 394, 395, 396, 397, 398, 401, 404, 408 symplectic, 169, 185, 200, 221, 222, 247, 257, 268, 272, 273, 274, 275, 279, 281, 296, 306, 325, 326, 327, 328, 329, 330, 331, 333, 334, 335, 336, 338, 339, 340, 342, 343, 345, 347, 447, 460, 462, 464, 482, 483 synchronous, 46, 48 systems, xi, 19, 123, 125, 166, 186, 238, 284, 285, 286, 288, 292, 294, 302, 304, 305, 306, 312, 325, 326, 327, 335, 341, 347, 348, 369, 402

T Taiwan, 351 teachers, 1 temporal, 46, 49 tensor field, 50, 453, 460, 461, 462 tetrad, x, 199, 200, 201, 202, 203, 220, 221, 222, 226, 233, 235, 236, 241, 255 Texas, 263 textbooks, 63, 65 theory, vii, viii, ix, x, xi, xii, 2, 5, 8, 9, 28, 35, 36, 37, 50, 53, 58, 121, 122, 123, 124, 129, 137, 139, 140, 141, 142, 144, 157, 171, 175, 187, 188, 195, 199, 200, 201, 202, 203, 221, 247, 249, 253, 254, 255, 256, 257, 264, 284, 306, 308, 313, 325, 326, 327, 337, 339, 347, 348, 351, 366, 369, 394, 396, 401, 402, 408, 447, 486, 552, 558, 559, 560, 563, 564, 577, 599, 600 third order, viii, 61, 62, 81, 182, 184 three-dimensional, 65, 263 time, x, 2, 3, 8, 10, 11, 27, 28, 30, 34, 36, 45, 46, 47, 48, 49, 51, 52, 62, 73, 74, 76, 77, 78, 80, 81, 82, 83, 86, 91, 107, 108, 113, 114, 117, 170, 176, 178, 180, 182, 195, 200, 205, 209, 247, 250, 252, 257, 258, 285, 289, 294, 303, 327, 353, 354, 365, 366, 368, 369, 371, 372, 373, 374, 376, 378, 380, 391, 486, 487, 507, 509, 512, 523, 528, 541, 553, 565, 571, 575, 577, 592 title, vii, 4, 10 Toda model, 55 Tokyo, 382 topological, x, xiii, 10, 11, 199, 203, 224, 256, 267, 291, 351, 352, 353, 354, 447, 449, 451, 454, 455, 456, 486, 489, 490, 491, 495, 496, 497, 500, 510, 511, 514, 515, 517, 519, 520, 521, 524, 525, 527, 529, 530, 532, 534, 537, 540, 541, 551, 554, 557, 558, 559, 563, 564, 565, 567, 568, 569, 570, 571, 573, 574, 575, 576, 577, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595 topological invariants, x, 199, 203, 224, 256 topology, 188, 190, 257, 258, 271, 279, 285, 286, 302, 303, 353, 354, 378, 448, 451, 456, 464, 465, 469, 470, 471, 472, 473, 474, 477, 479, 481, 483, 496, 503, 515, 534, 537, 542, 543, 554, 557, 558, 559, 561, 564, 565, 567, 568, 570, 573, 574, 575, 576, 585, 586, 588, 589, 590, 591, 592, 598

611

torus, ix, 277, 294, 296, 302, 332, 333, 478, 481, 497, 498, 499 total energy, 256 traction, 524 tradition, 15 trajectory, 12, 185, 286, 287, 293, 294, 304 trans, 507 transformation, vii, viii, 36, 50, 51, 52, 53, 61, 62, 63, 68, 71, 72, 73, 74, 78, 82, 83, 84, 85, 86, 123, 124, 125, 126, 129, 131, 137, 140, 145, 147, 156, 157, 158, 159, 160, 162, 164, 171, 203, 204, 208, 209, 210, 212, 213, 214, 215, 216, 221, 222, 227, 230, 231, 232, 234, 235, 236, 237, 238, 242, 243, 255, 258, 294, 371, 408, 448, 450, 460, 466, 472, 474, 478, 480, 482, 532, 533, 552, 555, 564 transformation matrix, 234 transformations, viii, ix, x, 17, 24, 25, 50, 51, 52, 65, 78, 123, 124, 125, 126, 127, 128, 130, 131, 134, 145, 199, 202, 203, 205, 208, 209, 210, 214, 215, 230, 231, 232, 233, 234, 235, 236, 239, 243, 250, 251, 256, 259, 269, 305, 333, 371, 390, 391, 415, 416, 417, 422, 423, 424, 425, 429, 430, 447, 448, 450, 459, 460, 463, 465, 466, 467, 471, 472, 473, 474, 476, 480, 481, 482, 560 transition, xii, 33, 35, 114, 204, 205, 208, 236, 354, 361, 362, 376, 377, 379, 401, 489, 496, 502, 512, 526, 546, 569 transition rate, 35 transitions, 34 translation, 68, 83, 131, 138, 201, 230, 259, 352, 354, 391, 457, 474, 478, 480 translational, 201, 210, 214, 215, 218, 219, 221, 222, 223, 231, 255, 256 transparent, 376 transport, 33, 485, 486, 490, 491, 493, 494, 495, 506, 507, 508, 509, 511, 513, 516, 517, 522, 535, 537, 542, 554 transpose, 3, 47, 48, 353, 358 travel, 258 trees, 402 trend, 50 triangulation, 548 triggers, 46 two-dimensional, 57, 122, 160, 161, 178

U uncertainty, 5, 48, 257, 258 unconditioned, 117, 118 unification, 2, 35, 202, 203, 227, 247 uniform, 92, 111, 112, 120, 129, 277, 315, 317, 319, 320, 322, 359, 366, 488, 490, 491, 492, 493, 508, 552, 554, 555, 557, 564, 567, 568, 569, 570, 573, 576, 577, 587, 592 unions, 568 universe, 10, 49, 258, 598 USSR, 347

612

Index

V vacuum, 49, 255 validation, 309 validity, 257 values, viii, 61, 62, 82, 84, 85, 86, 134, 170, 208, 251, 254, 278, 284, 287, 289, 304, 375, 404, 409, 413, 421, 423, 424, 425, 426, 427, 429, 430, 458, 471, 506, 512, 523, 524, 530, 539, 542, 555, 567, 572, 575, 591, 598, 599 variable, 114, 133, 134, 200, 248, 296, 306, 332, 356, 375, 378, 386, 388, 394, 395, 512, 556, 563 variables, 62, 91, 92, 104, 108, 114, 124, 125, 128, 129, 131, 132, 133, 134, 135, 136, 145, 155, 163, 188, 198, 200, 228, 229, 231, 233, 253, 270, 297, 310, 316, 325, 333, 337, 356, 366, 373, 374, 388, 389, 395, 396, 486, 556, 559, 560, 565, 566, 579, 594, 595, 596, 598 variance, 91, 102, 104, 108 variation, 80, 112, 114, 137, 215, 225, 226, 227, 228, 229, 230, 231, 232, 243, 244, 245, 246, 358, 359, 412 vector, viii, ix, x, xi, 4, 7, 12, 13, 17, 24, 27, 28, 37, 46, 47, 51, 52, 53, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 93, 94, 96, 97, 106, 139, 140, 141, 142, 143, 144, 145, 148, 149, 150, 152, 156, 157, 165, 166, 171, 175, 176, 178, 179, 181, 183, 190, 192, 196, 199, 201, 205, 207, 215, 220, 221, 227, 234, 236, 244, 250, 259, 260, 261, 268, 271, 277, 278, 281, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 307, 308, 309, 310, 311, 312, 313,

315, 316, 317, 318, 319, 320, 322, 323, 326, 329, 330, 331, 334, 335, 336, 337, 339, 340, 342, 343, 352, 354, 355, 359, 371, 374, 375, 380, 392, 393, 394, 395, 396, 397, 403, 453, 457, 458, 462, 463, 464, 470, 471, 473, 477, 487, 490, 515, 524, 525, 529, 530, 532, 555, 560, 565, 567, 593, 594, 595, 597, 600 velocity, x, 49, 50, 52, 62, 63, 64, 74, 76, 77, 78, 82, 83, 157, 158, 159, 178, 250, 283, 284, 285, 287, 290, 291, 292, 293, 294, 301, 302, 303, 330, 340 Victoria, 186 visible, 365 voice, 39 vortex, x, xi, 283, 284, 285, 291, 292, 293, 295, 296, 302, 306 vortices, 285

W war, 104 wave equations, 2, 55 wave number, 46 wave propagation, 185 weakness, 39 Weinberg, 37, 264 workspace, 188 writing, 1, 29, 231, 395, 396

Y Yang-Mills, x, 200, 201, 202, 256, 402 yield, 90, 124, 128, 456, 471