Integrable Systems on Lie Algebras and Symmetric Spaces (Advanced Studies in Contemporary Mathematics)

INTEGRABLE SYSTEMS ON LIE ALGEBRAS AND SYMMETRIC SPACES By A.T. Fomenko and V.V. Trofimov Faculty of Mechanics and Mathematics, Moscow State University, Moscow, USSR

Translated from the Russian by A. Karaulov, p.o. Rayfield and A. Weisman

5J/UwT~ fYcpcp. ~~UU

I

Gordon and Breach New York

London

Paris

Sci~ce

Montreux

U(

1'1 r0

Publishers Tokyo

Melbourne

© 1988 by OPA (Amsterdam) B.V. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A. Gordon and Breach Science Publishers Post Office Box 786 Cooper Station New York, New York 10276 United States of America Post Office Box 197 London WC2E 9PX England 58, rue Lhomond 75005 Paris France Post Office Box 161 1820 Montreux 2 Switzerland 3-14-9 Okubo Shinjuku-ku, Tokyo Japan Private Bag 8 Camberwell, Victoria 3124 Australia Library of Congress Cataloging-in-Publication Data Fomenko, A. T. Integrable systems on Lie algebras and symmetric spaces. (Advanced studies in contemporary mathematics, ISSN 0884-0016 ; v. 2) Translation of: Integriruemye sistemy na algebrakh Li i simmetricheskikh prostranstvakh. Bibliography: p. Includes index. 1. Hamiltonian systems. 2. Lie algebras. 3. Symmetric spaces. I. Trofimov, V. V., 1952II. Title. III. Series. QA614.83.F6613 1987 512'.55 87-26798 IS BN 2-88124-170-0 No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed in Great Britain by Bell and Bain Ltd, Glasgow.

Contents

Introduction

Xl

1. Symplectic Geometry and the Integration of Hamiltonian

Systems 1.

1.1. 1.2. 1.3.

1.4. 1.5.

Symplectic manifolds Symplectic Structure and its Canonical Representation. Skew-Symmetric Gradient The Geometric Properties of Symplectic Structures Hamiltonian Vector Fields The Poisson Bracket and Hamiltonian Field Integrals Degenerate Poisson Brackets

1 1 1 4 8 11

15

Symplectic Geometries and Lie Groups 2. 2.1. Summary of the Necessary Results on Lie Groups and Lie

17

Algebras 2.2. Orbits of the Coadjoint Representation and the Canonical Symplectic Structure 2.3. Differential Equations for Invariants and Semi-Invariants of the Coadjoint Representation

17

3.

3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

Liouville's Theorem Commutative Integration of Hamiltonian Systems Non-Commutative Lie Algebras of Integrals Theorem of Non-Commutative Integration Reduction of Hamiltonian Systems with Non-Commutative Symmetries Orbits of the Coadjoint Representation as Symplectic Manifolds The Connection between Commutative and NonCommutative Liouville Integration

22 27 30 30 32 34 36 46 47

vi

CONTENTS

4.

Algebraicization of Hamiltonian Systems on Lie Group Orbits 4.1. The Realization of Hamiltonian Systems on the Orbits of the Coadjoint Representation 4.2. Examples of Algebraicized Systems 5.

Complete Commutative Sets of Functions on Symplectic A1anifolds

2. Sectional Operators and Their Applications

6. 7.

7.1. 7.2. 7.3. 7.4.

9.

Sectional Operators, Finite-Dimensional Representations, Dynamic Systems on the Orbits of Representation Examples of Sectional Operators Equations of Motion of a Multi-Dimensional Rigid Body with a Fixed Point and Their Analogs on Semi-Simple Lie Algebras. The Complex Semi-Simple Series Hamiltonian Systems of the Compact and the Normal Series Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Ideal Fluid Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible Ideally Conductive Fluid

~=

I.B =

(~

x

~)/~

63

67 71

71 75

79 89

100

Construction of the Form Fe and the Flow ~ in the Case of a Symmetric Space The Case of the Group Spaces of Type II)

52 58

67

3. Sectional Operators on Symmetric Spaces 8.

52

100

(Symmetric

10. The Case of Type I, Ill, IV Symmetric Spaces 10.1 Symmetric Spaces of Maximal Rank 10.2. The Symmetric Space S·-l = SO(n)/SO(n - 1) (The Real Case) 10.3. Hamiltonian Flows XQ, Symplectic Structures Fe and the Equations of Motion of Analogs of a Multi-Dimensional Rigid Body 10.4. The Symmetric Space S·-l = SO(n)/SO(n - 1) (The Complex Case) 10.5. Examples of Flows Xg on S·-l (The Complex Case)

105

107 107 111 120 121 131

CONTENTS

4. Methods of Construction of Functions in Involution on Orbits of Coadjoint Representation of Lie Groups 11. Method of Argument Translation 11.1. Translations of Invariants of Coadjoint Representation 11.2. Representations of Lie Group s in the Space of the Functions on the Orbits and Corresponding Involutive Sets of Functions 12.

Methods of Construction of Commutative Sets of Functions Using Chains of Subalgebras

Vll

136 136 136 138

143

Method of Tensor Extensions of Lie Algebras Basic Definitions and Results The Proof of the General Theorem The Application of the Algorithm (Ill) to the Construction of S-Representations 13.4. Algebras with Poincare Duality

147 147 151

14. Similar Functions 14.1. Partial Invariants 14.2. Involutivity of Similar Functions 15. Contractions of Lie Algebras 15.1 Restriction Theorem 15.2. Contractions of Z2 -Grad ed Lie Algebras

167 167 168

13. 13.1. 13.2. 13.3.

5. Complete Integrability of Hamiltonian Systems on Orbits of Lie Algebras Complete Integrability of the Equations of Motion of a Multi-Dimensional Rigid Body with a Fixed Point in the Absence of Gravity 16.1. Integrals of Euler Equations on Semi-Simple Lie Algebras 16.2. Examples for Lie Algebras of so(3) and so(4) 16.3. Cases of Complete Integrability of Euler's Equations on Semi-Simple Lie Algebras 17. Cases of Complete Integrability of the Equations of Inertial Motion of a Multi-Dimen~ional Rigid Body in an Ideal Fluid 18. The Case of Complete Integrability of the Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible, Ideally Conductive Fluid 18.1. Complete Integrability of the Euler Equations on Extensions Q(G) of Semi-Simple Lie Algebras

160 162

171

171 174

179

16.

179 179 185 189

194

198 198

viii

CONTENTS

18.2. Complete Integrability of a Geodesic Flow on T*O(fj) 18.3. Extensions of O(G) for Low-Dimensional Lie Algebras

Some Integrable Hamiltonian Flows with Semi-Simple Group of Symmetries 19.1. Integrable Systems in the 'Compact Case' 19.2. Integrable Systems in the Non-Compact Case. MultiDimensional Lagrange's Case 19.3. Functional Independence of Integrals

203 204

19.

The Integrability of Certain Hamiltonian Systems on Lie Algebras 20.1. Completely Involutive Sets of Functions on Singular Orbits in su(m) 20.2. Completely Involutive Sets of Functions on Affine Lie Algebras

205 205 208 212

20.

Completely Involutive Sets of Functions on Extensions of Abelian Lie Algebras 21.1. The Main Construction 21.2. Lie Algebras of Triangular Matrices

214 215 219

21.

22. 22.1. 22.2. 22.3.

22.4.

Integrability of Euler's Equations on Singular Orbits of Semi-Simple Lie Algebras Integrability of Euler's Equations on Orbits 0 Intersecting the Set tH R, tEe Integrability of Euler's Equations x = [x, CPabD(X)] for Singular a Integrability of Euler's Equations x = [X,CPabD(X)] on the Subalgebra Gn Fixed Under the Canonical Involutive Automorphism (J:G-.G for Singular Elements aEG Integrability of Euler's Equations for an n-Dimensional Rigid Body

Completely Integrable Hamiltonian Systems on Symmetric Spaces 23.1. Integrable Metrics dS;bD on Symmetric Spaces 23.2. The Metrics dS;b on a Sphere S" 23.3. Applications to Non-Commutative Integrability

224 224 232

236 236 244

247 253

23.

24.

Morse's Theory of Completely Integrable Hamiltonian Systems. TopOlogy of the Surfaces of Constant Energy Level of Hamiltonian Systems, Obstacles to Integrability and Classification of the Rearrangements of the General Position of Liouville Tori in the Neighborhood of a Bifurcation Diagram

254 254 257 263

266

CONT ENTS

24.1. The Four-Dimensional Case 24.2. The General Case

Bibliography Index

IX

266 270 281 293

1 1 1 1 1 1 1 1

Introduction

There are at present quite a few integrability problems known in dynamics. The solution of these problems is based on the existence of n independent first integrals in involution, n being the dimension of the configuration space (which is equal to the number of degrees of freedom) of a mechanical system. Henceforth, these sets offunctions will be referred to as complete involutive sets. In these cases, according to Liouville's theorem, the equations of movement are integrated in quadratures. We know that the existence of a complete involutive set of first integrals implies a consequent qualitative picture of the behaviour of trajectories in 2n-dimensional phase space. Every phase space can be stratified by congruent surfaces of the level of first integrals into closed n-dimensional invariant manifolds. If these manifolds are compact and connected, then they are n-dimensional tori and the motion through them is quasiperiodic. This book sets out some new methods for integrating Hamilton's canonical equations. Common to all these methods is one overall idea: the realization of canonical equations in Lie algebras or symmetric spaces. Basically, the book sets out new results obtained by the authors and by participants in the scientific research seminar Contemporary Geometry Methods, run at Moscow University under the direction of A.T. Fomenko. For the reader's convenience, classical information on Hamiltonian systems is included in the first chapter. In classical mechanics the most widespread method for integrating Hamilton's equations is the Hamilton-Jacobi approach. We know that Hamilton-Jacobi equations enable us to solve the classic problem of finding geodesics in a triaxial ellipsoid. In contrast to the Hamilton-Jacobi method, instead of a non-linear equation in the partial derivatives (as/at) + H(t,q,(as/oq)) = 0, in applying the methods which are to be described in this book, we must solve the system of linear partial differential equations of the first order Lk.j C~jxk(aF/ax) = O. At the same time, as is well known, there exists an effective algorithm for solving such systems. This leaves us with a purely algebraic procedure for finding the first integrals of canonical equations: if the FV solution of the system V is Lk.jC~jXk(aF/ax) = 0, then F(x + Aa) is the first integral for any AE IR, while all such integrals are found to be in involution. 'three fundamental themes are examined in this book. First and foremost we are concerned with constructing the algebraic embeddings in Lie algebras of the Xl

Xli

INTRODUCTION

Hamiltonian systems which are so well known in mechanics. We shall state that these systems allow an algebraic representation. It has been shown that the general construction of a sectional operator for an arbitrary linear representation of a Lie group, permits us to realize many physically interesting mechanical systems on the orbits of the representations. Within the framework of the theory of sectional operators a construction is offered for symplectic forms (non-invariant under the action of the group) on symmetrical spaces, with respect to which the systems constructed are Hamiltonian. The second theme of the book concerns effective methods for constructing complete sets of functions in involution on orbits of coadjoint representations of Lie groups. The third and final theme of the book is the proof ofthe full integrability, after Liouville, of a fairly wide range of many parameter families of Hamiltonian systems that allow algebraic representation in the sense mentioned. One important fact is that these systems happen to include some interesting mechanical systems, e.g. the equation of motion of a multi-dimensional rigid body with a fixed point in the absence of gravity, the inertial motion of a rigid body in a fluid, as well as certain finite dimensional approximations of the equations of magnetic hydrodynamics. The basic difficulty which arises here is the proof of the functional independence of the first integrals.

1

Symplectic geometry and the integra tion of Hamiltonian systems

1. SYMPLECTIC MANIFOLDS 1.1. Symplectic structure and its canonical representation. Skewsymmetric gradient

We shall begin by studying an important class of smooth manifoldsthe so-called symplectic manifolds. They appear in many applied problems, for example in problems of classical mechanics, and it is therefore absolutely essential that they should be studied in order to solve many specific problems. One of the ways of introducing additional structure on a smooth manifold is to define a skew-symmetric scalar product which depends smoothly on the point. This leads us to symplectic manifolds, whose geometry is substantially different from that, for example, of Riemann spaces. Since the skew-symmetric scalar product (in the tangent spaces) is -defined by a second-degree skewsymmetric tensor it is sufficient to define an exterior differential form of the second degree. 1.1 A smooth even-dimensional manifold M 2n is called symplectic if it has defined on it the external differential second-degree form W = Li<jwijdx i /\ dx i , so that (1) this form is non-degenerate, i.e. the matrix of its coefficients II wiix) II is non-degenerate at each point, (2) this form is closed, i.e. dw = 0, where d is the operation of exterior differentiation (see [24], [120]). This form W is sometimes called a symplectic structure on a manifold. DEFINITION

It is clear that form W defines in a tangent space ~M2" a nondegenerate skew-symmetric scalar product w(a, b) = Li
2

A. T. FOMENKO AND

v. V. TROFIMOV

from algebra, there exists a transformation of coordinates in the tangent space T"oM, generated by a regular transformation of coordinates Xl, ... , x2n in the neighborhood of the point x o, so that the matrix Ilwij(x o) II takes the canonical form:

o

o

D

o This scalar product evidently defines a canonical identification of the tangent space TxM with the cotangent space Tx* M (on this operation see [24J, [120J for more detail). Let us bear in mind that the dual space Tx* M consists of covectors-linear forms in tangent space. The canonical identification of tangent and cotangent spaces, generated by the form w, enables us to define an important operation which is an analog ofthe operation for constructing a gradient vector field grad f on a manifold, provided the symmetric scalar product is defined on M (i.e. a Riemannian metric, see [24], [166]). 1.2 Let f be a smooth function on M and w be a symplectic structure. The skew-symmetric gradient s grad f of the function f (the "skew gradient") is the name given to the vector field on M, uniquely defined by the relation w(v, s grad f) = v(f), where v is an arbitrary smooth vector field on M and v(f) is the value of v considered as a differential operator on the function f DEFINITION

In other words, the first task is to examine the covector

ofI ' ... , oxen Of) E T,,* M ( ax and then, relying on the canonical identification T"M and T,,* M arising from the form w, to construct the corresponding vector field for this covector field: the resulting vector field will be the skew-symmetric gradient. If we had used Riemann's metric for this identification, we would have obtained the vector field grad f The simple nature of the s grad f construction is a consequence of the non-degeneracy of w. Usually local coordinates on a symplectic manifold are denoted by PI' ... ,Pn' q I ' . . . , qn where these coordinates are chosen in such a way

3

INTEGRABLE SYSTEMS ON LIE ALGEBRA

that at a fixed point Xo the matrix Ilwij(xo)11 is written as:

(~E ~).

where E is the (n x n) unit matrix. But if the coordinates are arranged in the following order Pl' ql' P2, q2 • ... , Pn' qn, then the matrix may be written in block-diagonal form, as shown above. Given this special choice oflocal coordinates Pl' ...• Pn' q l ' . . . , qn' the vector field s grad f at the point Xo can be written particularly simply. In fact, since

then

(:f ,... ,;f)EYxoM' s grad f = (:f ,... ,;f ,_ :f ,... ,_;1) . Pl

qn

ql

qn

Pl

Pn

In the coordinates Pl' ql' ... , Pn' qn we have: s grad f(xo) =

(:1 ,_:1 ,... ,;1 ,_;1). ql

Pl

qn

Pn

Obviously, the Riemannian metric can also be reduced at a point to a canonical (diagonal) form by choosing appropriate local coordinates. In this sense both structures, Riemannian and symplectic, are similar. They do, however, exhibit one serious difference which becomes evident the moment we proceed to examine the neighborhood of the point Xo' As is known (see [24], [48], [166]), Riemann's metric cannot generally be reduced to diagonal form in an entire neighborhood by means of coordinate transformations, as a non-zero Riemann curvature tensor can preventthis. A symplectic struCture, on the other hand, can always be reduced to canouical form by transforming coordinates in a small neighborhood of the point (the size of the neighborhood being defined by the properties of the form). Let W be a symplectic structure on M2n. Then for any point Xo EM there exists an open neighborhood with the local coordinates Pl, ... , Pn' ql' ... , Q2' so that in these coordinates the form W can be written in the simplest canonical way: W = Li'=l dpi 1\ dqj. PROPOSITION

].1

For proof see, for example [1], [46], [117], [120]. This theorem proves to be useful in many calculations connected with symplectic structures. Local coordinates, whose existence is affirmed in the theorem and which reduce the form W to canonical form, are sometimes called symplectic coordinates. It is clear that by covering the manifold M with

4

A. T. FOMENKO AND V. V. TROFIMOV

open neighborhoods of the form mentioned in Proposition 1.1 we shall obtain an atlas for M (see the definition of an atlas, for example in [24], [120]) an atlas that is also sometimes called symplectic. The simplest example of a symplectic manifold is the Euclidean space [R2"(P1' ... ,qn)' provided with the form w = dP1 A dq1 + .,. + dp" A dqn. 1.2. The geometric properties of symplectic structures

Unlike a Riemannian structure, a symplectic structure cannot be given on all smooth manifolds. As we noted in Definition 1.1, the most elementary limitation is that any symplectic structure must be evendimensional, since any skew-symmetric matrix of odd order is degenerate. A more substantial limitation is that any symplectic manifold is orientable. To prove this statement it is sufficient to define on the symplectic manifold M2k of dimension 2k a nowhere zero differential form of degree 2k. Let w be a symplectic structure on M 2k , then n = w A ... A •.. A W (the product is taken k times) gives a form of degree 2k on M2k. We shall show that n # 0. Indeed, in every tangent plane TxM2k of M2k there can be found a system of coordinates Xl, ... , Xk, y1, ... , yk such that W = Xl A y1 + ... + Xk A yk. Then, clearly,

.n =( _.1)[k(k-1»)/2k!xl _ 0'"

A Xk A)11 A'"

A

l

# 0..

We should note that a symplectic structure does not exist for all evendimensional orientable manifolds. PROPOSITION 1.2 There is no symplectic structure on the evendimensional sphere s2n in the case of n > 1.

Proof Let there be on S2" (n > 1) the differential form w, which defines a symplectic structure on S2". Then dw = and since the space of cohomologies H2(SP) = where P ~ 3, then w = dex for a certain one dimensional differential form ex. It is obvious that w A ... A W = d(a A dlY. A ••. A dlY.) (the external product is taken n times). On the other hand, as we saw above, n = W A .•• A W is not zero at any point on the sphere S2n. Therefore n = f· vol where f is some smooth function and vol is the volume form on s2n. Thus, using Stokes' theorem (see, for example, [120]), we have SS2. n = Sd(ex A dex A'" A dex) = 0, but ISs2. f vall ~ m IS 2. vol = mv(S2") # 0, where v(S2n) is the volume of sphere S2" and m is some constant, m > 0. This proves the proposition.

°

°

INTEGRABLE SYSTEMS ON LIE ALGEBRA

5

REMARK It is clear that the proof of Proposition 1.2 works for any compact manifold M such that H2(M, IR) = O. For example, on a compact semi-simple Lie group there are no symplectic structures, since H2(G>, IR) = 0 for the compact semi-simple Lie group G>. We shall now pass on to examples of symplectic manifolds which arise in various geometrical and mechanical constructions. The first source of symplectic manifolds is smooth orientable closed Riemann surfaces, i.e. spheres with handles. Here we may take as a symplectic structure the standard two-dimensional Riemann volume form which is a closed non-degenerate exterior two-form. If the surface is given parametrically r = r(u, v), then the form of the volume has the aspect OJ = JEG - F2 du A dv, where E = (r u , ru )' F = (ru, r v), G = (rv' rv) and ru, rv denote partial derivatives of the radius vector r along u and v respectively (see, for example [24]). Any sphere with handles allows an explicit parametric definition, for example, the equation

i= j) where B is sufficiently small, describes a sphere with eight handles. An analogous equation can be written for a sphere with n

ai

i=

aj (i

handles. The second source for obtaining symplectic structures is from cotangent bundles. As a rule, the position space of a mechanical system is a smooth manifold M. This is what we call a mechanical system's configuration space. From the mathematical point of view, phase space coincides with the cotangent bundle T* M of the manifold M (see [120]). Points of the cotangent bundle T* M are pairs (x, ~), where x E M, ~ E Tx* M, i.e. ~ is a covector at the point x. It is not hard to verify that T* M is a smooth 2n-dimensional manifold, n = dim M. The natural projection p: T* M -+ M is defined thus: p(x,~) = x. It is clear that T*M is the total space of a vector bundle, its base being the initial manifold M, while the fiber p -1 (x) over the point x is the cotangent space Tx* M. We shall define a symplectic structure (see for example [120]) on manifold T* M. To do so we shall first construct on T* M the smooth I-form oP), Let a E T,( T* M) be a tangent vector of the cotangent bundle T* M at the point y E 1'x* M, see Figure 1. The differential mapping p: T* M -+ M maps the vector a into vector p*a, tangent to the manifold M at point

6

A. T. FOMENKO AND V. V. TROFIMOV

Fig.!.

x = p(y) = p(X, ~). Now we shall define the differential I-form on the space T* M in the following way: uP)(a) = y(p*a), i.e. the value of the form is equal to that of the covector y on the vector p*a. Finally, for the 2-form we are seeking let us take the external differential form uP), i.e. w = dw(l). The form we have constructed is closed and nondegenerate, i.e. T* M is transformed into a symplectic manifold. We now adduce a coordinate description of the symplectic structure created. Let U be a coordinate neighborhood in M. Using the mechanical interpretation, we denote the coordinates in U by ql , ... ,q", n = dim M. Let us examine 0 c T* M; these are covectors whose point of apposition is in U. We can examine the basis fields O/oql, . .. ,O/oq". Let the covector values on these fields be PI' ... ,p": we may take (ql , ... , q", PI' ... ,Pn) as the coordinates in T* M in the neighborhood w, the 2form constructed in these coordinates, has the classical form:

o.

w = dpi /\ dql + ... + dplI /\ dqn

(see [120]). As an immediate corollary we find that T* M is an orientable manifold for any M. Not all symplectic manifolds can be obtained by this method. (a) T* M is not a compact manifold, and (b) the form w which gives a symplectic structure is exact, i.e. w = det. for a certain I-form IX. The third source of symplectic manifolds are Kiihler manifolds. Let M211 be a complex manifold (see [24], [48]), on which a Hermitian scalar product (~, 1]) is given. Let us examine w(~, 1]) = Im(~, 1]). It is apparent that w is a skew-symmetric non-degenerate 2-form. In order to obtain a

INTEGRABLE SYSTEMS ON LIE ALGEBRA

7

symplectic structure on M 2n, it is essential to have the equality dO) = o. This requirement is not met in an arbitrary complex manifold with a Hermitian metric. 1.3 A complex manifold, provided with a Hermitian metric, is called a Kahler manifold if the imaginary part 0) of the scalar product (~, t,) is a closed differential form (dO) = 0).

DEFINITION

Thus, all Kahler manifolds are symplectic. The converse, however, is not generally true (see [188J, [182J). One classic example of a Kahler manifold can be found in the complex projective space cpn. We have the natural holomorphic mapping n: c n+ 1 \0 -+ cpn. We may examine cn + 1 \0 the covariant 2-tensor

where

Zo, ...

,Zn are the standard coordinates in Cn + 1 .

1.3 There exists on cpn the Kahler metric F such that where F is the form defined abore.

PROPOSITION

n* F

= F,

This statement is derived from the following four evident properties of tensor F: a) The restriction of F to the mapping fiber of n: en + 1 \0 -+ cpn equals zero. b) The tensor F is invariant in relation to the natural action of the group C* = C\O on cn + 1 \0: z(zo, ... ,Zn) = (ZZo, ... ,ZZn), Z E C*. c) The restriction of F to the orthogonal complement of a fiber with respect to the flat metric in Cn + 1 is positive definite. d) The differential 3-form d(Im F) on cpn is invariant under the mappings induced by unitary transformations A of the space cn + 1, AE U(n + 1). The metric F constructed on cpn is called the Fubini-Studi metric (for details see, for example, [1]). We are now able to construct a rich store of symplectic manifolds.

8

A. T. FOMENKO AND V. V. TROFIMOV

DEFINITION

1.4 Any subset of the form

V(fU'" IN) = {P = (a o: ... : an) E cpn:

It (ao:· .. : an) = ... = IN(aO:' .. : an) = A}, where {fu' .. ,IN} is any set of homogenous polynomials in the ring C[X 0' ... ,XII] is called an algebraic variety in cpn. If grad /; (i = 1, ... ,N) are linearly independent, then the algebraic variety V(Il' ... ,IN) is a complex manifold embedded in cpn, this being an immediate consequence of the theorem on implicit functions. Let j: V(fl" .. ,IN) --+ cpn be the inclusion of the complex manifold V(fl' ... ,IN) into complex projective space cpn, Q V(fl, ... .JN) = j* 1m F where F is the Fubini-Studi metric. The differential form QV(J', ... .JN) gives a symplectic structure on the manifold V(fl"" ,IN)' This statement results from V(fl, ... ,fN) being a complex submanifold ofCP".

The construction that we have set out gives us examples of compact Kahler manifolds. 1.5 We shall call a bounded open connected subset in the space C a bounded domain.

DEFINITION N

Any bounded domain is a Kahler manifold (see [48]) and therefore a symplectic manifold.

1.3. Hamiltonian vector fields

1.6 A smooth vector field v on a symplectic manifold M with the form ill is called a Hamiltonian field if it has the form v = s grad F where F is some smooth function on M which is called the Hamiltonian. DEFINITION

In special symplectic coordinates (Pi' q;) the Hamiltonian vector field is written as (8F/8qi' -8F/8pJ (see above). Hamiltonian vector fields (sometimes called Hamiltonian flows) allow of another important description in the language of the one-parameter groups generated by them from diffeomorphisms of the manifold M. Let v be a Hamiltonian field and (f)v be a one-parameter group of diffeomorphisms of M, represented by translations along the integral trajectories of the field v. This means that group ffiV consists of transformations of 9t operating on

INTEGRABLE SYSTEMS ON LIE ALGEBRA

9

Fig. 2.

M thus: gt(x) = y, where x = y(O), Y = y(t), y is the integral trajectory of field v passing through point x at the moment of time t = 0, see Figure 2. In other words the diffeomorphism gt moves point x for time t along trajectory y. Since the form w is defined on M the diffeomorphism gt transforms this form into a new one (g:w)(x) = w(gt(x)). Consequently the derivative of the form w is defined along the vector field v, i.e. d/dt(g:w). 1.7 A vector field v on the symplectic manifold M is called locally Hamiltonian if it preserves the symplectic structure w on M, i.e. the derivative of form w in the direction of vector field v is equal to zero: d/dt(g:w) = O. To put it anothetway-,folfu w is invariant with respect to all transformations of type gt generated by field v, i.e. is invariant in relation to the operation of the one-parameter group (Dv. The term "locally Hamiltonian field" owes its derivation to the following: DEFINITION

1.4 A smooth vector field von a sYVlplectic manifold M is locally Hamiltonian if, and only if, there exists for any point x EM a neighborhood U(x) of this point and a smooth function H u defined in this neighborhood so that v = s grad H u, i.e. field v is Hamiltonian in the neighborhood of U with the Hamiltonian H u. PROPOSITION

For proof, see for example [IJ, [87]. It is clear that any Hamiltonian field on M is locally Hamiltonian. The reverse is not true, i.e. a field that allows a representation of the kind s grad Huon the neighborhood of U may fail to allow a global representation in the form s grad F where F is some smooth function which has been defined on the entire manifold. In other words, the local Hamiltonians H u defined on separate neighborhoods do not always "slot together" into one function F defined on all of M. In any case we shall be studying mainly the Hamiltonian fields defined on the entire manifold and having the form s grad F where F is a Hamiltonian defined on all of M.

A. T. FOMENKO AND V. V. TROFIMOV

10

One of the most important examples of Hamiltonian flows is a geodesic flow. Briefly recapitulating its definition, let M be a compact closed n-dimensional Riemann manifold, i.e. a covariant tensor field of degree two gij is given on M so that (a) gij = gji' (b) gii~i~j > 0, ~ #- 0 (see [24], [166]). We shall define gii by the requirement that gijgjk = c51. The Riemann metric gij defines a scalar product in the cotangent bundle (~, '1) = gij~i'1j' ~,'1 E Tx* M. We have a canonical symplectic structure on T* M. We can examine the Hamiltonian H(x) = 1- gi j Pi Pj on T* M and its corresponding Hamiltonian flow x = s grad H(x) with respect to the canonical symplectic structure on T*M. Insofar as H(x) is a first integral of this flow the unit cotangent bundle S = {x IH(x) = 1} is invariant under the flow s grad H.

L

DEFINITION 1.8 The restriction of the flow x = s grad H where H = 19iipi Pj to the invariant surface S is called a geodesic flow on the Riemannian manifold M. Metric gij gives a natural diffeomorphism T*M - TM, which is linear in each fiber (the classical raising of indices). The following theorem is valid: THEOREM 1.1 Under the natural isomorphism T* M ~ TM the geodesic flow trajectories map into trajectories which consist of vectors tangent to the geodesic lines in M. An individual transformation gt maps a pair (x o, Po) to a pair (x t , Pt) = gt(x o, Po) where the geodesic line must be taken through Xo E M in the direction Po in order to obtain Xt. X t will then be at distance t along the geodesic line from the point Xo whereas vector Pt is tangent to this line at X t and has the same direction as Po. Let M be a surface in [R3 given locally in the graph form z = f(x, y); (x, y) will then be a local system of coordinates on M. Let Px, Py be the corresponding coordinates in T* M. In this case the Hamiltonian of the geodesic flow has the following form: H = (1

+ f/(x,y))p;

2(1

T* M =

+ (1 + f/(x,y))p; + fx2 + 1/)

- 2fx/ypxp y

[R2(X, y)

EB [R2(px' py ).

In one sense geodesic flows are universal Hamiltonian systems: according to the Maupertuis principle any Hamiltonian flow with the Hamilton function H = 1- aij(q)pi Pj + V(q), where aij is a positive

L

INTEGRABLIO SYSTEMS ON LIE ALGEBRA

11

definite matrix, coincides with the geodesic flow on the manifold of constant energy H(q, p) = h for the metric ds 2 = (h - V(q))a ij dqi dqj (see for example [1], [29]). 1.4. The Poisson bracket and Hamiltonian field integrals

DEFINITION 1.9 The Poisson bracket of two smooth functions f and 9 on the symplectic manifold M is the name given to the smooth function {f, g} defined by the formulil

{f, g} = w(s grad f, s grad g) =

I

w;/s grad f)i(S grad g)j .

i<.j

In other words the skew-symmetric scalar product of the "skew gradients" of f and 9 must be calculated. In its explicit form, in terms of the partial derivatives of functions f and g, the Poisson bracket is written thus:

where Xi are the local coordinates and w ij the coefficients of the matrix inverse to the matrix Ilwull. We have here made use of the fact that (s grad f)i = Lj w O oflox i . As a result of the definition the following statement can easily be proved (see for example [1], [87]). PROPOSITION 1.5 The operation of taking the Poisson /bracket f, 9 -+ If, g} satisfies the relations: (1) the operation {f, g} is bilinear; (2) the operation {f,g} is skew-symmetric, i.e. {f,g} = -{g,f}; (3) the Jacobi identity {h, {f,g}} + {g, {h,f}} + If, {g,h}} = 0 holds for any functions f, g, h. Thus the infinite dimensional linear space COO(M) of smooth functions F on a smooth symplectic manifold M is naturally an infinite Lie algebra over the field IR. We should note that the emergence of this algebra is entirely due to the presence of the "skew gradient" operation, since the operation of taking the ordinary gradient does not generate a Lie algebra. This shows yet another distinct quality of skew-symmetric scalar products (symplectic structures) as opposed to symmetric ones (Riemannian metrics). In the pages that follow we shall often be concerned with various finite-dimensional subalgebras in the Lie algebra COO(M). We may construct a natural mapping ri. of the Lie

12

A. T. FOMENKO AND V. V. TROFIMOV

algebra COO(M) into the Lie algebra V(M) of all smooth vector fields on manifold M. This mapping can be defined thus: rx(f) = s grad f LEMMA 1.1 The mapping rx: COO(M) --. V(M) is a homomorphism of Lie algebras, i.e. rx{J, g} = [rx(f), rx(g)]. This means that the operation of taking the Poisson bracket changes, under the mapping rx, into the operation of taking an ordinary commutator of the two vector fields rx(f) , rx(g). The proof follows from Proposition 1.5, and the definition of the operation s grad. The image of the Lie algebra COO(M) in the Lie algebra V(M) is a subalgebra which we denote by H(M). Its elements are vector fields on M, which can be represented as s grad J, i.e. (in our terminology hitherto) Hamiltonian fields. Accordingly H(M) is a subalgebra consisting of Hamiltonian fields on M. This means in particular that the usual commutator of two Hamiltonian fields is once more a Hamiltonian field though the Hamiltonian is obtained as the Poisson bracket of the two initial Hamiltonians of the fields being commuted. We should note that the mapping rx: COO(M) --. H(M) is an epimorphism, but not a monomorphism, inasmuch as rx has a non-zero kernel. If the manifold is connected the kernel consists of functions which are constant on the manifold. Consequently Ker rx is onedimensional and H(M) ,.,. COO(M)/Ker rx ,.,. coo(M)/lRl. Of course the subalgebra H(M) of the algebra V(M) depends on the choice of symplectic structure on M. If in fact we change the form w, the subalgebra H(M) will also be changed: strictly speaking we ought to write it as H w(M), but we shall take the subscript w for granted and omit it. The Poisson bracket of functions f and g has the following simple interpretation {J, g} = (s grad f)g, i.e. it coincides with the derivative of function g along the vector field s grad f This follows from the definition w(v, s grad g) = v(g) = Li Vi of/ox i . This simple observation IS extremely useful when studying the integrals of Hamiltonian fields. DEFINITION 1.10 The smooth function f on a manifold M is called an integral of a vector field v if this function is constant along all integral trajectories of field v. In other words the derivative of function f in the direction of field v must equal zero, see Figure 3. PROPOSITION 1.6 Let v = s grad F be a Hamiltonian field on M, and f be a smooth function which commutes (in the Poisson bracket sense)

INTEGRABLE SYSTEMS ON LIE ALGEBRA

13

Fig. 3.

with the Hamiltonian of this field, i.e. {j, H} the integral of the field v.

= O. Then the function f

is

Proof If we calculate the derivatives of f along the field v, we have v(f) = w(v, s grad f) = w(s grad F, s grad f) = {F,f} = O. The proposition is proven.

The function F-the Hamiltonian-in particular is always an integral of the field v, since {F, F} 0 owing to the skew symmetry of Poisson's bracket. Thus any Hamiltonian field has at least one integral, this being the Hamiltonian. From this we get the following property of Hamiltonian fields: on a manifold the field's integral trajectory cannot be everywhere dense, for it always stays on the hypersurface which is defined by the equation F = const. Therefore if a field has an integral trajectory which is dense everywhere in a given open domain, this field cannot be Hamiltonian. In this way a Hamiltonian field preserves the foliation of the manifold n hypersurfaces F = const.

=

PROPOSITION 1.7 If f and 9 are two integrals of a Hamiltonian field v = s grad F then their Poisson bracket {f, g} is also an integral of the field. Proof From Jacobi's identity we obtain:

{F, {j, g}} = - {g, {F,f}} - {j, {g, F}} = O. This proposition does allow us in principle to arrive at new integrals of a Hamiltonian field provided that two such integrals are known. Nevertheless this method of constructing integrals all too frequently ends in failure since the function {f, g} may turn out to be functionally dependent on the initial functions f and 9 (i.e. we shall not find out anything new). Here we have run up against the problem of finding as

14

A. T. FOMENKO AND V. V. TROFIMOV

great a number as possible of integrals of a Hamiltonian field, The point is that each of these fields defines a system of ordinary differential equations of 2n order on a manifold M 2 n, since in the local coordinates h x 1 ,.,., x 2n th e fiIeld' v IS wn'tt en as x. i = vi(x 1 , ... , x 2n) ,were vi are th e components of v, while Xi are the derivatives with respect to time t. The integral trajectories of the field v are the solutions of the corresponding system. Thus, if a field v has an integral f we lower the system's order by one, restricting the field to the hypersurface f = const along which the flow v "runs." Let us suppose now that field v has two integrals f and g and that f and g are functionally independent on the manifold. It is convenient to formulate the condition of independent in the terms of the functions' gradients. What is actually happening is simple: if the two fields grad f and grad g are linearly independent at each point of a subset U c M which is open and everywhere dense, then f and g are functionally independent on the manifold. Moreover we shall later be dealing with polynomial functions on algebraic manifolds and for that reason we require the set U to be open and dense in M. When working with smooth functions it must be bome in mind that in this case a pair of functions may be functionally independent on one domain in M and yet dependent on another domain in M, see Figure 4. These situations are of course impossible for polynomials. If f and g are two functionally independent integrals then they define two different foliations of manifold Minto hypersurfaces: f = const and g = const where in the "general position" case these hypersurfaces intersect transversely in submanifolds Q of dimension 2n - 2. Since f and g are integrals of v, field v is tangent to the surfaces Q2n - 2, see Figure 5. Consequently, field v may be restricted to the family of invariant surfaces of dimension 2n - 2 and we have lowered the system's order by two. In the general case if we have found r functionally independent integrals of a field we have lowered the order of the initial system by r and reduced the problem of finding solutions of the system to a problem

Fig. 4.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

15

Fig. 5.

on surfaces of dimension 2n - r that are invariant with respect to the field. In the "ideal case" 2n - 1 functionally independent integrals ought to be found in order to find solutions of the system. In this case the common level surface of the set of the functions 11' ... , 12n _1 would be one-dimensional trajectories coinciding with the initial system's integral trajectories. One more function is formally required in order to give the movement of points over these trajectories. In other words we should have completely integrated the system in the sense that we should have represented all its solutions as the common level lines of the functions /1' ... ,/211-1 which we know. In real mechanical systems, however, such a situation is so rarely met with that it is not practical to count on such a set of independent functions existing. In this connection we are sometimes forced to make do with "partial integrability" of the system. It is desirable to find a set of independent functions /1'" . , f.. on a manifold so that their common level surfaces (over which the flow v runs) might be arranged simply enough, for example, for all, or almost all, of them to be diffeomorphic to some well-studied manifold. In fact the problem of describing the system's solutions can be broken down into two stages: (1) first we produce r integrals thus allowing us to describe their common level surface Q2n -'; (2) we then try to describe the system's motion (i.e. the motion of points along the integral trajectories) on these level surfaces. One remarkable fact is that many concrete systems have sets of integrals which do allow us to realize the theoretical program described above. One of the best known results of this sort is Liouville's theorem. 1.5. Degenerate Poisson brackets

So far we have been examining only non-degenerate Poisson brackets (if

A. T. FOMENKO AND V. V. TROFIMOV

16

are local coordinates, then det{ Xi, xi} --f. 0). Degenerate brackets are also of some interest. They are connected with foliating symplectic structures, as the following statement shows.

Xi

LEMMA 1.2 Let a (degenerate) Poisson bracket be given on manifold M. Let r: T*M --+ TM be the corresponding Hamiltonian operator:
Proof On the manifold R(M) are defined the structural differential forms Wi, w~ E n 1 {R{M»; 11:*(~) = wi(~)e;, where ~ E 1(x,e,)R(M) and if ~ = djdtlr=o(x(t), (ei(t))), then Ve;!dt = w{(~)ei where Vjdt is the covariant derivative arising from the affine connection on M. These forms satisfy the structural equations:

where S~q is the torsion tensor and Rtpq the Riemann curvature tensor (see [120]). The definition gives us the value w = Ii,i = 1 aiw; /\ wi E n2(R(M», ai = const. We claim that dw = O. In fact by using structural equations we arrive at: j dw = [aMw~ /\ wi - w~ /\ dw ]

I

= -1

I

.

P 2 " a.R'. ' ),pq w ,~

/\

.

(L: . .)

w q /\ w) - -21 . a·S) Wi. w P , 'pq )

Since S~ = 0 Ricci's identity R~,pq therefore dw = O.

+ R~,qj + R!,jp = 0

/\

w q.

is valid, and

REMARK An analogous structure may be carried out for any principal (jj bundle. U sing the form w to define the skew gradient s grad f we find s grad f

INTEGRABLE SYSTEMS ON LIE ALGEBRA

17

can be defined exactly modulo the elements from the form w's kernel. Form w gives a bundle mapping A: TM -+ T*M, A(s)(y) = w(s, y), s, y being sections of the bundle TM. Let f, g E COO(M) be smooth functions so that df, dg E 1m A, it will then be possible to define correctly their Poisson bracket {f, g} = w(s grad 1, s grad g). This definition does not depend on the choice of representatives for s grad f If df 1. 1m A then the definition gives us the value {f, g} = 0 for any g E COO(M). In this case, therefore, COO(M) is a Lie algebra and Lemma 1.2 can be applied to it. As the final result we obtain a foliation of space R(M) into symplectic su bmanifolds.

2. SYMPLECTIC GEOMETRIES AND LIE GROUPS 2.1. Summary of the necessary results on Lie groups and Lie algebras

2.1 Let I» be a smooth manifold on which the group structure is given. ffi is then called a Lie group if the mapping I» x I» -+ I» given by the formula (a, b) -+ ab -1 is smooth.

DEFINITION

Let v E Tel», then, dispersing v by means of left shifts over the entire group 1», we shall obtain a vector field on 1». To be more exact we can define La(g)=ag for aEI» and, since (L a)-l=La- l this is a diffeomorphism of the group 1». We may put ~ = (dLa)e(v), La: I» -+ 1». The vector field constructed is left-invariant: (dLa)b~b = ~ab for all a, bEl». The space of left-invariant vector fields is a Lie algebra G under the commutator (bracket) product of vector fields. This Lie algebra is finitedimensional: dim G = dim 1». It is called the Lie algebra of the Lie group 1». If ~ is a left-invariant vector field on 1», then ~ generates a certain globally defined group of diffeomorphisms. The smooth homomorphism ~: 1R1 -+ I» is called a one-parameter subgroup of the Lie group 1». It is easy to verify that the left-invariant field ~ generates a one-parameter subgroup of ffi. A Lie algebra can thus be defined by one offour equivalent means: (a) one-parameter subgroups; (b) tangent vectors at the unit of the group; (c) left-invariant vector fields; (d) left-invariant actions of the group IR. All one-parameter subgroups can be gathered into one universal mapping

18

A. T. FOMENKO AND V. V. TROFIMOV

Exp: G -+ ffi where Exp(tX): IRI -+ G - ffi gives a one-parameter subgroup with velocity vector X at the unit e E ffi. Group ffi acts on itself by means of the conjugation (91,92) -+ 91929),1. This operation then induces a linear operation on the Lie algebra Ad: ffi -+ GL(G), which is called the adjoint representation of the group ffi: Adg~ = d(Adg)e~'

DEFINITION

2.2

The group's adjoint representation induces that of the Lie algebra: ad = d(Ad)e: G -+ Hom(G). The equality adx(Y) = [X, Y] is valid, where [X, Y] is the commutator in the Lie algebra G. The requirements of Hamiltonian mechanics demand a different representation. Let G* be the dual space of G, i.e. the space oflinear mappings f: G -+ IR. We can define the coadjoint operation Ad*: ffi--+GL(G*): (Ad;f)(x)= f(Ad g-1 x). This representation's differential is called the Lie algebra's coadjoint representation: ad*: G -+ Hom(G*). The equality (ad~f)(y) = f([X, Y]), X, Y E G, f E G*. We shall be dealing with the orbits or coadjoint representation Ad*. There is a natural symplectic structure on these orbits and any homogenous symplectic manifold is the orbit Ad* of some Lie group (see [156J). We should note that orbits of Ad* and Ad are different in the general case. The simplest example we can cite of this phenomenon arising is as follows: Let ffi be the group of affine transformations of the straight line x -+ ax + b, a # 0, a, b E IR. This group allows a matrix realization:

There is no difficulty in showing that G

= Teffi = {(~

~)}

and

Ada~=a~a-l,

a2 ) 1 '

~ = (~ ~2)

If a

= (a~

then:

Ada~ = (~l al~ ~~la2)=(~1

~2)

°.

aEffi,

~EG.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

19

The explicit form of the coadjoint representation Ad gives {

lIt

= ~l

'12

= ~2al - ~la2

and the orbits are therefore constructed as indicated in Figure 6. We can now look closely at the orbits of coadjoint representation. We choose in G the basis

but in G* the conjugate basis It, 12: IJe) = J ij . In this basis we have the coordinates Xl' X2' Simple calculations show that Ad*

.ed

an

(X I ,X2)

= (Xl

-

X2a2,x2at),

therefore the orbits Ad * are arranged as shown in Figure 7. In particular group (f) gives an example of representations Ad and Ad* not being equivalent. 2.1 If there exists on G a non-degenerate scalar product (X, Y) so that (Ad g X, Ad g Y) = (X, Y) then the adjoint and coadjoint representations are equivalent, i.e. they have identical orbits.

PROPOSITION

2.

Fig. 6.

. . . . ... . . ~ .. '. .'. : . . . . . . . .. ... '.. . .. ....... '. . . " " . ••••••• . . .. . . . ..••' '

...

'

.

,

'.,

'~~~'.'::"

Fig. 7.

.

"

.

20

A. T. FOMENKO AND V. V. TROFIMOV

The theorem of the classification of such algebras may be found in [183], or in: C. R. Acad. Sci. Paris, 301, No. 10, Ser. I (1985), 507 510. In particular all the Lie algebras that we call semi-simple satisfy this condition. The bilinearform B(X, Y) = tr(ad x ad y ) is called the CartanKilling form. The characteristic property of semi-simple Lie algebras is that the form B(X, Y) is non-degenerate on G. Recall the structure theory of complex semi-simple Lie algebras (see for example [47], [50J). The maximal Abelian subalgebra H c G such that ad h for all h E H is a semi-simple linear transformation of G, called a Cartan subalgebra. If X EGis an arbitrary element then we use G(X, 0) to denote the subspace of elements in G that commute with X. The element X EGis called regular if dim G(X, 0) is minimal. If X EGis a regular element, then G(X,O) is a Cartan subalgebra in G; this subalgebra is denoted H(X). Regular elements form in G an open, invariant and dense subset. If G is a semi-simple Lie algebra then any Cartan subalgebra is commutative. Let G be a semi-simple Lie algebra over the field of complex numbers C. We fix a certain Cartan subalgebra H. A linear form lX(h) on His called a root if there is an element Ea E G, Ea # 0, such that [h, Ea] = lX(h)Ea for any hE H. Let Ga be an eigen-subspace corresponding to IX. Then G = H EB L,>,o G\ where the sign EB signifies the straight sum of linear spaces. In the semi-simple Lie algebra G all subspaces G' are one-dimensional in the case of IX # 0 (over the field q. We know that [G" GP] c Gdp, i.e. [Ea, Ep] = NapE a+ p. If IX + {3 # 0 then Ex and Ep are orthogonal with respect to form B(X, Y). Vectors Ea and E_ a are, however, not orthogonal. The restriction of the form B(X, Y) to H is non-degenerate, if r = dime H (the number r is called the rank of G) then there exist r linearly independent roots of the algebra G relative to H. The complete number of roots is generally speaking greater than r and the set of all roots is therefore not linearly independent. If IX, {3, IX + {3 are non-zero roots, then [G', GP] = Ga+ P; the only roots proportional to root IX # 0 are 0, + IX. Roots IX can be represented by vectors H~ EH. Since B(X, Y) is non-degenerate on H, for every IX E H* (H* is the dual space of H) there is a unique element H~ E H such that lX(h) = B(h, H~) for all hE H. Then, if IX # 0, then [Ea' X] = B(Ea' X)H~ for X E G -a and B(IX, IX) # O. We denote by Hoe H the subspace generated by all the vectors H~ with rational coefficients. Ho is a "real" part of H. It turns out that dimoHo = dime H = ! dimR H (here iQ is the field of rational numbers).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

21

Furthermore, the restriction of the form B(h, h') on H 0 is positive definite and takes rational values (h, h' E H 0); lX(h') E Q wherever IX # In particular lX(h') is a real number if h' E H o. In future we shall use A to denote a set of non-zero roots of G. Let H 1, ... ,H, be any fixed basis in H o. If A, fJ. are two linear forms on H 0 then it is said that A > fJ. if A(H;) = fJ.(H;) given i = 1,2, ... ,k and A(Hk + d > fJ.(H k+ d. We should not forget that if A, fJ. are roots, then A(h'), fJ.(h') are real numbers for any h' E H o. Thus a linear ordering is defined in the set A. The root IX E A is called positive if IX > 0, i.e. IX(Hi ) = given i = 1,2, ... ,k and IX(H k + 1) > 0. Root IX'S positivity means in itself that the first of its nonzero coordinates is positive. The linear ordering is not unique: from now on we shall suppose that the basis H l ' . . . , H, (r = rk G) is fixed. We shall denote the set of positive roots by A +. Then A = A + u A - where A + n A - = 0, and there is also a one to one correspondence between A + and A - which is given by the involution IX -+ -IX. It is clear that if IX E A + then (-IX) E A -. The positive root IX is called simple ifit cannot be represented as the sum of two positive roots. If r = rk G = dime H, there then exist exactly r simple roots 1X1' •.. , IX, which form a basis in Hover C and a basis in Hoover Q. Moreover each root 13 E A can be represented in the form 13 = m;lXi' where m; E 71 are integers of the same sign; if m i ~ then 13 E A + and if m i ~ then 13 E Ll - . The system of simple roots 1X1' ... ,IX, is usually denoted by ll. The system A + is defined uniquely by the system II. If we let V+ = L7>0 G\ V- = L,
o.

°

L

°

°

[E" E-J = -H~; [ E E] "

p

= {N,pE«+fJ' 0, B(h,H~)=IX(h),

°

IX + 13 # lX+f3#O

root non-root

hEH.

The vectors Ea E Ga may he chosen so that NaP = N -a-po The constants N,p may be taken to be real (after the appropriate normalization of

22

A. T. FOMENKO AND V. V. TROFIMOV

vectors Ea). See the algorithm for defining N. p through the system of roots in (122]. 2.2. Orbits of the coadjoint representation and the canonical symplectic structure

We examine the coadjoint representation Ad * of the Lie group ffi: Ad * : (f) -+ GL( G*), G* is the dual space of the Lie algebra G. If f E G*, then 0(1) = {Ad: I Ig E (f)} is called the orbit passing through the point f E G*. Our immediate aim is to prove the following theorem. THEOREM 2.1

(see [53J, [63J, [44J, [121J).

There is a natural symplectic structure which is invariant with respect to the coadjoint representation on the orbits of the coadjoint representation of any Lie group. The proof of this theo rem that we given is based on direct calculation. We may break it down into a number of lemmas (see also 4.5). LEMMA 2.1 Let O(f) be the orbit of the representation Ad* of the Lie group (f) passing through the point f E G*, then Tf o (f) = {ad!f I~ EG} c G*. Proof Any vector tangent to O(f) at the point f E G* has the fonn v = d/dtlt =0 Adtp,~ I E TfO(f) for a certain ~ E G. Let e1 , ... , en be the basis of G, and e1 , ..• ,en be the basis of G* conjugate to the ei , i.e. ei(ej) = bj;

if I

E

G*, then

1= /;e i , /; is a linear function

= 1( -[~,e;]) =

on G*. We have

-(adtf)(e;)

and therefore v = -adt f, q.e.d. DEFINITION 2.3 If ~,Y/ E TfO(f) then, according to what we have proven, it can be taken that ~ = adt f, '1 = ad~, f and we define w(~, '1) = f([~1' '11J)· We must check the correctness of the definition. Generally speaking,

23

INTEGRABLE SYSTEMS ON LIE ALGEBRA ~1,1]1

are not uniquely defined. Let the subspace Ann(f) = {~ E G Iad; f = O} be called f E G*, the annihilator of f Subspace Ann(f) is easily seen to be a subalgebra. It is obvious that ~ = adt f = adt, f if, and only if, ~ 1 - ~ 2 E Ann(f). We must show in order to prove the correctness that f([~ 1 + ~o, 1]1 + 1]0]) = f([~ l' 1]d), where ~o, 1]0 E Ann(f) and this holds due to the linearity of f and to the definition of the annihilator. It is stated that W is a non-degenerate form on 1jO(f). This means that either (a) detllwijll =I 0 or (b) if w(~, '1) = 0 for any ~ E Tf0(f), then '1 = O. We may check the second statement. Letw(~,'1) = f([~,'11]) = 0 for any ~1' where'; = adt, j; '1 = ad:, f Then (ad:, f)(~ d = 0 for any ~1 E G, therefore ad: f = 0 and consequently '1 = ad:, f = O. COROLLARY The dimension representation is even.

of each

orbit

in

the

coadjoint

LEMMA 2.2 The form W is invariant with respect to the coadjoing action d(Adt)wg = wf (see Figure 8). the following obvious equality Ad: adt, f = ad(Ad.~,)(Ad: f). The mapping Ad: is linear and therefore d(Adn = Adt; when the mapping is restricted to a submanifold the differential is restricted to the corresponding tangent plane, so that d(Adtg = Ad:';. Now the in variance ensues from the following calculation:

Proof We note

[d(Adn wg ]

(.;,

'1)

= wg(d(Adn~, d(Ad:M = wg{Ad: .;, Ad: 1]) = wg{Ad: adt f, Adt ad:, f)

= wg(ad!d.~, (Ad: f), ad!dh~' (Ad: f))

Fig. 8.

24

A. T. FOMENKO AND V. V. TROFIMOV

= g([Ad h ~l' Adh 171]) = g(Adh [~1' 'II])

= (Adt-

1

g)([~l,'Il])

= f([~l,'Il]) =

w/(~,'I),

Here Adt f = 9 and therefore f = Adt- 1 9 and ~ = adtl f, 'I = ad~l f Thus in order to give won the entire orbit it is enough to give w at one point and to distribute it over the whole orbit by means of the transformations Ad:, hE (i';. COROLLARY The form WE n2(0(f» constructed above is smooth, since the translation operation Ad* is a smooth transformation. We have thus constructed on each O(f) orbit a non-degenerate 2form w which is called Kirillov's form. To derive a symplectic structure from w all that remains is to check that dw = O. LEMMA 2.3 Let a non-degenerate differential 2-form w be given on a manifold M 2": with the help of w it will then be possible to define the Poisson bracket .. of og for f,gECOO(M) {f, 9} = WI' OXi Oxi (Xl, ... , x2n being a local system of coordinates on M2n), here w = Ilw u II and Ilwull = Ilwull- l . We assert that dw = 0 when, and only when, {j,g} satisfies Jacobi's identity:

{f,{g,h}}

+ {g,{h,J}} + {h,{f,g}} =0.

Proof We have found that {j,{g,h}} = -(sgradfHg,h} = -Lsgrad/{g,h}, where L~ is the Lie derivative. If 17 = sgradf, then the Lie derivative's properties give us: .. og Oh) L~{g, h} = L~ ( WI' oxi ox i .. og oh

.' .. o('Ig) oh

.. og o('Ih)

= (L~w )1] ox i ox i + WI' ox i ox i + WI' oxi ox i .. og oh . { } = (L~W)I' ox i ox i + {'1g,h} + g,'1h

= (L~W)ii

::i ::i -

{{f,g},h} - {g,{j,h}},

25

INTEGRABLE SYSTEMS ON LIE ALGEBRA therefore

{j, {g,h}} + {g, {h,f}} + {h, {j,g }}

.. og oh

= -(LI/w)') OXi OXj'

Further, L"w = I(y/) dw + dl(y/)W = I(y/) dw, see [46], inasmuch as 1'/ is a Hamiltonian vector field. Thus if Jacobi's identity is satisfied, then dw = 0, and, conversely, if dw = 0, then Jacobi's identity is valid. The Poisson bracket on the orbits can be joined into a single bracket on the space G*. If j, 9 E C'''(G*), then we get from the definition {f, g}(x) = {f IO(x), 9 I O(x)} O(X)(x). As G* is partitioned by the orbits Ad * this definition is correct. We can explicitly calculate this bracket (we shall call the bracket Berezin's bracket). LEMMA 2.4

Assuming

f

E

COO(G*), then (s grad f)x = adtJ(X)(x).

Proof The differential df(x) E G**, coming from the canonical isomorphism G** '" G allows us to consider that df(x) E G. According to the definition w(Y, s grad f) = df(Y) for any Y. We can check that this equality is satisfied if we put (s grad f)x = adjf(x)(x), xEG*. Let Y = ad: x for y E G, then w(Y,adtf(X)(x»

= w(ad: x,adtf(X)(x» = x([y,df(x)]).

On the other hand

dfAY) = dfAad: x) = (ad: x)(dfx) = x([y, dfx]) and inasmuch as w is non-degenerate, (s grad f)x can be defined uniquely. LEMMA 2.5 Let f, 9 be smooth functions on G*, then we have the expression {j,g} = -qjXk of/ox; og/ox i for Berezin's bracket, el , ... ,ell being here the basis of the Lie algebra G, e l , ... , e" the conjugate basis of G*, while we denote the corresponding coordinates by Xl , ... ,x" and Xl' ... ,XII; C~j is the structure tensor ofthe Lie algebra G in the basis e;: [e;, eJ = C~jek' The proof is evident from the following calculation:

{j,g}x = dfAadjg(x)(x»

= (adjg(x)x)(dfx)

= x([dgx,dfx]) =

k

og of -:1-; Ox; CXj

CijX k -

26

A. T. FOMENKO AND V. V. TROFIMOV

here we have used equalities

og

dg x = -;- ei , uX i

REMARK If we apply Lemma 1.2 to Berezin's bracket on G* we obtain orbits of coadjoint representation, on which the KiriIIov form was given, as integral submanifolds. Thus the bracket {f,g} = -C~jxkof/oxiog/oXj has been defined on space COO(G*). To prove that {f, g} on the orbits satisfies Jacobi's identity (which is precisely what had to be checked to prove that Kirillov forms are closed), we must show that Berezin's bracket on G* satisfies Jacobi's identity, then its restriction to each orbit also satisfies Jacobi's identity and this restriction coincides with the Poisson bracket with respect to the canonical symplectic structure on orbits Ad*. LEMMA 2.6 Berezin's bracket - {f, g} = qjxk of/oxi og/ox j on space G* satisfies Jacobi's identity. Proof Using the definition of Berezin's bracket we obtain:

{h,{j,g}} + {j,{g,h}} + {g,{h,f}}

The first item in this expression becomes zero by virtue of Jacobi's identity in the Lie algebra. We note that the tensor (qrC~j - qrC~q)XpXk is skew-symmetric in the indices r, i and that the second, third and fourth items are also equal to zero (the symmetric and skew-symmetric tensor convolution). Thus Theorem 2.1 has been proved in its entirety.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

27

2.3. Differential equations for invariants and semi-invariants of the coadjoint representation

Let ffi be a connected Lie group, G its Lie algebra, G* the dual space of G, Ad:ffi-+GL(G) the adjoint and Ad*:ffi-+GL(G*) the coadjoint representation. When constructing dynamic systems on G* we shall be using invariants and semi-invariants of the representation Ad*. We extract the system of differential equations which the (semi-)invariants satisfy. We use A( G*) to denote the space of analytic functions on G*. DEFINITION 2.4 A function f E A(G*) is called an invariant if for any 9 E ffi, x E G* we have f(Ad: x) = f(x) and it is called semi-invariant if f(Ad: x) = x(g)f(x) where X is a character of the Lie group ffi. Let U be an open subset in G*, V( U) be the space of vector fields on U; V(U) is a Lie algebra with respect to the Lie bracket. We have a representation of Lie algebras cp: G -+ V(U), which is defined on basis el' . . . , ell in G thus: i=l, ... ,n;

here C{k is the structure tensor of the Lie algebra G in the basis ej, while Xi are the coordinates in G* in the basis ei defined by ei(e;) = b{. Since the operations used here are tensorian in nature the representation obtained does not depend on the choice of basis in G. The vector fields Xi' as Jacobi's identity implies, satisfy the relation [Xi' Xj] = C~jXk' the result of which is that cp is a representation of the Lie algebras. We shall say that the operator Xi corresponds to the basis vector e j E G. LEMMA 2.7

Let function FE A(G*), then

d" dt" t=/(Ad~xPf~f) = [( -cp(m"F](f)· Proof The equality

d

of

d

(f) dt xi(Ad:xPt~ f) dt t -_0 F(Ad~xPt~ f) = ~ uX, t= 0

28

A. T. FOMENKO AND V. V. TROFIMOV

is valid since xi(Adrxpr~

f)

= (Adrxpr~ f)(ei) =

f(Ad ExP (-r~) e;).

Because f is a linear function and the differential of Ad is ad, we have

d dt r=o f(AdExp(-r~)eJ = f( -[~,ea) = - f(qi~jek) = -qi~iJic·· Thus:

d F(Ad* . k of dt r= 0 Expr~ f) = - CL~ 'Jj oX «() = ( - ({J«()F)(f), i i.e. when n = 1 the lemma is proved. For any n > 1 we have: [( -({J(eWF](f)

= :t r=o[( -({J(eW-1F](AdlxPt~ f) d

dn = dd n tr=o't'

1 1

t=o

F(Ad~xPt~Adtpr~ f)

s LEMMA 2.8

= t +'t'.

If the function FE A(G*) then

F(Adrxpr~ f) = F(f) +

f (- ((J(~»" F (f) . tn. n.

n=l

Proof results from the expansion of using Lemma 3.7. PROPOSITION 2.2

F(Adrxpr~

f) as a Taylor series

Let function FE A(G*), then

a) F is an invariant ofthe coadjoint representation of group (V if, and only if, XiF = 0, i = 1, ... ,n, n = dim G; b) F is a semi-invariant of the coadjoint representation of group (V corresponding to a character X if, and only if, XiF = -AiF, i = 1, ... ,n = dim G, Ai = dx(eJ and dX is the derivative of X at the group (V's identity element.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

29

Proof a) As F(Ad~xPt~ f) = F(f), so d/dtlt=o F(Ad~xPt{ f) = 0 and therefore according to Lemma 2.7, when n = 1 we find that XiF = 0, i = 1, ... , n. Conversely, if XiF = 0, then <(J(e)F = 0 and therefore [ -cp(e)]"F = 0, when according to Lemma 2.8. F(Ad~xpt~ f) = F(f). Since 6> is a connected group, F(Ad; f) = F(f) for any g E 6>. b) From the equality F(Ad~xPt~ f) = X(Exp te)F(f) it follows that

d d dt t = /(Ad't..Pt~ f) = dt t = /(Exp te)· F (f) and therefore [( - cp(e»FJ(f) = X*(e)F(f). Conversely, [( - cp(eWJF = [X*(~)]nF therefore, in accordance with Lemma 2.8:

F(Adtpl~ f) = [ 1 + 11~1 [x*~~)]" t" ] . F(f) and since X(Exp te) = Exp(tX*(e» our theorem is complete. Lastly, in order to find the invariants, the system of differential equations CtXk 3F/3x j = 0, i = 1, ... ,n must be solved, orfor the semiinvariants-C~jXk 3F/3xj = A;F, i = 1, ... ,n. For the methods of solving these systems see [106J, [117]. The semi-invariants' system's solution cannot generally be found for every character. In terms of the operators Xi a criterion of in variance of a subspace We A(G*) with respect to the operators Ad:, g E 6> can be given. 2.3 Let W be a finite-dimensional subspace in A( G*) and f E A(G*), then for any g E 6> (6) being a simply connected Lie group having Lie algebra G) f(Ad: x) E W if, and only if, XJ E W, i = 1, ... , n = dimG.

PROPOSITION

Proof If fEW then from the fact that f(Ad: x) E W for any g E 6> it follows that d/dtlr = 0 f(Ad~xPt~ x) E W since any finite-dimensional subspace is closed and then, according to Lemma 2.7 X;! E W. Conversely, it is enough to check that f(Ad: x) E W for g = Exp(t~) since the connected Lie group is generated by any neighborhood of the identity element, and a sufficiently small neighborhood of the element is generated by one-parameter subgroups. We have f(Ad~xpr{ x)

•

L (-cp(~W , f(x)' t" n. 00

= f(x) +

11=1

and since W is closed, f(Ad~xPt~ x) E

w.

30

A. T. FOMENKO AND V. V. TROFIMOV

REMARK Let p: ffi ~ End( V) be an arbitrary finite-dimensional representation of the Lie group ffi in a linear space V. A function F: V ~ ~ is called invariant if F(p(g)x) = F(x) for all 9 E ffi, x E V. Let dp(e;)jj = a~i.h where h is a basis of V and ei a basis of G (the Lie algebra of group ffi): F will be invariant if, and only if, atxk(oF/oxi) = 0, which is proved in exactly the same way as in the case of p = Ad*.

3. LIOUVILLE'S THEOREM 3.1. Commutative integration of Hamiltonian systems

DEFINITION 3.1 One says that two smooth functions I and 9 on a symplectic manifold are in involution if their Poisson bracket equals zero. As we have seen, full integration of a system requires that we should know 2n - 1 of the system's integrals. Actually for Hamiltonian systems it is enough to know only n functionally independent integrals (where 2n is the dimension of M) that are in involution. In this case each integral "can be reckoned as two integrals," i.e. it allows us to lower the system's order each time by two units at a go, instead of one. Moreover, in this case the initial system is integrated "in quadratures." ..

THEOREM 3.1 (Liouville) Suppose that a set of smooth functions 11' ... ,1;, in involution, i.e. {h, jj} = 0 when 1 ~ i,j ~ n, is given on a symplectic manifold M2n. Let M~ be the common level surface of the 1 ~ i ~ n}. Let us imagine functions (h), i.e. M~ = {x EM: hex) = that on this surface of the level all n functions /1' ... , In are functionally independent (i.e. the gradients grad h, 1 ~ i ~ n are linearly independent in all points of surface M ~). Then the following statements hold:

ei'

1) The surface M~ is a smooth n-dimensional submanifold, invariant under each vector field Vi = s grad h, whose Hamiltonian is the function

h· 2) If the manifold M ~ is connected and compact then it is diffeomorphic to the torus Tn or, in the general case, a connected nonsingular manifold (which need not necessarily be compact) being a quotient of Euclidean space ~n over some lattice of rank ~ n if all flows s grad h are complete.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

31

3) If the level surface M ~ is compact and connected (i.e. if it is an ndimensional torus), then in one of its open neighborhoods regular curvilinear coordinates SI' ... ,SIl' ({Jl' ... ,({J1l where ~ ({Jj < 2n (the socalled "angle coordinates"), can be introduced, such that (a) the symplectic structure w in these coordinates is written in the simplest way,i.e.w = dS I 1\ d({J1 + ... + dSIl 1\ d({Jll,whichisequivalenttosaying that functions SI' ... ,SIl' ({Jl' ... ,({J" satisfy the following correlations: {Sj,SJ = {({Jj,({JJ =0, {Sj,((Jj}=6jj; (b) functions SI,,,.,SIl are coordinates in the directions transverse to the torus and are functions of the integrals il' ... ,f", i.e. Sj = Sj(fl' ... ,f,,), 1 ~ i ~ n; (c) functions ({Jl' ... , ({J" are coordinates on torus T" = SI X ••• X SI where ({Jj is the angular coordinate on the i-th circle SI, ~ ({Jj < 2n; (d) each vector field v = Sgrad F, where F is anyone of the functions il' ... ,f" takes the form (pj = qj(~ l' ... ,~,,), when written in the coordinates ({Jl' ... , ({J" on torus Til: i.e. the field's components are constant on the torus and the field's integral trajectories describe the quasiperiodic motion of the system v, that is they give a "rectilinear helix" on the torus Til. Here functions qj, 1 ~ i ~ n are defined in the given neighborhood of the torus; on nearby surfaces we likewise have (pj = qj(Sj, ... , sn). Thus the initial system v = S grad F may be written in the neighborhood of the torus Til in the coordinates Sj, ... , ({J1l as oSj = 0, (pj = qj(SI, ... ,SIl)' 1 ~ i ~ n.

°

°

We see that if the functions il' ... ' f" are independent on M, then all non-singular level surfaces M~ (i.e. those whose function gradients are independent, in particular different from zero) are diffeomorphic to one and the same manifold-the n-dimensional torus. Of course, the embedding of this torus into the enveloping manifold may be rather complicated (see Figure 9), but this "complication" can be studied by starting from known functions ii' ... , J.,. Further, the vector field grad i is arranged on the torus itself with the maximum simplicity, since, with respect to the angle coordinates CPl' ... ,({J", it has constant components, i.e. it is wholly defined by giving the velocity vector at one point of the torus. In the "general position" case each of the field's trajectories describes a helix on the torus which is everywhere dense. The importance of this theorem stems from the capacity, as will be shown below, of many interesting mechanical systems and their analogs to undergo "Liouville integration." If we are given a concrete system v = S grad F where F is a given function on M, then we may say that this system is "completely (Liouville) integrable in a commutative sense" or

32

A. T. FOMENKO AND V. V. TROFIMOV

Fig. 9.

that "it allows complete commutative integration," ifthe set offunctions il = F, i2' ... ,In exists and satisfies the conditions of Theorem 4.1. In this case system v defines the quasiperiodic motion on tori of half the dimension. Later we shall also encounter the so-called "noncommutative integration." Thus, if Hamiltonian is fixed, the first problem is to find n - 1 more functions that will form together with F a functionally independent set, commutative with respect to the Poisson bracket. This is equivalent to including function F which we look at now as an element of the Lie algebra COO(M) (see above) in a commutative algebra of dimension n, whose additive basis consists of the functionally independent functions. We omit proof of Theorem 3.1, since it can be found in a number of textbooks (see for example [1], [97], [156]). We describe in Section 3.3 the "non-commutative Liouville theorem", which is based on the ideas of Cartan, Marsden, Weinstein. 3.2. Non-commutative Lie algebras of integrals

The key role in the Liouville theorem cited above is played by the commutativity of the set of functions i1' ... , I... In other words, the linear space G spanned by the i 1 , •.• ,In is a commutative Lie algebra of dimension n. Here the Hamiltonian of the system being integrated is contained in the Lie algebra as the element F = it. In many concrete situations, however, the system's Hamiltonians have a set of integrals

.

33

INTEGRABLE SYSTEMS ON LIE ALGEBRA

i1, ... ,.fie which do not form a commutative Lie algebra, i.e. are not in involution. That is why it would be useful to have a method available for integrating such systems. Let us imagine that i1" .. ,.fie are smooth functions on a symplectic manifold M 2n with form OJ and that they are functionally independent on an open set which is everywhere dense in M. If we examine the linear span G (over~) ofthe functions (.t;), then dimfl G = k. We shall suppose that the linear space G is closed with respect to the Poisson bracket, i.e. all the brackets {.t;, Jj} are linear combinations of the basis functions i1' ... ,.fie with constant (!) coefficients, i.e. {.t;, Jj} = = 1 C'fJq where C'fj E ~. This means that linear space G is a finite-dimensional real Lie algebra with respect to the Poisson bracket. One important particular case is when G is a commutative Lie algebra of dimension n.Here we end up in the situation of Liouville's "commutative theorem." We should bear in mind that the Lie algebra G is not necessarily compact. Indeed in the case of Liouville's theorem the algebra is commutative. Nevertheless it emerges that, if there is imposed one more simple condition on the algebra of integrals G, then the system v = s grad F, where F = j~ allows full integration permitting us to describe the system's trajectories just as simply as we can in the case of Liouville's theorem. We shall call the algebra constructed above the algebra of integrals G. We use (f) to denote a simply connected Lie group that corresponds to G. Then (f) = Exp G. Each element of the algebra G may be presented as a Hamiltonian field on M211. To do so one examines the mapping IX: i -+ s grad f Consequently the group (f) is represented as the group of the diffeomorphisms of the manifold which preserve the form OJ. These diffeomorphisms are called symplectic. Thus giving the algebras of integrals G defines a smooth symplectic action of the finitedimensional group (f) on M. We shall use the following notation: ~(X) = <~, X), where ~ E G*, X E G. In these notations
I:

34

A. T. FOMENKO AND V. V. TROFIMOV

3.3. Theorem of non-commutative integration

Using some ideas from Marsden and Weinstein [75] we shall formulate a theorem which naturally generalizes Liouville's theorem and which is proved by A. T. Fomenko and A. S. Mishchenko in [88]. THEOREM 3.2 Let a set of smooth functions 11, ... , fir., whose linear span is a Lie algebra G with respect to the Poisson bracket, i.e. {J;, Jj} = = 1 C[j /q where C[j are constants, be given on a symplectic manifold M211. Let M~ be a common level surface in general position of the functions (n, i.e. M ~ = {x EM: J;(x) = C 1 ~ i ~ n}. We shall suppose that on this level surface all k functions 11,' .. , fir. are functionally independent. We shall also suppose that the Lie algebra G satisfies the condition dim G + ind G = dim M, i.e. k + ind G = 2n. Then the surface M ~ is a smooth r-dimensional submanifold (where r = ind G), invariant under each vector field v = s grad h, where h E H~. Further, let v be one of the following Hamiltonian fields on M: (a) either v = s grad h, where the Hamiltonian h is an element of the algebra of integrals G and lies in the annihilator H~ of the covector which defines the level surface M~; (b) or v = s grad F is the Hamiltonian field on M for which all the functions in algebra G are integrals, i.e. 0 = {F, J} for all IE G. Then, as in the case of Liouville's "commutative" theorem, if manifold M ~ is connected and compact it is diffeomorphic to the rdimensional torus T r and on this torus the curvilinear coordinates CPt, ... 'CPr can be introduced, such that vector field v, being written in these coordinates on the torus, takes on the form (Pi = qi(~I" .. , ~k)' i.e. the field's components are constant on the torus and the field's integral trajectories define the quasiperiodic motion of system v, i.e. they give a "rectilinear helix" on the torus Tr.

L:

e

The proof will be given below. In the special case, when the Lie algebra of integrals G is commutative, the condition dim G + ind G = dim M becomes the condition that k + k = 2n, since the rank here is G = dim G = k. Thus k = n and we get Liouville's "commutative theorem." In many concrete examples the Lie algebra of the integrals turns out to be compact and non-commutative. As can be seen from Theorem 3.2, the system's motion proceeds through tori T', whose dimensional r is equal to the index of algebra G. In the semi-simple case the rank r of the algebra G (= ind G) is less than its dimensional and moreover in all fundamental cases it may be considered that r ~ jk,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

35

where k is the dimensionality of G. Thus, for example, in the case of the series All-I' when G = su(n), we have r = n - 1, k = dim G = n 2 - 1, i.e. rank G :::::: Jdim G. This means that r < k and since r + k = 2n, then /' < II = -1 dim M. In other words the motion of the system v = s grad F proceeds through tori whose dimension is less, and substantially less, than half that of the manifold. This shows that Hamiltonian systems with non-commutative symmetries, i.e. which have a non-commutative algebra of integrals in the sense we have outlined above, are very much degenerate; that is, their integral trajectories (in the general position case) wind densely everywhere round tori of low dimension r. This is what distinguishes such systems from those which satisfy the conditions of Liouville's "commutative theorem" whose motion proceeds through tori of half the dimensionality, i.e. r = n = -1dimM. Thus the "noncommutative theorem" 3.2 allows us to integrate systems with strong degeneracy, a degeneracy all the stronger, the smaller the index of the algebra of integrals of the system. Such types of system, being systems "with degeneracies" on the initial manifold, may turn out to be "Liouville type" systems on a given submanifold K in M. What is more, an interesting link can be found between commutative and non-commutative integration. For example, if a Hamiltonian system has a non-commutative algebra of the integrals (with the condition dim G + ind G = dim M, then in many cases it also has a commutative algebra of integrals of half the dimension. Furthermore, the following proposition proven in [88] holds. THEOREM 3.3 (Fomenko, A. T.; Mishchenko, A. S.) Let v = s grad F be a Hamiltonian system on a compact symplectic manifold M, and let it be completely integrable in the non-commutative sense, i.e. having a Lie algebra of the integrals G so that dim G + ind G = dim M. Then this same system will be completely integrable in the ordinary Liouville commutative sense, i.e. it also has a second commutative Lie algebra of .the integrals of G', for which dim G' = 1dim M. Here we are assuming, of course, that the additive generators of both algebras G and G' are functionally independent almost everywhere on M. It is clear that these algebras are not isomorphic if G is noncommutative. For this reason on a compact manifold the Hamiltonian system "with degeneracy" which is to be integrated has yet another commutative "general type" algebra of integrals. From the geometrical point of view such systems have an extremely simple structure. Let {T'}

36

A. T. FOMENKO AND V. V. TROFIMOV

be a family ofr-dimensional tori where r < n; over these tori the system's trajectories move, forming in general position helices on them which are everywhere dense. Then (see Theorem 3.3) these low r-dimensional tori can be organized into greater tori of dimension n, i.e. half the dimension of M, and the system's trajectories move over them. These greater tori Tn are level surfaces of the second algebra of integrals, now a commutative algebra, see Figure 10. We should note that such a system's trajectories cannot be everywhere dense on the big torus Tn, since this torus' stratification into tori T' oflow dimension is arranged locally as the direct product of a torus T' with a given complementary submanifold of dimensional n - r, see Figure 10. Theorem 3.3 can be proved from Proposition 3.9 by using the results of Chapter 5 and applying the classification of the (finite-dimensional) subalgebras of the Lie algebra COO(M) (with respect to the Poisson bracket) for the case of compact manifolds M; it is known that any finite-dimensional subalgebra in the Lie algebra C<Xl(M) with respect to the Poisson bracket) on a compact symplectic manifold is reductive. So far no analogous result has been proved for non-compact manifolds. Hypothesis: any Hamiltonian system on any symplectic manifold which is completely integrable in the non-commutative sense is also completely integrable in the Liouville commutative sense. The expansion of Theorem 3.3 to non-compact manifolds, it may be thought, arouses interest because many concrete Hamiltonian systems are realized in the form of flows on non-compact manifolds.

Fig. 10.

3.4. Reduction of Hamiltonian systems with non-commutative symmetries

We shall describe a simple and elegant construction which allows us to convert a Hamiltonian system which has a group of symmetries into a Hamiltonian system on a symplectic manifold of lower dimension (J. Marsden, A. Weinstein [7 5J). This procedure is called the reduction of a Hamiltonian system. As one of the applications, we shall prove Theorem 3.2.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

37

Let a Hamiltonian vector field v = s grad F with algebra of integrals G be given on a symplectic manifold (M 2n ,w), and let the algebra's additive generators be k (almost everywhere) independent smooth functions fl" .. ,k Let ffi be the corresponding simply connected group operating on M by symplectic diffeomorphisms (i.e. those that preserve w). For our purposes it is easier to look at the following mapping ({J. We shall identify each point x EM with the linearfunctional (the covector) ({Jx on the algebra G. We shall take the value ((JAf) = f(x), where f E G. Thus ({Jx is an element of the space G* dual to G. Consequently we have defined a smooth mapping
Proof According to the definition = =
Proof Let us examine the additive generators fl, ... , f,. in G and the gradients grad /1' ... , grad k Since M ~ is a non-singular surface they are accordingly transverse to M~, i.e. the k-dimensional plane V spanned by (grad J;) intersects with TxM~ only at zero, and V Et> T"M~ = T"M, (see Figure 11). We may conveniently take it that a Riemann metric is given on M such that the vectors (grad J;) are orthogonal to the surface M~. To prove the relation s grad f(x) E TxM~, it is enough to check that (s grad f)g 0 for any function 9 E G, i.e. that the derivative along

=

38

A. T. FOMENKO AND V. V. TROFIMOV

Fig. 11.

s grad f of any function g in G, which is constant on M~, equals zero. As it happens, (s grad f)g = (s grad f, s grad g), where (x, y) is the Riemann metric on M. Because the scalar products of s grad f with each vector (grad D equals zero, it follows that s grad f is orthogonal to the plane V, i.e. sgradfETxM~. Thus (sgradf)g= {f,g}. Further, {f,g}(x) = <~, {f, g}) =
3.2

The equality

(T"M~)

n (s grad f;f E G) = Hx = (s grad h; h E H~) is valid, (see Figure 13). Proof We have shown above that (s grad h; h EH~) c T,;M~ n (sgradf; fEG). We shall prove the converse. Let xET,;M~ and X = s gradf, where fEG. It must be proved that fEH~. We examine
=

=°

Fig. 12.

39

INTEGRABLE SYSTEMS ON LIE ALGEBRA

We also consider the group

f>~

with Lie algebra

H~,

i.e.

f>~ =

ExpH~c(fj.

The level surface M ~ is invariant relative to the action of . the group f>~ on the manifold M.

COROLLARY

We examine form OJ on M and assume that ill = OJIM~ is its restriction to the surface M~. The action ofthe subgroup f>~ on M ~ generates at each point x E M ~ the plane H x c T"M~, formed by vectors s grad f, where f E H~, see Figure 13. In other words plane H x is generated by the subalgebra H~.

Fig. 13.

3.3 The kernal of the form ill (the restriction of the form coincides with the plane Hx c T"M~.

PROPOSITION OJ

to

M~)

Proof We shall first prove that Ker ill =:J H x' Let X = s grad h where h E H~, X E H x c T"M ~. We need to prove that X lies in the kernel of the form OJ, i.e. that OJ(X, y) = 0 for any vector Y in the plane T"M~. In fact, OJ(X, y) = w(s grad h, Y) = Y (h) = 0 since vector Y is tangent to the

level surface, while function h, being an element of the algebra of integrals G, is constant on the level surface. We shall now prove the converse, i.e. that KerillcH x • Let w(X,y) =0 for any vector y E T"M~. The vector X must be represented in the form X = s grad h for a given function h E H~. We then consider the form w as a skewsymmetric scalar product on the tangent plane TxM and we use (TxM~).l to denote the orthogonal complement to plane TxM~ in TxM relative to the form w. Since the form is non-degenerate, the equality dim(T~M~).l = dim M - dim T"M~ = dim V = k is valid. We should bear in mind that in the case of skew-symmetric scalar products the space TxM need not necessarily decompose into the direct sum of T"M ~ and (T"M~)\ since these planes can have a non-zero intersection. It is clear that Kerill= T"M~n(TxM~).l. We shall prove that (sgradf;

40

A. T. FOMENKO AND V. V. TROFIMOV

f E G) = (TxM~)l.. In fact, if Y E TxM~ then w(s grad f; Y) = Y(f) = 0, since f = const on M £. Thus (s grad f; f E G) c (Tx M ~)l.. Further, dim(s grad f; f E G) = k = dim G. This equality is a consequence of the linear span of the gradients (grad f; f E G) having dimension k (see definition of G); the skew-symmetric scalar product is non-degenerate and the skew gradients' linear span also has dimension k. Lastly we note that dim(J:M~)l. = k, therefore (s grad f; f E G) = (J:M~)l., see Figure 12. The argument has been proven. Let us now bring all these facts together and study the geometric picture ofthe mutual interaction of the submanifolds we have described. The fundamental objects are: (a) the level surface M~, dim M ~ = 2n - k; (b) an orbit ffi(x) of a point x, dim ffi(x) = k; (c) an orbit f)~(x) of a point x under the action of the subgroup f)~ = Exp H~. It is clear that J:ffi(x) = (sgradf;fEG), J:f)~(x) = Hx = (sgradh;hEH~).Itfollows from this that ffi(x) n M~ = f)~(x), see Figure 14. We note that the dimension of orbit f)~(x) equals that of f)~ and is equal to r.

Fig. 14.

Let us look at the action of the group ffi on M and assume that in a small neighborhood of tbe surface M ~ this action has a single type of stabilizer subgroup, i.e. that all orbits of the group ffi close to an orbit ffi(x) are diffeomorphic to it. Let us examine the projection p: M --+ M Iffi of manifold M on the orbit space M/(fj = N. This space need not be a smooth manifold and it may have singularities. What is important to us is that space M/(fj is a smooth manifold of dimensionality 2n - k in a small neighborhood of the point pffi(x) EMIffi. In reality, if ffi is for example a compact group and operates smoothly on M then the union

INTEGRABLE SYSTEMS ON LIE ALGEBRA

41

of the set of orbits in general position which are diffeomorphic to each other is an open and every where dense submanifold in M; therefore space N is a 2n - k-dimensional manifold everywhere, with the exception of a subset of measure zero. We should bear in mind that the space (manifold) N need not necessarily be symplectic since it may be, for example, odd-dimensional. The projection p restricted to the surface M~, projects it onto the surfaces Q~ = Mdf:J~. Therefore space N is stratified by surfaces Q~, see Figure 15. Here our argument is based on Proposition 3.2. 1.

; , (x)

Fig. 15.

PROPOSITION 3.4 The manifolds Q~, i.e. the quotient manifolds of the level surfaces M~ under the action of the subgroup f:J~, are symplectic manifolds with a non-degenerate closed form p, which is the projection of the form OJ on M ~ under the mapping p: M ~ -+ Q~. Here p* p = OJ = wIM~.

The proof ensues from Proposition 3.3, since the kernal ofthe form cO on TxM~ coincides with the plane Hx c TxM~. Let us now go back to study Hamiltonian systems on M. Let v = s grad F be a system with algebra of integrals G, i.e. {F, G} = O. Since the Hamiltonian F commutes (in the Poisson bracket sense) with all elements of G, F is therefore invariant with respect to the group ffi. In actual fact (s grad f)F = {f, F} = 0, f E G. In particular, the subgroup f:J~, in acting on M, also maps function F to itself. Thus we have defined a natural projection of the vector field s grad F onto the space N = M Iffi. At the

42

A.

T. FOMENKO AND V. V. TROFIMOV

same time the vector field s grad F is tangent to the surface M ~ and is also projected onto a certain field E(F) on the quotient Q~, since the field s grad F is invariant with respect to ~~. Thus space N is stratified by symplectic manifolds Q~ and a vector field E(F), which is tangent to all surfaces Q~, see Figure 16, is defined on N. Finally the triad (M 2n , S grad F, w) corresponds to a new triad (Q~, E(F), p).

~,(X)

E(F)

Fig. 16.

PROPOSITION 3.5 The vector field E(F) is Hamiltonian with respect to the symplectic form p on the manifold Q~ for the Hamiltonian function ft, which is equal to the projection of function FIM~ onto the manifold Q~, i.e. E(F) = s grad p(p* FIM ~). The proof follows from Proposition 3.4 and the invariance of Hamiltonian F under the group action. The correspondence (M, s grad F, w) --. (Q~, E(F), p) constructed above is what we call the reduction of the initial Hamiltonian system s grad F. With this reduction we get a new Hamiltonian system on the manifold Q~ of dimensional 2n - k - r, lower than that of the initial manifold M, viz. 2n, while dim Q~ < dim M ~ = 2n - k. It could emerge that the reduced system on Q~ turns out to be simpler than the initial system on M. We shall assume that the reduced system has been successfully integrated. This will then allow us to increase the number of integrals which the initial system s grad F on M had, by "pulling" these integrals back from the manifold N onto the manifold M.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

43

3.6 Let G be a finite-dimensional algebra of integrals of system s grad F on M, satisfying all the conditions enumerated, and let E(F) be the reduced system on the manifold N = U~ Q~, a Hamiltonian system on each submanifold Q~. Let G' be a linear space of functions on the manifold N, such that their restrictions to the submanifolds Q~ form a finite-dimensional algebra of integrals of the flow E(F). The space of functions G EB Gil, where Gil = p*G', i.e. Gil = {gp, 9 E G'}, p: M - t N is then a Lie algebra of integrals of system s grad F, while [G, Gil] = o. PROPOSITION

Proof Let 9 be a given function on space N; its inverse image gp under the mapping p: M - t N is then a function on M, obviously invariant with respect to the action of GJ on M. But this means that the function gp

is in involution with the whole of the initial (integrals) algebra of functions G. Thus every new function which happens to be an integral of the reduced flow E(F) on N gives a complementary integral gp of the initial Hamiltonian flow s grad F on M. That these complementary integrals are independent ofthe functions of algebra G follows from their gradients being non-zero in the direction of the submanifolds Q~ which lie (locally) in the level surface M~; at the same time the gradients ofthe functions in G are orthogonal to M~. The proposition has been proved. Proof of Theorem 3.2

Let v be one of the systems outlined in formulating the theorem, i.e. either v = s grad F, {F, Q} = 0, or v = s grad h, where hE Ann ~ = H~. Let us examine the reduction described above. Insofar as the supplementary condition dim G + ind G = dim M, i.e. k + r = 2n, has now been met, the dimension of the surface M ~ equals r. The dimension of the orbit i)~(x) which is contained in M ~ is also equal to r (according to the definition), see Figure 15. From this it follows at once that M ~ = i)~(x), i.e. in the terms of Theorem 3.2 the level surface M ~ is the orbit of a point x under the action of i)~, and its Lie algebra is the annihilator of the covector ~ which defines the given level surface. In particular, dim Q~ = 2n - k - r = O. In the given case, therefore, the reduced system's structure is particularly simple. Since Q~ is a point, flow E(F) is zero, see Figure 17. Here space N has dimension n; since M~ is a level surface of the algebra G of integrals of the flow v, this flow is tangent to it in both the cases (a) and (b) (see the formulation of the theorem), i.e. M~ is a rdimensional submanifold which is invariant with respect to all fields of the type s grad h, h E Ann(~) and s grad F, {F, G} = o. It remains to be proved that the level surface is an r-dimensional torus in the case when

44

A. T. FOMENKO AND V. V. TROFIMOV

M~

is compact and connected. To do so we shall need an auxiliary argument. 3.7 Let ~ E G* be a covector in general position. Then its annihilator Ann ~ is commutative and so too, in particular, is the subgroup i)~ (see for example [26]). Let us examine the coadjoint operation of the group G) = Exp G on the coalgebra G*. We denote by O*(~) the orbit which passes through point ~ E G*. Since dim H ~ = rand dim G* = k, dim O*(~) = k - r. Since the covector ~ is in general position the orbits close to O*(~) are also diffeomorphous to it, it may also be taken that a sufficiently small neighborhood U of point ~ is fibered into homeomorphic sheets, see Figure 18. We will denote a local section of the fibration of U by the orbits of the action of G) by X o' We PROPOSITION

Fig. 17.

take advantage of U being representable as tpe direct product ofthe base and a fiber (i.e. a part of the orbit), see Figure 18. Let h(Yf) be a smooth function on U which is constant on the orbits. We argue that a(~, dh(~» = 0, where dh (the differential of h) is interpreted as an element of the dual space (G*)* ~ G, i.e. dh(~) E G. In other words, we are arguing that dh(~) E Ann(~). We must verify that a(~, dh(m(g) = 0 for any 9 E G. We have a(~, dh(~»)(g) =
gJ>

= -
= 0,

since the covector aR, g) = ad: ~ lies in the plane ~O* tangent to the orbit 0* of the point ~, while function h is constant on the orbits and constant, in particular, on the orbit 0*, see Figure 18. If we take section X o , which is a smooth surface of dimension r transversely intersecting the orbits close to the orbit O*(~), we can consider on it the set of r independent functions hI' ... , hr and extend them into smooth functions on the entire U, by extending them from the section X °with values that are constant along the orbits 0*. We get a(~, dhi(m = 0, 1 ~ i ~ r. Thus

45

INTEGRABLE SYSTEMS ON LIE ALGEBRA

x:

""

'-

f

-i

11

-

O(rf -Fig. 18. dhi(~) E Ann(~),

1~ i

~ r.

Since functions (h;) we chosen as independent, all differentials (dhi(~)) are independent in Ann ~ and the number equals r, i.e. exactly coincides with the dimension of Ann~. The differentials dhi(~) therefore form a basis in Ann ~ and all that is needed to prove the commutativity of the annihilator is to prove that the commutator of any two of these differentials is zero, i.e. [dhi(~)' dhi~)] = o. Since a(~, dhJ~)) = 0, the equality b(~)

=

a(~,dhi(m(dhi~))

=0

holds. Hence b(~)

(a(~,dhi(~),dh)~)

=

=

(~, [dhi(~),dh)~)])

= o.

'1

We may consider the arbitrary direction in the neighborhood U and differentiate the function b(~) 0 along this direction, i.e. we examine

=

d

d

o = d~ b(~) = d'1 (~, [dhi(~)' dhi~)]) =

('1, [dhJ~), dhiO]) + (~, [:'1 dhi(~)' dhj(~)J) + (~, [dhi(~)' -

=

:'1 dhi~)

J)

+

('1, [dhJ~)' dhj(~)])

(a(~, dhim, :'1 dhJ~)) - ( a(~, dhi(m, ~ dhj(~))

('1,

[dhJ~),dhi~)])

= 0,

46

A. T. FOMENKO AND V. V. TROFIMOV

since a(~,dhl~»=a(~,dhi(~»=O. Since t'fEUcG* is arbitrary, it therefore ensues from the equality (t'f, [dhi(n dhi~)]) = 0 that [dhi(~)' dhj(~)] = O. The proposition is proved. We now return to the proof of Theorem 3.2. We have proved that the surface M ~ is an orbit of the group ~~ and, since dim M ~ = dim ~~, then M~ is the quotient group of ~~ over a discrete lattice r: Since group ~~ = Exp Ann(O is commutative, in the compact and connected case M ~ is an r-dimensional torus. The remaining arguments can be proved just as for Liouville's theorem and we refer the reader to the textbooks we have already mentioned. One of the ways of applying the "non-commutative theorem," therefore, is this: if v = s grad f is a Hamiltonian system on M 2n, a noncommutative algebra G must be sought, such that the Hamiltonian f is contained in the annihilator of a covector ~ E G* in general position. If this algebra G has been found, then provided that dim G + ind G = 2n, the flow v moves along the r-dimensional torus Tr which coincides with the level surface M~, where r = the index of G.

3.5. Orbits of the co adjoint representation as symplectic manifolds

By applying the reduction construction we shall now construct the canonical symplectic structure which we explored above on the orbits of a Lie group's co adjoint representation. Let G be a finite-dimensional Lie algebra, G* be the space dual to G (a space we shall for brevity's sake call the coalgebra) and let Ad~: ffi - GL(G*) be the coadjoint representation of group ffi = Exp G on G*. Then G* is partitioned by the orbits 0*. Each orbit is a smooth submanifold contained in the linear space G*. A symplectic structure turns out to be naturally defined on each orbit. It will be remembered that this structure (Kirillov's form) is defined as follows. Let x E G* be an arbitrary point and ~l' ~2 E TxO* be two vectors tangent to the orbit. Each vector ~ tangent to the orbit can be represented in the form ad: x = a(x, g), since TxO* = ad~ x. Therefore elements gl,g2EG exist, such that ~i=a(x,g;), i= 1,2. Although this representation may not be unique, this does not affect the rest of the construction. We shall define the value of the form wx(~l' ~2) in point XEO* on vectors ~1'~2 tangent to orbit 0*, so W(~1'~2) = (x, [gl ,g2]), where x E G*, gl,g2 E G.

47

INTEGRABLE SYSTEMS ON LIE ALGEBRA

PROPOSITION 3.8 Fonn w defined above has the following properties: (1) the fonn is bilinear and its value does not depend on the choice of representatives g l' g 2; (2) the fonn is skew-symmetric and defines an exterior 2-fonn on the orbit; (3) the fonn is non-degenerate on the orbit; (4) the fonn is closed on the orbit. If the algebra G is compact it can be canonically identified, by using the Killing fonn, with G*. A fonn w on orbits 0 of the coadjoint representation is then defined thus: W"(~1'~2) = B(X,[gl,g2])' where B(x, y) is the Killing fonn and x, ~i' gi E G. We should note that in the general case fonn w may be written thus Wx(~1'~2)

=



=

<~l,gl>

=

-<~2,gl>·

Proof of Proposition 3.8

We shall apply the reduction technique worked out above. Let us consider the cotangent bundle M . T*(f) of the group m as the direct product M = T*(f) = TE*(f) x (f) = G* x (f). Here we shall give the action of (!) on this direct product in such a way that (f) operates on the second factor (f) by left shifts and does not change the coordinates for the first factor G*. We shall look at the algebra of integrals V which correspond to the left operation of group (f) on the phase space T*(f). It is clear that this algebra of integrals is isomorphic to G. Every function f E V is right-invariant and takes values according to thefonnulaf(~,g) = ,f E V = G, ~ E G*,g E(f). Thus the surface M { which corresponds to ~ E V* consists of all pairs (Y/, g) such that f(y/,g) = <~,f>, i.e. Ad:- 1 (y/) = ~ or Y/ = Ad:~. Insofar as (~, E) EM~, M~ = {(Ad: ~,g); g E (f)}. This tells us that surface M{ is a fibration with base O*(~) and fiber ~~, where 0* is an orbit of (f) on G*, and ~{ is the maximal torus corresponding to the subalgebra H~ which fixes the covector ~. We have represented a general position orbit of the coadjoint representation as a quotient-manifold M{/~{, where H{ is the annihilator of~. From Proposition 3.4 it follows that 0* is a symplectic manifold. Direct calculation shows that the symplectic fonn which arises on 0* coincides with the canonical fonn described above. 3.6. The connection between commutative and integration

non~ommutative

Liouville

If a Hamiltonian system on a symplectic manifold M is completely Liouville integrable in the non-commutative sense, i.e. if it admits a

48

A. T. FOMENKO AND V. V. TROFIMOV

finite-dimensional Lie algebra V of functionally independent integrals, and at the same time dim V + ind V = dim M and the algebra V is noncommutative, then the natural question arises as to whether this Hamiltonian system is completely integrable in the ordinary commutative sense, i.e. whether a commutative algebra Vo of functionally independent functions exists so that 2 dim Vo = dim M. Although the invariant surfaces have a dimension greater than dim M~, should such an algebra Vo exist, nevertheless it is useful to have an answer to the question, if only because of the great simplicity of the picture of movement along the integral trajectories. Actually, for a broad class of Lie algebras V the answer turns out to be in the affirmative. We shall examine the relevant Lie algebra that satisfies the following condition: (FJ) there exists a finite-dimensional space of functionally independent functions F 0 defined on the dual space V*, where any pair of functions f, 9 E F 0 is in involution on all orbits of the coadjoint representation with respect to the canonical symplectic structure and dimF 0 = !(ind V + dim V). PROPOSITION 3.9 (Fomenko, A. T.; Mishchenko, A. S.) Let M be a symplectic manifold, V the Lie algebra of functionally independent integrals of a Hamiltonian dynamic system, dim V + ind V = dim M. If algebra V satisfies the condition (FJ), then there is another commutative Lie algebra Vo of functionally independent integrals, where 2 dim Vo = dimM.

Proof (see [88]) Algebra Vo will be sought among those functions which are functions of the integrals in algebra V. Let Xl, ... , X" E V be a linear basis in algebra V. Then algebra Vo will be sought among functions of the form f(x l , ... ,x") where f is a smooth function of n independent variables. We should note that the Poisson bracket of two functions f(x 1 , ••• ,x") and g(Xl, ... ,x") is given by the formula: _ '\' of 1 n ~ 1 "i j { f,g } - ~axi(X , ... ,x)ox(x , ... ,x)[x,x]. l,J

(1)

)

On the other hand every function f of n independent variables defines a functionf* on the dual space V*; f*(~) = f«~, Xl), . .. , <~, x"». Then the Poisson bracket of the pair of functions f*(~), g*(~) on an orbit of

INTEGRABLE SYSTEMS ON LIE ALGEBRA

49

the coadjoint representation is given by the fonnula:

{J*,g*}(~) = ~ ;~i «~,Xl>, ... , <~,xn» '.J

= L ~f.

·.uX' I,J

~

« , Xl >, ... ,

<~ ,Xn » u~ ~g. « ~ , Xl >, ... , <~ , Xn >)

Fonnulas (1) and (2) show that the correspondence f --+ f* is an isomorphism of infinite-dimensional Lie algebras. If we take into account the property (FJ) we get the proof of our proposition. There are fairly abundant series of examples of Lie algebras which satisfy the condition (FJ). For example, as we shall see below, all semisimple Lie algebras V belong to them; for the commutative algebra F 0' the functions f(~ + Aa), ~ E V* are to be taken, where f is any function constant on the orbits of the coadjoint representation. Broad classes of soluble Lie algebras and certain semi-direct sums of Lie algebras also satisfy the condition (FJ) (see op. [89], [134], [126], [127], [10], [129], [123], [105]). Finite-dimensional Lie algebras of integrals V may be used likewise, when they are not functionally independent, so long as the action of annihilator ~ in the coadjoint representation has only one orbit type (see also 23.3). One example we can cite is a geodesic flow in the phase space oflinear elements on sphere sn with standard Riemann metric (see [88]). It will be easiest to consider that the sphere is contained in Euclidean space [Rn + 1, so that the space oflinear elements L(sn) consists of pairs of vectors (x, y), Ixl = 1, x.l Y, i.e. (x, y) = O. It is apparent that the dynamic system indicated is invariant under orthogonal transfonnations A E SO(n + 1). The system's equations may be written in the following fonn: x = y, y = -x. Let (x o, Yo) E T*sn, then the tangent vector ~ E 1(x o,Yo)(T*sn) may be given as a pair ~ = (x, y), x .1 xo, y.l Yo. Here the symplectic fonn w on the pair of tangent vectors ~1 = (X 1,Yl)' ~2 = (X 2,Y2) takes the valuew(~1'~2) = (X 1'Y2)(y l ' X2)' Then the function algebra V '" so(n - 1) corresponding to the action of the group so(n + 1) consists of functions of the fonn:

50

A. T. FOMENKO AND V. V. TROFIMOV

CEso(n + 1),

fc(x,y) = (X,CY), {fc"

(3)

fcJ = k,.C2]·

Later (see §16) we shall show that the Lie algebra so(n) satisfies the condition (FJ).1t is thus possible to construct polynomials Pk(fc) which are pairwise in involution. It is therefore enough to make explicit the maximal number of polynomials Pk(fc) which are pairwise functionally independent on the manifold T*sn. Formula (3) gives a mapping cp: T*S" --+ V*, which when written as a matrix has the form cp(x,y) = xl - yxt, where x, yare understood as column vectors, and the operation y --+ l is matrix transposition. The mapping cp is equivariant to the action of the group SO(n + 1). Space T*sn is foliated into submanifolds-the orbits of the action of the group SO(n + 1), which orbits may be parametrized by the length of the vector y, (x, y) E T*S" alone. The mapping cp maps the different orbits of space T*S" into different orbits, as the non-zero matrices A and .leA are not equivalent when .Ie #- 1, .Ie> O. We shall show that the mapping cp, restricted to an orbit O(x, y) c T*S" has a Jacobian matrix of rank equal to 2n - 2. To do so we have only to take the point (xo = (1,0, ... ,0), Yo = (0,1,0, ... ,0) and calculate the rank of the Jacobian matrix of the mapping at this point. The matrix cp(x,y) has the form II cpij I = cp(x,y), cpu = xiyj - yixj. Then, given 2 < i < j, the partial derivative functions cpu at the point (xo, Yo) equal zero. Further, OCPlj _

oyk -

°CP2j _ _ {}j

ox

k

-

Jj

2

k,

OCP2j _

ol -

k'

~

2

0,

k

~

~

k

n + 1, 2

~

~j ~

n + 1, 3

n + 1,

~j ~

n + 1.

Consequently the rank of the Jacobian matrix of the mapping cp equals 2n - 1. On the orbit O(x o, Yo) the rank of the Jacobian matrix then falls by one and is consequently equal to 2n - 2. In this way, if ~ = cp(x o, Yo) E V* then the mapping cp is a fiber bundle of cp: O(x o, Yo) --+ O(~). Insofar as so(n) satisfies the condition (FJ) (see §16), there exists for each orbit O(~) a set of functions PI' ... , P k on the dual space V* which are pairwise in involution and whose gradients on the orbit are linearly independent, and 2k = dim O(~) = 2n - 2. Then the compositions PI cp, ... , Pk cp are independent functions on orbit O(x o, Yo) and are in pairs in involution. By adding to the functions set C

0

INTEGRABLE SYSTEMS ON LIE ALGEBRA

51

PI lp, ... , Pk lp one more integral, the Hamilton function itself which in our case coincides with the length of vector y, we get k + 1 = n independent integrals which are pairwise in involution. Consequently, the geodesic flow of an n-dimensional sphere is completely Liouville integrable in the commutative sense. To provide one more example of applying non-commutative methods of integration, we shall look at the so-called point vortex system. Let U

0

M = T*lRn = IRn(p) , ... ,Pn) E8lRn (q), ... ,qn)' Pi be the impulses, and qi the coordinates in a configuration space IRn. We shall put Zk = Pk + iqb 1 ~ k ~ n. The system of n point vortices is a Hamiltonian system with Hamiltonian n

H(p, q) =

L k,/

KkK/lnlzk -

z/I.

The coefficient

Kk

is called the vortex

=)

intensity. LEMMA 3.2 A point vortex system has three functionally independent' first integrals: n

n

13 =

L (pf + qf). k=t

These functions in C'X) (M)j {const} form a Lie algebra isomorphic to the Lie algebra of the group Euclidean plane motions. Proof is obtained by direct calculation of the Poisson bracket. The arbitrary element of the Lie algebra E(2) of the group of planar motions can be represented in the form of a pair

The isomorphism is given by the correlations:

where it, i 2 , i3 are the images of functions 11 ,1 2 ,1 3 in the quotientalgebra C'X) (M j {const}. LEMMA 3.3 Given n = 3, the points vortex system is integrable in the non-commutative sense.

52

A. T. FOMENKO AND V. V. TROFIMOV

Proof We should note that the argument of the theorem of noncommutative integrability is retained if we move on to the subalgebras in COCJ(M)j {const}. In our case, as an example of such a subalgebra we can take the subalgebra generated by h, it, i 2 , i 3 , where h is the image of Hamiltonian H in COCJ(M)j{const}. This Lie algebra is isomorphic to the direct sum IR EB E(2) = V, dim(1R EB E(2)) = 4, ind(1R EB E(2)) = 2, and the condition of the non-commutative Liouville theorem has therefore been met: dim V + ind V = dim M. PROPOSITION 3.10 Given n = 3, the points vortex system is completely integrable in the Liouville commutative sense.

Proof It has only to be shown that V = !hi: EB E(2) satisfies the condition (FJ). For the family F 0 we can take thefunctions {h, 4ii + ii, i2Q 2 + i3 Q 3}' It is easy to confirm that they are in involution and are functionally independent on (!hi: EB E(2»*. REMARK At the present moment it has been proven that beginning with n = 4 a points vortex system is not a completely integrable one. 4. ALGEBRAICIZATION OF HAMILTONIAN SYSTEMS ON LIE GROUP ORBITS 4.1. The realization of Hamiltonian systems on the orbits of the coadjoint representation

DEFINITION 4.1 Let M and N be smooth manifolds and cp: M -+ N be a smooth mapping. We shall assume that at any point P E M the mapping cp has the maximal possible rank; then, if dim M ~ dim N, cp is called a submersion. DEFINITION 4.2 Let M and N be smooth manifolds and cp be a smooth mapping from the manifold Minto N. Let X and Y be vector fields on M and N respectively. The fields X and Yare called cp-connective if dcpp(X p) = Y
53

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Proof The vector field s grad 1 h is defined by the condition that w 1(X,s grad 1 h) = dh(X) for any vector field X E V(M d. We must check the identity wz(X, f.(s grad 1 (h f) = dh(X). Since f is a submersion, X = f.(Y) for some Y. Therefore 0

W2(X, f.(s grad 1 (h f» = wz(f.(Y), f*(s grad 1 (h f) 0

0

= (f·wz)(Y, s grad 1 (h = d(h

0

0

f)

= W1(Y, s grad 1 (h

0

f)

f)(Y) = dh(f*(Y» = dh(X),

q.e.d. 4.3 We call the mapping f:(Ml,wd -+ (M 2,w Z) a morphism of symplectic manifolds, if (a) f is a submersion and (b) f*w z = w 1· PROPOSlTJON 4.1 Let cp be a morphism of symplectic manifolds (M l' wd, (M z, w z ) and h1' h z E C""(M 2): we claim that {h1' h z } cp = {h1 ocp,h z cp}. DEFINlTJON

0

0

Proof The proposition is obtained from the following calculation: {h1,hz}z = wz(sgrad2hl,sgradzhz)

= wz(f*(s grad l (h l f», f*(s grad l (h2 f))) 0

= (f*wz)(s

0

grad 1 (h1 f), s grad l (h z f) 0

0

= W1(s grad l (h l f), s grad t (h1 f) 0

0

= {h 1 of,hzof}. Let cp be a morphism of symplectic manifolds from (M l' wd to (M 2, wz ) and let h l , ... ,hp be a set ofinvolutive functions on M z, then h1 cp, ... ,hp cp is a set of involutive functions on M 1.

COROLLARY

0

0

Thus, if there is a morphism M 1 -+ M z and a set of involutive functions is given on M 2 we can also construct on M 1 a set of involutive functions. We now want to discard the condition that (M 2 ,w Z) is a symplectic manifold; we shall instead assume that (M 2, w z ) is the union of homogenous symplectic manifolds-of the orbits of a Lie group's coadjoint representation. Naturally, in so doing, we shall have to discard the premise f·w z = w 1, insofar as there is no symplectic structure on the union of orbits, though there is on such a space a Berezin-yoisson bracket. Relying on Proposition 4.1 we shall give the foHowjrig definition. ! !

, !

I

I

I

54

A. T. FOMENKO AND V. V. TROFIMOV

4.4 A symplectic manifold (M, w) permits realization in a Lie algebra G, if in the space G* there exists a submanifold N c G* which is invariant in relation to Ad~ (i.e. N is comprised of orbits of Ad~) and if there exists a smooth mapping I: M -+ N such that {h 1,h 2h oI= {hI 0J,h 2 oI}1 for any h 1,h 2 EC""(N) where {J,gh is the Poisson bracket induced by the KiriIIov form on Nand {h, g}1 is the Poisson bracket generated by the symplectic structure w. DEFINITION

If h1' h2 are in involution on all orbits of the coadjoint representation, then hI 0 I, h2 0 I are in involution on the symplectic manifold (M, w).

COROLLARY

4.5 Let a symplectic manifold (M, w) permit realization I in a Lie algebra G and, furthermore, let a Hamiltonian system x = s grad H be given on M. We shall say that the system is realized in G with the help of J, ifthere exists on the submanifold N c G*, on which M is realized, a function such that H = HI 0 f

DEFINITION

If for the system x = s grad H 1 /0 we can find first integrals 11' ... ,1. pairwise in involution, then we can give an involutive set offirst integrals setfor the system x = s grad H: II J, ... ,1. J, since {H,,h J} = 0 by virtue of the equality {H,,h J} = {HI I,,h J} = {H 1 ,,h} I = 0, since 11""'1. are first integrals for H l ' The involutivity of II 0 J, . .. ,1. 0 I can be confirmed analogously. Mechanical systems are usually given as systems of ordinary differential equations on a Euclidean space. For example, the classical equations describing the motion of a three-dimensional rigid body with a fixed point (in the absence of gravity) are written in the threedimensional Euclidean space [R3(X, y, z) 0

0

0

0

0

0

0

where ,.1,1' ,.1,2, ,.1,3 are real numbers. Some Hamiltonian systems given on [Rn turn out to have a latent algebraic structure which, when revealed, allows them to be integrated. There are rather a lot of examples, but we shall single out a structure connected with Lie algebras. In its simplest form this means that the system we are studying preserves the orbits of the (co )adjoint representation of a given Lie group, whose Lie algebra is identified with the Euclidean space on which the system is given. DEFINITION

4.6

We shall say that the dynamic system Von a manifold

INTEGRABLE SYSTEMS ON LIE ALGEBRA

55

M n allows an embedding into a Lie algebra if M n can be identified with a submanifold N c G* in the dual space of the Lie algebra G of some Lie group ffi in such a way that (a) N = Uf O(f) is invariant with respect to the coadjoint representation of the Lie group ffi, i.e. N is the union of orbits O(f) the coadjoint representation; (b) the vector field v (after this identification) is tangent to the orbits O(f) of the coadjoint representation of group ffi on G* (i.e. all the orbits O(f) are invariant in relation to v); (c) the vector field v on N turns out to be Hamiltonian on the orbits O(f) with respect to the canonical symplectic form OJ and is realized as v = s grad f where f E COO(O(t». In this definition it is possible that N = G* in which case condition (a) is naturally satisfied automatically. REMARK

This class of systems contains important mechanical examples. Of course, it is not by any means every system that allows an embedding into a Lie algebra, considering that conditions (a) and (b) put rather severe restrictions on the structure of the field. At the same time it is clear that such an embedding's existence permits us to apply the welldeveloped apparatus of Lie algebra theory to the integrals of the system. If a system is given on IRn, one of the ways of integrating it is to seek to embed it into an appropriate Lie algebra. The element of choice that occurs in realizing this programme must be noted. Firstly, IR" may be identified with the Lie algebras of many different Lie groups, which means that all the algebras in the given dimension have to be considered. The number of such algebras increases with n. Secondly, in some cases it turns out to be enough to represent system v as Hamiltonian on one or another orbit 0* but not on all orbits. As a given Lie algebra has many orbits-(a) general position orbits; (b) singular orbits-this gives us opportunities for variation when trying to embed a given system into a Lie algebra. Let f E COO(O(t», t E G*, O(t) be an orbit of Ad*, then the Hamiltonian equations x = s grad f on the orbit O(t) relative to Kirillov's form on O(t) take the form x = ad:f(x)(x). Iff E COO(G*), then on each orbit there arise the Hamiltonian equations x = ad:!(x)(x), x E G*, f E Coo (G*). This system of equations on G* can be called Euler's equations. Euler's equations on G* have the very remarkable property that the corresponding vector field is tangent to all the orbits of Ad * and on each orbit these equations are Hamiltonian. Euler's equations are at their most interesting when the Hamiltonil"l~ f is a quadratic function on G*. /1

/

(

I

,I

A. T. FOMENKO AND V. V. TROFIMOV

56

We look more closely at this case in detail. If there is a linear operator C: G* --. G then a quadratic function can be constructed on G* : f(x) = !(x, C(x» and, obviously, the converse is true. Operator C defines symmetrical quadratic form when, and only when, (y, C(x» = (C(y),x), X,YEG*. From now on we shall always assume that C is a self-adjoint linear operator: (y,C(x» = (C(y), x). In this case it is apparent that dfx = C(x) if f(x) = !(x, C(c». Thus each self-adjoint linear operator C: G* --. G defines on G some system of non-linear differential equations x = adt(X)(x). This system of equations on G* is Hamiltonian on all orbits of Ad*; as a Hamiltonian, the restriction of f(x) = !(x, C(x» to the orbits of Ad* may be taken. We can express Euler's equations in coordinates. Let el , ... , en be a basis of G, and el, ... ,e" the dual basis in G*, xEG*, x=x1ei , C: G* --. G, C(ei) = aijej then Euler's equations take the following form: . -- a ijCkjsXiXk, (1) s = 1, ... , n = dim G Xs where

C7j

is the structure tensor of the Lie algebra G in the basis el , . . . , en' Thus, for each Lie algebra G and self-adjoint operator C we can give a non-linear system of differential equations Xi = rrXpXq where = apjC]i' In hydromechanics the tensor is also called the dynamic tensor. System (1) always has as its first integral T = !aiSxixs the energy integral.

rr

rr

LEMMA 4.2 Any function F that satisfies the system of equations in partial derivatives: k

of

CijXk-;-=O, uX j

(2)

i=1, ... ,n

(i.e. any invariant of the coadjoint representation, see 3.3) is integral of system (1). The proof is obvious, for the vector field s grad T is tangent to the orbits of Ad*. We can also cite the formal calculation:

of of

of,,

ut

uXs

--;- = -;- Xs = -;- a'lC)sxiXp uXs

= a'l"(Cl!lS X-X· OF) = 0 P::l uXs

I

•

This lemma gives us what amounts to an analog of Jacobi's method.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

57

When solving Hamiltonian equations by Jacobi's method we have to solve the equation in partial derivatives (os/ot) + H(t, q, os/oq) = 0 (non-linear, in general). In our method we likewise pass on from ordinary differential equations (1) to partial differential equations (2), but unlike Jacobi's method in dealing with linear equations, we have an algorithm for finding the solutions of such equations. Later we shall point out a special class oflinear operators for which a knowledge of the full set offunctionally independent solutions of equation (2) allows us to reconstitute the full set of first integrals of equation (1): in doing this a purely algebraic procedure of the argument translation is used. We shall write out Euler's equations for certain Lie algebras in explicit form. Let us look at Lie algebra E(2)--the Lie algebra of the group of plane motions: 1R2: E(2) = {e p e2, e3} and Eel' e2] = -e3' Eel' e3] = e 2· In this case system (1) has the following form: Xl = -a13x1x2 + a12xlx3 - a23x~ + a22x2x3 - a33x3x2 + a 32 xL X2 = -a11x1x3 - a21x2x3 - a31x~, X3 = a 11 xlxl + allx~ + a31x3xl'

! Li,j aiiXiXj'

F = x~ + x~. This system's Liouville tori are depicted in Figure 19 for the case of a positive

It has the following integrals: T =

Fig. 19.

definite kinetic energy T. Let us look at Lie algebra so(3): [e 1 , e1 ] = e3, [e 1, e3] = - e2, [el' e3] = el . In this case Euler's equations will be written thus:

58

A. T. FOMENKO AND V. V. TROFIMOV

+ a23(x~ - X~), 33 X2 = - a 23 x 2X I - a )XI X 3 + aI3(x~ - xD, ll l2 a )X1X2 + a X3 = a 32 x I X 3 (xi - x~). This system's first integrals have the form: T = ! Lj,j ajjxjx j , Xl =

a

13

x l X2 -

+ (a 33 + a 2l x2 X 3 + (all 22 a3lx3X2 + (a a

12

xlX3

a22)X2X3

xi + x~ + x~.

F=

The Liouville tori are depicted in Figure 20 (see also Figure 31) for the case of positive definite kinetic energy T.

Fig. 20.

In the general case, naturally, there are not enough of the integrals indicated for full integrability of the Euler equations. In order to obtain full integrability the type of linear operator must be restricted. 4.2. Examples of algebraicized systems

A) The realization of equations of the inertial motion of a rigid body on the Lie algebra so(3). In this case the embedding of the system [R3 which we are seeking exists and is simply arranged. It is enough to identify [R3 with the space dual to the Lie algebra of orthogonal group SO(3), i.e. with the space so(3) of skew-symmetric real 3 by 3 matrices. This Lie algebra is isomorphic to 1R3 with the vector product serving as the Lie operation. The isomorphism is given by the formula:

o

-W3

W2

o -W2

WI

0

As we know, the orbits Ad~o(3) are spheres; the ring of invariants is generated solely by the function xi + x~ + x~. Let e l , e2' e 3 be a basis of

INTEGRABLE SYSTEMS ON LIE ALGEBRA

59

1R3 = so(3), et , e2 , e3 be the dual basis, while D: (1R 3 )* --+ 1R3 is the linear operator, D(e i ) = dije j • We shall assume that d ij = d ji • Let us suppose that operator D is diagonal:

0

llA D

=

liB

o

I/C

then Euler's equations corresponding to the Lie algebra so(3) and this operator have the form (see the example in 4.1):

x

1

=

(~-~)x X C B 2

3

x2 = (~~)x A C 1 X3 X3 =

(~ - ~)XtX2.

(3)

LEMMA 4.3 Equations (3) on G* = so(3)* = (1R 3 )* are equivalent to the equations of motion of a three-dimensional rigid body with one fixed point, where the equivalence is given by the operator D: (1R 3)* --+ 1R3. Proof If we rewrite the equations (3) with the help of the operator Don so(3). We have

therefore X t = AYt, X2 = BY2, X3 = CY3, then equations (1) in the variables Yi will take the following form:

AYt = (B - C)Y2Y3 BY2

= (C -

A)YIY3

CY3 = (A - B)YtYz. If we put (Yt, Yz, Y3) = (p, q, r) we shall obtain classical Euler's equations for the motion of a solid body (see [140J): Aft = (B - C)qr

Bq =

(C - A)pr

cr =

(A - B)pq

(4)

A. T. FOMENKO AND V. V. TROFIMOV

60

where A, B, C are the body's moments of inertia with respect to the axes Ox, Oy, Oz respectively. For integrating these equations enough integrals have previously been presented: Fl = xf + x~ + x~ is the invariant of Ad*, while

is the energy integral. In mechanics textbooks there is usually a change to coordinates Yi' ending with the equations (4) having the following integrals: F = A2yi + B2y~ + C2y~ = A2p2 + B2q2 + C 2r2 is the integral of the moment of momentum, while the energy integral is F 2 = 1(Ayi + By~ + Cy~) = 1(A p 2 + Bq2 + Cr 2). We have thus constructed a realization of Euler's equations of inertial motion of a rigid body in a Lie algebra. Its uses are evident in this example: the very fact that it is realizable makes it possible to achieve integrability. Equations (2) can be rewritten on so(3) in the familiar commutator form. LEMMA 4.4 Equation (4) on so(3) is equivalent to the following system qJX = [qJX, X], X E so(3) and qJ(X) = IX + XI,

and

Al =

-A

+B+C 2

1

'

_A+B-C

11.3 -

2

Proof is to be found by direct calculation. In this form the equations may be generalized to the n-dimensional case. DEFINITION 4.7 We shall take the XESO(n) and qJ(X) = XI + IX, where

equations

qJX

= [qJX, X],

61

INTEGRABLE SYSTEMS ON LIE ALGEBRA

I=

while Ai + Aj 1= 0 for any i, j, to be the equations of motion of an ndimensional rigid body with a fixed point. Later we shall give an invariant description of the operator cp(X) = XI + IX (see 7.2). B) Realization ofToda's chain. First we shall show that the canonical symplectic space 1R 2n (qi' p;) together with (J) = dPI /\ dql + ... + dPn /\ dqn will allow realization in the Lie algebra of upper-triangular matrices. We introduce the following notations:

are the upper-triangular matrices,

the lower-triangular matrices. We have a natural non-degenerate pairing a: T + x L --+ IR, a(XY) = tr XY. This pairing lets us establish a canonical isomorphism (T+)* '" Land (L)* ~ T+. We shall be examining the Lie algebra of upper-triangular matrices T +, then (T+)* '" L. Let PI

0

SI

P2

0

A=

c L

0

It is easy to see that A

=

Pn-l

Uc A

c'

where

0

= (T+)*.

62

A. T. FOMENKO AND V. V. TROFIMOV

Ac =

PI

+ ... + Pn = C

o and Ac is the orbit of coadjoint representation of the Lie group corresponding to T+. We construct a mapping f: 1R 2n -+ A:

where

Pi

PI

0

81

P2

0

o

Pn-l

0

8n- l

Pn

= Yi,8j = eXj-Xj+ 1, i = 1, ... ,n,j = 1, ... ,n -

1.

LEMMA 4.5 The mapping f: 1R2n -+ A gives the realization of a symplectic manifold (1R 2 n, dPI /\ dq 1 + ... + dPn /\ dqn) in the Lie algebra T+ of upper-triangular matrices. Proof The equality {hI' h 2 } of = {h 1 c f, h2 J} can be checked by direct calculation. 0

By a Toda chain we mean a Hamiltonian system on T*lRn = IRn(Yt, ... ,Yn) Et> IRn(x l ' . . . ,xn) with the Hamiltonian: 1 n n-l H =-

L

2 k =1

yf +

L

ak eXk -

X

k+l.

k=1

The space T*lRn was realized by us in the Lie algebra T+. This realization turns out to provide a realization of the Toda chain as well. To do so it is sufficient to take the function HI (F) = ! tr(F + a)2 where F E A and

o a=

o as the function H 1 on A. Direct calculation shows that H = HI

a

f The

INTEGRABLE SYSTEMS ON LIE ALGEBRA

63

first integrals of the system s grad H 1 on orbits of Ad* are given, for example, in [107J: F k = tr(F

+ a)k .

5. COMPLETE COMMUTATIVE SETS OF FUNCTIONS ON SYMPLECTIC MANIFOLDS

As the preceding material shows, the following is a natural statement of the problem: how does one find a commutative set of independent functions n in number on a symplectic manifold M 2n ? In other words, how does one find on M 2n an algebra which is commutative (with respect to the Poisson bracket) and of dimension n, whose additive basis would consist of smooth functions, functionally independent almost everywhere on M? For brevity's sake we shall call these sets offunctions F(M) complete commutative sets. For the time being we are not concerned with integrating any Hamiltonian systems. If a complete commutative set is discovered on M, we automatically get a series of completely integrable Hamiltonian systems on M: it is enough to look at the fields s grad f where f is a function from set F(M). Then the ndimensional space of functions F(M) is a set of integrals for a system with Hamiltonian f This approach to the problems of Hamiltonian mechanics raises the basic problem of constructing the greatest possible number of various complete commutative sets on symplectic manifolds. The greater the store of such sets, the more examples we get of completely integrable systems (in the commutative sense). In the class of smooth functions, as A. V. Brailov has noted, the task of constructing a complete involutive family is solved very simply. To be more precise, the following proposition is valid. PROPOSITION 5.1 There exists on any symplectic manifold a complete involutive family of smooth functions which are functionally independent almost everywhere on the manifold.

Proof Each point has a neighborhood U(Pi' qi) such that w has a canonical form W=dp1"dq1+· .. +dPn"dqn' Let ii=pr+qf, It is apparent that {ii, fj} 0 for any i,j. Further, let f = LI= 1 ii = L (pr + qr)· We shall now define the smooth function g(t): IR -+ IR such that g(t) 1 when t ~ 6 1 , g(t) 0, t ~ 62 and g(t) > 0 is a monotone function of t when 61 < t < 6 2 , see Figure 21. We shall put h(p, q) = g(f(p, q))-this

=

=

=

64

A. T. FOMENKO AND V. V. TROFIMOV

Fig. 21.

function is now defined on the entire manifold (8t, 82 are chosen sufficiently small). As {h, i;} = 0, it is obvious that {hi;, hij} = 0 for all i, j. The functions hi; are defined on the entire manifold and they are zero outside a certain disk. By covering manifold Mn with a denumerable number of disks we can now stitch the functions hi; constructed above together into a set of functions k l' . . . ,kn on M n, so that each k i is distinct from zero on an open subset that is everywhere dense. We claim that hk t , . .. ,hkn are functionally independent. If F(hkt, ... ,hkn) = 0 then

F(g(k t

+ ... + k n )k t , ... ,g(k1 + ... + kn)kn) = 0

and, if function

F(g(xt

+ ... + xn)x t , ... ,g(x t + ... + xn)xn) ¥= 0

we arrive at kl' ... ,kn being functionally independent. Thus we are now left with only one thing to check, that

+ ... + xn)x t ,··· ,g(x t + ... + xn)xn ¥= O. If this function is identically zero, this means that g(x t + ... + xn)x 1 , ... , g(x t + ... + xn)xn are functionally dependent. We shall write out F(g(xt

the Jacobian of this system of functions:

o J

= det

+g 1

INTEGRABLE SYSTEMS ON LIE ALGEBRA

= 9n + - 9n -

1(

Xl

65

+ . • . + Xn ) OX' og

insofar as all invariants of the matrix

=

apart from the trace are zero. If J 0 we shall find that 9 satisfies the differential equation 9 + x(cg/cx) = 0, i.e. 9 is a linear function, which is untrue. There is one important way of making the original question bear more precisely on the problem: does a complete commu tati ve set consisting of algebraic (rational) functions exist on any algebraic symplectic manifold? There are probably topological obstructions that prohibit our constrllcting such a set on an arbitary algebraic symplectic manifold. The most natural class of functions in which to seek complete commutative sets on algebraic manifolds is the class of polynomial, rational or algebraic functions (see also §24). At the present time complete commutative sets have been discovered on an important class of symplectic manifolds-on the orbits of the coadjoint representation of many Lie groups. In [88J, [91J by A. T. Fomenko, A. S. Mishchenko hypothesis A has been formulated: let G be an arbitrary finite-dimensional Lie algebra, in which case there exists on the coalgebra G* a linear space of smooth functions whose restrictions to the general position orbits of the coadjoint representation of the group (f) = Exp G on the coalgebra G* form a complete commutative set F( 0*) on these orbits 0*, i.e. any pair of functions i, 9 E F( 0*) is in involution with respect to the canonical form w; the additive basis il' ... ,J" in F(O*) consists of functions which are functionally independent almost everywhere on 0*, and the dimension of the space F(O*) equals half that of the general position orbit, i.e. k = 1 dim 0* = 1(dim G - ind G). This hypothesis has been proved for all semi-simple and reductive Lie algebras by A. T. Fomenko and A. S. Mishchenko in [89J, [90J and also for many classes of non-compact real Lie algebras (see for example [126J, [127J, [1OJ, [134J). The complete commutative sets that were

66

A. T. FOMENKO AND V. V. TROFIMOV

discovered in verifying this hypothesis turned out to contain the Hamiltonians of important mechanical systems, which makes it possible to integrate them fully. We shall dwell on this in greater detail below. The significance of hypothesis A is not limited to the opportunity to produce integrable systems. Its validity would lead to that of Theorem 3.3 not only for compact but for non-compact manifolds as well, i.e. in this case any Hamiltonian system which is integrable in a noncommutative sense would automatically be integrable in a commutative sense too. To be more precise, the following argument is valid, as we proved in Section 3.6. Let a Lie algebra G of functionally independent functions, where dim G + ind G = dim M (see Theorem 3.2) be given on a symplectic manifold M; then, if hypothesis A has been tested for algebra G, another commutative algebra of independent functions of G' will now be found, so that dim G' = 1- dim M. We now go back to systems v, for which an embedding into a finite-dimensional Lie algebra G exists (see Definition 4.6). If hypothesis A is valid for algebra G then there is a complete commutative set of functions on orbits in general position (see above) 0* c G*. Furthermore, ifthe Hamiltonian f, where v = s grad f, belongs to the family F(O*), we then get a complete commutative set of integrals for v. Knowing the complete commutative sets on orbits in G* allows us in principle to integrate Hamiltonian systems embedded in G*. One of the ways of integrating systems on 1);l1" is thus the following: (a) we try to represent the system as Hamiltonian on orbits in G* given an appropriate choice of algebra G; (b) if there exists such an embedding of the system we try to find a complete commutative set of functions, that contains the system's Hamiltonian, on 0* c G*. There are a number of methods that allow us to construct complete commutative sets on the orbits. One extremely effective means of doing so has turned out to be a method based on the idea of shifting the invariants of the coadjoint representation along a covector in general position. At the present time there is a great deal of active work carrying on Poincare's classic researches on the questions of the nonexistence of supplementary first integrals. On this topic see op. [59], [60]. REMARK

2

Sectional operators and their applications

6. SECTIONAL OPERATORS, FINITE-DIMENSIONAL REPRESENTATIONS, DYNAMIC SYSTEMS ON THE ORBITS OF REPRFSENTATION

As we have shown in Section 4, it is possible to construct a system of non-linear differential equations corresponding to a Lie algebra G and a linear operator C: G* -+ G on G*. These equations represent wellknown mechanical systems as special cases of the construction. Besides, we can apply to this system, viz. x = ad~(x)(x) the analog of HamiltonJacobi theory (demonstrating the fact that this system is based on group theory). The system of equations constructed is a Hamiltonian one on all orbits of the coadjoint representation. If we do not have any conditions on the operator C, we cannot say anything about complete integrability ofthe equations x = adt(xix). For complete integrability of the system, we need a special construction for operators C. We shall describe this in the present section. For a Lie algebra so(n) we can obtain an invariant description (i.e. in terms of the Lie algebra so(n» of the "rigid body" operators cp(x) = XI + IX. As an example, let us construct the "rigid body" operators for some semi-simple Lie algebras first mentioned by A. T. Fomenko, A. S. Mishchenko in [89]. In [89] the use of the root expansion G = H Et> V+ Et> V- and of invertibility of the operator ada on the space V+ Et> V- was essential. If a Lie algebra is non-compact, there are not any natural analogs of the root expansion. Therefore, we need to introduce some new ideas which would enable us to include the case of non-compact Lie algebras. It turns out that in the non-compact case there are analogs of "rigid body" operators CPa,b,D (see [32,33]). Let us consider them. The following general idea of "sectional operator" was introduced by A. T. Fomenko. 67

68

A. T. FOMENKO AND V. V. TROFIMOV

Let H be a Lie algebra, Dbe the corresponding group; p: H --+ End( V) be a representation of H in the linear space V; oc: D --+ Aut(V) be the corresponding representation of the group; let O(X) denote the orbits of the action of the group Don V, X E V. If we introduce the linear operator which we shall call a "sectional operator" Q: H --+ V, the vector field XQ = p(QX)X will arise on the orbits. Note that the vector field X = p(Q(X»(X) is defined for any smooth mapping. However, here we shall not consider such a general case because for all applications Q is taken to be a linear operator. Also, having defined such opertor, we sometimes can define a symplectic structure on the orbits. The special class of those sectional operators which form a many-parameter set with two basic parameters a E V, bE Ker q,a; q,ah = (ph)a is important for applications. For example, in the particular case H = so(n), p = ad the field XQ coincides with the equations of motion of a multi-dimensional rigid body with a fixed point (in the absence of gravity). Thus, let a be any point in general position, i.e. the orbit corresponding to a has the maximal dimension. Let K c H be the annihilator of the element a, K = Ker q,a, q,a being defined above. If a is a point in general position, then the dimension of K is minimal. Let b E K be an arbitrary element. Let us consider the action of pb on V. Let us denote Ker(pb) c V by M. Let K' be any algebraic complement to K in H, i.e. H = K + K', K n K' = o. The choice of K' is not unique and hence, the set of parameters in the construction follows from the possibility of varying this complement. It is clear that a EM. From the definition of K' it follows that the mapping q,a: H --+ V transforms K' into some plane q,a(K') c Vmonomorphically. Since q,aK' = q,aH, the plane q,aK' does not depend on the choice of K', being defined uniquely by the choice of the element a and the representation p. Let us suppose that there exists an element b such that V can be expanded as the sum of two subspaces M and Im(pb), i.e. V = M EEl Im(pb). For example, we can take the semi-simple elements in K as b. Let us denote the planes which are formed by the intersection of q,aK' with M and Im(pb) by Band R' respectively. Thus, we have obtained a decomposition of q,aK' as a direct sum of three subspaces B + R' + P, Band R' being uniquely defined. At the same time, the complementary subspace P can be chosen in several ways, introducing a new set of parameters. Let us consider the action of pb on lm(pb). The pb maps Im(pb) into itself isomorphically (see Figure 22 where pb is invertible on Im(pb». Let (pb) -1 be the

INTEGRABLE SYSTEMS ON LIE ALGEBRA

69

1:

Q.

Fig. 22.

operator which is inverse to pb on Im(pb). Assuming R = (pb)-1 R' we obtain pb: R -+ R'. The space R is uniquely defined. Let us consider in Im(pb) the space Z which is the algebraic complement of Ron Im(pb). Then Im(pb) = Z + R'. Let T be the complement of Bin M. Thus we have constructed a decomposition of the space Vas a direct sum of four planes V = T + B + R + Z; R, B, M, Im(pb) being uniquely defined, whereas the choice of Z and T is ambiguous and therefore introduces a new set of parameters. Given a scalar product in V, Z and Tare uniquely defined as orthogonal complements. Since K' is isomorphic to ¢aK', K'

= fj + R + P, fj = 'A.f-1 B' R = tA.p -1a R' P = '+'a' A. -1 P 'a

Thus, we have defined a many-parameter decomposition of the algebra H as a direct sum of four subspaces K + fj + R + P. Let us define the sectional operator Q: V -+ H, Q : T + B + R + Z -+ K + fj + R + P by setting

o Q= o o

o

o

¢a- 1

0

0

¢a- 1 p(b)

o

o

o 0

K being any linear operator, ¢a- 1 : B -+ fj being the operator inverse to ¢a on the subspace B, ¢;;1p(b):R-+R, p(b):R-+R', ¢a- 1 : R' -+ R, D': Z -+ P (see Figure 22). Thus the operator Q is of the form Q(a, b,D,D'). Let us construct the dynamic system XQ = p(QX)X, X E V. We choose a as the point in general position in V because in this case the dimension of K' is maximal, i.e. the operators ¢a- 1 p(b) and ¢a- 1

D: T

-+

70

A. T. FOMENKO AND V. V. TROFIMOV

are defined in a maximal space. Let us note some important examples of the construction defined. If V= H*, p = ad*:H ~ End(H*), then cp;;lp(b) = cp;;l ad:. For example, taking as H the non-compact algebra so(n) EB IR" which is the Lie algebra of the group of motions in the Euclidean space, we find (see below) that the system XQ = ad~(x)(X) transforms into the equations of inertial motion of a rigid body in an ideal fluid. Given the obvious identifications of Hand H*, we obtain K = K*, Z = i = 0, R = R. Let m/5 be a compact symmetric space. Then, the Lie algebra G may be expanded as a sum of spaces H + V, H being a stationary subalgebra; V being a subspace tangent to (fj/5; the subalgebra H acting on V coadjointly. Then, the expansion V = T + B + R + Z which defines the sectional operator, is such that T is a maximal commutative subspace in V, aET, R=R', Z=O, bEK, ¢aK'+T=V= T + B + R. Provided that C: V ~ H is a sectional operator, we obtain the exterior 2-form F dX, ~,IJ) = B(CX, [~, IJ]); ~,IJ E 7',;0 on the orbits O(X) c V.

There are wide ranges of symmetric spaces and sectional operators for which this form defines (almost everywhere on the orbit) a symplectic structure which is non-invariant to the action of the group. Let us return to the general case. Let~, IJ E 7',;0 be tangent vectors. Then, the vectors IJ',~' E K'(X) such that p~' = ~, PIJ' = IJ exist and are defined uniquely. Let us take the sectional operator C: V ~ H. Let us define the bilinear form Fe = B(CX, [~', IJ']), [~', IJ'] E H, CX E H. This form is defined on the orbits and is skew-symmetric. Also, we have defined the flow XQ on the orbits. PROBLEM A For what operators C is the form degenerate on the orbits?

Fe

closed and non-

PROBLEM B For what C and Q is the flow XQ a Hamiltonian one with respect to the fo rm Fe? It turns out that in the case of symmetric spaces there are complete

answers to these questions (A. T. Fomenko). For example, let us consider the symmetric space SU(3)/SO(3). Then, the equations XQ on the plane B coincide with the Euler equations of motion for a three-dimensional rigid body with a fixed point and arbitrary inertia tensor. Note that among the systems XQ = ad~(x) X which we have constructed, there are the Hamiltonian equations of

INTEGRABLE SYSTEMS ON LIE ALGEBRA

71

motion of a rigid body with a fixed point (for any n, not only for n = 3). To confirm this, it is enough to take the semi-simple group ~ as a symmetric space. Then, it can be written in the form ~ x N~; the involution u: ~ x ~ ..... ~ x ~ is defined as a(x, y) = (y, x). The corresponding expansion of the Lie algebra G = H + H is of the form V = (X, -X), X EH; H = (X, X), X E H (both H and its image in G are denoted by the sameletter), uV = - V, uH = H. It is easy to verify that the form constructed above is transformed into a canonical symplectic structure on the orbits of the coadjoint representation; the field XQ (D' = 0) being transformed into the "rigid body" equations which we wanted to obtain. Thus, we have found a "multi-dimensional" series of dynamic systems which contains the equations under study. They are of interest to us due to the fact that they are abo defined on non-compact Lie algebras, being at the same time the natural analogs of "rigid body"like systems.

7. EXAMPLES OF SECTIONAL OPERATORS 7.1. Equations of motion of a multi-dimensional rigid body with a fixed point and their analogs on semi-simple Lie algebras. The complex semi-simple series

Let G be a semi-simple Lie algebra, B(X, Y) be the Cartan-Killing form, f be a smooth function on G. Let us associate with this function a dynamic system on the cotangent bundle T*(!) ofthe group by extending f to a left-invariant function F defined on the whole space T*(jj. Since T*(!) is a symplectic manifold, we can obtain a Hamiltonian system on T*(!) taking F as the Hamiltonian. This system is left-invariant and can be divided into two systems, one of which is defined on the cotangent space at the identity element of the group which is isomorphic to the Lie algebra G and is usually called a system of Euler equations. These equations can be described simply. Let grad f be the field on G which is dual to the differential df, i.e. B(grad f, e) = eU). Then, the Euler equations can be written in the commutator form X = [X, grad f(X)]. The case of geodesic flows for the left-invariant metrics on (!) is of particular interest. Here, f is a non-degenerate form on the algebra G; grad f(X) is defined by a linear operator in G, i.e. grad f(X) is of the form qJX; qJ: G -+ G being self-conjugate.

72

A. T. FOMENKO AND V.

v. TROFIMOV

Let G = so(n) be the Lie algebra of an orthogonal group, let the diagonal real matrix

o 1=

o

,1.2

Let us consider on so(n) the operator t/I(X) ~ IX + XI. Then, we call the equations t/lX = [X, t/lX] the equations of motion of an ndimensional rigid body. Let us write them in an explicit form using standard coordinates in so(n). Let us represent so(n) as the algebra of skew-symmetric matrices (real-valued) X = (xu) then t/I(X) = «Ai + A)x;). It is clear that

Assuming n = 3 we can obtain

As we have explained above, so(3) can be identified with [R3. Then, these equations coincide with the classical equations of motion of a 3dimensional body (see above). It is for this reason that we have called the equations t/lX = [X, t/lX] (for any n) the equations of motion ofa multidimensional rigid body. Let us choose I in such a way, that Ai + Aj "# 0 for any i,j. Then, the operator t/I is invertible on so(n) and the inverse operator t/I-1 = q> is of the form (q>X)ij = [l/(Ai + ..1)]Xij' Let us substitute the coordinates Y = t/lX in t/lX = [X, t/lX] then, the equations of motion of rigid body are transformed into Y = [t/I-l Y, Y]. Multiplying it by (-1) and denoting Y by X, we obtain the Euler equations X=[X,q>X],

q>: so(n)

-+

so(n)

being a linear self-conjugate operator. In what follows, we shall consider namely this form of the equations. In coordinate notation, we have

INTEGRABLE SYSTEMS ON LIE ALGEBRA

73

Above, having assumed n = 3 we obtained the embedding of this system in the Lie algebra so(3). A similar embedding exists for any n. •

1

7.1 The vector field X = [X, cpX], cP = "'- , '" X = I X + X I is tangent to the orbits of the coadjoint representation of the group so(n) on its Lie algebra so(n). This field is Hamiltonian on all orbits. PROPOSITION

Proof Let 0 be an orbit, the point X E so(n) = so(n)* belonging to the orbit. Then, TxO = ([X, y], Y E so(n)} therefore [X, cpX] E TxO. We have X = sgradF, F(X) = <X,cpX). This completes the proof.

Let us define the analogs of the equations of motion of a rigid body on an arbitrary semi-simple Lie algebra. We have a many-parameter set of the operators cp: G --+ G not only for the complex semi-simple Lie algebras but also for their real compact and normal forms. Thus, all systems X = [X, cpX] are completely integrable on the orbits in general position and hence their integrals define complete commutative sets of functions on both semi-simple Lie algebras and on their real forms. Let G be a complex semi-simple Lie algebra; G = T EB V+ EB V- be the root expansion of the algebra. Let a, bET, a # b be two arbitrary regular elements of the Cartan subalgebra. Let us consider the operator ada: G --+ G.ltisclearthatadal T = O,ada: V+ --+ V+ ,ada: V- --+ V- ,i.e. this operator preserves the root expansion of Gover C. Indeed, ada E, = oc(a)E, for any oc E~. We assume that a and b are in the "general position," i.e. oc(a) # 0, oc(b) # 0. Then, the operators ada and ad b are invertible on V+ EB V- = V, namely, ad a- 1 E, = E.joc(a). Let us define the linear operator CPa,b,D: G --+ G as follows (according to the general rules in Section 7): CPa,b,D(X) = CPa,bX' + D(t) = ad a- 1 ad b X' + D(t), X = X' + t being the uniquely defined extension of X to V and T; D: T --+ T being any linear operator symmetric with respect to the Killing form on T. The parameters a, band D belong to the operator CPa,b,D' It is clear that CPa,b,DE, = [oc(b)joc(a)]E,. In the Weyl basis (E., E _.' H~) the operator cP is defined by the matrix

74

A. T. FOMENKO AND V. V. TROFIMOV

0

),1

Ea Aq

= qJa,b,D

Al E_~

Aq

0 H'a qJab:

q

V --+ V;

A "

= lX(b) . lX(a) ,

= dim V± = (the number of the roots IX > 0).

The operator qJa,b,D is symmetric with respect to the Killing form if a, b, D satisfy the conditions given above.

PROPOSITION

7.2

Proof Let us denote the Weyl basis in V by (eJ It is enough to verify that B(qJei , e) = B(e i , qJe) for any i, j. We can assume i oF j. Remember that the plane T is orthogonal to the plane V = V+ EB V-. As qJ transforms Vand T into itself and D is symmetric on T, it is enough to check that qJa,b is symmetric on V. Since Ea (IX oF 0) are the eigenvectors

of qJ,

B(:~!~ Ea,Ep) = B( Ex, ~~!~ Ep) = 0 B(Ea,Ep O. If IX +

(if IX

+ f3 oF 0),

f3 = 0, then lX(b) (-IX)(b) lX(a) ( -1X)(a) .

This completes the proof.

In the "general position" case, the operator qJ on V has q distinct eigenvalues which are multiples of two. The operator qJ: V --+ V is an isomorphism of V with itself. Remember that V+ is a nilpotent subalgebra. Since V+ is generated by the vectors Ea , IX > 0, qJl v+ is symmetric with respect to the Cartan-Killing form B(X, Y). The eigenvalues of this operator in the "general position" case are distinct:

75

INTEGRABLE SYSTEMS ON LIE ALGEBRA

AI' ... ,Aq. We shall call this series the normal nilpotent series of the operator cp: v+ -+ V+. According to our construction, each complex series corresponds to one normal nilpotent series. The operator cp:G-+G maps the subalgebra V+ EBT into itself, CPIV+$T being an isomorphism of the space with itself. All eigenvalues of the operator cpl v+ $T are distinct and the operator is symmetric. This series of the operators is called a normal solvable series. In the Weyl basis, the operators cpl v+ and cpl v+ $T are expressed by the matrices Al

0

0

cpl v+ =

cpl v+ $T = 0

Aq

0

0

Aq

0

Thus, we have constructed Hamiltonian systems X = [X, CPa,b,D(X)] for each semi-simple Lie algebra (A. T. Fomenko, A. S. Mishchenko). These systems are analogs ofthe equations of motion of a rigid body and are completely integrable (see below). In particular, we obtain the equations on the Lie algebra so(n) (S. V. Manakov).

7.2. Hamiltonian systems of the compact and the normal series

We shall consider the set of Hamiltonian systems on the arbitrary compact real Lie algebra using real forms of the complex simple Lie algebras. Every complex semi-simple Lie algebra G has the compact form Gu (see e.g. [47,50]). Remember that Gu

= {E~ + E_~,

i(E~ - E_ a ), iH~}

=

W+ EB iTo.

As in the previous section, we define the symmetric operator cp: Gu -+ Gu which defines the Hamiltonian system X = [X, cpX] on Gu which preserves the foliation of Gu by the orbits of the coadjoint representation. Let a, b E iTo be elements in general position. Since (a

ada E~ = o:(a') being real. Hence,

ia(a')E~,

= ia', a' E To),

76

A. T. FOMENKO AND V. V. TROFIMOV

ada(E~

+ E_~) =

Ot(a')(i(E~ - E_~)),

ada(i(E~ - E _~)) = - Ot(a')(E~

+ E _~).

Thus, the operator ada: W+ -+ W+ rotates the vector E. + E_. into a vector which is proportional to i(E~ - E _.) and vice-versa. The operator ad b acts similarly; it is invertible on W+ due to the choice of a E iTo. Then, all vectors E~ + E_~, iCE. - E _.) are eigenvectors of the operator ({Ja,b = ada- 1 ad b : W+ -+ W+ with the eigenvalues Ot(b)/Ot(a) = Ot(b')!Ot(a'), a = ia', b = ib', a', b' E To. Similarly for the operators on the subspace W-. Let us define the operator ({Ja,b,D: Gu -+ Gu as follows: ({JX = ({J(X' + t) = ((Ja,b(X') + D(t) = ada- 1 ad b X' + D(t), X = X' + t being the uniquely defined extension of X to Gu = W+ EB iTo, X' E W+, t E iTo; D: iTo -+ iTo being an arbitrary linear operator which is symmetric on iTo. In the basis ((E~ + E _.), iCE. - E _.), iH~) the operator ({J is defined by the matrix

o o

o o

iH~ A.~ =

o

= ({Ja,b,D,

D

Ot(b)/Ot(a) being real; q = dim W+.

7.3 The operator ({J: Gu -+ Gu is symmetric if a, b, D satisfy the conditions given above. PROPOSITION

Proof The arguments are similar to the proof of Statement 7.2. The

only point we need to check is the orthogonality of the basis chosen in W+. Remember that iTo is orthogonal to W+. Then, we have B(E. + E_., iCE. - E-J) = O. The orthogonality of the rest of the vectors is known. In the general position case, the operator ({Ja,b: W+ -+ W+ has distinct eigenvalues which are multiples of two. Let us construct a similar set of Hamiltonian systems on some simple compact real Lie algebras which correspond to the classical normal compact subalgebras in the complex semi-simple Lie algebras. In any

77

INTEGRABLE SYSTEMS ON LIE ALGEBRA

compact form let us consider the subalgebra Gil which is called a normal compact subalgebra. This subalgebra is generated by the vectors Ea + E _" il EA. Since all these vectors are eigenvectors corresponding to the operator ({J of the compact series, we get the normal series if the operators are restricted by the subalgebra Gil. These operators coincide with ({Jab: Gu -+ Gu, ({JX = ad a- 1 ad b X, X E Gil; a, b E iTo, ilea) i' 0, il(b) i' 0. In the basis (E, + E-a) the operators ({J are defined by the matrices

q

({Jab =

= dim

W+.

Note that here a, b rt G u , i.e. to define the operators of the normal series we need the elements of some extended algebra. This is the difference between the normal series and the complex and compact ones, in the latter case the elements a and b belong to the algebra itself. Not any compact semi-simple algebra can be represented as Gil in some compact real form Gu c G. A complete list of all these simple Lie algebras is given below. The algebra Gil coincides with the fixed points of the automorphism r: G -+ G, rX = X when the latter is restricted to G u • Let P c Gu be a subspace which is orthogonal to Gil in Gu , r = -1. Then, the following commutativity relations are evidently valid: [G n , GIIJ c Gil' [P,PJ c Gn, [Gn,PJ c P. Thus, the symmetric space ij\/GJ II is defined. Then, the space P is identified with the tangent space of GJulGJ II , the latter is embedded canonically in GJ u as the Cartan model (see [48, 68J). Let us write down all the normal forms using the standard notations for the corresponding symmetric spaces (see [48J). AI G = sl(n, C), G u = su(n), Gn = so(n), (JX = X, n> 1. The algebra Gn is given in Gu as the subalgebra of real skew-symmetric matrices. FORM

BDI G = so(p + q, C), so(p, q) is the Lie algebra of the component of the unit of the group SO(p, q). The algebra so(p, q) is FORM

2),

+ q, IR)

by the matrices (X l X all Xi being real, X 2 X3 Xl' X 3 being skew-symmetric with the order p and q, X 2 being arbitrary. Then, realized in sl(p

G u = so(p

+ q)

::::>

so(p)


p> 1,

q> 1,

p

+ q "# 4.

78

A. T. FOMENKO AND V. V. TROFIMOV

The normal forms correspond to the following values p = q and p = q + 1, i.e. Gn = so(p) EB so(p) and Gn = so(p) EB so(p + 1).

X2)

. (Xl 1,sp(n, /R)Isthealgebra t ,Xi X3 -Xl . being real with order n, X 2 and X 3 being symmetric. Then, Gu = sp(n), G" = u(n), the embedding Gn --+ Gu is given as follows:

FORM

CI

A + iB

--+

G = sp(n,q,n

(A

~

B), A + iB E u(n), A and B being real.

-B A

These are all the normal forms G" c Gu where G u is a classical simple Lie algebra, i.e. one of the forms A", Bn, C", D". Apart from these forms, there are also several normal forms which are generated by the special Lie algebras (these we omit). In conclusion, we show that among the Hamiltonian systems of the normal series there are the classical equations of motion of a multidimensional rigid body with a fixed point (see 7.1). Let us consider the algebra so(n) which we represent as the normal form in the algebra su(n) (see above). Let us embed su(n) in u(n) in a standard way and consider two regular elements a, b of the Cartan subalgebra iTo in u(n) (not in su(n)!). Let

o)

= diag(ia l , . . . , ian)'

la"

o ) = diag(ib l , ... , ib,,), ib" /R; a i i= + a; bi i= acts as follows:

ai' bi

E

+ bj (i i= j). Then, the operator

({Jab:

Gn

--+

Gn

Since each root a is given by a pair of indices (i, j), Le. !X = !Xij (see e.g. [11]), each eigenvector E2 + E - 2 which corresponds to the pair (i, j), is multiplied by the eigenvalue Au = (b i - bN(a i - a). Thus, when ({J acts, the basis skew-symmetric matrices

79

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Eij

= 7;j - 1j;

=( 0 ... 01) -1

are multiplied by Aij. Therefore, the Hamiltonian system is of the form .

xij

~ = i...

X;qXqiAqj -

Aiq)

~ = i...

q= 1

Let

a

= - ib 2 , i.e.

XiqX qj

q= 1

ap

(b q aq

-

bj aj

-

X=

[X,
bi - bq ) a; - aq

.

= b;. Hence,

xij = L. n

q=1

XiqX qj

(1+ aj

aq

-

ai

1) + aq

.

Then, when a = ib 2 we obtain the system of equations of motion of a rigid body with a fixed point which is already familiar to us (see 7.1). Moreover, among the operators <Pab of the normal series there is the classical operator !/IX = IX + XI, I being a real diagonal matrix. Indeed, assuming b = - ia 2 , we obtain

I.e. <PabX = IX + XI

(I = -ia).

Thus, we have included the classical Hamiltonian system ofthe motion of rigid body with a fixed point (without potential) in a many-parameter set of similar Hamiltonian systems which are naturally defined on the simple compact Lie algebras. 7.3. Equations of inertial motion of a multi-dimensional rigid body in an ideal fluid

We shall give explicitly an embedding of such equations in the noncompact Lie algebras of the group of motion of Euclidean space. Then, the system will be a Hamiltonian one on the orbits in general position and will be completely integrable (see works of the authors [123], [124], [125]). We will construct completely integrable linear operators using the general construction of the sectional operators from Section 6. Let C(n) be the Lie group of rigid motions of 1If'. It is known that C(n)

80

A. T. FOMENKO AND V. V. TROFIMOV

is the semi-direct product of the group SO(n) and the commutative subgroup :! of translations, the latter being a nonnal divisor in the group C(n) and isomorphic to n-dimensional Euclidean space. A matrix representation of the group C(n) is of the fonn

SO(n)

o ...

0

1

The action of the group SO(n) on the nonnal divisor:! coincides with the standard action of SO(n) in [Rn. Hence, the Lie algebraE(n) of the group C(n) is a semi-direct sum so(n) (fJ", [Rn, q>: so(n) --+ End [Rn is the differential ofthe standard representation ofSO(n) in [Rn; [Rn is regarded as a commutative subalgebra. In a matrix notation E(n) is of the Conn

so(n)

o .. ,

0

The commutation operation in the Lie algebra E(n) is of the fonn [x

+ ~,Y + r,rJ = [x,yJ + x(r,r)

- y(~),

x(r,r), y(~) being the results of the action of the matrices x and Y on the vectors r,r and ~ respectively; so(n) acting on [Rn in the standard way. Let us identify the space E(n)* which is dual to E(n) with E(n). Let us define a non-degenerate scalar product in E(n) (non-invariant). We have E(n) = so(n) (fJ [Rn as linear spaces. Let B(X, Y) be the Killing fonn on so(n); (X, Y)e be the Euclidean scalar product in [Rn. We can assume «X 1 'Yl),(X2,h» =

B(X 1,X2) + (Yl'YZ)e,

X1,xzEso(n), Yl,Y2E[Rn. Let us regard all subspaces in E(n)* as subspaces of E(n) using the canonical identification which was given above. Let us calculate explicitly the fonn of the operation ad* under the isomorphism E(n)* = E(n). Remember that we denote adt(x), ~ E G, x x G* as a(x, ~).

81

INTEGRABLE SYSTEMS ON LIE ALGEBRA PROPOSITION 7.4

~Eso(n),

Let

S E so(n)* = so(n),

XE[RU,

ME ([RR)* = [Ru, ~

+

XE

so(n) EBcp [Ru

= E(n),

8 + ME E(n)* = (so(n) EB IRU )* = so(n) EB IRn. Then, a(8

+ M, ~ + x)lso(n) = [8,~] + t(Mx t a(S + M,~ + x)lnn = -~M.

-

xM

t

),

M, a E IRU are expressed as column vectors, (t) denoting transposition. The prooffollows directly from the definition of the operation a(x, ~). Let G be an arbitrary Lie algebra. Remember that the Euler equations on G* are the system of differential equations x = a(x, C(x», C: G* -+ G being a linear operator, a(x,~) being the linear functional described above. The flow x = a(x, C(x» flows along the orbits of the co adjoint representation Ad* of the group Gl. On these orbits, the Euler equations are the Hamiltonian ones with respect to the canonical symplectic structure. Let us apply the above construction to the construction of sectional operators for the coadjoint representation of the algebra E(n). We obtain a many-parameter set of sectional operators Q: E(n)* -+ E(n) for which the Euler equations are completely integrable Hamiltonian systems on the orbits of the coadjoint representation of the group Io'(n). Let us consider in E(n)* (and hence, in E(n» the subspace [(u+ 1)/2]- 2

K =

EB

k=O

IR(E 2k + 1 ,2k+2) EB IRen c E(n),

being elementary skew-symmetric matrices, ei being the standard orthonormal basis in IRn. The corresponding subspace in E(n)* we denote by K*.

Eij

PROPOSITION 7.5 The subspaces K and K* allow the invariant description K = Ann(x 1) = {~E G, a(x,~) = OJ, G = E(n) and K* = {~' E G*, a(~', X2) = 0; Xl E G* and X2 EKe G being elements in general position. The proof follows from Proposition 7.4. Let us denote the orthogonal complement to the subspace Win E(n) or E(n)* with respect to the scalar product R(X, Y) + (Z, R)e by W-".

A. T. FOMENKO AND V. V. TROFIMOV

82

LEMMA 7.1 Let a E K*. Let us consider the mapping <Pa: E(n) --+ E(n)*, x--+a(a,x)EE(n)*. Then, <PaK1.cK*\ K1. being the orthogonal complement to K, KU being the complement to the subspace K*. The proof is obvious. If a is in general position, then K = Ker <Pa and <Pa: K 1. --+ KU is an isomorphism. Therefore, the inverse mapping <Pa- 1 : K*1. --+ K 1. is defined. We have the splitting as a direct sum of linear spaces E(n) = K1. EB K and E(n)* = K*1. EB K1.. According to the general method, let a E K*, bE K, a being in general position. If z = x + YEE(n)*, x E K*1., YEK*, then Q(a,b,D)z = <Pa- 1 adt x + D(y); D:K* --+ K being arbitrary. Now, we can write down the general equations XQ = ad~x(X) on G* = (so(n) EB IR")* ~ so(n) (f) 1Rn, Q(a, b, D) being the sectional operator which we have constructed and which is a non-compact analog of the operators CPa,b,D which describe the motion of a rigid body. In our case Q(a, b, D): E(n)* --+ E(n). Therefore, XQ is defined on the space G*. The equations XQ = ad~(x)(X) can be written explicitly as

s.= [S, ~J + 1(Mx' {M= -~M

xm' )

( 1)

IRn being functions of the elements S, M, their dependence being defined by the operator Q(a, b, D), i.e . .; + x = Q(a, b, D)(S + M). Here S + ME E(n)*, .; + X E E(n). Since the operator Q(a, b, D) is defined explicitly, it is easy to obtain an explicit expression for the dependence of'; and x on S,M,a,b,D. .; E

so(n), x E

PROPOSITION 7.6 (Fomenko, A. T.; Trofimov, V. V.) The system of differential equations x = ad~(a,b,D)X(X) on the coalgebra E(n)* (written explicitly as the system (1)) is a Hamiltonian one on the orbits of the coadjoint representation Ad*(C(n)). Moreover, when n = 3, the system coincides with the equations of inertial motion of a rigid body in an ideal fluid. Thus, this system allows an embedding in the noncompact Lie algebra E(n) in the sense of Definition 4.6. Therefore, we can say that the equations (1) describe the inertial motion of a multi-dimensional rigid body in an ideal fluid for any n. Before we prove Proposition 7.6, let us remember the classical equations of inertial motion of a 3-dimensional rigid body in an ideal fluid (for details, see e.g. [39J). Let us introduce a moving frame of reference. Let U, be the components of the linear velocity of origin a moving frame and

INTEGRABLE SYSTEMS ON LIE ALGEBRA

83

Wi be the components of angular velocity of rotation of rigid body. Then,

the kinetic energy of the rigid body and fluid system is of the form T = !(Aijwiwj

+ Bijuiu j ) + Cijwiu j ,

A ij , Bij , Cij being constants which depend on body form and density of body and fluid; the summation from 1 to 3 done over subscripts which occur twice. For example, let the body have three mutually orthogonal axes of symmetry. Then, it is easy to check that all factors Aij which have i not equal to j are equal to zero and T = 1(A11

u; + }'22U; + ).33U; + A44W; + A55W; + ).66W;),

Foraspherewithradiusb:).ll = 2npb 3 /3,A22 = A.55 = ).66 = O. Thus, in this case 3

npb 2 T= -3-(ux

).33

= 2npb 3 /3,A 44 =

2 + uy2 + uJ.

Let N = (Yl' Y2, Y3); Yi = OT/OWi, K = (Xl' X2' X3), Xi = oT/ouu • Then, inertial motion of a rigid body in an ideal fluid can be described by the equations dN

-=Nxw+KxU dt dK -=Kxw dt '

(2)

The kinetic energy of a rigid body is an arbitrary positive definite homogenous quadratic form in six variables ui, Wi' Consequently, it is defined by the 21 factors (Aij' B u , C i) = (Ai)' The equations (2) have three classical Kirchhoff integrals in the general case. Therefore, for complete integrability of the equations (2) to be attained, we need one additional integral, functionally independent from them. Let us remember the classical Kirchhoff integrals. Taking the scalar product of the first equation in (2) by K, we obtain K(oK/ot) = O. Consequently, const = K2 = K; + K; + K;. Taking the scalar product of the second equation by N and of the first by K and adding them, we obtain N(oK/ot) + K(oN/ot) = O. Therefore, NK = KxNx + KyN y + KzN z = const. Evidently, the third integral is the energy integral T = const. Indeed, taking the scalar product of the

84

A. T. FOMENKO AND V. V. TROFIMOV

second equation by U and of the first by wand adding them, we obtain

oK

aN

U-+w-=O. at at

(3)

On the other hand, T is a quadratic form in Ux ' uy,u., w x , Wy, Wz . Therefore, according to Euler's theorem of homogeneous functions we have

= UK +wN.

Differentiating this equation with respect to t and using (3) we obtain dT au ow 2-=K-+N-. dt at at We have dT

aT DUx

aT oU

aT oU

- = - - + - -y+ - -z dt oUx at oUy at oUz at aT aw oW x at

aT oW oW y at

aT oW oW z at

+ - -x + - -y + - -z au ow =K-+N-. at at Thus, 2(dT/dt) = (dT/dt). Hence, dT/dt = 0, which completes the proof. Since the dimension of orbits in general position of the coadjoint representation of the Lie algebra ~(n) is equal to 4, four independent integrals (two of them define the orbit and two are not constant on the orbits) are enough for complete integrability of the system. Recall the three classical cases when this fourth additional integral is available (for a complete review see e.g. [39]). The first general solution of the equations of motion of a rigid body in a fluid was obtained for the function T (kinetic energy), using which it was possible to obtain the fourth integral, given the first three. Halfen suggested a solution of the first case of Clebsch, i.e. when the integral is, generally speaking, a linear homogeneous function of the variables Xi' y;. In 1978 Weber studied the second case of Clebsch, i.e. when the fourth integral is a homogeneous quadratic function of Xi' y;, with some

INTEGRABLE SYSTEMS ON LIE ALGEBRA

85

particular conditions on the arbitrary constants. Ketter obtained the general solution ofthe latter problem. V. A. Steklov discovered the third kind of kinetic energy with equations (2) integrable explicitly. Using variables Xi' Y;. the kinetic energy can

1. First case of Clebsch

be expressed as

T = 1bll(xi

+ xD + 1b33X~ + b I4 (XIYI + X2Y2) + b36 X3 Y3 + 1b 44 (yf + Y~) + 1b66Y~'

The equations of motion (2) allow the fourth linear integral with the variables Xi Yi' With this energy, the body does not change its form rotated around the z-axis through an angle of n12. Assuming b l4 = b 36 = 0, we have a rotating body. The equations of motion with the kinetic

2. Second case of Clebsch

energy T

= bllxi + b22X~ + b33X~ + b44 yi + b55Y~ + b66Y~

and the expression for the coefficients b

- b b44

22 33 + -=c=------==

b33

-

b55

bll

+

bll - b 22 0 = b66

allow the fourth integral which is a homogeneous quadratic function. The rigid body considered is symmetric with respect to three mutually orthogonal planes. 3. Case of Steklov of the form 2T =

The equations (2) are defined by the kinetic energy

L bllXI + 20" L b55b66XI Yl + L b44 y;,

bii being defined by the relations bll

=

0"2b44(b~5

+ b~6),

b 33

= 0"2b66(b~4 + b~5)'

Apart from the three classical mtegrals, these equations allow the natural homogeneous integral of the second order with respect to Xi' Yi' Here, 0" is an arbitrary constant, the symbol L means summation over the three expressions obtained by cyclic permutations of the groups of subscripts 1,2,3; 4,5,6.

86

A. T. FOMENKO AND V. V. TROFIMOV

Let us prove Proposition 7.6. LEMMA 7.2 Let us consider the mapping l/!: so(3) --+ 1R3 which maps the element X = XE12 + yE 13 + ZE 23 to the point (z, -y,x). Then, l/![X, Y] = -l/!(X) x l/!(Y), l/!(X) x l/!(Y) being the vector product in 1R 3 , Further, l/!(Mxt ~M

xMt)

-

=

= M x x,

-l/!(~)

x M,

Proof If Z = ~ + X E E(3), Z = S Proposition 7.4, a(Z, z) = (y, X); y=

[S,~]

+ 1(Mxt -

M,x ~E

1R 3 ,

so(3).

+ ME E(3)*

xMt) E so(3);

E

then, as shown

X =

10

-~M,

The vectors M and x being written as column vectors, the Lie algebra so(3) is given by the skew-symmetric matrices. This completes the proof. Let us write down the operators Q(a, b, D) constructed above for the simplest three-dimensional case when E(3) = so(3) EB 1R3 (in this case many higher-dimensional effects are absent which facilitates writing in an explicit form).

K~K'~ KJ. = K*J. =

(;,

(~ -X2

Let

87

INTEGRABLE SYSTEMS ON LIE ALGEBRA

o o o o al

o

0

o

o

b U2

2 al

--u b I 2

$(

0

Finally Q(a, b, D)

~(_Ofl \

-f2

0 -

-all - {3u 3 a

l --u 2 b2

afl

+ {3u 3 0

at

b u2

2

2 al

al

-u 2

b2

b2

al --u b2 I

0

E!1

f _ 3

-2 a l f b2 2

UI

_

yfl

b2a2

u2

+ 2b l a l ~ b~

b 2a2 + 2b lal b~

+ bU 3

a, {3, y, b being the constants which define the operator D: K* --+ K. The kinetic energy is of the form <X, Q(a, b, D)(X». The matrix of this quadratic form is as follows

)

88

A. T. FOMENKO AND V. V. TROFJMOV

y

-2a

o

o

o

o

o

o

o

o

o

0

0

a1

o

o

0

-l b2

2a

o

l -b2

--p 2

0

y

a

b1

-b 2 a1

+ 2b l a l

o

b~

al b2

o

0

o

-

o

It is clear that

det A = -a ib2'4(

--p 2

o

o o

r

G- P

+ 2c5a)'

i.e. we can change the sign of det A by the appropriate variation of e.g. the operator D. Writing A. = b;2(b 1 a2 + 2b l a l ) we obtain the diagonal form of the form T

from which we can see th~t the form T is not positive definite. The system of differential equations on E(3)* for the kinetic energy T = X(Q(a, b, D)(X)) is integrable explicitly since it is of the form

Xl = -ax 2 Y3 -

(p + ~JX2X3'

89

INTEGRABLE SYSTEMS ON LIE ALGEBRA

X3 = 0,

Yt = yz

=

(f3 + at) (at -"21') 1'2: ) + (at) + (at b f3 + b b

-OCY1Y3 -

OCYtYJ

Y2 X 3 -

1

b

2

XtY3

-

1

2

Y3 =

X 2 Y3

Yt X 3 -

(b

+ "2 +"2A) X2 X 3,

(A2: +"2b) X t X3,

0, A = (b2al

+ 2b 1 a 1 )/bL

Xi

= aT/aui ,

being the linear velocity, Wi being the angular velocity of rotation of the rigid body, U

i

Y3

X

E

E(3)*: X =

(_oYJ

Y2

°

If the dimension is greater than three, the explicit formulas become

too complicated and therefore we do not consider them.

7.4. Equations of inertial motion of a multi-dimensional rigid body in an incompressible ideally conductive fluid

Let us consider the classical equations of magnetic hydrodynamics for a non viscous ideal1y conductive fluid (see e.g. [64]).

av

- + (rot v) x v = p -t(rot H) x H - grad II;

at aH at = rot(v x H),

II = II(x, t) being a uniquely defined function in the bounded region D with a smooth boundary. II is defined by the condition that av/at be a non-divergent vector field on D; the field being tangent to the boundary of D. As shown in [137], the simplest finite-dimensional analog of these equations is ~ =

en, M] -

{H = [n,H]

[H,J]

(4)

90

A. T. FOMENKO AND V. V. TROFIMOV

defined on the Lie algebra so(n) of skew-symmetric matrices. n = (R g -,) * g is a right-translation to the unit of the group SO(n) of the velocity vector g E ~SO(n); J = Ad:-, j, j being flow density in the body; H = Adg h, h being the tension of magnetic field in the body, M being the kinetic moment in space. Let us give explicitly the embedding of the system (4) into the Lie algebra. Then, the system will be the Hamiltonian one on all orbits (the results of this section are given in [131]). Let G be any Lie algebra over the field K. Let us construct a new Lie algebra na,/I(G), a, /3 E K which contains G as a subalgebra. As a linear space, n~,/I(G) is the direct sum G EB G. Let us denote the elements ofthe first summand by a E G, of the second one by eb, bEG, i.e. any element of n~,/I(G) is of the form x + ey, X,YEG. Let us define on na,/I(G) the product [Xl

+ eYI,x 2 + eY2] =

[X I ,X2]

+ /3[YI,YZ] + e([YI,x2] + [X I ,Y2] + a[Yl,Y2])'

Note that n~,o(G) is an extension of G since there is an epimorphism of Lie algebras f: n",o(G) -+ G, f(x + ey) = x. LEMMA 7.3 The linear space n,,/I(G) = G + eG over the field K, the product [Xl + eYI, X2 + eY2] defined above, is a Lie algebra for any

a,/3EK. The proof follows from the definition of the commutator [Xl X2

+ eYI,

+ eY2].

LEMMA 7.4 If 4{3 + a2 = 48 + 1]2 = 0, then the Lie algebras n,,/I(G) and n~,6(G) are both isomorphic, being isomorphic to the standard extension no,o(G) as well.

Proof Let us define the isomorphism cp: n",/I(G) --+ no,o(G) = G + (jG, n~,/I(G) = G + eG, e2 = ae + {3 «(j2 = 0) by the formula cp(x + ey) = (x + (a/2)y) + (jy. Evidently, [cpX, cpY] = cp[X, Y]. In what follows, we shall denote the extension n"o(G) by n'(G). For the application to Hamilton mechanics we need the Lie algebra no,o(G) which we denote by n(G) = G + eG (e 2 = 0). We can define the Lie algebra n(G) in a different way. Let us consider the semi-simple sum G + G, where G acts on G by the coadjoint representation ad, the second summand being regarded as an Abelian Lie algebra. Then, the resulting direct sum coincides with n(G).

91

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Let us consider the Lie algebra Q(G) = G + eG, e2 = O. We can assume Q( G)* - G* + (eG)* 3 (fl,i2) defining the isomorphism as follows XEG,

yEG.

Let us denote f E Q(G)* by f = x* + ey*; x*, y* E G*, ey* E (eG)*. We have denoted already the coadjoint representation in the Lie algebra G by a(x, y) = ad;(x), x E G*, y E G; a(x, y)(z) = <x, [y, z]), z E G, (x, y) being the value of the covector x E G* on the vector y E G. PROPOSITION

7.7

Let X+eYEQ(G) and x* +eY*EQ(G)*. We state

that

a(x* + ey*,x + ey)

= a(x*,x) +

a(y*,y) + ea(y*,x);

x*, y* E G* and a(g, f) being calculated in the Lie algebra G. Proof Let x E G, x* E G*, y E G. Then, a(x*, x)(y) = x*([x, y]) = a(x*, x)(y) and

a(x*, X)(ey)

= x*([x, ey]) = 0

since x*if.G = O. Let x E G, x* E G*, y E G, then

a(ex*, x)(y)

= eX*([X, y]) = 0

since Ex*iG = 0,

= eX*([X, ey]) = x*([x, y]) = a(x*, x)(y).

a(eX*, X)(ey)

Let y E G, x* E G*, g E G, then

a(x*,ey)(g)

= x*([ey,g]) = 0

a(x*, ey)(eg)

= X*([ey, eg]) = 0

since x*i'G = 0 and

because [ey, eg] = O. Let y E G, x* E G*, g E G; then

a(eX*,ey)(g)

= eX*([ey,g]) = x*([y,g]) = a(x*,y)(g)

and

a(ex*, ey)(eg) = X*([ey, eg])

= O.

92

A. T. FOMENKO AND V. V. TROFIMOV

Therefore, we finally obtain a(x*

+ ey*, x + ey) = =

+ a(ey*, x) + a(x*, ey) + a(ey*, ey) (a(x*,x),O) + (O,a(y*,x» + (0,0) + (a(y*,y),O)

a(x*, x)

= a(x*,x) + a(y*,y) + ea(y*,x). This completes the proof. Let us consider the linear operator C: n(G)* - neG) (we call it a sectional operator). Then, in the space n(G)* we can give the system of non-linear differential equations a = a(a, C(a», a E n(G)*. This system appears when we integrate geodesic flows ofthe left-invariant metrics on the Lie group m. Let a = (X, Y) E G* + eG*; C(a) = (x, y) E G + eG. Then, from Proposition 7.7 we obtain the following corollary. PROPOSITION 7.8 The system of Euler equations a E n(G)* is of the form

~ _ a(X, x) + a(Y, y)

{ Y-a(Y,x).

a = a(a, C(a», (5)

Our next aim is to construct the linear operators C: n( G)* - n( G) for which the Euler equations (5) are completely integrable in the Liouville sense on all orbits in general position of the co adjoint representation of the Lie group n(m) which is associated with the algebra neG). PROPOSITION 7.9 The Euler equations on n( G)*, G being a semisimple algebra, coincide with the finite-dimensional approximations (4) of the equations of magnetic hydrodynamics in the case when G = so(n). Proof Identifying G* '" G with the use of the Cartan-Killing form, we

evidently obtain a(y, x) = [x, y]. Then, the system of the equations of magnetic hydrodynamics (4) follows from the Euler equations (5). Let G be a semi-simple complex Lie algebra, for which the nondegenerate symmetric scalar Cartan-Killing product S(X, Y), X, Y E G is given. We have the representation p = ad*: neG) - End(n(G)*). Let a En(G)*, then the mapping cPa:n(G)-n(G)*, cPa (x) = ad; a, x E neG), a E n(G)* is defined. In the case ofthe semi-simple Lie algebra G we shall identify G* with G using the Cartan-Killing form S(X, Y): G* '" G as we did before. It is easy to check that in this

93

INTEGRABLE SYSTEMS ON LIE ALGEBRA

isomorphism ad: y maps into the ordinary commutator [x, y]. In this case a = a l + Ga2 EQ(G)* - G + cG and we can assume al,a2 EG. 7.10 Let a E Q(G)* be an element in general positipn, i.e. the orbit of the coadjoint representation of the Lie group Q«f)) which corresponds to a has maximal dimension. Therefore, we can take a from cG. Then, a will be an element in general position in G. In this case, Ker ¢a = H + cH, H being a Cartan subalgebra in the semi-simple complex Lie algebra G. PROPOSITION

Proof Let x =

Xl

+ X 2c E Ker ¢a' Then, a(al

+

Ga2, Xl

+

CX 2) =

°

if and only if a(a2' xd = 0, the case of G we obtain the following condition [Xl' a l ] + [X2' a2] = 0, [Xl' a2] = 0. Let us prove the first assertion of the proposition. We have

In

If we take y in G (in general position) and assume e.g.

= 0, we obtain

X

dim On(G)(O, y) = 2 dim 0G(y) = 2 dim 0G' Since in a semi-simple Lie algebra the codimension of an orbit is equal to the number of functionally independent invariants of the coadjoint representation (which is equivalent to the adjoint one because there exists a non-degenerate invariant scalar product-the Cartan-Killing form), (0, y) is an element in general position. This completes the proof of the first assertion of the proposition (for the calculation of the invariants of the coadjoint representation of the Lie group Q(3"") see [132] and Theorem 13.1). Let us calculate Ker ¢a' From the equation [Xl' a 2 ] = it follows that Xl belongs to the Cartan subalgebra H which contains the element a2 (see [47]). Let a l = al,h + L~"o al,aea' x 2 = X 2 ,h + x2,~ea; G = H + L~,.o iCe, being the root expansion of the Lie algebra with respect to Hand al,h' X 2 ,h E H, ea being the root vector which corresponds to the root IX E H*. Then,

°

La,.o

94

A. T. FOMENKO AND V. V. TROFIMOV [XI,a l ]

+ [X 2,a 2] = L (al"a(xl)e, -

X 2.,IX(a 2)e,)

= O.

'''0

We have al .• IX(xd - x2 .• a(a2) = 0, Xl' a 2 E H for any root IX "# O. Since a2 is in general position, lX(a2) "# 0 for any IX "# 0, a E~. Therefore, according to the first part of the proposition, a l can be taken as zero. Then, a l •• = for any IX "# 0, and x 2 •• = 0 for any IX "# 0, i.e. X2 E H. Therefore, Ker 4>a c H + eH. The inverse expression can be checked easily. Thus Ker 4>a = H + eH.

°

PROPOSITION 7.11 Let b = bl + eb 2 E Ker 4>a = H + eH c Q( G), H be a Cartan subalgebra of G. Let b l be in general position. Then, Ker p(b) = H

+ eH

c Q(G)* - G

+ eG

(the isomorphism is given by the Cartan-Killing form). such that a(x i + BX 2 , bl + eb 2 ) = O. Calculating ad* for Q(G) we find that this is equivalent to the system a(x 2, bl ) = 0; a(x l , bl ) + a(x 2, b2 ) = o. By identifying G* - G using the Cartan-Killing form we obtain

Proof We need to find all

Xl

+ eX 2 EQ(G)*

Since bl is in general position, X 2 belongs to the Cartan subalgebra H. Then, since b2 , X2 E H, [b 2 , x 2 ] = O. Therefore, from the first equation we obtain [bl,x] = 0 from which it follows that Xl EH, i.e. Xl + BX2 EH + BH. This completes the proof. PROPOSITION 7.12 Let a be as in Proposition 7.10. Then, 4>a: Ker 4>/; --+ Ker p(b).L For any bE Ker 4>a (b is as in Proposition 7.11) we have p(b)(Ker p(b).L) c Ker p(b).L. Here V.L means orthogonal complement with respect to the direct sum of the Cartan-Killing forms on G: Ker 4>/; = V + B V and Ker p(b).L = V + B V, V = ICe, being an orthogonal complement to the Cartan subalgebra H of G with respect to the Killing form.

L,,,o

a = at + BQ 2 E Ker p(b) = H + eH provided that L. x.e. + B L. y.e. E (Ker 4>a).L, Proof Let

a(a 1

c

Q(G)*.

Then,

+ea2,~x.e.+e~y.e.) = L x.IX(al)e. + LY.IX(a2)e. + e L x.a(a 2)e, E Ker p(b).L,

95

INTEGRABLE SYSTEMS ON LIE ALGEBRA

as was claimed. Let b l eLaYaeaEKerp(b)-L. Then, a(

~ xae~ + e ~ y~ea' b

l

L~ x 2 ea +

+ eb 2)

= L x 2 o:(b l )e 2 ex

+ eb 2 E Ker 4>a = H + eH,

+L

Yao:(b2)e~

rl

+ eL

y~o:(bde~ E Ker p(b)-L

:X

as was claimed. Here ea is the root vector which corresponds to the root 0: EH*. Evidently, the mapping 4>a: Ker 4>; isomorphism of vector spaces. REMARK

--+

Ker p(b)-L is an

With these preliminary considerations over, let us construct the sectional operators for the complex semi-simple Lie algebras. We shall use the general method given in Section 6. Let D: Ker p(b) --+ Ker 4>a be any linear operator. DEFINITION

7.1

Let us define the operator C: Q(G)*

--+

Q(G) by the

matrix

C=C(a,b'D)=(4>a-~adt ~) in accordance with the splittings Q(G) = Ker 4>. + Ker 4>;, Q(G)* = Ker pCb) + Ker p(b)-L (see Figure 23). Considering what was said above, this definition is correct since all operators which are used here are well defined. As in the semi-simple case, we shall call the operators constructed operators of the "complex" series for the Lie algebra Q(G). Let H be the Cartan subalgebra of the semi-simple complex Lie algebra G, 0: EH* be a non-zero root. We shall define h2 EH using the condition o:(h) = B(h, h:) for any hE H, B(X, Y) being the Cartan-

..IHG) Fig. 23.

A. T. FOMENKO AND V. V. TROFIMOV

96

Killing form of the Lie algebra G. Let H 0 be the subspace in H which is generated by all vectors h~ with rational coefficients. We shall consider the standard compact real form G u of the Lie algebra G (see [47]). We can take vectors ih~, ea + e _" i(e x - LJ (i = as a real basis in Gu • Let us construct the sectional operators for Q(G u ). The operators constructed we shall call operators of the "compact" series for the algebra Q(G).

FI)

LEMMA 7.5 Let a = i{a 1 + W2) E Ker CPa n Q(G.). Then, the operator ad: for the Lie algebra Q(G.) can be described as follows e,

+ e_, --+ lX(adi(e,

-

LJ,

i(e. - e_,) --+ -1X(al)(e, + e_J,

+ e_ x) --+ lX(a 2 )i(e,

e(e.

- Lx)

+ elX(a 1 )i(e, -

La),

ei(ea - La) --+ -1X(a 2)(e. - La) - lX(ade(e a + La)·

The proof is obtained by direct calculation using Proposition 7.7. LEMMA 7.6 Let a = i(a 1 + W2) E Ker p(b) n Q(G u )*. Then the operator CPa for the Lie algebra Q(G u) can be described as follows ea i(e x

+ L , --+ lX(a1)i(e, -

La) --+

e(e,

- e- a )

+ elX(a2)i(ea -

-1X(ad(e x

+C

+ cal

lX(az)i(e, - e_J,

--+

a) -

ei(e, - e_,) --+ -1X(a2)(ea

Ker p(b) n Q(G u )*

elX(a 2 )(e x

e_,),

+ e_,),

+ L,),

~Ker p(b)

n Q(G u ).

The proof is obtained by direct calculation using Proposition 7.7. PROPOSITION 7.13 Let a=i(a 1 + W2)EKerp(b)nQ(G u)* be such that lX(a 2 ) =F 0 for any root IX =F 0, b = i(b 1 + eb 2 ) E Ker CPa n Q(G.). Then the operator CPa- 1 ad: for the Lie algebra Q(G u) can be described as follows

97

INTEGRABLE SYSTEMS ON LIE ALGEBRA

The proof follows directly from Lemma 7.5 and Lemma 7.6. Now we can construct the operator for n(G u) using the general technique given in Section 6. Using the Cartan-Killing form we identify n(G u)* with n(G u) = G u + €G u . Let

v" = LIR(ea + LJ + LlRi(ea

-

e_,) c:: G.

We have the splitting as a direct sum of linear subspaces n(GJ = (H~

+ €H~) + (v" + € v,,); H~

DEFINITION 7.2

=

L lRih~.

In accordance with the direct decompositions n(GJ = (H~

+ €H~) + (v" + d~)

n(G u)* = (H~

+ €H~) + (v" + €v,,)

and we define C

= C(a,b,D) = (~

4>a-~adt).

D: H~ + €H~ --+ H~ + €H~ being any linear operator, a E n(G u)*, bE n(G u) being chosen as described above. REMARK Remember that we can construct the "compact" series of sectional operators for the semi-simple Lie algebras as well. In the case of the semi-simple Lie algebras, the sectional operator C: --+ Gu is diagonal with respect to the canonical basis e, + e _a' i(e, - e -J. In our case, different from the semi-simple one, the operator 4>a- 1 c adt is not diagonal with respect to the canonical basis. Let us construct the sectional operators for n(G.), Gu being a normal compact subalgebra in the semi-simple complex Lie algebra G. By definition, Gu = o lR(e, + L,).

G:

L, ..

LEMMA 7.7

Let a E iH~ +

eiH~ c::

n(G u)*, b E iH~ + €iH~ c:: n(G u).

98

A. T. FOMENKO AND V. V. TROFIMOV

Then the operators of the "compact" series More precisely,

qa, b, D): Q(G n )*

'" Gn + 6G n

The isomorphism Q(Gn )* '" Gn Cart an-Killing forms.

+ 6G n

--+

qa, b, D) preserve

Q(Gn )

'"

Q(Gn ).

Gn + 6G n •

is given by the direct sum of

The proof follows directly from the explicit form of the operator rPa- l ad: in Proposition 7.12. Thus we have defined the operators qa, b, D) = C: Q(Gn )* --+ Q(Gn ). As in the semi-simple case, the elements a, b should not belong to Q(G n )* and Q(G.) respectively. We shall call these operators operators of the "normal" series for the algebra Q(G). In conclusion, note that in the case ofthe Lie algebra G = so(3) from the construction of the sectional operators we obtain the following Euler equations. Let

0

Xl

Xl

-Xl

0

X3

-Xz

-X3

0

+6

0

YI

Y2

-Yt

0

Y3

-Yz

-Y3

0

EQ(G)* = G* + 6G*,

6z = O. Then

Xl = -k1XlX3 + (k s - k z)YzY3 + (k s - k 3)x Z Y3 + (k6 - k 4 )X 3Yz, Xl = kt X I X 3 + (k z - kS)Yl Y3 + (k3 - k 6)X l Y3 + (k 4 - k 6)X 3YI'

Yt =

-k t X3Yl + (k6 - k 3)YlY3'

Yz = k t X 3Yt

+ (k3

- k 6)YtY3,

Y3 = O.

In this case, the kinetic energy H = <X, qa, b, D)(X» is of the form H

= ktx~ + ks(yi + yD + kzY~ + 2k 6 (X l YI + XlYl) + (k3 + k 4 )X3Y3'

Using the diagonal form of the energy, we conclude that the quadratic form H always has at least two "minuses" and four "pluses." Also, note that the energy H coincides (except for one term) with the kinetic energy in the first case of Clebsch (which describes inertial motion of rigid body in ideal fluid). More precisely, the kinetic energy constructed on Q(so(3»* is the limit (when b ll --+ 0) ofthe kinetic energy in the first case

INTEGRABLE SYSTEMS ON LIE ALGEBRA

99

of Clebsch (see notations in 7.3). Hence our case can be cited as an analog of the first case of Clebsch for non-positive-definite kinetic energy.

3

Sectional operators on symmetric spaces

8. CONSTRUCTION OF THE FORM Fe AND THE FLOW XQ IN THE CASE OF A SYMMETRIC SPACE

A. T. Fomenko's results given in this chapter can be found in [32J, [34J, [125]. Let m= mit) be a homogeneous space. Then we can assume that the Lie algebra G of the group m is split as a sum of linear subspaces: G = H + V, H = Tet) being the stationary subalgebra; V = 1'. m, i.e. it is identified with the space tangent to m. The subalgebra H acts on V via the adjoint action dp = ad, i.e. adw(X) = [H',X], H' EH, X E V. For the purpose of simplicity, we suggest that m, t) are semi-simple Lie groups; the space V is partitioned by the orbits 0 of the coadjoint action Ad!,';. Let T c V be a maximal commutative subalgebra of V (a Cartan subalgebra of V). Then V = T + T', T' being the orthogonal complement to T in V (the sum of the linear subspaces). We assume that on G = H + V the Cartan-Killing form is used, so that its restriction to V defines a non-degenerate scalar product. Let K c H be the annihilator of T, i.e. it is the subalgebra of H which consists of all the elements k E H such that [k, TJ = 0 ([k, tJ = 0 for any t E T). We assert that K is the annihilator of any element in general position t E T. This annihilator is the same for all elements in general position in T. Let H = K + K', K' being the orthogonal complement to K in H. Let us consider the subspace M = Ann K in V, i.e. mE M if and only if kJ = 0 for any k E K. It is clear that M is the annihilator of an arbitrary element in general position in K. Evidently T c M. Let Z c M be an orthogonal complement to T in M. Then M = Ann K = T + Z (see Figure 24). Let R be an orthogonal complement to M in V. Then V = T + Z + R = R + M.

em,

100

INTEGRABLE SYSTEMS ON LIE ALGEBRA

101

I I I

I I

, ,:H

,, I

K

Fig. 24.

For the purpose of simplicity, let us assume that m= (fi/i; is a symmetric space. This means that on G there is defined an involutive automorphism (J such that (J = + Ion Hand (1 = -1 on V «(12 = Ion G). Hence [H, H] c V, [H, V] c V (these relationships are valid for any homogeneous space), [V, V] c H (the latter is valid in the symmetric case). Let a E T be an element in general position in the subalgebra T. Then, the mapping ada: G --'> G allows us to identify the subspaces K' and T'. LEMMA 8.1 ForanyaETingeneralpositionthemappingada:H --'> V defines an isomorphism between K' and T' (G, H being semi-simple Lie algebras, m= (f)/i; a symmetric space). Proof Since K

= Ann T = Ann(a) II H

(a is the element in general because H = K + K',

position), Im(ad.) = ada K' c T' K = Ker(ada) II H. Conversely, ad. T' c K' c H because T = Ker(ad a) II V. It is clear that ad.: T' -+ K' is a monomorphism since if a vector t' E T' existed, such that ad.(t') = 0, it would mean that t' E V commuted with the element in general position a E T, i.e. [t', T] = 0 and hence T would not be maximal in V. Since dim H / K = dim K' = dim O(X) = dim T' (O(X) is an orbit in general position In V), ada: K' -+ T' is a linear isomorphism. The lemma is proved. Since ada: K' -+ T' is an isomorphism, the inverse mapping ad; 1: T' --'> K' is uniquely defined. The mapping ad.: T' --'> K' behaves similarly. It is clear that in general these mappings are different. Generally speaking, the compositions ad;: K' --'> K' and ad;: T' -+ T'

102

A. T. FOMENKO AND V. V. TROFIMOV

are not the identity mappings. Let R = ad; 1 R, Z = ad a- 1 Z, then K' = t + R. We can define the subspaces t and R in an alternative way. K acts as the stationary subalgebra on the space tangent to Sj/5l which is isomorphic to T. This action can be defined on the algebra H itself as the coadjoint action K on the space K' which is orthogonal to K in H and naturally identified with T. Hence Ker(ad K) n T = Z is isomorphic with Ker(ad K) n K'. Indeed Ker(ad K) n K' coincides with t = ada- 1 Z. The action of the mapping ada is illustrated in Figure 25. In particular, ada Z = t, ada t = Z, ada R = R, ada R = R, as proved below. PROPOSITION

8.1

Let us consider the coadjoint action adK:H -> H. Then ad K: K' -+ K' and Ker(add n K' = Z. In particular, R is the orthogonal complement to t in K' and both spaces Rand tare invariant under the action of ad;: K' -+ K'; a being an element in general position in T (see Figure 25). Proof Let k

E

K be an arbitrary element in general position in K. Then

Ker(ad K) n K' = Ker(ad K) n K'. Similarly Ker(ad K) n T

= Ker(ad K) n

T.

Let a E T be an element in general position in T. Then, [a, k] = 0 and ada: K' -> T is an isomorphism. Let us prove that ad a- 1 Z = t c Ker(ad K) n K'. If ZE t then [k, [a, i]] = 0 since [a, i] E Z. From the Jacobi identity for the algebra G we have

[k,[a,Z]] + [z,[k,a]] + [a, [i,k]]

= 0,

R , /,'

,.J

l'" I I

Fig. 25.

,"

" ,, I

INTEGRABLE SYSTEMS ON LIE ALGEBRA

103

i.e. ada[Z,k] = 0 (since [k,a] = 0). Since [k,i] EK' and ada: K' --+ Tis an isomorphism (see Lemma 8.1), [k, i] = 0 which was what we wished to prove. Conversely, let sEKer(adK)nK'. We need to prove that SEZ = ad;! Z, i.e. adasEZ, i.e. [k,adas] = 0, [k, [a,s]] = O. Since S E K' and [k, s] = 0, from the Jacobi identity and because [k, a] = 0, we obtain [k,[a,s]] + [s,[k,a]] + [a,[s,k]] =0.

Then [k, [a, s]] = 0 which completes the proof. The arguments for R and R reduce to studying the orthogonality relations. Let us prove that ada Z = Z = Ker(ad K ) n K'. If Z E Z, then [a, z] E K' and [k, [a, z]]

= - [z, [k, a]] - [a, [z, k]]

= 0

since [k,a] =0, [z,k] =0, zEZ=Ker(adk)nT'. Thus, adaZcZ. Since dim Z = dim Z, ada Z = Z. Therefore, ada R = R which completes the proof. The two mappings ada: Z --+ Z and ad a-! : Z --+ Z differ from each other by the transformation ad;: Z --+ Z which is a non-degenerate linear self-mapping of Z. REMARK

In order to construct the form Fe we need to define the linear mapping c: V --+ H. Let us construct the natural mapping C using the canonical properties of symmetric spaces, in particular, the relationship [V, V] c H. First we notice that we cannot construct the natural form on Vby using the restriction ofthe standard Kirillov form on V. Indeed, B(X, [~, IJ]) = 0 for any X, ~,IJ E V since [~, IJ] EH and H is orthogonal to V. Of course, we could consider Vwhich is a linear perturbation ofthe plane V (produced e.g. by the parallel translation of the plane by the vector h E H, h "# 0). Then, we could restrict the Kirillov form on V and obtain a non-trivial form on V (which is isomorphic to V = V + h). However, in general this form is not closed. Under such an approach we still need to consider the equations dF c = 0, XQFC = O. Since ad v : V --+ H, we need to construct the operator C: V --+ H using operators of the form ad v, v E V. At the same time it is natural to choose as the operator C a transformation which would be as close as possible to the identity in the case when e.g. m= (f> x f»/f> = f>. Since the mappings ad v : V --+ H are not identical for the case ~ = (~ x ~)/f>, the action of the operators ad v needs to be compensated by operators of the form ad - 1 • We should combine the operators of both forms, i.e. ad and

A. T. FOMENKO AND V. V. TROFIMOV

104

ad -1 , because, as we mentioned above, the space V can be mapped in H with the use of both ad and ad- 1 (see Lemma 8.1). All these considerations are given solely to clarify the general problem of identifying V with a subspace of H. Let us write down V and H as the sum of linear subspaces V = T + R + Z, H = K + R + :2 (see Figure 25). We assume that this expansion is fixed. DEFINITION 8.1 Let us define the operator C: V --+ H by using the following matrix C = ada' + ad a- 1 ad b + D: C

=

(

0 0)

ada' 0

ad a- 1 ad b

0

o

0

D

,

a, a' E T being arbitrary elements in general position in T, b E K being an element in K (not necessarily one in general position), D: T --+ K being any linear operator,' ad;! ad b : R --+ R; ada': Z --+ :2. Since Z = Ker(ad K ) n V, Ker(ad b : V --+ V) :::J Z (if b is in general position in K, then Ker(ad b : V --+ V) = Z) and ad b : V --+ R; ad;!: R --+ R (see Figure 25). We call the operators C: V --+ H sectional operators for the representation p: H --+ End( V), REMARK The sectional operator C: V --+ H in Definition 8.1 is constructed according to the general method given in Section 6. DEFINITION 8.2 Let us define the form Fe on the space V (and on the orbits 0 = O(X) c V) by the formula FdX,~,Yf) = B(CX,[~,Yf]), the operator C being as defined above C=C(a,b,a',D); X,~,YfEV; B(X, Y) being the scalar product. It is clear that the 2-form Fe is skew-symmetric on V (with respect to ~,Yf). Following the general method, we would have to consider the operation [~',Yf'] instead of [~,IJ] (~= [X,n, Yf = [X,Yf']). However, since we consider the semi-simple case only, the forms B(CX, [~, Yf]) and B(CX, [~', Yf']) are equivalent (as in the Kirillov case for the semi-simple algebra). DEFINITION 8.3 Let us define the vector field XQ on the space V by the formula XQ = [X,QX], the operator Q being defined above: Q = Q(ii,Ei,ii 1 ,D);Q:R + Z + T--+ R + t + K;Q = ada! + adi! ad 6 + D. It is clear that the field

action of tl on

v.

XQ is tangent to the orbits 0

of the coadjoint

105

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 8.2 Let f be a smooth function on V and the vector field s grad f E Vbe such that F ds grad f, Y) = Y(f) for any field Y E V, Y(f) being the derivative of the function f along the field Y. Then, grad f = [CX, s grad jJ. Proof We have Y(f) = B(Y,gradf). Hence B(Y, grad f)

= B(CX, [s grad f, YJ) = B(CCX, s grad jJ, Y); B(gradf - [CX,sgradjJ, Y)

= O.

Since Y E V is arbitrary we obtain [CX, s grad jJ - grad f = 0 which completes the proof. If the form Fe is non-degenerate at the point X (on V or on O(X)), then the field s grad f is uniquely defined by the equality F ds grad f, Y) = Y(f).

9. THE CASE OF THE GROUP SPACES OF TYPE II)

iI = m= (il x

~)/~

(SYMMETRIC

Let us consider the semi-simple group ~ as a symmetric space. Then, as we know (see e.g. [48J), this symmetric space can be written as (D x D)/D, the involution a: ~ x D --+ D x D being defined by lJ(x, y) = (y, x). The corresponding expression in the Lie algebra H + H is as follows: if V = (X, -X), X E H; H = (X, X), X E H (both H and its image in G = H EB H being denoted by the same letter), a V = - V, aH = H.

The following equalities hold: Z = i = 0, H V = T + R (a Cartan decomposition of the algebra H).

LEMMA 9.1

= K + R,

ret,

Proof Here T = -tn, tE yr, T r being a Cartan sub algebra in H and K = t)}, t E yr, so that clearly T and K are isomorphic to yr. Since V and H correspond to the same group, the coadjoint action ad H

ret,

on V is of the form ad(h.h)(X, - X) = ([h, X] - [h, X]) E V, i.e. it is identical with the action ad h X. Hence, to find Z it is sufficient to find the centralizer yr in H. Because we consider the semi-simple case, we have Z = 0, i.e. the centralizer coincides with yr. As a consequence of the semi-simplicity the orthogonal complement R to T (and R to K) is generated by the root subspaces of the algebra H. This means that the decompositions V = T + Rand H = K + R are isomorphic to the

106

A. T. FOMENKO AND V. V. TROFIMOV

Cartan decompositions (linear for V). Note that although ~ is diffeomorphic to ~, v is not embedded in (Ij = ~ x ~ as a subgroup. We can assume that H = {(X, X)} and V = {(X, - X)} are identified by using the natural mapping IX: (X,X) = (X, -X). In particular, the orbits 0 c V coincide with the orbits of the standard coadjoint action of ~onH.

9.1 The form FdX;~,tI), C(a,a,O,E) = C (i.e. a = b, a' = 0, D = E being the identity operator) is identical with the Kirillov form on the algebra H (which is isomorphic as a linear space to V). In particular, it is non-degenerate and closed (and invariant) on the orbits o of the coadjoint action of ~ on V. PROPOSITION

+ R, X = nX + Y, ex = DnX + ad; 1 ad b Y =

Proof Since V = T

nX E T, Y E Rand nX

+Y =XEH.

Thus, FdX,~,tI)=B(X,[~,tI]). In the semi-simple case this form actually coincides with the Kirillov form, up to a linear transformation at each point. The forms B(X, [¢, tI]) and B(X, [¢', tI']) are invariant on V with respect to Adfl. Therefore, it is sufficient to compare them at just one point in the general position Xo in V; ~ = ad x3', tI = ad xo tI'; B(Xo, [ad xo ¢',adxotl'J) and B(Xo, g', tI']) differ from each other by the non-degenerate linear transformation ad xo which maps the tangent space Tx o0 into itself. Hence the form Fein general is not closed on V (in this example, Fe is closed on the orbits only). It is in this example (~ = ~) that we can see the role of the operator ad; 1 ad b which enables us to identify R and Ii in a natural way: the action ad b is compensated by the action ad a- 1 from which we can obtain the identity operator E assuming a = b. Using only one operator ad b (or ada) we would not obtain the operator E as a particular case of the sectional operator e (because on the space T the set of the roots is a redundant basis and hence, the system of equations lX(t) = 1, IX runs over all roots, generally speaking, does not have solutions on T). Let us consider the field XQ = [X,QX], Q = Q(a,b,O,D): T+R~K+R, D:T~K, ada-ladb:R~R. By using the identification of VandH, we find that the field XQ on V (on H) is given as follows X = [X,qJa,b,D(X)], the operator qJa,b,D: V ~ H (i.e. H ~ H) being identical with the multi-dimensional rigid body operator

INTEGRABLE SYSTEMS ON LIE ALGEBRA

107

introduced in [92]. In our case, D: T -> T, ad a- 1 ad b : T' -> T, T' being the subspace generated by the eigenvectors of the operators ad l , tEO T, H = T + T' being the Cartan decomposition. As follows from [89], all flows of the form ({la,b,D (a, bET are elements of the general position in T), being Hamiltonian oneS with respect to the form Fe (Kirillov form) on 0, are completely integrable on the orbits of the general position in H. Thus, the important case of completely integrable "rigid-body" like systems is one of the examples of the pair Fe, XQ.

10. THE CASE OF TYPES I, III, IV SYMMETRIC SPACES 10.1. Symmetric spaces of maximal rank

Let us consider the space mj~ with rank (i.e. dim T) equal to the rank of m (i.e. the dimension of the Cartan subalgebra of G). Example: m= SU(n)jSO(n); the embedding of SO(n) in SU(n) being the standard one. If dim T = rk G, then T c V is a Cartan subalgebra of G (not only of V), i.e. K = (it is impossible to extend T by including T in a larger commutative subalgebra since the Cartan algebra is maximal). Hence Z + T = V, i.e. R = R = (the annihilator K = in V coincides with V). The following lemma holds.

°

°

°

10.1 If the space m= mj~ is of maximal rank, then the 2-form Fe on V is generated by the curvature tensor of the symmetric space v: 4B(a',R(X,~),t/)=FdX,~,t/), R being the curvature tensor, a'ET being a fixed vector. LEMMA

Proof Sincc K=R=R=O, b=O, X=nX+X', nXET, X'EZ; = C(nX + X') = DnX + ada' X' (a' E T). Since K = 0, D = 0, i.e. CX = ada' X',

cx

FdX; ~,rf)

= B(CX, [~, t/]) = B([a', X], = B(a', [X,

[~, t/]])

[~,t/])

= 4B(a', R(X, ~)t/),

R being the curvature tensor (see e.g. [48]).

Since K = 0, 0 ~ ~, i.e. the orbits of maximal dimension in V are diffeomorphic to the group ~. Thus, the 2-form Fe on 0 ~ ~ is obtained by the restriction of the 2-form B(a', R(X, ~)t/) to ~ c V, the field

A. T. FOMENKO AND V. V. TROFIMOV

108

R(X, ~)'1 being tangent to the orbits 0 '" of the form

V. The flow XQ =

x = [X, [X,a]] = adi(a),

[X, QX] is

a E T.

On G = su(n) consider the involutive automorphism a(X) = X, X E G. Then H = so(n) and V consists of all symmetric imaginary matrices of order n, with trace equal to zero. The diagonal matrices in V constitute the maximal Abelian subspace T. In this case, K = 0, M = V and Z is the orthogonal complement to T in V (i.e. R = 0). Let Iij be the n x n matrix on n, with the only non-zero element being equal to one and occupying the location OJ) (i is the row, j the column), li,jl = Ii j + ~i' Simple calculations show that if X=iLp
Ii

adaY = L (a q p
-

ap)xpqlTpq - Tqp) Eso(n);

therefore, the sectional operator C: V -+ H is defined completely. We will write down an arbitrary element X E V in the form X = i L xpqlp,ql p
+ i L"

p= 1

xpTpp-

LEMMA 10.2 The equations X = [CX, X] for the symmetric space of maximal rank SU(n)/SO(n) are of the following form Xpq =

L (2a j - a p - aq)xpjxjq p<j
+

L (2aj - a p - aq)xpjxqj j>q

+ L (2a j - ap - aq)xjqxjp + (a q - qp)(Xq - xp)xpq, j
Xi =

L (aj -

a;)x&,

p
i=l, ... ,n

j

and C is the seasonal operator defined above for the symmetric space SU (n )/SO(n). Proof Let us consider the decomposition ofthe Lie algebra su(n) as the direct sum of linear spaces su(n) = so(n) + V, V consisting of the symmetric imaginary n x n matrices with trace equal to zero. In V, the diagonal matrices T form a maximal Abelian subspace. Evidently, K = O. Therefore, the element

109

INTEGRABLE SYSTEMS ON LIE ALGEBRA

o X=

=h

+

+Y

iXnn

is mapped into

0

iXpq

by the sectional operator C. Since [C(h

+ y),h + Y] = [ada Y,h + YJ = [ada Y,h] + [ada Y, YJ,

the equations X = [CX, X] are of the form shown in Lemma 10.2. The orthogonal projection onto the plane Z of the vector field X = [CX, X] which is constructed for the symmetric space SU(3)jSO(3), coincides with the Euler equations of the motion of rigid body with a fixed point and an arbitrary inertia tensor in ~3. PROPOSITION

10.1

Proof Remember the form of the equations of motion of a rigid body

dp

A dt = (B - C)qr,

dq B- = (C - A)rp dt

'

dr C dt = (A - B)pq.

In our case

XZ3 = (2a 1 - az - a3)x 13 x 12'

It is easy to see that the system of linear equations for the moments of inertia A, Band C:

A - B

= (2a 1

-

a z - a3)C

always has a solution, i.e. we can obtain an arbitrary inertia tensor. Thus, the statement is proved. 10.2 Let C be the sectional operator for the symmetric space SU(n)jSO(n) which is given by the element a E T. Then the external 2-form Fe is closed on Vand hence it is closed on the orbits 0: dF c = O. PROPOSITION

Proof Let us remember F dX; ~,'1) = B(CX, [~, '1]).

110

A. T. FOMENKO AND V. V. TROFIMOV

Let us define the matrix-valued differential form d~ = II dX ij lion the space V We can take the exterior product of this matrix with itself IX = d~ 1\ d~. If ~,'1 E V, then (d~ 1\ d~)(~, '1) is the matrix with the location (i,j) occupied by the element Cij = Lk (~ik'1kj - '1ik~kj)' On the other hand, let us consider [~, '1]. Evidently, [~, '1J ij = C ij' i.e. (d~ 1\ d~)(~, '1) = [~, '1]. Therefore, we can write down the differential form Fe using the Cartan-Killing metric explicitly: F dX) = tr[C(X)(d~ 1\ d~)]. Let the matrix elements d~ 1\ d~ be of the form n

wij = L dX ik

1\

dx kj ·

k=l

Then, d~

1\

d~ =

I

L dXik

1\

dXkj Iij·

i ~j k

The operator C is of the form C(X) = L (Uj - u;)xjj(Tij - 1jJ E so(n). i<j

If

x=

i L xpq(Tpq P.q

+ Yqp) E V,

then Fe =

I

p,q

C(X)pqWqp

and

C(X)pq = (U q

-

ap)xpq

provided that Xpq = Xqp' Thus,

p,q,i

p,q.,i

+ L. (uq - ap) dx pq

1\

dXqj

1\

dXiP]

P.q,l

=

I

(a p - Uj + Uj - aq + aq - up)dXjp

p,q,i

This completes the proof.

1\

dxpq

1\ dXqj

= O.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

10.2. The symmetric space

sn-I

111

= SO(n)/SO(n - 1) (the real case)

Let us consider the Grassmann manifolds SO(n)/SO(n - p) x SO(p). For simplicity, let us consider the case p = 1. We shall give a complete description and classification of the Hamiltonian flows XQ and symplectic structures F c. Let us consider the sphere S,,-1 = SO(n)/SO(n - 1) and the corresponding decomposition of the Lie algebra so(n) = G = V + H; H = so(n - 1) (using the standard embedding), V = 7;,S,,-I. This decomposition is generated by an involution a: G -+ G. In the algebra G = so(n) we choose the standard basis formed by the skew-symmetric matrices Eij (the location (ij) is occupied by + 1; the location (ji) by - 1; the rest of the elements are zero). Here i denotes row number andj column number. For simplicity, let us denote E;j by (i, j). Then the basis in G consists of those (i, j) for which i < j. Evidently (i, j) = - (j, i), [(i, j), (p, q)] = 0, provided that in the quadruple i, j, p, q all the numbers are different, and [(i, IX), (IX, q)] = (i,q). A basis for V is the set (l,k), 2~k~n, i.e. if XEV, X= L~ ~ 2 x k ( 1, k), X k being real numbers (see Figure 26). Then

"If

H~

o '" .....

so(n- :1)

0

0

. . : : .. 0':'

o

:

T

=>'(12.)

..

>.:. .

'

'.'

K = SO(n-2)

.... .... .:.', .. '.

:. '..... .. '

.." : . . .. .

K' . : ..

....

.... .. .'.

....

Fig. 26.

'

'.'

..

,,':

.

A. T. FOMENKO AND V. V. TROFIMOV

112

L

= so(n - 1) =

H

h,p(a{J);

2~a.
= A(12);

T

K

T'= LXk(lk); k=3

L

= so(n - 2) =

y,p(afJ);

k=3

°

Thus K = so(n - 2) and hence Z = (it is easy to check that T = Ann K (1 V). Therefore V = T + Rand H = K + R (2 = 0). Hence, the operator e: V ~ H is of the form

ex = C(nX + Y) = DnX + ada- 1 ad b Y,

D: T

~

K.

Since dim T = 1, any element a E T is of the form A( 12). Therefore, we can take (12) as a. The operator D: T ~ K is uniquely defined by the single element d = D(12). So we have the following statement. LEMMA 10.3 The sectional operator for SO(n)/SO(n - 1) is determined by b, dE K = so(n - 2).

the symmetric space the two elements

= (12), bE K = so(n - 2) be an arbitrary element, d=D(a)=D(12)EK. Let b= \\b,p\\, d= \\d u\\. Then, the operator LEMMA lOA

Let a

e: V ~ H, qa, b,O, D) = e is of the form

ex = Xz L.

dij(ij) -

3~z<J~n

Proof Since X

±(±

k=3

x,b,k)(2k).

1:;:::3

= nX + Y, YET'; nX E T;

ad b X = ad h Y =

bikXk[(ik), (lk)] +

L

L

bkjXk[(kj), (lk)]

3~i
=

-

kt (J3

X.b,k)(1k).

Evidently ad(1z,(2k) = (lk), i.e. ad(l~): (lk) ~ (2k), i.e. ad; l(R ~ -ada(R ~ R). Therefore ad a-

1 b kt Ct3 ad Y = -

Since X = Lk= 3 xk(lk), nX proves the lemma.

=

R) =

X.b,k}2k).

X2( 12), DnX = Xz

L3

'i;i<j'i;"

dij(ij) , which

113

INTEGRABLE SYSTEMS ON LIE ALGEBRA

10,5 The form FdX;e,I]), C = C(a,b,O, D), is given on the space V = T.,sn -1 by the formula LEMMA

Fe = -X 2

L

dijdxi

dXj +

1\

3~i<j~n

i (i X,b,k)dX

k=3

2 1\

dXk

a:=3

(see Figure 27), The orbits 0 of the adjoint action SO(n - 1) V= T.,sn-l are the spheres S~-2 defined by the equation

= i) on

n

L

x~

= R2

(R

= const),

k=2

The restriction Ie of the form Fe to the orbit

L

Ie =

w ij dX i

1\

S~ - 2

dx j ,

3~i<j~"

X.1.

.•. :........1 - - -

,,

,

if(X) ,,, , ,,

x' 2.:, ,

,,

o

0

o

0

0

CY .'. "

F= c

'. ,

'.

.. .

, '.

.'

o -Cy .:'., ... :x~c( ',,: .... :' :', .. :', .

.

.'

'

,

....

-8. dE K = so (n- 2) YEt< Fig. 27.

is of the form

114

A. T. FOMENKO AND V. V. TROFIMOV

'\'11 2. t th esp here S"-2 X22_R2 -L.,k=3Xk,X3, ... ,XII b· emgcoord·maeson R •

Prooi Since the embedding K c: H c: so(n - 1) is the standard one

(see Figure 26), II

O=S~-2:

L xl=R2, k=2

X2, ... ,XII being Cartesian coordinates in V = 1R"- I . Let ~ = Lk=2 ~k(1k), IJ = Lk=2IJk(1k). Then [~,IJ]ij = -~ilJj + ~jlJi· By definition F dX;~, IJ) = B(CX, [~, IJ]). Using Lemma 10.4 we obtain Fe = trace(DnX Fe = -X2

+ ada-I ad b Y)[~, IJ] ;

dijdx i /\ dXj +

L 3~i<j~11

±(± X7b~k)dX2

k=3

/\ dx k ·

:x=3

Let us obtain the restriction Ie to the form Fe on o. Since on 0: x~ = R Z - Lk=2 xl, Lk=2 ~kXk = 0 for ~ E T"sn-Z; ~ = -(1/x2) Lk=3 ~kXk· Let us assume Ie = 11+ 12;

12

Then 12 =

=

11 = B(DnX,[~,IJ]),

B(ad a- 1 ad b Y, [~, IJ])·

J3 Ct X~b~k}~2IJk -1J2~k) + (xr + xDbik}~ilJk

Let us calculate the form II on O. We have

-lJi~d·

115

INTEGRABLE SYSTEMS ON LIE ALGEBRA

L

- -X2

dij(~i'1j - ~j'1J.

3:('i<j:('n

THEOREM to.l The form Fe (C = C(a, b, 0, D), a = (12), bE K = so(n - 2» is closed on V = YeS· -1 (in particular, on the orbits 0 = S'R - 2 as well) if and only if d + 2b = 0. In this case, the form Fe is given by dl, the I-form I being defined by the vector field leX) = (0, X2b(y», Y = (X3' ... ,x.), bE so(n - 2). The form Fe is not invariant on V or on 0 with respect to the action of the groups K = SO(n - 2) and i) = SO(n - 1). If the orbit 0 = S'R- 2 is even-dimensional and the skewsymmetric matrix bE K = so(n - 2) is non-degenerate (e.g. b is in general position in K), then the form fe = F elo (i.e. the restriction of the form Fe to 0) is non-degenerate on 0 almost everywhere. Thus, if d = -2b, b is non-degenerate (in general position) and n is even, then the form fe defines a symplectic structure (not invariant under the action of the group) on the open everywhere dense subset of 0 = S'R- 2 . Proof Let us prove that Fe is closed on V provided that d Indeed, since

.L.

Fe = -X2

dijdX i

A

dX j

+

3~l<J~n

dF e = -

~.

dijdX2

L

dij dX 2

A

dX i

A

dX j +

A

dX i

A

dX j

-

3:(,i<j:('n

L

(dij

°

X.b. k)dX 2

A

dXk'

1X=3

f (f b.kdX.)

k=3

3:(,t<J:(,n

3~i<j:(,n

±(±

k=3

+ 2b = 0.

A

dX2

A

dXk

A

dx.

A

dXk

,2;;3

L

2

b.kdx2

3~a
+ 2bij) dX 2 A

dX i

/\

dx j •

°

Thus dF e = if and only if dij + 2bij = for all i, j. Evidently (see Lemma to.5 and the explicit expression for the forms), the forms Fe and fe are not invariant. Let us prove that the form fe is non-degenerate on S·-2 almost everywhere. Let us consider the point Xo ES'R- 2 , Xo = CR, 0, ... ,0), i.e. x 2 = R, x k = 0, 3 ~ k ~ n. At the point X 0 on TxoS'R- 2 the form is given by the skew-symmetric matrix 112Rbij II. This follows from the fact that the 2-form

116

A. T. FOMENKO AND V. V. TROFIMOV

vanishes at the point (R, 0, ... ,0). Since R t= 0 the non-degeneracy of the form Ie is equivalent to the non-degeneracy of b = Ilb;ill. Since bE K = so(n - 2) and dim TS~- 2 = n - 2, provided that n is even, we obtain the proof (in a small neighborhood of the point X 0 on the sphere S" - 2 the form is non-degenerate and the coefficients of the form are algebraic functions of X3, ... ,xn). The form Fe is non-degenerate on S" - 2, however it is degenerate on V = T.,sn -1 . Moreover, if n is odd, then Fe is degenerate on vn -1 . Since the form Fe is closed on V, Fe is exact. The I-form 1is such that dl = Fe is of the form 1 = X2 L3 ,,~,k<;;n bakxa dXk' The vector field I which corresponds to the I-form I is of the form I(X) = (0, X2bY), Y = (x 3 , • •• , XII)' bE K = so(n - 2). The projection of the field Ion the OX 2-axis is equal to zero; in particular, I is tangent to the spheres S~;3 C sn-2 = 0 (the latter are given by the equation X2 = r = const; see Figure 27). The field I on S~; 3 is given by the standard action of the operator bESO(n - 2) on S"-2 C Rn-2(X3,'" ,XII)' We still need to check the expression

Fe = dl;

dt

=

J3 Ct3

b. kXX )dX 2

+

1\

dX k

x2(I

.
because d

+ 2b = O.

bakdx.

1\

dX k +

L bakdx.

1\

dx k ) = Fe

.>k

This completes the proof.

We can explain the fact that the symplectic structure Ie on S~- 2 is not invariant under the action of the group by the absence of symplectic structures on the spheres (with dimension greater than 2) (see 1.2). Let us calculate the flow X = [C' X, X] on the orbits O. The operator C': V ..... H is defined by the parameters a', b', D' (see above). Since dim T = 1, a' = A' (12). Therefore we can assume that a' = a = (12). The element b' E K = so(n - 2) is arbitrary; the operator D': T ..... K is uniquely defined by the element d' = D'(12). LEMMA 10.6 Let a = (12); b', d' E K = so(n - 2). Then the flow X = [C' X , X] is given on V(x 2, . . . , XII) as follows

117

INTEGRABLE SYSTEMS ON LIE ALGEBRA n

X2

= 0,

Xk = X2

L

X~(b~k - d~k)'

3~k

~

n.

a=3

This means that the flow X preserves the decomposition of the sphere o = S~ - 2 into the spheres S~; 3 = {X2 = const} (see Figure 27). If X=(X2,Y), Y=(X3, .. ·,Xn)ERn- 2, then the flow X on the sphere S~; 3 coincides with the flow Y = X2 . pY, x 2 = const, p = b' - d ' E K = so(n - 2), pY being the image ofthe vector Y under the standard action of so(n - 2) on R(x3, ... , Xn). Proof From Lemma 10.4,

X=

C,,~j"n x 2d{iij) -

at

kt Ct x(lb~ pt )e2k),

=

3"i~j"n

=

-J3 (J3 Xab~k)[(2k)'X2(l2)] 3"i~j"n J3 + o(

x 2d;jx,[(ij), (let)] - kt3

X2 x ad(j [(ij), (let)]

-kt Ct3 Xab~k)Xk(12) kt +

-

L n

d~jX2Xa[(etj), (let)]

at3

L

+

Xp (l P)]

Ct3 X(lb~k)[(2k),

=

xk(lk)] .

3"i~j
x2x,b~k(lk) x 2x adia[(iCt), (lCt)]

n

+ L L

x2xab~k(1k)

k=3a=3

=

L.

X2 x ad ja(1j)

3~~<J~n

=

it J3 (X2

+

I

x 2x adja(lj)

+

3~JJS;;Cl~n

xadja )(1j) +

kt at Xab~k (X2

± ±Xab~k)(1k) (X2

k=3

)elk)

~=3

A. T. FOMENKO AND V. V. TROFIMOV

118

The lemma is proved. Note one obvious integral of the flow n

I

k=3

XkXk=XZ

I

X:

XaXkPak=O,

3~~k~n

i.e. the function Ik= 3 xi = const along the flow X (we noted this before since X = [C' X, X] preserves S'R - 2). Evidently the equations X = [C' X, X] are integrable explicitly. Thus, both the flow X and the field I(X) are of the same form: x1qY, q E so(n - 2), Y E R(x 3, . .. , xn), X2 = canst on the sphere S~; 2 = S'R- 2 (') (X 2 = const). We obtain the form Fe on V as dl, the I-form I corresponding to the field I. Although the flow X is completely Liouville-integrable explicitly (in the classical commutative sense), some very simple integrals of this flow are of interest. Since the integral X z = const of the flow X is trivial, it is enough to consider the restriction of the flow X to the sphere S"- 3 = (X2 = const). Since p E so(n - 2), rf = - p; X2 • pX is tangent to sn-3. Let us consider the linear operator

I: Rn-2(X3"'" xn)

--+

R n - 2 (X3"'" xn)

and construct the function Q/(X) = B(X, I X) on R n -

2.

LEMMA 10.7 The function Q/(X) is an integral of the flow X = pX provided that [I,p] = 0 and Q/(X) to on S"-3. Proof Q/(X)

=

B(X,IX)

+ B(X,IX) =

-B(X,x2PIX)

+ B(X,Ix 2 PX) = 0

if [I,p] = O. The condition B( X, I X) t 0 on S" - 3 means that the vecto r I X is not tangent to the sphere sn - 3 at the point X. PROPOSITION 10.3 Let P E so(n - 2) be an element of the general position in so(n - 2). Let us consider a uniquely defined Cartan subalgebra S(p) c so(n - 2), P E S(p), dim S(p) = r = [en - 2)/2]. Let P = t 1 , t 2 , · · · , tr be linearly independent vectors in S(p) (e.g. an orthonormal basis). Then the functions Q;(X) = B(tiX,pX), I ~ i ~ r are functionally independent integrals of the flow X = x 2 PX = (0, x 2 py) at points in general position on the linear space R n - 2(X3,"· ,XII) = {X2 = const}, these integrals being in involution.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

119

Also, the functions Q l' . . . , Qr are functionally independent integrals of the flow X on the sphere S· - 2 = 0 which has the coordinates (X3' ... ,x.) (ifthe projection is orthogonal, then S" - 2 -+ vn- 2). We can take the coordinate X2 (preserved by the flow X) as one of these integrals. If n = 2r + 2 is even, then the form Ie is non-degenerate on S·-2 = 0 (see Theorem 10.1). Then the flow X is completely Liouvilleintegrable; the number of the integrals Q1"'" Qr being equal to r = (the dimension of 0 divided by two). If P E so(n - 2) is in general position, then the system defines the motion along the surface of the torus T' which is embedded into the sphere S2r in the standard way (an irrational helix on the torus Tr). If n = 2r + 3 is odd, then the number of the integrals Q1" •. , Qr is equal to ~(dim 0 - 1).

Proof Since the operators ti commute with p, from Lemma 10.7 it follows that the functions Qi are integrals of the flow X. Since the Q; are quadratic functions, grad Qi(X) = (t;p)X. Since all elements p = t 1 , t 2 , • •• ,tr belong to the same subalgebra S(p), we can assume that all these operators are simultaneously reduced to the canonical form in some basis f{J1' ... ,f{JZr (provided that n - 2 = 2r). Let the operator t; be (t;f{J1 = t;f{Jz, t;f{Jz = -t;f{J1' defined by a set of the numbers t;, ... etc.). If = 1 C;tfpkXk = 0 were a linear dependence relation (for all k), then Ii=l C;t~ = O. Since the t; are independent on S(p), wefind C; = 0, t ~ i ~ r. Thus, the independence ofQ;(X) is proved (almost everywhere on W- Z). Let us consider in R"-Z(X3,'" ,XII) the sphere S"-3, n = 2r + 2. Restricting the linear space L of integrals (where we take linear combinations with constant coefficients) to S"-3, we obtain a (r - I)-dimensional space, i.e. one of the integrals is constant on the sphere S·-3. Indeed, the system of linear equations A = Ii=l C;t7Pk' 1 ~ k ~ r has a unique non-trivial solution {C;} (since the system is in general position). Geometrically, the solution can be interpreted as a vector in L which is orthogonal to the sphere S" - 3. This vector generates Kern where n:L -+ TS"- 3 is the orthogonal projection. Since the projection of the gradient of the function X z (defined on S"-2) onto the plane (X3' ... , x.) coincides with this vector in Ker n, we can regard the function X2 as an integral. Geometrically, the motion of the trajectories of the flow X on s2r = S"- 2 can be interpreted very simply. The flow X preserves s·-3 = s2r-1 C R 2r(X3,' ",X2r+2)' In the standard basis 2r f{J1,' .. ,f{JZr in V (see above) the flow X = (0, X2PY) is defined explicitly as follows: X(t) = Li= 1 1jjeiX2tPJ (rotations within each of the 2-dimensional planes of the standard basis). Evidently, in the general

Li

,tr

120

A. T. FOMENKO AND V. V. TROFIMOV

position case, the closure of the trajectory X(t) in sn - 3 = s2r-l is the torus Tr which is embedded in s2r-l C s2r. Thus, the tori of the commutative Liouville theorem (T' in M 2r = S2r) are generated by the standard embeddings of the standard tori into s2r. This completes the proof. 10.3. Hamiltonian flows XQ, symplectic structures Fe and the equations of motion of analogs of a multi-dimensional rigid body

For simplicity, let us consider the compact symmetric space lB = miS, where m, S are semi-simple Lie groups. The construction below can be extended to the complex semi-simple case in a trivial way. Let us consider in G = H + V the arbitrary orbit m(X) ::> m(X) n V. LEMMA 10.8 The intersection of the orbit m(X) with the plane V coincides with the orbit O(X) c seX) c V. Proof Evidently, O(X) c m(X) since Scm. Since K = Ann T, dim TO = dim K' = dim SIK. Because [V, V] c H, O(X) c m(X) n V. This completes the proof.

Let a, bEG be two commuting elements in general position. Let us consider a Cartan subalgebra Sea, b) c G which contains a, b (we know that such Sea, b) exists). Let us construct the operators ({la,b,D: G -+ G which were defined and studied above (see [89], [90]): ({la bD = ad; 1 ad b + D. Then, the flow X = [X, ((la,b,D(X)] describes the ni~tion of the analogs of a multi-dimensional rigid body. The flow is tangent to the orbits m(X) and is completely Liouville-integrable (in the commutative sense); see [89], [90] and Chapter 5. As a, b, D vary ([a, b] = 0, D: Sea, b) -+ Sea, b)), the flow X varies on O(X) as well. Let K oF 0, K c H, K = Ann T, let the elements a, bEG (which were initially elements in general position in G) approach V and H and eventually (becoming, in general, singular) come to Tand K, a E Tc V, b EK c H. Then, the flow X = [X, ((la,b,D(X)] reverses on the orbit m(X) and, approaching a "limit," preserves the intersection m(X) n V = 0= seX). On 0 this extremal "rigid body" flow can be regarded as coinciding with the flow XQ' Q = ada' + ad a- 1 ad b + D (see Figure 25). Of course, the flow XQ is not the limit of the rigid body flow literally because if a E T is singular in G, then for some roots oc we have oc(a) = 0, i.e. the operator ada annihilates some plane E, in G and the operator

INTEGRABLE SYSTEMS ON LIE ALGEBRA

121

ada-I is not defined on E~. Since the flow XQ is the limit of the orthogonal projections of the rigid body flow on the plane which is orthogonal to E~, it is clear that this plane coincides with V. Since the rigid body flow is completely integrable with the use ofthe integrals f(X + Aa), f(X) being constant on the orbit functions, we can obtain the integrals of the flow XQ by restricting the functions f(X + Aa) from the orbit GJ(X) to the orbit ~(X) = 0 c GJ(X). From this point of view, the flow XQ (studied in 10.2) is the flow of a "singular rigid body" which corresponds to the compact series of the "rigid body" operators (general position). However, the theory of the integrals of multi-dimensional rigid body (see 7.2) involves one additional series, namely, the normal series. In the following section we shall study the analog of it which is the singular normal series on sn -I .

10.4. The symmetric space

sn-l

=

SO(n)/SO(n - 1) (the complex case)

Let us consider the sphere sn -I = SO(n)/SO(n - 1) and the corresponding decomposition so(n) = G = V + H (see 10.2). Let us embed so(n) in u(n) in the standard way (as the subspace of real matrices). In contrast to the case of the compact series, let us take the elements a, b not from the algebra G = so(n) but from the extended algebra u(n). Let us take the matrix il121 = i('1;.2 + T2d, i 2 = -1 as the element ii; the matrix 5 = i L3 ;;;p
Let ii = i1121, 5= i L3;;;p
122

A. T. FOMENKO AND V. V. TROFIMOV

ex = X 2 I

dJij) -

±(±

k~3

3<;p
X a bak )(2k).

a~3

Proof Evidently, ad,,(2k) = illkl, i.e. adilllki = -i(2k). Let X

E

V,

X = L~=2 xk(lk);

kt Ct kt Ct

ad/) X = [6, X] = - i

Xabak)i

1 adi ad"X = -

X a bak }2k).

lkl ;

In the complex case, the operator C differs from the corresponding operator C in the real case (see 10.2) only in the replacement of the skewsymmetric operator I bpq II by the symmetric one. These calculations are similar to the ones in Lemma 10.4. This completes the proof.

e

LEMMA 10.10 The form Fe, = ad,,-1 ad" + D (the elements ii and 6 being as given above) is given on the space V = TeS" -1 by the formula:

L

F e =F 1 +F 2 =-X 2

dijdxi/\dxj

3~i<j~1I

The form F 2 is exact on the space V: F 2 = dl, the I-form I is given by the formula X2 L;'"k) = 3 b'k X' dXk· The orbits 0 of the coadjoint action are the spheres S~ - 2. The restriction fe of the form Fe to 0 is of the form

fe

= fl + f2 = L wijdx i /\ dXj - -X 2

I

dij dXi /\ dXj

II

X~

= R2 -

L xf. k~3

We can write this in another way:

INTEGRABLE SYSTEMS ON LIE ALGEBRA X3' ...• Xn

123

are coordinates on the sphere S~ - 2.

Proof Evidently. the restriction of the form fl

= B(DnX; [~,17]) =

d.·I] dx.l /\ dx·J

-X2 3~i<j~n

to the orbit 0 is of the form

L

-Xl(X 3 .···,Xn )

dijdx; /\dx j .

3:f:i <j ~n

Then f2 = B(adil ad6X,[~,1J]) = (repeat the proof of Lemma 10.5)

- kt3 Ct3 x,b,k) dX2

/\ dx

k·

The restriction fe of the form Fe to 0 is calculated as in Lemma 10.5:

L

fe=-x 1

dijdx;/\dxj

3~i<j~n

Rewriting the form f2, we obtain II

L

x,(x;b,j - xjb,;}

,=3

= =

L

x,(x;b,j - xjb,;}

L

(x;b,j - xjb,;) + xixj(b jj - bi;) + bij(X i2

,.,. (i,j)

+ Xi(xib ij - xjbi;} + Xj(xib jj - xjb j;}

, .,. (i,j)

Let us prove that F 2 = dl on V. Let us consider dt =

L b'k X, dX2

/\ dXk

+ X2 L b'k dx, /\ dX k "k

,
',k II

L

b"k X, dX2 /\ dX k = F 2'

(',k) = 3

The lemma is proved.

-

xJ).

124

A. T. FOMENKO AND V. V. TROFIMOV

REMARK Similarly to the real case (see 10.2), the I-form I is generated by the vector field 1 = (0, X2 . bX), b = I bij II: R(X3,"" Xn) -+ R(X3,' .. ,xn ) being a symmetric operator (rather than a skewsymmetric operator in the real case). In contrast to the real case, the field Ion R n- 2 is not tangent to sn-3(X2 = const); Ion vn-l:1 E TxO. THEOREM 10.2 The form F c, C = ad,,-1 ad 6 + D (the elements ii and Ei being as described above, d = D(12) E K = so(n - 2», is closed on vn -1 = I; S· - 1 and, in particular, on the orbits 0 = S~ - 2 if and only if d = O. In this case, the form Fe on V· - I is exact and can be written as dl, 1= x2Lk=3 (L:=3b~kXJdxk' On V·- 1 and on 0, the form Fe

=

dl

=

kt3 (t3

Xab~k)dX2

1\

dXk

is not invariant under the action of the groups K = SO(n - 2) and [) = SO(n - 1). If the orbit 0 = S·-2 is even-dimensional (n = 2r + 2, • r = rk SO(2r» and the symmetric operator b = I b pq I is in general position (being the element Ilib pq II, bpq = bqp of the Lie algebra u(n - 2», then the form fe = Felo is non-degenerate almost everywhere on O. Thus, if d = 0 (n is even and the operator bW = b) is in general position), then the form IE defines a non-degenerate symplectic structure (not invariant under the action of the gre>up) on open everywhere dense subset of the sphere 0 = S·-2. Prool Let us obtain a necessary and sufficient condition for the form Fe to be closed on V. Since FE = F 1 + F 2, dF E = dF 1 because dF 2 = d 21 = 0 (see above). Let us calculate dFt. If Fl = Li<j wij dX i 1\ dx j , then

dF 1 =

L

(OiWjk

+ OjWki + 0kWi)dxi

1\

dXj

1\

dx k ·

i<j
In our case,

because x~ = R2 - L:=3X;, Thus, dF e = dF 1 = 0 if and only if Xidjk - Xjd ki + xkd ij = 0 for all i, j, k, i.e. d = O. As follows from explicit formulae, the forms Fe and fe are not invariant. Let us prove that if n is even and b is in general position, then Ie is non-degenerate on O. As follows from Lemma 10.10, on the sphere sn-2 the form Ie with the

125

INTEGRABLE SYSTEMS ON LIE ALGEBRA

coordinates X3, . .. 'X n can be regarded as a form on the linear space R n- Z(X3' ... ,X,,), iij

=

-~ X

z

[± x~(x;b.j

-

~= 3

xjb~;) + x;x/bjj -

b;;)

+ bij(x;Z

- xJ)].

It is enough to prove that for some operator b, the matrix II.t;jll is nondegenerate. Let b be close to a diagonal matrix, i.e. bij for i #- j. Then, the matrix II.t;j II becomes the matrix .t;j = (b ii - bjj)(XiXj/Xl)· Let us assume bii = bi. Then, it is enough to prove that the matrix II (b; - b)(xix)xz)11 is non-degenerate provided that the vector b = (b 3, ... , bPI) is in general position. Thus it is enough to prove that the matrix I (b i - bj)11 is non-degenerate (when n is even) considering in W- l (X3' . .. ,xn) the point with the coordinates X3 = ... = x" = 1.

°

Using the condition that the vector (b 3 , • •• ,bn ) is in general position, let us assume b. = b·, b #- 0. Then 1 - b Zr - 1 1- b 1 - b2

°

°

b- 1

1 - b2r - l

1- b

bZr-l_l

°

As b approaches zero, we find that it is enough to check that the matrix <j (i, j) is non-degenerate. Considering the corresponding system of linear equations, we obtain

Li

° + Xl + + °+

-Xl

- Xl -

Xz

-

° ° + °=°

+ ... + Xlr = X3 + ... + Xlr = X3

X3 - .•.

Hence x 2 + X3 + X4 + ... + XZr = 0, x 2 + X3 = 0, X3 Xk + Xk+l = 0, ... , X2r-1 + Xlr = 0, i.e.

+ x 4 = 0,

... ,

n

L (x ls + XZ s+l) + xlr =

0.

8=1

°

Therefore X2r = and consequently the system has only the trivial solution. This completes the proof. Evidently, provided that for some pair (ij): bi = bj,theform II(b i - b)(xixj/xz)11 is degenerate on Rn-l.Let us calculate the flow X Q = [X, QX] on o. The operator Q: V --+ H is of

A. T. FOMENKO AND V. V. TROFIMOV

126

the form ad a-: 1 ad b, + D'. Since we suppose a' = Ai 1121 = Aa, let us assume a' = a, b' = C = i L3.;p<j';" cpjlpjl; Cpj = cjp being a symmetric operator C in general position. The operator D': T --. K = so(n - 2) is uniquely defined by the element d' = D'(12). From Theorem 10.2, on 0 the element d is not involved in the construction of the symplectic structure fdd = 0). Therefore, we denote d' by d when constructing XQ• LEMMA 10.11 Let the elements a, c(c t = c), dE K = so(n - 2) be chosen as described above. Then the flow XQ = [QX, X], Q = adi 1 ad" + D, is given on the space Vex 2' ... ,X n ) by n

L

X2 = -

Xk = X2

X.XkC. k ;

3 ~.z,k ~n

L

.=

X.(cak - d:zk), .

3

3 ~ k ~ n. In this case, ct = C, d t = -d. Having x~ + ... + x; as an integral, the flow Xo is tangent to the orbits 0 = S'R- z c V (and, in contrast to the real case, does not preserve the decomposition of the sphere sn-2 into the spheres sn-3(X2 = const). Proof Let X = L~ = 2 Xk(1k); nX = x2(l2). From Lemma 10.9 we

obtain

x = L';i~j.;n X2 dij(ij) -

kt3

(J3

XaCak)c2k),

pt

Xi1,P)]

= (see the proof of Lemma 10.6)

=

-ktCt

XaC,k)Xk(12) + kt[X2

J3

Xa(Cak -dak)}lk)

n

= X = x2(12) + Hence,

x2 = - k

L

k=3

±(± = 3

xk(lk).

a= 3

n

There is one obvious integral x~ n

XZ X2 +

L k= 3

Xk = X2

XaC:Zk)Xk;

L Xa(cak -

a= 3

dak )·

+ ... + x; of the flow XQ because

XkX k = - Xz L XaXkC. k .,k

a,k

.,k

127

INTEGRABLE SYSTEMS ON LIE ALGEBRA since c t

= C, d t =

-d. This completes the proof.

It is useful to write the first expression as

x2 = - L"

c~.x; - 2

L C.kX,Xk =

-B(CY, Y).

~
:t= 3

Let us consider the problem whether the flow XQ is Hamiltonian with respect to the symplectic structure it on the orbit 0 = S" - 2. THEOREM 10.3 Let the closed 2-form it = Fe lobe given on the orbit 0 by the operator c = ad,,-l ad/1 = c(a, 5,0,0), the elements a, 5 being as described above; 5 = ib, bt = b. Let the flow }(a on 0 be given by the operator Q = adi! ad" + D, c = ic, ct = C, d = D(12)EK. Let us consider together with the symmetric operators c, b (which map t R,,-2(X3' ... ,x 4 ) into itself) the symmetric operator h = sb + bs ; s = C - d, i.e. h = (bc + cb) + [b,d]; ht = h, h:R"-2 -+ R"-2. Let us consider on Rn - 2 four vector fields cX = X, hX, bX, cX and the corresponding I-forms EX

L"

=

Xa

dx ,;

,=3

HX

= I"

n

BX

=

I

,= 3

(hX)a dx ,,;

a= 3 /I

(bX)~ dx,;

CX =

L (cX),dx

,=3

a•

Then on the orbits 0 the form it is invariant under the flow Xa (i.e. Xait = 0) if and only if the I-forms E,H, B, C satisfy the relationship E 1\ H = 2B 1\ C. The symbol 1\ denotes exterior product. If n = 2r + 2 is even and the operator b is in general position, then it is a non-degenerate symplectic structure almost everywhere on s2r and the condition E 1\ H = 2B 1\ C describes all such Hamiltonian flows XQ with respect to fc. In particular, ifthe flow Xa is fixed, varying band d in such way that E 1\ H = 2B 1\ C, we obtain a series of different symplectic forms for which the flow XQ is Hamiltonian. If the operators band c are both diagonal, b=I?=3b;Tij, C=Lt=3C;T;;, then the condition E 1\ H = 2B 1\ C becomes [b, d] = 0, [b, C] = 0, C = cij cij = C; + cj • This means that if bk "# bp , then C k + cp = and d pk = 0; if bk = bp , then Ck + c p and d pk are arbitrary.

°

I II,

REMARK The vector fields eX, hX, bX, cX on R"-2 can be identified with the gradients of the following quadratic functions ie = B(X, X), J;. = E(hX,X), j" = B(bX,X), f; = B(cX,X). Then a necessary and

128

A. T. FOMENKO AND V. V. TROFIMOV

sufficient condition for the flow XQ to be Hamiltonian (which was formulated in Theorem 10.3) is E

(BC

1\

+ CB +

[B,d])

= 2B

1\

C.

Thus we can vary the flow XQ so as to preserve the form fe (for a fixed fe) since the factorization of the form w = E 1\ H = 2B 1\ C in the Lie algebra of exterior forms on Rn-2(X3' .. . , xn) is not unique. The more such factorizations we have, the more flows XQ exist which are Hamiltonian with respect to the form fe. Proof of Theorem 10.3 V n - 1 is

Let us assume t =

+ =

d. Let us calculate

C -

XQF e = ~ (~X,bock ) dx z

As follows from Theorem 10.2, the form Fe on

1\

t (~x,b"k)dX2

1\

k

ex

1\

dX k

dXk 1\

dXk

+ ~ (~x,bock )d( -

1\

dXk +

(XqC sq dx s + XsC sq dxq)

1\

dX k

s,q

I (I k

a

X, bock)

(I s,q

XSXqC Sq )

X2t(~x,bock)~tskdX2

xz[~ {(~ b,k(~ xsts,) + ~ ( ~ x,b,. )tkS } dX 2 - 2

E

1\

dX k

d(X2~Xstsk)

X2~(~bock(~Xst.,))dX2 - I (I x,b,k) I

=

1\

~ (~X,b'k) dX 2

~ (~b,k(~ X2Xst.,)) dX2 + ~(~X2b'k)dX2

=

dXk +

XqCsq dX s)

1\

dX k

1\

dxkJ

1\

dX s)

129

INTEGRABLE SYSTEMS ON LIE ALGEBRA

=

X2

L: k

[:L xs(b>.ktsa + bsatka)] dX2

A

dX k

a,s

-2~[(~X,bak)(~XqCsq)]dXs

A

dXk.

This calculation was for vn -1. Evidently, on V we have Fe #- 0, i.e. in general the flow XQ is not Hamiltonian with respect to Fe. Let us consider the orbit o. Let us restrict the derivative t c to o. Using the relationship

we obtain

Fclo = - L: k,q

= -

[:L xsxq(baktsa + bsatka) + 2(:L Xabak)(:L XScqs )] dXq a,s

:L [L: {xsxibaktsa + b.,tka) -

q
+ = -

xsxk(boqtsa

+ bsatqa )}

a,s

2(~Xabak)(~XSCqs) 2(~x,b.q)(~ -

:L [L: xs(xqhks -

q
Xkhqs)

bqaXa)(

XSCkS) ]dXq A dXk

+ 2(L: bk.Xa)(L: CqSXs) :x

s

-2(~

~ CksXS)] dXq

s

A

dxk.

We are also using

h = bt T + tb, a

S

Hence,

Fclo

=

ic =

A

S

0;

-

q~k [Xq(~ hkSXS) - Xk(~ hqsXs) - 2{ (bX)q(cXh - (CXMbX)d] dX q A dX k

dX k

130

A. T. FOMENKO AND V. V. TROFIMOV

= -

I

q
= -

{I

q
= -die Thus

2B

[(xq(hXh - xk(hX)q) - 2(dJ;, (die

1\

1\

dh

dj~)qk dXq

+ 2dib

1\

1\

dXk - 2

1\

I

q
dfc)qk] dXq (dib

1\

1\

dX k

die)qk dXq

1\

dXk }

dfc·

Jc = -dfc 1\ dh + 2dJ;, 1\ dfc, i.e. Jc = 0 if and only if E 1\ H =

C. This completes the proof of the first part ofthe theorem. Let the operator b be diagonal and consider the equations for the operators C and d. Since bij = (jijb i , the above system becomes 1\

s

s

XkXq(hkk - 2bqCkk - hqq + 2b kcqq )

+ x;(h kq +

I

s",,(k,q)

- 2bhg) - xt(h qk - 2bkcqk)

xs[xq(h ks - 2bqCkS) - xk(h qs - 2bkcqS )]

= O.

Thus we obtain

hkk - hqq - 2bqCkk + 2bkCqq = 0, hkq - 2bqCkq = 0; If s =F (k, q), then hks - 2b qcks diagonal, i.e. cij = (jijc i , then

= 0; hqs

hqk - 2b kcgk = O. - 2b kcqs

hkk - hqq - 2c kbqq + 2c qbkk hkq + 2cqbqk = 0;

= O. If the operator C is

= 0;

hqk + 2bkq Ck = 0

and if s =F (k, q), then hks + 2b sk cq = 0; hqs + 2bsqCk = O. If both operators band C are diagonal, then hkk = 2bkCk; and if k =F s

hks

= bkt.k + tksbs =

-bkdks

+ dksb s = (b s -

bk)dks '

Thus

hkk - hqq

+ 2bh + 2bkcq =

(Ck

+ cq)(bk -

bq).

Since bks = Cks = 0 for k =F S, hks = 0 for k =F s. Therefore (b s - bk)dks = O. Finally if b, C are diagonal then we obtain the equations

131

INTEGRABLE SYSTEMS ON LIE ALGEBRA

+ cj)(b; dij = 0; C; = (C;

bj )

= 0,

-Cj.

(b; - bj)dij = 0 for h, c, d. Thus if h; =1= hj then This means [b, d] = 0 and [b, C] = 0, = C; + Cj.

cij

Thus the theorem is proved completely. REMARK The equations for b, c, d can be derived from the condition XQl = 0 (Fe = dl on V) as well. Let us consider examples of the forms Fe and the flows above. Some of the flows we already know.

10.5. Examples of flows

Xo on S·-1

XQ obtained

(the complex case)

EXAMPLE 1 Let us consider XQ, Q = ad; 1 ad" + D. Let C = 0, dE K = so(n - 2). Then X2 = 0, Xk = - X2 I~= 3 Xadak' i.e. the flow XQ coincides with the flow of the form Xc (see 10.2). This flow is generated by the standard action of an arbitrary skew-symmetric operator on the sphere (see 10.2). The flow is Hamiltonian and completely Liouvilleintegrable. Thus for C = 0 the "real" and "complex" flows coincide on the sphere. EXAMPLE 2 Let us consider XQ assuming d = 0, C;j = buc; (i.e. c is 3 ckxf; Xk = X2CkXk' 3 ~ k ~ n. This flow is diagonal). Then, X2 = integrable explicitly on the sphere sn -1. The complete set of integrals (obtained by A. V. Brailov) is very simple: !;j(X3' ... , xn) = (xij)!(xj;); 3 ~ i <j ~ n; f = x~ + x~ + ... + x;. Calculating directly, we can check that X!;j = O. Evidently the integrals f34, f45, ... , In-l,n are functionally independent at the points in general position on the orbit 0= sn-2. These integrals are n - 3 = dim 0 - 1 in numbers, i.e. the flow X is integrable explicitly. Note that by a trivial substitution 2Yk ; Yk = In Xk, 3 ~ k ~ n the flow X Q can be reduced to X2 = 3 cke Yk = CkX2; 3 ~ k ~ n.

IL

Ik=

In Examples 1 and 2 we pointed out two "extremal" flows; the flow XQ is of a general nature and has the properties of both these flows. Among these "mixed" flows, there is a classical rigid body flow on the Lie algebra so(3); 0 = S2 (see 7.1).

m= S2 = SO(4)!SO(3); we shall define the flow . usmg . the operator Q-= ad; ado + D, c = (C30 c0) -= . E su(2), XQ EXAMPLE 3

Let n = 4;

1

' C

4

IC

132

l.e.

A. T. FOMENKO AND V. V. TROFIMOV

C3+ C4=O.

Let dEK=so(n-2)=so(2), d=(

0

A).

We

-A 0 3 = z. Then the flow XQ on V = 1/S2 is given

assume X 2 = X, X3 = y, X4 by x = -cyZ + cz 2 , y = x(cy + AZ), i = x( -cz - Ay). Assuming c = 1 we obtain x = - y2 + zZ, y = x(y + AZ), i = x( - Z - Ay). Let us transform this flow into a more convenient form by performing an orthogonal rotation through the angle n/4 in the plane (y, z), i.e. we introduce new coordinates (x + y)/ j2 = u, (z - y)/ j2 = v. Then x = (z - y)(z + y); (z + y)' = X(A - 1)(z - y); (z - y)' = -X(A + 1)(z + y), i.e. we obtain x = 2uv, = - (A + 1)xu in the variables x, u, v.

v

U = (A - 1)xv,

Let us recall the general form of the equation of motion of a classical three-dimensional rigid body x = UV(A 1 3 - A23 ); U = XV(AZ3 - A12 ); v = X4(A 12 - Al3), Aij = (b j - bj)/(a j - aj) = a j + aj since b = a 2 (see 7.2). Therefore x = uv(a 1 - a z)' U = xv(a3 - ad, v = xu(a2 - a3); a1 , a 2, a3 being arbitrary. Comparing this with our system, we derive the conditions a 1 - a2 = 2, az - a 1 = A - 1; az - a 3 = A. - 1. Therefore, the relationships a 1 =
10.4

For n = 4 the "complex flow" XQ defined by the

133

INTEGRABLE SYSTEMS ON LIE ALGEBRA

operator

Q=

ad,,-l ad"

+ D,

ii = ill21,

c=

ic E su(2), c =

(~ c~).

+ C4 =

0 coincides with the equations of motion of a threedimensional rigid body with a fixed point (a!, a2' a3 being arbitrary). Moreover, this flow XQ is Hamiltonian with respect to the nondegenerate symplectic structure Ie on 0 = S2 defined by the operator c = adi! ad b (with an appropriate choice of the operator Ii = ib) according to the general construction given in 10.4. C3

Proof Above we already have proved the first part ofthe theorem. We need to choose the closed non-degenerate form fe which is invariant under XQ• For this, let us consider the form Fe = dl given by the 34 arbitrary operator Ii = ib, b = bb ) (not necessarily symmetric) on V = R 3 •

(:3343 44 Then the explicit expression for Fe is Fe

Ie to

The restriction

= kt3

(t3 X~b~k)dX2

S2 is given by

Ie = Iij dX i "

k

=

-~ [ L x2

~ '" (i,j)

dx j ,

Xa(Xi b7j - xjbai ) + xixj(b jj

In our case, all fij

"dx k ·

-

bii ) + xrbij - XJbji].

= 0 except

2 2 1 f34 = - - [X3 X4(b 44 - b 33 ) + X3 b 34 - x 4b43 ]· X2

Since Fe = dl, this form is closed on Vand 0 for any b. Since 134 #- 0, i.e. b #-

Jl( ~ ~). Jl E R, the form Ie is nondegenerate on S2(X3' x4 ). Thus it

is enough to verify that the equation XQ fe = 0 (regarded as the equation in the operators b, d) has a solution such that d#-O (i.e. A #- 0),

b i=

JlG ~).

Indeed, since c is diagonal, the equation XQIe = 0 is

equivalent to the following system (see Theorem 10.3) h33 - h44 - 2c 4b 33 - 2C3b44 = 0 h34 + 2b34C3 = 0

134

A. T. FOMENKO AND V. V. TROFIMOV

h = bt T

+ tb =

bc

+ cb + [b,a].

Hence,

h=(

2C3b33 - Jc(b 34 + b43 ) b 34 (C3 + c4) + Jc(b 33 - b44 b43 (C 3 + C4) + Jc(b 33 - b44 ) 2x 4b44 + Jc(b 34 + b43 )

»)

+ bd + (c 3 + c4)(b 33 - b44 ) = 0 b 34 (3c 3 + c 4) + Jc(b 33 - b44 ) = 0 -Jc(b 34

b43 (C 3 + 3c 4) + Jc(b 33 - b44 ) = O.

Since

C3

+ C4 =

0, from the system we obtain

Thus, b=

p

q

-q

2c p - -q

,

det b # O.

Jc

Since Jc # 0 and p, q are arbitrary real numbers, the theorem is proved. If we are looking for a solution of the equation X Qie = 0 in the class of symmetric operators b, from Theorem 10.3 we obtain 3 d = 0, (c 3 + c 4)(b 3 - b4) = 0, i.e. b = (b o b0), ie is closed and nonREMARK

4

degenerate on S2. In this case, due to the condition d = 0 (i.e. Jc = 0) it is guaranteed that such a rigid body flow is Hamiltonian only if J.l. = 3, i.e. at = 3, a 2 = 1, a3 = 2, i.e. the inertia ellipsoid is oftheform (1,2,3); the inertia ellipsoid is not arbitrary (though the axes are of different lengths). The statement formulated in Proposition lOA can be proved not only for the rigid body operators (i.e. ic E su(2), i.e. C 3 + C4 = 0) but also for any diagonal c, C3 + C4 = O. In this case, we have the relationship

Jc2 = (C3 + c 4)(3c 3 + C4)(C3 + 3C4)/2(c4 - c 3 ) between Jc, c 3, C4. Therefore, the equation

XQ ie = 0 can be solved.

135

INTEGRABLE SYSTEMS ON LIE ALGEBRA

4 Let us construct a more simple case: n = 3. Then, b is defined by the single number b = b, c = c, d = 0. We have the following formulas for XiJ: X2 = -cx~, X3 = CX2X3. Hence, x = 2cxx, x = cx 2 + A; let c > 0. Then, if A > 0, the solutions are given by the functions EXAMPLE

)crtg(jH ~ if A < 0, by the functions

~arcth( J

1

-J / +

o

t,

Ixl<~~;

= t,

Ixl>H·

(fl· TX )

arcth -

AC The image in the phase space is given by T

2

V = -

x = y, y =

f~ f(~) d~ = _(C2

f(x) = 2C 2 X 3

+ 2dx

t;

X4

= !(X)2

+ 2h2 +

= !y 2;

J1}

(see Figure 28).

A. V. Brailov suggested the following interpretation of some forms of the type Fe in the case of the symmetric space sn -1 in V. Let us consider the vector bE K cHand the parallel translation of the plane V along vector b. We obtain the plane v" = b + V; here the restriction of the Kirillov form is not trivial. Further, we obtain the form Wb which is closely linked with the forms of the type Fe (at least, in the real case on REMARK

sn-l ).

u(x)

>-<

0

Fig. 28.

4

Methods of construction of functions in involution on orbits of coadjoint representation of Lie groups

11. MET HOD OF ARG UME NT TRAN SLAT ION 11.1. Translations of invariants of coadjoint representation

The most efficient method of construction of functions in involution on the orbits of the coadjoint representation of Lie groups is the method of argument translation. Originally, this meth od was used in the theory of the Korteweg--de Vries equation (see [74], [28], [92]). It is possible to construct the set of functions fa(x) = f(x + .lea), .Ie ER, a E V for the function f(x) on a linear space V Generally speaking, this process does not give new functions. For example, let us consider on the plane [R2(X, y) expressed as a function of f(x, y) = eX. Then fa(x, y) = eX Ha = eAa'eX , i.e. !a(x, y) is expressed functionally by the initial function f(x, y). Taking a polynomial in general position as f, we obtai n (translating !a(x) with a fixed) N new functions (N = deg f is the degree of the polynomial f): j~(x)

N

= f(x + .lea) =

L Pi(X, a).le i .

In this case, general position to all appearances implies that repeated factors are absent in the expansion of f in irreducible factors. For example, the following statement holds: LEMMA 11.1 Let f(x, y) = ax 2 + bxy translation of the function f:

136

+ cy2. Let us consider the

137

INTEGRABLE SYSTEMS ON LIE ALGEBRA

d

(f)(x)

Then,

= dA ;.=/(X + Aa).

f and (f) are functionally dependent if and only if A = b2

-

4ae = O.

The proof follows from the explicit expression of the Jacobi matrix J of the functions f = ax 2 + bxy + ey2; (f) = 2aAx + bBx + bAy + 2eBy, the translation vector being of the form a = (A, B) i= O. J=(2ax+b Y 2aA + bB

bx+2e y ). bA + 2eB

The determinant J is equal to zero if and only if (4ae - b 2 )B = 0 and (b 2 - 4ae)A = 0 which gives the proof of the statement. A more general example (when the shifts are functionally independent) is given by invariants of the coadjoint representation of semi-simple Lie groups (see Chapter 5). The following theorem validates the use of argument translation in the theory of Hamiltonian systems.

11.1 (see [88], [89], [90], [91]) Let F and G be two functions on the space K* which is dual to the Lie algebra K. These functions are constant on the orbits OCt) of the coadjoint representation (invariants of coadjoint representation of the Lie group .;(" which corresponds to the Lie algebra K), a E K* being a fixed covector, A, J.1. E II\I! being arbitrary fixed numbers. We claim that the functions F).(t) = F(t + Aa) and all(t) = G(t + J.1.a) are in involution with respect to the standard symplectic Kirillov structure on all the orbits OCt) (for any A,J.1. E II\I!). THEOREM

Proof Since s grad t F

= ad:F (t), we need to prove that I


~

I

= -t([dFt,dar])·

Let A i=

J.1.

and t = !X(t

+ Aa) + P(t + J.1.a), f3 =

AI(A - J.1.),

!X

= J.1.1(J.1. - A).

138

A. T. FOMENKO AND V. V. TROFIMOV

Then
= a«t + Aa), [dF(t + Aa), dG/l(t)]) + f3«t + f.W), [dF.. (t), dG(t + Ita)]) = a
because the functions F and G are constant on the orbits Ad*. If A = It, then the result is zero because of the continuity. This construction allows us to obtain a complete involutive set of functions for a wide class of the Lie algebras including the semi-simple Lie algebras. The idea of translation of invariants at first appeared in [74J (S. V. Manakov) for the case Lie algebra so(n) and was then developed by A. T. Fomenko and A. S. Mishchenko in [89J in the case of arbitrary semi-simple algebras. THEOREM 11.2 (Bolsinov, A. V.) Let G be the complex Lie algebra, x E G* be the regular covector, a E G*. We denote the space {df(x + Aa) I AEC, fEI(G)} by SI. The equality dim SI = 1(dim G + ind G) holds if, and only if, the element a + AX is regular for any complex number AE C. COROLLARY The translation of the invariants of the coadjoint representation is the complete commutative set if, and only if, the inequality dim M sing > 1 holds, where M sing is the set of the singular points for coadjoint representation. This theorem was proved in 1986 and generalized many previous results. 11.2. Representations of Lie groups in the space of the functions on the orbits and corresponding involutive sets of functions

There is a general construction of argument translation for functions from the finite-dimensional representations. This construction includes ~ll known methods of translation (see [126]). Let W be a subspace of the space of analytic functions A(G*) on the space G* which is dual to the Lie algebra G and is such that (a) dim W < OC!; (b) for any g E (f), f E W:f(Ad; x) E W Let us choose a basis f1' ... ,is in W Then, on the group (f) which corresponds to the Lie algebra G, each function defines a set offunctions

139

INTEGRABLE SYSTEMS ON LIE ALGEBRA

cj:f(Ad; x) = cj(g).t;(x) and a set oflinear functionals c~(f) on G, i.e. c~(f) EG*: c~(f)(()

d. . cj-(Exp t() = dcj-«() , dt t=O

=-

c~(f) being the differential of the mapping cj at the unit of the group.

Under all the above assumptions, we claim that (s grad f)(x) = C~(f)f1 (x) + ... + c~(f)f.(x).

PROPOSITION

11.1

Proof Let us differentiate the equation (with respect to t for t

=

0)

= c 1 (Exp t()fdx) + ... + c'(Exp ()f.(x):

f(Ad~xPt~ x)

C~«()f1 (x)

= :t t=O

f(Ad~xPt~ x)

= f*(adt x)

=

f*.x(:t 1=0

+ ... +

c~«()f.(x)

Ad~xPI~ x)

= (adt x)(f*.x) = x([(,J*.x])

= -x([f*,x,(]) = -
OJ(c~/;(x), a(x,

en = OJ( -a(x, f*,x), a(x, en = <x, - [f*.x, (J) = <x, [(,J*.xJ) = a(x, ()(f*.x) =
11.2 Let Xi = C~jXk 8/8x j be differential operators in the space COO(G*), Xi being coordinates in the basis ei E G* adjoint to the basis e i E G, C~j be the structure tensor G in the basis ei' Then,

PROPOSITION

a) if W is an invariant subspace in A(G*), then for any fEW, Xi E. W. Le0"] Wand fl ' .. ,j. be J~e basis in W. Then Xi f = c{ jj; J 6) If Xi f - ci jj, then c*(f) Ci e .

r

Proof From the equality f(Ad~xPt~ x)

= ci(Exp t()/;(x)

140

A. T. FOMENKO AND V. V. TROFIMOV

it follows that d -d t

f(Ad~xpt~ x)

=

.

C~(~).t;(X),

t=O

but

d dt f(Ad~xPt~ x)

= dfAadt x) = (adt x)(dfx)

= (x, [~,dfxJ) = -(adffJx»(~) = - [(XJ)ei](~).

Therefore (XJ)e i = - .t;(x)c~. Let us prove claim (a). Provided that W < A(G*) is a finitedimensional subspace of A(G*) invariant with respect to Ad* and since, as a consequence, W is closed, we obtain

d -d

t t =0

f (AdExPt~ * x) E W.

The validity of claim (b) follows from the equation ( - .t;c~) = [X;(f)]e i = c{ fje i . LEMMA 11.2 Let f, g be smooth functions on G*. Then, {f, g} = 0 on all orbits Ad* if and only if C~jXk of/oxi og/OXj = 0; ei being a basis in G, G~j being the structure tensor for G in the basis ei , ej being the adjoint basis in G*, Xi being the coordinates in the basis ej .

Proof We have the expression s grad fx = adff,(x) for the skew gradient. Therefore, {f,g}x = w(sgradfx,sgradgJ = (dfx,sgradg x )

= (dfx, addyx(X» = [addg)X)](dfx) = (x, [dg x, dfx]) og of

k

og of

= x([ei' ej]) -;--;= CijXk -;~; UX i uX j UX i UX j og dg x = -;- (x)e;, uX;

of dfx = -;- (x)e;. uX;

THEOREM 11.3 (Trofimov, V. V.) Let W be a finite-dimensional subspace of A(G*) invariant under the coadjoint representation of the group (D which corresponds to the Lie algebra G. Let fl' ... , f. be a

141

INTEGRABLE SYSTEMS ON LIE ALGEBRA

basis in W. Let us define in W the functions hi' ... , hp E W such that 1 ~ i, j c~*

=

~

p,

c~(hj) with respect to the basis fl""

1~ k

~

s,

,J..

We assert that (a) {hi' hj} = 0, 1 ~ i,j ~ P on all orbits of the coadjoint representation of the Lie group GJ; (b) the translates of hi are in involution on all orbits of the coadjoint representation Ad*(GJ): {hi(x + Aa), hj(x + Ila)} 0, 1 ~ i, j ~ p, A, Il E IR, a E G*.

=

Proof Let us check that

aha

d;'jXk-;;-(X UXj

ah

+ Aa)::l

P (x

uX j

+ Ila) = O.

(1)

142

A. T. FOMENKO AND V. V. TROFIMOV

If we consider expression (1) as a limit, the proof follows for v = J.I. - A= O. Definition 11.1 follows from Theorem 11.3. DEFINITION 11.1 We shall call the finite-dimensional subspace W of A(G*), G being a Lie algebra together with a set of functions hi' ... , hp E W, an S-representation of the group
= 0 on G*.

REMARK 1 Evidently, the condition
INTEGRABLE SYSTEMS ON LIE ALGEBRA

143

Let G be a Lie algebra, (J be an involutive automorphism of G, G = Go EB GI be the expansion ofG with respectto (J, G* = G~ EB GT be the expansion of G* with respect to 0"*. Since (J is an automorphism of G, [G i , Gj ] C Gi + j; i, j = 0, 1 being taken modulo 2. Let f be an~ smooth function on G*, we denote the restriction of the function by f LEMMA 11.3 Let (J be an involutive automorphism of the Lie algebra G, G = Go EB GI and G* = G~ EB GT be the corresponding decomposition of G and G*, a E GT, A., J.l E IR; j, 9 be invariants of G. Then the restrictions of the translates ofthe invariants La and gila are in involution on all orbits of the coadjoint representation of G~ (the invariants are meant to be invariants of the coadjoint representation). Proof We have dJ.,(x) = !(dJ.,(x) + d]_a(x)), x E G6 for the translate of any smooth function f by the element a E GT, ](x) = f(J*(x». If j E J(G), then ] E J(G) (I(G) being the invariants of the coadjoint representation of G). Therefore {La,glla}(x)

= i<x, [djAa(X) + d]-Aa(x),d9Ila(x) + dg- Ila (x)]) = i<x, [dfAa(X), d9Ila(x)]) + i<x, [dfAa(X), dg -Ila(x)]) + i(x, [d]-;.a(x),d9Ila(x)]) + i<x, [df-;'a(x),dg- Ila (x)]) = Hha,glla}(x) + i{!;'a,g-Ila}(x) + iU-;.a,g/l,}(x) + iU-;.a,g-lla}(x) = O.

The! above expression follows from Theorem 11.1. Since f, g, ], g are invariants of the coadjoint representation Ad*(f).

12. METHODS OF CONSTRUCTION OF COMMUTATIVE SETS OF FUNCTIONS USING CHAINS OF SUBALGEBRAS

The results of this section were obtained by V. V. Trofimov [127]. Let el , • •• ,en be a basis of the Lie algebra G; el , ..• , e" be the conjugate basis of G*; Xl, ... , xn and x I, ... , Xn be the corresponding coordinates. Let H be a subalgebra of the Lie algebra G. We have the projection G* -+ H* which is defined by the restriction of linear functionals. In this case, the functions defined on H* can be lifted to G*. LEMMA 12.1 Let f and 9 be functions on H* which are in involution on all orbits of Ad*(~). Let us consider the extension ofthese functions

144

A. T. FOMENKO AND V. V. TROFIMOV

to G*. They remain in involution on all orbits of the coadjoint representation Ad* of the Lie group (f) which corresponds to the Lie algebra G. Proof Let e1 , •.• ,e. be a basis for H. Let us extend it to a basis for e1 , ... , e., e. + 1 , ... ,en in G. Let i = 1, ... ,s; IX = S + 1, ... , n; a = 1, ... ,n. We shall use these notations throughout this section. In this case, the lifting offunctions from H* to G* does not depend on the coordinates Xa' Let us check the involutivity using Lemma 11.2. We have

_ k

al ag

= C"Xk-ax; aXj = O. I}

Considering ag/axp = 0 and [e;, ej] E H, we obtain clj = O. Finally, we obtain zero because we have the condition that {f, g} = C7jXk ai/ax; ag/axj 0 on all the orbits of Ad*(~).

=

LEMMA 12.2 Let G ~ H be a chain of subalgebras. Iff is an invariant of the coadjoint representation Ad*(f)), and 9 is the lift of a function on H*, then I and 9 are in involution on all orbits of the coadjoint representation of the group (f) which corresponds to G. The proof follows from the equality

k -al -a ag _ (k a/) -a ag -_ O. cijx k aXi Xj - CijXk-a Xi Xj LEMMA 12.3 If H c G', G' being the derived algebra of the Lie algebra, I being a semi-invariant of Ad*(f), ag/axa = 0, then {j, g} 0 on all the orbits of the coadjoint representation Ad*(f)).

=

Prool The function I is a semi-invariant of Ad* if and only if C7jXk af/ax j = ).,J Therefore

INTEGRABLE SYSTEMS ON LIE ALGEBRA

145

but A; = 0, since the character on the derived algebra equals zero. Thus, for each chain of subalgebras G ::::J H 1 ::::J H 2 ::::J ... ::::J H. such that G::::JG'::::JHi::::JH~::::JH2::::JH2::::J···::::JH~_1::::JHs we can construct functions in involution on G* using a large number of the The latter functions may be obtained by functions in involution on e.g. translation of invariants, with the subsequent addition of semiinvariants of the coadjoint representation of the group t); which corresponds to the Lie algebra H;. We can add not only semi-invariants but also functions from some representations.

H:.

LEMMA 12.4 Let K be a subalgebra of G and V c A(G*) be a finitedimensional subspace of the space of analytic functions on G* invariant with respect to Ad&. A function f E V defines co vectors c~ E G* when we choose a basis in V. If the extension of a function 9 E A(K*) is constant along these covectors, then it is in involution with the function f on all orbits of the coadjoint representation of the group ffi. \

\ The proof is similar to that of Lemma 12.3. REMARK It is enough to use the procedure described in Lemma 12.4 for a construction of a complete involutive set of functions on Borel subalgebras in the semi-simple Lie algebras (see [126], [127]). The simplest example of the construction described above can be obtained if H. is a maximal Abelian su balgebra in G'. We can take as an invariant of an Abelian subalgebra any element of the subalgebra if we regard it as a function on dual space. In general, the number of invariants in the maximal Abelian subalgebra is not enough to construct completely integrable systems. Therefore we need to use some subalgebras. LEMMA 12.5 If there is a chain of subalgebras G ::::J H 1 ::::J H 2 such that G::::J G'::::J Hi ::::J H~ ::::J H 2, H2 = 0, [H i ,H 2J = 0, then any function f E A(Ht) and elements H 2 (regarded as functions on H!) are in involution on all orbits of Ad&(G*).

146

A. T. FOMENKO AND V. V. TROFIMOV

Proof As follows from the condition [H I ,H 2 ] = 0, the operators Xd do not involve derivatives with respect to the coordinates conjugate to the coordinates in HI and the function f does not depend on the rest of the coordinates. Therefore,

Finally, we have a theorem. THEOREM 12.1 Let V=> S be a chain of subalgebras. If the functions f and g on S* are in involution on all orbits of Ad('lJ) , 'lJ being the Lie group which corresponds to the Lie algebra S, then the functions J and gare in involution on all orbits of Ad~; m being the Lie group which corresponds to the Lie algebra V; J = f 0 n, g = go n, n: V* --+ S* being the restriction mapping. If f is an invariant of the coadjoint representation of the group mand g is the lifting of g E A(S*), then {f, g} 0 on all the orbits of Adt. If the chain V=> S is such that V => V' => S, f being a semi-invariant of Ad~ and g being the lifting of g E A(S*), then {f, g} = 0 on all the orbits of the coadjoint representation Ad* of the group m.

=

REMARK The technique used in this section is a generalization of the construction of M. Vergne (see [134]). In [134] for a construction of global symplectic coordinates on the orbits of maximal dimension of the representation Ad* in a nilpotent Lie algebra chains of ideals were used such that G I C G 2 c··· c G; dim GdGi-1 = 1. Invariants of the representation Ad* of the ideals Gi (i = 1, ... ,dim G) were lifted to G* by means of the natural projection G* --+ Gt which provided the coordinates needed.

In conclusion, we give two statements in which the operations of argument translation and function lifting are used simultaneously. Let a E G* be an element of the dual space of the Lie algebra G; let Ga = {X E G: (ad X): = O} be the annihilator of a with respect to the coadjoint representation of G on G*. LEMMA 12.6 Let a E G* be an element of the space dual to the Lie algebra G; G be the annihilator of a; f EI(G) be the invariant of G; g be a smooth function on (GQ)*. Then the function fa(x) = I(x + a) and g (regarded as a function on G*) are in involution on all orbits of the Q

147

INTEGRABLE SYSTEMS ON LIE ALGEBRA coadjoint representation of the group algebra G) on the space G*.

m (m

corresponds to the Lie

Proof We have {fa,g}(x)

= (x, [dla(x),dg(x)]>

= - (a, [dfa(x), dg(x)] > = -(addg(x)):,dfa(x» = 0 because dg(x) E G*. The lemma is proved. LEMMA 12.7 Let (1 be an involutive automorphism of the Lie algebra G, G = Go EEl G 1 and G* = G~ EEl Gr being the corresponding decompositions of G and G* (see Section 11); a E Gr, I EI(G), g be a smooth function on (G )* which is the space dual to the annihilator of a in G. Then, the function fa (the restriction to G~ of the translate of the invariant I by the element a) and the function g (regarded as a function on G~) are in involution on G*.

o

Prool We have {fa,g}(x)

= (x, [dfa(x),dg(x)J>

= 1(X, [dfa(x) + d]-a(x), dg(x)J> = -1(a, [dfa(x) - d]-a(x), dg(x)J>

= 1«(addg(x)):,dfa(x) because dg(x) E

d]_a(x»

= 0,

Go. The lemma is proved.

13. METHOD OF TENSOR EXTENSIONS OF LIE ALGEBRAS

13.1. Basic definitions and results

The results of this section are given in the work by V. V. Trofimov [129]. Let us consider an arbitrary r-dimensional Lie algebra Gr over the field k. We denote the ring of polynomials in the variables Xl' ... , Xn over the field k by k[Xl"'" x n]. Let fl'"'' f. E k[Xl' ... ,xn ] be such that 11 (0) = ... = f.(0) = O. Let iii be the quotient-ring iii = k[x 1, ... , x n]/ (fl' ... , f.); (fl' ... , f.) being the ideal in k [x l ' . . . ,xn ] generated by the polynomials 11"'" f.; let 1[: k[Xl' ... , Xn ] -+ iii be the natural

148

A. T. FOMENKO AND V. V. TROFIMOV

projection. We denote the image of Xi under the mapping 7r. by ei • Then, IK is multiplicatively generated by the elements e1' ... ,e" E IK over the field k. Let us extend the ring of scalars of the Lie algebra G to IK, i.e. consider the Lie algebra GK= Gr ® IK over the ring IK. The field k is contained in the ring K Therefore, we can consider the Lie algebra G';.; over the field k. We denote this Lie algebra by OAG), 1 = (f1' ... ,/.) being a polynomial mapping I: IRn -+ IRs such that I{O) = O. We claim that 0 f{G) is an extension of the Lie algebra G. It is enough to construct an epimorphism of Lie algebras g: Of (G) -+ G. Note that any element y E Of (G) can be represented as

Let us set g(y) = xo ..... o' Evidently this is an epimorphism of Lie algebras. The Lie algebra Of (G) is a split extension of G. Let us explain this. Since G is embedded in Of (G) as a subalgebra, Of (G) is the semidirect sum of G and some Lie algebra M according to some representation p: G -+ Der{M), Der{M) being the Lie algebra of derivations of M. In this case, M is the ideal in G ® IK generated by the subspaces e1 G, ... , en G. Then 0 f{ G) = G + M is a direct sum of linear spaces. The semi-direct sum of G and M is mapped into the Lie algebra of derivations of M via the natural homomorphism X -+ ad x on G. Let us consider an arbitrary polynomial mapping IK = IR[XJ/(X2) in the construction described above. Then, we obtain the Lie algebra O(G) for which the algorithm for calculating the invariants of the coadjoint representation (given the invariants of the coadjoint representation of the Lie algebra G) is given in [132]. REMARK

Let us consider an arbitrary polynomial representation n I: IR -+ IRS omitting the condition I{O) = O. Then, we obtain a Lie algebra Of (G) which contains G as a subalgebra. REMARK

Let us consider the Lie algebras Of (G) where

1=

(xi'

+1, ... ,x~n+1)E(k[X1""

,xnJ)".

We denote Of (G) by Om, ..... mn{G). Let e1"'" er be a basis of the Lie algebra G. Then the vectors e'1' ... e~nej; 1 ~ j ~ r; 0 ~ lXi ~ mi; 1 ~ i ~ n form a basis of the Lie algebra Om, ..... m,(G). Let us denote the space dual to the space G by G*. If ei E G (i = 1, ... ,r) is a basis of the space G, then we denote the dual

INTEGRABLE SYSTEMS ON LIE ALGEBRA

149

basis in G* by e i ; it is defined by the relationship (ei,e j ) = Jj;
2:

=

e'1' ... e:'X(Pl, ... ,f3n)i'

(1 )

7j+Pj=mj

)=l, ... ,n

Description of the algorithm (m). Let F(x 1 , • •. , x.) be an analytic (e.g. polynomial) function on the space G*. Let us consider the function F(Zl' ... ,z.) on the space !lm, ..... m.(G)* with the values in the ring IK. Expanding F(x 1 , ••• ,x.) in a Taylor series and substituting Zi from (1), we obtain a finite sum because the higher powers of ei are equal to zero. Thus,F(zl" .. ,zr) is a well-defined function on !lm, ..... mJG)* with values in IK. We can express this function as F(Zl"'" zr)

=

2:

o ~lXj~mj

e'1"" e~'F7, ..... 7JX(f3l'···' f3n};}·

j=l, ... ,11

We shall call the functions F 7, ..... 7.: !lm, ..... m,( G)* --+ k the homogeneous components of the function F(z 1, .•. ,zr)' By definition, the algorithm (m) transforms the function F on the space G* into the set of homogeneous components '!((F) = {F 7,..... a.} of the function F. The algorithm (m) has the following properties (see [129J): THEOREM 13.1 Let F 1> ••• ,FN befunctionally independent invariants of the coadjoint representation of the simply-connected Lie group <» which is associated with the Lie algebra G. Then functions of the set {m(Fd} u· .. u {m(F N)} are invariants of the coadjoint representation of the simply-connected Lie group !lm, ..... mJ<») associated with the Lie algebra ~, ..... m.(G); the invariants {m(Fd} u··· u {m(F N)} being functionally independent on the space nm, .....m.(G)*. THEOREM 13.2 Let the functions F 1(x), ... ,FN(X) defined on G* be in involution with respect to the Kirillov form on all orbits of the coadjoint representation of the Lie group <» which is associated with the Lie algebra G. Then, after the algorithm (m) is applied, all functions {m(Fd} u·· . u {m(FN)} are in involution with respect to the Kirillov form on all orbits of the coadjoint representation of the Lie group !lm, ..... mJ<») which is associated with the Lie algebra !lm" .... m,(G). In

150

A. T. FOMENKO AND V. V. TROFIMOV

addition, if F l' . . . , FN are functionally independent on G*, then all functions of the set U~~ 1 ~(Fi) are functionally independent on Qm ...... mJG)*. We shall prove this theorem in the following section. DEFINITION 13.1 Let F l ' . . . ,FN be a complete set of functionally independent invariants of the coadjoint representation of the Lie algebra G. We say that the invariants separate the orbits if the maximal dimension of an orbit of the coadjoint representation of the Lie group (f) (corresponding to the Lie algebra G) is equal to dim G - N. REMARK The class of such algebras is not empty. For example, all semi-simple Lie algebras G and their Borel subalgebras in G belong to this class. The following statement is a corollary of Theorem 13.1 ("ind G" means "index of the algebra," i.e. the minimal codimension of an orbit; for a semi-simple Lie algebra G: ind G is precisely equal to the rank of G).

THEOREM 13.3 Let the invariants of the Lie algebra G separate the orbits. Then, ind Qm" ... ,m.(G) = dimk iii ind(G). In this case, the algorithm (~) transforms the complete set of the invariants of Ad*(f) into the complete set of the invariants of Ad*(Qm" ... ,m.(f)). REMARK The algorithm from [132] is a special case of the algorithm (~). The following statement is a corollary of Theorems 13.2, 13.3 and

[IOJ, [89J, [126J, [127J. THEOREM 13.4 Let the invariants of the Lie algebra G separate the orbits. Then, the algorithm (~) transforms a complete involutive set of functions on G* into a complete involutive set of functions on the space Qm" ... ,mJ G)*. In particular, if G is a semi-simple complex Lie algebra or a compact, normal compact or a real Borel subalgebra thereof, then the complete involutive set offunctions on the space dual to the Lie algebra

may be constructed explicitly by means of the algorithm (21). Let us give explicit expressions for functions F from algorithm some rings iii = k[Xl, ... ,Xn ]!Ul, ... ,f.).

~

and

EXAMPLE 1 In [132J invariants of a Lie algebra (which we denote by

INTEGRABLE SYSTEMS ON LIE ALGEBRA

151

0 1 (G)) are constructed. This Lie algebra corresponds, evidently, to the ring IK = lR[x]/(x 2 ). Let us consider the ring lR[x]/(x 3 ). We have the coordinates x(Ob x(1b x(2)j and the variables Zj = X(O)je 2 + x(1)ie + x(2)j (e 3 = 0) in the space Q 2 (G)*. Using the Taylor formula, we obtain F(ZI' ... , zr) = F o(z) + F 1 (Z)e + F 2(Z)E 2 , F o(Z) = F(x(2);); r

j~1

F 1(z) =

8F(x(2);) 8x(2)j x(l)j;

__~ ~ 8 F(x(2);) - 2 p,;: 1 8x(2)p 8x(2)q x(l)px(l)q 2

F 2(Z)

~ 8F(x(2);) 0 8x(2h x( )k'

+ k~l

2 Let us consider the Lie algebra which corresponds to the ring IK = IR[X 1 ,X 2]/(xi,xD. Then 0l,l(G) = G + E1G + E2G + E1E2G. We denote the coordinates in Ql,l(G)* by x(O,O);, x(l,O);, x(O,l);, x(l, l)j in accordance with the decomposition of 01,1 (G) as a direct sum of linear subspaces. In this case, the variables are Zj = El E2X(0, O)i + EIX(O, l)j + E2x(1,0)j + x(1, l)j. Using the Taylor formula, we obtain EXAMPLE

Fo(z) = F(x(l, 1);); r

F1,o(z) =

I k=l

=

I k=l

r

F O,I(Z) r

F1,l(Z) =

I

p,q =

+

1

8F(x(I,I);) 8 (11) x(O,lh;

x,

k

8F(x(1,I)j) 8 (1 1) x(I,Oh;

x,

k

82F(x( 1, 1);) 8 (1 1) 8 (1 1) x(I,O)px(O,l)q

i k= 1

X

,

p x

,

q

8F(x(l, 1);) x(O,Oh. 8x( 1, l)k

13.2. The proof of the general theorem

In what follows, we shall use the tensor notation, e.g. ajb j means be the structure tensor of the Lie summation with respect to i. Let algebra G in the basis e1 , ••• ,er ; let c7j be the structure tensor of the

ct

152

A. T. FOMENKO AND V. V. TROFIMOV

algebra nm" ... ,m.(G) in the basis e~' ... e~"ej' In the space COO(G*) of smooth functions on G* the differential operators Xi = X(e;) = CtXk a/ax j are defined. We shall call them operators of infinitesimal translation of the Lie algebra G in the basis e1 , •.. , er • 13.1 The operators of infinitesimal translation of the Lie algebra nm" ... ,mJG) in the basis ei"" e~"ej, 0 ~ IXj ~ mj, 1 ~j ~ n, 1 ~ i ~ r (e i being the basis of G) are given by the formulas PROPOSITION

The proof follows directly from the definition of the commutator in the Lie algebra nm" ... ,m.(G). 13.2 Let F be an invariant of the coadjoint representation of the Lie group
PROPOSITION

Proof The function f is an invariant of the coadjoint representation if and only if X(e'i' ... e~"ej)f = 0 for alllXj,j (see 2.3). Since the operators of

infinitesimal translation are linear, to check if all f it is enough to check the condition

E ~(F)

are invariant

X(e'i' ... e;'e)F(zl' ... ,zr) = 0

in the ring !R[x 1, ...

,xn]/(x~,+I, ... , X::,' + 1).

yve have

X(el' ... e:'e;)F(z) " L.,

o"'~ "'m p """ p +P p..... 1

k

C jj X(1X 1

aF(zl,···,Zn)

+ /31" .. , IXn + /3nh a (/3 X

/3 ).'

1"'"

(2)

n)

~p~n

the z/s being substituted by their expressions (1). From (1), it follows that

a _____ .= aX(/31, ... ,/3n)j

Using (3) we transform (2) X(ei' ... e:'e;)F(z)

a

em,-fJ, .,. em.- fJ . 1

n

az j '

(3)

INTEGRABLE SYSTEMS ON LIE ALGEBRA k Cij X(", "'1

of(z) + 1'1' R ", + R) om,-/1, •.. om,-/1,----:-_ .•. ''''n I-'n k"l "nO Z·,

O~ap tpp~mp 1 ~p~n

L

k of =C .. , IJ

OZj

x(","'1

+ 1-'1""''''n R ", + R) om, -/1, •.. om,-/1, I-'nk" "n'

O';;~p+'!P.,;;m

1

~p""","

Note that since Pi + O(i ~ mi, mi "divide" by e~' ... e~', i.e.

x(","'1

x

Denoting

(Xi

+ Pi

153

Pi

~

(Xi'

Therefore it is possible to

+ 1-'1' R ", + R) om, - /1, -~, •.. om, - /1" -~, ... , "'n I-'n k"l "n .

by Yi we obtain

X(el' ... e~"e;)F(z)

Let us add the summands X(Y1" .. , Ynhe';"-Y' ... e:;"-Y, (Yi < O(i for at least one subscript i) to the sum ~>(Y1"'" Yn)ke';"-Y"" e:;"-Y,. These summands do not affect the result since from the condition Yi < O(i it follows that O(i + (mi - Yi) > O(i + mi - O(i = mi , i.e. after multiplication byel' ... e~' each new summand gives zero (i,e. has eti with tS i > mi which is zp.rol Fin,,!h,

X(et' ... e~'ei)F(z) = el' .. , e~'C~j ~F

L

X(Y1" .. , Yn)ke';" -y, ' .. e:;'n-Y,

Zj O~Yp~mp 1 ~p~n

the latter expression is zero since F is an invariant of Ad*«fj) and therefore C~jZk of(Z)/OZi = 0 (see 2.3). Proposition 13,2 is proved. PROPOSITION 13.3 Let F l' . . . ,FN be functionally independent functions on the space G* which is dual to the Lie algebra G. Let' nm, ..... m,(G) be an extension of G constructed by means of the ring !R[x 1 , ... , xnJ/(x';'l + 1, ... , x:;" + 1). Then all functions from the set

154

A. T. FOMENKO AND V. V. TROFIMOV

{m:(Fd} u··· u {m:(F N)} are functionally independent on the space nm, ..... mJG)*. Proof Let us consider the function F(ZI"'" z,) on the space nml ..... mJG)*; the z;'s being given by (1). Let us express Zj as follows:

Let us apply the Taylor fonnula to F(z; evidently, we obtain F(ZI' ... ,z,) = F(x(m 1 ,

+

I.

•••

L

P= 1

,mn)I' ... ,x(m 1 ,

~

+ x(m 1 , ... ,mn);).

•••

Then,

,mn),)

P

v F(x(m 1 , ••• ,mn);) (il ..... jp) p. vx(ml' ... ,mn)j, ... , vx(m 1 , ••• ,mn)jp

x [(~>~,

... f.~"x(Pl' ... ,Pn);)

x .. .

(5)

The summation symbol means summation over allcx;, Pj ?: 0 such that CX; + pj = mj, 1 ~ i ~ n, cxi + ... + cx; t= O. Let us separate the variables into groups in such a way that within each group PI + ... + Pn = const. Let us arrange the groups according to increasing order of PI + ... + Pn and arrange the elements within each group lexicographically. From (5) we obtain that the variables corresponding to the p-th derivative have sum greater than that of the variables corresponding to the first derivative. Then the Jacobi matrix of homogeneous (with respect to the variables x(cx 1 , ••. ,cxn );} components o~dered as described is a triangular matrix with diagonal elements Aij = vF j(x(m 1 , · · · ,mn)p)/vx(m 1 , •• • ,lnn)j:

o J= *

Thus Proposition 13.3 is proved.

155

INTEGRABLE SYSTEMS ON LIE ALGEBRA

The proof of Theorem 13.1 follows immediately from Propositions 13.2 and 13.3 Let F l' . . . ,Fs be a complete set offunctionally independent invariants of the coadjoint representation of the simplyconnected Lie group (\j which corresponds to the Lie algebra G. Then, dim OG = n - s; OG being an orbit of maximal dimension of the coadjoint representation Ad* (\j of the Lie group (\j. The algorithm (m) gives s dimk 1K functionally independent invariants of the Lie algebra Proof of Theorem 13.3

nm, ..... m,(G): {m(F 1 )} u··· u {m(Fs)}'

This may not be a complete set of invariants, i.e. the number of the invariants nm, ..... m.(G) is not less than s dim K Therefore, dim Om" .... mn(G) ~ dim G dim k IK - s dim k 1K

= dimk 1K(dim G - s) = dim k IK dim OG'

(6)

because the invariants G separate the orbits. In the above expression, Om, ..... mJG) is the orbit of maximal dimension of the coadjoint representation for the extension nm, .....mJG). On the other hand, from the explicit form of the operation of commutation in nm, ..... mn(G) we obtain

=

rk(* Bij

= dim k 1K dim OG(x(m 1 ,

• .•

,mn)iei),

0G(x) being the orbit corresponding to the point x E G*; Bij = k ( cijx m 1 ,· •• ,mnh· Let us choose Y = x(m 1 ,· •• ,mn)ie'. such that the dimension of 0G(Y) is maximal. Then,

dim Om, ..... m.cG) ~ dim Om, .....m,CG)(z) = dimk IK dim OG because of the element YEn"" ..... m.cG)*;

z=

L

lZi+···+(.(;~o

a(~l , ... , ~")ie2' ... e~'ei

+ y.

156

A. T. FOMENKO AND V. V. TROFIMOV

Comparing this with (6) we obtain dim Om, ..... m,(G) = dim k IK dim 0 0 from which our assertion follows. To prove Theorem 13.2 we need some preliminary considerations. Let F, H be two functions with their values in IR defined on the space G* dual to the real Lie algebra G. Then these functions are in involution with respect to the Kirillov form on all orbits ofthe coadjoint representation Ad*«!j) ofthe Lie group (!j which corresponds to the Lie algebra G ifand only if C~jXk of/ox, oG/ox j = 0 (see Lemma 11.2). Let us consider on the space Om( G)*, m = (m l , ... , mn ) functions F, H with values in the ring IK = IR[XI' ... ,xnJ/(xi' + 1, ... , + l). Let us define a new bracket {F,H}® on such functions F,H. The function {F,H}® is defined on Om (G)*, m = (m l ' . . . ,mn ) and has the values in the IR-algebra

x:'

IK (8)~ IK

= lR[x l , ... ,xnJ/(xi' +1, ... ,X:,+l) (8)~ IR[X!> ... , x 4J/(xi' +1, ... ,X:,,+l).

DEFINITION

13.2

By definition, we set

{F,H}

®

-k aF(z) aH(z) =CijOJk a a ElK (8) IK, OJ i OJ j

qj being the structure tensor of Om(G), m =

(m l , . . . ,m.) in the basis sl' ... s:"ei and OJ = (x(cx l , ... , IX.») running through those elements of Om(G)*, m = (m 1 , • •• , mn ) comprising the basis dual to sl"" s:'e i ; Zj = ZiOJi) being the function of the variables OJ i defined by equation (1). We assume

aF(OJ)

-~.~EIK®{1} c

aOJi

IK®IX

and

aH(OJ) aOJ

,

E

{1} ® IK

c IK

® IX.

j

Let us denote Si ® 1 by (i ElK (8) IX; 1 ® Si by ~i ElK ® IK. Note that the bracket {F, H} ® is linear with respect to each argument but is not skew-symmetric. Let us consider in the ring IK ® IK the element

A=( PROPOSITION

L (f~1)( L (~~~) ... ( L (~~:).

p+q=m,

13.4

p+q=m2

p+q=m,

The following equality holds in the ring IK ® IK

157

INTEGRABLE SYSTEMS ON LIE ALGEBRA

AC J;:~mi 'I' ... ':'X(Pl' ... ,P.h) 1 ~i~n

=

ACJ;~mi ('1"" (~'X(Pl"'"

Pn)k) ,

1 ~i~n

i.e. A(1 ® Zk) = A(Zk ® 1) for any k. Proof Let us consider the sum

°"""p1

L

X(lX

l + Pl" .. ,IX. + p.h,m, -" ... ':"-"('l' - P' ... (:,.- p'.

+{J."",m.

:S;;p~n

Let us denote IXj + Pj by Yj and and the sum (7) becomes

I

X(lX

mj -

IXj

by

rj.

l + Pl' ... ,IX. + P.h'i' -', ... ':'-"~i"

Then, -~,

Pj = Yj + rj -

(7) mj

... ~::,.-~.

!X.i,(J i

=

(~X(Yl'" . ,Y.h~~'

Denoting

I

X(IX I

IXj

+ Pi by Yj

~::"-'-)A.

-y, '"

and

mj -

Pj by r i , we obtain for formula (7)

+ PI' ... ,IX. + P.h 'i' -', ... ':'-"~~' -p ... ~::,. -{J.

CXj,{J j

=

(~X(Yl""

,Ynh''l'-Y'"

.,:"-")A.

Comparing this with the previous expression for the sum (7), we obtain the proof of Proposition 13.4. COROLLARY Let F be a real-valued analytic function of the variables Xl' ... ,Xr • Let us assume

Vi = 1(Xi ® 1 + 1 ® x;) E IK ® IK.

A. T. FOMENKO AND V. V. TROFIMOV

158 Then AF(v 1,··· ,Vr)

=

AF(X1 01, ... 'X r (1)

=

AF(10x 1,···, 10xr)'

Proof From Proposition 13.4 we obtain that A(!(x; 0 1 + 10 x;))P = A(x; 0

W=

A(10 x;)p.

(8)

Expanding F as a power series, we obtain the proof from (8). PROPOSITION 13.5 Let F, H be analytic functions defined on the space G*. Let us consider F(z 1, ... , Zr), H(z l' ... , Zr) which are functions on nml ..... m,( G)* with values in the ring IK. Let {F, H} G be the Poisson bracket on the space G* arising from the Kirillov form. We state that {F, H}® = A{ F, H} G(v 1, ... , Vr), A ElK ® IK being the element defined above and Vi = t(Z1 ® 1 + 10 zJ Proof From the definition of the commutator in the Lie algebra nm, ..... m,(G) it follows that {

F,H }

®

i3F

i3H

UWj

uW j

'k

= CijWk~®~

®

i3H OX(f31' ... ,f3n)j

,

(i3F/i3w i ) ® (aH/aw) means the product in the ring IK ® IK of the elements (aF/i3w i ) ® 1, 1 ® (aR/aw j ). Using equation (3) we obtain { F,

H}

®

=

k

i3F

Cij -;uZj

i3H ® -;UZj

x

x(","'1

+ f3 1,···,"'"n N + f3) rml -a, ... rm,-a. nk':>1 '>n (9)

Using Proposition 13.4 we obtain

L O~tJ.p+lJp~mp

1

~p~n

x(oc 1 + f31"'" OCn + f3n)k'f ' -"'" ,::,,-a'ef , - fJ,

••• e~,-{J,

159

INTEGRABLE SYSTEMS ON LIE ALGEBRA

ACJ~:=mi (11 ... (~·X(Pi'··· ,Pnh)

=

1

~i~n

C+t=mi ~1'

=A

1

~

... ~~"X(f3i' ... ,f3nh) .

(10)

i::;;n

In the light of (10) we can rewrite (9) as

{F,H}

®

A

k

=-cij(zk@I+I@zk)

2

of(z ® 1) aH(1 ® z)

a

a

~

~

(we use the obvious relationship)

aF(z) ® 1 = aF(z ® 1) .

1®

o(z ® t)i '

aZi

aH(z)

=

aZi

aH(1 ® z) 0(1 ® Z)i

.

Applying the corollary, we obtain

{F,H} Vk

=

1(Zk @

®

=

k

ACijVk

aF(v) aH

aVi -a ' Vj

1 + 1 ® Zk). Thus, Proposition 13.5 is proved.

Proof of Theorem 13.2 In Proposition 13.3 we proved that the u· .. u {'lI(F N)} are functionally functions in the set (j = {'lI(F independent. Let us prove that all functions in the set ~ are in involution. Let F, H E~, then

in

fJ 1 ~ ••• ,/1"

Because the bracket {F,H}® is linear, we obtain

{F , H}® =

"{F H p).· ..• Pn }(a,1 ... (>"~~I ... ~~" i..J :xl,. ..• ::.!,,' n 1 n • :Xl' ···.(III

fJ I'·· .• p"

From Proposition 13.5 we obtain

{F,H}® = A{F,H}cC1(Zi ® 1 + 1 @zJ) = 0 since according to our condition {F, H} G = 0 and therefore all the brackets {F~, ..... x.,HtJ" .... pJ are equal to zero which was what we had to prove.

A. T. FOMENKO AND V. V. TROFIMOV

160

REMARK Theorem 13.1 follows from Theorem 13.2 since F is an invariant of the coadjoint representation if and only if {F, xd = 0 for all coordinate functions. The algorithm (Ill) transforms the coordinates in G* into the coordinates in Qm" .... m,(G)*. REMARK The algorithm (Ill) is also applicable to functions defined on some open set U c G*. Then the functions from Ill(F) are defined on some open subset -0 c Qm, ..... m,(G)*; in this case Theorems 13.1 and 13.2 hold as well. Such a situation arises if we consider e.g. rational invariants of the coadjoint representation. 13.3. The application of the algorithm (W) to the construction of S-representations

The algorithm (Ill) allows us to construct S-representations (see Definition 11.1) systematically from the simplest S-representations. THEOREM 13.5 Let (hI' ... ,hp; W) be an S-representation of the Lie group (f). Then, (~l(hd, ... ,~((hp); ~(W)); Ill(W) = {Ill(b), bE W} is an S-representation of the Lie group QA(f)) associated with the Lie algebra Qa(G).

REMARK If hI"'" hp are semi-invariants of the coadjoint representation Ad *(Ill), W = EBr= I Ih\:h j , then (hI' ... ,hp; W) is always an S-representation. This follows from [127J, [89J, [12J, [26J, [27J, [116]. If we apply the algorithm (Ill) to this S-representation, then we obtain the S-representation of the group Qa«(f)) which, in general, cannot be reduced to semi-invariants. A complete set of semi-invariants for some classes of Lie groups is described in [10J, [126J, [118J, [132]. PROPOSITION 13.6

Let FE A(G*); then

X(e~' ... e~'ei)(F 11, .....11)

F/l, ..... fJ,

E

= (X(ej)F)/l, -., ..... /l.-.,'

Ill(F) being the homogeneous component of the function F.

Proof From the definition of the commutator in Qa(G) it follows immediately that X(e~l ... e~'e;)

=

"

L"

O~xp+pp~mp 1

k cijx(ct 1 + /31' ... , ct n + /3n)k:"l

uX

(/3 a /3 ). ' 1"",

n)

~p~n

Cfj being the structure tensor ofthe Lie algebra G in the basis e to obtain from this equation that

j•

It is easy

161

INTEGRABLE SYSTEMS ON LIE ALGEBRA X(I>'i' .. '1>~'ei)F(ZI" .. ,Z.)

= 1>1' .. '1>~'(X(ei)F)(ZI" .. ,Z.);

x(e i ) being the operator in COO(G*) which corresponds to the basis vector ei E G and Zi being substituted by its expression (1). Applying to the

function F(z l' . . . ,Z.)

=

L

1>1' .. 'I>~'F a, ..... aJX(Pl' ... ,Pn);)

O:::;cxl:!:;mi

1 :::;):E:;n

the operator

X(I>~' ... I>~'ei)

and using the equation

X(l>l' ... 1>:'e;)F =

1>1' ... 1>:·(x(e;)F)

the proposition is clearly proved. COROLLARY If W is an invariant subspace in A(G*) with respect to Ad*(ffi), then meW) = {m(b), b E W} is an invariant subspace with respect to Ad*(Qa(ffi». PROPOSITION 13.7 If W is a finite-dimensional invariant subspace in A(G*), then meW) is a finite-dimensional invariant subspace in A(Qa(G)*).

Proof Let us prove that if fl , ... ,!. is a basis in W, then {m(fl)} u ... u {m(!.)} is a basis in meW). This follows from the explicit form for the homogeneous component of ];.a, ..... a. which is easy to obtain by expanding ];(ZI' ... ,zr) in a Taylor series (see (5». Let (hI" .. ,hp; W) be an S-representation of the Lie group ffi and X(e;)hk = c~(hd.fj (see [126]). Then since meW) is an invariant finitedimensional subspace, X(l>a, .. 'I>a.e)h - ci.tJ, I" • 1 n i k"Pl,""y,. * .....tJ.(hk.}'t ..... }',. ) i,Cll •... ,a,. Jj,p1,. .. ,p,.' hk.y, ..... y. E 21(h k ) is the homogeneous component hk E W

PROPOSITION 13.8

The following equality holds

ci.tJ, ..... tJ·(h

*

Di,

k.yl.···.y,.

).

t.::cl,. ••• iX li

= cilh *' k,).DP,)II

being the Kronecker symbol.

Proof From Proposition 13.6 it follows that

-lit

... DI!..

Yn-ex,.'

(11)

162

A. T. FOMENKO AND V. V. TROFIMOV

Comparing this expression with (11), we obtain the proof of Proposition 13.8. Proof of Theorem 13.5 From the corollary it follows that the space 'lC( W) is invariant. Let us check the condition (b) of Definition 11.1. We rewrite this condition using the coordinates k k ohp
c~j(h)

=

c~(h)(ej) being the coordinates of a covector c~(h) in the basis

ej E G. Multiplying

by B~' ... B~" and adding, we find that it is enough to prove that the expressIOn

is equal to zero (or, obviously, to the expression ck*"yp"""Y"fJ (h . , ,

1,···,

11

oh(z)

,)Bm,-fJ,. ··Bm."-fJ" _ _

J J~l''''''?n

From Proposition 13.8 it

follo~s

).

;':J

UZj

that the last expression is

Bm,-fJ, ... em"-fJ"DY" ... DY" ck 1 n ~,-fJ, ~"-fJ"

(h) oh(z)

*

J'

OZ., '

which is obviously zero because, as is given, C~(hj)i oh(z)/ozj = O. This completes the proof (see also [130J). 13.4. Algebras with Poincare duality

Let A be a graded commutative algebra, A = Ao Et>. , . Et> An' AjAj C A j + j (i + j ::s; n). We assume that dim A. = 1. Let 0 be a linear functional on A which is identically zero on Aj if i ~ n - 1 and is not zero on An' Let {3(a, b) = O(ab) be a symmetric bilinear form on A. If {3 is a non-degenerate form, then the algebra A is called an algebra with Poincare duality.

163

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Evidently, for such algebras Ao = k. Let 61 = 1. Let us choose the arbitrary basis 62"'" 6h in AI' then choose the arbitrary basis 6 + l' . . . , 6h in A 2 , etc. up to degree n12. In An/2 we choose a basis such h that the matrix of the form {3 is diagonal. In spaces A j , i > n/2 we choose the bases conjugate with respect to {3 to the ones already chosen in A(n/2) -i' As a result we obtain 6 1" .. , 6 N , a homogeneous basis of the algebra A self-conjugate with respect to {3; {3(6*j,6) = bij, * being a permutation. Let el ' . . . , ern be a basis in G*; XI" •• , Xrn be the corresponding coordinates. It is natural to regard the linear functions on G* as the elements of the Lie algebra G. Conversely, the elements Xi ® 6j of the Lie algebra GA = G ® A (regarded as a Lie algebra over k) can be regarded as linear coordinate functions on G!. Let xl = Xi ® 6j be coordinates on

G!. For a polynomial function P(x) on G* co

P(x 1 ,

••• ,

X >II )

="L. "L..

p.'I""",. x·'I .. , x·"

=p[x[

(12)

k=O (il, ... ,i!)

with 1 ~j

~ N

let us define a polynomial function p(j)(x) on G!: co

P(j)(xt, ... , x~) =

L L

P il , ... ,i.(6j, 6jl ..... 6jJX~jl ... x~j,

k=O (il, ... ,i,)

U1, .. ·,i,r.>

(13) (6j ,6jl

••.••

6j ) being the projection of the element 6jl ..... 6j , on the

basis vector 6 j • Let xEG*, Ann(x) = {gEG, (adg)*x = O} be the annihilator of the element x. The dimension of the annihilator of the element x E G* in general position is called the index of the Lie algebra G, ind(G) = inf{dim Ann(x), x E G*}. Let F be a set of polynomial functions on G*. The set of polynomial functions on G~ = (G ® A)* of the form pUl, P EF, 1 ~j ~ N we denote by FA. THEOREM 13.6 (A. V. Brailov [19J) (i) If F is an involutive set on G*, then FA is an involutive set on G! = (G ® A)*. (ii) If F is a complete involutive set and the number of independent polynomial invariants ofthe Lie algebra G is equal to the index of the Lie algebra, then FA is a complete involutive set on G~ = (G ® A)*. (iii) If a set F contains a non-degenerate quadratic form, then the set FA also contains a non-degenerate quadratic form.

164

A. T. FOMENKO AND V. V. TROFIMOV

We deduce a corollary. THEOREM 13.7 Let PI"", P r be a set of independent polynomial invariants of the Lie algebra G with index r. Then, (i) ind( GA) = dim A . ind(G); (ii) P\l), ... , P~N) is a complete set of independent invariants of the Lie algebra GA' REMARK Let A be the commutative Frobenius algebra. Then the. equality ind G ® A = dim A ind A holds (Brailov, A. V.). Let us denote by an asterisk the linear operator *ei = e*i, *2 = 1. Let (X, Y) be the scalar product corresponding to the basis e1 , ••• , eN in the algebra A. It is easy to check that for any elements a, b in the algebra A the equality (*a, b) = (eN' ab) holds. We have another corollary. LEMMA 13.1 We have (*a, be) = (*b, ae), a, b, e E A. Let ~ E A ® A, ~= (*8;) ® 8;. In what follows, we omit the summation symbol for repeated indices. The element ~ is an exact analog of diagonal cohomology class (see [69]).

Li

LEMMA 13.2

We have

~(1

® a) =

~(a

® 1), a E A.

Proof The following equations hold ~.

(1 \CJ 'X' a)

= *e·,\CJ 'X' 8·a = (*e·J' 8.a)*8. 'X' ® e·J VY *8·J = (*8' 8.a)*8. J I

J

= 8j a ® *8j =

~

. (a ®

I'

I

I

1).

Let PEk[x 1 , . . . ,x nl We define a polynomial PEA[x 1 , . . . ,xm ], p(x) = p[8 J xf, the multi-index notations are clear (see formulas (12), . (13)). The following lemma explains the role of the polynomials with the "tilde. " LEMMA 13.3

We have {P(i), Q(i)} = (e i ® ej , ~{P, Q} ® 1).

Proof Since { p(i) , Q(i)}(x) =

(e. e. aP(x) ax:. to.

,\CI

l'

1)(1 \CI

aQ(x) {x' x p}) ax: "q ,

it is enough to prove that

o:(x) ® O~(X) x'r x qP

{x~,XqP} = ~. {P,Q}(x) ® 1.

(14)

The left-hand side we denote by {P,Q}®. The operation {P,Q}®

INTEGRABLE SYSTEMS ON LIE ALGEBRA

165

introduced by us is bilinear and if one argument is fixed, it becomes a derivation of the other one. This means

i\ ® I{P 2 ,Q}® + P2 ® I{P 1,Q}®

(15)

{P,Q1Q2}® = 1 ®Q1{P,Q2}® + 1 ®Q2{P,Q1}®'

(16)

{P 1P 2 ,Q}® =

Other properties of the left-hand side are A· {P1 P 2,Q} ® 1::= P 1 ® 1· (A, {P";:'Q} ® 1)

-

-

+P 2 ®I(A'{P 1,Q}®I) A·

(17)

{P,Q1Q~} ® 1 = 1 ® <21' (A' {~} ® 1)

-

+1®Q2·(A·{P,Qd®I).

(18)

The property (17) follows from the fact that the operation P --+ P is multiplicative and from the properties of the standard bracket {J, g}. The property (18) follows from Lemma 13.2. Comparing (15), (16) and (17), (18) we obtain that it is enough to prove the equality (14) forlinear P and Q. {X"XqrS = *S. ® *sp{x~,xn = (Sk'S, . sp)*s.

® *spC~qX~

= (*Ek, s.· sp)*s.

® *Epc~qxfk

= (*s., Sk . sp)*s.

® Epc~qxfk

10. t *k -_ EkEp 'CI *EpC,qX t

= A . Ek ®

1 . C~qXfk

=

A . {~} ® 1.

LEMMA 13.4 Let P 1" .. ,P, E k[x 1 , ••. , xm] be polynomials which are independent at the point Y E G*, Y = (Y1, ... , Ym). Then, the polynomials P\l), ... ,p~N) are independent at any point x E G~ such that the highest coordinates x~ = Yi' 1 ~ i ~ m.

Proof Let us suppose that the polynomials P 1, ... , P r are independent with respect to the first r coordinates, det

a~j(Y) "# O. Xi

Let us consider the matrices Msix), 1 ~ s, q ~ N

166

A. T. FOMENKO AND V. V. TROFIMOV

1~i,j~r.

If s > q, then (e q, a . e.) = O. Hence, if (eq,el) i= 0 then ax1 J/axt· = O. Therefore, when s > q the matrix Msq(x) consists of zeros. When s = q, we obtain ax?J a *s = (e q, e])P I a *. Xi Xi

ap~q)(x) J

_ L PJ'IXi, *1 *1 ... ,Xi 1,1

'1

I-I

s axt *1 aX.*S X" 1+1

...

*1 _

Xi '" -

I

aPj(y) ay. . I

Let us consider the matrix M sqij(X) constructed with the matrices M sq(x). This matrix is block-triangular, the diagonal being occupied by nondegenerate matrices aPj(y)/aYi' Therefore, it is non-degenerate and hence, the polynomials Pi1 ), ... ,p(N) are independent at the point x. Proof of Theorem 13.7 Let P(x) be an invariant. It is equivalent to {P,x.} = 0, 1 ~ s ~ m. From Lemma 13.3 we obtain {p(i) ,x: j } = {P(il,x~j)} = {ei ® ej'~' {P ,xs} ® I} = O. Hence the pWs are invariants of the algebra GA = G ® A. If PI' ... ,Pr are independent invariants, then from Lemma 13.4 it follows that rN copies ofthe invariants PP), ... ,P~N) of the Lie algebra GA = G ® / are independent. Therefore, ind GA ~ rN. The converse inequality can be obtained as follows. Let Lsix) be a square matrix (Lsix»ij = {xf, xr}(x). If s > q, then L.q = O. If s = q, then (Lsq(x»ij = xt'c~j' If Y E G* (Yt = x~, t = 1, ... ,m) is a regular point, then the rank of the matrix bij = xt' C~j is m - ind G. Therefore,

-

ind G A

~

mN - rank L.ix) = mN - (m - ind G)N = rN.

Proof of Theorem 13.6

(i) From Lemma 13.3 it follows that FA is involutive; (ii) Let PI"'" P b E F be independent and form a complete commutative set. In this case, k = !(m + r), r = ind G. From Lemma 13.4 it follows that kN copies of the polynomials Pil ), ... ,piN) (which are in involution) are independent. For this set to be complete it is enough that kN = !(dim GA - ind GA) (this follows from Theorem 13.7: !(dim GA

+ ind GA ) = -t(mN + rN) = kN

).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

167

(iii) Let Q E F be a non-degenerate quadratic form. We can assume Q(x) = qi(Xi)2, qi t= 0 (1 ~ i ~ m). Then QN(X) = qix{xfi is also nondegenerate quadratic form. It is easy to see that in the real case this form is always indefinite. The results of 13.1 in the polynomial case can be obtained from 13.4 if we take ~[Xl' ... , xn]/(x~l + I , ... , x;:'" + I) as the algebra A. For the non-commutative case of Theorem 13.6 see [200]. REMARK

14. SIMILAR FUNCTIONS 14.1. Partial invariants

The results of this section were obtained by A. V. Belyayev. Let H be a semi-simple subalgebra of the Lie algebra G. Let us consider the adjoint representation of the algebra H in G. The algebra G becomes an Hmodule. Let V be an H -submodule G (not necessarily irreducible, but containing H). We have a representation ofthe algebra H in the space V. If a basis Xl' ... 'X n is fixed in V, this means that there is an (n x n)matrix Wof 1-forms on H. Let us denote the value of this matrix on a vector h E H by vv" (thus vv" is the matrix of the transformation ad h : V -> V in the basis Xl' ... ,xn ). We denote the n-tuple (Xl" .. ,xn ) which consists of the vectors in the basis for the space Vby X. Then [h,X] = XW , h

(1)

[h, X] stands for the line ([h, Xl]' ... , [h, X n ]), hE H. The elements wij of the matrix Ware I-forms on H. Since the algebra H is semi-simple, we can assume that they belong to H (using the identification H ~ H* given by the Cartan-Killing form). Let [h, W] be the matrix, the (i, j)-th location of which is occupied by adt W ii = [h, wij]. Then, [h, W] = [W,

vv,,].

Here on the right-hand side is the matrix commutator. Indeed, for any vector h' E H: giving

(2)

A. T. FOMENKO AND V. V. TROFIMOV

168

B(X, Y) being the Cartan-Killing form. This is true for any h'. Therefore adt W = [h, W] = [W, w,;]. From (1), (2) it follows that [h,xwr] = [XWw,;].

Let E be an r-form invariant under the action of the algebra H on ntuples (h: Y --+ YW ). SUC? forms ex!st b~cause H is. a s~mi-sim~le h algebra. Then, E(XWl, ... ,XW') IS an H-mvanant, I.e. [h, E(XWe., .. . , X we,)] = 0 if we consider E(XW e1 , .. . , XW e,) as the element of a symmetric algebra of tensors S(G) over G. If we regard the vectors G as linear forms on G*, then the above statement can be expressed in a different way. PROPOSITION 14.1 Functions of the form E(XW e1 , .. . ,XWe,) on G* are in involution with the linear functions OJ ij , 1 :::;; i, j :::;; n. It is possible that the H-module G has several sub modules V', V", . .. , VIm) which are isomorphic to V. In this case, it is possible to choose basis n-tuples X', X", ... , x(m) on which H acts in the same way as on the n-tuple X.

COROLLARY The functions E(XWe1,x,w e" ... , x(m) we,) involution with all linear functions OJij; 1 :::;; i, j :::;; n.

are in

DEFINITION 14.1 Let us call functions of the form E(xwe f , X' we" ... , x(m)we,) canonical H - i n v a r i a n t s . · · · 14.2. Involutivity of similar functions

According to the classical Cayley-Hamilton theorem, "-1

W" =

L

(3)

a;(W)Wi.

i=O

DEFINITION 14.2

In

the

form

of the

canonical

H-invariants

E(XWe" X'W e" ... , x(m)we,) the powers of the matrix Ware given

explicitly. We shall call two canonical H -invariants similar if one ofthem can be obtained from the other by the replacement of the matrix W" by Wi; the powers i being such that the function aJW) in the expansion (3) is not identically zero. Let

E be the matrix

of a bilinear form E. Then,

E(X'W e" XW e,) = X'We'E(xwe,)T = (_1)e,x,w e , +e'Ex T

169

INTEGRABLE SYSTEMS ON LIE ALGEBRA because WE

+ EW T

=

o.

DEFINITION 14.3 Suppose G is a Lie algebra and H is its subalgebra. We will say that the pair (G, H) satisfies the M -condition if the following conditions are satisfied. We will assume that G = H EEl V; EEl· .. EEl VR is an expansion ofthe H -modules G in irreducible submoduli. It is required that a) H be a simple Lie subalgebra in G of the classical type An' Bn, Cn, Dn; b) All V; are either trivial H -moduli or the representation of the Lie algebra corresponding to submodulus V; (i = 1, ... ,k) is equivalent to the minimum one. THEOREM 15.1 Suppose G is a Lie algebra and H is its subalgebra. We will assume that the pair (G, H) satisfies the M-condition, and Fi = X'WiEX T is canonical H-invariant. Then the Poisson bracket of the similar functions {Fi,Fn} (X can be identical with X') is zero. The proof of this theorem will be given below. Let {w ij , x t } = x.A stij • We shall prove several auxiliary statements. LEMMA 14.1

We have the equality

Proof From the equation

{wi,x t } = (XWw,)t = x.(w.pw i) it follows that A.tij = Aij•t. LEMMA 14.2 Let us denote Moreover {h, W} = [W, w,,]. Proof Let U =

Wst = A.tij Hji.

Then,

[W,

W]

= o.

1S p( W2) be the invariant of the algebra Lie H. Then

{h,{U,x t }} = {U,{h,x t }} = {U,x.}(w,,).t· Moreover {U, x t } = x.A. tii lfij- Hence XSA. kij Hji(w"ht

= Xk(w,,)k.Astij Hji - X.A.tik{w"h j Uji + X~A.tki lfiiw,,)jk· Since this equality is valid for all for w". Lemma 14.2 is proved.

Xt,

h, we can omit

Xt

and substitute W

170

A. T. FOMENKO AND V. V. TROFIMOV

Proof of Theorem 14.1 First, we prove the theorem for the special case when {xlP),x~q)} = 0 for any i,j,p,q. We have

0= {F",F,,} =

{Fn,~ai(W)Fi} = ~ai(W){FII,Fi}.

The following equation follows from the fact that (H, H) satisfies the Mcondition p,q

Hence we have the expression for {Fn' FJ Clearly {F II , Fd = 0, since otherwise

L:aJW){Fn,FJ =f. O. i

We will now consider the general case. Equation (4) follows from the Mproperty. We have

{Fn,Fp}

=

{x;,xj}~;EXTU:J~EXT + X'WnE*iX'WPE*j{Xi,Xj}

+ x'wnE*i{Xi,xi}U:J~EXT - X'WP~*i{Xi,xi}U:J~EXT. \

(5) Let us show that the last two summands in (5) cancel each other. For this, it is enough to prove that the matrix N with (ij)-th element given by Eij{x i , xi}, commutes with W-or, which is equivalent, N(h) commutes with l¥" for any h EH. Note that {h,N} = [N, l¥,,]. If {xi,xi} EH, then {h,N}(h) = {B([h,M],h) = O. Thus, it follows that [N, l¥,,](h) = 0, which we wanted. Since {Fn,Fp} = !({Fn,Fp} - {Fp,Fn}),

{x;, xi} ~;EXTU:J* EXT =1(XW"E~i{X;, xi} U:J~EXT - XWPE~i{X;,xi} U:J~EXT) =

o.

We may transform the second summand in (5) similarly. Thus, the theorem is proved. Let us consider the H-module V + V' assuming that V and V' are isomorphic. The vectors (Xl" .. ,Xn, X~, ... ,X~) = Y form a basis for V + V'. The actions of H on the basis 2n-tuple Y and the basis 2n-tuple Y' = (AX I , ... ,AXn , Xl'.'" Xn) are isomorphic. Tht..efore we can construct H-invariants for the module V + V'. If the representation of H

INTEGRABLE SYSTEMS ON LIE ALGEBRA

171

in V is given by the matrix of I-forms W, then in V + V' the corresponding matrix W is of the form

(~ ~). Then,

y'WiEyT = ;'XWiEX T + X'WiEX T .

Using this example, it is easy to obtain a corollary. COROLLARY the form

Linear combinations of similar canonical H-invariants of F

P

="L..

(I. ..

IJk

X(i)WPE.X(k)

J'

are in involution.

15. CONTRACTIONS OF LIE ALGEBRAS

The results of this section were obtained by A. V. Brailov (see [197]). 15.1. Restriction theorem

Let (M, w) be a symplectic manifold, H be a smooth function on M; s grad H be the vector field dual to the differential dH with respect to w. Also, we shall consider complex integrals of the system s grad H assuming that in our case the Poisson bracket is linearly extended to smooth complex functions. Let L be a group which acts on M by symplectic diffeomorphisms, H be a function which is constant on the orbits ofL. Then, we shall say that the group L is a group of symmetries of the Hamiltonian system (M, w, H). PROPOSITION 15.1 (Restriction theorem) Let L be a compact group of symmetries of the Hamiltonian system (M, w, H), ~ be a L-invariant algebra of integrals, ~!: be a L-fixed subalgebra, N be a manifold of fixed points, ~ be the set of restrictions of the integrals in ~ to N. Then (a) (N, ro) is a symplectic manifold, ro = wiN; (b) ~ is closed with respect to the Poisson bracket {j; g} N; the restriction mapping is an epimorphism of Lie algebras {j,g} N) -+ {k, eh); (c) iY is the algebra of integrals of the Hamiltonian system (N, ro, fl); fl being the restriction of H to N.

m!:,

m,

A. T. FOMENKO AND V. v. TROFIMOV

172

LEMMA 15.1 Let (M,w) be a symplectic vector space; p:!: --+ End(W) be a completely reducible symplectic representation of the group !:, Wo be a fixed subspace. Then, wi Wo is a non-degenerate form. Proof Let ~

Wo and ~.l be its orthogonal complement. Since ~ E Wo, ~.l is invariant under !:. Hence, there exists an invariant onedimensional subspace 1R'1 such that W = ~.l EE> IRI'/. From the definition of 1'/ it follows that w(~, '1) "# O. Given that 1R'1 is !:-invariant, we deduce 1'/ E WOo E

LEMMA 15.2 If F

E

~I;' then

s gradM F

= s gradN F.

Proof Let q EN. Because F is invariant, we obtain s gradM F(q) E ~N.

Therefore, to obtain the proof of the lemma it is enough to check the equality w(s grad M F, X) = w(s grad N

F, X)

(1)

for any vector X on M. Both sides of (1) are equal to X(F) which pt?ves the lemma. Let q EN be a fi~d point. Then, the \ correspondence a --+ dqa: Y'qN --+ ~N is a symplectic representation of !: on the space ~N. Since, under the hypothesis of the proposition,!: is compact, this representation is completely reducible. Applying Lemma 15.1, we obtain assertion (a) of the proposition. From the definition of the Poisson bracket and equality (1) it follows that the restriction to ~I: is a homomorphism and the image of ~I: is closed in ~ under the Poisson bracket. We obtain the epimorphism given that the restriction of the function F to N coincides with the composition of restriction and averaging over the group !:: Proof of Proposition 15.1

.u(~)

L

F a d.u(a) , 0

which evidently belongs to ~I:' Thus we have proved assertion (b) of Proposition 15.1. Let us prove assertion (c). Indeed, let F E ~ be the integral (M,w,H). Then {H,F}M = O. Applying (b) we obtain {fj'p} N = O. Assertion (c) is proved which completes the proof of Proposition 15.1. Let us give a simple example of a symplectic group action. PROPOSITION 15.2

Let !: be a compact group which acts by

173

INTEGRABLE SYSTEMS ON LIE ALGEBRA

automorphisms on the Lie algebra G, Gn be a fixed subalgebra, x E G: c G*, 0G(x) be the orbit of the representation Ad* of the Lie group (which corresponds to G) containing the point x; let OG (x) be the " orbit ofG.. Then, (a) L acts by symplectic diffeomorphisms on Odx); (b) OG (x) is open in the manifold of L-fixed points in the orbit OG(x); (c) if " W, Wn are the Kirillov forms of the Lie algebras G, Gn respectively, then Wn

= wiG" First let us prove two lemmas.

LEMMA 15.3 Let L,Odx) be as in Proposition 15.2; 0' E L, 0'*: G* -+ G* be the linear mapping conjugate to 0'. Then O'*(Odx» c OG(x).

Proof Let x

E

G*. Let us define the neighborhood ~, =

~,

in 0G(x')

{Exp(Ad;)(x/), g E G}.

(2)

Applying 0'* to (2), we obtain O'*(~,)

= {Exp(ad:-1(g»(0'*x/), g E G}.

(3)

Hence, (4)

Thus the lemma is proved in the local case. To prove it in the general case, it is enough to notice that the orbit 0G(x) is a connected set and Odx) n O'*-I(OdO'*(x» is both open and closed as follows from (4). LEMMA 15.4 Then wx(~' 1])

=

Let x

E

G*,

~,I] E

TxOG(x),

W

being the Kirillov form.

wO"x(O'*~, 0'*1]).

Proof By definition of the vectors such that ~

~

and 1], there are vectors ~/, r(

E

= adt(x),

G

(5)

then, by the definition of the Kirillov form, wx(~' 1]) = (x, [~/, 1]']). Applying 0'* to (5), we obtain O'*~

= ad:-l~'(O'*x),

0'*'1

= ad:-1q,(0'*x).

(6)

Using (6) it is easy to calculate the Kirillov form at the point O'*x: wO"(X)(O'*~,

0'*'1) = (O'*x, [0'-1~/, 0'-11]]) = (O'*x, O'-l[~','1I]) = wX<~,'1)

Proof of Proposition 15.2

Assertion (a) follows from Lemmas 15.3 and

174

A. T. FOMENKO AND V. V. TROFIMOV

15.4. Let us prove assertion (b). Let N be the manifold of fixed points. From (3) it follows that in the neighborhood of x, N is given by equations Exp(ad:- 1(g»)(x) = Exp(ad;)(x),

a E 1:.

(7)

From (7) we obtain ad;(x) E T,;N <=> ad; (x) = ad;(9)(x),

a E 1:.

(8)

I

Let Ann(x) = {u EG ad~(x) = O},g EG,g = g' + g",g' EG n + Ann(x), g" E W, Wbeing a 1:-invariant complement to G n + Ann(x) in G. Then, ad;,,(x) = ad:g,,(x). Hence g" = O. Indeed, if g" '# 0, there exists a E 1: such that g" '# ag", therefore, 0 '# g" - (Jg" E W which contradicts the fact that g" - ag" E Ann(x). Hence we can strengthen (8): ad;(x) E TxN <=> g" = O. This means that

T,;N = TxO G (x).

(9)

"

Since evidently OG (x) eN, assertion (b) follows from (9). Assertion (c) " of Proposition 15.2 follows from the definition of the Kirillov form trivially. Thus Proposition 15.2 is proved. 15.2. Contractions of Zrgraded Lie algebras

Let G be a Lie algebra. Let us suppose that G = H E9 V, [H,H] c H, [II, V] c V, [V, V] c H. Then G is called a &:2-graded Lie algebra. Let H*( V*) be the subspace of all co-vectors which annihilate V (respectively H) in G*. Then G* = H* E9 V*. Below, for any x E G* and 9 E G, XH and Xv stand for the II* and V* components of x; gH and gv stand for the H and V components of g. Let us define a new commutator [x,yt on G:

[g,g'];. =

[gH,g~]

+ [gv,g'v] +

[gH,gV]

+ A.[gv.g~].

(10)

The commutator [x, y];. satisfies the Jacobi identity. We denote the Lie algebra obtained by G;.. DEFINITION

15.1

Let Gbe a &:2-graded Lie algebra, G;. be the series of Lie algebras constructed above corresponding to the commutators [x,Y];.. We shall call the Lie algebra Go a contraction .of the &:2graded Lie algebra G. DEFINITION

15.2

Let G be a &:2-graded Lie algebra. Let us denote the

INTEGRABLE SYSTEMS ON LIE ALGEBRA

175

Poisson bracket for the contraction of G by {x, Y}o and the annihilator of x by Anno('x). Let G be a semi-simple tE 2 -graded Lie algebra, Go the correspondin~ con!raction. Then the index of Go is equal to the rank of the Lie algebra G (= index of G).

THEOREM 15.1

LEMMA 15.5 Let G be a tE 2 -graded Lie algebra, x E G*, 9 E G, G = H $ V, then

{x,g}o = {XH,gH}

+ {xv,gv} + {Xv,gH}'

(11)

Proof If g' E G, then

<{x,gL,.,g') = <x, [g,g'];.)

= <x, [gH,g~] +

+ [gH,g~] + tl[gv,g~]) = <XH' [gH' g~]) + <xv, [gv, g~] + [gH' g~]) + tl<x H, [gy, g'y]) = <{XH,gH} + {xv,g~} + {XV,gH} + tl{XH,gV},g'). [gv,g~]

Assuming tl = 0, we obtain (11). Let G be a semi-simple Lie algebra, B(X, Y) be the Cartan-Killing form on G, 9 E G. Let us denote the covector dual to the vector 9 with respect to B(X, Y) by g*. If 9 EGA., then g* E G! is defined by the formula g*(u) = B(g, u), u EGA., the form B(X, Y) being on G;. the same as on G (remember that G and GA. are identical as vector spaces). If W eGis a subspace, then W* = {g* 9 E W}.

DEFINITION 15.3

I

We know that in G we can choose a Cartan subalgebra T such that the following conditions are satisfied: (a) T = TH $ Tv, TH = Tn H, Tv = Tn V; (b) if 9 E V and [g, Tv] = 0, then 9 E Tv; (c) TH is a Cartan subalgebra in K, K being the centralizer of Tv in H; (d) Tv is a reductive subalgebra in G. Proof This can be found in e.g. [26].

LEMMA 15.6 Let G be a semi-simple Lie algebra, T be a Cartan subalgebra in G such that conditions (aHd) are satisfied; let T* c G* be the subspace of G~ dual to T in the sense of Definition 15.3. Then, for x E T* (in general position) Anno(x) Proof Let

x=

XH

+

xv,

XH

E H*,

Xv

E V*

=

T.

and

u E Anno(x),

176

A. T. FOMENKO AND V. V. TROFIMOV

U = UH + UV' UHEH, UVE V. From (11) it follows that

+ {xv, Uv } = 0 (12) { {xv, UH} - o. By definition, x = g* for some gET, 9 = gH + gv, gH E H, gv E V. Since II is orthogonal to V, gt = XH' gt = xv. Therefore the equations (12) can {XH, UH}

be rewritten as [gIl, UIl] { [gv,u ] H

+ [gv, u v] = 0 = O.

(13)

For gv (in general position) from the second equation (13) it follows that UHEK since gHETHcK and [gH,UH]EK. Therefore, taking the bracket of the first equality (13) with gv, we obtain (14) because Tv is reductive in G, from (14) it follows that [gv, u v] = O. From this equation and the first equation of (13) we obtain [gH' UH] = O. Hence, using the relationship [gv, u v] = 0 we deduce for gH and gv (in general position) that Uv E Tv, uH E TIl' As a result, U E T. Thus we have proved that Anno(x) c T. The inverse inclusion can be easily obtained from (13). LEMMA 15.7 If G is an arbitrary Zz-graded Lie algebra, and Go is its contraction, then ind(G) ~ ind(G o). Proof Let G.. denote one of the Lie algebras introduced in the definition of contraction. The correspondence gH

(F

+ gv -+ gH + .yygv

is an isomorphism G,\" -+ G,\, if A', A" =f. O. Therefore, ind(G ,J = const(A) if A =f. O. Let us prove that if A = 0, then the index can only increase. Indeed, the index is equal to the rank of the matrix Ilct(A)xk II, c7P) being the structure tensor of the Lie algebra G,\; X k being the coordinates of a covector in generaf position. If A varies slightly, then the rank of the matrix Ilc7P)xk II can only increase. Therefore, the index of G,\ can only decrease, Hence,

A=f. O. Proof of Theorem 15.1

From Lemma 15.6 we obtain

INTEGRABLE SYSTEMS ON LIE ALGEBRA

177 .

ind(G o) :::; dim Ann(t*) = dim T = rank(G) = ind(G); t E T being an element in general position in a Cartan subalgebra T. On the other hand, from Lemma 15.7 it follows that ind(G o) ~ ind(Gd = ind(G). Therefore ind(G o) = ind(G). This completes the proof.

Let G = H EB V be as before a Z2-graded Lie algebra. Similarly to the above definition of the contraction of the commutator [x, y] -+ [x, y]o we can define an analog of the contraction for certain functions F on G*. DEFINITION 15.4 Let us suppose that the expansion of F in exterior powers on V terminates FO(x H) + F'(xH,x V) + ... + P(xH,X V); Fk(XH'X V) being a k-form Xv for fixed XH. Let us define FA(x) = An/2 F(XH,A- 1 / 2 xv). Note that FA(x) is defined for A = 0: Fo(x) = Fn(XH,XV) as well.

PROPOSITION 15.3 Let G = H EB V be a Zz-graded Lie algebra, F, F' befunctions on G* such that (a) {F, F'} = 0; (b)FA and F';. are defined (see Definition 15.4; e.g. if F and F' are polynomials). Then, we assert that {FA' F';.} = 0 in particular, {Fn, F'm} 0 = 0; P, F,m being the terms of highest degree of the expansion into homogeneous components on V. Proof The correspondence gH

+ gv -+ gH + A-l/Z gv

is a homomorphism of Lie algebras G -+ GA if A =F O. Under this F maps into A-n/ZF A. Therefore {FA' F';.} A = 0 if A =F O. This equation holds for A = 0 as well because both the commutator and the functions depend on A continuously. Proposition 15.3 can be used to construct involutive sets of functions on G6. Let us show how we can use the proposition to determine the invariants of the contracted Lie algebra Go. PROPOSITION 15.4 Let G = H EB V be a Zz-graded Lie algebra, F be a polynomial on G*, invariant under the coadjoint representation; let F 0 be the contraction of F (see Definition 15.4), Go be the contraction of G. Then F is an invariant of the coadjoint representation of Go. Proof Let

Go, then u can be regarded as a linear function on Gt. Evidently, here the contraction u - U o is equal to U v if U v =F 0 and equal to UH in the opposite case. We can express the fact that F is Ginvariant as follows U

E

178

A. T. FOMENKO AND V. V. TROFIMOV

( 15)

From the remark made above it f.:>llows that {F 0' UH} = 0, {F 0, uv}o = O. These equations imply that the contraction F 0 is Goinvariant. EXAMPLE The algebra so(4) is defined by the equations [e ij , ejkJ = eik , i,j,k taking values from 1 to 4; eij = -eji . The subaJgebra H = so(3) is generated by the vectors el2 , e l3 , e23 ; let V be the linear subspace (e 14 , e24' e34)' It is easy to check that the decomposition so(4) = H EB V gives a Z2-grading, the corresponding contraction Go being a semidirect sum so(3) EB [R3 (in other words, the Lie algebra of the group of motions of the Euclidean space [R3). We know that the functions F = Li.j and F' = X 12 X 34 - X l3 X 24 + X23Xl4 are invariants of so(4). Contracting F and F' we obtain the invariants of the Lie algebra so(3) EB [R3; Fo = Xf4 + X~4 + X~4' F~ = F'.

x5

5

Complete integrability of Hamiltonian systems on orbits of Lie algebras

16. COMPLETE INTEGRABILITY OF THE EQUATIONS OF MOTION OF A MULTI-DIMENSIONAL RIGID BODY WITH A FIXED POINT IN THE ABSENCE OF GRAVITY 16.1. Integrals of Euler equations on semi-simple Lie algebras

The Hamiltonian systems given in Section 7 allow embedding in Lie algebras and, in addition, are completely Liouville-integrable (in the commutative sense); the semi-simple case is given in [89, 90, 92]. In particular, for these cases we obtain positive answer to Hypothesis (a) (see Section 5) because we have complete commutative sets of functions on the orbits in general position for the semi-simple and compact Lie algebras. The integrals of these Hamiltonian systems are very simple. To construct them, it is enough to know the invariants of the Lie algebra, i.e. a set offunctions which are constant on orbits in general position. In outline, the construction ofthe integrals can be described as follows. Let fbe some invariant ofthe algebra which is a function on G* (or on G for the compact and semi-simple case). Let a E G* be a covector in general position. Let us translate the argument ofthe function f(x), i.e. consider the function f(x + Aa), A E C or IR. Since in our cases the functions fare polynomials, we can expand the function f(x + Aa) in powers of the formal variable A and obtain an expansion of the form f(x + Jia) = Lk Pk(x, a)Ak. Note that in all the above-mentioned cases it is these resulting polynomials Pk(x, a) (or, which is the same, the functions f(x + Aa» which form complete commutative sets of functions (integrals). We call this technique of the construction of integrals the method of argument translation. It is a development of the idea suggested in [74J for the case of the algebra so(n). We already know that 179

180

A. T. FOMENKO AND V. V. TROFIMOV

translates of invariants are in involution. Therefore here we shall mainly prove the functional independence of the translates of invariants. The method of argument translation gives a positive answer to Hypothesis (a) for many non-compact Lie algebras as well. We can obtain complete commutative sets of functions on orbits in general position by applying this method not only to the invariants of an algebra (sometimes those are not enough for obtaining the sets needed) but also to so-called semi-invariants (i.e. functions which are multiplied by the character of the representation under the (co )adjoint action of the group on the orbit). Naturally, invariants are examples of semi-invariants because the former are fixed points of the action on the space of functions. Examples of semi-invariants can be found in [to], [127], [126]. Let us consider construction of commutative sets of integrals on the orbits in general position in semi-simple Lie algebras. We shall prove that these sets are complete in other sections. We shall mainly consider the complex semi-simple Lie algebras together with the Euler equations ofthe form x = [x, q>x]; the operators q> = q>abD define the Hamiltonians of the complex series (see Section 7). Let us consider the coadjoint action of the complex semi-simple Lie group (f) on the corresponding Lie algebra G (we assume G = G*). The group partitions the algebra G into orbits. We set g E (f).

LEMMA 16.1 Any smooth function f(x), x E G which is invariant under the coadjoint action of the group (i.e. it is constant on the orbits) is an integral of the Euler equation x = [x, q>xJ; q>: G ~ G being an arbitrary self-conjugate operator. The proof follows directly from TxO = {[x, yJ}, the y vector running over the whole algebra G. Note that in the complex case there exist the elements which do not belong to the orbit OCt), t E H; H being a fixed Cartan subalgebra. Let us consider the set of all complex vectors grad f(x), f E I G, I G standing for the ring of invariant polynomials on the algebra G. Let H(X) be a subspace of G which consists of all elements commuting with x. If x E Reg G, then H(x) is a Cartan subalgebra, and in a semi-simple algebra -any two Cartan subalgebras are conjugate. In particular, if x E Reg G, then H(x) = goH(a, b)go 1 for some ga E (f), H(a, b) being the Cartan subalgebra containing a, b. Evidently, H(x) belongs to the subspace generated by grad f(x), f E IG and if x E Reg G, then H(x) =

INTEGRABLE SYSTEMS ON LIE ALGEBRA

181

{grad f(x), f E IG}. This follows from the fact that the Killing form is non-degenerate and the space H(x) is orthogonal to the orbit's tangent space (see Figure 29). LEMMA 16.2 A smooth function f is constant on the orbits of an algebra if and only if [x, grad f(x)] = 0 for any x E G. We denote by grad f(x) the value of the field grad f at the point x E G.

Proof Remembering that TxO = {[x, ~]}, ~ running over the Lie algebra G, we obtain (grad f(x), [x, ~J> = 0 for any ~ because [x, ~]f(x) = o. Since the operator adx is skew-symmetric, ([grad f(x), x], 0 = 0 and because the Killing form is non-degenerate this means that [grad f(x), x] = O. The converse assertion can be checked similarly. The equation [x, grad f(x)] = 0 is obtained from dtjXk of/OXj = 0 (see 2.3) by using the isomorphism G* ~ G given by the Cartan-Killing form. REMARK

PROPOSITION 16.1 Let f E IG, i.e. the function is invariant and constant on the orbits. Then the complex functions h,,(x) = f(x + -ta) are (for any -t) integrals of the equation = [x, qJabD(X)] , qJabD being the operator of the complex series. The function F(x) = (x, qJx) is an integral as well.

x

Proof Let us check the identity 0 along the trajectories of the flow (grad h,,(x), x) = O. We have

=

(djdr)h,,(x), r being the parameter

x. This is equivalent to checking that

Fig. 29.

182

A. T. FOME NKO AND V. V. TROFIMOV


=
= <[grad f(x + ita), x + ila], cpx) - A
+ ita), [b, x']) = it
= - ),< [grad f(x + ita), x + ita], b) = 0 taking into account Lemma 16.2 and the identity [b, t + ita] = O. Thus (d/dt)h .. (x) = 0 along x. Furth er d dt F(x) = <x, cpx)

+ <x, cpx) = 2<[x, cpx], cpx) = 0

because

cp

IS

symmetric and the opera tor ad is skew-symmetric. Let us consider the model example sl(n, q. Evidently, the stand ard symmetric polynomials on the eigenvalues of the matrix x are integrals constant on the orbits of the algebra. We manipulate the equation as follows (x + ita)" = [x + ita, cpx + itb]. Indeed, expanding the comm utato r and making trivial calculations, we obtai n the initia l equation x = [x, cpx]. We have used the fact that [a, b] = 0 and [x, b] + [a, ipx] = 0 because of the definition of CPabD' Thus the equation has not changed but we notice a new series of integrals: the symmetric polynomials of the eigenvalues of the matrix x + ita. These integrals can be presented in two ways: (1) Given the expansion of the polynomia l det(x + ita - fLE) = La.P P apit7fLP in powers of it and fL, all polynomia ls P aP(x, a) are integrals of the equat ion; (2) Using the functions Sk = trac(x + Aa)k and their expansions in powers of it: Sk = Q~k)(X, lX)it7 • The link between the Newton polynomials and the symmetric polynomials (1; also interrelates the Q~k) and Pap.

La

INTEGRABLE SYSTEMS ON LIE ALGEBRA

183

Let us construct the integrals of the compact series. Let G u be a compact form of the Lie algebra G. Let x E Gu , a, b E Hu, x - Aa E Gu if A. is real. Let us consider the action of ffiu on G u. Unlike the complex case, the union of the orbits defined by a Cartan subalgebra Hu = Hu(a, b) coincides with Gu. Let cp: Gu --+ Gu be an operator of the complex series. LEMMA 16.3 Any smooth function f invariant under the coadjoint action of ffiu (i.e. constant on the orbits) is an integral of the Euler equation x = [x, cpx], cp: Gu --+ G u being an arbitrary self-adjoint operator. The proof is trivial. Let I Gu be a ring of invariant polynomials on Gu. Let us present explicitly the multiplicative structure of the ring I Gu. Let N be the normalizer of H u in Gu; then, N/~u = ¢ is the Weyl group. Let t E H u' then the orbit O(t) is orthogonal to the algebra H u. This orbit returns to H u intersecting H u in a finite number of points which are the images of the element t under the action of the Weyl group. The ring IG u is identical with a ring of polynomials in Hu which are invariant with respect to the action of the Weyl group. This ring can be described simply: if ffiu is connected then the ring I Gu is a free algebra on r generators where r = the rank of Gu among which it is possible to choose homogeneous, algebraically independent polynomials P k 1 , · · · , P k , where k; = deg P k. For simple Lie algebras of number k;, the degrees of • the polynomials P k are as follows: •

A,,:2,3,4, ... ,n,n+ 1; B,,:2, 4,6, ... , 2n;

C,,:2,4,6, ... ,2n; D,,:2,4,6, ... ,2n - 2,n; G 2 :2,6; F4:2,6,8,12; E 6 :2,5,6,8,9,12;

E7: 2,6,8, 10, 12, 14, 18;

E8:2,8, 12, 14, 18,20,24,30. The polynomials P k , can be given explicitly. Consider the linear representation of the algebra Gu of minimal dimension with matrices of

184

A. T. FOMENKO AND V. V. TROFIMOV

the size (m x m), let AI' ... ,Am be the weights of the representation, i.e. the linear functionals in Hu corresponding to the eigenvectors of the operators in Huon the representation space. The coordinates AI' ... ,Am in H u can be linearly dependent. The polynomials Pk , have the form: n+l

A PI'. "~ A kj,.' j=1 n

Brz'. "~ A/k .•' j=1

n k ,. Cn'. "L. A J'

j=1 n

. " A ki',. D1J0l..J j=1

k i = 2,4,6, ... ,2n - 2; P~

= AIA2 .. , An·

If Gu is a particularly simple algebra then P k , = Ij= 1 A~'. It is clear that all the rings IG u are sub rings of the ring of symmetric polynomials S(Al' ... ,Am). All of the indicated functions are of the form Trace Xk, =

Ij=1 A~'; except for the series Dn in which a further polynomial JDet x is added. PROPOSITION 16.2 Let f E IG u , i.e. the function is constant on the orbits of the algebra Gu . Then the functions h;,(x) = f(x + Aa) are (for any A) integrals of the equation x = [x, cpx] where cp is the operator of the compact series x + Aa E Gu , ). E IR. The function F(x) = <x, cpx) is also an integral. The proof proceeds in the same way as the proof of Proposition 16.1. See [90] for details. Now consider the integrals of the normal series. Consider the embedding of Gn to G u • The operators CPab: Gn ..... G" are generated by the vectors a, b E H u' in particular a, b ~ G" and hence x + Aa ~ G", if x E G", AE IR. PROPOSITION 16.3 Let f E IG u , i.e. the function f is constant on the orbits of the algebra Gu . Consider the functions g,,(x) where A E IR,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

185

Gn C Gu which are the restriction of the functions hi.(x) = f(x + .lea) to Gn c Gu• Then the functions g;. are integrals of the equation x = [x, CPabX] where CPub is the operator of the normal series. The function F(x) = (x, cpx> is also an integral. X E

16.2. Examples for Lie algebras so(3) and so(4)

For a clear illustration of this consider some examples of the series of integrals constructed above for very simple Lie algebras. In particular, we find that there are well-known classical integrals contained among these integrals. Let G u = so(3), we can consider so(3) as su(2), by making use of the well-known isomorphism. Let us include the algebra su(2) into the compact real form Gu of the algebra G = sl(n, C); then su(2) coincides with the stationary points of the involution o"x = - xt. It is clear that Gu decomposes into the sum of three one-dimensional subspaces generated by vectors of the following form: E+ = Ex + E_"

where Eo EHu = iHo, i Eo = ( 0

0 )

-i'

The operator ofthe compact series cP: su(2) way:

--+

su(2) acts in the following

where Ao #- 0 is an arbitrary real number, b = A+a, A+ #- 0, lX(a) #- 0, i.e. A+ = lX(b)jIX(a). Finally, cpE+ = A+E+, cpE_ = A+E_, cpEo = AoEo and differs from zero. For cp in general position A+ #- Ao. In the case where Gu = su(2) the operators cp of the compact series form a twoparameter family (A +, Ao). If

x then

=

iz . ( -x + lY

x + iY) .

-IZ

E

su(2)

186

A. T. FOMENKO AND V. V. TROFIMOV

It is clear that

<x, x) = 0, i.e. the velocity vector x is tangent to an orbit

of the adjoint action of SU(2) on su(2). The orbits are two-dimensional spheres centered on the point 0 together with the point 0 itself. All the orbits apart from the point 0 are orbits in general position. Let us fix an arbitrary orbit in general position, then the integral trajectories of the flow x on the sphere coincide with the trajectories of the points of the sphere during its rotation around the axis Eo, see Figure 30. The following functions should be the integrals of the flow: Trace(x + ..1.a)k. We have 1

x+lI.a=

(i(Z + q,l.) -x

+ iy

x + iy ) -i(z+q..1.) ,

where a = qED, q -:f O. Hence Sl = 0, S2 = -2(x 2 + y2 + Z2 + 2zq,l. + q2,l. 2). The integrals are the coefficients of each power of ,l., i.e. Ql(x,a) = x 2

+ y2 + Z2,

Q2(x,a) = zq,

Q3(x,a) = q2,

i.e. in reality the functions z = const, x 2 + y2 = const. The integral trajectories are the intersection of the spheres with the planes z = const. We have obtained one of the simplest classical cases of the movement of a solid: the integral Ql(x,a) = Ql(X) is the integral of kinetic moment, the integral z = const is equivalent to the integral of energy in the common case in which the invariants 11' 1 2 , 13 are connected by the relationship II = 12 (ellipsoid of rotation). Let us consider once more the algebra so(3) and let us write it now in the usual form, i.e. let us study the integrals of the normal series for so(3). Let G = sl(3, q, Gu = su(3), Gn = so(3), (JX = - i T , LX = i, Gn is the manifold of stationary points of the involutions (J and r. The subalgebra Gil = so(3) is generated by the three vectors Eij = E~ + E_2'

Fig. 30.

187

INTEGRABLE SYSTEMS ON LIE ALGEBRA

E12 =

C[0) -1

0

0

,

ED =

0 0

000

E 23 =

U ~} 0

C ~

°o 0) 1 .

-1

0

Let a, b E H u = (diagonal purely imaginary matrices 3 x 3 with zero trace). Then the operators ({Jab: G" ~ Gil take the form:

The manifold {({Jab} for the normal series forms a three-parameter family compared with the two-parameter one for the compact series. Not one of the compact operators is normal. Let us put Aij = (b j - b)/(aj - a) then .

x = yf3(A l 3 -

A23)EI2

+ Cq(A23 - Ad E 13 + ex.f3(A12 - A13 )E 23

where x = ex.E12 + f3E D + yE 23 . It is clear that (x,x> = 0, i.e. the vectors x are tangent to the spheres with centers at point O. Let us recall that for so(3) = su(2) the Killing form coincides with a Euclidean scalar product. We have

x

+ Aa =

ex.

13

-ex.

iAa2

y

-13

-y

iAa3

The integrals are defined by the functions Trace(x

Fig. 31.

+ Aa)k, 1 ~ k

~

3.

188

A. T. FOMENKO AND V. V. TROFIMOV

Computation gives P(x) = 0(2 + p2 + 1'2, Q(x,a) = 0(2(a l + a 2) + pl(a t + a 3) + 1'2(a l + a3). These integrals coincide with the classical P = M2-the integral of kinetic moment and Q = E-energy. The integral trajectories are depicted in diagram 31. As opposed to the previous case, this depicts the level-energy ellipsoid E = const and its intersection with the spheres Ml = const. The Euler equation is fully integrable for any a, b E H U. The flow x of the compact series is obtained by passage to the limit from the flow of the normal series. In this classical case the Liouville tori permit a differential-geometric description. This can be confirmed as follows: Consider the curve of intersection of an ellipsoid and a sphere

+ Byl + Cz 2 = { x 2 + y2 + Z2 = a 2 AX2

1

then it can be proved that at all points on this curve V4/ PtP2

Pt

+ P2

(ABC)3/4 -

2

ABCa - BC - CA - AB

= const,

where Pt, P2 are the principal radii of curvature of the ellipsoid. This can be verified by direct calculation. It would be interesting to obtain a similar description of Liouville tori in higher dimensions also. For our next example let us analyse the flows of the normal series for Gil

= so(4)

c Gu c su(4) c G

= sl(4, q.

The algebra so(4) can be represented in G as the skew-symmetric matrices spanned by the vectors in standard form Eij = E2 + E-r Let us write x E so(4) in the form

x = O(E 12 + PE13 + 1'E14 + DE23 + pE24 + 8E 34 where all the coefficients are real. We recall that the rank of so(4) = 2 and the four dimensional manifolds S2 x S2 are orbits in general position. Let a, b E H 4 c su(4) then b

({Jab X

= 0( atl

-

bt

b2

- a2

E12

+P

-

b3

at - a3

E13

+ l'

bt

-

b4

a 1 - a4

Et4

INTEGRABLE SYSTEMS ON LIE ALGEBRA

189

For every pair a, b in general position we obtain a flow x on S2 x S2. The integrals will be the functions Trace(x + .lea)k, 1 ~ k ~ 4 where

x +.lea =

.leal

0(

p

Y

-0(

.lea z

fJ

p

-f3

-fJ

.lea 3

8

-'I

-p

-8

Computations give four integrals: hI = Trace x 2 , h z = Trace X4, h3 = Trace x 2 a, h4 = 2 Trace x 2a 2 + trace xaxa. The integrals hi and h z are constant on the orbits and have the form hI = (X2 + f32 + '12 + <>2 + p2 + 8 2 , h2 =

hi + 4(f3<>yp -

(X<>Y8

+ O(pf38) -

2 2 2(0( 8

+ f32p2 + yZb Z).

In fact hz is the square of the second degree integral q (after computing the function hf from h2 ) where q = 0(8 - pp + 0(0. Thus, the two quadratic integrals hI and q are the generators ofthe ring I so(4), i.e. any polynomial which is constant on the orbits can be decomposed in terms of hI and q. It is easy to prove that hI and q are independent. The equations hI = p, q = t where p, t are constants define orbits in general position. These integrals, in particular q, were discussed in [65]. The integrals h3 and h4 are no longer constant on the orbits and have the form: I

+ az) + f3 Z(a l + a3) + yZ(a l + a4) + o2(a z + a3) + p2(a z + a4) + 82 (a3 + a4), h4 = O(Z(af + alaZ + a~) + p2(af + a l a3 + a~) + y2(a; + a 1a4 + a~) + b2(a~ + a Za3 + a~) + pZ(a~ + a Za 4 + ai) + 8Z(a~ + a3a4 + a~). h3 = (J(Z(a l

It is easy to prove that the integrals hi' q, h 3 , h4 are functionally independent and that the integrals h3 and h4 are in involution on all

orbits. 16.3. Cases of complete integrability of Euler's equations on semi-simple Lie algebras

Here we shall give a short sketch of the proof while the technical details can be found in [89] and [90].

190

A. T. FOMENKO AND V. v. TROFIMOV

THEOREM 16.1 (A. T. Fomenko, A. S. Mishchenko) (1) Let G be a complex semi-simple Lie algebra and let x = [x, CPabD(X)] be Euler's equations with an operator of the complex series. Then this system is fully integrable (by Liouville) on orbits in general position. Let f be any invariant function in the algebra. Then all the functions h;.(x,a) = f(x + A.a) are integrals of the flow xfor any A.. Any two integrals h;.(x, a) and g/l(x, a) arising from the functions f, 9 E IG are in involution in orbits. The Hamiltonian F = <x, cpx) of the flow x also commutes with all integrals of the form h;.(x, a). From the set of these integrals one can choose integrals, functionally independent on orbits in general position equal in number to half the dimension of the orbit. The integral F may be expressed as a function of integrals oftheform h;.(x, a). (2) Let G u be a compact real form of a semi-simple Lie algebra and let x = [x, cpx] be a Hamiltonian system determined by a compact series operator cpo Then the set of the functions of the form f(x + A.a) where f E IG u forms a complete commutative set on orbits in general position in the Lie algebra Gu' (3) Let Gn be a compact normal subalgebra in the compact algebra Gu and let x = [x, cpx] be a Hamiltonian system of the normal series. Then the set of functions of the form f(x + A.a) where fEIGn forms a complete commutative set of functions on orbits in general position. The involutivity of the translation of the invariants stems from the results of Section 11. Let us reproduce this proof for the case of a semisimple Lie algebra. We can calculate in its explicit form s grad f for any smooth function f of G, expressing s grad f in terms of grad f. LEMMA 16.4 Any smooth function s grad f(x) = [grad f(x), x].

f of G satisfies the identity

Proof Let ~ be a vector in T"O, then ro(s grad f,~) = U(x) = . We have proved the following assertion. LEMMA 16.5 Any smooth functions {j, g} = <x, [grad f, grad g]).

f and

9 on G fulfil the identity

191

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 16.6 Let f and g be smooth functions on G which are constant on each orbit. Then [grad f, grad g] = o. Reg G initially. As f and g are constants on each orbit their gradients are orthogonal to the orbit, i.e. they both lie in H(x) and, consequently, commute. As regular elements are everywhere dense the lemma is proved.

Proof Let x

E

This lemma can also be obtained from the results of Proposition 2.3. PROPOSITION 16.4 Let f and g be smooth functions of G which are constant on each orbit. Let us consider the functions h;.(x, a) = f(x + Aa), d/l(x, a) = g(x + fla) where a E H(a, b). Then the integrals h). and d ll commute. Moreover, {F,hA} = 0 for any fEIG.

Proof Let us recall that the functions h;. and d/l by Proposition 16.1 are integrals of the flow x = [x, cpx]. By Lemma 16.5 it is sufficient to prove that

<x, [grad hA' grad d/l]) = 0, i.e.

+ fla)]) = O. Aa, x + fla = Y + va

<x, [gradf(x + ).a),gradg(x

Let us put Y = x + Aa then x = Y v = fl - A. Let us suppose initially that v -:f. O.

where

z =
+ va»

- <[Aa,gradg(Y

+ Aa),gradf(y)]).

As f El G then [Y, grad fey)] = 0 by Lemma 16.2. As g E IG then by the same lemma

[Y + va, grad g(Y + va)] = O. Hence

[Y, grad g(Y

+ va)]

= - v[a, grad g(Y

+ va)].

Substituting in z we obtain Z

A

= -- <[Y, grad g(Y + v

A

va)], grad f(Y»

= -
=

0

as f EIG. The assertion is proved for A -:f. fl. If A = fl then

192

A. T. FOMENKO AND V. V. TROFIMOV

+ Aa),gradg(x + Aa)]) by Lemma 16.6. It remains to be proved that {F,h;J = O,i.e. to compute L = <x, [
L

= <[Y - Aa,
= <[Y,
- A<[Y, <pa], grad feY»~

- A<[a,
=

°

as in the first term [Y, grad f(Y)] = 0, similarly in the second, in the fourth cpa = Da E H(a, b), i.e. [a, <pa] = 0, in the third [a,
°

The same line of reasoning gives the proof of the involutivity of the integrals of the normal series. Let us move on to the proof of the completeness of the commutative sets of functions which have been presented. This final part of the proof is technically more subtle and, therefore, we shall restrict ourselves to setting out only the plan of the constructions. Let G be a complex semi-simple algebra, x E Reg G. Let It , ... ,1. E I G be the complete set of invariants of the algebra. At the point x there arises a set of complex vectors grad hA,k where hA,k(X, a) = hex + Aa). Let Vex, a) be the subspace of G generated by vectors grad hA,k(x, a). We must find a lower bound for dim Vex, a). Let us consider the decomposition hA,k = L{~i/ P~Ai where qk + 1 = deg k Let h increase with increasing degree. Let N + 1 = q, + 1 = deg 1. be the highest degree among the generators k All the polynomials hA,k can be considered as polynomials of degree N + 1 and of these certain coefficients in high degrees in A are equal to zero. We have grad hA,k = L1~o UPi where U~(x, a) = grad p1(x, a) are polynomials of degree i in x and a.It is clear that UZ'(a) does not depend on x as pZ' is linear in x. The vectors U{' generate the Cartan subalgebra H independent of the choice of x. It is clear that V(x,a) is generated by the vectors U~.

193

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 16.7 For each k the following recurrence relationships on the vectors U~ are satisfied: [Ur,x] =0 [Ui-x]

+ [Ur, a] =0

[U:,x]

+

[U:- 1 ,a] = 0

[U:,a] =0.

Proof By Lemma 16.2 [x, grad A(X)] = O. Applying this identity to the functions h;..k we obtain [x + Aa, grad h;. ,k(X, a)] = 0, i.e.

[x + Jo Aa,

UiAi] = O.

Our assertion follows from this. As H(x) is a Cartan subalgebra, it is possible to construct the root decomposition G relative to H(x) and to select a Weyl basis. LEMMA 16.8

If a EH(x) EB V+(x) then

grad h;.,k(X, a) E H(x) EB V+ (x), i.e. U~ EH(x) EB V+(x). Further, let us consider that a E H(x) EB V+ (x). Let us consider the simple roots a 1 , .•. ,a" then every positive root can be given in the form a= = 1 mja j where mj ~ 0 and mj are whole numbers. The whole number k = k(a) = = 1 mj is called the order or the height ofthe root a. We shall denote the subspace of V+(x) generated by the vectors Xa for which k(a) = k by v,,+(x). Then, it is evident that V+(x) = V1+ EB'" EB V. + and V1+ is generated by x", ... ,xa ,' i.e. by simple roots. Let us and a = 1 VjX'i specify a choice of a a E H(x) EB V+ (x). Let a E where all Vi"# 0,1 :'S; i:'S; r. Then [v,.+ ,a] c v,.";.1' [H(x),a] c J-t+.

Ir

Ir

vt

LEMMA 16.9 Let x, a be chosen as shown above. Then and ada:H(x) -+ Vi+ is an isomorphism.

Ir=

J-t+ =

[H(x), a]

194

A. T. FOMENKO AND V. V. TROFIMOV

LEMMA 16.10 Let [J);+,a] i.a. ada: v,.+ LEMMA 16.11

v;+

Ef) ...

X,

--+

a be the vectors shown above. Then J);-t;.l

J);-t;.l is an epimorphism.

The following

relations: ~ r.

EB V;+ are valid for 1 ~ k

LEMMA 16.12

U~ EH(x ),

=

ul EH(x ) EB

Vectors U~,j ~ k generate the entire subspace H(x) Ef) V;+ Ef) ... EB V;+.

LEMMA 16.13

Let a be an element in general position. Then

dime Vex, a) ~ dim H(x) Ef) V+ (x) = ~(dim G

+ rank G),

where Vex) is a complex subspace generated by all the vectors grad h;'.k for all points x E G from an open subset everywhere dense in the Lie algebra G. In order to complete the proof it is enou gh to note that the elements x and a have been used symmetrically in all the previous statements, as f(x

+ Aa) = AtIfG x + a).

where f EIG is a polynomial of degree q. The proof of the completeness of the comm utativ e sets, constructed above, in the cases of compact and norm al series can be obtai ned from the mentioned scheme, taking into account the involutions defining those two series. For details see [89], [90]. 17. CASES OF COM PLET E INTEGRABILITY OF THE EQUATION S OF INERTIAL MOT ION OF A MULTI-DIMENSIONAL RIGID BODY IN AN IDEAL FLUI D

Let us consider the embedding of the type of system mentioned in the title, in the non-c ompa ct Lie algeb ra of the group of motio ns of Euclidean space (this has been done in the autho rs' works [123], [124] , [125]). It turns out that in this case too the meth od of argum ent trans lation makes it possible to const ruct a complete comm utativ e set of integrals on the orbits in general position. LEMMA 17.1 Let f be an invar iant of the coadj oint representation of the group of motio ns of the Euclidean space ~n. Then the functions

INTEGRABLE SYSTEMS ON LIE ALGEBRA

195

f(x + Aa) for any real A are the integrals of the Euler equations x = ad~x x, where Q(a, b, D) is the family of sectional operators Q: E(n)* --+ E(n), constructed above. Proof It is sufficient to check the equality (a(x, Qx», df(x + Aa» = 0 where (x, 0 is the value of the functional x on the vector ~. Obviously: A

= (a(x, Qx), df(x + Aa» = - (Qx, a(x + Aa, df(x + Aa») - A(a(Qx, a), df(x + Aa».

As f is invariant, the first term is zero. Using the definition of the sectional operator Q(a, b, D) we obtain

-~A

= (a(cp;;l adt xl,a),df(x + A.a» + (a(Dx2,a),df(x + Aa» ,

where Xl EKU, x 2 EK*. The second term is zero, as DX2 EK, a EK*. The first term is equal to
+ X 2 + Aa),b),df(x + Aa» = (a(x + Aa,df(x + Aa», b) = 0 because f is invariant. This concludes the proof of the lemma. (a(xl

THEOREM 17.1 (A. T. Fomenko, V. V. Trofimov) (1) The differential equations x = ad~x x where Q = Q(a, b, D) on E(n)* is completely integrable on the orbits in general position. (2) Let f be an invariant function on E(n)*. Then the functions h;.(x) = f(x + Aa) are motion integrals any number A. Any two integrals h;. and gil are in involution on all the orbits of the representation Ad* of the Lie group 6"(n), while the number of independent integrals of this type is equal to half the dimension of the orbit in general position. Also, if 0 is the maximal dimension of an orbit in general position of the coadjoint representation, then codim 0 = [en + 1)/2]. Proof The fact that the given functions are integrals has been checked in Lemma 17.1. Their involutivity was, in fact, shown in Chapter 11. The only thing to check now is that the shifts of the invariants f(x + Aa) comprise a complete commutative set on the orbits in general position. The statement codim 0 = [en + 1)/2] can be checked by standard means (see 2.3). We shall give the complete set of invariants of the algebra. For this, we write E(n)* in the matrix form:

A. T. FOMENKO AND V. V. TROFIMOV

196

0 E(n)*

SO(n)

=

0 Y1 ." Yn

0

Let us call the minor of a matrix x at the intersection of rows ii' ... ,i. and the columns jl"" ,js M~::::::~:(x) where 1 ~ il < ... < i. ~ n, 1 ~jl < ... <js ~ n. Then the functions

are the invariants of the algebra. The functions with even numbers are equal to zero and the functions with odd numbers represent the complete set of invariants, as can be found from direct calculations. Let (.t;) be the full set of polynomial invariants, then N,

.t;(x + Aa)

=

L Pi.(X, a)AS, .=0

d.t; E E(n)**

=

E(n). Suppose N,

d.t;(x

+ Aa) = L:

uis(x, a)AS ,

.=0

where Uis

E

E(n).

LEMMA 17.2

The following recurrence relations hold: a(x, uiO )

=0

a(x, un) + a(a, UiO) = 0;

a(x, UiN ) + a(a, Ui,N,- d

=0

a(a, UJ' ,N ,) = O.

Let n = 2s + 1. Consider the complexification eE(n). The Lie algebra so(n, q is simple. Let so(n,q = H E9

L Gt EEl L G;i;.1

i;.1

where the subspaces Gi± are spanned the root vectors ea with the weight

197

INTEGRABLE SYSTEMS ON LIE ALGEBRA

ofthe root a: equal to +iandH = Et>~(~~)]-l CE2k+l,2k+2·Considerthe graded subspace of E(n). E(n) +

= (H Ef> Cen)EE)(Gi Ef> B1 ) Ef> •.• Ef> (G s+ Ef> Bs) Ef> L k~s+

Gk+ 1

i~O

where B s + l - j = C(e2j-l + ie2j) c Cn,j = 1, ... ,s. The subspace E(n)+ with this grading may be considered to lie in E(n)*. We choose x, a E CE(n)* in the following way: x E K* in general position, a E Gi + Bl such that all the components in decomposition upon the root basis in G i and the component of the base vector en _ 2 + ien_ 1 E cn . are nontrivial. LEMMA 17.3 Let aEGi Ef>Bl c E(n)* be the element mentioned above. Then, for H; c E(n) we have a(a,H;) c H;+l c E(n)* for i ~ O. LEMMA 17.4 Let x, a E E(n)* be chosen as indicated above, then the mapping Hi -+ H i + 1 c CE(n)* defined by y -+ a(a,y), YEH i c CE(n) is an epimorphism. LEMMA 17.5

The relations (a) ujO E H 0' (b) ujk E H k hold for any j.

LEMMA 17.6 The vectors U jk generate the entire subspace H k . We conclude that the dimension of the subspace, generated by df(x + Ita) is at least dim E(n) + = S2 + 2s + 1; but for complete integrability we need codim 0 + -t(dim E(n)* - codim 0) = S2 + 2s + 1 functionally independent integrals on G*. The theorem, therefore, has been proved in the case n = 2s + 1. The case n = 2s could be examined in a similar way. We shall not dwell here on the technical details.

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A. T. FOMENKO AND V. V. TROFIMOV

18. THE CASE OF COMPLETE INTEGRABILITY OF THE EQUATIONS OF INERTIAL MOTION OF A MULTIDIMENSIONAL RIGID BODY IN AN INCOMPRESSIBLE, IDEALLY CONDUCTIVE FLUID 18.1. Complete integrability of the Euler equations on extensions Q(G) of semi-simple Lie algebras

The results given here have been obtained by V. V. Trofimov. Consider the embedding of the equations of magnetic hydrodynamics into the Lie algebra Q(so(n)) described in 7.4. It turns out that in this case the method of tensor extensions of Lie algebras makes it possible to construct a complete commutative set of integrals on the orbits in general position. We shall study first the Euler equations on Q(G)* with the "complex" series' sectional operators, constructed earlier. Let G be a complex semi-simple Lie algebra and ty(G) the set of functions on G* representing shifts of invariants F of the coadjoint representation G, i.e. it consists of the functions h(x) = F(x + Aa), A E C, a E G* fixed covector. Applying algorithm (~l) from Theorem 13.1 to the functions h(x) E mG), we can construct functions h(y), h(x, y) on Q(G)* and obtain a set of functions mQ( G)) on space Q( G)*. We remind the reader that the construction of Section 13, applied to the Lie algebra Q(G) = G + 8G, 8 2 = 0 enables us to construct functions j\(x,y), ... , Fs(x, y) E COO(Q(G*)), with Fi(x, y) = (8F i(y)/8y)x j (Xi coordinates in G* and Yi in 8G*), using functions F 1 (x), ... ,Fs(X)ECOO(G)*. If Fi is in involution on all the orbits of the coadjoint representation of the Lie group (f) associated with Lie algebra G, then the functions F 1 (y), ... ,Fs(y), F 1 (x, y), ... ,Fs(X' y) are in involution on all the orbits of the coadjoint representation of the Lie group Q«f)) associated with Lie algebra Q(G). If Fi are functionally independent on Q(G)*, then Fi(y), Fix, y) are functionally independent on Q(G)* too. (See Section 13, Theorem 13.2). THEOREM 18.1 Let a function h be functionally dependent on the family of functions ty(Q(G)), then the Euler equations x = a(x, dh x ), x E Q(G)* are a completely integrable Hamiltonian system on all the orbits in general position ofthe coadjoint representation Ad* of the Lie group Q«f)) associated with Q(G).

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

proof The involutivity of the given functions follows from Theorem 13.2. Let F 1 (x), ... ,FN(X) be a complete set of involutive functions on . G* (constructed above), then F 1 (y), ... ,FN(Y), F1 (x, y), ... ,FN(X, y)

are functionally independent. In order to achieve complete integrability it is necessary to have s integrals, where s = 1(dim Q(G) + ind Q(G» = 1(2 dim G + 2 ind G)

= 2 [1(dim G + ind G)] = 2N according to Theorem 13.3. This concludes the proof. THEOREM 18.2 Let G be a complex semi-simple Lie algebra, x = a(x, C(a, b, D)(x» the Euler equations on Q(G)* for x E Q(G)* with the complex series operator, then this system is completely integrable in the Liouville sense on all the orbits in general position of the coadjoint representation of the Lie group Q(m), associated with Q(G). Or, more precisely, let F(x) be any smooth function on Q( G)*, invariant with respect to the coadjoint representation of the group Q(ffi), then all the functions F(x + Aa), A E C are first integrals of the Euler equations for any A E C. Any two of those integrals F(x + Aa), H(x + Ita), A, J1 E Care in involution on all the orbits with respect to the Kirillov form. It is possible to choose from the given set of integrals a set of functionally independent integrals equal in number to half the dimension of a general position orbit of the coadjoint representation of the Lie group Q(ffi). Proof The involutivity of the shifts of the invariants is a well-known fact (see Section 11). We shall prove that the shifts of the invariants are first integrals of the Euler equations x = a(x, C(a, b, D)(x», x E Q(G)*.

To do that it is enough to check that
+ Aa), a(x, C(x») =

The first term is equal to zero, as F is an invariant, thus


+ Aa), a(x, C(x»)

+ Aa), a(a, cPa- 1 ad: x' + Dh»

= -

A
= -

A
as Dh E Ker cPa, where x = x'

+ h, x' E V + e V,

hE

H

+ eH, H

IS

a

200

A. T. FOMENKO AND V. V. TROFIMOV

Cartan subalgebra in G and V is an orthogonal complement with respect to the Cartan-Killing form, therefore


+ }.a), a(x, qx») = - A
+ Aa), adt(x ' + h + }.a»

= -A
(1)

Let us examine now the invariants Ad* for Q(G): F 1 (Y),···,F N (y),

8F 1(y) 0Yi

of N(Y) Xi"'"

8 Yi

Xi

(2)

we should take shifts of these invariants along a vector a where a is taken from 8G*, then after shifting functions (2) along the vector a, we, obviously, obtain functions (1) and the latter, as we know, are functionally independent. The theorem has been proved in full.

In the same way as for the complex operators the following theorems can be proved. THEOREM 18.3 Let function h be a function which is functionally dependent on the family offunctions mQ(G u than the Euler equations x = a(x, dhJ, x E Q(GJ* are a completely integrable Hamiltonian system on all the orbits in general position of the coadjoint representation of the Lie group Q(ffi u), associated with Q(G.).

»,

THEOREM 18.4 Let Gu be a compact form of a comple1' semi-simple Lie algebra, x = a(x, qa, b, D)(x» the Euler equations on Q(G u )*, with compact series operators qa, b, D): Q(GJ* --+ Q(G u ). Then this system is completely integrable in the Liouville sense on all the orbits in general position of the coadjoint representation Ad* of the Lie group Q(ffiu),

INTEGRABLE SYSTEMS ON LIE ALGEBRA

201

associated with the Lie algebra Q(G u). Or, more precisely, let F(x) be any smooth function on Q(G u)* invariant with respect to Ad*(Q(GJ u)), then all the functions F(x + Aa) are integrals of the Euler equation for any AE IR. Any two such integrals F(x + Aa) and H(x + fla) are in involution on all the orbits with respect to the Kirillov forms. From the mentioned set of integrals one can choose functionally independent integrals equal in number to half the dimension of an orbit in general position of Ad*(Q(ffiu))' We shall examine now the construction of integrals of the Euler equation with the "normal" operator series. PROPOSITION 18.1 Let f(x) be a function, invariant with respect to the coadjoint representation of the Lie group Q(GJ.). Consider functions h)x) = f(x + Aa)lu(G")*' Then the functions h). are, for any AE~, first integrals of the Euler equation x = a(x, qa, b)(x)) on Q(G II )*, where qa, b) is a "normal" series operator, a E iH~, b E iH~ + BiH~. Proof Suppose that we have a differential df(x + Aa) in Gil' then dfk(x + Aa) is an orthogonal projection df(x + Aa) on Gil' In the case Q(G,,) = G" + BG" we obtain df(x + Aa) = v1 + w 2 , Vi E Gil (i = 1,2) and dflu'G")(x + Aa) = n(v 1 ) + en(v2) where n is an orthogonal projection on Gil with respect to the Cartan-Killing form. Then
+ Aa), a(x, Cx» = (dflu(G"ix + Aa), [Cx, x]) = (n(vd + t:n(v2), [Cx, x]) = (v 1

+

BV2,

[Cx, x]);

as [Cx, x] E Gil' it is possible, therefore, to add any term to the first factor, orthogonal to G". We have thus:
+ Aa), a(x, Cx» = (df(x + Aa), [Cx, x]) =
as has been proved already in the case ofthe "compact" series operators. Proposition 18.1 has been proved. PROPOSITION 18.2 Any two integrals of Proposition 18.1 "normal" series type are in involution on all the orbits Ad* ofthe Lie group Q(GJ n ) with respect to the Kirillov form. Moreover, the number offunctionally independent integrals, given by Proposition 18.1 is equal to half the

202

A. T. FOMENKO AND V. V. TROFIMOV

dimension of an orbit in general position ofthe coadjoint representation Ad* of the Lie group n(m n ), associated with Lie algebra n(G n ). Proof We reduce all cases to that of a semi-simple "normal" series of operators. We have G" c Gu • Let f(x) be an invariant of Gu , then on Gn the complete set of functions in involution (see [89], [90]) can be obtained as the restriction to Gn of functions f(x + Aa). Applying to these functions the (~l) algorithm, one finds a complete set of functions in involution on n(Gn )*. These functions have the form fey + Aa)/Gn (as the operations of restriction and substitution are commutative) and (ofk(Y + Aa)/oy;)x;. Let us check that those functions are identical to the functions mentioned in the proposition. According to Theorem 13.1 the n(G u) invariants have the form fey) and (af(y)/8y;)x i • It is clear that the shifts of these functions along a suitable a after restriction to n(G.) will give the necessary results. Thus, the algorithm (~) leads to the necessary set of functions, which proves the proposition. The results are summarized in the following theorem. THEOREM 18.5 Let Gn be a normal compact subalgebra in a complex semi-simple Lie algebra G; x = a(x, C(a, b)(c)). The Euler equations on n(G.)* with the operators of the "normal" series C(a, b): n(G.)* --> n(G n ) are a completely integrable system in the Liouville sense on all the orbits in general position of the coadjoint representation of the Lie group n(m n), associated with the Lie algebra neG,,). Or, more precisely, let F(x) be any smooth function on n(Gu )*, invariant with respect to Ad*(nCmJ), then all the functions F(x + Aa)1 Q(G,) are integrals of the Euler equations for any A E IR. Any two such integrals F(x + Aa), H(x + Ita) are in involution on all the orbits Ad* n(m,,). One can choose from the indicated set of integrals a set of functionally independent integrals equal in number to half the dimension of an orbit in general position of the representation Ad* n(m n ). In a way similar to the complex case the following theorem can be proved. THEOREM 18.6 Let function h be functionally dependent on the functions of the ty(n(Gu))IQ(G,) family, then the Euler equations x = a(x, dh x ), x E n(G.)* are a completely integrable Hamiltonian system on all orbits in general position of the coadjoint representation Ad* of the Lie group nem n), associated with Lie algebra n(G n).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

203

18.2. Complete integrability of a geodesic flow on T*Q(ffi)

Let G be a complex semi-simple Lie algebra, with compact form, Gu and G. a normal compact subalgebra in G. Let n«(I)), n«(I)u), n«(I).) be Lie groups corresponding to the Lie algebras neG), n(G u), n(G.). Consider the cotangent bundles T*n«(I), T*n«(I)u)' T*n«(I).). Let quadratic forms <
Proof Let us use the non-commutative form of Liouville's theorem (see 3.2). First we need a lemma. LEMMA 18.1 The Hamiltonian flow on T*n«(I), T*n«(I).), T*n(<».) corresponding to a quadrat~c form H, is completely Liouville-integrable in the non-commutative sense.

Proof Let i) be n«(I), n«(I)u) or n«(I).). Then, since i) acts symplectically on Wl = T*i), a finite-dimensional Lie algebra of integrals Vi exists on T* i) which is isomorphic to Tei). Above we pointed out that the commutative Lie algebra of integrals Va (dim Va = 1(dim T"i) - ind Tei) exists for the operators C(a, b, D) on the orbits of Ad*(i). Extending these functions to left-invariant functions on T* i) in a manner similar to 3.2 we obtain a commutative Lie algebra of functions Va on T* i), which commutes with Vt. The proof of the lemma can now be obtained by verifying that dim T*i) = dim(Vo + Vi) + ind(Vo + Vi). As we proved in 3.6, the commutative Liouville theorem follows from the non-commutative one, provided that a complete involutive set of functions exists on (Vo + Vi)*. Above we have constructed a complete

204

A. T. FOMENKO AND V. V. TROFIMOV

involutive set offunctions on vt; hence, such a set exists on (Vo + J.t)* as well. Since Vo is an Abelian Lie algebra, we can select any basis in Vo as a complete involutive set offunctions on Vo* (any element of Vo is a linear function on Vo*).

18.3. Extensions of il(G) for low-dimensional Lie algebras

In this section we shall consider Lie algebras over the field IR with dim G ::::;; 5, as well as nilpotent Lie algebras with dim G ::::;; 6. We call these Lie algebras low-dimensional Lie algebras. Our aim is to show that on nu(G)* it is possible to construct a complete involutive set of functions. The full classification of such Lie algebras is known and the list of them, together with the invariants of their co adjoint representations is given in [106] (we shall use the notation given in [106]). 18.3 Let G be a low-dimensional Lie algebra; then the orbits of G are separated by invariants in the sense of Definition 13.1.

PROPOSITION

The proof can be obtained by a calculation of the rank of the matrix II qjXk II with the use of the tables given in [106]. 18.4 Let G be a low-dimensional Lie algebra; then it satisfies Condition (FJ) from 3.6 .

PROPOSITION

. Proof We can construct a complete involutive set of functions on G*

using one of the following methods: (a) as shifts of the invariants of the representation Ad* (see Section 11); (b) as semi-invariants of Ad* (see Section 11); (c) through chains of subalgebras of the Lie algebra under consideration (see Section 12). Shifts of the invariants give the complete set of functions in involution for the following Lie algebras: A 3 ,4' A 3 ,6' A 3 ,s, A 3 •9 , A 4 ,1' A 4 ,s, A 4 ,10' A S ,I' A S ,2, A S ,4' A S ,10' A S ,40' A 6 ,1' A 6 ,2' A 6 ,3' A 6 ,16' A 6 ,20' A 6 ,22' We do not have enough shifts of invariants for the rest of Lie algebras. For those Lie algebras A6,d1 ::::;; i ::::;; 22) which are affected by this lack of the invariants, we can use the chain of subalgebras H c G (H is an Abelian subalgebra). For the rest of the Lie algebras we need to use semiinvariants. We omit minor details of the calculations because of their complexity. This completes the proof.

205

INTEGRABLE SYSTEMS ON LIE ALGEBRA

THEOREM 18.8 Let G be a low-dimensional Lie algebra and let f be any function which is functionally dependent on the functions constructed by applying the algorithm (m:) of Theorem 13.2 to the functions on G* which were given in Proposition 18.4. Then on the space Qa(G)* there exists a complete involutive set of functions, giving 83 infinite series of Lie algebras for which the Euler equations = a(x, dfJ, x E Qa(G)* are completely integrable in the classical sense on all orbits in general position of the representation Ad* QAGl).

x

The proof follows directly from Theorem 13.2 and Proposition 18.3.

19. SOME INTEGRABLE HAMILTONIAN FLOWS WITH SEMISIMPLE GROUP OF SYMMETRIES 19.1. Integrable systems in the "compact" case

Let us take a canonical H-invariant (or a linear combination of such functions) as the Hamiltonian V: G* --+ R, see Section 14. Then, any linear function on H* is an integral. Besides, functions similar to V are integrals, if they satisfy the conditions of Theorem 14.1. These integrals may be enough for "complete integrability. THEOREM 19.1

Ifwetake All: (su(n), su(n - 2));

B II : (so(2n

+ 1), so(2n-

1));

ell: (sp(2n), sp(2n - 2)); D II : (so(2n), so(2n - 2))

as a pair (G, H), together with the Cartan-Killing form as the invariant 2-forms, then the corresponding Hamiltonian systems x = s grad V, V = L IXii WP X(i), XUl)kk are completely integrable. REMARK For pairs of Lie algebras (so(n) EB 1Rn, so(n - 1), (so(n + 1), so(n - 1)), (su(n) EB C", su(n - 1)), (su(n + 1), su(n - 1)), the following more general result holds, namely the functions

L

[IXij(W2PX(i),XUl)

i,j= 1,2

form an involute set.

+ Pij(W 2p- 1 X(i),XUl)]

206

A. T. FOMENKO AND V. V. TROFIMOV

A similar statement holds for the pairs (su(n), su(n - 1)), (so(2n), so(2n - 1)), (so(2n + 1), so(2n)). However, these examples are of minor interest because, first, there is a "redundancy" of the integrals, i.e. the first integrals which are independent in Theorem 19.1, are now functionally dependent and, second, the resulting Hamiltonian systems are linear ones. REMARK

The results of this section were obtained by A. V. Belyayev.

REMARK

The series Dn, Bn. We consider a pair (so(n), so(n - 2)). The adjoint action ofthe algebra so(n - 2) on the algebra so(n) splits as a direct sum of invariant subspaces: so(n - 2), IR" - 2, IRn - 2, IRI, bases of the latter we denote by X, X', Z. Taking the restriction of the Cartan-Killing form as an invariant 2-form, we obtain four kinds of canonical invariants, in matrix notation xw2eXI, x,w 2eX'I, XW 2e Xt, X'W 2e + I X t (the same invariants can be obtained using the form P = {Xi' X;}; in what follows it will become clear why it is necessary to use this form). Note that matrix W is skew-symmetric. In these expressions for the canonical invariants the parity of the exponent of the matrix W is essential because, in the expansion of wn in lower powers, only powers of W with the same parity as n are present. In any case, Theorem 14.1 proves involutivity only for the kinds of invariant described above. It can be verified that the Poisson bracket of canonical invariants which are not similar (for example, X'W 2e X t and X'W 2e + 1 X') is not equal to zero. Thus, the following set of functions is an involutive one:

Li

Vo = V2p --

(X

(Xl

I XW

Lxf + (X2 LX? + (X3 LXiX; + (X4L {x;,x;},

2PX t

+ (X 2 X'W 2PX + (X 3 X'W 2PXI + (X 4 X'W 2 p-1 x t ft

(similarly with L {x;, x;}, the functions are of the form X'W 2 P-I XI because Fo = L {X;, x;} = X'PX t and hence, F

= 2p

X'W 2PPX t

=

XW 2p-1

'"

L. i

{W. x.})

*""

The number of similar invariants which are functionally independent on the orbit (so(n)) (as well as of the invariants which are independent, with linear integrals w ij , 1 ~ i, j ~ n - 2) is [en - 1)/2]. The proof of the mnctional independence is given below in Theorem 19.4. The dimension of the orbit of the general position of the algebra so(n) is dim so(n) [n/2]; the rank of the algebra of first integrals which is isomorphic to the

INTEGRABLE SYSTEMS

207

ON LIE ALGEBRA

direct sum of so(n - 2) and [R[(n -1)/2] is n - 2. According to Section 3, we have an integrable case of a Hamiltonian system since dim so(n) -

[~J = dim so(n -

2)

+ [n ;

1]

+n-

2

(the right-hand side is the sum of dimension and index of the algebra of integrals). Series All' C n • The expansion of the algebra su(n) as an su(n - 2)module to submodules in an appropriate basis is given by the following expression: Xl

X~

Xn - 2

Xn - 2

Z

Z'

W

-X n -2

-Xl

-,

-,

-X n - 2

-Xl

-,

-z

,

Esu(n).

(1)

Z

Provided that (1) holds, the irreducible invariant subspaces (apart from su(n - 2» are V= {Xl'''' ,Xn -2}, V' = {X~, ... ,X~-2}, {z},{z'},{z"} plus their complex-conjugates. Canonical invariants are of the following form: XWeX, X'WeX', X'WeX, XWeX'. Since in the expansion W" = Li ai Wi, an -1 = 0 (from which it follows that a priori it is unknown whether, for example, {X wn X, X X n -1 X} = 0), the involutive character of the invariants differing from each other only by a power of the matrix W, does not follow directly from Theorem 14.1. To prove this, we need to use a non-standard expansion all wn = W"-2 - Li~l ai Wi. From similar integrals which are functionally independent on the orbit, we can select n - 1 (see Theorem 19.4) which, in tum, are functionally independent with integrals wij (1 ~ i, j ~ n - 2). We can easily get an integral z + z" considering that Tr(W) = 0 and the whole matrix (1) has a trace equal to zero. PROPOSITION 19.1 {z + z", F} = O.

Let

F

be

a

canonical

invariant,

then

Let us show that for integrability to be complete, the first integrals already available are enough. The dimension of an orbit of the algebra su(n) is n 2 - n; the dimension of the algebra of integrals is

208

A. T. FOMENKO AND V. V. TROFIMOV

dim su(n - 2) + (n - 1) + 1 and the index of the algebra of integrals is rk su(n - 2) + n = 2n - 3. Thus, we have a complete integrability following from the non-commutative Liouville theorem. REMARK Since the dimension of orbits in general position for the coadjoint representation of the algebras su(n) and u(n) coincide, we have constructed above the integrable set for the pair of algebras (u(n), su(n - 2)). In the case ell we have similar arguments, though we should add that the dimension of the orbit of the general position of the algebra sp(2n) is dim sp(2n) - n; (dimension) + (index of algebra of integrals) = dim sp(2n - 2) + 3n - 1 and dim sp(2n) = dim sp(2n - 2) + 4n - 1. REMARK According to 3.6, for Liouville integrability to be attained, it is enough to show that the algebra of integrals satisfies Condition (FJ). If from the algebra of integrals we can split off a commutative direct summand, it is enough to prove the fulfillment of Condition (FJ) for the non-trivial summand. For all completely integrable sets (in the noncommutative sense) which have been considered in this section, the algebra H of the pair (G,H) is semi-simple and hence satisfies (FJ) (see Section 16 where the entire involutive family of functions on H* is constructed) and is a non-trivial summand. 19.2. Integrable systems in a non-compact case. Multi-dimensional Lagrange's case Theorem on isomorphism of algebra pairs

For non-compact algebras H, the volume form and the symplectic form for the Lie algebra sp(2n, IR) are natural invariant forms. To be able to apply the previous constructions to non-compact Lie algebras we need the following statement to be valid: THEOREM 19.2 Let (G 1 , H 1) and (G 2 , H 2) be pairs of real Lie algebras, such that G1 ® IC and G2 (8)1C are isomorphic and that the isomorphism induces an isomorphism. H 1 ® IC ~ H 2 ® IC. Then, a complete involutive set of one pair (G i , HJ maps into a complete involutive set of the other pair. Proof We might consider the problem of involutive and functional dependence of polynomial functions on (G i )* in terms of the Lie algebra

INTEGRABLE SYSTEMS ON LIE ALGEBRA

209

Gi ® IC but from this point of view G I ® IC and G2 ® IC would be indistinguishable. Let us consider as an example the pair (sl(n, IR), sl(n - 2, IR)) together with the pair (su(n), su(n - 2)) it satisfies the condition of Theorem 19.2. The splitting of the Lie algebra sl(n, IR) into sl(n - 2, IR)-submodules can be expressed by the following matrix:

x'I

Xl

W (2)

Xn - 2 YI

Yn-2

z

Z'

y~

Y~-2

Z",

Z"

Comparing (1) and (2) we obtain involutive sets YWeX, Y'WeX', Y WeX ' , Y' weX. If we did not consider the "compact" case, it would be difficult to get these, because the bilinear transformation B: (X, Y) -+ Y X is not a bilinear form on the invariant subspace generated by the vectors Xl' ... ,X. _2' YI, ... , Y. _ 2. Note incidentally, that the complex involutive sets are quite useful because they allow us to obtain the real ones; for example in this case Y' we X, Y we X' follows from X'WeX, XWeX'.

Volume form

Let H = sl(n, IR), V, V', ... , v(m) be H -submodules; isomorphic to each other and corresponding to a minimal representation of sl(n, IR) in IRn, G = sl(n, IR) EB (V + V' + ... + v(m). Then, the functions det(X Wi!, ... , X' Wi., ... , X" Wit) are H -invariants. det(XW\ . .. ,X'Wi" . .. ,X"Wi" .. .). Let us write W·· ..• W (k times) instead of the matrices W k in the expression for the function F I, enumerating all the resulting copies of W We then enumerate all row vectors x(m) in the expression for F 2. Let F I be X WY (a); X = X(k) W', k and r being dependent on a; W being the matrix with the number a; y(a)=det(X,X', ... ,X(k), ... ,W) is a column; F 2 = x(m)z(fJ), x(m) being the row vector with the number /3; Z(/l) = det(X, X', ... , X(k), ... , W) is a column. Then {F 1,F 2}

=

LXi Yj { w ij , xlm)}Z)/l) ~./l

=

I x./l

Xi yr) Asrijx~m)zl/l)·

210

A. T. FOMENKO AND V. v. TROFIMOV

Direct computation gives ASlij

= 15s/jli.

{F 1,F2} =

I

Thus,

X Z(fl) x(m)y(»

•

>,(l

Now it is easy to verify that Statements 19.2, 19.3 and 19.4 are valid. Thefunctionsdet(X,Xw, ... ,XWk"XI,X'W, ... , , 2) are in involution on 2, X', . .. , x H EB (V + V') provided that kl < k2 < [(n - 1)/8]. A similar statement can be easily obtained for H EB (V + V' + V"). PROPOSITION

19.2

X'W e,) and det(X, Xw, ... ,XWk

we

19.3 Functions Fp = det(X, Xw, ... , XWP, X', X'W, ... , xwn- -2), p = 0, ... , [(n - 2)/2J together with linear functions w ij ' x;, 1::::; i, j ::::; n form a complete non-commutative set on H EB (V + V'). PROPOSITION p

19.4 Functions det(X, ... , X Wio, ... , X', ... , X' Wi" ••• , x(m) Wi .. ) are invariants of the algebra H EB (V + ... + v(m») .;rovided that is = [n/(m + 1)J + (1 + 1)/2, Ls is + m + 1 = n.

PROPOSITION

The proof is a trivial one and thus can be omitted. Multi-dimensional Lagrange case

The equations of the motion of multi-dimensional rigid body can be written in Hamiltonian form in the dual space of the algebra so(n) + ~ •. Let us take as coordinates on (so(n) + ~n)* linear forms w ij ' x; which are bases of so(n) and ~n respectively. Let us choose these linear forms so that {w ij , Wjl} = WiI' {w ij , xj} = xj. Then the Hamiltonian in terms of these variables is w?I) V = -1" L. + "L. rix;. 4 i,j (oc i + ocj ) i

(3)

The vector field s grad V is then given by

The Lagrange case follows from (3) provided that ri

= 0,

OC i

= C,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

211

1 :::; i :::; n - 1, a" = kc, r" #- O. The corresponding Hamiltonian system involves a set of linear integrals OJij' 1 :::; i, j :::; n - 1, the latter form a Lie algebra which is isomorphic to so(n - 1). The Hamiltonian for a symmetric body is V = 21 Tr(W2) + 22 L xf + A3 z , where Xi = OJ ni , x~ = Z and the factors 2i depend on C, k and r". Evidently, the Hamiltonian is a linear combination of canonical so(n - I)-invariants (z = 1/(n - 1) Li {x;, xJ), and hence we can write down an involutive series of the first integrals Fp = ..1.1 Tr(W 2p+2) + 2 2XW 2PX t + 2 3 X'W 2P-l XI. The proof of the conclusion that the integrals obtained form a completely integrable set is given in [13]. The motion of a multi-dimensional rigid body in an ideal fluid

As we know already, the equations of such motion are given on (so(n) EB !R")* as a quadratic Hamiltonian. The following canonical invariants are available: XW 2e X t , X'W 2e X'I, X'W 2e Xr, X'W 2e +1 XI and hence, for the family of Hamiltonians V{.<;} = 21 Tr(W2) + 22 L xf + 23 L X;2 + 24 L X;Xi we can obtain involutive sets of first integrals. Formally we can extend this family V{.<;} =

21 Tr(W2)

+ 22 I

xf

+ 23 LX? + 24 LXiX; + 2SZ2

since I~ ~ f x? + z2 is an invariant of the coadjoint representation. The proof of the completeness of this set is the same as in the previous case. We can obtain the integrable case for the series given in [103J assuming 24 = O. REMARK

Theorem of integrability for non-semi-simple Lie algebras

We recognize a similarity between the examples of integrating the pairs (so(n + 1), so(n - 1» and (so(n) EB !R", so(n - 1»; and this is no surprise: the "contraction" construction (see Section 15) transforms the algebra so(n) into a direct sum and "commutes" with the construction of involutive sets according to the given algorithm. Therefore, a transformation like

A. T. FOMENKO AND V. V. TROFIMOV

212 (so(n

+ 1), so(n

- 1))

--+

(so(n) EB !Rn , so(n - 1))

is practicable for other Lie algebras as well, e.g. (su(n + 1), su(n - 1)) (u(n

+

1), su(n - 1))

(sl(n + 1, !R), sl(n - 1,!R)

--+

(su(n) EB en, su(n - 1))

--+

(u(n) EB C", su(n - 1))

--+

(sl(n,!R) EB !R", sl(n - 1, !R».

For all right-hand pairs we can construct entire involutive sets of functions. Let us formulate this in a way similar to Theorem 19.1. THEOREM 19.3 Let 0: G --+ GB be the contraction of the algebra G. If we take (so(n)EBW,so(n-l)), (su(n)EDcn,su(n-l», (u(n) ED cn, su(n - 1», or (sl(n,!R) EB !Rn, sl(n - 1,!R)) as a pair (G B, H), then the Hamiltonian systems are given by x = s grad 0* V, the Hamiltonian V = L (IXij(X(i)WP, XU»kk + PijP(X(i)W(p), X(j» being constructed for the pairs of the algebras (so(n + 1), so(n - 1», (su(n + 1), su(n - 1», (u(n + 1), su(n - 1», (sl(n + 1, !R), sl(n - 1, !R». We can prove this theorem following the standard technique given above and using as well the dimension of the orbits in general position of coadjoint representation of semi-direct sums, in connection with this see [118].

19.3. Functional independence of integrals

Note that the functions in the set ¢ = {Fi' i = 1, ... , rk W, /; E (H*)*, 1 ~ i ~ dim H}. where the Fi are similar functions are functionally independent on the space G* dual to the algebra G. Functional independence can be proved in two ways: either (a) by showing that the invariants defining the orbits are functionally independent of the functions in the set ¢; or (b) if this cannot be done, for obvious reasons, by proving that the skew gradients of the set of functions ¢ are linearly independent almost everywhere. The semi-simple case

In 19.2 examples of Hamiltonian systems on semi-simple algebras are

l

j

j i

INTEGRABLE SYSTEMS ON LIE ALGEBRA

213

considered. As is known, in this case the orbits of the (co )adjoint representation of the group Exp G are given by traces of powers of the (co )adjoint representation of Lie algebra G. It is enough, then, to show functional independence within the set cp, when the Fi are canonical Hinvariants. It is clear that then the similar linear combinations of invariants included in the set cp as the functions Fi will comprise a functionally independent set. THEOREM 19.4 The following sets of functions cp are functionally independent on orbits in general position of the coadjoint representation of the corresponding Lie algebras: (1) so(2n): cp = {elements so(n - 2) viewed as linear forms on so(n - 2)* and either (a) XWZPX t, or (b) X'WzpX't, (c) X'WzpX't or (d) X'W 2p - 1, o ~ p ~ n - 1}; (2) so(2n + 1): cp = {elements of so(2n - 1) viewed as linear forms on so(2n - 1)* and either (a) XWPX' or (b) X'WzPX't, (c) X'WzPX t , 0 ~ p ~ n or (d) Z,X'WZp-1X t , 0 ~ p ~ n - 1}; (3) su(n): cp = {elements of su(n - 2) viewed as linear forms on su(n - 2)* and either (a) Z, XWPX or (b) Z",X'WPX', (c) Z' + Z', XWPX' + X'WPX, o ~ p ~ n - 3}; (4) sp(2n): cp = {elements of sp(2n - 2) viewed as linear forms on sp(2n - 2)* and either (a) XWZX or (b) XW Zi + 1X}. LEMMA 19.1 Let Vi: H* ~ IR (1 ~ i ~ rk H) be functionally independent invariants of the coadjoint representation of the group Exp H. Then the projections of the gradients grad Vi' 1 ~ i ~ rk H = r on the subspace ltj = {Wli' ... ,wnJ c H are linearly independent at points in general position.

In order to prove (1)(a-d), (2)(a-d) and (3)(a--c) it is enough to apply Lemma 19.1 to enveloping algebras of pairs (G,H). Statement 3 of Theorem 19.4 can be proved by direct calculation of skew gradients of functions from set cp.

The non-semi-simple case

As an example we shall consider the non-commutative set of functions, corresponding to the volume form. Direct calculation leads to sgrad Fp = {Fp,X} = Li':g(P+Z) JiX'W, 0 ~ p ~ [(n - 4)/2]. Vectors X, Xw, ... ,XWi are, in general, linearly independent, therefore the s grad F p are linearly independent too. It is not difficult to see also that the F P are functionally independent on an orbit in general position in

214

A. T. FOMENKO AND V. V. TROFIMOV

sl(n, ~) EB (~n EB ~n) with linear integrals w ij ' x;, 1 ~ i, j ~ n. And,

finally,

dim(sl(n,~) EB (~n + ~)) -

(1 + n - 2[~J)

= dim(sl(n, ~) EB ~n) + [n-2

2J + 1 + [n-2-- 2J

(the left-hand side ofteh equation is the dimension of an orbit in general position in sl(n, ~) EB (~n + ~n) and the right-hand side is the sum of the dimension and the index of the algebra of integrals). Note that the number oflinearly independent skew gradients gives a lower bound the dimension of an orbit of the coadjoint representation of the algebra sl(n, ~) EB (~n + ... + ~n) and Proposition 19.4 gives an upper bound for this dimension.

20. THE INTEGRABILITY OF CERTAIN HAMILTONIAN SYSTEMS ON LIE ALGEBRAS

Let G be a Lie algebra, G* its dual space, I(G*) the set offunctions on G* invariant under Ad* (invariants of Lie algebra G). For any element a E G* we shall define a set of functions la = la(G*) = {J;.,/;.(x) = f(x + Aa), fE/(G*)}. As we know, la is an involutive set with respect to the standard bracket {f, g} on G* (see Section 11): {f, g}(x) = <x, [dxf, dxgJ>. In the case ofa reductive Lie algebra there is an invariant identification of G with G*. Thus the notation la = la(G) makes sense for a E G as well. Let 0 c G* be an orbit of the representation Ad* of the group (fj associated with G. We shall call an involutive set of functions on completely involutive if after restriction to an orbit 0 in general position it gives! dim 0 independent functions. It was proved in [89], [90] that for the compact form Gu of a complex semi-simple Lie algebra G the set la is completely involutive when a E Gu is an element in general position. In addition, if Gn is a normal compact subalgebra in Gu , the set [a for an element a E Gu with a.l Gn is completely involutive for an element a in general position, where [a = la I Gn· The sets la and t;, are called the compact and normal series integrals. The Euler equations (1)

215

INTEGRABLE SYSTEMS ON LIE ALGEBRA

are completely Liouville integrable for Hamiltonians f functionally dependent on the sets la or fa as these sets are completely involutive. It will be shown below (Theorem 20.2) that, in the case of G u = su(m) for any element a the sets la and fa can be extended to complete involutive sets. Therefore the system (1) is integrable for singular elements a too. It is interesting to note that for many affine Lie algebras complete involutive sets on G* may be constructed using translates of singular invariants (Theorem 20.4), whereas for elements a E G* in general position it is impossible to extend the involutive sets la(G*) to complete involutive sets. We prove, as an auxiliary result, that the restriction of the set la(su(m)) to any singular orbit 0 gives!- dim 0 independent functions for a certain fixed element a E su(m) (independent of the orbit). This result is similar to the result ofDao Chong Thi [21] in whose proof, however, there are . . some maccuracles. The results of this section are due to A. V. Brailov. 20.1. Completely involutive sets of functions on singular orbits in su(m)

Let Iii E gl(m, C), (Ii) .. = J;rJjs, P~,q be homogeneous polynomial functions, defined uniquely by the following identity: (J=l)S Tr(x

+ a)S = L

(_l)np~,q(x),

n+q=s

= n. Let su(m) = {x EgI(m, C), x = _xt, trx = O}; u(m) = {x EgI(m, C), x = -xt}; so(m) = {x EgI(m,C), x = x = -xT

degp:,q

THEOREM 20.1 Let a Eso(m), a=L;n=-llaa(Ii,i+l-Ii+l,i)' ai,#O (1 :::; i :::; m - 1). Then the restrictions of polynomials p:,q to any orbit 0 in su(m) give !-dim 0 independent functions on it. Proof It is possible to select x E 0 for any orbit 0 in such a way that

x = diag(J=1x l , . , . ,J=1xm), where Xl :::; . , . :::; x m ' G~ c gl(m, C) be the subspace, generated by the vectors E],i+q

= (J=1)q+l(1},i+q + (-1)q1}+q,),

Gq

= G\

Et)

Let

G~.

For any real or complex subspace W c gl(m, C) we define the following linear subspaces in gl(m, C): wq = W n Gq, W! = W n G~, Let also G± = E9 q ;;.o Gi, Gi = E9o<;p<;q Gi and for an-y W: W± - W n G±,

216

A. T. FOMENKO AND V. V. TROFIMOV

~±

= W n Gi. Let T = 1',,0 be the tangent space to the orbit O. The direct calculation of the differential ofthe polynomial p:,q at a point x as a function of (su(m»* leads to: m-q

"L. Cqja ( )Uqn-t,j d Xj+q,j + + Rn,q a'

dx P an,q --

j~t

where Xj~ is the linear function on (su(m))* arising from the element E/j Eu(m); q(a) = (q + l)aj'''aj+q+t; Urj = Lko+ "'+kq~iXJ~"'XJ:; R:,q E Gq-t. We shall show that dxp:,q E su(m): . First let x be a point in general position. Then by considering the sequence of identities + t x] = [d x pn,q x] we obtain dx pn,q E su(m)q. + For singular x we [dx pn,q a' a' a obtain dxP:,q E su(m)q+ from continuity. As a consequence R:,q E su(m);_t. Let d~P:,q be the projection of dxP:,q on We shall show below that the d~P:,q generate T't for q ~ 1. Taking into account that dxP:,q E su(m)q+ we obtain that d[P:,q, the projection of the differentials dxP:,q onto T, generates T+. As dim T+ =-!-dimO, the theorem is proved.

n.

Let us prove in addition that the differentials dip:,q generate (q ~ 1). As R:,q Esu(m)q+_t,

n

m-q

dip:,q = L q(a)U~_t,jEj:j+q,

q(a) '# O.

j~t

It is enough therefore to show that the matrix vq, V;j = U'l,nU) is nonsingular, where n(1) < n(2) < ... < n(mq) is a maximal sequence of indices such that xnV} < xnV} + q (it follows from the definition of the sequence n(j) that mq = dim T't).

LEMMA 20.1 Let Yt"",YN be indeterminates, for 1 ~j, p ~N, j + p ~ N, let ~f(Y) be the homogeneous component of i in the formal series Uj.P: JtjP(y) = (1 - y)-t"'(I_ Yj+p)-l. Let WP be the (N - p) x (N - p) matrix of these polynomials 0 ~ i ~ N - P - 1, 1 ~ j ~ N - p. Then det WP(y) = (Y P+2 - Yt)(Y p+3 - Yz) ... (YN - YN_p_ddet Wp+l(y). Proof We carry out on the matrix WP the following elementary transformations: we subtract the second column on the right from the rightmost one, then the third on the right from the second on the right, etc. As a result we obtain a matrix WP in which-with the exception of the first column-all the columns consist ofterms (from top to bottom)

217

INTEGRABLE SYSTEMS ON LIE ALGEBRA of the series

l-liJ + 1 = {term

of degree i of the series WjP},j ~ 1. As (W01 ' W 02 " ' " Wo,N-p-d = (1,0,0, ... ,0) and WjP = (Yj+p+1 - Y) WjP + 1 , we have

WjP+ 1 WjP+ 1 -

det

WP

H,jP+ 1 -

= det

-

WjP,

WP

=

(Y p +2 -

Y1)'"

(YN -

YN_p_ddet

WP

+1

.

LEMMA 20.2 Let Xl ~ ... ~ xm; 1 ~ q ~ m; n(l) < ... < n(mq ) be a maximal sequence of indices, such that XlIV) < xnU)+q and r(l) < ... < r(m~) be a maximal sequence of subscripts amongst sequences satisfying the condition x'U) = X'U+1) = ... = x,u+s) ~ S ~ q. Then x"U)+s = xrU+s) where j = 1, ... , mq , S = 0, ... , q.

Proof Let k(O) = 0, Xk(1)+l

= ... = k(i)

Xl

Xk(2)

= ... = x k(1) = A1 , =

A 2 ,··· ,Xk(t-1)+1

= min(q, k(i) - k(i -

= ... =

1»,

xk(t)=m

=

At;

i= 1, ... ,t; i

k(O) =0,

kU) =

L

k(i).

s= 1

We can check directly that xrU+s) = Ap, where the subscript p has been determined from the inequality f(p - 1) < j + s ~ f(p). On the other hand if ~ s ~ q, X"U)+S = /c p • Whence x"U)+s = xrU+s)' The lemma is proved.

°

Now everything is ready for calculating the determinant of the matrix V • Let Yj = xr(j)' then V;j = U'!,nU) = {the component of degree i of the formal series (1 - x,,(j) -1 ... (1 - XliV) +q) -1} = {the component of degree i oftheformal series (1 - Xr(j» -1 ... (1 - XrU+q» -1} = W;j(y). As q det wm -1 = 1, applying induction, we obtain det v = det wq #- 0. The theorem is proved. q

THEOREM 20.2 (i) For any element X E su(m) the set IAsu(m» can be extended to a completely involutive set. (ii) For any element X E su(m) orthogonal to so(m) the set [Aso(m» can also be extended to a completely involutive one.

Proof (i) Let Gu = su(m), Gil = so(m), G~ = {g E Gu ; [g, x] = O} the centralizer of x in Gu , in the same way let G~ be the centralizer of x in Gn . As x is a semi-simple element, G~ is a reductive subalgebra in Gu and the rank of G~ is equal to the rank of Gu • We may take

218

A. T. FOMENKO AND V. V. TROFIMOV

As G~ is a reductive Lie algebra, there can be found on G: an involutive set fl' ... 'h compnsmg k = -!(dim G: + rank G:) independent functions. Extend the functions h on G u by taking them to be constant along vectors orthogonal to These extended functions we shall denote by h too.

G:.

LEMMA 20.3

The set I x is in involution with the set {II" .. ,h}.

Proof Let fey) = J(y + AX), J EI(G.). Then {f, J;}(y) = (y, [dyf, dyh])

= -A(x, [dyf,dyh]) = l([x,dyh],dyf) = 0 as dyh E G~. It is known that dxl a = dal x (for any set offunctions ~, by definition dx ~ = {linear span dx f, I E ~}). Let 0 be an orbit passing through x, as follows from Theorem 20.1 it is possible to select an element a in such a way that dim nT(dxl a) = t dim 0 and therefore dim nT(dal x) = tdim 0

= t(dim Gu - dim G~)

too. Therefore we can choose from the set I x functions h + l ' . . . , h +s where s = !(dim Gu - dim G~), so that nT(daD (i = k + 1, ... , k + s) generate T +. As T+ C T = T"O 1. G~ and dah E G~ (i = 1, ... , k) all the functions II' ... 'h +s are also independent. Their number is equal to k

+s=

!(dim G~

+ rg G~ + dim Gu -

dim G~) = !(rg Gu

+ dim Gu ).

Therefore this is a completely involutive set on G u • Statement (i) of the theorem is proved. (ii) W~dd=Ei1G~I+l. For any subspace WcG let W~dd = W" GO;:d. It has been proved in Theorem 20.1 that a vector a E su(m) can be chosen from so(n) such that the projection of the differential dxl a on T is nT(dxla) = T+. As dx1a = da1x

nT(da1x) = T+.

219

INTEGRABLE SYSTEMS ON LIE ALGEBRA LEMMA 20.4

We claim that -t(dim G~ + rg G~) = dim(G~)'1d.

Proof As G~

= soCk!) $ ... $ so (k t ) ,

G~

=

(G~)~d

s(u(k 1 ) $ .. ·$u(kt

= (su(kd EB· .. $

»,

sU(kt»~dd

it is enough to show that i(dim soCk) + rg so(k» = dim(su(k»~dd but this was proved in [90]. The lemma is proved. It is possible to construct on G~ an involutive set fr, ... ,1". consisting of k = -t(dim G; + rg G;) independent functions. In the same way as in (i) we shall extend the functions J: to Gn taking them to be constant along vectors orthogonal to G;. As in (i) we have {Ix,];} = o. And indeed,

{kJ:}(y)

= (y, [dyJ:,dyJ;J> _

1.

- (y, [z(dyf;.

+ dyf-.J,dyJ;])

= -A(X, [-t(d y!;. -

dyf-;.),dy/J)

= A([x,dyJ:J,-t(dyf" -

dyf-;)

=0

as dyJ: E G~. It has been shown above that it is possible to select from set functions 1". + 1> •.• ,1". +s in such a way that

In

7tT ,odd(da C{" + l' . . . , he +s}) = T~dd. As T~dd 1- G: all the functions fr, ... ,1". +s are independent. Their number is k + s = dim T~dd + dim(GX)~d = dim(G u)'1d. But, as was noted above, dim(Gu)~dd = !(dim Gn + rank Gn ). Therefore the set fl, ... ,he+s together with the set I X U{1".+l' ... '1".+s} form a completely involutive set. The theorem is proved.

20.2. Completely involutive sets of functions on affme Lie algebras

Let G be a Lie algebra and let G = H EB V where H is a subalgebra and V is a commutative ideal. Let p = ad v H be the adjoint representation of H on V. G in fact is the split extension of Lie algebra H determined by the representation p. Such Lie algebras are called affine Lie algebras.

220

A. T. FOMENKO AND V. V. TROFIMOV

For any representation p of a Lie algebra H in a vector space V the number ind p = {the codimension of an orbit in general position} (i.e. of an orbit ofthe action ofthe group ~ corresponding to the Lie algebra H) is called the index of the representation. The index of the coadjoint representation ind G = ind ad * is called the index of the Lie algebra G. Let G be a Lie algebra, W c: G a vector subspace, x E G* an element of the space dual to G. We define the vector subspace WX = AnnO'V, x) = {g E W, ad; x = O} c: W. If W is a subalgebra in G, then W X is a subalgebra too. We shall need, when calculating the index of an affine Lie algebra. THEOREM 20.3 (see [118J) Let G be an affine Lie algebra which is the split extension of Lie algebra H determined by a representation p of the algebra H on V. Then for an element x E G* in general position the equality ind G = ind H X + ind p* holds, where p* is the representation of H on V*, dual to p.

Proof Let x E G*, x = X H + xv, X H E H*, XV E V be an element such that the following conditions are satisfied: (a) ind p* = dim H dimH x ; (b) indHxv=inf{indHY,YEV*}; (c) dim Ann(HXV,xiwv) = ind HXv. All such elements x constitute a non-empty Zariski-open set in G*. Thus: the general position elements in G* satisfy the conditions (a}-(c). Therefore, in order to prove the theorem it is enough to check that the equation dim GX = ind H X + ind p* follows from (a}-(c). Let 9 = 9H + 9v E G X , then

+ ad:v(x v) = 0 ad:H(x V ) = O.

(2)

ad;H(xH)

(3)

It follows from (3) that gH E H X ' . Consider the restriction of equation (2) to HXV: (ad:v(xv),HXV)

=

-(ad~'v(xv),gv)

= 0;

(ad:H(x H), HXV) = (x H, ad:H(HXV» = (xHiw v, ad9H(HXV» = (ad;H(xHi wv), HXV);

so, restricting (2) to HXv we find (ad:H)(xiw v ) = O. Let 7tH be the projection from G onto H along V. We have proved that 7tH(G X ) c: Ann(HXV, xiw v). On the other hand, let gH E Ann(HXV, xiwv). Then

INTEGRABLE SYSTEMS ON LIE ALGEBRA

221

equation (3) holds automatically. We shall show that it is possible to choose gv E V in such a way that equation (2) holds too. LEMMA 20.5 If YEH* and Ylw v = 0, then there is gv E V such that Y = ad:v(xv). Proof We have shown above that (ad yv )* xvlw v = 0, therefore it is sufficient to check that dim(ad v)* Xv = dim{y E H*, ylw" = o}. Clearly dim{y E H*, Ylw v = O} = dim H/Hxv. On the other hand, dim(ad v)* Xv = dim V/Vxv. Let L(x, y) = (xv, [x, yJ) be a bilinear skew-symmetric form on G. It is easy to check that H and V are isotropic subspaces under the form Land HXV EB VXV = Ker L, from which it follows that dim V/Vxv. , , The lemma is proved.

As «ad yH )* xH)lw v = 0 we can choose, on the strength of the lemma, gv in such a way that (ad yH )* XH + (ad y )* Xv = O. Hence it follows that gH E 7tH(G X) and 7tH(G X ) = Ann(HXv, xlw). We now evaluate the dimension of a fiber of the projection 7tH : GX -+ 7tH(G X ). It follows from equation (2) that dim(7tH)-l(gH) = dim Vxv, From which dim GX

= dim Ann(HXV, xl w,.) + dim Vxv.

Note now that dim Vxv = dim V - dim V/vxv. It was proved above that dim V/Vxv = dimH/Hxv, therefore dim VXV = dim V - dim H + dim HXv = ind p* . Now from the condition (c) we obtain dim GX = ind HXV + ind p*. The theorem is proved. Let a E G* and al v = O. As V is an ideal, V c: Ga. As V is a commutative ideal, it follows from Lemma 20.3 that IaCG*) u V is an involutive set on G*. The following theorem gives conditions sufficient for this set to be a completely involutive set of functions. THEOREM 20.4 Let G be an extension of Lie algebra H determined by a representation p of H on V. Suppose that the following conditions are satisfied: (a) the number of independent polynomial invariants of the Lie algebra G is equal to its index ind G; (b) for an element y E V* in general position Lie algebra HY also has a number of independent polynomial invariants in I(HY*) equal to its index ind HY and for an element a' E HY* in general position the set I a.(HY*) is completely involutive. Then for an element a E H X in general position the set I a( G*) u V is completely involutive.

222

A. T. FOMENKO AND V. V. TROFIMOV

Proof We shall show further that dim(d,Ja(G*) + V) = !(dim G + ind G) from which it follows, of course, that Ia(G*) u V is a completely involutive set of functions. We shall need the following algebraic redefinition of the differential dx1a(G*).

LEMMA 20.6 Let G be any Lie algebra satisfying condition (a) of the theorem. Let I = I(G*), x,aEG*, Ia = Ia(G*) and qJxa:G ~ G be the partial many-valued operator qJx.a = ¢;I¢a' where ¢a(~) = ad: a. Then for a sufficiently large N and for x E G* in general position, dx1a = (qJx.at(G X). Proof Let f be a polynomial invariant. We shall define a polynomial .faq by equation deg!

fq(x

+ a) = I

faq(x),

deg faq

= deg f -

q.

q=O

For the polynomials f q = faq the identity (ad dxf q+ 1)* X = (ad dxfq)*a holds and, therefore, dxfq + 1 E qJx.a(dx fq). Whence dxfq E (qJx,a)q(G X ) and dxlac(qJx.a)N(G X ) for N~max{degIl, ... ,degf2}' where 11' ... , f,. is any complete set of polynomial invariants. The reverse inclusion (qJx,a)N(G x) C dx1a follows from the fact that for x E G* in general position, dxI = GX and that the ambiguity of the operator qJx.a coincides with GX and, consequently, with dxI too. The lemma is proved. LEMMA 20.7

Let xEG*, aEH*, x' = xlw v' a ' = alw v' Then 7r H(dJa(G*» = dx,Ia,((HXV)*).

Proof Let Ia = Ia(G*), I~, = Ia,((HXV)*), qJ = qJx.a' qJ' = qJx',a" It is enough to show, on the strength of Lemma 20.6, that for k ~ 0:

(4)

7rH(qJk(GX» = (qJ')k(Ann(HXV, x'».

By virtue of Lemma 20.6 it is enough to show that for k 7rH(qJk(GX» = (qJ')k(Ann(HXV, x'».

~

0:

(5)

It has been shown above that 7r H ( GX) = Ann(HXV, x'), hence the equation

(5) is true for k = O. Let us suppose that we have proved (5) already for some k ~ 0 and

l

E

qJk(G X )

C

(qJ')k(Ann(HXV, x'» EEl V,

gk = g~ + gi, g~ E (qJ')k(Ann(HXV, x'», gi E V. We want to prove that

223

INTEGRABLE SYSTEMS ON LIE ALGEBRA

7r:H(CP(gk» = cp'(g~) a consequence of which is the equation (5) for k According to the definition of the operators cp and cp' we obtain cp(gk) = {gk+l, (adl+ 1)* x = (adg k )* a}, cp'(g~)

=

{g~ +1,

+

1.

(ad H g~+ 1)* x' = (ad H gk)* a'}.

Let gk+ 1 E cp(gk), which means that the following equations hold: (adg~+l)*

XH

+ (adgVl)* Xv =

(adg~+l)*

Xv

(adg~)*a,

(6)

= O.

(7)

We obtain from (7) g~+1 E HXv, from which it follows that «adg~+l)* xH)lw v = (adg~+l)* x'. As for g~, g~ E HXV so that «adgt)*a)lwv=(adgt)*a'. In addition, «adgt+ 1)*x v )iw v =0, therefore restricting elements of G*, which appears in (6) to HXv, we get (ad g~ +1)* x' = (ad g~)* a'. Thus, if gk +1 E cp(gk) then 7r:H (gk+l) = g~+l Eo/(g'). On the other hand, let g~+l Ecp'(gk), then it is possible to choose gk +1 E o/(gk) in such a way that 7r: H(gk +1) = g~ +1 . And indeed, it follows from g~+ 1 E cp'(gk) that «adgt)* a - (ad gt+ 1)* xH)lwv =

o.

Therefore (Lemma 20.5) there is an element g} +1 such that (adg}+l)* Xv = (adgt)*a - (adgt+l)* x H . Hence gk +1 = g~ +1 + g} + 1 E cp(l) and 7r: H(l +1) = gt +1 . Thus 7r: H(cp(gk)) = cp'(g~) and equation (5) has been proved for k + 1. The lemma is proved. We obtain from Lemma 20.7 therefore dim(dJa

+ V) = dim dx,I~, + dim V =

~(dim

=

~(ind HXv

HXV + ind HXV) + dim V

+ dim G + dim HXV + dim V

- dim H).

Note now that ind p* = dim V - dim H/Hxv, therefore dim(dx1a

+ V) = t(ind HXv + dim G + ind p*) = !Ond G + dim G)

224

A. T. FOMENKO AND V. V. TROFIMOV

in accordance with Theorem 20.3. That means that fa completely involutive set on G*. The theorem is proved.

+

V is a

Let G be the extension of a compact Lie algebra H determined by a representation p: H -+ gl(V). Then for an element a E H* in general position the set fa(G*) u V is a completely involutive set. COROLLARY

Proof For any y E V* the subalgebra HY is reductive. Therefore, fa.(HY) is a completely involutive set for an element a' E (HY)* in general

position. That polynomial invariants exist follows from the fact that [G, G] = G.

The stabilizers HY of elements y in general position have been studied, for example, in the work [31]. This makes it possible to build completely involutive sets on extensions of some non-compact semisimple Lie algebras too. The possibilities of using Theorem 20.4 are not restricted to extensions of semi-simple Lie algebras only. It gives the existence of completely involutive sets on some Lie algebras with noncommutative radicals too. REMARK

21. COMPLETELY INVOLUTIVE SETS OF FUNCTIONS ON EXTENSIONS OF ABELIAN LIE ALGEBRAS 21.1. The main construction

In this section we give the results of Le Ngok Tyeuyen. We will use here the method of constructing involutive sets given in Section 13. Let mbe a connected Lie group, G its Lie algebra and G* the dual space G. We shall use, for the sake of simplicity, the notations Ad: f = 9 x f, gEm, f E G*, adt f = ~ x f, ~ E G, f E G*. The number r = ind G = dim G* - SUPjEG* dim O(f) is called the index of Lie algebra, where O(f) is the orbit of the coadjoint representation passing through f E G*, r = ind G = inf dim G(f) where G(f) = g E G Iadt f = ~ x f = o}. The point f E G* is called a point in general position if dim O(f) = dim G* - r or, equivalently, dim G(f) = r. Let the Lie algebra G be decomposable as the direct sum of an ideal Go and an Abelian subalgebra H: G = Go + H, let (fjo and ~ be the connected Lie subgroups of (fj corresponding to Go and H. We obviously have (for a given decomposition) that G~ is isomorphic to the subspace G~

INTEGRABLE SYSTEMS ON LIE ALGEBRA

225

of G*, G~ = {f E G* I flH = O} and H* is isomorphic to jj* = {h E G* I hi Go O} c G*. We can therefore consider G~ and H* as subspaces of G*. The representations Ad*: mo --+ GL(G~) and ad*: Go --+ End(G~) are defined. We introduce the notations Ad: f = g ® f if g E mo, f E G~ and adt f = e ® f if e EGo, f E G~. Thus, if fEG~cG*, gEffiocffi, eEGocG then exfEG*, e®fEG~, g x f E G*, g ® f E G~ and, generally, e x f and e ® f as well as g x f and g ® f do not coincide in G*. Let 7to be the projection of G* onto G~ along H* (7tH the projection of G* onto H* along G~), then we obtain the following simple relations

=

fEG~,

(1)

LEMMA 21.1 Let hE H*, f E G*, then O(f + h) = O(f) + h, i.e. the orbit of the coadjoint representation passing through the point f + h can be obtained by a translation of the orbit passing through f along the vector h.

Proof Wehaveg x (f+h)=g x f+g x h,gEffi,fEG*,hEH.As H is a commutative subalgebra and Go an ideal in G, g x h = h for all gEm, i.e. O(h) = {h}. It follows from that fact that g x (f + h) = g x f + h which was to be proved. COROLLARY 1 Let the space HJ in G* be obtained by translating space H* along the vector f, i.e. f' E Hy if and only if 7t o(f') = f Then O(f) n Hy is a subgroup with respect to addition in HJ. Indeed, if f + hi E O(f), hi E H*, i = 1,2 then it follows from the lemma that f + hI + h2 and f - hI belong to O(f). COROLLARY 2 It follows from the lemma that f E G* is a point in general position if and only if 7t o(f) is a point in general position for G*. Because the set of all points in general position for any Lie algebra is open and everywhere dense, it is not difficult to deduce, using this fact, that there is an open and everywhere dense set Wo in G~ such that for any f E Wo, f is, at the same time, a point in general position both in G* and G~. Let f E Wo c G~, i.e. f is a point in general position both for G* and G~, and let Oo(f) be an orbit of the coadjoint representation of ffio on G~, i.e. Oo(f) = {ffi ® fig E ffio} = ffio ® f Then the tangent space to Oo(f) at the point fis the space TfO(f) = {e ® fl e EGo} = Go ® f On the strength of equation (1) we have

226

A. T. FOMENKO AND V. V. TROFIMOV Oo(f) = 1t0(u(f)),

(2)

where

u(f) Go ® f

= {g

x fig E (£)o} == (£)0 x f,

= TfOo(f) = 1to({ ~ x f, ~ EGo}) = 1t O(G O x f)·

LEMMA 21.2 Let f E Wo C G~, h EH*. Then if ~ x for all the t E IR: (Exp t~) x f = f + tho

f = h, ~ E Go then

Proof If h = 0, then our assertion follows from the definition; we can thus assume h =I O.{ In addition, it is enough to prove the lemma for the case when dim H = 1, as the general case can be reduced to this by supposing G* = GT + IR . h (i.e. by considering a new decomposition of G as the direct sum of an ideal G 1 and a one-dimensional subalgebra). Let y(t) = Exp(t~) x f be a curve in G*. We have to prove that y(t) = f + tho It follows from the condition ~ x f = h, ~ E Go and equation (2) that ~ ® f = 1t o(h) = 0, therefore (Exp t~) ® f = f, t E 1R1, Exp t~ E (£)0. But 1to[(ExO t~) x f] = (Exp t~) ® f, therefore y(t) = (Exp to x f E Hi. The latter means that y(t) = f + rx(t)h, rx(t) E IR. Let t 1, t2 E IR, then y(tl

+ t 2) = [EXP(tl + t2)~]

= (Exp tl~)

x

f = (Exp tl~) x [(Exp t2~) x f] [f + rx(t 2 )h] = f + rx(tl)h + rx(tz)h X

because (Exp tI~) x h = h. Therefore rx(ti + t 2) = rx(td + rx(t2)' Moreovery(t)lt=o = hord(t)lt=o = 1, therefore rx(t) = t,which was to be proved. COROLLARY Let f E Wo C G6, then (£)0 x f => {f where ffio is the Lie group corresponding to Go.

+ (Go

x

f

n H*)}

LEMMA 21.3 There is a subspace Vo* c H* such that Vo* = Go x f n H*, f E J¥t where J¥t is open and everywhere dense in G6' In particular, Vo* c OU) n Hi for any f E WI·

Proof We shall prove the lemma using the method of induction. Let G be decomposable as a direct sum of an ideal Go and an Abelian su balgebra H, dim H = 1: G = Go + H. Consider the restriction of the coadjoint representation on G* to (£)0' then G* is partitioned into the orbits of this action(£)o x f, f E G*. Similarly to Corollary 2 of Lemma 21.2 we have: f E G* is in general position «£)0 x f has maximal dimension) if and only if 1to(f) is in general position for this action. This means that there is in G6 an open everywhere dense set W' of points in

227

INTEGRABLE SYSTEMS ON LIE ALGEBRA

general position for the action of &0 on G*. Let W1 = Wo n W'. It is obvious that J.ti is open and everywhere dense in G~ also. Let us take any two vectors i1, 12 E W1, then 11,12 are in general position both for G~ and G*, i.e. dim(G o x Id = dim(G o ® 12) and dim(G o x Id = dim(G o x 12)' Suppose that Go x 11 n H* f= 0, thus G 1 x 11 n H* = H* as dim H* = 1. According to (2) we have: l£ o(G o x 11) = Go ® I implying dim Go ® 11 = dim Go x 1- 1, therefore dim Go 0

12 = dim Go 0 11 = dim Go

x

11 -

1 = dim Go x

12 -

1,

but Go 012 = l£ o(G o x 12), thus Go x 12 n H* = Go X 11 n H* = H*, which means that there is a subspace Vo* c H* (here either Vo* = 0, or Vo* = H*) such that Go x In H* = Vo* for all the IE W1 · Suppose we have proved the lemma for the case dim H = m. Consider the decomposition G = Go + H, Go an ideal, H an Abelian subalgebra, dim H = m + 1. Let us choose in G a basis e 1 , e2 , ••• , ejo' ejo+l, ... ,ejo+m+l for H. We denote the conjugate basis by G* as el, ... ,eio+m+l where ei(e) = oj, i,j=l, ... ,jo+m+1. Consider G 1 = Go + (eio +1 " " ,ejo+m> = Go + Hl where (eio +1 " " ,eio +m is the subspace of H with basis eio +1 "" ,eio +m' It is obvious that G1 is a Lie algebra which can be decomposed as the direct sum of an ideal Go and an Abelian subalgebra H 1, dimB 1 = m. In accordance with the inductive hypothesis, there is Vo* in H* = (e io + 1 , ••• , eio +m such that Go x (ld n H! = Vo* where ~ x (1) I denotes the action of the vector ~ E Go on the vector I E ~ c G6 under the representation of G1 in G!, where ~ is an open and everywhere dense set in G6. According to the construction, we have a decomposition

>

>

where G 1 is an ideal in G, H 2 an Abelian subalgebra, dim H 2 = 1. In the same way as when dim H = 1, we find in G6 an open and everywhere dense subset W' such that dim(G o ® Itl = dim(G o ® 12), dim (Go x 11) = dim(G o x 12) for all the 11,12 E W', but Go ®;; = l£ o(G o x /d, i=1,2, therefore dim(G o x/lnH*)=dim(G o x/2nH*) for all 11'/2 EWe G6· Write J.ti = W' n ~, then w;, is the open and everywhere dense set in G6. Let 1£1 be a projection of G* on G! along H!, then from relation (2) we obtain:

A. T. FOMENKO AND V. V. TROFIMOV

228

Go x Il)/; = (GO

X

o)D n Ht = 1t 1[(G O X

1tl (GO X /;),

/;)

n H*],

i = 1,2, 11'/2 E

»t.

We have the following equations: dim[(G o

X

It> n H*] = dim[(G o x 12) n

[(Go X(1)/;) nHt]

= 1t1[(G O x

/;) nH*],

H*]

i = 1,2;

dim[G o x (1)/1) n Hi] = dim[(G o x (lJi2 n Ht] = dim Vo* for any 11,12 E »t. But H* = Ht + H! and dim H! = 1, therefore it follows from these equations that there is a subspace Vo* of H* such that Go x In H* = VO* for all I E WI' which was what we had to prove. We introduce the notation k = dim Vo*, where Vo* = Go x In H*, IE »t. REMARK It follows from Lemma 21.3 and Corollary 1 of Lemma 21.1 that for lEw;. there is a basis e1, ... ,ejo ' ejo+ 1, . . . ,en of G such that Go = (e u e2"'" ejo H = (ejo+1"'" en> and

>'

s(f)

k

ffio x In H*

=

I +

L

lRe1o+1

+

;=1

L

Ze1o+k+S

s=l

wherek = dim V*doesnotdependon/and:L7=1IRejo+1 = Vo*.Let,as before, G be expanded as the direct sum of an ideal Go and an Abelian subalgebra H:G = Go + H, let ffi,ffio,f) be the connected Lie groups corresponding to these Lie algebras. Consider the restriction of the coadjoint representation ffi to G6, so that ffi acts naturally on G6. We denote by ffi ® I for IE G6 the orbit of the point I under this action. Obviously, ffio ® I contains the orbit Oo(f) = ffio ® I of the coadjoint representation of <»0 on G6. According to equation (2) we have ffi ® 1= 1to(ffi ® I). There is in G6 an open and everywhere dense set W2 of points in general position for the action of ffi on G6. We denote its intersection with WI by W: W = WI n W2 • Then W is also an open and everywhere dense set in G6. In a way similar to Lemma 21.3, the follbwing lemma can be proved. LEMMA 21.4 There is in the space H* a subspace V*(V* dim V* = k + I such that for all lEW we have:

·G x In H*

=

V*.

=:J

Vo*) ,

(3)

It follows, obviously, from Lemma 21.4 that V* may be written as a direct sum of two subspaces V* = Vo* + l'1*, where Vo* is as constructed

229

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Vr

in Lemma 21.3, dim = l. Let G be any Lie algebra, n = dim G, r = ind G. Let m smooth functions be defined on G*: F I (f), . .. ,Fm(f), Fj(f) E COO(G*), 1 ~ i ~ m. We remind the reader that such a set of functions is called a completely involutive set for G, if the functions F 1 , .•• ,Fm are mutually in involution on all orbits, i.e. {F i (!), Fj(f)} 0, 1 ~ i, j ~ m, rk(dF I (f), ... , dF m(f) ~ (n + 2)/2 on all orbits in general position. Let G = Go + H, and let functions F(f) be defined on G6 c G*, then P(f) can be extended to functions F(f) on G* by setting F(f + h) = F(f), where! E G6, h E H* (see Section 12). The following theorem is the main theorem of this Section.

=

THEOREM 21.1 Let G be a Lie algebra decomposable as a direct sum of an ideal Go and an Abelian subalgebra H: G = Go + H. Let FI(f), ... ,Em(f) be a completely involutive set of functions on Go. These functions FI' . . . , Fm can be naturally extended to functions F I ' ... , F m on G* (see Section 12). Then we claim that the class of functions F l ' . . . ,Fm is a completely involutive set for the Lie algebra G. Proof According to Lemma 21.4 there are in H* subspaces V* and Vo* such that V* =:J Vo*, dim Vo* = k, dim V* = k + 1= s (V* = Vo* + VI*' dim VI* = I), G x f n H* = V*, Go x f n H* = Vo* for any fEW, where W is an open and everywhere dense subset in G6' As the functions Fl ' ... , Pm are mutually in involution, their extensions F I ' ... , Fm to G*, as proved in [127] (see also Section 12) are in involution on G*, i.e. 0 = {Fj(f), Fj(f)} for all i,j. Obviously, dFj(f) = dFj(f) E Go for any f E G6, 1 ~ i ~ m, therefore rk(dF I (f), ... ,dFm(f» = rk(dF l (f), ... ,dFm(f» = q. According to the hypothesis of the theorem we have g ~ Uo + ro)/2 = M 0, where jo = dim Go and ro = ind Go. In order to prove the theorem it is necessary to check the inequality

rk(dF 1 (f),··· ,dFm(f))

=q~

n+2 2 - (n - jo - s)

- n+r-

2n

+ 2j 0 + 2s

2

-

2j 0

+ 2s + r -

n

2

for which it is enough to show that

M0 =

jo

+ ro 2

~

2jo + 2s + r - n --'-'--2---

(4)

We shall prove the inequality (4) using induction. Let us prove it first for

230

A. T. FOMENKO AND V. V. TROFIMOV

S = 0 and S = 1. Let S = 0, i.e. G x III H* = 0 for any lEW As lEW, I is in general position both for G~ and G*. For linear subspaces Go ® I and Go x Iwe obtain from equation (2): Go ® 1= 7t o(G o x I)(G o ® f is the tangent space to the orbit Oo(f) of the coadjoint representation of (fjo in G~). As Go x III H* = 0, we have dim Go ® I = dim Go x f and dim Go ® I = dim Go x I ~ dim G x 1= n - r. Therefore Mo =jo

= 2jo - dim Go ® I ;:: 2jo + r - n 2 2 2

+r

and the inequality (4) is proved. Let s = k + 1 = 1; consider the two cases: (a) k = 1, 1 = 0 and (b) 1 = 1, k = O. In the first case dim(G o x III H*) = 1, but Go ® 1= 7t o(G o x I), therefore dim(G o ® I) = dim Go x 1- 1 ~ n - r - 1. In the second case dim(G x III H*) = 1, dim (Go x III H*) = 0, therefore dim Go x I ~ dim G x 1- 1. And, as a consequence, dim Go ® 1= dim Go x I ~dim G x 1- 1 = n - r - 1. Thus in both cases (a) and (b) we have Mo = !(2jo - dim Go ® I);:: !(2jo but !(n

+ r)

+ r + 1-

n),

is an integer, therefore, !(n - r) is an integer too, hence Mo;:: -t(2jo

+r +2

- n) = !(2jo

+ r + 2s -

n)

and in this case the inequality (4) is proved also. Suppose that the inequality (4) has been proved for all s < So (so;:: 2) and let us prove it now for s = So. Let So = ko + 10 ; consider the two cases: (a) ko '" 0, (b) ko = 0, 10 = So '" O. The first case. We choose for G a basis e 1 , e2, ... , ejo' ejo +1, ... , en such that G = Go + H, H = (e jo + l ' ... , en), Go = (el, ... ,ejo )' Let H*=(ejo+I,· .. ,e:), e:EGox/IlH*. Write G?=(el, ... ,ejo,ejo +1 ' ... , en-I)' HI = (en), G = G I + HI then GI may be decomposed as a direct sum of an ideal Go and an Abelian subalgebra H 2 = (ejo+l,·.·,en-I); G I =G O +H 2 • Let G I x(1d be the coadjoint representation of Lie group (fjl in GT. In accordance with equation (2): Go x (1) 1* = 7tG! (Go x I), where 7tG! is the projection of G* on GT along HT and Go x(1d IlH! = 7t G!(G o x III H*). The construction is such that H! = (e:) c Go x III H*, therefore dim(G o x (1) III

Hn = dim(G o x III H*) -

1 = ko - 1.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

231

Similarly, from the fact that G1 x (1) f = n G , (G 1 X f) it follows that dim[(G I X(I)f) n

Hn =

10

+ ko

- 1 = So - l.

Therefore, according to the inductive assumption, we have Mo;;::: -t(2jo + 2(so - 1) + '1 - nd, n1 = dim G 1 , '1 = ind G 1 but n 1 = n - 1; -dim(G l x(1)f);;::: -dim(G x f) + 1, therefore '1 = nl - dim GI x(1)f;;::: ,. Thus, Mo;;::: !(2jo + 2(so - 1) + r - n) +!. Having noticed that M 0 and !(2jo + 2(so - 1) + r - n) are integers we obtain M 0 ;;::: !(2jo + 2s + r - n). Likewise, for the second case, when ko = 0, 10 = So: Ht = <e:> c G x f nH*, dim(G o x(1d nH 2 ) = ko = 0 and dim(G l x (1) f n H!) = 10 - 1 = So - 1. The inequality (4) is proved. COROLLARY Let G be a Lie algebra of the "radical" type, i.e. G may be decomposed as the direct sum of a nilpotent ideal Go and an Abelian subalgebra H: G = Go + H, then there is a complete commutative set of functions on G.

Proof It has been shown in [134] that for any nilpotent Lie algebra there is a completely involutive set of functions F 1 (f), ... ,Fm(f) (m = !(n o + ro»; the corollary follows from this assertion and from our theorem. Let G be any Lie algebra of dimension n, r = ind G, and let 11' ... ,lr be a complete set of invariants for G (i.e. Ii is invariant under the coadjoint representation). It is obvious that for any F(f) E CO(G*): {F(f),IJJ} = 0, i = 1, ... ,r. This means that if a set of smooth functions F 1 (f), ... ,Fm(f) on G* is complete and commutative, then the set F I' . . . ,Fm' II'" . , Iris commutative. From Theorem 21.1 immediately follows the theorem: THEOREM 21.2 Let the Lie algebra G be decomposable as the direct sum of an ideal Go and an Abelian subalgebra H: G = Go + H and let 11 (f), ... ,1r(f) be a complete set of invariants for G(r = ind G). Let F1 (f), ... , Fm(f) be a completely involutive set of smooth functions on G~. Then the set F 1 (f), ... ,Fm(f), 11 (f), ... ,I,(f) is a completely involutive set on G, where F 1 (f), ... ,Fm(f) are the liftings of functions Fl ' . . . , Fm to G*. COROLLARY 1 Let G be decomposable as the direct sum of a nilpotent

A. T. FOMENKO AND V. V. TROFIMOV

232

ideal and an Abelian subalgebra: G = Go + H.lfthere is a complete set of invariants for G, then a completely involutive set of functions on it exists. COROLLARY 2 Let G = Go + H, Go a nilpotent ideal, H an Abelian subalgebra and r = ind G = 0, then a completely involutive set of functions on G exists. COROLLARY 3 Let BG be a Borel subalgebra in a semi-simple complex Lie algebra G. It is obvious that BG = Go + H is the direct sum of a nilpotent ideal Go and an Abelian subalgebra H. It follows from Corollary 2 that there always is a completely involutive set offunctions on G for Borel subalgebras BG. THEOREM 21.3 (Trofimov, V. V., [126], [127]). Let G be the semisimple complex Lie algebra, BG = EEli !Rh i EB La>o !Rea be a Borel subalgebra in G, Wo be the element of the group Weyl of the maximal height (see [11]). If 0 be the orbit of the maximal dimension of the coadjoint representation, then codim 0 = ! card A, where A = {a E Lli ( - wo)a #- a}; Ll be the set of the simple roots of the Lie algebra G. Remark The completely involutive set of functions for Borel subalgebras of semi-simple Lie algebras is given explicitly in [126], [127]. The set of functions constructed here differs from the set of functions given in those papers. 21.2. Lie algebras of triangular matrices

Let F be a smooth function on the dual space Lie algebra G. We recall that F is called semi-invariant if F(Ad: f) = X(g)F(f) for any 9 E 6j, f E G* where X(g) is a character of the group 6j and Ad* is the coadjoint representation of the group 6j in G*. Recall also the main properties of semi-invariants from [10]. If F is a semi-invariant for G, then s grad F(f) = - F(f) dX for any f E G*, dX E G* (see Section 11). Therefore, any function ¢(f) is in involution with a semi-invariant F(f) if and only if (dX, d¢(f» 0 for all f E G* (d¢(f) E G). Let the semiinvariants F(f) and ¢(f) of the algebra G be in involution {F(f), ¢(f)} 0, f E G*, then for any hE G* and any A, J.l E!R the functions FA,h(f) = F(f + Ah), ¢I',h(f) = ¢(f + J.lh) are also in involution {F A,h(f), ¢/l,h(f)} o.

=

=

=

233

INTEGRABLE SYSTEMS ON LIE ALGEBRA

G7

Let {el' ... , ell} be a basis in G; let j be the structure tensor of the Lie algebra G with respect to this basis. Denote by (/1 ,f2, ... , f,,) the system of coordinates in G* relative to (e 1, e2, ... , e") where ei(ej ) = £5~, i, j = 1, ... , n (e 1 , • •. , en) is the conjugate basis in G*. Then for any vector f E G* we have dim 0(/) = rk II ct.h II ' where O(n is the orbit of i the coadjoint representation passing through f, and f = 1 J;e gives f with respect to the basis (e 1 , e2 , ••• , e"). Let M(n, Iffi) be the space of all matrices with n rows and n columns. Define the matrices Ii ojo by Iioio = (£5 iio £5 jjo )' i,j = 1,2, ... , n. Then the Ii j , i,j=1,2, ... ,n form a basis for the space M(n,Iffi): M(n, Iffi) = L7,j=1 IffiIij' Let 7;, be the space of all upper-triangular matrices 7;. = Ll ~i~j~1I IffiIij' 7;. is a Lie algebra. Using the scalar product (x, y) = T,.(x, y), x, Y E M(n, Iffi) it is possible to identify the space 7;,* with the space of all lower-triangular matrices Ll ~j~i~1I IffiIi j . We shall consider therefore that 7;.* = Ll ~j~i";l1 IffiIi j . Let x E M(n, Iffi) be the

I7=

matrix, let

x( ~1' i.2" .. ,i:)

be the minor at the intersection of the rows

it.]2,···,jp

numbered i 1 , . . . , ip and the columns numberedj1"" ,jp in the matrix X. Consider the following functions on 7;.*

S (x)= p,k

"

i..J

k
x(i

1 , ...

,i p ,n-k+1, ...

12k . ,

,'0',

.

,ll,o .. ,lp

,n)

,

p~O,k;::O.p+ 2k~n

So,o(x) = 1, x E 7;.* then Sp,k(X), p = 0, 1, 0 ~ k ~ [en - p)/2J are semiinvariants of the coadjoint representation and are mutually in involution (see [10]). Moreover, if A E 7;.* is a point in general position (in particular A = Xo = 7;,,1 + 7;,-1,2 + ... + 7;.-[11/2]-1,[<1/2]) then the coefficients of the terms in A of the polynomials Sp,k(X + AA), p = 0, 1, o ~ k ~ [en - p)/2J form completely involutive set of functions (see [IOJ). Consider Lie algebras of the following form: L = V + Ll ~i<j~,. IffiIi j , where V is any subspace of the n-dimensional space of diagonal matrices VeL? = 1 IffiIii' It is obvious that L is a Lie subalgebra of 7;. and L* = V + Ll ,,;j
234

A. T. FOMENKO AND V. V. TROFIMOV

functions; (b) Pi(X ap ) = Xii + xjj,i = 1,2, ... , [(n - 1)/2J,j = n + 1 - i, Pk(X ap ), k > [(n + 1)/2] does not depend on the variables Xaa ' IX = 1,2, ... ,no Proof By properties of the Poisson bracket, we have {F l, F IF 3} = {Fl,F2}F3 + {Fl,F3}Fl for any three functions F l ,F 2 ,F3' It follows from this that if F l' F 2, ... , F m is a completely involutive set offunctions on T,.* then the set of functions F 1, ... ,Fm where Fi = Fi for ~ny i =1= io, i = 1, ... ,m and either Fio = Fi o + D{a)e{l,2, .... m) Fa or Fi o = Fi o - D{ale{l. 2 •... ,m) Fa will satisfy the same condition. Let us take Xo = T,.,l + ... + T,.-[n/2]+l,[n/l]· The coefficients of terms in A. of the polynomials Sp,k(X + Axo), P = 0, 1,0::;:; k ::;:; [(n - p)/2] have the desired properties. The lemma is proved.

Let L be a Lie algebra of the form L = V + LI <>i<j<>n IJU;j' where V is a subspace of the space of diagonal matrices V c Ii= 1 lRT;i' Then there is on L* a set of polynomials ql, ql,' .. ,qm, m = -t(dim L + ind L) which are mutually in involution on all the orbits of the coadjoint representation of the Lie group belonging to the Lie algebra and they are independent at points in general position in L*. THEOREM 21.4

Proof According to Lemma 21.5 we have a set of polynomials P = {Pi(X ap )} on T,.*, such that Pi = Xii + Xjj, i = 1,2, ... , [(n + 1}/2J, j = n + 1 - i and Pk(XaP)' k > [(n + 1)/2J does not depend on the variables X aa ' IX = 1,2, ... ,n. In addition, {Pi' Pj} 0 for all i,j = 1,2, ... ,M = -t(dim T,. + ind T,.). Consider in the subalgebra L of the algebra T,. the element Xo = T,..I + ... + T,.-[n/l]+I,[n/2] EL*. As this element Xo E Tn* is in general position in this space, ind 7;, = cork II C~Lox~v II where C~p,yo are the structure constants of 7;, with respect to (T;), 1 ::;:; i ::;:; j ::;:; n; letting (x~J denote the coordinates of Xo in 7;,* we have the following formulas:

=

C~p,yo

CaP, yO

= «(jpy(j~(j~

- (jao(j~(jp)(1 - (jpy(jaO)

= C~LoX~v = «(jpy(j:tJ

(5)

- (jaO(j~:J)(1 - (jpy(jao)·

We shall prove the theorem for an even n = 2m, say the case of odd n can be proved in the same way. Let L = V + Lhi<>j<>n lRT;j, dim V = K and 'ta =

Ii=l ~ai T;i' IX = 1,2, ... ,k form a basis in

V. As Xo EL* one may

235

INTEGRABLE SYSTEMS ON LIE ALGEBRA

consider the matrix II C~P.y6X~V II, where C~P.Y6 are the structure constants of L with respect to the basis (T;j' 1 ~ i < j ~ n, fx~' IX = 1, ... ,k), rkIIC~il.Y9x~vll = dim O(x o), where O(xo) is an orbit of the coadjoint representation of the Lie algebra L. It follows from (5) that

corkIIC~p.y6X~vll where the matrix A

= corkllC~p,y611 = cork A, has the form (n = 2m): 1- 1

T1.1

1- 1

~."

0

0

T2 • 2

1- 1

0

1

Tm,.. A = Tm+l ,m+l T2m ,1

1- 1 1- 1

0 0 1- 1

0

1

Tm+1.m

It is obvious that cork A = ind T" = m. In addition cork II C~P.y6X~v I corkll Cap ,y611 = cork B where the matrix B has the form:

ft.l B=

0

C

-c

0

=

t,k T2m , 1

Tm + 1 • m

and C = (C"'!.t,Oy), 1 ~ IX ~ k, m + 1 ~ b ~ 2m, y = 2m + 1 - b. As T;,,, = =1 ~~i T;i' each row s of the matrix C is a linear combination of rows

L:'

e

of the matrix D, where A = ( set g~i}' 1 ~ IX

OlD)

with coefficients taken from the -D 0 ~ k, 1 ~ i ~ n. Let sl , " " Os, (I = rk C) be the 1 linearly

e

236

A. T. FOMENKO AND V. V. TROFIMOV

independent rows of this matrix (1 ~ Sl" .• ,Sl ~ k). Let ((x. p ), 1 ~ P < Cl ~ n, (x .. ), Cl = 1,2, ... ,k) be a system of coordinates in L* and for ~ S ~ k, S =1= Si' ~ i ~ 1 suppose Os = CliOs, By dint of the construction of the matrices D and C we have:

1

1

I

,=I

1

Cli(eS,i o - esJ )

I:=l

=e

SiO -

e sjo '

(6)

For each Os consider on L* the first degree polynomial qo, = Cl,Xs,s, - xss then it follows from (6) that

I:=l

m

qo, =

I

Pi(X ii

+ Xn+l-i.n+l-i)'

(7)

i= 1

As the polynomials Pk(X.p) for k > [(n + 1)/2] = m on T,,* do not depend on the variables x •• , 1 ~ Cl ~ ~, one may take these polynomials to be defined on L*: Pk(X.P) = qk(X.P)' Thus we have defined on L* a set of polynomials {qo,(Xll' ... , Xnn)' S =1= Si' i = 1,2, ... ,1; qk(X.P)' k> [(n + 1)/2] = m}. It follows from (7) that these polynomials are mutually in involution on L*, that they are independent at all points in general position and that there are !(dim L + codim O(xo» of them. But !(dim L

+ codim O(x o»~ 1(dim L + ind L).

Thus the theorem is proved.

22. INTEGRABILITY OF EULER'S EQUATIONS ON SINGULAR ORBITS OF SEMI-SIMPLE LIE ALGEBRAS

This section is devoted to some more precise results and extensions to those of Section 16. These results are due to A. V. Brailov (see [199],

[201]). 22.1. Integrability of Euler's equations on orbits 0 intersecting the set tH R , t EC

Let G be a semi-simple Lie algebra. An element a EGis called semisimple if the endomorphism ada is semi-simple. In this case G = [G,G] EEl Ga, where Ga is the centralizer of a in G. The restriction adal [G.G] is invertible. Thus the mapping ad a- 1 : [a, G] --+ [a, G] is well

INTEGRABLE SYSTEMS ON LIE ALGEBRA

237

defined. Let bE Ga , then (adb)([a, G]) c [a, G] and therefore the mapping ada- 1 ad b : [a, G] -+ [a, G] is defined. Let D: GU -+ Ga be an operator symmetric with respect to a non-singular Ad-invariant form Q. We define operators CfJabD by the matrix

0)

1

CfJabD

_ (ad a- adb 0 D

(see Section 7) with respect to the decomposition G PROPOSITION 22.1

The decomposition G

=

= [a, G] ffi Ga.

[a, G] EB GU is orthogonal.

PROPOSITION 22.2 Let Q be a non-singular symmetric bilinear invariant on the Lie algebra G, a E G being a semi-simple element. Then the restriction of Q to the centralizer GU of a is also a non-singular form. PROPOSITION 22.3 Suppose that the hypotheses of Proposition 22.2 are satisfied, and that bE Ga and D: Ga -+ Ga is an operator symmetric with respect to Q, then the operator CfJabD: G -+ G is symmetric with respect to Q also. LEMMA 22.1 Let G be a Lie algebra, a EGa semi-simple element, Q an invariant symmetric bilinear form on G, Ga the centralizer of a, Cent Ga its center, D: Ga -+ Ga an operator symmetric with respect to Q, Q the restriction of Q to GU , and 11 (y), ... ,.h(y) integrals (in involution) of the equation of motion Y = [Y,D(y)], Y EGu. Then (a) if bEG a , Euler's equations of motion x = [x, CfJabD(X)] , X E G have integrals J(x + Aa) where J El(G), A E IR; (b) if bE Cent GU, the functions heX) = heY), where Y is the projection of x onto Ga along [a, G], are integrals of the equation of motion; (c) all the above-mentioned integrals of the equation of motion x = [x, CfJabD(X)] , X E G are in involution with respect to the Kirillov bracket transferred to the Lie algebra G using Q. Let (J be an involutive automorphism of the Lie algebra G, G* the space dual to G, (J* the automorphism of G and G* arising from (J and (J* (see Section 11). Let <X, Y) be a non-singular Aut(G)-invariant symmetric bilinear form on G. Then <X, Y) is invariant, in particular with respect to (J and, using <X, Y) to identify G with G*, Gt; is identified with Go, GT with G1 , where G 1 is the orthogonal complement of Go. LEMMA 22.2 Let G be a Lie algebra, <X, Y) a non-singular symmetric bilinear form on G invariant under Aut(G), (J an involutive automorphism of G, G = Go EB G 1 the decomposition of G arising from

238

A. T. FOMENKO AND V. V. TROFIMOV

let f, 9 be invariants of G, a E G 1 , A, Il E IR; let h,u(x) = f(x + Aa), gll,u(x) = g(x + Ila) be their translates, h,u, gll,u the restrictions of h,u, gll,u to Go, Ga the centralizer of a, Cent Ga its center, Go = Ga n Go, D: Go ---+ Go an operator symmetric with respect to the restriction (X, Y) of (X, Y) to G~, fl(Y),'" ,,h(Y) integrals in involution (under the Kirillov form transferred to Go by (X, Y) ) of the equation of motion Y = [Y,D(Y)], Y EGo. Then we claim that (a) if bEGa n G1 , then functions J(x + Aa), J E J(G), A E IR are integrals of the equations of motion i=[X,IPubD(X)], XEG o; (b) if bEGa n G 1 , the functions J;(x) =J;(Y), where Y is the projection of x onto G~ along [a,G 1 ] are also integrals of x = [x, IPabD(X)], x E Go; (c) all the above-mentioned integrals of the equation x = [x, IPubD(X)]: J(x + Aa) and /;(x) (for bE (Cent(G n Gd) are in involution on Go under the Poisson bracket, which corresponds to the restriction of (X, Y) to Go. CT;

U

)

The proof of these statements is standard and we omit it. We note a further property of the center of the centralizer Ga. LEMMA 22.3 Let G be a real or complex semi-simple Lie algebra, a E G semi-simple element, G the centralizer of a, Cent Ga its center, (X, Y) an invariant non-singular symmetric bilinear form which we use to identify G with G*, fl""',h generators of J(G), the algebra of invariants. Then the differentials dfl (a), ... , d,h(a) generate Cent GU. U

Proof (1) The complex case. Let H be a Cartan subalgebra of G. Then, as is well known, the restriction mapping j: J(G) ---+ S(H) where S(H) is the algebra of polynomial functions on H, is an embedding j(J(G» = S(H) W where S(H) W is the subalgebra of S(H) comprising the elements invariant under the Weyl group W. Let bE Cent G wa the stabilizer of a in W, W b the stabilizer of b in W. We have then wa C W b . Let {aI' a1 , . . . , an} be an orbit of a under the Weyl group. We choose a positive function 9 on H in such a way that dg(a) = band dg(a;) = 0 for ai =I- a. Let g = n LWEWg' w. Then dg(a) = band g E S(H) w, Therefore, f = rl(g) is an invariant ofG such that df(a) = b, Thus we have shown that if fl""',h are generators of the invariants of the algebra G, f = P(fl' ... , j,J for a suitable polynomial p, Therefore, b = L~=1 oP/o/;d/;(a), which was what we had to prove, (2) The real case. Let G be a real Lie algebra. Consider the complexification Gc of G. Then G is a real form of Gc ; let CT be the conjugation on G. Let r = rk G and fl"'" fr be generators of U

,

239

INTEGRABLE SYSTEMS ON LIE ALGEBRA

J(G). Let gl = 11 + 11 a, ... ,gr = fr + fr a, gr+l = (~)-I(fl - 11 0 a), ... , g2r = (j=1)-1(f2 - fr 0 a), where the line denotes complex conjugation. Then 9 l' . . . , 9 2r E J (G c) and all of these are real on G, therefore their restrictions to G, which we also denote by gl" .. , g2" are invariants of G. Let 9 be an invariant of G. Let gc be the complex extension of 9 to Gc . Then 9c is an invariant of Gc and gc = P(fl,'" .J2) for a suitable polynomial P. As 11 = 91 + j=1gr+1,"" fr = 9r + vI-lg 2r' gc and 9 are polynomials in 9 l ' . . . , 9 2r' Let G~ be the centralizer of a in Gc and Cent G~ its center. Then Cent G~ = Cent G" + vI-l Cent Ga and any element bE Cent Ga is a linear combinations of differentials dfI (a), ... , dfr(a) with complex coefficients by dint of (1), therefore, b is a linear combination of differentials dg 1 (a), . .. , dg 2r (a) with real coefficients which was what we had to prove. Let G be a semi-simple Lie algebra over the field k = ~ or C; H a splitting Cartan subalgebra of G; R = R(G, H) a root system of the split Lie algebra (G,H). Forany root a ERlet G a = {x E G: [h,xJ = a(h)x for all the hE H}; the dimension of each of the vector spaces Ga , [G a, G -aJ is equal to one. For any root a E R the space [Ga, G - aJ is contained in H an element H a E [Ga, G - aJ is uniquely defined by the condition a(H,) = 2. We define the real subspace H ~ in H in the following way H~ = ~Ha' Note that in case of k = ~ we have H" = H. algebra

0

0

LER

THEOREM 22.1 Let (G, H) be a split semi-simple Lie algebra over the field k = ~ or C; 0 an orbit of G, intersecting the .set tH ~ where t E k; let a E G be a semi-simple element, Ga its centralizer, b E Ga, Q a nonsingular invariant symmetric bilinear form on G, D: Ga --+ Ga a symmetric operator with respect to Q. Then Euler.s equation of motion

x = [x, lPabD(X)J,

XEO

(1)

has integrals in involution J(x + .!ca), J E J(G), .!c E ~, from which it is possible to choose independent functions on the orbit 0 equal in number to half of its dimension for any general position element a in G. We need, further, the following result, due to B. Kostant (see [26J). LEMMA 22.4 Let G be a semi-simple Lie algebra with rank r, H a splitting Cartan subalgebra, R = R( G, H) the root system, B a basis of R, h an element of H such that a(h) = 2 for any a E B. Suppose h = LaEB a,H a' For any root a E B denote by ba and Ca scalars such that

A. T. FOMENKO AND V. V. TROFIMOV

240

let x,EGa, x_,EG-a, where [x"x_,]=H" U = LOEB b,x" v = L,EB c,x_" S = ku + kh + kv. We claim that (a) [h,u] = 2u, [h,v] = -2v, [u,v] = h, with dimG u = dimG v = r; (b) consider G as an S-module under the adjoint representation. Let G = Al $ ... $ An be some decomposition of this module as a direct sum of simple S-modules of dimensions VI + 1, ... , vn + 1, where VI ~ ••• ~ Vn . Then n = r; (c) let J I' . . . ,Jr be homogeneous algebraically independent generators of the algebra of invariants J(G) of degrees m 1 + 1, ... ,mr + 1, where ml ~ .•. ~ mr • Then Vi = 2m, for any 1 ~ i ~ r; (d) differentials of functions J 1, ... ,Jr are linearly independent at any point in the set u + GV. For the element h of this lemma all the eigenvalues of the endomorphism ad h are even. For an integer n let G n be an eigenspace of adh for the eigenvalue 2n. This subspace is called the n-th diagonal ofthe Lie algebra G (with respect to basis B). We have b,c,=a,

and

(2) Let R+(B) (and, respectively R_(B» be the set of positive (negative) roots in the basis B. Let a E B; the height of the root a in basis B is the number lal = LpEB mp, where mp are integers such that IX = LPEB mp . p. From the definition of the diagonals of the Lie algebra G it follows that for any integer n '1= 0 we have Gn = EB1'I=n G'. For any element x of the Cartan subalgebra H of G and basis B of the root system R we define the following subsets of R: RO(x)

= {IXER: IX(X) = a},

BO(x) R~(x,B)

=

B n RO(x) ,

= R + (B) n

RO(x) ,

R'(x) B'(x)

=

=

R - RO(x),

B n R'(x) ,

R~(x,B)

= R±(B) n

R'(x).

Let C = {x E H R: IX(X) ~ 0 for all a E B}, the closure of positive Weyl chamber, tEk and xEtC. Then any root IX in R~ = R~(x,B) is an integer linear combination of roots in BO = BO(x) and for R'+ = R'+ (x, B) we have ail embedding (R'+

+ B) nRc R'+ .

(3)

LEMMA 22.5 Let (G, H) be a split Lie algebra over k, R the root system, B a basis of R, t E k, C the closure ofthepositive Weyl chamber, x EtC, 0 an orbit in G passing through x, T = T"O the tangent space, Tn = Tn Gn the intersection of T with Gn, the n-th diagonal of Lie

INTEGRABLE SYSTEMS ON LIE ALGEBRA

241

algebra G. Then: (a) T = ffi nEZ Tn; (b) ad x : Tn ~ Tn is an isomorphism; (c) (adaHTn) c Tn + 1 for a = L~EB x~ and elements x~ as in Lemma 22.4.

Proof (a) The equality T = ffinEz Tn follows from the fact that T = [x, G], x EH = GO and from formula (2). (b) adA Tn) c Tn as a consequence of formula (2). As the endomorphism ad x is semi-simple, ad x : T ~ T is an isomorphism. It follows from this that ad x : Tn ~ Tn is also an isomorphism. (c) This follows from formula (3). The lemma is proved. Let J l, . . . , J r be algebraically independent generators of the algebra of invariants leG), m l + 1, ... ,mr + 1 their degrees, ml ::::; ... ::::; m r • The numbers ml , . . . , mr are called the indices of Lie algebra G. Let a be an element of G. We define polynomial functions J{,a (i = 1, ... ,r; j=0, ... ,mi+1): mi+ 1

JJx + Aa)

= L

AiJ{jx).

(4)

i= 0

As J l' . . . ,Jr are invariants

[x

+ Aa,gradJi(x + Aa)]

= O.

(5)

We obtain from (4) and (5) mi+ 1

L

Ai([X, u{] + [a, u( l])

= 0,

(6)

j=O

where u{=gradJl.a(x) (i= 1, ... ,r; j= 1, ... ,m;+ 1), U;-l =0 (1 ::::; i ::::; r). As J~t lex) does not depend on x, U,!,i + 1 = 0 and we obtain, as a result, the following chain of equalities (see [89], [90]):

[x,u?] = 0 [x,u;]

+ [a,u?] = 0 (7)

[x,u,!,']

+ [a,u,!,,-l] = 0

[a, u,!,,] = O. LEMMA 22.6 Let (G, H) be a split semi-simple Lie algebra, R its root system, B a basis for R, G l the first diagonal of the Lie algebra G, x E H, a E Gl ; J l, ... ,Jr homogeneous algebraically independent generators of the algebra of invariants leG) of degrees m1 + 1, ... ,mr + 1, where

242 m1 ~

A. T. FOMENKO AND V. V. TROFIMOV ~ mr

are the indices of the Lie algebra G, J{,a homogeneous polynomials from the decomposition JJx + Aa) = Lj~~1 AiJI.Ax) (i = 1, ... ,r); u{ = grad J{,a(x) their gradients, v" = v,,(x, a) the linear span of the u{ such that i = 1, ... , r,j = 0, ... ,po Then Vp C GO + ... + GP. .•.

Proof Suppose, first, that x

is a regular element of G. In this case the centralizer GX ofthe element x is equal to H. As u? = grad Ji(x) E GX , Vo C GX = H = GO and for p = 0 the lemma is proved. We proceed by induction. Suppose that Vp -1 C GO + ... + GP-1 . As a consequence of Lemma 22.5(c) [v,,-1' a] C G1 + ... + GP, consequence offormulas (7) and Lemma 22.5(b) we obtain

v" c

EH

(Kerad x)

+ (G 1 + ... + GP).

As x is regular, by hypothesis, v" c GO + ... + GP. Thus for regular x the lemma is proved. For arbitrary x E H the lemma follows from the continuous dependence of the gradients grad J{,a(x) on x. The lemma is proved. Note that Dao Chong Thi proves (and then uses) the assertion that v" c GP which is not, in general, true (see [21J). LEMMA 22.7 Under the hypotheses of Lemma 22.6 suppose that a = LaEB xa where for any 0( E B the element Xa #- 0 and Xa E Ga. Let Gp = GO + .. , + GP, Ga be the centralizer of a, G; = Gp n Ga. Then

G; c v".

Proof Note that U'('i = grad J~~(x) = grad J;(a). Let ~ be the linear span of those U i for which m i ~ p. As a consequence of Lemma 22.4(a)(d) we have dim Ga = r and the gradients grad Ji(x) generate Ga, therefore the gradients U'('i = grad Ji(a) (1 ~ i ~ r) are linearly independent. Therefore dim ~/~-1 = m{p) where m(p) is the number of indices mi of the Lie algebra G equal to p. Let A l' ... , Ar be simple modules as in Lemma 22.4, where S = ku + kh + kv and u = a, Al = A1 n GP, ... , Af = Ar n GP. As GP is an eigenspace of the endomorphism ad h , GP = Al EB" . EB A~ and GP n Ga = EB~; dAf n Ga). Note that Af n Ga #- 0 only if mi = P and then dim(Af n Ga) = 1. Hence dim(GP n G") = m(p). Therefore, dim ~ = dim G;. By dint of Lemma 22.6 Wp c G;, whence Wp = G~. Finally, G; = ~ c v". The lemma is

proved. LEMM'A 22.8 Let us assume the notations and hypotheses of the previous Lemmas 22.5, 22.6, 22.7. Then TV c v".

243

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Proof If p is greater than all the indices m1 , ••• , m, then TP = 0 and the lemma is proved. Let r:x be a root of height p and r:x(x) # O. As a consequence of Lemma 22.5(a) for any x~ E Ga we have [x~, a] E TP+ 1. As a consequence of Lemma 22.5(b) there is a U E p+ 1 such that [u,x] + [x"a] =0. Suppose, by induction that p+l c v;,+1' then ,"p+l j j h j I d epend'mg on u. U smg . u = '", L."i = 1 L."j = 1 Ci U i , were Ci are sca ars formulas (7) we obtain [u,x]

+

Ltl :t:

C{utl, a]

= o.

Therefore, x~ - Li = 1 LJ~ l c{ut 1 EGa. As x~ E P and ut 1 E G P for all the i = 1, ... , r andj = 1, ... , p + 1 (by dint of Lemma 22.6) then also x~ - Li = 1 LJ~ l ul- 1 E As a consequence of Lemma 22.7: c Vp. Hence x> E Vp. Therefore TP c v;,. The lemma is proved.

G:.

G:

Proof of Theorem 22.1 Let B be a basis of the root system R = R( G, H), (X')'EB a set of non-zero elements x~ E G>, a = L>EB X" X E tH R nO. As the Weyl group acts transitively on the Weyl chambers, we can assume, without loss of generality, that x EtC where C is the closure of the positive (with respect to B) Weyl chamber. Let J 1, ... , J, be homogeneous algebraically independent generators of the algebra of invariants of G of degrees m 1 + 1, ... ,m2 + 1, JI,uCx) the functions found in the decomposition Ji(x + Aa) = Lj~"ol AjJI,.(x) for i = 1, ... ,r and j = 0, ... , m i ; let V(x, a) be the linear span of their gradients grad JI,.(x) (1 ~ i ~ r, 0 ~j ~ mJ, T = TxO the tangent space; for any integer p: TP = Tn GP, T+ = EBp>o TP. As a consequence of Lemma 22.8 we have T+ c V(x, a). As dim T+ = ! dim 0, it is possible to choose independent functions on the orbits 0 equal in number to half its dimension among the functions J(x + Aa), where J is any invariant of G, A E ~. The assertion about the independence of a sufficient number of these integrals for a general position element a in G follows from the algebraic dependence ofthe functions J(x + Aa) on the parameter a. The rest of the theorem follows from Lemma 22.5(a). The theorem is proved. THEOREM 22.2 Let Gu be a compact semi-simple Lie algebra, Q a nonits singular bilinear symmetric invariant form on Gu , element a E Gu , centralizer, bEG:, D: G~ --+ G~ an operator symmetric with respect to Q, o an orbit in Gu • Then Euler's equations

G:

x=

[x, IPabD(X)] ,

XEO

(8)

244

A. T. FOMENKO AND V. V. TROFIMOV

are Hamiltonian on the orbit 0 under the Kirillov symplectic structure, given by form Q and have the motion integrals in involution J{x + Aa), where J E J(G u ), A E IR. For a general position element a in Gu it is possible to choose among these integrals independent functions, equal in number to half the dimension of the orbit O. Proof Let Gc be the complexification of the Lie algebra Gu' H a Cartan subalgebra of Gc , R = R(G c , H) the root system, (XaLER a Weyl basis for Gc modulo H. As all the compact real forms Gc are isomorphic, we assume, without loss of generality, that

The space Hu = J=1HIl is a Cartan subalgebra of Gu • Therefore, the orbit 0 intersects Hu' Let Oc be the orbit ofInt(Gd, containing O. As 0 intersects H u' Oc intersects H u = J=1H II' According to Lemma 22.1 we can choose 1 dime Oc = 1 dim II 0 independent functions among the complex-value polynomial functions J(x + Aa), where J E J(G), A E C for any element a E Gc from a Zariski-open non-empty subset of Gc. Since manifold 0 x Gu is a real form of the manifold Oe x Gc (more exactly: o x Gu is a connected component of the set of fixed points in Oc x Gc under conjugation, defined by the real form GuofGd it is also possible to choose among the functions J(x + Aa), where J E J(G.), A E ~,! dim II 0 independent on 0 functions for any element a E Gu in a non-empty subset of Gu open in the Zariski topology. The assertions that J(x + Aa) where J E J(G.), A E R are motion integrals of equation (8) and that they are in involution follow from Lemma 22.1. The theorem is proved. Equations (1) and (8) are Hamiltonian with the Hamilton function equal to habD(X) = 1Q{x, qJabD(X)), The quadratic form habD{X) in Theorem 22.1 in the case of the field ~ is neither positive nor negative definite if b =f. 0, and for regular semi-simple b there are always at least ~1(dim G - rg G) "minuses" and as many "pluses." In Theorem 22.2 the form habD(X) may be positive definite. REMARK

22.2. Integrability of Euler's equations

x=

[x, fPabD(X)] for singular a

Let G be any Lie algebra, Q a non-singular invariant symmetric bilinear form on G. The integer rk G equal to the codimension in G of an orbit 0

INTEGRABLE SYSTEMS ON LIE ALGEBRA

245

in G of maximal dimension is called the rank of the Lie algebra G. An involutive set of functions (relative to Q) on G is called complete if it contains independent functions, equal in number to -t(dim G + rk G). THEOREM 22.3 Let G be a compact semi-simple Lie algebra, Q a nonsingular invariant symmetric bilinear form on G; a E G, Ga the centralizer of a in G, Cent Ga its center; D: Ga ~ Ga an operator symmetric with respect to Q, f1 (y), ... ,J,.(y) an involutive (under the restriction of Q to Ga) set of motion integrals of the equation

y=

a) For any bE Ga the functions J(x motion integrals of Euler's equation

x=

(9)

[y, D(y)] ,

[x, lP.bD(X)] ,

+ Aa), where J E J(G), A E IR are xEG.

(10)

b) For any b ECent Ga , 1 :::::; i :::::; k the functions g;(x) = .t;(y), where y is the projection of x onto Ga along [a, G] are motion integrals of the equations (10). c) The above-mentioned integrals of the equations of motion (10) form an involutive set (in relation to Q) and, d) this set is complete, if the set f1' ... ,J,. is complete. Proof The assertions (a), (b) and (c) follow from the statements (a), (b) and (c) of Lemma 22.1. In order to prove statement (d) let us consider the orbit 0 passing through the element a. Let T = T. 0 be the tangent space to the orbit. As T = [a, G], we have a Q-orthogonal splitting G = T EB Ga (because of Proposition 22.1).

LEMMA 22.9 (see [89], [90]) Let x E G; let V(x, a) be the linear span of gradients grad J(x + lea), where J is any invariant of G, A E IR an arbitrary number. We have then V(x, a) = V(a, x). Proof As the algebra of invariants J(G) is generated by homogeneous invariants, we may assume that J is a homogeneous invariant of degree n. We have

where for the function cjJ(x, a) in two variables x and a (x, a E G), grad x cjJ(x, a) (respectively grad.
246

A. T. FOMENKO AND V. V. TROFIMOV

4J(x, a) viewed as a function of x (respectively a) alone. Hence Vex, a) = Yea, x). The lemma is proved.

As a consequence of Theorem 22.2 (the role of element a in Theorem 22.2 being played here by the element x) for a general position element x E G the projection VT(a, x) of the space Yea, x) onto T along Ga has dimension dim VT(a, x) = ! dim T. By Lemma 22.9 yea, x) = vex, a), therefore, for a general position point x E G in G it is possible to select -t dim T independent functions on the set x + T among the functions J).,a(x) = J(x + A.a), J E leG), A. E IR. Let these be the functions gk + l' ... , gk +s' where s = ! dim T. As the gradients of functions gl' ... , gk at any point G belong to GU and Ga is Q-orthogonal to T, the functions gl"" ,gk+s are functionally independent on G. Let us compute their total number k + s. As fl, ... , h. are independent on Ga we have, according to the definition of a completely involutive set, k = !(dim Ga + rk Ga ), As any Cartan subalgebra H, containing the element a is a Cartan subalgebra of Ga, rk Ga = dim H = rk G. Therefore k + s = -t(dim Ga + rk Ga) + 1 dim T = 1(dim Ga + rk G) + 1 dim T = 1(dim Ga + dim T + rk G) = 1(dim G + rk G). We have therefore proved that gl, ... , gk +s is a completely involutive set on G. The theorem is proved. COROLLARY 1 Let G be a reductive Lie algebra, G = Cent(G) EEl S, where Cent(G) is the center of G, S = [G,G] a semi-simple ideal. Suppose that the ideal S is a compact semi-simple Lie algebra. Then there is a completely involutive set of polynomial functions on G. Proof Let Q be an invariant form on G, Xl'" . , Xk a basis for Cent(G), .fl,'" ,j~ linear forms on G, .h(Y) = Q(xi,y) (i = 1, ... ,k). Let

be a completely involutive set on S, s = -t(rk S + dim S). As rk G = rk S + k and dim G = dim S + k, then !(rkG+dimG)=k+s. Therefore .fl""'h.+s is a completely involutive set on G. h.+l""'h.+s

INTEGRABLE SYSTEMS ON LIE ALGEBRA

247

22.3. Integrability of Euler's equations x = [x, CP.bD(X)] on the subalgebra Gn fixed under the canonical involutive automorphism (J: G ...... G for singular elements a E G

Let G be a complex semi-simple Lie algebra, H a Cartan subalgebra in G; let R be the root system; G",H"H'i/, as in 22.1. Let (X')'ER be a Weyl basis for G modulo H. We shall define Go

= H'i/, EB (EB

2ER

u

G = -i=lH'iI.

EB (

EB

lR(x,

~ER!

+

IRX,) ,

L,)) EB (EB ~-i=l(x, ~ER+

- x -a)),

where R+ is a set of positive roots in some basis B of the system R, Gn = Go n G u , V = Go n -i=lG u • We have Go = G. EEl V the Cartan decomposition. Let a E V. Then [a, Gn ] C V, [a, V] c Gn and for the centralizer Go of a in Go we have Go = G~ EB va, where G: = Go n G., v" = V n Go. Whence we obtain the following decompositions Go =

G~

EB [G n , a] EB v· EB [V, a] ,

G. = [a, V] EEl G~.

(11)

Let Q be a non-singular symmetric bilinear form on Go, invariant under all automorphisms of Go. Then Vis the Q-orthogonal complement of G. in Go and Go is the Q-orthogonal complement of [a, Go] in Go (Proposition 22.1). Whence it follows that all the subspaces involved in expansion (11) are mutually Q-orthogonal and the restriction of the form Q to them is non-singular. In the following theorem the restriction of the form Q to G n and G: is used to determine the involutivity of the functions on G. and G~. THEOREM 22.4 Let Go be the above-mentioned real semi-simple Lie algebra, H'i/, a Cartan subalgebra, Q an invariant form on Go, a E H'i/" Go the centralizer of a in Go, Cent(Go) its center, bE Cent(G'O) n V, D: G~ -+ C~ an operator symmetric with respect to Q, f1 (Y), ... ,j~(Y) a completely involutive set of independent motion integrals of Euler.s equation

y=

[y, D(y)] ,

Then Euler's equations on Gn

YEG~.

(12)

248

A. T. FOMENKO AND V. V. TROFIMOV

x=

[X, <"PabD(X)] ,

XEG"

(13)

have motio n integrals J(x + Aa), where J E leG), A E IR and motio n integ rals gi(X) = .fly) where Y is the proje ction of x onto G~ along [a, V). These integrals comp rise a completely involutive set. LEMMA 22.10 Let B be a basis of the root system R = R(G , o H ~), aEH~; X±= L2EB X±" x=x + +x_ ; y;= x +(-1 )1,l x _, for any 2 root E R of heigh t For any integ er p ~ 0 we define F~=E8121=plRy;, F~=O, F~=H~, F:=F~EB···EBF~. Let J l ' . . . ,J, be homo geneo us algebraically indep enden t gener ators of the algeb ra of invar iants of Go of degrees m + 1, ... ,my + 1, m :::; •.. 1 1 :::; m 2; functions J{,a (1:::; i :::; rand 0:::; j :::; m;) are defin ed by the expan sion Ji(a + h) = L;;" 1 AjJ{j a) for any integ er p ~ 0; let ~ = ~(a, x) be the linea r span of the gradi ents grad JUa) such that 1 :::; i :::; rand 0 :::; j :::; p. Then Vp C F; .

a

lal.

Proo f Supp ose, first, that a E H ~ is a regul ar eleme nt of the Lie algeb ra Go. For such a we have Vo c Go = H ~ = F O. We proce ed by induc tion. Supp ose that Vp - 1 c F;-l ' As x E F~ it follows that [x, V - ] c As p 1 a E H ~ = F~, we obtai n Vp C F; using formu las (7) from 22.1. For regul ar a the lemm a is proved. For singu lar elements a E H ~ the lemm a follows from the conti nuou s depen dence of gradi ents grad J{,x(a) on a. The lemm a is prove d.

F;.

LEMMA 22.11 Let (G,H ) be a split Lie algeb ra, R = R(G, H) the root system, B a basis of R, for any integ er n let G" be the n-th diago nal of G. Let Y E G, y = ~ co Yi' where Yi E Gi is called the diago nal comp onen t of Y of degre e i. If Yi :f. 0 and Yj = 0 for allj > i (respectively < i), then Yi is called the maxi mal (respectively minim al) diago nal comp onen t of y and the numb er i the maxi mal (respectively minimal) diago nal degree of y. We suppo se furth er that all the hypo these s of Lemm a 22.10 are satisf ied. Let D~ = grad Jf x(a) - grad Jf x (a). Then eithe r the maxi mal ± (respectively minimal) diago nal degree of D~ (respectively D~) is no more (respectively no less) than p - 1 (respectively -(p - 1» or D p± -- 0 .

L

-

•

I

Proo f Let us prove the lemm a for D~. The proof for D~ is the same as for D~ after chang ing at appro priate places ( + ) for ( - ) and vice versa . By formu las (7) from 22.1 we have [gradJ~+l(a),a]

+

[gradJ~(a),x]

=0,

INTEGRABLE SYSTEMS ON LIE ALGEBRA [gradJ~:l(a),a]

+ [gradJ~t(a),x+]

249

=0.

Subtracting one equation from the other we obtain [D~+l,a] = -[D~,x+] - [gradJ~(a),L].

(14)

Suppose, by induction, that the maximal diagonal degree of D~ does not exceed p - 1 (for p = 0: D~ = 0 and the lemma is proved). As a consequence of Lemma 22.10 the maximal diagonal degree of grad J~(a) does not exceed p, therefore the maximal diagonal degree of [grad J~(a), L] does not exceed p - 1. By induction the maximal diagonal degree of D~ does not exceed p - 1, the maximal diagonal degree of [D~, x +] does not exceed p. Thus, the right-hand side of the equation (14) consists of two terms, each of maximal diagonal degree not exceeding p. Therefore, this is also true of the left-hand side of (14) which is [D~+l ,a]. Suppose that a EHIR is a regular element of Go, then Ker ada = Hiland ada (Gi) = Gi (for j -:f. 0), therefore, D~+ 1 has maximal diagonal degree no greater than p. By continuity, we deduce that the same is true for any a E H IR. By induction, we see that the lemma is true for any p ~ o. LEMMA 22.12 We keep the notations of Lemmas 22.10 and 22.11. Let T = [a, bo], for any integer p ~ 0 let V/ be the projection of Vp = Vp(a, x) onto T along Go. We have then Tn F; = V/. Proof For p = 0 we have Tn F; = 0 and ~T = 0 and the lemma is proved. Suppose, inductively, that F;-l n T = V/- 1 • By Lemma 22.8 Tn GP c Vp(a, x +). Therefore, the maximal diagonal components (of degree p) ofthe gradients grad Jf.xJa), where 1 ::::; i ::::; r generate T n GP. By Lemma 22.11 the same is also true of the gradients Jf.Aa), where 1 ::::; i ::::; r. Therefore, for any root ex of height lexl = p with ex(a) -:f. 0 we have x, = C~gradJf.Aa) + M, where M, has maximal diagonal degree no greater than p - 1. As a consequence of Lemma 22.10 Jf.Aa) E F;, therefore the minimal diagonal degree of the sum C~gradJf.x(a) is equal to (-p) and the minimal diagonal component of this sum (of degree (- p)) is equal to - ( -1)px _,. Therefore the minimal diagonal degree of M a is also equal to - p and the minimal diagonal component of M, (of degree - p) is equal to (-1)Px_ a • Recall that y,- = x, - (-l)px_,. It follows from the above that y,- = = 1 C~ grad Jf.x(a) + N a where N, E F;_ 1 . Projecting all the components of this equation onto T along Go, we obtain y; = C~gradT Jf.x(a) + N;, where grad T Jf.x{a) (respectively N;) is the

Lr=l

D=l

Li

Lr=l

250

A. T. FOMENKO AND V. V. TROFIMOV

projection of grad Jf,Aa) (respectively of N a) on Talong G~. As F;-I = (F;-I nT)EB(F;-I nG and NaEF;-I' N;EF;_I nT. By induction Tn F;-I = ~-r:.I(a,x), therefore N~ E ~~I(a,x) and Y; E V/(a, x). As we supposed at the beginning that IX was any root of height p such that lX(a) #- 0, it follows that Tn F; t C ~T(a, x). As a consequence of Lemma 22.10 l';.(a, x) c F;t therefore Tn F; = ~T(a, x). By induction the lemma is proved for any integer p ~ O.

o)

LEMMA 22.13 Let Go and Gn be Lie algebras as in Theorem 22.4, r = rk Go, rk Gn be the rank of Gn; let m t , ... ,mr be the indices ofthe Lie algebra Go. Then the rank of Gn is equal to the number of odd indices among m t , ... ,mr •

Proof It is enough to prove the lemma for simple Lie algebras Go only. Suppose, first, that Go is a simple Lie algebra with the root system of type AI' Br (r ~ 2), C r (r ;;::: 2), Dr (r is even and r ~ 4), E 7 , E 8 , F 4 or G 2 . Then all the indices m t , ... , mr are odd (see [11]). Therefore an automorphism of HR equal to (-1) belongs to the Weyl group [11]. Therefore the canonical automorphism (J: Go ~ Go equal to ( -1) on H R and mapping Xa into x -a for any root IX is an inner automorphism. As Gn coincides with the set of fixed points of (J, it follows that rk Gn = rk Go = r. As all the indices mI , ... ,m. are odd, for these Lie algebras Go the lemma is proved. Let us consider the remaining simple Lie algebras Go case by case. The series of roots of A. (r ;;::: 2). Indices: 1,2, ... ,r; Gn = so(r + 1). For even r the number of odd indices is equal to r/2 and rk so(r + 1) = r/2. For odd r the number of odd ipdices is equal to (r + 1)/2 and rk so(r + 1) = ~(r + 1). The series of roots of Dr (r is odd, r;;::: 3). Indices: 1,3,5, ... ,2r - 3 and r - 1. The number of odd indices is equal to r - 1, Gn = so(r) EB so(r) and rk Gn = r - 1. The series of roots E 6 • Indices: 1,4,5, 7,8, 11; Gn = sp(4) number of odd indices is equal to 4 and rk Gn = 4. The lemma is proved. The following lemma supersedes the argument in [90]. LEMMA 22.14 Under the hypotheses of Theorem 22.4, assume (Go, H R) is a real split semi-simple Lie algebra. Let R = R(G o , H R) be the root system, B its basis, R+ the system of positive roots, ROdd the set of

INTEGRABLE SYSTEMS ON LIE ALGEBRA

251

roots of odd height, R~d = R + n Rodd. For any finite set M let Card M be the number of its elements. Then !(rk G. Proof We have G.

+ dim Gn) =

= EElaER+

~(xa

Card(R~d).

+ x-a)

and dimG. = CardR+.

Therefore it is enough to prove that rk G. = 2 Card(R~d) - dim Gn

= 2 Card(R~d) - Card(R +) = =

Card(R~dd)

-

Card(R"~en)

L m(2i + 1), i~O

where Re;en is the set of roots of even height and m(2i - 1) the number of indices of the Lie algebra Go equal to 2i + 1. This is exactly what was proved in Lemma 22.13. The lemma is proved. LEMMA 22.15 Again under the hypotheses of Theorem 22.4 we remind the reader that a is an element of H II , Go its centralizer, G~ = Gn n Go. Let R = R(G o, H II ) be the root system of the split Lie algebra (Go, H II) and B a basis such that for any root a E B we have a(a) ~ O. Then t(rk G:

+ dim G:) = Card(RO n

R~d),

where RO is the set of roots a E R, equal to zero on a and R~d the set of roots a E R positive (with respect to B) and of odd height. Proof As the element a is semi-simple in Go (see [26]) Go is a reductive

Lie algebra. Let Cent Go be its center, S = [Go, Go] is a semi-simple ideal in G~. Then G~ = (Cent G~) EE> S. We have HI! C G~ and the normalizer of H II in Go coincides with H II' Therefore Cent Go c Hiland Hs = HII n S is a split Cartan subalgebra in S. Let R = R(S,Hs) be the root system of the split Lie algebra (S, H s), R' the set of roots a E R not equal to zero on a. We have G~=HI!EE>(EElaERoGa) and S = Hs EE> (EElaERo Ga). Hence it follows that associating with each root aERo its restriction to Hs we obtain a one-to-one mapping ofthe set RO onto R. Let BO = B n RO = (a?, ... ,a~), aERo. Then for suitable integers nl" .. , nk we have a = L~= 1 nja? Let Y E Rand y = a/H s. Then y = L~ = 1 n;{3 j where /3i = a? /H s for every 1 ~ i ~ k. Therefore jj = (/31' ... , /3,,) is a basis of R and the height of root aERo with respect

252

A. T. FOMENKO AND V. V. TROFIMOV

to B is equal to the height of the root y = C1./H s E R with respect to B. Let Sn = S II Gn. By Lemma 22.14 -t(rk Sn + dim Sn) = Card R~d, where R~d is the set of roots PER of odd height and positive with respect to B. It follows from these considerations that Card(R~d) = Card(RO II R~d). Note that Sn = EBaER~ ~(xa + La) = G~, where R~ is the set of positive roots C1. in RO. Therefore -t(rk G~ + dim G~) = Card(RO II R~d). The lemma is proved. Proof of Theorem 22.4 The functions specified in the theorem are integrals in involution for equation (13) as a consequence of Lemma 22.2. Let V(x, a) be the linear span of the gradients grad J(x + Aa) where J E /(G o), ). E R and the functions J(x + Aa) are viewed as functions of the variable x E Go, let V(x, a) be the linear span of the gradients gradn J(x + Aa) where J E/(G o), A E lR and the functions J(x + All) are viewed as functions of the variable x E Gn- Then for any x E G.: V(x, a) is the projection of V(x, a) onto G. along V. As a consequence of Lemma 22.9 V(x, a) = V(a, x). Let VT(a, x) be the projection of V(a, x) onto T= [a, Go] along G~ and VT(a,x) = JIT(x,a) the projection of V(x,a) = V(a, x) onto Til Gn = [a, V] along G~, F- = F;, where the space F; is defined in Lemma 22.10 and p is greater than all the indices of the Lie algebra Go. As a consequence of Lemma 22.12 we have VT(a, x) = Til F- for x = L~EB (xa + x-a), given the basis B such that for any C1. E B the value C1.(a);;::: O. Therefore, VT(x, a) = VT(a, x) = Til FOdd, where FOdd = LO:ER,:dd lRY2 (for the definition of Ya see Lemma 22.10, here R".;'d is the set of positive roots of odd height). We have, further, dim pdd = Card(R' II R".;'d) where R' is the set of roots C1. E R such that C1.(a) -# O. As (fl' ... ,j,,) is a completely involutive set on G~, then k = -t(rk G~ + dim G~). Since grad gi(X) E G~ (1 ~ i ~ k) we have

dim( V(x, a) EB

C~ ~ grad 9i(X»)) + -t(rk G~ + dim G~) = Card(R' II R~d) + Card(RO II R~dd) =

Card(R'

II R~d)

= Card(R~d) = -t(dim Rn + rk G.). Here we use Lemmas 22.14, 22.15. The set of integrals, therefore, is complete. The theorem is proved.

253

INTEGRABLE SYSTEMS ON LIE ALGEBRA 22.4. Integrability of Euler's equations for an n-dimensional rigid body

Consider an n-dimensional rigid body with fixed point outside the zone of any forces. Let Xl' . .. , X" be the coordinates in a rigid system of coordinates connected with it rigidly; let 7;j denote the elementary n x n matrix with zeros everywhere except for a one at the intersection of the jth row andj-th column; let Eij denote the elementary skew-symmetric matrix E'j = 7;j - 1j,. Let us suppose that a rigid body revolves with an angular velocity 0 in a moving system of coordinates which has a corresponding elementary skew-symmetric matrix E'j. In this case the p(x)(x; + xJ)d"x, kinetic energy T is given by the formula T = 1 where p(x) is the density function of the distribution of the mass in the body, and d"x is the n-dimensional volume element. We introduce the constants a, = S· .. Sp(x)x;dnx (1 ::::; i ::::; n) which depend only on the distribution of mass in the body. Then the kinetic energy of our rigid body revolving with an arbitrary velocity 0 = L'<j OJijE,j equals 1 OJ~(a, + aj ). Hence we find an expression for the kinetic moment M = A(O), A(O) = 10 + 01, where I = diag(a 1 , ... , an). The Euler equations have the following form, M = [M, OJ, where M = A(O). The matrix I is called the inertia tensor of the rigid body.

J... J

L

THEOREM 22.5 Let Q be an invariant symmetric bilinear form on so(n) with respect to which the basis (Eij)l ",<j"n has been made orthonormal, let I = diag(a 1 , •.• , an) be the inertia tensor of the rigid body, and M = A(O) be the kinetic moment of the body in a moving system of coordinates where A(O) = 10 + 01, a = 12, b = I, so(nt is the centralizer of a in so(n). Then, a) for an appropriate operator symmetric with respect to Q, D: so(n)a ...... so(n)a we have A-I (M) = CfJabD(M) b) the functions tr(M + ).J2)k (k = 2, ... ,n, A E IR) are the integrals of motion of the Euler equation M = [M, A-I (M)] c) the set of these functions can be extended into a complete involutive (with respect to Q) set of integrals of motion of the equation M = [M, A-I (M)] by including polynomial functions which depend only on the Q-orthogonal projection of M onto so(nt. Proof Without loss of generality, we assume that a 1 ~ ~ an > O. Let Y1, ... , Y. be numbers such that

a2

~

...

254

A. T. FOMENKO AND V. V. TROFIMOV

a1 = ... = aq, = Yl > aq, +1 = ... = aq, = Y2 > ... >aqs-l +1 =···=aq, =y s'

Yl > ... > Ys'

Then so(n)" = so(pd EEl' .. EEl sO(Ps) where PI = ql' P2 = q2 - Q1' ... , P. = Q. - q. -1' Let the operator D: so(n)a --+ so(n)a be multiplication by I'ion SO(PI) (i = 1, ... ,s). Then it is easily verified that A -1 (n) = Cf'abD(n). Let G = sl(n, ~), Gn = so(n) in Theorem 22.4. The functions tr Mk (2 ~ k ~ n) are algebraically independent homogeneous generators of the algebra of invariants of sl(n, ~). Thus, if we give a completely involutive set of integrals of the equation

Y=

[Y, D(Y)] ,

YEso(nY,

(15)

Theorem 22.5 is proved as a corollary of Theorem 22.4. Since [Y, D(Y)] = 0 for any Y E so(n)", any set of functions on so(n)a is a set of integrals of motion of equation (15). Therefore we can take any completely involutive set on so(n)a as a completely involutive set of integrals of motion of equation (15). Such sets exist, in view of Corollary 1 of Lemma 22.9. The theorem is proved.

23. COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS ON SYMMETRIC SPACFS

The first part of this section gives the constructions of some completely integrable geodesic flows on symmetric spaces, flows which are generated by sectional operators Cf'ab (see Sections 6 and 8). The second part is devoted to a generalization of non-commutative complete integrability (see Sections 4 and 3). All the results set out here are the work of A. V. Brailov (see [198], [199]). 23.1. Integrable metrics dS;bD on symmetric spaces

Let the Lie group ij) with the Lie algebra G act smoothly on a manifold M. In this case to each element 9 E G corresponds vector field g on M. The action of ij) on M induces a symplectic action of ij) on the cotangent bundle T* M of M. This symplectic action corresponds a moment mapping P: T*M --+ G* (see [1]), which in the given special case is defined as follows:

INTEGRABLE SYSTEMS ON LIE ALGEBRA

255

= <x,g(m» where x E T':(M) and <X, Y> on the left-hand side denotes the pairing of G* and G, while on the right-hand side it denotes the pairing of T': M and TmM. As was noted in [88] every moment mapping P has the following property which is very important to us: for any two smooth functions I and 9 on G* we have the equality

{foP,goP}

= {J,g}oP

(1)

where the braces {X, Y} on the left-hand side of the equation denote the standard Poisson bracket on T":M and those on the right-hand side denote the standard Poisson bracket on G* corresponding to the Kirillov form on the orbits of the coadjoint representation of (fj in G*. Lete there be on G a non-degenerate invariant symmetric bilinear form Q. By identifying G* with G with the help of Q we shall examine the moment mapping

Q(P(x), g) = <x, g(m» where 9 E G, X E T': M. It is clear that for the moment mapping P = P Q the property (1) also holds if we use the Poisson bracket {X, Y} corresponding to the form Q, on the right-hand side of the equation. Given the fixed point mE M the moment mapping P is linear along x E T': M. Therefore, ifthe operator l.{JabD has been defined on G (a and b may be elements of the Lie algebra which contains G), then HabD(X) = 1Q(P(x), l.{JabD(P(X))) is a quadratic function on T*M. If a quadratic function HabD(X) is a positive definite non-degenerate quadratic form, then it induces a corresponding Riemannian metric dS;bD on M (this correspondence is defined in the general case and is called the Legendre transformation (see for example [1])). The metrics dS;bD are called the metrics of the moment mapping P. THEOREM 23.1 Let M be a globally symmetric Riemannian manifold, rk M its rank, (fj the connected component of the unit in the isometries of M, G its Lie algebra, where Q is an Aut(G)-invariant symmetric nondegenerate bilinear form on G. We shall assume that (fj is a compact semi-simple Lie group. a) If rk M = rk G, l.{JabD' is an arbitrary positive operator on G (a and b may be elements of a larger Lie algebra), and if 11 (x), ... ,f,.(x) is the completely involutive set of independent integrals of Euler's equation

256

A. T. FOMENKO AND V. V. TROFIMOV

x = [X, IPabD(X)]

(X E G), k = ~(rk G + dim G) then k = dim M and the functions j~ (P(x)), ... ,J;.(P(x)) are independent integrals in involution of the geodesic flow on T* M of the Riemannian metric dS;bD of the moment mapping P: T*M -> G

b) If rk M < rk G, a, bEG, IPabD is a positive operator, then the geodesic flow on T* M of the Riemannian metric dS;bD of the moment mapping PQ : T*M -> G has an involutive set of integrals of motion J(P(x) + Aa), J EI(G), A E ~ and for an element a E G in general position in G we may choose from this set functions that are independent on T*M equal in number to the dimension of M. Proof (a) We fix any point mE M. An isometry of M which leaves point m fixed and which moves the geodesics passing through m, defines an involutive automorphism of (I) (see [48]). This automorphism of (\j defines an automorphism of the Lie algebra G, which we denote by a. Let G = H EB V be the decomposition of G such that the automorphism a equals Ion Hand ( - 1) on V. For the point m we have P(T':(M)) = V. Since rk M = rk G, we can choose a Cartan subalgebra K in G so that K c V. In view of the compactness of the Lie algebra G it follows that every orbit of G> in G intersects V. As is well known, the action of G> on T* M under the moment mapping corresponds to the coadjoint action on G (see for example [1]); therefore G = P(T*M). Consequently, the functions fl (P(x)), ... ,J;.(P(x)) are independent on T* M. Their involutivity is implied by formula (1). Thus all that remains to be proved is that dim M = ~(dim G + rk G). Let a be an arbitrary element of V, where Ga is the centralizer of a in G, H a = H n Ga, va = V n Ga. Since [H,H]cH, [V,V]cH, [H,V]cVand aEV, Ga=Ha+va. We define on G a skew-symmetric form La(X, Y) = Q(a, [X, Y]). Since Q is invariant with respect to Aut(G), it is also invariant, in particular, with respect to a. Therefore H and V are Q-orthogonal complements of each other in G. It follows from this that H ~ = V + Ha, V ~ = H + va where H~ and V~ are skew-orthogonal complements to H and V in G with respect to La' Consequently the quotient La on GIG" = HIH a EB VI V" is non-degenerate and realizes a pairing of the spaces HIHa and Viva. Consequently, dim HIH a = dim Viva. Let K be a Cartan subalgebra in G such that K c V and a E K is an element regular in G. Then Ga = va = K, H a = O. We have:

257

INTEGRABLE SYSTEMS ON LIE ALGEBRA

!(dim G + rk G) = l(dim G + dim Ga)

= l(dim H + dim V + dim va) = l(dim

H - dim H a + dim V - dim va) + dim va

= dim V I va + dim

va

= dim V = dim M .

b) Let K be a Cartan subalgebra in G such that K v = K n V is the maximal commutative subspace in V. In this case dim Kv = rk M. For an element a e Kv in general position in K v we have va = K v , therefore for the orbit 0 of the group (fi in G which passes through point a, we have

1- dim 0 = 1- dim GIGa = !(dim HIH a + dim

VIVa)

= dim Viva.

It follows from Theorem 22.1 that the set of functions (l(x + Aa», 1 e /(G), AE IR contains! dim 0 functions independent on O. Since the invariants of G are constant on 0 but are not constant on K v, then by using Lemma 22.3 on the image of the mapping P, i.e. on P(T*M), we shall get ! dim 0 + dim K v independent functions in the set 1 A,a' le/(G), AEIR. Consequently, there are at least !dimO+dimK v functions independent on T*M among the functions l(P(x) + Aa). Further:

1- dim 0

+ dim K v

= dim VI va + dim

va

= dim V = dim M.

The functions l.l.,a(x) are integrals of motion in involution of the Euler equation x = [x, <PabD(X)] , which is Hamiltonian on the orbits of (ij in G with the Hamiltonian habD(X) = 1Q(x, <PabD(X», In view of formula (1) we find that the functions l(P(x) + Aa) are integrals of motion in involution of the Hamiltonian system on T* M with the Hamiltonian HabD(X) = habD(P(X». Since the Hamiltonian HabD is a Hamiltonian of the geodesic flow on T* M of the metric dS;bD' and since the set of integrals of motion l(P(x) + Aa) contains functions independent on T* M equal in number to the dimension of M, the theorem is proved.

23.2. The metrics dS;h on a sphere S"

Let S" = {y E IRn + 1 : yi + ... + y; + 1 = I} be a sphere; let SO(n + 1) be the special orthogonal Lie group consisting of (n + 1) x (n + I) matrices g such that ggt = E and det g = 1 where gt denotes the transposed matrix, E = diag(I, ... , 1); let so(n + l)betheLiealgebraof

258

A. T. FOMENKO AND V. V. TROFIMOV

the group SO(n + 1), consisting of (n + 1) x (n + 1) matrices x, such that x = _Xl. In the standard way the group SO(n + 1) acts on the sphere sn: the element 9 E SO(n + 1) sends the point Y E sn to the point (9IY, where 91 is the matrix product of matrix 9 and of the matrix 1 which is the transpose of the row matrix Y = (Yl' ... , Yn + d. THEoREM23.2 Let a,bEgI(n+l,IR), a=diag(al, ... ,a n +d, b=diag(b 1 ,· .. ,bn + 1 ), a 1 > .. ·>an + 1 >0, b1 > .. ·>bn + 1 ; where Q(X, Y) = -1 tr(XY) is the invariant symmetric non-degenerate bilinear form on glen + 1, IR), positive definite on so(n + 1) c glen + 1); let ({Jab = ad a- 1 ad b be symmetric with respect to Q; the group SO(n + 1) acts in a standard way on the sphere sn = {yf + ... + + 1 = I}, P = PQ : T*sn --+ so(n + 1) is the corresponding moment mapping; dS;b is the Riemannian metric on sn that corresponds (under the Legendre transofrmation) to the quadratic Hamiltonian H ab , where Hab(q) = 1Q(P(q), ((Jab(P(q))) for q E T*sn. Then:

Y;

a) the geodesic flow on T*sn of the Riemannian metric dS;b has n independent quadratic integrals in involution H l ' . . . , Hn, where Hk(q) = 1Q(P(q), ((Ja,a.(P(q))), ak = diag(a~, ... , a~ + d and q E T*sn; - b = diag(a 1 1 , . . . , a;+\), then with the substitution Yi = xd (i = 1, ... , n + 1) the metric dS;b becomes a metric b) if

fi.

[2a 2+ ... + anx 2+ J-1 (dX1 + ... + dx Xl

n+ 1 -2-

1

1

2

2 n + 1)

that is conformally equivalent to the standard metric dxi of the ellipsoid

+ ... + dx; + 1

Proof (a) Let c = diag(c1' ... , Cn+ d, hac(X) = !Q(x, ((Jac(X)) for X E glen + 1, IR). As follows from the results of [89], the quadratic function hac(X) is a linear combination of the functions J(X + A.a) where J is the invariant of glen + 1, IR}, A. E IR. As a corollary of Lemma 11.3, the

259

INTEGRABLE SYSTEMS ON LIE ALGEBRA

quadratic functions hac> where hac is the restriction of hac to so(n + 1) c gl(n + 1, 1R1) are pairwise in involution for any diagonal matrix C E gl(n + 1, 1R1). As an involutive automorphism a, whose set of fixed points is so(n + 1), we may take the automorphism a(X) = - Xr. As a corollary ofthe fundamental property ofthe moment mapping (formula (1) from 23.1), the functions H 1" .. , Hn are pairwise in involution on T*sn. In order to prove the independence of these functions we shall need to have their explicit calculation in local coordinates. Let sn+ be the hemisphere given by the inequality Yn +1 > 0. The functions Y1' ... ,Y. are local coordinates on S"t, while Yn +1 = yf + ... + Let Z1' ... ,Zn be the corresponding impulse variables on T*S"+. Then (Z1"" ,zm Yl"" ,Yn) is a system of coordinates on T*sn+ and the standard symplectic structure is ill = I7=1 dZi 1\ dYi.The matrix X = Ilxijll Eso(n + 1) has a corresponding vector field on S" which in the local coordinates Y1' ... ,Yn equals

J

n-1"

(a

a)

y;.

a

n

.I . ~ Xij Yi~ - YFa-. +.I Xi.n+lYn+l~· ,=1 )=,+1 Y y, ,=1 y, Thus, P(z,y) = IIPij(z,y)ll, where Pi}z,y) = ZiYj - ZjYi provided that J

i, j ~ n, and Pi,n + 1 (z, y)

= Zi Yn + 1 provided that 1 ~ i ~ n. Hence we

find that

~ Ci L... + i=1 a Let Yo = (0, ... ,0, 1). Then

From this we find the Jacobian

-

cn + 1

i -

an + 1

2 2

]

Zi Yn+1 .

260

A. T. FOMENKO AND V. V. TROFIMOV

=~(l

•...• I)

1 at

1

+ an + 1

ai + a 1a,,+1 al"+ a n-l 1 a,,+1

a Z +a,,+1

z

+a;+1

az

+ ... +" all + 1

a"2

+

2 + aZa,,+1 + a,,+1

n 1 a 2 - a ,,+1

+ ... + a"11+1 1

a" + an + 1 2

+ a" a" + 1 + an2 + 1 an"+ an,,-1 a" + 1 + . . . +" all + 1 all

To calculate the determinant of this matrix we make the following (n - 1) elementary transformations. The first transforml'tion consists of subtracting the (n - l)-th row, multiplied by all + l ' from the n-th row. The second transformation consists of suh~racting the (n - 2)-th row, multiplied by a" + 1> from the (n - l)-th row, and so on. The last transformation consists of subtracting the first row, multiplied by an + 1, from the second row. We carry out all these transformations in the order shown. The result is the matrix: 1 1 1 a1

az

a"

a"1

a"2

a" "

................

whose determinant, the well-known Vandermond determinant, is nonzero, in view of the fact that a 1 > ... > an' Thus, the involutivity and independence of the functions HI"'" HII has been proved. These functions are integrals of the geodesic flow of the metric d:);h' since the Hamiltonian Hah which corresponds to this metric is a linear combination of HI' ... ,H" for any diagonal matrix b. b) Let -b=diag(al1, ... ,a,~)d. Then I

H ah (z'Y)=-2

[,,-1L

/I

a:- 1

_

L ) _

i~1 j~i+l

ai

a.- 1 I

aj

(ZiYj-ZjYi)Z

261

INTEGRABLE SYSTEMS ON LIE ALGEBRA

-1

2

-1 2 ]

+ an+1Yn+l i~l ai II

Zi

_ 1[n -1 2 ) 2 -I 2 n -I z·a· "a.I-1 z·~2 (" "a.) y. - " y.I 2 ,l..J ,i..J ) .~ a·I I I 1=1

J=I n

1=1

n

n

- I L

(aiaj)-lziZjYiYj +

i=lj=l

-I

2

L (aia;)-lziZiYiYi i=1

-1

2

n ] + all + 1 Yn + 1 i~l ai Zi

= !Z(A

- y)zt,

where Z = (Zl' ... , ZII) is a row matrix, A = diag(al 1e, ... , all-Ie), C = Lj~f aj-1yJ, Y = IIYijll, Yij = (aiaj)-lYiYj' Whence we obtain the expression for the Lagrangian 2(y, y) corresponding to the Hamiltonian Hab: 2(y, y) = !y(A - Y) -1 Ywhere.V = (Yb' .. '.Vn)' But this is, obviously, the Lagrangian of a free particle on a sphere with metric dS;b = (dy)(A - Y) -l(dyy, where (dy) = (dYl" .. ,dYn)' The only expression left to calculate is (A - y)-l. We have (A - y)-l = (A(E - A-1y))-1 = (E - A-1y)-lA- 1

=A- 1 +A-1YA- 1 +A-1(YA-1)YA- 1 +A-1YA-1YA-1YA- 1 + .... Note that

262

A. T. FOMENKO AND V. V. TROFIMOV n

(Y A-I Y)ij =

L

k=1

(aia k) -1 Yi Ykakc -1 (aka j) -1 Yk Yj

where L = c- 1 Lk=1 ak- 1yf. From which we obtain (A - Y) -1 = A-I

+ A-I Y A-I + LA -1 Y A-I + IJ A-I Y A-I + ...

=A- 1 +

1 A- 1YA- 1 . l-L

Proceeding further:

1

1

1 _ "n L..,k=1 ak

-- -

--~----

-1

1- L

2

Yk

-

"~+1 a:-ly~

L..,J = 1

-1

J

2

an+1Yn+l

J

.

"~+1 a:-ly~

L..,J=1

J

J

Hence we obtain

dS;b=c-l[ia;(dYi)2+a~+1 i

As (dYn+l)

=

d

=1

Yn + 1

±

YiYjdYidYj]'

i.j = 1

J Yt +"'+Yn = Yl dYl + ... + Yn dYn , Jyi + ... + Y; 2

2

we obtain n

an+l(dYn+l)2 = a;+1 L YiyjdYidYj' Yn+l i,j=1

The change of variables Xi = metric dS;b taking the form

Ja; Yi (i = 1, ... ,n + 1)

leads to the

INTEGRABLE SYSTEMS ON LIE ALGEBRA

conformally equivalent to the metric (dXl)2 ellipsoid

+ ... + (dx. + 1)2

263 on an

2} Xl2 X.+ l -+ ... +--= 1 . { al

a.+ l

23.3. Applications to non-commutative integrability

Let (M,w) be a symplectic manifold, OJ a Lie group with a Poissonian action on M (see, for instance [IJ); let P: M --+ G* be the corresponding moment mapping. Let ~ E G*, and let OJ~ stabilizer of ~ under the coadjoint representation; denote by M~ = P-l(~) level surface. As the moment mapping P maps the Poissonian action of OJ on M onto the coadjoint action on G* , and the stabilizer OJ~ leaves the level surface M ~ invariant. If the level's surface M ~ is a smooth manifold and the factor set N~ = M~/OJ~ has the structure of a smooth manifold such that the canonical projection n: M~ -+ N~ is a smooth fiber bundle (such a smooth structure is uniquely defined), then the symplectic structure w induces a symplectic structure w~ on the manifold N~. The symplectic manifold (N~, w~) is called a reduced symplectic manifold. Let H be a Hamiltonian on M, invariant, under OJ, and let G be the algebra of the Hamiltonian system (M, w, H). Let X H be a Hamiltonian vectorfield, related to the Hamiltonian H, i.e. X H = s grad H; let XH/M~ denote the restriction of this vector field to M ~ (note that X H is tangent to M~). As the vector field XH/M~ is invariant with respect to OJ~, it is projected onto a uniquely defined vector field X H on N~ which, as one can easily check, is Hamiltonian on N~ with Hamiltonian H~(y) = H(x), where Y E N~ is the projection of x. Let OJ~ be the subgroup OJ~ leaving each point X E M ~ fixed. OJ~ is, obviously, a normal subgroup in OJ~. Let OJ~IT = OJ~/OJ~ be the effective stabilizer of ~. It can be shown that for a general position point ~ E P(M) in P(M) the connected component ofthe unit of the group OJ~IT is commutative (for compact Lie algebras G the entire stabilizer OJ~ and, therefore, OJ~ also are connected Lie groups). From which it follows that the reduced Hamiltonian system (N~, w~, H~) is quadrature equivalent to the initial Hamiltonian system (M, w, H) for any point ~ E P(M) in general position in P(M) (the mapping n is considered to be given by known functions). Suppose that the stabilizer OJ~ acts locally transitively on a surface M ~ in general position in M. Then for a general position point ~ E P(M) in

264

A. T. FOMENKO AND V. V. TROFIMOV

P(M) we have dim N~ = 0 and the reduced Hamiltonian system (N~, w~, H~)

is trivially integrable. The initial system, therefore, (M, w, H) is integrable in quadratures for general position initial conditions in M. It is said in this case that the system (M, w, H) is noncommutatively integrable with integral algebra G. If the integral Lie algebra G is commutative, then the non-commutative integrability with integral algebra G is the normal full integrability in the Liouville sense. REMARK This definition of non-commutative integrability coincides with the definition in [84] and is somewhat weaker than that used earlier in this book (see Chapter 3 and also [88]) where it was required in addition that the linear generators of integral algebra G be functionally independent. This non-commutative integrability with a Lie algebra of functionally independent integrals we shall call non-commutative integrability in the strong sense. One special case of non-commutative integrability in the weak sense (i.e. with functionally dependent integrals) was examined in Section 3. In Section 3 we discussed in detail the connection between non-commutative integrability in the strong sense and full integrability in the Liouville sense. THEOREM 23.3 Let the Hamiltonian system (M, w, H) be noncommutatively integrable (in the weak sense) with compact integral Lie algebra G; let P: M --+ G be the corresponding moment mapping. Then the Hamiltonian system (M, w, H) has motion integrals in involution of the form J(P(x) + Aa), J EI(G), A E IR and it is possible to select among these functions independent functions equal in number to half the dimension of M for a general position element a E G.

Proof We shall show that the codimension of a general position orbit o in P(M) is equal to the dimension of M~, EO. Let ~(P(M))1. be the intersection of the kernels of all functionals 1"/ E ~(P(M)); let ~ 01. be the

e

intersection of the kernels of all functionals 1"/ E ~ O. The codimension of o in P(M) is equal to dim ~01./~(P(MW. On the other hand, ~01. is the Lie algebra of ffi~, and ~(P(M))1. is the Lie algebra of ffib. Since ffi~ acts locally transitively on M ~ we have

r = dim ~ 01. /~(P(M))1. = dim ffi~ff = dim M ~. By Theorem 22.2 the set of functions J(x + Aa), J E J(G), A E IR for a general position element a E G in G contains ! dim 0 independent functions on o. Adding r more invariants G (the existence of r independent invariants on P(M) follows from Lemma 22.3) we obtain

INTEGRABLE SYSTEMS ON LIE ALGEBRA r

+ ! dim 0 independent functions in the set J(x + Aa), J E J(G),

265 A E IR!

on P(M). Hence we have the same number of independent functions in the set J(P(x) + Aa), J E J(G), A E IR!. We have only to count their number. We have ! dim 0

+ r = !(dim P(M)

- r)

+ r = !-(dim P(M) + r)

= !(dim P(M) + dim M~) = !- dim M. The theorem is proved.

In conclusion we give now a simple and~from the physical point of view~natural condition for the Lie algebra of integrals of a Hamiltonian system (M, OJ, H) to be compact. Note that this condition is somewhat weaker than the Lichnerowicz condition, where the compactness of the entire manifold M is the condition for compactness of integral algebra (the proof is based on invariance of a positive definite certain scalar product (f, g) = JM fgdi, where k = ! dim M). THEOREM 23.4 Let (M, OJ, H) be a Hamiltonian system, G an integral Lie algebra. Suppose that for any h the iso-energetic surface M h = {x EM: H(x) = h} is compact. Then G is a compact Lie algebra. Proof Let g E G, and let

be the corresponding Hamiltonian vector field on M. The vector field Xg is tangent to Mh for any h and is, therefore, complete on M (i.e. the integral trajectories of the field can be extended indefinitely). Therefore an action of the connected simple-connected Lie group G> belonging to the Lie algebra G on the manifold M is defined. This action is Poissonian as, by the definition of vector field x g , it is a Hamiltonian vector field with Hamiltonian g. Let P: M -+ G* be the corresponding moment mapping, P(x)(g) = g(x). As G is the integral Lie algebra of the Hamiltonian system (M, OJ, H), the iso-energetic surfaces are invariant under the action of G> and their images P(M h) are invariant under Ad* for any h. As, by hypothesis, Mh is compact, P(Mh) is compact too. Let g E G be a nilpotent element in the Lie algebra G. Then ad; is a nilpotent endomorphism of G* and the mapping t -+ Exp(ad~) x is polynomial for any x E G*. Since P(M h) is invariant for x E P(Mh) this mapping is a mapping from IR! to P(Mh). As P(M h) is compact and the mapping t -+ Exp(ad~) x polynomial, it is constant, Exp(ad~) x = x for all t. Therefore, for any x E P(M) we have ad; x = O. Therefore any nilpotent element g lies in Z the center of G. Let R be the solvable radical of G. Then [R, R] consists of nilpotent elements and, therefore, Xg

266

A. T. FOMENKO AND V. V. TROFIMOV

[R, R]

c Z. As a consequence the entire radical R consists of nilpotent elements and, therefore, R = Z. Thus, G is a reductive Lie algebra. Let S = [G, G] be its semi-simple ideal. As S (1 Z = 0, there are no non-zero

nilpotent elements in S. Therefore, S is a compact semi-simple Lie algebra. Therefore, G is a compact Lie algebra. The theorem is proved. Theorem 3.3 is a consequence of Theorem 23.4.

24. MORSE'S THEORY OF COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS. TOPOLOGY OF THE SURFACES OF CONSTANT ENERGY LEVEL OF HAMILTONIAN SYSTEMS, OBSTACLES TO INTEGRABILITY AND CLASSIFICATION OF THE REARRANGEMENTS OF THE GENERAL POSITION OF LIOUVILLE TORI IN THE NEIGHBORHOOD OF A BIFURCATION DIAGRAM

In this section we briefly discuss the elements of the new "Morse-type theory" of integrable Hamiltonian systems, which has recently been constructed by A. T. Fomenko. (See details in [149], [150].) 24.1. The four-dimensional case

Recently many new cases of the Liouville integrability of important Hamiltonian systems in the symplectic manifolds M 2n have been discovered. In this connection the problem of detecting stable periodic solutions of integrable systems is particularly urgent. It is found that when n = 2, on the basis of, at most, the data on the group H 1 (Q, Z) of one-dimensional integral homologies (or the data on the fundamental group), using the fixed three-dimensional surface Q3 of constant energy in which this system is integrable, we can sometimes guarantee the existence of at least two stable periodic solutions of the system on this surface Q3 c M4. These solutions can be effectively obtained by examining the minima and maxima of the additional (second) integral, defined on a separate constant-energy surface. Thus, this result not only gives the existence of two stable solutions, but also enables them (in principle) to be obtained. This statement follows from A. T. Fomenko's more general classification statement on the canonical representation of the surface Q in the form of an amalgamation of the

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elementary manifolds of the four simplest types. At the same time it is assumed that the system v has a second smooth "Morse-type" integral in Q, i.e. such that its critical points on the surface Q3 are organized into non-degenerate smooth critical submanifolds. In this connection Fomenko develops Morse's specific theory of integrable systems, which differs from Morse's usual theory and which uses the well-known Bott theory of functions with degenerate critical points (these functions could be called "Bottian" or Bott functions, see R. Bott, Non-degenerate critical manifolds, Ann. of Math., 60 (1954), 248-261). At the same time there is also a natural development of some of the important ideas of S. P. Novikov [96J, V. V. Kozlov [59J, R. Bott (R. Bott, Nondegenerate critical manifolds, Ann. of Math., 60 (1954),248-261), D. V. Anosov (D. V. Anosov, Typical problems of closed geodesics, Izv. AN SSSR, Ser. Mat.,46, no. 4 (1982)) and S. Smale (S. Smale, Topology and mechanics, Invent. Math., 10, no. 4 (1970), 305-331; The planar n-body problem, Invent. Math., 11, no. 1 (1970),45-64). It appears later than the non-singular surfaces of constant energy of integrable Hamiltonian systems have specific properties which isolate them from all the threedimensional manifolds. Hence we obtain new topological barriers to the integrability of Hamiltonian systems in a class of Morse-type functions. Thus, suppose the Hamiltonian system v = s grad H is specified in M4, where H is a smooth Hamiltonian. Consider the fixed non-critical surface Q3 of constant energy, i.e. Q = {H = const} and grad H :f. 0 in Q. Suppose the system v is integrable on Q using the second independent smooth integral f, which commutes with H on Q, but generally does not necessarily commute with H outside Q. In other words, if Q = {H = O}, then {H, f} = A.H, where A. = const. This equation is more common than {H,f} = O. 24.1 The integral f is called Morse-type (or Bottian) in Q if its critical points form non-degenerate critical submanifolds in Q, i.e. the Hessian d 2f is non-degenerate in the subspaces that are normal to these submanifolds. DEFINITION

The class of these integrals is wider than the class of analytic integrals. Accumulated experience of investigating specific mechanical systems shows that most ofthe integrals which have already been discovered are Morse-type. 24.2 Suppose y is a closed integral trajectory ofthe system von Q3 (i.e. a periodic solution). We will say that y is stable if some of its

DEFINITION

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tubular neighborhood in Q as a whole is stratified into two-dimensional tori which are invariant with respect to the system v. The integrable system cannot have stable periodic solutions. Example: the geodesic flow of a Euclidean two-dimensional torus. It appears that a simple connection exists between the following three items: (a) the Morse-type integral f on Q, (b) the stable periodic solutions of the system v on Q, and (c) the group of integral homologies H 1 (Q, Z) or the fundamental group 1t1 (Q). THEOREM 24.1 (A. T. Fomenko) Suppose v = s grad H is a Hamiltonian field in the smooth symplectic four-dimensional manifold M4 (compact or non-compact), where H is a smooth Hamiltonian. We will assume that the system v is integrable on some kind of single nonsingular compact three-dimensional surface of the level Q of the Hamiltonian H using the Morse-type integral f on Q. Then, ifthe group of homologies H 1 (Q, Z) is finite cyclic, v has no less than one stable periodic solution on Q; and if H 1 (Q, Z) is finite and integral f is orientable (see below), v has no less than two stable periodic solutions. At the same time f reaches a local minimum or maximum in each of these trajectories. This criterion is effective, since a verification of the Morse-type character of the integral f and a calculation of the rank H 1 (Q, Z) is usually easy. In specific examples the surfaces Q of constant energy (or thei'r reduction) are often diffeomorphic either to the sphere S3, the projective space IRp 3, or S1 x S2. For example, after appropriate factorization, for the equations of motion of a heavy solid in a zone of large velocities we can assume that Q ~ S1 X S2. In the problem of the motion of a four-dimensional solid with respect to inertia with a fixed point we have Q ~ S1 X S2. In the integrable (three-dimensional) Kovalevskii case, we can' assume that some Q ~ S1 X S2. If the Hamiltonian H has an isolated minimum or maximum point in M 4 , all the rather close surfaces of level Q are spheres S3. PROPOSITION 24.1 Suppose the system v = s grad H is integrable using the Morse integral f on some single surface of constant energy Q, homeomorphic either to S3 or IRp3, or to S1 x S2. Then the system v has at least two stable periodic solutions on S3 and at least one stable periodic solution on IRp3, S1 x S2. In the case orientable integral we' have at least two such solutions in all three cases.

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In particular, as we shall see, the integrable system has two stable periodic solutions on not only the small spheres surrounding the minimum or maximum point H, but also on all the "remote" expanding surfaces of the level, while they are geomorphic to S3. The criterion of Theorem 24.1 is accurate in the sense that examples are known when the system has exactly two (and no more) stable periodic solutions on Q = IRp3. Suppose R = rank 11:1 (Q), that is the least possible number of generatrices of the fundamental group of the surface Q. THEOREM 24.2 (A. T. Fomenko) Suppose the system v is integrable on some non-singular compact surface Q3 of constant energy in M4 using the Morse integral f If R = 1, then v has no fewer than one stable periodic solution on Q, on which f reaches a local minimum or maximum. Ifthe rank of the group HI (Q, 2) ~ 3, then v can generally not have stable periodic solutions in Q. In the case of the integrable geodesic flow of a plane torus T2 we have: Q = T 3 , the rank HI (Q, 2) = 3 and all the periodic solutions of this system are unstable. From the well-known results of Anosov, Klingenberg and Takens (see D. V. Anosov, The typical properties of closed geodesics, Izv. AN SSSR, Ser. Mat., 46, no. 4 (1982), and W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, 1978, Grundlehren des Mathematischen, Wissenschaften, 230) it follows that an open and everywhere dense subset of flows without stable periodic trajectories exists in the set of all the geodesic flows in smooth Riemannian manifolds. Thus, the property of the flow does not have stable trajectories-a property of the general position. COROLLARY 1 Consider a two-dimensional manifold which is diffeomorphic to a sphere with a Riemannian metric of the common location, i.e. without stable closed geodesics. Then the corresponding geodesic flow is non-integrable in the class of smooth Morse integrals on each separate surface of constant energy. QUESTION Can any three-dimensional manifold be a surface of constant energy of an integrable system? COROLLARY 2 Not every three-dimensional smooth compact closed orientable manifold can play the role of a surface of constant energy of a Hamiltonian system, integrable using the Morse integral (on this surface).

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We can give a clear meaning to the words "not every." We shall not discuss this here. Thus, the topology of the surface Q serves as an obstacle to integrability. All the results follow from the general Theorem 24.3 (see below). If f is a Morse integral on Q, then the separatrix diagram peT) is connected to each of its critical submanifolds T, i.e. the set of integral trajectories of the field grad f, which enter Torleave T. We will call the integral I orientable if all its separatrix diagrams are orientable. Otherwise we will call the integral non-orientable. Consider the following simplest three-dimensional manifolds, whose boundaries are the two-dimensional tori T2. (1) The complete torus Sl x D2. (2) The cylinders T2 x DI. (3) The direct product (we shall call it the oriented saddle) N 2 x SI, where N 2 is a disk with two holes. (4) Consider the non-trivial fibration A3-+N 2 Sl with the base SI and the fiber N 2 • The boundary of the manifold A 3 is the two tori T 2 • It is clear that A 3 (we will call it a non-oriented saddle) is implemented in [R3 in the form of a complete torus, from which the second (thin) complete torus, which twice passes around the axis of the large complete torus (dual winding), is drilled. (5) Consider the non-trivial fibration K3 ~Dl K2 with the base K2 = Klein bottle and the fiber DI = interval. The boundary of K3 is the torus T2. THEOREM 24.3 (A. T. Fomenko) (Fundamental classification theorem in dimension 4) Suppose v = s grad H is a Hamiltonian system which is integrable on some single non-singular compact threedimensional surface of constant energy Q3 c M4 using the Morse . integral f Suppose m is the number of periodic solutions of the system v on the surface Q, on which the integral I reaches a local minimum or maximum (then they are stable). Then Q = m(SI x D2) + p(T2 X DI) + q(N 2 X SI) + seA 3) + r(K 3 ), i.e. Q is obtained by splicing m complete tori, p cylinders, q orientable saddles, s non-orientable saddles and r non-orientable cylinders using some diffeomorphisms from the boundary tori. If the integral I is orientable, then s = r = 0, i.e. there are no non-orientable saddles and cylinders. 24.2. The general case

Suppose v = s grad H is a smooth integrable system in M 2 n and F: M 2n ~ [Rn is a mapping of the moment, i.e. F(x) = (ft (x), . .. , f,,(x) , where J; are commuting smooth integrals and II = H. The point x E M is regular if the rank dF(x) = n and it is critical otherwise. Suppose

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N c M is a set of critical points and 1: = F(N) is a set of critical values

(bifurcation diagram). If a E 1R"\1:, then the compact fiber Ba = F -l(a) c M2n consists of Liouville tori. For the deformation a outside 1:, the fiber Ba is transformed by means of the diffeomorphisms. If the curve y, along which a moves, meets 1:, then the fiber Ba undergoes topological rearrangement. Problem: describe these rearrangements. It appears that a complete solution of the problem exists in the case of the common location. If dim 1: < n - 1, then all the fibers Ba, where a E IRn \1: are diffeomorphic. The basic problem is when dim 1: = n - 1. Consider five types of (n + I)-dimensional manifolds. (1) We shall call the direct product D2 x r -1 a dissipative complete torus. Its boundary is the torus Tn. (2) We will call the product Tn X Dl a cylinder. Its boundary is the two tori Tn. (3) Suppose N 2 is a two-dimensional disk with two holes. We shall call the direct product N 2 x r -1 an oriented torus saddle. Its boundary is the three tori r. (4) Consider all the nonequivalent fibration Aa -+N2 Tn -1 with base the torus Tn-l, with a fiber N 2 • They are classified by the elements (XEH 1 (T n - 1 ,Z2) = ZZ-l. N 2 X Tn -1 when (X = 0 is a special case. If (X =F 0, the fibration Aa is nontrivial and all manifolds Aa are diffeomorphic. The manifolds Aa when (X =F 0 will be called non-orientable torus saddles. They have a boundary-the two tori r. (5) Let us consider the manifolds = Tn /G a' where Tn is the torus, (X = 0, 1, and Ga is the group of the transformations defined as follows (this action was introduced by A. V. Brailov and V. T. F omenko):

K:

_ {( -al' a 2 + 1, a3, ... , an), Ra (a) 1 ) (a2,a 1,a 3 +"2,a 4 , ... ,an ,

(X=O , (X= 1,

where a = (a 1, . .. , an) E IR"/zn = Tn. Here n ~ 2 in case (X = 0 and n ~ 3 in case (X = 1. Then K: = Ko x r- 2 , K~ = Ki x T n- 3 and Ki = K2 x Sl. Let us consider the two-fold covering p: P -+ K: and let K;+l = K;,:l is the cylinder of the map p. It is clear that OK;+l = Tn. We will describe five types of rearrangement of the torus P. (1) The torus P, implemented like the boundary of the dissipative complete torus D2 X Tn-l, contracts to its "axis," the torus T n-l (we will put P -+ Tn -1 -+ 0). (2) The two tori Tt and Ti-the boundaries of the cylinder P x Dl moves in opposite directions and merge into one torus Tn (i.e. 2P -+ r -+ 0). (3) The torus Tn-the lower boundary or the oriented torus saddle N 2 X Tn -1 rises upward and, in accordance with the topology N 2 x P -1 , splits into two tori Tt and Ti (i.e. Tn -+ 2 Tn).

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(4) The torus Tn-one of the boundaries Aa rises with respect to Aa and is rearranged in its "middle," becoming once more a single (twice wound) torus (i.e. Tn --+ Tn). These rearrangements are parametrized by the nonzero elements (X E HI (Tn -1 ,1'2) = 1''2- 1 • (5) Let us realize the torus Tn as the boundary of K; + 1. Let us deform Tn in K n+ 1 and collapse Tn on K~. We obtain p: Tn --+ We shall fix the values ofthe last n - 1 integrals 12" .. ,In and shall consider the resulting (n + I)-dimensional surface xn + 1 . Limiting in it 11 = H, we obtain the smooth function I in xn + 1 . We will say that the rearrangement of the Liouville tori, which generate the non-singular fiber Ba (assumed compact), is a rearrangement of the common location if, in the neighborhood ofthe rearrangement the torus Tn, the surface xn + 1 is compact, non-singular and the restriction I of the energy 11 = H on xn + 1 is a Morse function in the sense of Section I in this neighborhood. In terms of the diagram L, this means that the path y along which a moves, transversally intersects L at the point C, whose neighborhood in L is a smooth (n - 1)-dimensional submanifold in IRn , and the last n - 1 integrals 12" .. ,In are independent on xn + 1 in the neighborhood of the torus Tn.

K:.

THEOREM 24.4 (A. T. Fomenko) (Theorem of the classification of the rearrangements of Liouville tori) (I) If dim L < n - I, then all the non-singular fibers Ba are diffeomorphic. (2) Suppose dim L = n - 1. Suppose the non-degenerate Liouville torus Tn moves along the common non-singular (n + I)-dimensional surface of the level of the integrals 12" .. ,In, which is entrapped by the change in value of the energy integral It = H. This is equivalent to the fact that the point a = F(Tn) E IRn moves along the path y in the direction ofL. Suppose the torus Tn undergoes rearrangement. This occurs when and only when Tn meets the critical points N ofthe mapping of the moment F (i.e. the path y at the point C transversally pierces the (n - I)-dimensional sheet L). Then all the possible types of rearrangement ofthe common location are exhausted by the compositions of the above five canonical rearrangements 1,2,3,4,5. In case I (the rearrangement Tn --+ T n- t --+ 0) as the energy H increases the torus Tn becomes a degenerate torus Tn-I, after which it disappears from the surface of the constant energy H = const (the limiting degeneration). In case 2 (the rearrangement 2Tn --+ Tn --+ 0) as the energy H increases the two tori Tr and T2 merge into one torus Tn, after which they disappear from the surface H = const. In case 3 (Le. Tn -+1% 2Tn) as H increases the torus "penetrates" the critical energy level and splits into two tori 7;," and T2

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on the surface H = const. In case 4 (i.e. Tn --+a Tn) as H increases the torus Tn "penetrates" the critical energy level and once more becomes the torus Tn (a non-trivial transformation of a double coil). In case 5 the torus P merge into the manifold and disappears from the surface H = const. Changing the direction of the motion of the torus rn, we obtain five inverse processes: (1) the production of the torus T" from the torus Tn, (2) the trivial production of the two tori Tt and T2" from one torus T", (3) the non-trivial merging of the two tori Tt and T; into one torus P, (4) the non-trivial transformation ofthe torus Tn into the torus P (double coil), (5) the transformation of into the torus P. The previously known rearrangements of two-dimensional tori in the Kovalevskii case and in the Goryachev-Chaplygin case (see M. P. Kharlamov, A topological analysis of classical integrable systems in solid body dynamics, DAN SSSR, 273, no. 6 (1983), 1322-1325) are special cases (and compositions) of the rearrangements described in Theorem 4. When changing H, the torus Tn drifts along the surface xn + 1 of the level of the integrals f2'" . ,f". It can happen that Tn contracts to the torus T" -1. These limiting degenerations emerge in mechanical systems with dissipation. If we introduce small friction into the integrable system, we can assume, to a first approximation, that the energy dissipation is modelled using a decrease in the value Hand causes, consequently, a slow evolution (drift) of the Liouville tori along X n + 1. An answer to the question-What kind of topology is the topology of the surfaces X" + 1 ?-is given by the following theorem.

K:

K:

THEOREM 24.5 Suppose M 2n is a smooth symplectic manifold and the system v = s grad H is integrable using the smooth independent commuting integrals H = f1' f2' ... ,f". Suppose X" + 1 is any fixed nonsingular compact common surface of the level of the last n - 1 integrals. Suppose the restriction H on xn + 1 is a Morse function. Then

+

L s.(Aa) + L r.K~.~

,,"0

1,

"

i.e. a splice of boundary tori (using some ditTeomorphisms) of the following "elementary bricks" is obtained: m dissipative complete tori, p cylinders, q torus oriented saddles, s = Sa torus non-oriented saddles and r = ro + r 1 non-oriented cylinders. The number m equals

L."o

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the number oflimiting degenerations of the system v in xn+I, in whichH reaches a local minimum or maximum. Theorem 24.3 follows from Theorem 24.5 when n = 2. All the above results also hold for Hamiltonian systems permitting "noncommutative integration." In these cases the Hamiltonian H is included in the noncommutative Lie algebra of G functions on M2n, such that the rank G + dim G = dim M2n. Then the trajectories of the system move with respect to the tori yr, where r = rank G. When proving the above results we use the following statements. LEMMA 24.1 Suppose in the singular fiber Be there is exactly one critical saddle torus Tn-I. (1) Suppose the integral f is orientable in X n + I and a < c < b, where a and b are close to c. Then C b = (f ~ b) is homotopically equivalent to Ca = (f ~ a), to which the manifold Tn -1 X Dl is attracted with respect to the two non-intersecting tori 71n,; 1 and T;,; 1. (2) Suppose the integral f is non-orientable. Then Cb is homotopically equivalent to Ca, to the boundary Ba of which, using the torus Tn -1 , is attracted the n-dimensional manifold yn which has the boundary T" -1 and which is a fibration ~D' T"-I, which 1 corresponds to the nonzero element IX EZ'2- =H 1 (T n - 1 ,Z2)' (3) Further, each of the tori Tt,;I, T2~;I, 7;,n-l always realizes one of the generatrices in the group of homologies H n -1 (7;,n, Z) = zn -1 . If any of these (n - I)-dimensional tori are attached to one and the same Liouville torus 7;,n, they do not intersect and they realize one and the same generatrix of the group of homologies H.-l (7;,., Z), and therefore they are always isotopic in the torus 7;,n.

y:

We will provide one more description of the three-dimensional surfaces Qof constant energy of the integrable (using an oriented Morse integral) systems on M4. Let us suppose that all critical manifolds of integral fare orientable. Suppose m is the number of stable periodic solutions of the system in Q, on which f reaches the minimum or maximum. Consider the two-dimensional connected closed compact orientable manifold M; of the genus g, where g ~ 1 (i.e. a sphere with g handles) and take the product M; x SI. We shall separate an arbitrary finite set of non-intersecting and self-non-intersecting smooth circles lXi in Mi, among which there are exactly m contractible circles (the remainder are non-contractible in M;). In M; x S2 the circles lXi determine the tori I? = lXi x SI. We will cut out M; X SI with respect to all these tori, after which we will inversely identify these tori using

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some diffeomorphisms. As a result we obtain a new three-dimensional manifold. It appears that the surface Q has precisely this form. PROBLEM Find an explicit convenient corepresentation of the group 7r1 (Q), where Q3 is the surface from Theorem 24.3. Give an explicit classification of the surfaces of constant energy of the integrable systems of arbitrary dimension. How can we make an upper estimate of the number of complete tori (i.e. stable periodic solutions) in Q3, in terms of the topological invariants Q (homologies, homotopies) in the general case. Discuss the complex analytical analog of the Morse theory of integrable systems constructed above. Does an integrable foliation to the two-dimensional (in a real sense) complex tori exist in the analytical manifold M4? Probably, we can obtain these obstacles in explicit form in examples of surfaces of the K3 type. 24.3. New topological invariant of integrable Hamiltonians

In this section we describe the topological invariant, which was introduced by A. T. Fomenko on the basis of his Morse-type theory of Bottian integrals. Let M4 be a symplectic manifold, v be a Hamiltonian system with Hamiltonian H; v is completely integrable on the compact regular isoenergetic surface Q3 = (H = const); f: Q --+ R is a second independent Bottian integral on Q. The critical submanifolds of fare non-degenerate in Q. The Hamiltonian H will be called non-resonance if the set of Liouville's tori with irrational trajectories of v is dense in Q. The set f - 1(a) is the set of tori in the case when a E R is regular. THEOREM 24.6 (A. T. Fomenko) There exists a one-dimensional graph Z(Q, f), two-dimensional closed compact surface P(Q, f) and the embedding h: Z(Q,f) --+ P(Q, j), which are naturally and uniquely defined by the integrable non-resonance Hamiltonian H with the Bottian integral f on Q. The triple (Z, P, h) does not depend on the choice of the second integral f This means that if f and l' are two arbitrary Bottian integrals of a given system, the graphs Z and Z' are homeomorphic, the surfaces P and P' are homeomorphic, and the diagram h:Z--+P

2 h': Z'

--+

l P'

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is commutative. Consequently, the graph Z(Q), surface P(Q) and the embedding h(Q):Z(Q) --+ P(Q) are the topological invariants of the integrable case (of Hamiltonian H) proper. The triple Z(Q), P(Q), h(Q) allow us to classify the integrable Hamiltonians corresponding to their topological types. In particular, we can now demonstrate the visual difference between the invariant topological structure of the Kovalevskaya case, Goryachev-Chaplygin case and so on. The subdivison of the surface P(Q) into the sum of the domains is also the topological invariant of the Hamiltonian H and describes its topological complexity. The graph Z*, which is dual to the graph Z on the surface P, has the vertices of the multiplicity no more than four. The collection of the graphs Z(Q), surfaces P(Q) and embeddings h(Q) is the total topological invariant (topological portrait) of integrable Hamiltonian H. Let us construct the graph Z(Q, f). If a is a non-critical regular value for J, then fa is a union of a finite number of Liouville's tori. Let us represent these tori by the points in R 3 lying on the level a. Changing the value of a (in the domain of regular values), we force the points to move along the vertical in R3. Consequently, we obtain some intervals, viz. the part of the edges of our graph Z. Let us suppose that the axis R is oriented vertically in R3. If the value a becomes critical (we denote such values by c), the critical (singular) level of the integral f becomes more complicated. Let !c be a connected component of a critical level surface ofthe integral. We denote by Nc the set of critical points of the integral f on !C. Let us consider two cases: (a) Nc = !C, (b) Nc c!c. In Section 24.2 A. T. Fomenko gives the complete description of all cases and the topological structure of !C. (See [149], [150].) Let us consider case (a). Here only three types of critical sets are possible. The "minimax circle" type. Here Nc =!c and this set is homeomorphic to a circle Sl. The integral f has a minimum or maximum on S1. The circle Sl is the axis of the filled torus which foliated into non-singular two-dimensional Liouville's tori. We represent this minimax circle by the black point (a vertex of the graph) with one edge (interval) entering the point or emerging from it. The "torus" type. Here Nc = !C. This set is homeomorphic to a twodimensional critical torus. The integral f has a local minimum or maximum on this torus. The tubular neighborhood of this torus in Q is

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homeomorphic to the direct product T2 x Di. We represent this minimax torus by the white point (the vertex ofthe graph). Two edges of the graph Z enter this vertex or emerge from it. The "Klein bottle" type. Here Nc = !c. This set is homeomorphic to a two-dimensional Klein bottle K2. The integral f has the local minimum or maximum on this manifold. The tubular neighborhood of K2 is homeomorphic to the skew product of K2 and the interval Di. We represent this minimax Klein bottIe by a white disc with a black point at the centre (the vertex of graph Z). One edge of the graph Z enters this vertex or emerges from it. Let us consider case (b). Here N c c: !c and N c ¥- !c. Here N c is a union of non-intersecting critical circles in !c. Each of these circles is a saddle circle for f We shall call the corresponding connected component !c a saddle component. Each saddle component !c is represented by a flat horizontal square in R3 on the level c. Some edges of the graph Z enter the square from below (when a -+ c and a < c). Some other edges of graph Z emerge upwards from the square (when a> c). Finally we define some of graph A which consists of the regular edges described above. Some edges enter the vertices like the three types described above. Graph A is a subgraph in graph Z. Graph A was obtained from the union of the edges which are the traces of the points representing the regular Liouville tori. Let us define the graphs 7;. We consider a vector field w = grad f on Q. Let us call by separatrices the integral trajectories of w which enter the critical points on critical submanifolds (or emerge from them) and call their union the separatrix diagram of a critical submanifold. Then we consider the local separatrix diagram of each saddle critical circle Si . Let us consider two regular values c - e and c + e which are close to c. They define the regular Liouville tori above and below !c. The separatrix diagrams of critical circles meet these tori and intersect them along some smooth circles. These curves of intersection divide each torus into the sum of two-dimensional domains which will be referred to as regular. Each inner point of a regular domain belongs to the integral trajectory of the field w, which is not a separatrix. The trajectory goes upwards and leaves aside the critical circles on the level !c. Then the trajectory meets some torus on the upper non-singular level !c+,. We obtain a certain correspondence (homeomorphism) between regular domains on the levels !c-, and !c+ •. Let us consider the orientable case, when all separatrix diagrams are orientable. Since each regular torus is

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represented by a point on graph A, we can join the corresponding points by arcs which represent the bundle of parallel integral trajectories. Consequently we obtain some of graph 7;,. All edges of the graph 7;, represent the trajectories of single regular domains of Liouville's tori. The tori break down into the sum of single pieces, then these pieces are transposed and joined into new tori again. Each upper torus is formed from the pieces of lower tori (and conversely). The ends of the edges of the graph 1'.: are identified with some ends of the edges of the graph A. Graph 7;, demonstrates the process of transformation of lower tori into upper tori after their intersection with a saddle critical level of the integral. Let us consider the non-orientable case when we have the critical circles with non-orientable separatrix diagrams. Let us consider all Liouville's tori which are in contact with the level surface fc with a nonorientable separatrix diagram of some critical circles on fc. Let us mark by asterisks all regular domains on these tori which are in contact with non-orientable separatrix diagrams. We mark by asterisks the corresponding edges of the graph. Finally we double all edges of the graph (preserving the number of its vertices) and denote the resulting graph as 7;,. Finally, we define the graph Z as the union Z = A + 7;" where {c} are the critical values of f Let us construct the surface P(Q, f). This surface is obtained as the union P(A) + P(7;,) (here {c} are the critical values of f) where P(A) and P(7;,) are two-dimensional surfaces with boundary. Here P(A) = (A x SI) + I D2 + I 11 2 + LSi X Dl. Here A = Int A, I D2 denotes the non-intersecting 2-discs corresponding to the vertices of the graph A, which have a "minimax circle" type; I 11 2 denotes the non-intersecting Mobius bands, corresponding to the vertices of the graph A, which have a "Klein bottle" type; I SI x Dl denotes the non-intersecting cylinders, corresponding to the vertices of the graph A, which have a "torus type". The corresponding boundary circles of Ax SI are identified with the boundary circles of D2, 1l2, SI X Dl by some homeomorphisms. Let us construct the surfaces P(7;,). Let us consider the orientable case. Fomenko proves (see Section 24.2 and [149], [150]) that in this case the surface fc is homeomorphic to direct product Kc x SI, where Ke is some graph. The graph Ke is constructed from several circles, which are tangent in some points. Such circles can be realized as a cycle on the

Ie

Ie

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torus contained in !C. This cycle intersects with a critical circle on !c only in one point. The surface !c is obtained as a two-dimensional cell-complex by the union of several species of two-dimensional tori along some circles. The tori stick together along the critical circles realizing a non-trivial cycle on the tori. The critical circles do not intersect and they are homologous in !C. They cut !c into the sum of flat rings. Consequently, the circle y (non-homologous to zero) is uniquely defined on a critical level surface !C. We can choose the circle a which is a generator on the torus contained in !C. The circle a is complementary to y. We obtain the set of circles a which are tangent to one another at points on critical circles. Each circle a will be called oval. The ovals can be tangent to one another at several points. The graph Kc is the union of all ovals. The surface P(T.,) can be realized as "normal section" of a small neighborhood of a critical level surface!c in Q. The intersection of P(T.,) with !c is the graph Kc. To realize the surface P(T.,) in Q, we must consider the small intervals on the integral trajectories of the field w = grad J, which intersect the graph Kc. This definition is correct in all non-critical points on Kc. Let us consider the vertices ofthe graph K" i.e. the critical points of the integral f on !C. Then we consider the small squares orthogonal to the critical circles on !C. The surface P(T.,) is the union of these squares and the bands, which are formed from the small intervals defined above. Finally, we identify the boundary circles of the surface P(A) with the boundary circles of the surfaces P(T.,). The graph Kc is embedded in the surface P(T.,). We obtain some graph K as the union of all graphs Kc and all boundary circles, described above. THEOREM 24.7 (A. T. Fomenko) The graph Z(Q,J) is conjugate to the graph K(Q,f) in the surface P(Q,J). Consequently, the graph Z(Q,f) is embedded in the surface P(Q, f). The surface P(Q, f) does not embed (in general case) in the surface Q. The construction of the triple Z, P, h is finished. Theorem 24.6 states that this triple does not depend from the choice of the Bottian integral f PROPOSITION Let f and l' be two arbitrary Bottian integrals of a system v. Then the homeomorphism h:Z(Q,f) -+ Z(Q, 1') (see Theorem 24.6) transforms the subgraphs T., into the subgraphs T.,'. The asterisks of the graph Z(Q, f) are mapped into the asterisks of the graph

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Z(Q, /'). The vertices of the types "minimax circle" and "Klein bottle" on the graph Z(Q, f) are mapped into the vertices of the same type on the graph Z (Q, /'). The vertices of the "torus type" on the graph Z (Q, f) may change their type and be mapped into the usual inner points of some edge on the graph Z(Q, /,). Conversely, some usual inner points of the edges on the graph Z(Q, f) can be mapped into the vertices of the "torus type" on the graph Z(Q, 1'). This event corresponds (from the analytical point of view) to the operation f --+ f2 (square offunction) or, (square root). conversely, to the operation f --+ If a non-resonance Hamiltonian H is fixed, we can consider all its nonsingular isoenergetic surfaces Q. This set consists (in concrete cases) usually of a finite number of triples (Z, P, h). We formulate the new definition based on Theorem 24.6.

J7

We shall call the triple Z(Q), P(Q), h(Q) an invariant topological portrait of a non-resonance integrable Hamiltonian H on a fixed isoenergetic surface Q. The discrete set of all triples {Z, P, h} will be called the total topological invariant portrait of the integrable Hamil tonian. We shall obtain the following corollary from Theorem 24.6. If two integrable systems have non-homeomorphic topological portraits, then there exists no transformation of coordinates which would realize the equivalence of these systems. So, the systems with non-homeomorphic topological portraits are non-equivalent. On the other hand, nonequivalent systems with homeomorphic topological portraits do exist. Practically all the results listed above are also valid in the multidimensional case. These results will be described in a separate paper by Fomenko. DEFINITION

Bibliography

[lJ Amol'd V. I., Mathematical Methods in Classical Mechanics, Nauka, Moscow, 1974 (English translation, Springer-Verlag, Berlin and New York, 1978). [2J Adler M., van Moerbeke P., Completely integrable systems, Euclidean Lie algebras and curves, Adv. in Math., 38, no. 3 (1980),267-313. [3J Adler M., van Moerbeke P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math., 38, no. 3 (1980), 318-379. [4J Avez A., Lichnerowich A., Diaz-Miranda A., Sur I'algebre des automorphises infinitesimaux d'une variete symplectique, J. Differential Geometry, 9, no. 1 (1974), 1-40. [5] Adler M., On a trace functional for formal pseudo-differential operators and the symplectic structure for Korteweg~e Vries type equations, Invent. Math., 50 (1979),219-248. [6] Adler M., Some finite dimensional systems and their scattering behaviour, Comm. Math. Phys., 5S (1977). [7] Adler M., On a trace functional for formal pseudo-differential operators and the Hamiltonian structure of Korteweg~e Vries type equations, Lecture Notes in Math., no. 755 (1979), 1-16. [8] Adler M., Moser J., On a class of polynomials connected with the Kortewegde Vries equation, Comm. Math. Phys. (1978), 1-30. [9] AirauIt H., McKean H. P., Moser J., Rational and elliptic solutions of the Korteweg~e Vries equation and related many body problem, Comm. Pure Appi. Math.,30 (1977), 95-148. [IOJ Arkhangel'sky A. A., Completely integrable Hamiltonian systems on a group of triangular matrices, Mat. Sb., 108, no. 1 (1979), 134-142 (in Russian). [llJ Bourbaki N., Groupes et Aigebres de Lie, Chapters IV-VI, Hermann, Paris VI, 1968. [12] Borho W., Gabriel P., Rentschler R., Primideal in Einhiillenden Auflosbaren LieAlgebren, Lecture Notes in Math., no. 357 (1973), Springer-Verlag, Berlin. [13J Belyayev A. V., On the motion ofa multi-dimensional body with a fixed point in a gravitational field, Mat. Sb., 114, no. 3 (1981),465-470 (in Russian). [14] Bogoyavlensky O. J., New algebraic constructions of integrable Euler equations, Reports USSR Acad. Sci., 268, no. 2 (1983),277-280 (in Russian). [15] Bogoyavlensky O. J., On perturbation of the periodic Toda lattice, Comm. Math. Phys., 51 (1976),201-209.

281

282

A. T. FOMENKO AND V. V. TROFIMOV

[16] Burchnall J. L., Chaundy T. W., Commutative ordinary differential operators, Proc. London Math. Soc., 81 (1922),420-440; Proc. Royal Soc. London (A), 118 (1928),557-593; Proc. Royal Soc. London (A), 134 (1931),471-485. [17] Berezin F. A., Perelomov A. M., Group-theoretical interpretation of the Korteweg-de Vries type equation, Comm. Math. Phys. [18] Brailov A. V., Some cases of complete integrability of Euler equations and their applications, Dokl. Akad. Nauk SSSR, 268, no. 5 (1983), 1043-1046 (in Russian). [19] Brailov A. V., Involutive sets on Lie algebras and expansion of the ring of scalars, Moscow Univ. Vestnik, Ser. 1, Math. Mech., no. 1 (1983),47-51 (in Russian). [20] Calogero F., Solutions of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 12 (1971),419-436. [21] Dao Chong Thi, The integrability of Euler equations on homogenous symplectic manifolds, Mat. Sb., 106, no. 2 (1978), 154-161 (in Russian). [22] Dubrovin B. A., Theta functions and non-linear equations, UMN, 35, pt. 2 (1981), 11-80 (in Russian). [23] Dubrovin B. A., Completely integrable Hamiltonian systems, which are connected with matrix operators, and Abelian manifolds, Functional Anal. Appl., 11, pt. 4 (1977),28-41 (in Russian). [24] Dubrovin B. A., Novikov S. P., Fomenko A. T., Modern Geometry-Methods and Applications, Nauka, Moscow, 1979 (in Russian), (English translation, SpringerVerlag, Berlin and New York, 1984). [25] Dikiy L. A., A note on Hamiltonian systems connected with a rotations group, Functional Anal. Appl., 6, pt. 4 (1972), 83-84 (in Russian). [26] Dixmier J., Algebres Enveloppantes, Gauthier-Villars, Paris, Brussels, Montreal, 1974. [27] Dixmier J., Duflo M., Vergne M., Surla representation coadjointe d'une algebre de Lie, Compositio Math. 25 (1974), 309-323. [28] Dubrovin B. A., Matveyev V. B., Novikov S. P., Non-linear equations of the Korteweg-de Vries type, finite-zone linear operators and Abelian manifolds, UMN, 31, pt. 1 (1976),55-136 (in Russian). [29] Eisenhart L. P., Continuous Groups of Transformations, Princeton University, 1933. [30] Elashvili A. G., The canonical form and stationary subalgebras of general position points for simple linear Lie groups, Functional Anal. Appl., 6, pt. 1 (1972), 51---{j2 (in Russian). [31] Elashvili A. G., Stationary subalgebras of general position points for irreducible linear Lie groups, Functional Anal. Appl., 6, pt. 2 (1972), 65-78 (in Russian). [32] Fomenko A. T., On symplectic structures and integrable systems on symmetric spaces, Mat. Sb., 115, no. 2 (1981), 263-280 (in Russian). [33] F omenko A. T., Group symplectic structures on homogeneous spaces, Dokl. Akad. Nauk SSSR, 253, no. 5 (1980), 1062-1067 (in Russian). [34] Fomenko A. T., The algebraic structure of certain classes of completely integrable systems on Lie algebras, in collection of scientific works, Geometrical Theory of Functions and Topology, Kiev 1981, Institute of Mathematics of Ukrainian SSR Academy of Sciences, pp. 85-126 (in Russian). [35] Fomenko A. T., The complete integrability of certain classic Hamiltonian systems, in collection of scientific works, One-valued Functions and Mappings, Kiev 1982,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

[36]

[37] [38] [39]

[40] [41]

[42]

[43] [44] [45]

[46] [47] [48] [49J [50] [51] [52] [53]

283

Institute of Mathematics of Ukrainian SSR Academy of Sciences, pp. 3-19 (in Russian). (a) Fomenko A. T., Algebraic properties of some integrable Hamiltonian systems. Topology, Proceedings, Leningrad 1982, Lecture Notes in Math., No. 1060 (1984), 246-257, Springer-Verlag, Berlin. (b) Fomenko A. T., Algebraic structure of certain integrable Hamiltonian systems. Global analysis-Studies and applications. I, Lecture Notes in Math. no. ll08 (1984), 103-127, Springer-Verlag, Berlin. (c) Fomenko A. T., The integrability of some Hamiltonian systems, Annals Global Anal. Geometry, 1, no. 2 (1983), 1-10. (d) Fomenko A. T., Topological Variational Problems, Moscow State University Press, Moscow, 1984. Flaschka H., On the Toda lattice I, Phys. Rev. B, 9 (1974), 1924--1926. Flaschka H., Toda lattice II, Progr. Theor. Phys., 51 (1974),703-716. Gorr G. V., Kudryashova L. V., StepanovaL. A., Classical Problems of Rigid Body Dynamics. Development and Contemporary State, Scientific Thought, Kiev, 1978 (in Russian). Gel'fand I. M., Dikiy L. A., Fractional degrees of operators and Hamiltonian systems, Functional Anal. Appl., 10, no. 4 (1976), 13-19 (in Russian). Gel'fand I. M., Dorfman I. Ya., Hamiltonian operators and the algebraic structures connected with them, Functional Anal. Appl., 13, pt. 4 (1979), 13-30 (in Russian). Gardner C., Korteweg--
284

[54] [55]

[56]

[57] [58] [59]

[60] [61] [62] [63] [64] [65]

[66] [67]

[68] [69] [70] [71]

[72] [73]

[74]

A. T. FOMENKO AND V. V. TROFIMOV

Lectures in Modem Analysis and Applications, Lecture Notes in Math., no. 170 (1970), Springer-Vedag, Berlin. Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34, no. 3 (1980), 19S-338. Kulish P. P., Reymann A. G., The hierarchy of symplectic forms for Schrodinger and Dirac equations on a straight line, Records Sci. Seminars LOMI, 77 (1978), 134-147 (in Russian). Kazhdan D., Kostant B., Sternberg S., Hamiltonian groups: actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., 31, no. 4 (1978), 481-507. Krasil'shchik I. S., Hamiltonian cohomologies of canonical systems, Reports USSR Acad. Sci., 251, no. 6 (1980), 1306-1309 (in Russian). Knorrer H., Geodesics on the ellipsoid, Invent. Math., 59, no. 2 (1980),119--144. Kozlov V. V., Qualitative Analysis Methods in Rigid Body Dynamics, MGU, Moscow, 1980 (in Russian). Kozlov V. V., Onishchenko D. A., The non-integrability of the Kirchoff equations, Reports USSR Acad. Sci., 266, no. 6 (1982), 1298-1300 (in Russian). Kupershmidt B. A., Wilson G., Conservation laws and symmetries of generalized Sine-Gordon equations, Comm. Math. Phys., 81 (1981), 189-202. Kupershmidt B. A., Wilson G., Modifying Lax equations and the second Hamiltonian structure, Invent. Math., 62 (1981),403-436. Kirillov A. A., Unitary representations of nilpotent Lie groups, UMN, 17, pt. 4 (1962), 57-110 (in Russian). Landau L. D., Lifshits E. M., Continuous Medium Electrodynamics, Fizmatgiz, Moscow, 1959 (in Russian). Langlois M., Contribution id'etude du mouvement du corps rigide a N dimensions autour d'un point fixe, in These presentee la faculte des sciences de l'universite de Besan,on, Besanl.;on, 1971. Lax P. D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21, no. 5 (1968),467-490. Lebedev D. R., Manin Yu. I., The Gel'fand-Dikiy Hamiltonian operator and the coadjoint representation of a Volterra group, Functional Anal. Appl., 13, pt. 4 (1980),40-46 (in Russian). Loos 0., Symmetric Spaces, vol. 1,2, Benjamin, New York and Amsterdam, 1969. Milnor J., Stasheff J., Characteristic Classes, Princeton University Press, NJ, 1974. McKean H., Integrable systems and algebraic curves, in Global Analysis, Lecture Notes in Math., no. 755 (1979), 83-200, Springer-Verlag, Berlin. McKean H., Trubowitz E., Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math., 29 (1976), 143-226. van Moerbeke P., Mumford D., The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154. Meshcheryakov, M. V., The integration of equations of geodesic left-invariant metrics on simple Lie groups with the help of special functions, Mat. Sb., 117, no. 4 (1982), 481-493 (in Russian). Manakov S. V., Note on the integration of Euler equations of the dynamics of a ndimensional rigid body, Functional Anal. Appl., 10, pt. 4 (1976), 93-94 (in Russian).

a

INTEGRABLE SYSTEMS ON LIE ALGEBRA

285

[75] Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Reports Math. Phys., 5, no. 1 (1974), 121-130. [76] Mozer J., Various aspects of integrable Hamiltonian systems, in Proc. CIME Conf., Bressanone, Italy (June 1978). [77] Mozer J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16, no. 2 (1975), 197-220. [78] Mozer, J., Geometry of quadrics and spectral theory, in The Chern Symposium 1979, pp. 147-188, Springer-Verlag, Berlin, 1980. [79] Mozer J., Finitely many mass points on the line under the influence of an exponential potential: an integrable system, Lecture Notes in Physics-Dynamical Systems, Theory and Application, ed. J. Mozer, 28 (1975),467-498. [80] van Moerbeke P., The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81. [81] van Moerbeke P., The Floquet theory for periodic Jacobi matrices, Invent. Math., 37 (1976), 1. [82] Moody B., A new class of Lie algebras, J. Algebra, 10 (1968),211-230. [83] Mishchenko A. S., Integrals of geodesic flows on Lie groups, Functional Ana/. App/., 4, pt. 3 (1970), 73-77 (in Russian). [84] Mishchenko A. S., The integration of geodesic flows on symmetric spaces, Math. Notes ("Zametki"), 31, no. 2 (1982),257-262 (in Russian). [85] Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162. [86J Mumford D., An algebra-geometric construction of commuting operators and of solution to the Toda lattice equation and related nonlinear equations, Proc. Int. Symp. on Algebraic Geometry, Kyoto, 1977, pp. 115-153. [87] Mishchenko A. S., Fomenko A. T., A Course of Differential Geometry and Topology, MGU, Moscow, 1980 (in Russian). [88] Mishchenko A. S., Fomenko A. T., A generalized method for Liouville integration of Hamiltonian systems, Functional Anal. Appl., 12, pt. 2 (1978), 46-56 (in Russian). [89] Mishchenko A. S., Fomenko A. T., Euler equations on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR (Math. Ser.), 42, no. 2 (1978), 396-415 (in Russian). [90] Mishchenko A. S., Fomenko A. T., The integrability of Euler equations on semisimple Lie algebras, Works of the Seminar on Vector and Tensor Analysis, pt. 19 (1979), 3-94 (in Russian). [91] Mishchenko A. S., Fomenko A. T., The integration of Hamiltonian systems with non-commutative symmetries, Works of the Seminar on Vector and Tensor Analysis, pt. 20 (1981), 5-54 (in Russian). [92] Mishchenko A. S., Fomenko A. T., On the integration of Euler equations on semisimple Lie algebras, Reports USSR Acad. Sci., 231, no. 3 (1976), 536-538 (in Russian). [93] Mishchenko A. S., Fomenko A. T., Symplectic Lie group actions, Lecture Notes in Math. Algebraic Topology, Aarhus, 1978, no. 763, 1979, pp. 504-538. [94] Novikov S. P., The Korteweg-de Vries periodic problem 1, Functional Anal. Appl., 8, no. 3 (1974), 54-66 (in Russian). [95] Novikov S. P., Hamiltonian formalism and the many-valued analogue of Morse's theory, UMN, 37, no. 5 (1982), 3-49 (in Russian).

286

A. T. FOMENKO AND V. V. TROFIMOV

[96] Novikov S. P., Shmel'tser I., Periodic solutions of Kirchoff equations of free movement of a rigid body in an ideal fluid and the expanded LusternikSchnirelman-Morse (LSM) theory. I, Functional Anal. Appl., 15, pt. 3 (1981, 54-66; II, Functional Anal. Appl., 15, pt. 4 (1982), J-49 (in Russian). [97] Nekhoroshev N. N., Action-angle variables and their generalization, Works of MMO, vol. 26,1972, pp. 181-198, MGU, Moscow (in Russian). [98] Olshanetsky M. A., Perelomov A. M., Explicit solutions of certain completely integrable Hamiltonian systems, Functional Anal. Appl., 11, pt. 1 (1977),75--76 (in Russian) [99] Olshanetsky M. A., Perelomov A. M., Geodesic flows on zero curvature symmetric spaces and explicit solutions of the generalized Calogero model, Functional Anal. Appl., 10, pt. 3 (1976),86--87 (in Russian). [100] Olshanetsky M. A., Perelomov A. M., Explicit solutions ofthe classical generalized Toda models, Invent. Math., 56 (1976), 261-269. [10 1] Olshanetsky M. A., Perelomov A. M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math., 37 (1976),93-109. [102] Poincare H., Methodes Nouvelles, 1, Chapter 5, 1892. [103] Perelomov A. M., Some observations on the integration of the equations of motion ofa rigid body in an ideal fluid, Functional Anal. Appl., 15, pt. 2 (1981) (in Russian). [104] Perelomov A. M., The simple relation between certain dynamical systems, Comm. Math. Phys., 63 (1978), 9-11. [105] Pevtsova T. A., The symplectic structure of orbits ofthe coadjoint representation of Lie algebras of the type Exp G, UMN, 37, no. 2 (1982),225--226 (in Russian). [106] Patera J., Sharp R. T., Wintemitz P., Zassenhaus H., Invariants of real lowdimension Lie algebras, J. Math. Phys., 17, no. 6 (1976), 986--994. [107] Reyman A. G., Semyonov-Tyan-Shansky M. A., Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, Invent. Math., 54, no. 1 (1979), 81-100. [108] Reyman A. G., Semyonov-Tyan-Shansky M. A., Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II, Invent. Math., 63, no. 3 (1981), 423-432. [109] Reyman A. G., Semyonov-Tyan-Shansky M. A., Flow algebras and nonlinear equations in partial derivatives, Reports USSR Acad Sci., 251, no. 6 (1980), 1310--1314 (in Russian). [110] Reyman A. G., Semyonov-Tyan-Shansky M. A., Frenkel' I. I., Graduated Lie algebras and completely integrable dynamic systems, Reports USSR Acad. Sci., 247, no. 4 (1979), 802-805 (in Russian). [111] Reyman A. G., Integrable Hamiltonian systems connected with graduated Lie algebras. Differential geometry, Lie groups and mechanics III, Notes LOMI Sci. Sem.,95 (1980), 3-54 (in Russian). [112] Ratiu T., The C. Neumann problem as a completely integrable system on an adjoint orbit, Trans. Amer. Math. Soc., 264, no. 2 (1981),321-329. [113] Ratiu T., The motion of the free n-dimensional rigid body, Indiana Univ. Math. J., 29, no. 4 (1980), 609-629. [114] Ratiu T., On the smoothness of the time t-map of the K dV equation and the bifurcation of the eigenvalues of Hill's operators, Lecture Notes in Math., no. 755 (1979), 248-294.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

287

[115] Ratiu T., Euler-Poisson equation on Lie algebras and N-dimensional heavy rigid body, Proc. Nat. Acad. Sci. USA Phys. Sci., 78, no. 3 (1981), 1327-1328. [116] Rentschler R., Vergne M., Sur Ie semi-centre du corps enveloppant d'une algebre de Lie, Ann. Sci. Ecole Norm. Sup., 6 (1973), 380-405. [117] Rashchevsky P. K., Geometrical Theory of Equations with Partial Derivatives, Ogiz, Gostekhizdat, 1947 (in Russian). [118] Rai"s M., Indices of semi-direct product Exp ffi, C. R. Acad. Sci. Paris, 287, no. 4 (1978), Ser. A, 195-197. [119] Symes W., Systems of Toda type, inverse spectral problems and representation theory, Invent. Math., 59, no. 1 (1980), 13-53. [120] Sternberg S., Lecture on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1964. [121] Souriau J. M., Structure des systemes dynamiques, in Maitrises de Mathematique, Dunod, Paris, 1970. [122] Tits J., Sur les constantes de structure et Ie theoreme d'existence des algebres de Lie semi-simples, Publ. Math. Paris, 31 (1966),21-58. [123] Trofnnov V. V., Fomenko A. T., Methods for constructing Hamiltonian flows on symmetric spaces and the integrability of some hydrodynamic systems, Dokl. Akad. Nauk SSSR, 254, no. 6 (198Q), 1349-1353 (in Russian). [124] Trofimov V. V., Fomenko.-\. T., Group non-invariant symplectic structures and Hamiltonian flows on symmetric spaces, Works of the Seminar on Vector and Tensor Analysis, pt. 21 (1983),23-83 (in Russian). [125] Trofnnov V. V., Fomenko A. T., Dynamic systems on orbits of linear representations of Lie groups and the complete integrability of some hydrodynamic systems, Functional Anal. Appl., 17, no. 1 (1983),31-39 (in Russian). [126] Trofimov V. V., Finite-dimensional representations of Lie algebras and completely integrable systems, Mat. Sb., 3, pt. 4 (1980),610-621 (in Russian). [127] Trofnnov V. V., Euler equations on Borel subalgebras of semisimple Lie algebras, Izv. Akad. Nauk SSSR (Math. Ser.), 43, no. 3 (1979), 714-732 (in Russian). [128] Trofnnov V. V., Euler equations on finite-dimensional soluble Lie groups, Izv. Akad. Nauk SSSR (Math. SifT.), 44, no. 5 (1980), 1191-1199 (in Russian). [129] Trofnnov, V. V., Completely integrable geodesic flows ofleft-invariant metrics on Lie groups, connected with commutative graduated algebras with Poincare duality, Dokl. Akad. Nauk SSSR, 263, no. 4 (1982), 812-816 (in Russian). [130] Trofnnov V. V., Methods for constructing S-representations, M GU "Vestnik" Ser. Math. Mekhanika, no. 1 (1984),3-9 (in Russian). [131] Trofnnov V. V., Expansions of Lie algebras and Hamiltonian systems, Izv. Akad. Nauk SSSR (Math. Ser.) (1983) (in Russian). [132] Takiff S. J., Rings of invariant polynomials for a class of Lie algebras, Trans. Amer. Math. Soc., 160 (1971), 249-262. [133] Vinogradov A. M., Kupershmidt B. A., The structure of Hamiltonian mechanics, UMN, 32, pt. 4 (1977), 175-236 (in Russian). [134] Vergne M., La structure de Poisson surl'algebre symmetrique d'une algebre de Lie nilpotente, Bull. Soc. Math. France, 100 (1972), 301-335. [135] Veselov A. P., Finite zone potentials and integrable systems on a sphere with quadratic potential, Functional Anal. Appl., 14, pt. 1 (1980),48-50 (in Russian). [136] Vinogradov A. M., Krasil'shchik I. S., What is Hamiltonian formahsm?, UMN,

288

A. T. FOMENKO AND V. V. TROFIMOV

30, pt. 1 (1975), 173-198 (in Russian). [137] Vishchik S. V., Dolzhansky F. V., Analogues of Euler-Poisson and magnetic hydrodynamics equations, connected with Lie groups, Reports USSR Acad. Sci., 238, no. 5 (1978), 1032-1035 (in Russian). [138J Wilson G., Commuting flow and conservation laws for Lax equations, Math. Proc. Cambridge Phi/os. Soc., 86 (1979), 131. [139J Wilson G., On two constructions of conservation laws for Lax equations, Quart. J. Math., 32, no. 128 (1981). [140] Whittaker E. T., Analytical Dynamics, Cambridge University Press, Cambridge, UK,196O. [141] Wintner A., Celestial Mechanics, Princeton University Press, Princeton, 1947. [142] Zakharov V. E., Faddeyev L. D., The Korteweg-iie Vries equation-a completely integrable Hamiltonian system, Functional Ana/. App/., 5, pt. 4 (1971), 18-27 (in Russian). [143] Atiyah M. F., Convexity and commuting Hamiltonian, Bull. London Math. Soc., 14 (1982), 1-5. [144] Atkin C. J., A note on the algebra of Poisson brackets, Math. Proc. Camb. Phil. Soc.,96 (1984),45--60. [145] Bogoyavlensky O. I., Integrable Euler equations on and their physical applications, Comm. Math. Phys., 93 (1984),417---436. [146] Conley C., Zehnder E., Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37, no. 2 (1984),207-255. [147] Date E., Jimbo M., Kashiwara M., Miwa T., Landau-Lifshitz equation quasiperiodic solutions and finite dimensional Lie algebras, J. Phys. A.: Math. gen., 16 (1983),221-236. [148] Dhooghe P. F., Backlund equations on Kac-Moody-Lie algebras and integrable systems, J. of Geometry and Physics, 1, no. 2 (1984),9--38. [149] Fomenko A. T., Morse theory of integrable Hamiltonian systems, Dokl. Akad. Nauk SSSR, 287, no. 5 (1986), 1071-1075 (in Russian). [150] Fomenko A. T., Topology of the constant energy of integrable Hamiltonian systems and obstacles to the integrability, Izv. Akad. Nauk SSSR, no. 6 (1986) (in Russian). [151] Fomenko A. T., Topology of three-dimensional manifolds and integrable mechanical Hamiltonian systems. Tiraspol 's V . Symposium on general topology and applications. Kischinew (1985),235--237 (in Russian). [152] Fordy A., Wojcechowski S., Marshall I., A family of integrable quartic potentials related to symmetric spaces, Physics Letters, 113A, no. 8 (1986), 395--400. [153] Gibbons J., Holm D., Kupershmidt B., Gauge-invariant Poisson brackets for chromohydrodynamics, Physics Letters, 9OA, no. 6 (1982),281-283. [154] Golo V. L., Nonlinear regimes in spin dynamics of superfluid 3He, Letters in Math. Phys.,5 (1981), 155--159. [155] Godman R., Wallach N., Classical and quantum mechanical systems of Todda lattice type. II, Comm. Math. Phys., 94 (1984), 177-217. [156] Guillemin V., Sternberg S., Symplectic Techniques in Physics, Cambridge University Press, Cambridge, London, Melbourne, Sydney, 1984, 468 pp. [157] Guillemin V., Sternberg S., Convexity properties of the moment mapping, Invent. Math.,67 (1982),491-513.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

289

[158] Guillemin V., Sternberg S., Multiplicity fry space, J. of DifJ. Geom. (1983). [159] Haine L., Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263, no. 4 (1984), 435-472. [160] Haine L., The algebraic completely integrability of geodesic flow on SO(4), Comm. Math. Phys., 94, no. 2 (1984),271-282. [161] Holm D., Marsden J., Ratiu T., Nonlinear stability ofthe Kelvin-Stuart eat's eyes flow, Proc. AMS. SIAM Summer Seminar, Santa Fe, July, 1984, 1-16. [162] Holm D., Kupershmidt B., Levermore D., Hamiltonian differencing of fluiddynamics, Adv. in Appl. Math., 6 (1985), 52-84. [163] Holm D., Kupershmidt B., Noncanonical Hamiltonian formulation of ideal magnetohydrodynamics, Physica, 7D (1983), 330--333. [164] Holm D., Kupershmidt B., Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Physica, 6D (1983), 347-363. [165] Kac V. G., Infinite Dimensional Lie Algebras, Birkhiiuser, Boston, Basel, Stuttgart, 1983. [166] Klingenberg W., Riemannian Geometry, Walter de Gruyter, Berlin, New York, 1982. [167] Kupershmidt B. A., Discrete Lax equations and differential---{}ifference calculus, Asterisque, no. 123 (1985). [168] Kupershmidt B. A., Deformations of integrable systems, Proc. Royal Irish Academy, 83A, no. 1 (1983),45-74. [169] Kupershmidt B. A., On dual spaces of differential Lie algebras, Physica, 7D (1983), 334-337. [170] Kupershmidt B. A., Supersymmetric fluid in a fry one-dimensional motion, Lettere al Nuovo Cimento, 40, no. 15 (1984),476-480. [171] Kupershmidt B. A., Odd and even Poisson brackets in dynamical systems, Letters in Math. Phys., no. 9 (1985), 323-330. [172] Kupershmidt B. A., Mathematical aspects of quantum fluids I. Generalized twococycles of 4He type, J. Math. Phys., 26, no. 11 (1985),2754-2758. [173] Kupershmidt B. A., Super Korteweg---{}e Vries equations associated to super extensions of the Virasoro algebra, Physics Letters, I09A, no. 9 (1985),417--423. [174] Kupershmidt B. A., Mathematics of dispersive water waves, Commun. Math. Phys., 99 (1985), 51-73. [175] Kupershmidt B. A., Odd variables in classical mechanics and Yang-Mills fluid. Dynamics, Physics Letters, 112A, no. 6, 7 (1985),271-272. [176] Kupershmidt B. A., Ratiu T., Canonical maps between semidirect products with applications to elasticity and superfluids, Commun. Math. Phys., 90 (1983), 235-250. [177] Marsden J. E., Weinstein A., The Hamiltonian structure of the Maxwell-Vlasov equations, Physica, 4D (1982), 394-406. [178] Marsden J. E., Morrison P. J., Noncanonical Hamiltonian field theory and reduced MHD, Contemporary Mathematics, 28 (1984), 133-1SO. [179] Marsden 1. E., Ratiu T., Weinstein A., Reduction and Hamiltonian structures on duals of semidirect product Lie algebras, Contemporary Mathematics, 28 (1984), 55-99. [180] Marsden J. E., Weinstein A., Coadjoint orbits, vortices, and Clebsch variables for

290

A. T. FOMENKO AND V. V. TROFIMOV

incompressible fluids, Physica, 7D (1983),305-323. [181] Marsden J. E., Weinstein A., Ratiu T., Schmidt R., Spencer R. G., Hamiltonian systems with symmetry, coadjoint orbits and plasma physics. Proc. IUAMISIMM Symposium on "Modem Developments in Analytical Mechanics," Turin (June 7-11,1982), Atti della Accademia della Scienza di Torino, 117 (1983), 289-340. [182] McDuff D., Examples of simply-connected symplectic non-Kiihlerian manifolds, J. Diff. Geom., 20, no. 1 (1984),267-277. [183] Medina A., Structure orthogonale sur une algebre de Lie at structure de Lie-Poisson associee. Seminar de Geometrie difJerentielle, 1983-1984, Universite des sciences et techniques du Languedoc, 113-121. [184] Montgomery R., Marsden J., Ratiu T., Gauged Lie-Poisson structures, Contemporary Math., 28 (1984), 101-114. [185] Mulase M., Cohomology structure in soliton equations and Jacobian varieties, J. Diff. Geom., 19, no. 2 (1984),403-430. [186] Ness L., A stratification of the null cone via the moment map, Amer. J. Math., 106, no. 6 (1984), 1281-1325. [187] Ratiu T., van Moerbeke P., The Lagrange rigid body motion, Ann. de /'Institute Fourier, 32, no. 1 (1982),211-234. [188] Thurston W., Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc.,55 (1976),467-468. [189] Trofimov V. V., The symplectic structures on the groups of aut omorph isms of the symmetric spaces, MGU "Vestnik" Ser. Math. Mekhanika, no. 6 (1984), 31-33 (in Russian). [190] Trofimov V. V., Deformations of the integrable systems, Analysis on Manifolds and DifJerential Equations, Voronez, VGU (1986), 156-164 (in Russian). [191] Trofimov V. V., Fomenko A. T., The Liouville integrability of the Hamiltonian systems on the Lie algebras, UMN, 39, no. 2 (1984), 3-56 (in Russian). [192J Trofimov V. V., Fomenko A. T., The geometry of the Poisson brackets and the methods of the Liouville integration of the systems on the symmetric spaces, Sovremen. Probl. Math. VINITI,29 (1986), (in Russian). [193J Weinstein A., Symplectic manifolds and their Lagrangian submanifolds, Adv. in Math., 6 (1971), 329-346. [194J Weinstein A., Symplectic geometry, Bull. Amer. Math. Soc., S (1981), 1-13. [195] Weinstein A., The local structure of Poisson manifolds, J. of DifJ. Geom., 18, no. 3 (1983), 523-557. [196] Le Hong Van, Fomenko A. T., Lagrangian manifolds and Maslov's index in the theory of minimal surfaces, Dokl. Akad. Nauk SSSR (1987) (in print). [197] Brailov A. V., Some constructions of the complete sets of the functions in involution, Works of the Seminar on Vector and Tensor Analysis, pt. 22 (1985), 17-24 (in Russian). [198] Brailov A. V., Construction of the completely integrable geodesic flows on compact symmetric spaces, Izv. Akad. Nauk SSSR (Math. Ser.), SO, no. 4 (1986), 661-{i74 (in Russian). [199J Brailov A. V., Some cases of the complete integrability of the Euler equations and applications, Dokl. Akad. Nauk SSSR, 268, no. 5 (1983), 1043-1046 (in Russian). [200] Brailov A. V., The complete integrability of some geodesic flows and complete systems with non-commutative integrals, Dokl. Akad. Nauk SSSR, 271, no. 2

INTEGRABLE SYSTEMS ON LIE ALGEBRA

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(1983),273--276 (in Russian). [201] Brailov A. V., The complete integrability of some Euler's integrals. Applications of the topology in modern analysis, Voronez VGU (1985). 22-41 (in Russian).

Selective key to the notation used

Ill-the set of all real numbers. C-the set of all complex numbers. Z-the set of all integers. Q-the set of all rational numbers. [Rn-n-dimensional real linear space. W t 1\ w 2 -the exterior product of differential forms W t and W 2 . Xl, XT-the matrix transpose of a matrix X. Tij-the elementary matrix: (7;)pq = (c5 i/jjQ ). Eij = 7;j - 1}i-the elementary skew-symmetric matrix. li,jl = 7;j + 1}i-the elementary symmetric matrix, adt(x) = (ad ~)* x = a(x,~) where ~ E G, X E G* is the coadjoint representation of the Lie algebra G in the space G* dual to the Lie algebra G. R(X, Y}--the Cartan-Killing form. <X, Y )-pairing between the space V and the space V* that is dual to it. COO(M}--the space of all smooth functions on smooth manifold M. H(M)-the space of all Hamiltonian vector fields. A(W)-the space of analytical functions on space W. V(M)-the Lie algebra of all vector fields on a smooth manifold M under the commutator of vector fields. Exp-the exponential mapping Exp: G -+ ffi of the Lie algebra G into the Lie group ffi. F(M)-the full commutative set of functions on a symplectic manifold M. Reg(G)-regular elements of Lie algebra G. Exp G-the Lie group corresponding to the Lie algebra G. Qk(M)-the space of the differential k-forms on the manifold M.

j j j j j j j j j j j j j

Index

Adjoint representation, 18 Affine Lie algebras, 219 Algebra with Poincare duality, 162 Algebraic variety, 8 Argument translation, 137 Bifurcation diagram, 271 Bounded domain, 8 Canonical H-invariants, 168 Cartan-Killing form, 20 Cartan subalgebra, 20 Case of Steklov, 85 Chain subalgebras, 144 Coadjoint representation, 18 Compact series, 75 Complete torus, 270 Complex semi-simple series, 71 Condition (FJ), 48 Configuration space, 5 j-connective vector fields, 52 Contraction of the Lie algebra, 174 Cylinders, 270 Dissipative complete torus, 271 Dynamic tensor, 56 Embedding of the dynamic system into a Lie algebra, 55 Equations of magnetic hydrodynamics, 89 Euler's equation, 55 First case of Clebsch, 84 Fubini-Studi metric, 7

Functions in involution, 30 Geodesic flow, 10 Hamiltonian field, 8 Index of the Lie algebra, 33 Integral, 12 Invariant, 27 Kiihler manifold, 7 Kirchhoff integrals, 83 Lagrange case, 210 Lie algebra, 17 Lie group, 17 Locally Hamiltonian vector field, 9 M-condition, 169 Morphism of symplectic manifolds, 53 Morse-type integral, 267 n-dimensional rigid body, 61 Non-oriented saddle, 270 Non-resonance Hamiltonian, 275 Normal nilpotent series, 75 Normal series, 75 Normal solvable series, 75 One-parameter subgroup, 17 Oriented saddle, 270 Poisson bracket, 11

293

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Realization in a symplectic manifold, 54 Reduction of a Hamiltonian system, 37 Regular element, 20 Root, 21 Second case of Clebsch, 84 Sectional operator, 68 Sectional operators on symmetric spaces, 104 Semi-invariant, 27 Similar functions, 168

Simple root, 21 Skew-symmetric gradient, 2 S-representation, 142 Stable trajectory, 267 Submersion, 52 Symplectic atlas, 4 Symplectic coordinates, 3 Symplectic manifold, 1 Symplectic structure, 1 Toda chain, 62