Integrable systems on Lie algebras and symmetric spaces

INTEGRABLE SYSTEMS ON LIE ALGEBRAS AND SYMMETRIC SPACES

Advanced Studies in Contemporary Mathematics A series of books and monographs edited by R. V. Gamkrelidze, V.A. Steklov Institute of Mathematics USSR Academy of Sciences, Moscow, USSR Volume 1

Geometry of Jet Spaces and Nonlinear Partial Differential Equations I.S. Krasil'shchik, V.V. Lychagin and A.M. Vinogradov Volume 2 Integrable Systems on Lie Algebras and Symmetric Spaces AT. Fomenko and V.V. Trofimov Additional volumes in preparation

Lagrange and Legendre Characteristic Spaces V. A, Vasilyev Some Classes of Partial Differential Equations A. V, Bitsadze ISS,N:0884-0016. This book is part of a series. The publishers will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

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.D. Rayfield and A. Weisman

Gordon and Breach Science Publishers New York

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Q 1988 by OPA (Amsterdam) By. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A. Gordon and Breach Science Publishers

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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 ISBN 2-88124-170-0

87-26798

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

1. Symplectic Geometry and the Integration of Hamiltonian Systems 1.

Symplectic manifolds

1.1.

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

1.2. 1.3. 1.4. 1.5.

2.

2.1.

Symplectic Geometries and Lie Groups Summary of the Necessary Results on Lie Groups and Lie Algebras

xi

1

1

1

4 8 11 15 17 17

2.2. Orbits of the Coadjoint Representation and the Canonical

Symplectic Structure Differential Equations for Invariants and Semi-Invariants of the Coadjoint Representation

22

3.

Liouville's Theorem

30

3.1. 3.2. 3.3. 3.4.

Commutative Integration of Hamiltonian Systems Non-Commutative Lie Algebras of Integrals Theorem of Non-Commutative Integration Reduction of Hamiltonian Systems with Non-Commutative

30

Symmetries

36

2.3.

Orbits of the Coadjoint Representation as Symplectic Manifolds 3.6. The Connection between Commutative and NonCommutative Liouville Integration

27

32 34

3.5.

46 47

Vi

CONTENTS

Algebraicization of Hami Ionian 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 4.

5.

Complete Commutative Sets of Functions on Symplectic Manifolds

2. Sectional Operators and Their Applications 6.

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 7.2. Hamiltonian Systems of the Compact and the Normal 7.

52 52 58 63

67

67 71

7.1.

Series

71

75

Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Ideal Fluid 7.4. Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible Ideally Conductive Fluid

7.3.

3. Sectional Operators on Symmetric Spaces 8. 9.

79 89

100

Construction of the Form Fc and the Flow XQ in the Case of a Symmetric Space

100

The Case of the Group S _'$ Spaces of Type II)

105

x Sj)/.5 (Symmetric

The Case of Type 1, III, IV Symmetric Spaces 10.1 Symmetric Spaces of Maximal Rank 10.2. The Symmetric Space Sn' = SO(n)/SO(n - 1) (The Real 10.

Case)

10.3. Hamiltonian Flows XQ, Symplectic Structures Fc and the Equations of Motion of Analogs of a Multi-Dimensional Rigid Body 10.4. The Symmetric Space S"-' = SO(n)/SO(n - 1) (The Complex Case) 10.5. Examples of Flows X, on S"-' (The Complex Case)

107 107 111

120 121 131

CONTENTS

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

11.

12.

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

vii

136 136

136

138

143

Method of Tensor Extensions of Lie Algebras 13.1. Basic Definitions and Results 13.2. The Proof of the General Theorem 13.3. The Application of the Algorithm (21) to the Construction of S-Representations 13.4. Algebras with Poincare Duality

147

Similar Functions 14.1. Partial Invariants 14.2. Involutivity of Similar Functions

167

Contractions of Lie Algebras 15.1 Restriction Theorem 15.2. Contractions of 7L2 -Graded Lie Algebras

171

13.

14.

15.

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

147 151

160 162

167 168

171 174

179

16.

17.

Cases of Complete Integrability of the Equations of Inertial Motion of a Mufti-Dimensional Rigid Body in an Ideal Fluid

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 f2(G) of Semi-Simple Lie Algebras

179

179 185 189 194

18.

198

198

Viii

CONTENTS

18.2. Complete Integrability of a Geodesic Flow on 18.3. Extensions of fl(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.

20.

The Integrability of Certain Hamiltonian Systems on Lie Algebras

205

205 208 212

214

20.1. Completely Involutive Sets of Functions on Singular Orbits in su(m)

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

215 219

21.

22.

224

224 232

Integrability of Eider's Equations on Singular Orbits of Semi-Simple Lie Algebras

22.1. Integrability of Euler's Equations on Orbits 0 Intersecting the Set tHa, teC 22.2. Integrability of Euler's Equations x = [x, (J1abD(x)] for Singular a 22.3. Integrability of Euler's Equations z = [X,(QabD(x)] on the Subalgebra G. Fixed Under the Canonical Involutive

Automorphism a:G-+G for Singular Elements aeG 22.4. Integrability of Euler's Equations for an n-Dimensional Rigid Body Completely Integrable Hamiltonian Systems on Symmetric Spaces 23.1. Integrable Metrics dspbD on Symmetric Spaces 23.2. The Metrics dsob on a Sphere S" 23.3. Applications to Non-Commutative Integrability

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

66

CONTENTS

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

lx

266 270

Bibliography

281

Index

293

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 of functions 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 quasipcriodic. 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/aq)) = 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 Y_k.J C'Jxk(aF/ax;) = 0. 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 >k,j

0, then

F(x + Aa) is the first integral for any Ac l8, 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 xi

xii

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 of the 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 integration of Hamiltonian systems

1. SYMPLECTIC MANIFOLDS I.I. 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.

A smooth even-dimensional manifold M2 is called symplectic if it has defined on it the external differential second-degree form to = Y-; <j coi; dxi n dx', so that (1) this form is non-degenerate, i.e. the matrix of its coefficients I wij(x) jj is non-degenerate at each point, (2) this form is closed, i.e. dw = 0, where d is the operation of exterior DEFINITION 1.1

differentiation (see [24], [120]). This form co is sometimes called a symplectic structure on a manifold.

It is clear that form w defines in a tangent space TAM2" a nondegenerate skew-symmetric scalar product w(a, b) = Ei,j wi;aW where a = (a'), b = (hi), a, b e TAM. If the point xo is fixed, then, as is apparent I

2

A. T. FOMENKO AND V. V. TROFIMOV

from algebra, there exists a transformation of coordinates in the tangent

space Tx0M, generated by a regular transformation of coordinates x1, ... , x2n in the neighborhood of the point x0, so that the matrix 1w1i(xo)II takes the canonical form:

where D = 0

D

This scalar product evidently defines a canonical ide tangent space TxM with the cotangent space T *M (on this operation see

[24], [120] for more detail). Let us bear in mind that the dual space

T*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 of the 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]). DEFINITION 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.

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

(fix

' 8x

") e T*M

and then, relying on the canonical identification TxM 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 Co. Usually local coordinates on a symplectic manifold are denoted by , q,, where these coordinates are chosen in such a way PI,. , p", 41, .

3

INTEGRABLE SYSTEMS ON LIE ALGEBRA

that at a fixed point x0 the matrix Ilw;;(xo)II is written as:

0

0,

(-E

where E is the (n x n) unit matrix. But if the coordinates are arranged in then the matrix may be the following order pt, qI, P2, q2, ..., p,,,

written in block-diagonal form, as shown above. Given this special choice of local coordinates pI, ... , pn, q,,. . . , qn, the vector fields grad f at the point x0 can be written particularly simply. In fact, since

(L.3f\ EToM, x

aPt

then

s grad f =

aqn

of of - of of agl,...'agn' apt,..., aR,

In the coordinates pl, q1, ../. , p,,, qn we have: s grad f (xo) =

of , - of ... , of

\\0q1

- f)

aqnOpn apt 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 x0. 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 prevent this. A symplectic structure, on the other hand, can always be reduced to canonical form by transforming coordinates in a small neighborhood of the point (the size of the neighborhood being defined by the properties of the form). PROPOSITION 1.1

Let o) be a symplectic structure on Men. Then for any

point xo e M there exists an open neighborhood with the local coordinates pt, ... , pn, qt, ... , q2, so that in these coordinates the form to can be written in the simplest canonical way : w = i= t dpi A dq;.

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 182n(p1, provided with the form co = dpl A dql + + dp" A dqn.

,

qn),

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 even-

dimensional, 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. n Aw Let w be a symplectic structure on M2k, then 0 = w A (the product is taken k times) gives a form of degree 2k on M2k. We shall show that f) # 0. Indeed, in every tangent plane TXM2k of M2k there can

be found a system of coordinates x' Co = x1 n yl +

xk y1

,

yk such that

+ xk n yk. Then, clearly,

f j = ! _ 1)[k(k-1)]/2kl x1 A A xk A y1 A ... A yk

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 even-

dimensional sphere Stn in the case of n > 1. Proof Let there be on Stn (n > 1) the differential form co, which defines

a symplectic structure on S2n. Then dw = 0 and since the space of cohomologies H2(S°) = 0 where p >, 3, then co = dot for a certain one A CO = dimensional differential form a. It is obvious that CO n A dot) (the external product is taken n times). On the d(a n da n

A co is not zero at any point other hand, as we saw above, 0 = CO n on the sphere Sen. Therefore f2 = f vol where f is some smooth function and vol is the volume form on Sen. Thus, using Stokes' theorem (see, A dot) = 0, but for example, [120]), we have j s2. 0 = f d(a A da A IfS=" f volt >, m J S2. vol = mv(S2") # 0, where v(S2") is the volume of sphere S2" and m is some constant, m > 0. This proves the proposition.

5

INTEGRABLE SYSTEMS ON LIE ALGEBRA

It is clear that the proof of Proposition 1.2 works for any compact manifold M such that HZ(M, l) = 0. For example, on a REMARK

compact semi-simple Lie group there are no symplectic structures, since

H2(6, R) = 0 for the compact semi-simple Lie group 6. 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 co = EG - FZ du A dv, where E = (r,,, r.), F = (r,,, r.), G = (ru, rv) and r,,, r 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 3

3

2

z2 + Lye + (a' - xz) fl (x -a i)2 fl (x + a.)2x2J = Ez i=1

a,

i=1

a, (i A j) where s is sufficiently small, describes a sphere with eight

handles. An analogous equation can be written for a sphere with n 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 T ,*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 T*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 1-form co"), Let a e Tq(T *M) be a tangent vector of the cotangent bundle T *M at the point y e T ,*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

A. T. FOMENKO AND V. V. TROFIMOV

6

P_ f

Fig. 1.

x = p(y) = p(x, ). Now we shall define the differential 1-form on the space T*M in the following way: co("(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 co"),

i.e. w = da ' The form we have constructed is closed and non.

degenerate, 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 q', ... , q", n = dim M.

Let us examine U c T *M; these are covectors whose point of apposition is in U. We can examine the basis fields a/aq',... , a/aq". Let the covector values on these fields be pl, ... , p.: we may take (q1, ... , q", pt, ... , p") as the coordinates in T*M in the neighborhood U. co, the 2form constructed in these coordinates, has the classical form:

w=dp1 Adq'+...+dp" Adq" (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 co which gives a symplectic structure is exact, i.e. co = da for a certain 1-form a. The third source of symplectic manifolds are Kahler manifolds. Let MZ" be a complex manifold (see [24], [48]), on which a Hermitian scalar product rl) is given. Let us examine 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 MZ", it is essential to have the equality dw = 0. This requirement is not met in an arbitrary complex manifold with a Hermitian metric. DEFINITION 1.3

A complex manifold, provided with a Hermitian

metric, is called a Kahler manifold if the imaginary part co of the scalar product 1) is a closed differential form (dw = 0). Thus, all Kahler manifolds are symplectic. The converse, however, is not generally true (see [188], [182]). One classic example of a Kahler manifold can be found in the complex projective space CP". We have the

natural holomorphic mapping n: c"+'\0 - CP". We may examine C""\0 the covariant 2-tensor

f

Ek"=0 / 1zkzk)z

J\ > ZkZkE k

dzk ®dZk/

k

where z0, .

. .

(zkdzk)®( zk®dzk/},

, z" are the standard coordinates in C"+I

There exists on CP" the Kahler metric F such that 7r*F = P, where f is the form defined above. PROPOSITION 1.3

This statement is derived from the following four evident properties of

tensor F:

a) The restriction of P to the mapping fiber of n : C""\0 -+ CP" equals zero.

b) The tensor P is invariant in relation to the natural action of the group C* = c\0 on Cn+1\0: z(z0, ... , z") = (zz0.... , zz"), z E C*. c) The restriction of P to the orthogonal complement of a fiber with respect to the flat metric in C" is positive definite. d) The differential 3-form d(Im F) on CP" is invariant under the mappings induced by unitary transformations A of the space C"+1, AEU(n+ 1). The metric F constructed on CP" is called the Fubini-Studi metric (for details see, for example, [1]). We are now able to construct a rich store of symplectic manifolds.

A. T. FOMENKO AND V. V. TROFIMOV

8

DEFINITION 1.4

Any/ subset of the form

V(f1.... JN) _ {P = (ao:...: a.) E CP": f1(aO:...: a") _ ... = fN(aO:...: a") =

o},

where { f1, ... , fN} is any set of homogenous polynomials in the ring C[X0, ... , is called an algebraic variety in CP".

If grad f (i = 1, ... , N) are linearly independent, then the algebraic variety V(f1,... , fN) is a complex manifold embedded in CP", this being an immediate consequence of the theorem on implicit functions. Let

j: V(f1, ... , f,) - CP" be the inclusion of the complex manifold V(f1,... , f ,) into complex projective space CP", II v(f,,.,,, f.^,) = j * Im F where F is the Fubini-Studi metric. The differential form SZ v( f '...'fN)

gives a symplectic structure on the manifold V(f1,... , fN). This statement results from V(f1i .

.

.

,

fN) being a complex submanifold

of CP". The construction that we have set out gives us examples of compact Kahler manifolds.

DEFINITION 1.5 We shall call a bounded open connected subset in the space CN a bounded domain.

Any bounded domain is a Kahler manifold (see [48]) and therefore a symplectic manifold. 1.3. Hamiltonian vector fields

DEFINITION 1.6

A smooth vector field v on a symplectic manifold M

with the form co 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. In special symplectic coordinates (p;, qt) the Hamiltonian vector field

is written as (8F/8q;, -8F/8p;) (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 ( be a one-parameter group of diffeomorphisms of M, represented by translations along the integral trajectories of the field v. This means that group 6" consists of transformations of g, operating on

INTEGRABLE SYSTEMS ON LIE ALGEBRA

9

3c=,V(O) Pxo

Fig. 2.

M thus: g,(x) = y, where x = y(0), 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 co is defined on M the diffeomorphism g, transforms this form into a new one (g*w)(x) = w(g,(x)). Consequently

the derivative of the form w is defined along the vector field v, i.e. d/dt(g*w). A vector field v on the symplectic manifold M is called locally Hamiltonian if it preserves the symplectic structure co on M, i.e. the derivative of form win the direction of vector field v is equal to zero: d/dt(g*w) = 0. To put it another-way+-,form w is invariant with respect to DEFINITION 1.7

all transformations of type g, generated by field v, i.e. is invariant in relation to the operation of the one-parameter group (i°. The term "locally Hamiltonian field" owes its derivation to the following: A smooth vector field v on a symplectic manifold M is locally Hamiltonian if, and only if, there exists for any point x E M a neighborhood U(x) of this point and a smooth function Hv defined in this neighborhood so that v = s grad Hv, i.e. field v is Hamiltonian in the neighborhood of U with the Hamiltonian Hu. PROPOSITION 1.4

For proof, see for example [1], [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 Hv on 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 Hv 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.

10

A. T. FOMENKO AND V. V. TROFIMOV

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 0, 0 degree two g, is given on M so that (a) g1, = gji, (b) S. (see [24], [166]). We shall define g'i by the requirement that g'Jg;k =k The Riemann metric g;j defines a scalar product in the cotangent bundle p) = n e T*M. We have a canonical symplectic structure on T *M. We can examine the Hamiltonian H(x) =,L g''p; p, 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 H(x) = 1} is invariant under the flow s grad H.

The restriction of the flow z = s grad H where

DEFINITION 1.8

H = 129''p; pj to the invariant surface S is called a geodesic flow on the Riemannian manifold M.

Metric g;j 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 (xo, po) to a pair (xt, pt) = gt(xo, po) where the geodesic line must be taken through x0 e M in the direction po in order to obtain xt. xt will then be at distance t along the geodesic line from the point x0 whereas vector p, is tangent to this line at x, and has the same direction as po. Let M be a surface in 1183 given locally in the graph form z = f(x, y); (x, y) will then be a local system of coordinates on M. Let p, py, be the corresponding coordinates in T*M. In this case the Hamiltonian of the geodesic flow has the following form:

H_(1+ f2(x,Y))Ps-2TsfP1p +(1+fZ(x,Y))Py 2(1 + fz + f2) T *M = l 2(x, y) © R2(Px, pr)

In one sense geodesic flows are universal Hamiltonian systems: according to the Maupertuis principle any Hamiltonian flow with the Hamilton function H = i Y al(q) p; pj + V(q), where a'' is a positive

INTEGRABLE SYSTEMS ON LIE ALGEBRA

11

definite matrix, coincides with the geodesic flow on the manifold of constant energy H(q, p) - It for the metric ds2 = (h - V(q))a;; dq' dq' (see for example [1], [29]). 1.4. The Poisson bracket and Hamiltonian field integrals

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

w;;(s grad f)'(s grad g)'.

{f, g} = w(s grad f, s grad g) _ i<;

In other words the skew-symmetric scalar product of the "skew gradients" of f and g must be calculated. In its explicit form, in terms of

the partial derivatives of functions f and g, the Poisson bracket is written thus: Y_

i<j

Of d9 8x` 8x'

where x' are the local coordinates and w'' the coefficients of the matrix inverse to the matrix II w;; II We have here made use of the fact that (s grad f)' = E; w'' 8f/8x'. 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, g -> { f, 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 }} + { f, { g, h}} = 0 holds for any functions f, g, h. Thus the infinite dimensional linear space C°°(M) of smooth functions F on a smooth symplectic manifold M is naturally an infinite Lie algebra

over the field R. 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 C°°(M). We may construct a natural mapping at of the Lie

12

A. T. FOMENKO AND V. V. TROFIMOV

algebra C°°(M) into the Lie algebra V(M) of all smooth vector fields on

manifold M. This mapping can be defined thus: a(f) = s grad f. The mapping a: C°°(M) -+ V(M) is a homomorphism of Lie algebras, i.e. a{ f, g} = [a(f ), a(g)]. This means that the operation of LEMMA 1.1

taking the Poisson bracket changes, under the mapping a, into the operation of taking an ordinary commutator of the two vector fields a(f), a(g).

The proof follows from Proposition 1.5, and the definition of the operation s grad. The image of the Lie algebra C°°(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 f, 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 a: C°°(M) - H(M) is an epimorphism, but not a monomorphism, inasmuch as a has a non-zero kernel. If the manifold is connected the kernel consists of functions

which are constant on the manifold. Consequently Ker a is onedimensional and H(M) = C°°(M)/Kera = C'(M)/W. 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 co, the subalgebra H(M) will also be changed: strictly speaking we ought to write it as H.(M), but we shall take the subscript co for granted and omit

it. The Poisson bracket of functions f and g has the following simple interpretation If, 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) = E; v` af/ x`. 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.

Let v = s grad F be a Hamiltonian field on M, and f be a smooth function which commutes (in the Poisson bracket sense) PROPOSITION 1.6

INTEGRABLE SYSTEMS ON LIE ALGEBRA

13

Fig. 3.

with the Hamiltonian of this field, i.e. { f, H} = 0. Then the function f is the integral of the field v. -

Proof If we calculate the derivatives of f along the field v, we have The v(f) = w(v, s grad f) = w(s grad F, s grad f) = {F, f } = 0. 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.

If f and g are two integrals of a Hamiltonian field v = s grad F then their Poisson bracket { f, g} is also an integral of the PROPOSITION 1.7

field.

Proof From Jacobi's identity we obtain:

IF, { f,g}} _ -{g, {F, f}} - {f, {g, F)) = 0. 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 g (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 M2 , since in the local coordinates x', ... , xZ" the field v is written as x' = v'(x', ... , xZ"), where v' are the components of v, while z' 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 borne 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 M into 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 Q211 - z, 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 fl,... , would be one-dimensional trajectories coinciding with the initial system's integral trajectories.

One more function is formally required in order to give the move-

ment 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 f1,. , f2,,- I 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 fl, ... , 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 Q2, `'; (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

16

A. T. FOMENKO AND V. V. TROFIMOV

are local coordinates, then det{x' , x'} 0). Degenerate brackets are also of some interest. They are connected with foliating symplectic structures, as the following statement shows. X`

Let a (degenerate) Poisson bracket be given on manifold TM be the corresponding Hamiltonian operator: = {'p, o}(m), m e M. Then (1) the distribution m -> I',.(T*) in M is integrable and (2) the operator F induces a symplectic structure on the distribution's integral manifolds. LEMMA 1.2

M. Let F: T*M

Here is an example of a degenerate Poisson bracket arising in a concrete geometrical situation (see [189]). THEOREM 1.2

Let M be a manifold having a torsion-free affine

connection and R(M) be the principal frame bundle over M, while n: R(M) -+ M is its projection, then an n-parameter family (n = dim M) of closed 2-forms w e W(R(M)), dw = 0 exists on R(M).

Proof On the manifold R(M) are defined the structural differential forms w', w; E f21(R(M)); n*(5) = w'(4)e;, where E TX,Q,) R(M) and if = d/dtJ,_0(x(t), (e;(t))), then De;/dt = where V/dt is the

covariant derivative arising from the affine connection on M. These forms satisfy the structural equations: dw' + wk A wk = zS,gw° A wq A CO", dw; + ws n wj = where Sam, is the torsion tensor and R' pq the Riemann curvature tensor

[120]). The definition gives us the value co a;w A w E f22(R(M)), aj = const. We claim that dw = 0. In fact by using structural equations we arrive at: (see

dw =

[a;(dwj A wj - w; A dwJ]

= 2 Z a, Ri,. wp A wq A w' -

1\

2

ai S'nq wj l w° A CO.

Since Spq = 0 Ricci's identity RJ,pq + R,.qi + Rq jp = 0 is valid, and therefore dw = 0. REMARK An analogous structure may be carried out for any principal

0 bundle. Using the form co to define the skew gradient s grad f we find s grad f

17

INTEGRABLE SYSTEMS ON LIE ALGEBRA

can be defined exactly modulo the elements from the form w's kernel. Form co gives a bundle mapping A: TM -p T *M, A(s)(y) = co(s, y), s, y being sections of the bundle TM. Let f, g c C°°(M) be smooth functions so that df, dg e Im A, it will then be possible to define correctly their Poisson bracket {f, g} = w(s grad f, s grad g). This definition does not depend on the choice of representatives for s grad f If df 0 Im A then the definition gives us the value t f, g} = 0 for any g e C°°(M). In this case, therefore, C°°(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 submanifolds.

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

Let (h be a smooth manifold on which the group structure is given. (f) is then called a Lie group if the mapping 6 x (5 -+ (5 given by the formula (a, b) -+ ab-' is smooth. DEFINITION 2.1

Let v c TQ(5, then, dispersing v by means of left shifts over the entire group 6, we shall obtain a vector field on (5. To be more exact we can define Lo(g) = ag for ac-T9 and, since (La)-1 = La this is a diffeomorphism of the group Cf). We may put _ (dL0)e(v), La: (5 _+ 6.

The vector field constructed is left-invariant:

bab

for all

a,bcCC).

The space of left-invariant vector fields is a Lie algebra G under the commutator (bracket) product of vector fields. This Lie algebra is finite-

dimensional: dim G = dim (5. It is called the Lie algebra of the Lie group 6. If is a left-invariant vector field on (5, then generates a certain globally defined group of diffeomorphisms. The smooth homomorphism

: R' -+ 6 is called a one-parameter subgroup of the Lie group 6. It is easy to verify that the left-invariant field generates a one-parameter subgroup of (5. A Lie algebra can thus be defined by one of four 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 R. All

one-parameter subgroups can be gathered into one universal mapping

18

A. T. FOMENKO AND V. V. TROFIMOV

Exp: G -+ 6 where Exp(tX): R --+ G - 6 gives a one-parameter subgroup with velocity vector X at the unit e e 6. Group 6 acts on itself by means of the conjugation (91,92) -' 91929,.1 . This operation then induces a linear operation on

DEFINITION 2.2

the Lie algebra Ad: 6 -+ GL(G), which is called the adjoint representation of the group 6: Ad9 = The group's adjoint representation induces that of the Lie algebra: ad = d(Ad)e: G -+ Hom(G). The equality ad,(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 of linear mappings f : G - R. We can

define the coadjoint operation Ad*: 6 - GL(G*): (Ad9 f)(x) _ f (Ad. _ 1x). This representation's differential is called the Lie algebra's ad*: G - Hom(G*). The equality coadjoint representation: (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 [156]). 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 6 be the group of affine transformations of the straight line x -+ ax + b, a 0, a, b E R. This group allows a matrix realization: a # 0, a, b e R1.

There is no difficulty in showing that

G= TQ6=

ac6, EG.

and

If

a= (0

1),=(0

02)

then:

Ada = CY1 0

alb

q1 q2 SIa2)=(0 0).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

19

The explicit form of the coadjoint representation Ad gives Jnl =FY q2 = b2a1 - 1a2

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 1

el

0

0

0)'

e2 =

0

1

0

0)

but in G* the conjugate basis fl, f2 : f (e) = 5;;. In this basis we have the coordinates x1, x2. Simple calculations show that Ad* (0

(x1,x2) _ (x1 - x2a2,x2a1), 12)

therefore the orbits Ad* are arranged as shown in Figure 7. In particular

group 6 gives an example of representations Ad and Ad* not being equivalent.

If there exists on G a non-degenerate scalar product (X, Y) so that (Adg X, Adg Y) = (X, Y) then the adjoint and coadjoint representations are equivalent, i.e. they have identical orbits. PROPOSITION 2.1

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. 1 (1985), 507 510. In particular all the Lie algebras that we call semi-simple satisfy this condition. The bilinear form B(X, Y) = tr(adx ad,.) 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], [50]). The maximal Abelian subalgebra H c G such that adh for all h e H is a semi-simple linear transformation of G, called a Cartan subalgebra. If X e G is an arbitrary element then we use G(X, 0) to denote the subspace of elements in G that commute with X. The element X e G is called regular if dim G(X, 0) is

minimal. If X E G is a regular element, then G(X, 0) 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 a(h) on H is

called a root if there is an element E. e G, E, # 0, such that [h, E,] = a(h)E, for any h E H_ Let G2 be an eigen-subspace corresponding to a. Then G = H ED Y-,#o G1, where the sign Q signifies the straight sum of linear spaces. In the semi-simple Lie algebra G all subspaces G' are one-dimensional in the case of a # 0 (over the field C). We know that [G', GO] c G'+0, i.e. [E., EO] = N,8E,+0. If a + f # 0 then E. and E. are orthogonal with respect to form B(X, Y). Vectors E,

and E_, 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 a, fi, a + fi are non-zero roots, then [G', GO] = G'+p;

the only roots proportional to root a # 0 are 0, ±a. Roots a can be represented by vectors Ha EH. Since B(X, Y) is non-degenerate on H, for every a e H* (H* is the dual space of H) there is a unique element

H' e H such that a(h) = B(h, Ha) for all h e H. Then, if a # 0, then [E X] = B(E X)H' for X e G and B(a, a) # 0. We denote by Ho c H the subspace generated by all the vectors HQ with rational coefficients. Ho is a "real" part of H. It turns out that dimo Ho = dime H = 2 dimR H (here 0 is the field of rational numbers).

21

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Furthermore, the restriction of the form B(h, h') on Ho is positive definite and takes rational values (h, h' E Ho); a(h') E 0 wherever a # 0. In particular a(h') is a real number if h' e Ho. In future we shall use A to denote a set of non-zero roots of G. Let H,,. . . , H, be any fixed basis in

Ho. If A, ,u are two linear forms on Ho then it is said that A > p if 1(H;) = µ(H;) given i = 1, 2, ... , k and A(Hk+l) > u(Hk+l). We should not forget that if 2, y are roots, then 1(h'), µ(h') are real numbers for any h' e Ho. Thus a linear ordering is defined in the set A. The root a e A is called positive if a> 0, i.e. a(H1) = 0 given i = 1, 2, ... , k and a(Hk+1) > 0. Root a'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 H1..... 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 0+ and A- which is given by the involution a -+ -a. It is clear that if ac-A' then (-a) e A-. The positive root a is called simple if it cannot be represented as the sum of two positive roots. If r = rk G = dims H, there then exist exactly r simple roots al,... , a, which form a basis in H over C and a basis in Ho over Q. Moreover each root /3 E A can be represented in the form /3 = > m;a;, where m; e Z are integers of the same sign; if m; 3 0 then fl e 0+ and if m; c 0 then l E A-. The system of simple roots al , ... , a, is usually denoted by II. The system A + is defined uniquely by

the system 11. If we let V+ = >,>o G2, V- = E,
Ho (over 0) and in H (over 0). This basis can be supplemented by vectors E. E G2, a - 0, a e A. Vectors E. may be chosen so that B(E E_,) = 1. The commutation operation in G is then written as follows:

[h, E,] = a(h)E2,

h eH (a(h) a Q, if h eHO)

[E,,, E-s] = -Ha;

[E E0] = 10,

a+p#0 root a + p # 0 non-root

B(h, H') = a(h), h e H. The vectors E. E Gs may be chosen so that NP = N

The constants

N,, may be taken to be real (after the appropriate normalization of

22

A. T. FOMENKO AND V. V. TROFIMOV

vectors E J. See the algorithm for defining N,, 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 6: Ad*: (ti - GL(G*), G* is the dual space of the Lie algebra G. If f c- G*, then

O(f) = {Ad,f I g e 61 is called the orbit passing through the point f e G*. Our immediate aim is to prove the following theorem.

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

THEOREM 2.1

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 theorem that we given is based on direct calculation. We may break it down into a number of lemmas (see also 4.5). Let 0(f) be the orbit of the representation Ad* of the Lie then Tf 0(f) = {ad* f I e G} c G*. LEMMA 2.1

group 6 passing through the point f e G*,

Proof Any vector tangent to 0(f) at the point f E G* has the form v = d/dtJ, = o Ad*Ep,4 f E Tf 0(f) for a certain e G. Let e1,. .. , e" be the basis of G, and el, . . . , e" be the basis of G* conjugate to the e;, i.e. e'(ej) = S';

if f E G*, then f = f e', f is a linear function on G*. We have =of (AdExp`gf)

dt t=O (AdEXPf{f)(ei) d

dt

r

of

= f (-

(AdEXP(-r4) ei) = f (dt

r=o

ei)

e,]) = - (ad*f)(e1)

and therefore v = -ad* f, q.e.d. DEFINITION 2.3

If ,,eTfOff) then, according to what we have

proven, it can be taken that q) = f([ I, n,])

= ad*, f,

ad*1 f and we define

We must check the correctness of the definition. Generally speaking,

23

INTEGRABLE SYSTEMS ON LIE ALGEBRA

bl,gl are not uniquely defined. Let the subspace Ann(f) = { E G I ad4 f = 0} be called f e G*, the annihilator of f Subspace Ann(f) is easily seen to be a subalgebra. It is obvious that = ado, f = adf2 f if, and only if, 1 - 2 E Ann(f ). We must show in order to prove the correctness that f

qi + qo]) = .f([S 1, q1]), where

o, qo 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 Tf0(f). This means 0 or (b) if that either (a) detjjw1fJJ q) = 0 for any c TfO(f), then q = 0. We may check the second statement. Let q) = f ql]) = 0 for any 1, where = ad*1 f, q = ad,* f Then (ad* 0 for any bl e G, therefore ad* f = 0 and consequently q = ad*1 f = 0. dimension of each orbit representation is even. COROLLARY The LEMMA 2.2

in

the coadjoint

The form co is invariant with respect to the coadjoing

action d(Adh)w9 = wf (see Figure 8).

Proof We note the following obvious equality Ade* ad*, f = f). The mapping Adh is linear and therefore d(Adh) = Adh ; when the mapping is restricted to a submanifold the differential is restricted to the corresponding tangent plane, so that

d(Adh)g = Adh . Now the invariance ensues from the following calculation : q) =

d(Adh )q)

= w9(Adh , Adh q) = w9(Adh ad41 f, Adh ad*, f) = w9(ad**Ad,4,(Adn* f), ad** ,,,(AdI*, f))

Fig, 8.

24

A. T. FOMENKO AND V. V. TROFIMOV

= wg(ad a, , g, adZd,,, 9) = g(CAdh I, Ad,, n11) = g(AKKdh[4I,'TI))

=(Adh , Here Adh f = g and therefore f = Adh-, g and

wJ( ,q), = ad*, f,

ad*, f

Thus in order to give co on the entire orbit it is enough to give co at one

point and to distribute it over the whole orbit by means of the transformations Adh, h e (hi. COROLLARY The form co e 122(O(f )) constructed above is smooth, since

the translation operation Ad* is a smooth transformation.

We have thus constructed on each 0(f) orbit a non-degenerate 2form co which is called Kirillov's form. To derive a symplectic structure from co all that remains is to check that dw = 0.

Let a non-degenerate differential 2-form co be given on a manifold MZ": with the help of co it will then be possible to define the Poisson bracket LEMMA 2.3

{f,9}=w"afag ax` ax'

for

fgeC°°(M)

(x1,. .. , x2" being a local system of coordinates on M2"), here w = Ilwii II and II w" II = I w,j I

1. We assert that dw = 0 when, and only when,

{ f, g} satisfies Jacobi's identity:

{f,{g,h}}+{g,{h,f}}+{h,{fg}}=0. Proof We

have found that { f {g, h}} = - (s grad f){g, h} _ -Lsgrad f{g, h}, where L, is the Lie derivative. If g = s grad f, then the Lie derivative's properties give us:

L"{g, h} = L" w'j I

_ (L n

= (L" w)''

_

- (L"w) ,j

ag` ah

ax ax ag Oh + wij a(rl9) Oh + w,;

ax' axj

ax' ax' ag Oh

ax' ax'

Og

ax' ax'

+ {r19,h} + {g,r1h}

ag Oh

ax' axe -

{{f, g}, h} - {g,If,h}},

25

INTEGRABLE SYSTEMS ON LIE ALGEBRA

therefore f f, { g, h}} +I g, {h, f } } + {h, { f, g}}

a g`

ax

Further, LRw = t(q) dw + dt(q)w = t(q) dw, see [46], inasmuch as q is a

Hamiltonian vector field. Thus if Jacobi's identity is satisfied, then dw = 0, and, conversely, if do) = 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 f, g e C°°(G*), then we get from the definition { f, g}(x) = { f I O(x), g 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 Cm (G*), then (s grad f)x = add flxl(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 = add f(x)(x), x e G*. Let Y = ad* x for y e G, then w(Y, ad*f(x)(x)) = w(ad* x, addf(X)(x)) = x([y, df(x)]). y

On the other hand dfx(Y) = dfx(ad* x) = (ad* x)(dfx) = x([y, dfx])

and inasmuch as w is non-degenerate, (s grad f)x can be defined uniquely.

Let f g be smooth functions on G*, then we have the { f, g} _ -Cxk of/ax' ag/ax' for Berezin's bracket, e 1 , . .. , e" being here the basis of the Lie algebra G, e', ... , e" the LEMMA 2.5

expression

conjugate basis of G*, while we denote the corresponding coordinates by xl , ... , x" and x , , .. . , x"; C is the structure tensor of the Lie algebra G in the basis e;: [e;,ej] = Cek. The proof is evident from the following calculation : If, g}x = dfx(ad a(x)(x)) = (add'(x) x)(dfx)

= x([dgx, dfx]) = C xk ag

of

ax; axe

,

26

A. T. FOMENKO AND V. V. TROFIMOV

here we have used equalities

of

ag

e 8x;

dg.,,

ej. dfx = Oxj ---

If we apply Lemma 1.2 to Berezin's bracket on G* we obtain orbits of coadjoint representation, on which the Kirillov form was given, as integral submanifolds. REMARK

C Xk of/ax; ag/axj has been defined on space C°°(G*). To prove that If, g} on the orbits satisfies Jacobi's

Thus the bracket If, g}

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*. Berezin's bracket - If, g} = C xk of/ax; ag/axj on space G* satisfies Jacobi's identity. LEMMA 2.6

Proof Using the definition of Berezin's bracket we obtain: {h, If, g}} + { f, {g, h}} + {g, {h, f}} _ (Cq,C, + C? C;q + CCqi) Jr

4

+J

Jr

P

Xk -

+ (Cq,C - CJ C 9)XPxk

+ (Cq, C,J - C C1q)X PXk

Oh of Og aXq OXi aXJ

0 2f

Og Oh

OX, aXi aXi aXq 82g

Oh of

OX, ax; axi axq

a2h

of Og

aX, aXi aX j aXq

The first item in this expression becomes zero by virtue of Jacobi's identity in the Lie algebra. We note that the tensor (CP,C - C Ciq)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 6 be a connected Lie group, G its Lie algebra, G* the dual space of

G, Ad: (i - GL(G) the adjoint and Ad*: (ti -+ 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 g e 0, x e G* we have f (Ada* x) = f (x) and it is called semi-invariant if

f(Ad9 x) = X(g)f(x) where x is a character of the Lie group 6. 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 q : G -. V(U), which is defined on basis G thus:

qq(e,)=X`=X(e1)=Cikx -, axk

i

1,...,n;

here CJ, k is the structure tensor of the Lie algebra G in the basis e,, while xj

are the coordinates in G* in the basis e' defined by e'(e,) = &. 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 X,, as Jacobi's identity implies, satisfy the relation [X,, X3] = C'JXk, the result of which is that q is a representation of the Lie algebras. We shall say that the operator X; corresponds to the basis vector e; e G. LEMMA 2.7

Let function F e A(G*), then d"

dt" =o

F(AdEXpt f)

= [(-

Proof The equality d

F(Ad&W f) = dtt=0 ax

(f) dt

OF

d

xi(AdE*xpt4 f) t=O

= ax, (f)at t=o f (AdE=p(-rs) et)

28

A. T. FOMENKO AND V. V. TROFIMOV

is valid since Xi(AdExPr4 f) = (AdE*xp,4 f)(ei) = f(AdEXP(-r4) e.).

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

at r=o f (AdExP(- rc) ei) = f ( - [

,

eij) = - f

Cjt!jfk .

Thus: d

dt

r-O (AdExpeCf)

-Cki

(-(P()F)(f),

i.e. when n = I the lemma is proved. For any n > 1 we have: d

f)

t=o

do-1

d dt

d dt

0

dTn -

i=0

dn -1

F(Ad& p(,

r=odT n-1 t=0

d" ds" S=0

LEMMA 2.8

F(AdEx P TCAdEx Pr4 f)

1

F(AdEx., f ),

r)4

f)

s = t + T.

If the function F e A(G*) then

F(Ad* pf4 f) = F(f) +

(- (p(s))"F n=1

n!

(f).

t".

Proof results from the expansion of F(Ad*,,, f) as a Taylor series using Lemma 3.7. PROPOSITION 2.2

Let function F e A(G*), then

a) F is an invariant of the coadjoint representation of group % if, and only if, X1F = 0, i = 1, ... , n, n = dim G; b) F is a semi-invariant of the coadjoint representation of group 6

corresponding to a character x

if,

and only if, X,F = -AiF,

i = 1, ... , n = dim G, 2i = dx(ei) and dx is the derivative of x at the group 6's identity element.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

29

Proof a) As F(Ad Ex ,, f) = F(f), so d/dtl t=0 F(Ad*Ex P,{ f) = 0 and therefore according to Lemma 2.7, when n = 1 we find that X; F = 0, i = 1, ... , n. Conversely, if X,F = 0, then (p(i )F = 0 and therefore [-tp(i )]"F = 0, when according to Lemma 2.8. F(AdExpt4 f) = F(f ). Since Chi is a connected group, F(Ade f) = F(f) for any g e (5. b) From the equality F(AdExp,4 f) = X(Exp t)F(f) it follows that d

d dt

F(Ad,*pt4f)=dt t=o

t=o

X(Exp tj)'F Cr)

X*()F(f). Conversely, [(- (p(s))"]F = therefore, in accordance with Lemma 2.8:

and therefore [(-

F(Ad ExP, f) =

and since X(Exp ti;) =

11 +

n1

[X*()]" tnl . F(f) ni.

J

our theorem is complete.

Lastly, in order to find the invariants, the system of differential equations Cxk OF/8x; = 0, i = 1, ... , n must be solved, or for the semi-

invariants-C xk OF/ax; = 1F, i = 1, ... , n. For the methods of solving these systems see [106], [117]. The semi-invariants' system's solution cannot generally be found for every character. In terms of the operators X; a criterion of invariance of a subspace W c A(G*) with respect to the operators Ada , g E 6 can be given. PROPOSITION 2.3

Let W be a finite-dimensional subspace in A(G*) and

f e A(G*), then for any g e (5 (% being a simply connected Lie group having Lie algebra G) f (Ad,* x) e W if, and only if, X, f c- W, i = 1, ... , n = dim G.

Proof If f e W then from the fact that f (Ada x) E W for any g e 6 it follows that d/dtl t = 0 f (Ad* P,{ x) e W since any finite-dimensional subspace is closed and then, according to Lemma 2.7 Xi f e W. Conversely, it is enough to check that f (Ad* x) E W for g = 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 W (-cp(s))" f (Ad* P t4 x) = f (x) + n=I f (x) n! and since W is closed, f (AdE.p,4 X) E W.

A. T. FOMENKO AND V. V. TROFIMOV

30

REMARK

Let p: 6 - End(V) be an arbitrary

finite-dimensional

representation of the Lie group 6 in a linear space V A function F: V -* l is called invariant if F(p(g)x) = F(x) for all g c- 6, x e V. Let dp(e;) f; = a fk where f is a basis of V and e; a basis of G (the Lie algebra

of group 6): F will be invariant if, and only if, axk(8F/ax') = 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 f and g 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."

Suppose that a set of smooth functions fl , ... , f in involution, i.e. { f , f } - 0 when 1 < i, j < n, is given on a THEOREM 3.1 (Liouville)

symplectic manifold MZ". Let M, be the common level surface of the i.e. M, = {x e M: f(x) = i;;, 1 < i n}. Let us imagine functions are functionally that on this surface of the level all n functions fl,. .. independent (i.e. the gradients grad f , 1 < i n are linearly independent in all points of surface Me). Then the following statements hold :

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

2) If the manifold M, is connected and compact then

it is

diffeomorphic to the torus T" or, in the general case, a connected nonsingular manifold (which need not necessarily be compact) being a quotient of Euclidean space 68" over some lattice of rank < n if all flows s grad f are complete.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

31

3) If the level surface M, is compact and connected (i.e. if it is an n-

dimensional torus), then in one of its open neighborhoods regular curvilinear coordinates s1, ... , s", q,. .. , qp" where 0 < T; < 2n (the so-

called "angle coordinates"), can be introduced, such that (a) the symplectic structure co in these coordinates is written in the simplest + ds" A d(p", which is equivalent to saying way, i.e. w = ds1 A dcpl + that functions s 1 , ... , s,, cpl, ... , (p" satisfy the following correlations: {s;, sj} = {(p;, (pj} = 0, {s;, (pj} = d;j; (b) functions sl, . . . , s" are coordinates in the directions transverse to the torus and are functions of

the integrals fl, ... , f", i.e. s; = s,(fl, . . , f"), 1 < i < n; (c) functions x S1 where Ti is the (pl, ... , (p" are coordinates on torus T" = S' x angular coordinate on the i-th circle Si, 0 < T, < 2n; (d) each vector .

field v = s grad F, where F is any one of the functions fl,. .. takes the when written in the coordinates cpl, ... , q,, on form (p; = torus T": 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 T". Here functions q;, 1 < i <, n are defined in the given neighborhood of the torus; on nearby surfaces we likewise have ( p , = qi(s,, ... , s"). Thus the initial system v = s grad F may be written in the neighborhood of the

torus T" in the coordinates si,... , (p" as Si = 0, (pi = gi(sl,... , s"), 1

i<, n.

We see that if the functions fl, ... . f" are independent on M, then all non-singular level surfaces M4 (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 ff1, 1 , . . , f". Further, the vector field grad f is arranged on the torus itself with the maximum simplicity, since, with respect to the angle coordinates (pl, ... , (p", 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

A. T. FOMENKO AND V. V. TROFIMOV

32

Fig. 9.

that "it allows complete commutative integration," if the set of functions f, = F, f2,. . , f, 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 "non.

commutative 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 C'(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 f, , ... f,,. In other words, the linear space G spanned by the f1,. .. , f 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 = f,. In many concrete situations, however, the system's Hamiltonians have a set of integrals

INTEGRABLE SYSTEMS ON LIE ALGEBRA

33

fi , ... , fk 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 fl, ... , fk are smooth functions on a symplectic manifold M2 with form co and that they are functionally independent on an open set which is everywhere dense in M. If we examine the linear span G (over l) of the functions (f), then dim, G = k. We shall suppose that the linear space G is closed with respect to the Poisson bracket, i.e. all the brackets { f , f } are linear combinations of the basis functions fi, , fk with constant (!) coefficients, i.e. { f , f } _ E9=, Cg fq where C? a R. 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 = Ji 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 6 to denote a simply connected Lie group that corresponds to G. Then

(i = Exp G. Each element of the algebra G may be presented as a Hamiltonian field on MZ". To do so one examines the mapping a : f - s grad f. Consequently the group 6 is represented as the group of the diffeomorphisms of the manifold which preserve the form w. These diffeomorphisms are called symplectic. Thus giving the algebras of

integrals G defines a smooth symplectic action of the finitedimensional group t l on M. We shall use the following notation: (X) = <, X>, where c e G*, X e G. In these notations = Ad9 X> and = ad, X> = . Let c- G* be a given covector. We examine in the algebra G the subspace H4 = Ann(e), which consists of all vectors X so that ad* = 0. Subspace H is called the annihilator of the covector . We shall say that e G* is the covector of general position if its annihilator's dimensionality is minimal. We

shall call the dimension of the annihilator of a covector in general position (see [26]) the index of the Lie algebra G. If G is a semi-simple

algebra this definition coincides with the rank of the Lie algebra G (= the dimension of a Cartan subalgebra).

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].

Let a set of smooth functions , fk, whose linear span is a Lie algebra G with respect to the Poisson bracket, i.e. {J, f } _ Y9_, CQ f9 where C9 are constants, be given on a symplectic manifold

THEOREM 3.2

A_.

M2". Let M, be a common level surface in general position of the functions (f), i.e. M4 = {x e M : f(x) = i, 1 < i < n}. We shall suppose that on this level surface all k functions fl, ... , fk 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 H4 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, f } for all

f c 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' and on this torus the curvilinear coordinates cp, , ... , cp, can be introduced, such that vector field v, being written in i.e. these coordinates on the torus, takes on the form (p. = 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 P. 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 v",

INTEGRABLE SYSTEMS ON LIE ALGEBRA

35

where k is the dimensionality of G. Thus, for example, in the case of the

series A,,-,, when G = su(n), we have r = n - 1, k = dim G = nz - 1, i.e. rank G z dim G. This means that r < k and since r + k = 2n, then r < n = 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 = z dim M. 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' = i dim 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 of r-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

T" 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 T", since this torus' stratification into tori T' of low 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 C°°(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°°(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 [75]). 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 (Mzn, (o), and let the algebra's additive generators be k (almost everywhere) independent smooth functions fl, ... , fk. Let 0 be the corresponding simply connected group operating on M by symplectic diffeomorphisms (i.e. those that preserve (o). For our purposes it is easier to look at the following mapping cp. We shall identify each point x e M with the linear functional (the covector) co., on the algebra G. We shall take the value tps(f) = f (x),

where f e G. Thus q' is an element of the space G* dual to G. Consequently we have defined a smooth mapping qp:M -' G*. Let E G* be an arbitrary operator. Then its full inverse under the mapping (p, is a common level surface M4 of the integrals f1, ... , fk which generate algebra G. LEMMA 3.1

image

Proof According to the definition T

{x e M : f (x) = (f)},

where f e G. Since (f) is an additive basis in G, then f = E+=, a; f, i.e. If (f) _ I, 1 ,
for xEM,. Thus (I, ... , k)}. The lemma is proved.

Mapping q : M - G* need not necessarily be an "onto" mapping. For example, if M is compact, then the image cp(M) does not cover the whole coalgebra G*. We can introduce a sonvenient notation. If E G*, f E G, then we denote ad* by f ). Then _ = g), f ).. We have H, c G, H4 = {g E G : g) = 01,

i.e. H, = Ann(f). Let element f (a function) of algebra G lie in the

PROPOSITION 3.1

annihilator H, and let x e M where M, is a non-singular surface. Then s grad f (x) E TX M,. This means that the skew gradients of function in the

annihilator of the covector lie in the tangent plane of the level surface M,, the surface defined by this covector.

Proof Let us examine the additive generators .11,. .. , fk in G and the gradients grad fl, . , grad fk. Since M, is a non-singular surface they .

.

are accordingly transverse to M,, i.e. the k-dimensional plane V spanned

by (grad f) intersects with TM, only at zero, and V p TzM = 7 M, (see Figure 11). We may conveniently take it that a Riemann metric is given on M such that the vectors (grad f) are orthogonal to the surface M. To prove the relation s grad f (x) E it is enough to check that (s grad f )g = 0 for any function g 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 f) equals zero, it follows that s grad f is orthogonal to the plane V, i.e. s grad f e TxM4. Thus (s grad f )g = { f, g}. Further, { f, g}(x) _ f) = 0, f e Ann , q.e.d. f), g> = 0 since < , f f, g}> = All fields s grad f that are generated by elements f of the annihilator are thus tangent to the corresponding level surface M,. PROPOSITION 3.2 The equality (Tx M4) 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 eH4) c TxM, n

(s grad f ; f eG). We shall prove the converse. Let x e TxM4 and X = s grad f, where f oG. It must be proved that f oH4. We {f g}> - 0 for any g e G, since { f, g}(x) = 0. examine f ), g> = This last equality arises from the fact that t f, g}(x) = (s grad f )glx = X(g) = 0 since X e TM4, and all g e G are constant on the level surface f) = 0, f), g> - 0 given any g e G. This means that M. Thus

i.e. f e Ann = H4. The argument is proved, see Figure 12.

Fig. 12.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

39

We also consider the group Sj with Lie algebra Hf, i.e. Exp H4 c 0. COROLLARY The level surface M4 is invariant relative to the action of

the group 6 on the manifold M. We examine form co on M and assume that 6 = O) M4 is its restriction to the surface M. The action of the subgroup .5, on M, generates at each

point x e M, the plane Hs c TM,, formed by vectors s grad f, where f EH4, see Figure 13. In other words plane Hx is generated by the subalgebra H.

Fig. 13.

PROPOSITION 3.3

The kernal of the form w (the restriction of the form TxM.

w to W coincides with the plane Hx

Proof We shall first prove that Ker w

H. Let X = s grad h where

h e H,, X e Hx c T,,M,. We need to prove that X lies in the kernel of the form co, i.e. that w(X, Y) = 0 for any vector Yin the plane TxM. In fact, w(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 Ker w c H. Let w(X, Y) = 0 for any vector

YETxM,. The vector X must be represented in the form X = s grad h for a given function h EH,. We then consider the form co as a skewsymmetric scalar product on the tangent plane TxM and we use (TxM4)' to denote the orthogonal complement to plane TXM, in TxM relative to the form co. Since the form is non-degenerate, the equality dim(TTM,)1 = dim M - dim TM4 = dim V = k is valid. We should

bear in mind that in the case of skew-symmetric scalar products the space TM need not necessarily decompose into the direct sum of T^ and (T M4)', since these planes can have a non-zero intersection. It is clear that Kerco = T. M, n (T,, M,)'. We shall prove that (sgrad f;

40

A. T. FOMENKO AND V. V. TROFIMOV

f e G) = (TTM,) -. In fact, if Y E TM, then uo(s grad f; Y) = Y(f) = 0,

since f = const on M,. Thus (s grad f; f c- G) c (TXM,)1. 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 c- 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(TAM,)1 = k, therefore (s grad f; f e G) = (1 M,)1, see Figure 12. The argument has been proven.

Let us now bring all these facts together and study the geometric picture of the mutual interaction of the submanifolds we have described. The fundamental objects are: (a) the level surface M,, dim M, = 2n - k; (b) an orbit 05(x) of a point x, dim (5(x) = k; (c) an orbit S>,(x) of a point

x under the action of the subgroup .54 = Exp H,. It is clear that Tx6(x) = (s grad f ; f c G), T.-5,(x) = HX = (s grad h; h e H,). It follows

from this that (5(x) r M5 = SJ,(x), see Figure 14. We note that the dimension of orbit 1),(x) equals that of Sao and is equal to r.

Fig. 14.

Let us look at the action of the group 6 on M and assume that in a small neighborhood of the surface M, this action has a single type of stabilizer subgroup, i.e. that all orbits of the group 0 close to an orbit 6(x) are diffeomorphic to it. Let us examine the projection p: M - M/6 of manifold M on the orbit space M/(5 = N. This space need not be a smooth manifold and it may have singularities. What is important to us is that space M/6 is a smooth manifold of dimensionality 2n - k in a small neighborhood of the point p(5(x) E M/0i. In reality, if (fi 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 Q4 = M,/.5,. Therefore space N is stratified by surfaces Proposition 3.2.

see Figure 15. Here our argument is based on

Fig. 15.

The manifolds Q4, i.e. the quotient manifolds of the level surfaces M, under the action of the subgroup .5,, are symplectic manifolds with a non-degenerate closed form p, which is the projection of the form w on M, under the mapping p: MC - Q4. Here p*p = th _ PROPOSITION 3.4

w/M4.

The proof ensues from Proposition 3.3, since the kernal of the form w

on TM, coincides with the plane H,, c 1 M4.

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} = 0. Since the Hamiltonian F commutes (in the Poisson

bracket sense) with all elements of G, F is therefore invariant with respect to the group 6. In actual fact (s grad f )F = { f, F} = 0, f e G. In particular, the subgroup .5,, 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/Qi. At the

42

A. T. FOMENKO AND V. V. TROFIMOV

same time the vector field s grad F is tangent to the surface M4 and is also projected onto a certain field E(F) on the quotient Q4, since the field

s grad F is invariant with respect to .54. Thus space N is stratified by symplectic manifolds Q4 and a vector field E(F), which is tangent to all

surfaces Q4, see Figure

16, is

defined on N. Finally the triad

(MZ", s grad F, co) corresponds to a new triad (Qr, E(F), p). '

,f(x)

Al Fig. 16.

The vector field E(F) is Hamiltonian with respect to the symplectic form p on the manifold Q4 for the Hamiltonian function F, which is equal to the projection of function FI M4 onto the manifold Q4, i.e. E(F) = s grad p(p*FI Md. PROPOSITION 3.5

The proof follows from Proposition 3.4 and the invariance of

Hamiltonian F under the group action. The correspondence (M, s grad F, w) - (Qs, 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 Q4 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 Q4 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

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 = U4 Q4, a Hamiltonian Let Gbe a linear space of functions on system on each submanifold the manifold N, such that their restrictions to the submanifolds Q4 form a finite-dimensional algebra of integrals of the flow E(F). The space of functions G E) G", where G" = p* G', i.e. G" = {gp, g e G'}, p: M - N is then a Lie algebra of integrals of system s grad F, while [G, G"] = 0. PROPOSITION 3.6

Proof Let g be a given function on space N; its inverse image gp under the mapping p: M -* N is then a function on M, obviously invariant with respect to the action of (( 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 of the functions of algebra G follows from their gradients being non-zero in the direction of the submanifolds Q4 which lie (locally) in the level surface M4; at the same time the gradients of the 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 h c Ann = H,. Let us examine the reduction above. Insofar as the supplementary condition dim G + ind G = dim M, i.e. k + r = 2n, has now been met, the described

dimension of the surface M, equals r. The dimension of the orbit S5,(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, = .5,(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 Sj,, and its Lie algebra is the annihilator of the covector which defines the given level surface. In particular, dim Q4 =

2n - k - r = 0. In the given case, therefore, the reduced system's structure is particularly simple. Since Q4 is a point, flow E(F) is zero, see Figure 17. Here space N has dimension n; since M4 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(e) and s grad F, {F, G} = 0. It remains to be proved that the level surface is an r-dimensional torus in the case when

A. T. FOMENKO AND V. V. TROFIMOV

44

M, is compact and connected. To do so we shall need an auxiliary argument. PROPOSITION 3.7

Let c e G* be a covector in general position. Then its

annihilator Ann is commutative and so too, in particular, is the subgroup .5, (see for example [26]). Let us examine the coadjoint operation of the group 6 = Exp G on the coalgebra G*. We denote by the orbit which passes through point e G*. Since dim H, = r and

is in general

dim G* = k, dim 0*(c) = k - r. Since the covector

are also diffeomorphous to it, it may position the orbits close to 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 (b by X0. We

take advantage of U being representable as the direct product of the base and a fiber (i.e. a part of the orbit), see Figure 18. Let h(q) be a smooth

function on U which is constant on the orbits. We argue that a(c, dh(g)) = 0, where dh (the differential of h) is interpreted as an element of the dual space (G*)* = G, i.e. dh(c) e G. In other words, we 0 e Ann(e). We must verify that are arguing that for any g e G. We have g>

gj>

g), dh(f)) = 0,

g) = ad, lies in the plane TO* tangent to the orbit 0* of the point , while function h is constant on the orbits and since the covector

constant, in particular, on the orbit 0*, see Figure 18. If we take section X0, which is a smooth surface of dimension r transversely intersecting

the orbits close to the orbit O*(i;), we can consider on it the set of r independent functions h1, ... , h, and extend them into smooth functions on the entire U, by extending them from the section X0 with values that 0, 1 < i < r. Thus are constant along the orbits 0*. We get

45

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Fig. 18.

E Ann(e), 1 < i <, r. Since functions (h) we chosen as independent, are independent in Ann and the number equals all differentials

r, i.e. exactly coincides with the dimension of Ann l;. The differentials 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. 0. Since 0, the equality 0

holds. Hence

dhj(f)) =

dh;(f)]> = 0.

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

0 d b(c) = d dhf()]> +

_ +

[dq

dh;()] )

dh;()J) + <,1,

d-

dh,())

_ <, [dh(), dhO]> = 0,

dh() j /

46

A. T. FOMENKO AND V. V. TROFIMOV

0. Since 9 e U c G* is arbitrary, it

since

therefore ensues from the equality
dh;( )]) = 0 that

We now return to the proof of Theorem 3.2. We have proved that the surface M, is an orbit of the group .5, and, since dim M, = dim Sa,, then

M, is the quotient group of .5, over a discrete lattice F. Since group Sao = Exp Ann(e) 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 M2", 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 T' which coincides with the level surface M,, where r = the index of G.

3.5. Orbits of the coadjoint 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 coadjoint 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,*: 6 -* GL(G*) be the coadjoint representation of group Cfi = 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 c G* be an arbitrary point and b 1,b 2 e 'T,0* be two vectors tangent to the orbit. Each vector c tangent to the orbit can be represented in the form ada* x = a(x, g), since T,,O* = ad,* x. Therefore elements g1, 92 e G 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 w

in point x e 0* on vectors

b 1, S2

tangent to orbit 0*, so w( 1, b 2) _

<x, 191, 92]) , where x e G*, 91, 92 a G.

47

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Form co defined above has the following properties: (1) the form is bilinear and its value does not depend on the choice of representatives g,, g,; (2) the form is skew-symmetric and defines an exterior 2-form on the orbit; (3) the form is non-degenerate on the orbit; (4) the form is closed on the orbit. PROPOSITION 3.8

If the algebra G is compact it can be canonically identified, by using

the Killing form, with G*. A form co on orbits 0 of the coadjoint representation is then defined thus: wT( 1, Z) = B(x, [g1,92]), where B(x, y) is the Killing form and x, ti, gi e G. We should note that in the general case form w may be written thus w

zz

zz

= <51,91> = -<SZ,91>-

Proof of Proposition 3.8 We shall apply the reduction technique worked out above. Let us consider the cotangent bundle M ---:- T*T) of

the group 6 as the direct product M = T*6 = T, *T) x 6 = G* x 6. Here we shall give the action of 6 on this direct product in such a way that 6 operates on the second factor 6 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 6 on the phase space T*6. 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 the formula f (i;, g) = , f e V = G, E G*, g e 6. Thus the surface M, which corresponds to e V* consists of all pairs (q, g) such that f(q,g) _ f), i.e. Ada_1(q) _ or q = Ada . Insofar as (, E) e M4 = {(Ad* , g); g e 6}. This tells us that surface M4 is a 9 fibration with base 0* (fl and fiber fj,, where 0* is an orbit of 6 on G*, and .5, is the maximal torus corresponding to the subalgebra H4 which fixes the covector . We have represented a general position orbit of the coadjoint representation as a quotient-manifold where H4 is the annihilator of . From Proposition 3.4 it follows that 0* is a symplectic

manifold. Direct calculation shows that the symplectic form which arises on 0* coincides with the canonical form described above. 3.6. The connection between commutative and non-commutative Liouville integration

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 non-

commutative, 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 F0 defined on the dual space V*, where any pair

of functions f, g E Fo is in involution on all orbits of the coadjoint representation with respect to the canonical symplectic structure and dimF0 = '(ind V + dim V). 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 PROPOSITION 3.9 (Fomenko, A. T.; Mishchenko, A. S.)

algebra V satisfies the condition (FJ), then there is another commutative Lie algebra Vo of functionally independent integrals, where 2 dim Vo = dim M.

Proof (see [88]) Algebra Vo will be sought among those functions which are functions of the integrals in algebra V . Let x 1 , . .. , x" E V be a linear basis in algebra V. Then algebra Vo will be sought among functions of the form f (x' , ... , x") where f is a smooth function of n independent

variables. We should note that the Poisson bracket of two functions f(x',... , x") and g(x',... , x") is given by the formula:

{f,9} = I8x` (x',.. ,x") aOxj(g W',...,x")[xi,x']

(1)

On the other hand every function f of n independent variables defines a function f * on the dual space V*: f x' ), ... , x")). Then on an orbit of the Poisson bracket of the pair of functions f

f

49

INTEGRABLE SYSTEMS ON LIE ALGEBRA

the coadjoint representation is given by the formula:

{f*, 9*}()

_ i.iI az

x1>,

... ,

x a9 (<

= Yi,i ax

x1>,

... ,

x">)

,

xl>,

x">W , x`>,

x">) 9

xl>, .. x

,

xi>} x">)

[xi, x']>

(2)

Formulas (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 Fo, 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 fj 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 of linear elements on sphere S" with standard Riemann metric (see

[88]). It will be easiest to consider that the sphere is contained in Euclidean space R" + 1, so that the space of linear elements L(S consists of pairs of vectors (x, y), Ixl = 1, x 1 y, i.e. (x, y) = 0. It is apparent that the dynamic system indicated is invariant under orthogonal transformations A E SO(n + 1). The system's equations may be written

in the following form: x = y, y = -x. Let (xo, yo) E T*S", then the tangent vector E T xu.yu)(T *S") may be given as a pair = (x, y), x I x0,

y 1 yo. Here the symplectic form co on the pair of tangent vectors S2) = (xl, Y2) 1 = (X141), S2 = (X242) takes the value (yl,x2). Then the function algebra V- so(n - 1) corresponding to the action of the group so(n + 1) consists of functions of the form:

50

A. T. FOMENKO AND V. V. TROFIMOV

fjx, y) = (x, cy),

c c- so(n + 1), (3)

{ff,,f'2} = f[c,.cz]-

Later (see §16) we shall show that the Lie algebra so(n) satisfies the condition (FJ). It is thus possible to construct polynomials Pk(f) which are pairwise in involution. It is therefore enough to make explicit the which are pairwise functionally maximal number of polynomials

independent on the manifold T*S". Formula (3) gives a mapping cp: T*S" -+ V*, which when written as a matrix has the form qp(x, y) =

xy' - yx', where x, y are understood as column vectors, and the operation y - y` is matrix transposition. The mapping co is equivariant to the action of the group SO(n + 1). Space T*S" 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 2A are not

equivalent when 2 # 1, 2 > 0. We shall show that the mapping (p, restricted to an orbit 0(x, y) 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 9(x, y) has the form II cp" II = T (X' y), (p'' = x' y' - y'x'. Then, given 2 < i < j, the partial derivative functions (pU at the point (xo, yo) equal zero. Further,

ail'=0, cx k

a ki

ask'=61, cy

2i

=o,

2
Yk

Consequently the rank of the Jacobian matrix of the mapping cp equals 2n - 1. On the orbit O(xo, yo) the rank of the Jacobian matrix then falls by one and is consequently equal to 2n - 2. In this way, if [; = gp(xo, yo) E V* then the mapping cp is a fiber bundle

of (p:O(xo, yo) - O(ff). Insofar as so(n) satisfies the condition (FJ) (see §16), there exists for each orbit O(ff) a set of functions P,,.. . , Pk on the dual space V* which are pairwise in involution and whose gradients on

the orbit are linearly independent, and 2k = dim 0(t) = 2n - 2. Then the compositions p, o (p, ... , pk o (p are independent functions on orbit O(xo, yo) and are in pairs in involution. By adding to the functions set

51

INTEGRABLE SYSTEMS ON LIE ALGEBRA p1

cp, ... , pk , (p 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

M = T*l"= R"(p...... Pn)Ol "(q1,...,q"), p, be the impulses, and q, the coordinates in a configuration space l". We

shall put Zk = pk + iqk, 1 < k < n. The system of n point vortices is a Hamiltonian system with Hamiltonian

H(p, q) _ Y_ xkrc, lnlzk - z,l. The coefficient Kk is called the vortex k,l = 1

intensity. LEMMA 3.2

A point vortex system has three functionally independent'

first integrals: n

11 = L Pk, k=1

n 12 = L qk,

n

13 = y

k=1

(pk2

+ qk)

k=I

These functions in C°°(M)/{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

j\ Ox 0/

\u2) E E(2). 1

The isomorphism is given by the correlations:

a(il) = (0) ,

c'(12) = 1

O

(2I3)

where i1, i2, i3 are the images of functions 11, algebra C°°(M/{const}.

-

0

_0 0) in the quotient-

Given n = 3, the points vortex system is integrable in the non-commutative sense. LEMMA 3.3

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 C°°(M)/{const}. In our case, as an example of such a subalgebra we can

take the subalgebra generated by h, il, i2, 6, where h is the image of Hamiltonian H in C°°(M)/{const}. This Lie algebra is isomorphic to the direct sum l G+ E(2) = V, dim(18 $ E(2)) = 4, ind(tJ p E(2)) = 2, and the condition of the non-commutative Liouville theorem has therefore been met: dim V + ind V = dim M. Given n = 3, the points vortex system is completely integrable in the Liouville commutative sense. PROPOSITION 3.10

Proof It has only to be shown that V = R Q E(2) satisfies the condition (FJ). For the family F0 we can take the functions {h, 4i2 + i3, i2a2 + i3a3}. It is easy to confirm that they are in involution and are functionally independent on (l (D 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. Let M and N be smooth manifolds and co be a smooth mapping from the manifold M into N. Let X and Y be vector fields on M DEFINITION 4.2

and N respectively. The fields X and Y are called (p-connective if dcpp(X p) = YNcp) for any p c- M. LEMMA 4.1

Let f : (M1, w1) - (M2, w2) be a smooth mapping of

symplectic manifolds, such that (a) f is a submersion and (b) f *w2 = wl then s grad2 h and s grads (h o f) are f-connective vector fields for any h E C°°(M2); here s grad; h is the skew gradient with respect to form w;,

i = 1,2.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

53

Proof The vector field s grads h is defined by the condition that w1(X, s grads h) = dh(X) for any vector field X E V(M1). We must check

the identity w2(X, f*(s gradl(h o f)) = dh(X). Since f is a submersion, X = fl(Y) for some Y. Therefore

w2(X,f*(sgradl(h o f)) = a 2(f*(Y),f*(sgradl(h °f)) = (f*w2)(Y,sgradl(h o f)) = wl(Y,sgradl(ho f)) = d(h ° f)(Y) = dh(f*(Y)) = dh(X), q.e.d.

We call the mapping f : (M1, (01) - (M2,(02) a morphism of symplectic manifolds, if (a) f is a submersion and (b) DEFINITION 4.3

f*w2 = ('U1

-

PROPOSITION 4.1

Let (p be a morphism of symplectic manifolds

(M1, wl), W21(02) and hl, h2 E C'(M2): we claim that {hl, h2} ° (p = {h1 ° (p, h2 o (p) .

Proof The proposition is obtained from the following calculation: {hl, h2}2 = w2(s grad2 hl, s grade h2)

= w2(f*(s grads (hl ° f)), ,,(s grads (h2 ° f))) = (f *(02)(s grads (h l ° f), s gradl(h2 ° f ))

= wl(sgradl(h1 °f),sgradl(h1 °f))

={h1of,h2of}. COROLLARY

Let (p be a morphism of symplectic manifolds from

(M 1 , w 1 )

to (M2, (02) and let h,_., hp be a set of involutive functions on M2, then hl o (p, ... , h p o p is a set of involutive functions on M1. Thus, if there is a morphism M1 -+ M2 and a set of involutive functions is given on M2 we can also construct on M1 a set of involutive functions.

We now want to discard the condition that (M2, (02) is a symplectic

manifold; we shall instead assume that (M2,(02) 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*w2 = w1, insofar as there is no symplectic structure on the union of orbits, though there is on such a space a Berezinoisson bracket. Relying on Proposition 4.1 we shall give the definition.

A. T. FOMENKO AND V. V. TROFIMOV

54

A symplectic manifold (M, co) permits realization in a

DEFINITION 4.4

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 oforbits of Ada) and if there exists a smooth mapping f : M -, N such that {h1, h2}2 ° f = {hl ° f, h2 ° f }1 for any hl, h2 a C°°(N) where f f, g}2 is the Poisson bracket induced by the Kirillov form on N and {h, g} 1 is the

Poisson bracket generated by the symplectic structure co.

If h1, h2 are in involution on all orbits of the coadjoint representation, then h1 ° f, h2 ° f are in involution on the symplectic COROLLARY

manifold (M, co). DEFINITION 4.5

Let a symplectic manifold (M, co) permit realization f

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 f, if there exists on the submanifold N c G*, on which M

is realized, a function such that H = H1 ° f. If for the system z = s grad H1/0 we can find f i r s t integrals f1, ... , f, pairwise in involution, then we can give an involutive set of first integrals set for the system )i = s grad H: f1 ° f, ... f, ° f, since {H, f - f } = 0 by

virtue of the equality {H, f ° f } = {H1 ° f, f ° f } = {H1, f } -f = 0, since f1,. .. , f, are first integrals for H1. The involutivity of f1

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 l3(x, y, z)

_-12 Al

x

.l +1 2yz' 1

y

_ 13-l1 1 +A Xz, 3

1

z

_ 12-A3

.l2 +1

3

Xy,

where Al, 12, 23 are real numbers. Some Hamiltonian systems given on R" 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 V on a manifold

INTEGRABLE SYSTEMS ON LIE ALGEBRA

55

Mn allows an embedding into a Lie algebra if M" can be identified with a submanifold N c G* in the dual space of the Lie algebra G of some Lie group 6 in such a way that (a) N = U f 0(f) is invariant with respect to

the coadjoint representation of the Lie group 6, i.e. N is the union of orbits 0(f) the coadjoint representation; (b) the vector field v (after this identification) is tangent to the orbits 0(f) of the coadjoint representation of group % on G* (i.e. all the orbits 0(f) are invariant in relation to v); (c) the vector field v on N turns out to be Hamiltonian on the orbits 0(f) with respect to the canonical symplectic form co and is realized as v = s grad f where f c- COO (0(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 L", 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, 11" 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 C°°(0(t)), t e G*, 0(t) be an orbit of Ad*, then the Hamiltonian

equations z = s grad f on the orbit 0(t) relative to Kirillov's form on 0(t) take the form z = add* (,,)(x). If f E C°°(G*), then on each orbit there arise the Hamiltonian equations x = add f(x)(x), x E G*, f e C°° (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 Hamiltonian f is a quadratic function on G*.

56

A. T. FOMENKO AND V. V. TROFIMOV

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) = J<x, C(x)> and, obviously, the converse is true. Operator C defines

symmetrical quadratic form when, and only when, = , x, y c- G*. From now on we shall always assume that C is a

self-adjoint linear operator: < y, C(x)> = G defines on G some system of non-linear differential equations z = ad*c(x)(x). This system of equations on G* is Hamiltonian on all orbits of Ad*; as a Hamiltonian, the restriction of f(x) = i<x, C(x)> to the orbits of Ad* may be taken. We can express Euler's equations in coordinates. Let e , ,. . , e" be a

basis of G, and e',... , e" the dual basis in G*, x e G*, x = xle`, C: G* -+ G, C(e') = a`3ej then Euler's equations take the following form:

s = 1,...,n=dim G

(1)

where C is the structure tensor of the Lie algebra G in the basis el , ... , e". Thus, for each Lie algebra G and self-adjoint operator C we can give a non-linear system of differential equations .z; = I X pxq where

f;m = a"JC7t. In hydromechanics the tensor r" is also called the dynamic tensor. System (1) always has as its first integral T = Za'sxjx, the energy integral.

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

OF axi.

=0,

i=1,...,n

(2)

(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: aF

OF

aF

at = ax x' = ax s

a"Cip. x;xp

s

aFl (Cjp.xp ax,

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 (8s/et) + H(t, q, 8s/8q) = 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 of linear operators for which a knowledge of the full set of functionally 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: R2: E(2) = {e1, e2, e3} and [e1, e2] _ -e3, [e,, e3] = e2. In this case system (1) has the following form: )i

1- -a 13X X +a 12X 1

2

21

1

X3 -a a 23X22 + a22X2X 3

x2 = -a' X1X3 - a X2X3 - a 1

31

a33X3 X2 + a32x23,

2

X3,

x3 = a11X1X2 + a21X2 + a31X3X2.

It has the following integrals: T = Y_;,; a'Jx;x;, F = x2 + x3. This system's Liouville tori are depicted in Figure 19 for the case of a positive

Fig. 19.

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

A. T. FOMENKO AND V. V. TROFIMOV

58

JC1 = a13x1x2 - a12x1x3 + (a33 - a22)X2X3 + a23(X2 - X3),

X2

a23x2x1 + a21x2x3 + (a11 - a33)xlx3 + a13(X3 - xl),

X3 = a32x1x3 - a31x3x2 + (a22 - a11)X1X2 + a12(X1 - x2).

This system's first integrals have the form: T = i E;,j a''x;xj, F = X + x2 + x3. The Liouville tori are depicted in Figure 20 (see also 1

Figure 31) for the case of positive definite kinetic energy T.

49> 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 l3 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 R3 with the vector product serving as the Lie operation. The isomorphism is given by the formula:

f((J)1,(J)2,(0s) = I

0

-u)3

co3

0

-w2

(1) 1

w2

-w1 I

.

0

As we know, the orbits Adso(3l are spheres; the ring of invariants is generated solely by the function xl + x2 + x3. Let e1, e2, e3 be a basis of

INTEGRABLE SYSTEMS ON LIE ALGEBRA

59

= so(3), e', ez, e' be the dual basis, while D: (l83)* - l3 is the linear operator, D(e') = d'jej. We shall assume that d'i = d". Let us suppose that operator D is diagonal: ii

0

1/A

D=

1/B 1/C

0

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

rl -

1)

XI =

C B

Xz =

A C )X1X3

xzxa

C1-1

1

X3 =

B

11 - A)X1XZ.

(3)

Equations (3) on G* = so(3)* = (1183)* are equivalent to the equations of motion of a three-dimensional rigid body with one fixed LEMMA 4.3

point, where the equivalence is given by the operator D: (l')* -+ R3.

Proof If we rewrite the equations (3) with the help of the operator Don so(3). We have 1

1

1

Yl = AX1,Y2 =

B

X2,Y3 = C X3

therefore x1 = Ayl, xz = Byz, X3 = Cy3, then equations (1) in the variables y; will take the following form: Ay1 = (B - C)Y2Y3

Byz = (C - A)Y1Y3

If we put (y1, yz, y3) = (p, q, r) we shall obtain classical Euler's equations for the motion of a solid body (see [140]):

AP = (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: F, = xl + x2 + x3 is the invariant of Ad*, while ( X 12

FZ

x22

x3

2 A+B+C 1

is the energy integral. In mechanics textbooks there is usually a change to coordinates y;, ending with the equations (4) having the following integrals:

F = A2yi + B2y2 + C2y3 = A2p2 + B2q2 + C2r2 is the integral of the moment of momentum, while the energy integral is

+ Bq2 + Cr2). F2 = i(Ayi + By22 + Cy2) 3 = 1(Ap2 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

pX = [cpX, X], X e so(3) and qp(X) = IX + XI,

22

and

-A+B+C 2

,

'2 =

A-B+C 2

,

X13 =

A+B-C 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 equations qpX = [QpX, X], X e SO(n) and p(X) = XI + IX, where

INTEGRABLE SYSTEMS ON LIE ALGEBRA

61

0

while A, + A, :?6 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 q,(X) _ XI + IX (see 7.2). B) Realization of Toda's chain. First we shall show that the canonical symplectic space l 2n(q;, p.) together with co = dp1 A dq1 + + d p. A dq will allow realization in the Lie algebra of upper-triangular

matrices. We introduce the following notations:

are the upper-triangular matrices, 0

*

*

the lower-triangular matrices. We have a natural non-degenerate pairing a: T+ x T_ - 68, a(XY) = tr XY. This pairing lets us establish a

canonical isomorphism (T+)* = T_ and (T_)* = T+. We shall be examining the Lie algebra of upper-triangular matrices T+, then P

0

S

PZ

0

0

Pn-1

0

Sn-1

P.

It is easy to see that A = Uc A, where

62

A. T. FOMENKO AND V. V. TROFIMOV

P1 S1

0

P2

0 sn -1

P.

and A, is the orbit of coadjoint representation of the Lie group corresponding to T. We construct a mapping f : V?;Z" -+ A: PI

0

St

P2

0

0

Pn-1

0

Sn - I

Pn

f(x1,...,x,, Y1,...,Yn)

where

p1=Y1,sj=ex'-xj+1,1=1,...,n,j=1,...,n-1.

LEMMA 4.5

The mapping f : l2n -+ A gives the realization of a + dp" A dqn) in the Lie

symplectic manifold (l82n, dp1 A dq1 + algebra T+ of upper-triangular matrices.

Proof The equality {h1, h2} - f = {h1 - f, h2 - f } can be checked by direct calculation.

By a Toda chain we mean a Hamiltonian system on T*P" R"(Y1,

,

yn) O l"(x1,

, 1

xn) with the Hamiltonian: n-1

n

H = - Y_ Yk + Z akexk-xk+, 2k=1

k=1

The space T* l8" 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 H1 (F) = z tr(F + a)2 where F E A and

/0 a=

0

a1

0 ET,.

n-I 0

as the function H1 on A. Direct calculation shows that H = H1 o f. The

INTEGRABLE SYSTEMS ON LIE ALGEBRA

63

first integrals of the system s grad H1 on orbits of Ad* are given, for example, in [107]: Fk = 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 MZ"? In other words,

how does one find on MZ" 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 of functions

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 sgradf 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(p,, q,) such that w has a canoni-

cal form w=dp1 ndg1+ +dp" ndq". Let f =pi +q; . It is apparent that {J, f } - 0 for any i, j. Further, let f = °= f = (p? + q?). We shall now define the smooth function g(t): l - R such that g(t) 1 1

when t 1< E1, g(t) = 0, t 1> eZ and g(t) > 0 is a monotone function of t when E1 < t < e2 i see Figure 21. We shall put h(p, q) = g(f (p, q))-this

A. T. FOMENKO AND V. V. TROFIMOV

64

Fig. 21.

function is now defined on the entire manifold (el , ez are chosen sufficiently small). As {h, f } = 0, it is obvious that {hf , hf } = 0 for all i, j. The functions hf are defined on the entire manifold and they are zero

outside a certain disk. By covering manifold M" with a denumerable number of disks we can now stitch the functions hf constructed above together into a set of functions k1, . . , k" on M", so that each k; is .

distinct from zero on an open subset that is everywhere dense. We claim

that hk,, ... , hk" are functionally independent. If F(hkl.... , hk") = 0 then

F(g(k1 +...+k")kl,...,g(ki +...+k")k")0 and, if function

F(g(x1 +...+ x")xl,... g(xl +.+

0

we arrive at k1,. .. , k" being functionally independent. Thus we are now left with only one thing to check, that F(g(x1 + ... + x")xl,...,g(xl + ... + x")x" # 0.

If this function is identically zero, this means that g(xl + + x")xl, ... , g(x, + + x")x" are functionally dependent. We shall write out the Jacobian of this system of functions: -

1))

INTEGRABLE SYSTEMS ON LIE ALGEBRA

65

= gn + gn - 1(x1 + ... + xn) ag ,

insofar as all invariants of the matrix X2

n

...

x2

n

apart from the trace are zero. If J - 0 we shall find that g satisfies the differential equation g + x(ag/ax) = 0, i.e. g 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 commutative set consisting of algebraic (rational) functions exist on any algebraic symplectic

manifold? There are probably topological obstructions that prohibit

our constructing 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 [88], [91] 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 6 = Exp G on the coalgebra G* form a complete commutative set

F(O*) on these orbits 0*, i.e. any pair of functions f, g e F(O*) is in involution with respect to the canonical form o; the additive basis

fi,... , fk in F(O*) consists of functions which are functionally independent almost everywhere on 0*, and the dimension of the space F(0*) equals half that of the general position orbit, i.e. k = dim 0* _ z z(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 [89], [90] and also for many classes of non-compact real Lie algebras (see for example

[126], [127], [10], [134]). 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' = dim M. We now go back to

i

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, if the 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 Ii" 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. REMARK

At the present time there is a great deal of active work

carrying on PoincarFs classic researches on the questions of the non-

existence of supplementary first integrals. On this topic see op. [59], [60].

2

Sectional operators and their applications

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

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 = adc*(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 of the equations x = ad*c(X)(x). 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 Q V+ G) V- and of invertibility of the operator ada on the space V+ O 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 ipa,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, .5 be the corresponding group; p : H - End(V) be a representation of H in the linear space V; a: Sa -+ Aut(V) be the corresponding representation of the group; let O(X) denote the orbits of the action of the group S on V, X e V. If we introduce the linear operator

which we shall call a "sectional operator" Q : H - V, the vector field

go = p(QX)X will arise on the orbits. Note that the vector field 9 = 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, b e Ker 0a; &h = (ph)a is important for applications. For example, in the particular case H = so(n), p = ad the field )to 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 Oa, ¢q 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) V by M.

Let K' be any algebraic complement to K in H, i.e. H = K + K', K n K' = 0. 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 c M. From the definition of K' it follows that the mapping 0a: H -- V transforms K' into some plane (pa(K') c V monomorphically. Since OaK' = 4aH, the plane &&K' 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 Q 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 (.K' with M and Im(pb) by B and R' respectively. Thus, we have obtained a decomposition of O .K' as a direct sum of three subspaces B + R' + P, B and 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 Im(pb). The pb maps Im(pb) into itself isomorphically

(see Figure 22 where pb is invertible on Im(pb)). Let (pb)-' be the

INTEGRABLE SYSTEMS ON LIE ALGEBRA

69

Fig. 22.

operator which is inverse to pb on Im(pb). Assuming R = (pb) -'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 R on Im(pb). Then Im(pb) = Z + R'. Let T be the complement of B in M. Thus we have constructed a decomposition of the space V as 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 4aK',

K'=$+A

1B,R=Oa'R,P=Oa'P.

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

K+P+I +P by setting D 0

0 0 a1

0

0

0

0

0

0

ga1p(b)

0

0

0

0

D'

B - B being the operator D: T - K being any linear operator, inverse to 46, on the subspace B, /a 1p(b):R - R, p(b): R - R', Oa 1: R' -+ A, 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 X"Q = 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 to 'p (b) and 0. 1

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 0a -'p(b) _ a 1 adb . For example, taking as H the non-compact algebra so(n) Q R" which is the Lie algebra of the group of motions in the Euclidean space, we find (see below) that the system IQ = adQ)X)(X) transforms into the equations of

inertial motion of a rigid body in an ideal fluid. Given the obvious identifications of H and H*, we obtain K = K*, Z = 2 = 0, R = R. Let (5/.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 6/.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, a e T, R = R', Z = 0, b E K, qaK' + T = V = T + B + R. Provided that C: V - H is a sectional operator, we obtain , q) = B(CX, e T,T0 on the orbits

the exterior 2-form FC(X,

0(X)cV. 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 , q e Tx 0 be tangent vectors. Then, the vectors e K'(X) such

that pi;' = , pit = q exist and are defined uniquely. Let us take the sectional operator C: V -+ H. Let us define the bilinear form Fc = B(CX, [c', q']), [ ', q] a H, CX e H. This form is defined on the orbits and is skew-symmetric. Also, we have defined the flow x"Q on the orbits.

PROBLEM A For what operators C is the form FI. closed and nondegenerate on the orbits? PROBLEM B For what C and Q is the flow XQ a Hamiltonian one with respect to the form Fc?

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 IQ 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 IQ = adQ)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 Sa as a

symmetric space. Then, it can be written in the form Sa x sj/15; the involution a:.5 x .5 -+ .5 x Sa 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),XEH;H = (X,X),X E H (both H and its image in G are denoted by the same letter), aV = - V, aH = 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 9Q (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 also 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*(5 of the group by extending f to a left-invariant function F defined on the whole space T*(fi. Since T*6 is a symplectic manifold, we can obtain a Hamiltonian system on T*6 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, ) = (f ). Then, the Euler equations can be written in the commutator form I = [X, grad f(X)]. The case of geodesic flows for the left-invariant metrics on 6 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 cpX; cp: 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 0

A.#2j,

I=

i0j.

Let us consider on so(n) the operator O(X) = IX + X1. Then, we call

the equations i/iX = [X, >/iX] 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 = (xi,) then i(X) _ ((A, + 2)x1 ). It is clear that

_ A; - 2; + 2iI 4i

xjq xqj.

Assumingyn = 3 we can obtain

2 + Al X 13X 32

A 12

1

X 23

433

X13

_ 23-2j 3

+ % X12X23, 1

+ A22 X21 X13

As we have explained above, so(3) can be identified with l . 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 1yX = [X, OX] (for any n) the equations of motion of a multi-

dimensional rigid body. Let us choose I in such a way, that 2; + 2; # 0 for any i, j. Then, the operator >' is invertible on so(n) and the inverse

operator 0

1

= cp

is of the form (pX);; = [1/(2; + A;)]Xj. Let us

substitute the coordinates Y = OX in 09 = [X, OX] then, the equations of motion of rigid body are transformed into Y = [0 -'Y, Y]

Multiplying it by (-1) and denoting Y by X, we obtain the Euler equations

X = [X, cpX],

cp: 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

X. X A

q[=.,1 (A

Y_ xigxgj L

9=1

+ Aq)(2q + Ad

1

1

.+q

Aj + I,q

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, vector field I = [X, (pX], (p = OX = IX + XI 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 7.1

The

Proof Let 0 be an orbit, the point X a so(n) = so(n)* belonging to the orbit. Then, TXO = {[X, Y], Y E so(n)} therefore [X, (pX] a TXO. We have X = s grad F, F(X) = <X, (pX>. 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 (p: 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, pX] 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 Q V+ Q V- be the root expansion of the algebra. Let a, b a T, a : b be two arbitrary regular elements of the Cartan subalgebra. Let us consider the operator ada: G - G. It is clear that adalT =_ 0, ad.: V' -+ V', ad.: V- -+ V-, i.e.

this operator preserves the root expansion of G over C. Indeed, ado E. = a(a)E0 for any a a A. We assume that a and b are in the "general position," i.e. a(a) 0, a(b) 0. Then, the operators ad, and adb are

invertible on V+ Q V- = V, namely, ada 1 Ea = Ea/a(a). Let us define the linear operator (pa,b,D: G - G as follows (according to the general rules in Section 7): (pa,b,D(X) = cp,,,,X' + D(t) = ad-' adb 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, b and D belong to the operator (Pa,b.D It is clear that (pa,b,DE= = [a(b)/a(a)]E0. In the Weyl basis (Ea, E_ H,) the operator 9 is defined by the matrix

A. T. FOMENKO AND V. V. TROFIMOV

74

= Y'a,b,D

Aq

ab. V -' V;

a(b) na = a(a) ; 7

q = dim V± = (the number of the roots a > 0). The operator cpa,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 (ei). It is enough to verify that B(q ei, e) = B(ei, epee) for any i, j. We can assume i 0 j. Remember

that the plane T is orthogonal to the plane V = V+ Q V-. As W transforms V and T into itself and D is symmetric on T, it is enough to check that pa,b is symmetric on V. Since Ea (a 0) are the eigenvectors of (P, BI a(b) Ea, Ef I = BI Ez, fi(b) EP)

=0

(if a + #

0),

B(E,,, Eft 0.

Ifa+/3= 0, then _ (-a)(b) a(a) (-a)(a) a(b)

This completes the proof.

In the "general position" case, the operator qp on V has q distinct eigenvalues which are multiples of two. The operator gyp: V - V is an

isomorphism of V with itself. Remember that V + is a nilpotent subalgebra. Since V + is generated by the vectors E a > 0, cpl " is symmetric with respect to the Cartan-Killing form B(X, Y). The eigenvalues of this operator in the "general position" case are distinct:

INTEGRABLE SYSTEMS ON LIE ALGEBRA

75

AI, ... , AV 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 pp: G - G maps the subalgebra V+ ® T into itself, (pi v' ®T being an isomorphism of the space with itself. All eigenvalues of the operator (pi 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 q , and Cpl v' ®T are expressed by the matrices

(PI V+ EDT =

0

Thus, we have constructed Hamiltonian systems X = [X, cpa b p(X)] for each semi-simple Lie algebra (A. T. Fomenko, A. S. Mishchenko). These systems are analogs of the 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 Ga (see e.g. [47, 50]). Remember that

{E,+E_ , i(E,-E_,), iHQ} = W+Q+iT0. As in the previous section, we define the symmetric operator cp: G. -p G,'

which defines the Hamiltonian system X = [X, cpX] on G. which

of G. by the orbits of the coadjoint representation. Let a, b c iTo be elements in general position. Since preserves the foliation

ada E, = [a, E,] = i [a', E,]

(a = ia', a' E To),

ada E, = ia(a')Ea, a(a') being real. Hence,

76

A. T. FOMENKO AND V. V. TROFIMOV

ada(E,, + E_a) = a(a')(i(E, - E_a)),

ada(i(Ea - E_a)) = -a(a')(E2 + E_,). Thus, the operator ada: W+

W+ rotates the vector E. + E_2 into a

vector which is proportional to i(Ea - E_j and vice-versa. The operator adb acts similarly; it is invertible on W+ due to the choice of a e iTo. Then, all vectors Ea + E_a, i(E8 - E_a) are eigenvectors of the operator Pa,b = ada 1 adb: W+ -. W+ with the eigenvalues a(b)/a(a) _ a(b')/a(a'), a = ia', b = ib', a', b' e To. Similarly for the operators on the subspace W-. Let us define the operator tpa,b,D: G. - G. as follows: tpX = ip(X' + t) = tpa,b(X') + D(t) = ada 1 adb X' + D(t), X = X' + t being the uniquely defined extension of X to Ga = W+ $ iTo, Y e W+,

t E iTo; D: iTo - iTo being an arbitrary linear operator which is symmetric on iTo. In the basis ((E8 + E_a), i(E,, - E_8), iH') the 2

operator tp is defined by the matrix

/11

0

01

Ea+E_a 0

A9

A1

0

Y'a,b,D,

i(E2 - E_a) iH,

\

0

0

A

D/

a = a(b)/a(a) being real; q = dim W. PROPOSITION 7.3

The operator tp : G. - G. is symmetric if a, b, D

satisfy the conditions given above.

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(Ea + E_., i(E, - E_2)) = 0. The orthogonality of the rest of the vectors is known. In the general position case, the operator tpa,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 G which is called a normal

compact subalgebra. This subalgebra is generated by the vectors E, + E_ a c A. Since all these vectors are eigenvectors corresponding to the operator cp of the compact series, we get the normal series if the operators are restricted by the subalgebra G,.. These operators coincide with (p,b: G. - G., cpX = ado ' adb X, X e G,; a, b e iT0, a(a) # 0, a(b) # 0. In the basis (E, + E_,) the operators (p are defined by the matrices

1

0

(Pb=

,

q=dimW+.

0

Note that here a, b f G,,, 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 G. in some compact real form G. c G. A complete list of all these simple Lie algebras is given

below. The algebra

G,,

coincides with the fixed points of the

automorphism z : G -- G,,rX = X when the latter is restricted to G.. Let P c G be a subspace which is orthogonal to G in G,,, r = - 1. Then, the following commutativity relations are evidently valid: [G., G.] C [P, P] c G,,, [G., P] c P. Thus, the symmetric space 6 /6n is defined. Then, the space P is identified with the tangent space of the latter is embedded canonically in 6 as the Cartan model (see [48, 68]). Let us write down all the normal forms using the standard notations for the corresponding symmetric spaces (see [48]).

FORM Al G = sl(n, C), G. = su(n), G = so(n), aX = X, n > 1. The algebra G. is given in G. 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 X1 XZI, realized in sl(p + q, l8) by the matrices (X2 all X; being real, X3

X1, X 3 being skew-symmetric with the order p and q, X2 being arbitrary. Then, G. = so(p + q) D so(p) Q so(q),

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. G. = so(p) ® so(p) and G. = so(p) $ so(p + 1). FORM CI G = sp(n, C), n > 1, sp(n, l8) is the algebra \Xs

XX i

)x1

being real with order n, X2 and X3 being symmetric. Then, G. = sp(n), G. = u(n), the embedding G. - G. is given as follows:

A + iB -.

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

These are all the normal forms G. c G,, where G. is a classical simple Lie algebra, i.e. one of the forms A, B,., 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 ial

0

0

iaa

ibl

0

a=

= diag(ial, ... , ia ),

=

b = 0

a,, b; E l8; a,

±a; b,

ib,,

±bl (i

j). Then, the operator <pa,,: G,,

G

acts as follows: a(b)

(Pab(E,, + E-.,) =

a(a)

(E, + E-a).

Since each root a is given by a pair of indices (i, j), i.e. a = a;, (see e.g. [11]), each eigenvector E, + E which corresponds to the pair (i, j), is multiplied by the eigenvalue ti,j _ (b, - b;)/(a, - a;). Thus, when cp acts, the basis skew-symmetric matrices

79

INTEGRABLE SYSTEMS ON LIE ALGEBRA

0

1

-1

0

Eij=Tj - Tji= are multiplied by ),,j. Therefore, the Hamiltonian system X = [X, c,X] is of the form

_

"

Xij = L9=1 EXigX4j(Agj - Ai9) =

b4 - bj Xi4X4j

fl=1

aq - aj

bi - b9 ai - aq

Let a = -ibZ, i.e. ap = bp. Hence, "

1

Xij = E xigxgj (aj -+a., 9=1

1

- ai + aq

Then, when a = ibZ 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 cpab of the normal series there is the

classical operator iliX = IX + XI, I being a real diagonal matrix. Indeed, assuming b = -ia2, we obtain b; (PabEij =

0 = (Pa,_ia2;

ai -

bj

Eij = (ai + aj)Eij,

J.

(pabX = IX + XI

(I = -ia).

Thus, we have included the classical Hamiltonian system of the 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 '(n) be the Lie group of rigid motions of I?'. It is known that B(n)

80

A. T. FOMENKO AND V. V. TROFIMOV

is the semi-direct product of the group SO(n) and the commutative subgroup T of translations, the latter being a normal divisor in the group d° (n) and isomorphic to n-dimensional Euclidean space. A matrix representation of the group 41(n) is of the form

(x1,...,x")EI.

The action of the group SO(n) on the normal divisor 1 coincides with the standard action of SO(n) in R". Hence, the Lie algebra E(n) of the group d(n) is a semi-direct sum so(n) ®, III", cp: so(n) - End l" is the differential of the standard representation of SO(n) in III"; l8" is regarded

as a commutative subalgebra. In a matrix notation E(n) is of the form

0

0

The commutation operation in the Lie algebra E(n) is of the form

[x + , Y + q] = Cx, YA + x(q) - Y( ), being the results of the action of the matrices x and y on the x(q), vectors q and respectively; so(n) acting on 1B" 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) Q R" as linear spaces. Let B(X, Y) be the Killing form on so(n); (X, Y)e be the Euclidean scalar product in U8". We can assume ((x1,YI),(x2,Y2)) = B(xl,x2) + (Y1,Y2)e,

xl , X2 e so(n), y1 , Y2 E R". 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 form of the operation ad* under the isomorphism E(n)* = E(n). Remember that we denote ad* (x), e G,

x x G* as a(x, ).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

PROPOSITION 7.4

Let

E so(n),

x c: li',

81

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

ME(P")* = R",

+xeso(n)6),R"=E(n), S+MeE(n)*=(so(n)(DR")*=so(n)g I". Then,

a(S + M, + x)l,o(") = [S, f] + 2' (Mx' - xM`), a(S + M, c + x)JRn = -CM. M, a e 11" are expressed as column vectors, (t) denoting transposition. The proof follows directly from the definition of the operation a(x, g). 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 z = a(x, C(x)) flows along the orbits of the coadjoint representation Ad* of the group 0. 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 f(n). Let us consider in E(n)* (and hence, in E(n)) the subspace

K=

[(n+1)/21-2

O

l(E2k+1,2k+2) O Ire" C E(n),

k=0

E,, being elementary skew-symmetric matrices, e; being the standard orthonormal basis in R". The corresponding subspace in E(n)* we denote by K*. PROPOSITION 7.5

The subspaces K and K* allow the invariant

description K = Ann(x1) = { E G, a(x, ) = 0}, G = E(n) and K* = E G*,

x2) = 0; xI E G* and X2 E K

G being elements in general

position.

The proof follows from Proposition 7.4. Let us denote the orthogonal complement to the subspace W in E(n) or E(n)* with respect to the scalar product B(X, Y) + (Z, R)e by W.

82

A. T. FOMENKO AND V. V. TROFIMOV

LEMMA 7.1

x

Let a E K*. Let us consider the mapping 0.: E(n) --+ E(n)*,

x) E E(n)*. Then, &I( 1 c K#1, K1 being the orthogonal

complement to K, K*1 being the complement to the subspace K*. The proof is obvious. If a is in general position, then K = Ker ¢a and

0a: K1 -- K*1 is an isomorphism. Therefore, the inverse mapping K*1 - K1 is defined. We have the splitting as a direct sum of linear spaces E(n) = K1 p K and E(n)* = K*1 ® K1. According to the

general method, let a e K*, b c- K, a being in general position. If z = x + y e E(n)*, x e K#1, Y E K*, then Q(a, b, D)z = 0o I adb* x + D(y); D: K* - K being arbitrary.

Now, we can write down the general equations XQ = ad*,(X) on G* = (so(n) @ D8")* = so(n) $ p", 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` = IS, f] +

2(Mx' - xm')

11%M =

(1)

E so(n), x e R" 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 + M E E(n)*, + x c- 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.

PROPOSITION 7.6 (Fomenko, A. T.; Trofimov, V. V.) The system of differential equations ii = 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*('(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. [39]). Let us introduce a moving frame of reference. Let ni be the components of the linear velocity of origin a moving frame and

INTEGRABLE SYSTEMS ON LIE ALGEBRA

83

w, 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 = 2(Atl(otwf + B1Juiuj) + Cuuw1uj,

AlJ, B.J, C1J 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 A1J which have

i not equal to j are equal to zero and T = i(211uz + 222Uy + A33U2 + Aaawz + )55wr + A66wz)

For a sphere with radius b: All = 21rpb3/3, A22 = A33 = 27rpb3/3, A44 = 1.55 = A66 = 0. Thus, in this case 3

T=-3b (uX+uy+uZ). Let N = (yl, y2, y3); y1 = OT/8w1, K = (x1, x2, x3), x; = 8T/8u,,. Then, inertial motion of a rigid body in an ideal fluid can be described by the equations dN

dt =Nxw+Kx U dK

dt =

(2)

K x c o,

U = (U1, U2, U3),

w = (w1, w2, (0 3)

The kinetic energy of a rigid body is an arbitrary positive definite homogenous quadratic form in six variables u1i w1. Consequently, it is defined by the 21 factors (A,J, B1J, C.J) = (A1J). 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(8K/8t) = 0. Consequently,

const = K2 = KX + Ky + K. Taking the scalar product of the second

equation by N and of the first by K and adding them, we obtain N(8K/et) + K(8N/et) = 0. Therefore, NK = KXN., + KYNY + K=Nz = const. Evidently, the third integral is the energy integral T = const. Indeed, taking the scalar product of the

A. T. FOMENKO AND V. V. TROFIMOV

84

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

U a +w

a =0.

(3)

On the other hand, T is a quadratic form in ux, uy, uZ, o.

wy, wZ.

Therefore, according to Euler's theorem of homogeneous functions we have

aT OT aT aT aT aT 2T=ux-+uy+uZ+wz +wy awy -+wZaux NY

auZ

aWZ

awx

= UK + wN.

.

Differentiating this equation with respect to t and using (3) we obtain 2

di =K

at

We have dT

OT auz

aT auy

dt - aux at + NY at + aT awX

OT auZ

auZ at

OT awy

OT awZ

+ awx at + awy at + awZ at

=K 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 ca(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 x;, y;. In 1978 Weber studied the second case of Clebsch, i.e. when the fourth

integral is a homogeneous quadratic function of x,, y;, with some

85

INTEGRABLE SYSTEMS ON LIE ALGEBRA

particular conditions on the arbitrary constants. Ketter obtained the general solution of the latter problem. V. A. Steklov discovered the third kind of kinetic energy with equations (2) integrable explicitly.

1. First case of Clebsch be expressed as

Using variables xi, yi, the kinetic energy can

T = 2b11(xl + x2) + 2b33x3 + b14(x1Y1 + x2Y2) + b36x3Y3 2

+ 2b44(Yi + Yi) +

2b66Y3

The equations of motion (2) allow the fourth linear integral with the variables x; yi. With this energy, the body does not change its form

rotated around the z-axis through an angle of n/2. Assuming b14 = b36 = 0, we have a rotating body. 2. Second case of Clebsch energy

The equations of motion with the kinetic

T = b11xi + b22xi + b33x3 + b44Yi + bssyi + b66 Y3

and the expression for the coefficients b22

- b33 + b33 - b11 + b11 - b22 = 0 b55

b44

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 b11xI + 20

bssb66x1 Y1 + Y b44Y21

bii being defined by the relations

bll =

0,2 b44(b52 5

2

+ b66),

2

b22 = Orbss(b66 +

b42

4),

b33 = Q2b66(b44 + bss)

Apart from the three classical integrals, these equations allow the natural homogeneous integral of the second order with respect to xi, yi. Here, or is an arbitrary constant, the symbol E 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 >t': so(3) -* l83 which maps

the element X = xE12 + yE13 + zE23 to the point (z, - y, x). Then, fi[X, Y] = ->y(X) x ci(Y), qi(X) x >y(Y) being the vector product in O. Further,

fi(M)e - xM`) = M x x,

M=-

M, x E 111;3,

x M,

e so(3).

Proof If Z = + x e E(3), Z = S + M e E(3)* then, as shown in Proposition 7.4, a(Z, z) = (y, X);

X=

Y = [S, f] + Z(MX' - xM`) a so(3);

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) @ R3 (in this case many higher-dimensional effects are absent which facilitates writing in an explicit form).

K = K* _

K1=K*1=

0

a1

0

0

-al

0

0

0

0

0

0

2

/0

0

x2

ul

0

0

X3

u2

-X3

0

-X

2

Let

f=

0

0

f2

ul

0

0

f3

u2

0

0

-f2 -f3

)eK*`.

Then (-a, U2

0

0

- a1f3 + 2u1a2

0

0

a1f2 + 2u2a2

a1u1

0

0

a1f3 - 2u1a2 - f_a1 - 2u2a2

INTEGRABLE SYSTEMS ON LIE ALGEBRA

87

Let 0

b,

0

0

0®0

0

0

b= (_b1 0

0

0

X2

0

0

x3 ® y2

-x2

-x3

0

x=

,

bz

Then

b1x3 -j'bzy,

0

0

0

0

x36, - jb2 v1

-x2b, - jb2.v2

NX) =

.

0

0

-b2x2

-b1x2 - 3fb2y2 ® -b2x3 0

0

Therefore 0

0

4e ' adq

0

0

f2

0

0

f3 ® U2

z,

U,

=

0

0

-b

u, 1

z2

2

0

f

f2

0

0

-a1

a1 U2

'2

b2

1=

2a1 f3 + u1

z2 =

-2

-b2a2 2 2b,a,

62

bz

0

U1

b2

f3 - u2

b2a2 + 2b,a1 bzz

Finally Q(a, b, D)

0

f,

f2

-fi

0

f3 ® uz

-f2 -f3

0

U1

U3 a,

0

aft + flu3

a, 62

62

a1 U2

62

2 b2

f3 - u,

bZ

b2a2 + 2b1a1 -, u1 ® -2 a,fa22-

0

U3

b2a2 + 261a1,

a,

62 u2

U2

0/

62

b2

\

2

}'f1 + bu3

a, fi, 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. TROFIMOV

I -2a

0

0

0

0

0

Y

0

A

2

2a 1

0

0

0 b2

0

0

a,

0

al

0

0

0 v

-bzaz + 2b1a1

al bz

z bz

-2a1

al

0

b2

b2

2-P

0

0

bz

b2

0

0

- bzaz + 2b1a1

0 b2

0

0

0

S

l

It is clear that 1z

det A=

-a4b-4

r(T - f3J + 26a) ,

i.e. we can change the sign of det Abby the appropriate variation of e.g. the operator D. Writing A = bz z(bzaz + 2b1a1) we obtain the diagonal form of the form T

T= -2a(fi

-

-Au1-

2

Y

4a al

z

U3 +

/z

z

f3

b2A

z

+61 u3

2a1

al

- .i.Iu+bzA f2

\

z

+ (2a1)2 /

2

f3

z

(fz - 2b2) +

2ai(b1 - 1)

b z.lf3

1l2

)

2

f3

from which we can see that 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 = -Q:xzY3 - Y +

ab1

\l

Jxzx3, z

zz =

al

(b2

+

xlx3 + axlY3,

89

INTEGRABLE SYSTEMS ON LIE ALGEBRA

X3 = 0,

al at + b2) YzX3 (b2

yt = -aY2Y3 1

yz = aY1Y3 +

-y

2X1 Y3 +

R

al

+

A\

-y

2)XZY3

+2+2

J

xzx3,

a

Ylx3 - 2 + 2

X1X3,

Y3=01 x; = OT/au;,

A = (b2a2 + 2b1a1)/b22 ,

yi = aT/aw;,

u; being the linear velocity, a , being the angular velocity of rotation of the rigid body, 0

X cE(3)*:X =

-Y2

-y3

0

Yz

-Y

1

Yi

© x2

0

(XX,3

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 nonviscous ideally conductive fluid (see e.g. [64]). av

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

aH = rot(v x H), at

n= 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

M = [Q, M] - [H, J] I 1 _ Ell, H]

(4)

90

A. T. FOMENKO AND V. V. TROFIMOV

defined on the Lie algebra

so(n) of skew-symmetric matrices.

0 = (R9-,) * 4 is a right-translation to the unit of the group SO(n) of the velocity vector 4 e T;SO(n); J = AdY*-, j, j being flow density in the body; H = Ade 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 0.,,(G), a, fi E K which contains G as a subalgebra. As a linear space, 52,,,(G) is the direct sum G ® G. Let us denote the elements of the first summand by a e G, of the second one by eb, b e G, i.e. any element of 52,,,(G) is of the form x + By, x, y c- G. Let us define on 0.,, (G) the product

[x1 + ey1,x2 + EYz] = [x1,x2] + $[YI,Y2] + E([Y1,x2] + [x1,Yz] + a[Y1,Y2]) Note that S28,0(G) is an extension of G since there is an epimorphism of Lie algebras f : S2,,o(G) -+ G, f (x + By) = x.

The linear space 52,.,(G) = G + eG over the field K, the product [x1 + CY1, x2 + eye] defined above, is a Lie algebra for any LEMMA 7.3

a,feK. The proof follows from the definition of the commutator [x1 + By, xZ + EYz]

If 4$ + a2 = 40 +, 2 = 0, then the Lie algebras 52,,,(G) and 0,,.8(G) are both isomorphic, being isomorphic to the standard LEMMA 7.4

extension 00,0(G) as well. Proof Let us define the isomorphism cp: i2=,#(G) -+ Q(),o(G) = G + SG, S2,,R(G) = G + eG, e2 = ae + $ (SZ = 0) by the formula T(x + By) _ (x + (a/2)y) + by. Evidently, [cpX, cpY] = p[X, Y].

In what follows, we shall denote the extension 0,,o(G) by S22(G). For the application to Hamilton mechanics we need the Lie algebra !00(G) which we denote by Q(G) = G + eG (e2 = 0). We can define the Lie algebra Q(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 Q(G).

91

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Let us consider the Lie algebra S2(G) = G + eG, E2 = 0. We can assume (2(G)* = G* + (6G)* 3 (fl, f2) defining the isomorphism as follows

(fi,f2) -+ (.fi,f2)(x + y) = .fi(x) + f2(y),

x e G,

y e G.

Let us denote f e SZ(G)* by f = x* + Ey*; x*, y* E G*, By* 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 + By e S2(G) and x* + Ey* E S2(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 c- G. Then,

a(x*, x)(y) = x*([x, y]) = a(x*, x)(y) and

a(x*, x)(ey) = x*([x, By]) = 0 since x*IFG = 0. Let x c- G, x* e G*, y e G, then

a(Ex*, x)(y) = ex*([x, y]) = 0 since Ex*I G = 0,

a(Ex*, x)(ey) = ex*([x, By]) = x*([x, y]) = a(x*, x)(y) Let y e G, x* E G*, g c- G, then a(x*,Ey)(9) = x*([Ey,9]) = 0

since x*IEG = 0 and a(x*,Ey)(E9) = x*([Ey,E9]) = 0

because [Ey, Eg] = 0. Let y c- G, x* E G*, g c- G; then a(Ex*,Ey)(9) = ex*([Ey,9]) = x*([y, g]) = a(x*, y)(g)

and a(Ex*,ey)(E9) = x*([Ey,eg]) = 0.

92

A. T. FOMENKO AND V. V. TROFIMOV

Therefore, we finally obtain a(x* + ey*, x + ey) = a(x*, x) + a(ey*, x) + a(x*, ey) + a(sy*, ey)

= (a(x*, x), 0) + (0, a(y*, x)) + (0, 0) + (a(y*, y), 0)

= a(x*, x) + a(y*, y) + ea(y*, x) This completes the proof.

Let us consider the linear operator C: S2(G)* - S2(G) (we call it a sectional operator). Then, in the space S2(G)* we can give the system of non-linear differential equations a = a(a, C(a)), a e f2(G)*. This system appears when we integrate geodesic flows of the left-invariant metrics on

the Lie group 6. 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 = a(a, C(a)),

a e O (G)* is of the form

X = a(X, x) + a(Y, y)

(5)

{ Y = a(Y, x). Our next aim is to construct the linear operators C: S2(G)* --,. S2(G) for which the Euler equations (5) are completely integrable in the Liouville sense on all orbits in general position of the coadjoint representation of the Lie group n((5) which is associated with the algebra S2(G). PROPOSITION 7.9

The Euler equations on fl(G)*, G being a semi-

simple 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 B(X, Y), X, Y E G is given. We have the representation p = ad*: S2(G) -> End(Q(G)*). Let a e S2(G)*, then the mapping 0,: S2(G) - S2(G)*, fa(x) = ad* a, x e S2(G), a e S2(G)* is defined. In the case of the semi-simple Lie algebra

G we shall identify G* with G using the Cartan-Killing form B(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 = a1 + sae E fl(G)* = G + eG and we can assume aI, a2 E G. PROPOSITION 7.10

Let a c Sl(G)* bean element in general position, i.e.

the orbit of the coadjoint representation of the Lie group Q(6) which corresponds to a has maximal dimension. Therefore, we can take a from G. Then, a will be an element in general position in G. In this case,

Ker 0, = H + eH, H being a Cartan subalgebra in the semi-simple complex Lie algebra G.

Proof Let x = xI + x29 E Ker 0,. Then, a(a1 + eat, X1 + cX2) = 0

if and only if a(a2, x1) = 0, a(a1, x1) + a(a2, x2) = 0,

a, c- G* = G,

x, e G

(1 = 1, 2).

In the case of G we obtain the following condition [x1, a1] + [x2, a2] = 0, [x1, a2] = 0. Let us prove the first assertion of the proposition. We have dim OII(G)(x, y) = rk

CjXk Cit

C yk 0

)

If we take yin G (in general position) and assume e.g. x = 0, we obtain dim OO,G,(0, y) = 2 dim OG(Y) = 2 dim Oc

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 fl(.T) see [132] and Theorem 13.1). Let us calculate Ker 0,. From the equation [x1, a2] = 0 it follows that x1 belongs to the Cartan subalgebra H which contains the element a2 (see [47]). Let aI = al.k + Y:#o al.ae,, x2 = X2.k + >4:o x2.:ea; G = H + Y,:o Ce, being the root expansion of the Lie algebra with respect to H and al.h, x2,,, E H, e, being the root vector which corresponds to the root a e H*. Then,

94

A. T. FOMENKO AND V. V. TROFIMOV

[x1, all + [x2, a2] = Y (aI.=a(x1)e: - x2,.xa(a2)e.2) = 0. a*O

We have a1,,a(x1) - x2,,a(a2) = 0, x,, a2 E H for any root a # 0. Since

a2 is in general position, a(a2) # 0 for any a # 0, a e A. Therefore, according to the first part of the proposition, a, can be taken as zero. Then, a,,, = 0 for any a # 0, and x2,, = 0 for any a # 0, i.e. x2 E H. Therefore, Ker 1,, 9 H + H. The inverse expression can be checked easily. Thus Ker 0. = H + eH. Let b = b, + e62 E Ker 0. = H + eH c S2(G), H be a Cartan subalgebra of G. Let b, be in general position. Then, PROPOSITION 7.11

Kerp(b)=H+eH eS2(G)* =G+eG (the isomorphism is given by the Cartan-Killing form).

Proof We need to find all x, + ex2 e S2(G)* such that a(x, + ex2, b, + e62) = 0. Calculating ad* for f2(G) we find that this is equivalent to

the system a(x2, b1) = 0; a(x b1) + a(x2, b2) = 0. By identifying G* = G using the Cartan-Killing form we obtain [b,,x1] + [b2,x2] = 0,

[b1,x2] = 0.

Since b1 is in general position, x2 belongs to the Cartan subalgebra H. Then, since b2, x2 e H, [b2, x2] = 0. Therefore, from the first equation we obtain [b1, x] = 0 from which it follows that x1 E H, i.e. x1 + ex2 EH + rH. This completes the proof. Let Ker p(b)1

PROPOSITION 7.12

0a: Ker 0;

a

be

as

in

Proposition

7.10.

Then,

For any b e Ker ¢, (b is as in Proposition 7.11) we have p(b)(Ker p(b)1) c Ker p(b)1. Here V' means orthogonal complement with respect to the direct sum of the Cartan-Killing forms on G: Ker 0, = V + e V and Ker p(b)1 = V + 6V, V = >,#0 Ce, being an orthogonal complement to the Cartan subalgebra H of G with respect to the Killing form.

Proof Let

a = a1 + eat e Ker p(b) = H + eH c S2(G)*. provided that Y-, x. e. + e Y. yae, a (Ker Oa)1,

Then,

a(a, + ea2,Ex,e, +eY_ yae,) _

x,a(a,)e, +

y,a(a2)e, + e 2

x,a(a2)e, c Ker p(b)1,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

as

was

claimed.

Let

bl + eb2 a Ker 4 a = H + sH,

95

E, x,e, +

eY,y,e,EKerp(b)1. Then,

+eb2l

a/l

_

x,a(bi)e, + > y,a(b2)e, + e

y,a(b1)e, E Ker p(b)1 z

as was claimed. Here e, is the root vector which corresponds to the root a c- H*. REMARK Evidently, the mapping 0,: Ker 4 -+ Ker p(b)1 is an isomorphism of vector spaces.

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 & be any linear operator. DEFINITION 7.1

Let us define the operator C: S2(G)* -- S2(G) by the

matrix

C = C(a, b, D) =

(&1 adb 0

0

1

D)

in accordance with the splittings S2(G) = Ker 0. + Ker 0a , 12(G)* _ Ker p(b) + Ker p(b)1 (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 S2(G).

Let H be the Cartan subalgebra of the semi-simple complex Lie algebra G, a e H* be a non-zero root. We shall define h, e H using the

condition a(h) = B(h, h;) for any h e H, B(X, Y) being the Cartan-

Fig. 23.

96

A. T. FOMENKO AND V. V. TROFIMOV

Killing form of the Lie algebra G. Let Ho be the subspace in H which is generated by all vectors hz with rational coefficients. We shall consider the standard compact real form G. of the Lie algebra G (see [47]). We can take vectors ihz, e, + e_ i(e, - e_2) (i = as a real basis in G. Let us construct the sectional operators for 12(G,,). The operators

constructed we shall call operators of the "compact" series for the algebra Q(G).

Let a = i(al + ea2) E Ker 0, n S2(G.). Then, the operator ado for the Lie algebra f (G.) can be described as follows LEMMA 7.5

ex + e-, -+ a(a1)i(e, - e_,),

i(e, - e_) - -a(a1)(ea + e_2), e(e, + e_s) - a(a2)i(e, - e_,) + ea(a1)i(e, - e_a),

ei(e, - e_,) - -a(a2)(ea - e_,) - a(a1)e(ea + e_,).

The proof is obtained by direct calculation using Proposition 7.7. a = i(a1 + eat) E Ker p(b) n Then the LEMMA 7.6 Let operator ¢a for the Lie algebra fl(G,,) can be described as follows a(a1)i(e, - e_,) + ea(a2)i(e, - e_,),

e, + e_a

i(e, - e_,) -' -a(a1)(ex + e-2) - ea(a2)(e, + e-=), e(e, + e_,) - a(a2)i(e, - e_2),

ei(e, - e_,) - -a(a2)(e,, + e-,), Ker p(h) n f (G )* = Ker p(b) n The proof is obtained by direct calculation using Proposition 7.7. PROPOSITION 7.13

that a(a2)

Let a = i(a1 + sae) E Ker p(b) n S2(G.)* be such 0, b = i(b1 + eb2) e Ker 0, n S2(G,).

0 for any root a

Then the operator 0a 1 adb for the Lie algebra S2(G,,) can be described as follows e,

+ e_,

a(bI)

e ( e,

+ e_, ) ,

a(a2) i(e,

-

e-7!)

a( a

z) e

i( e,

- e_,

),

97

INTEGRABLE SYSTEMS ON LIE ALGEBRA

a(b2)

e(e,+e(o((a2) e_5)

a(b2)

(a(a2) -

a(b1)a(a1) a(az)

z

a(b1)a(a2)

a(a2)Z

E(es+e_,)+

a(b1)

a(az)

(e=+e ),

ei(e, - e-z + a(b1) Ilea - e-s a(a2)

The proof follows directly from Lemma 7.5 and Lemma 7.6.

Now we can construct the operator for

using the general

technique given in Section 6. Using the Cartan-Killing form we identify S2(G )* with S2(G,,) = G,, + eG,,. Let

(e,+e-,)+I Ri(e, - e_,) c- G. a

S

We have the splitting as a direct sum of linear subspaces

(Ho + eHo) + (V + n(Gu) = (Ho + eHo) + (V, + EVJ,

Ho =

Mhx. Q

In accordance with the direct decompositions

DEFINITION 7.2

(Ho + eHo) + (V + and

S2(G,,)* = (Ho + eHo) + (V,, +

we define C = C(a, b, D)

D C0

0

0a' adb J

D: Ho + eHo - Ho + eHo being any linear operator, a e S2(G,,)*, b c S2(Gu) 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: G,* - G. is diagonal with respect to the canonical basis e, + e_2, i(e, - e_,). In our case, different from the semi-simple one, the operator 0a 1 c ad,*, is not

diagonal with respect to the canonical basis. Let us construct the sectional operators for 12(Gn), G. being a normal compact subalgebra in the semi-simple complex Lie algebra G. By definition, G. _

Loo l8(e, + LEMMA 7.7

e_2).

Let a e iHo + eiHo c f2(G,,)*, b e iHo + eiHo c O(G.).

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A. T. FOMENKO AND V. V. TROFIMOV

Then the operators of the "compact" series C(a, b, D) preserve S)(G ). More precisely,

G+

G,, + eG -

C(a, b, D): f (G )*

The isomorphism S2(G )* = G. + eG is given by the direct sum of Cartan-Killing forms. The proof follows directly from the explicit form of the operator / 1 adb in Proposition 7.12.

Thus we have defined the operators C(a, b, D) = C: S2(G )* S2(G ). As in the semi-simple case, the elements a, b should not belong to i2(G.)*

and 12(G.) respectively. We shall call these operators operators of the "normal" series for the algebra S2(G). In conclusion, note that in the case of the Lie algebra G = so(3) from the construction of the sectional operators we obtain the following Euler equations. Let

1

0

x1

xz

-X1

0

-XZ

-X3

0

Y1

x3 J+e

-Y1

0

0

-Y2

-Y3

Yz

Y3 JES2(G)* = G* + eG*, 0

ez = 0. Then X1 = -klxzx3 + (k5 - k2)Y2Y3 + (k5 - k3)x2Y3 + (k6 - ka)x3Yz, X2 = k1x1x3 + (k2 - k5)Y1Y3 + (k3 - k6)x1Y3

+ (k4 - k6)x3Y1

,

X3=0, YI = -k1x3Y2 + (k6 - k3)Y2Y3, Y2 = k1x3Y1 + (k3 - k6)Y1Y3,

Y3=0. In this case, the kinetic energy H = <X, C(a, b, D)(X)> is of the form H = k1x3 + k5(Yi + yz) + k2Y3 + 2k6(x1Y1 + x2y2) + (k3 + k4)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 0(so(3))* is the limit (when b11 - 0) of the 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

S. CONSTRUCTION OF THE FORM F, AND THE FLOW XQ IN THE CASE OF A SYMMETRIC SPACE A. T. Fomenko's results given in this chapter can be found in [32], [34], [125]. Let Z3 = 6/t, be a homogeneous space. Then we can assume that the

Lie algebra G of the group 6 is split as a sum of linear subspaces: G = H + V, H = T, being the stationary subalgebra; V = TQ 13, i.e. it is identified with the space tangent to fly. The subalgebra H acts on V via the

adjoint action dp = ad, i.e. ad,. (X) = [H', X], H' e H, X e V. For the purpose of simplicity, we suggest that (5,J5 are semi-simple Lie groups; the space V is partitioned by the orbits 0 of the coadjoint action Ads.

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 Tin 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, T] = 0 ([k, t] = 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. m e M if and only if

[m, k] = 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

Min V. Then V=T+Z+R=R+M. 100

INTEGRABLE SYSTEMS ON LIE ALGEBRA

101

Fig. 24.

For the purpose of simplicity, let us assume that 'I3 = 6/5 is a symmetric space. This means that on G there is defined an involutive automorphism a such that a- = + 1 on H and a = -1 on V (a2 = 1 on G). Hence [H, H] c V, [H, V] c V (these relationships are valid for any homogeneous space), [V, V] = 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'.

For any a e Tin general position the mapping ada: H -> V defines an isomorphism between K' and T' (G, H being semi-simple Lie algebras, l3 = (5/55 a symmetric space). LEMMA 8.1

Proof Since K = Ann T = Ann(a) n H (a is the element in general position), H = K + K', because Im(ada) = ada K' c T' ada T' c K' c H because K = Ker(ada) n H. Conversely, T = Ker(ada) n V. It is clear that ada: T' - K' is a monomorphism since if a vector t' c- T' existed, such that ada(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 0(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 ada 1: T' - K' is uniquely defined. The mapping ada: 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 = ada ' R, Z = ada' Z, then K' = 2 + R. We can define the subspaces Z and R' in an alternative way. K acts as the stationary subalgebra on the space tangent to fj/R 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(adK) n T' = Z is isomorphic with Ker(adk) n K'. Indeed Ker(adK) n K' coincides with Z = ada ' Z. The action of the mapping ada is illustrated in Figure 25. In

particular, ada Z = 2, ada Z = Z, ada R = R, ada R = R, as proved below. PROPOSITION 8.1

Let us consider the coadjoint action adK: H -+ H.

Then adK: K' -. K' and Ker(adK) n K' = Z. In particular, R is the orthogonal complement to 2 in K' and both spaces R and 2 are invariant under the action of adQ : K' - K'; a being an element in general position in T (see Figure 25). Proof Let k c- K be an arbitrary element in general position in K. Then

Ker(adK) n K' = Ker(adK) n K'. Similarly

Ker(adK) n T' = Ker(adK) 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 ada 'Z = Z c Ker(adK) n K'. If i c-,Z then [k, [a, z']] = 0 since [a, z'] E Z. From the Jacobi identity for the algebra G we have

[k, [a, -]] + [z, [k, all + [a, [z, k]] = 0,

Fig. 25.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

103

i.e. ada [i, k] = 0 (since [k, a] = 0). Since [k, z] E K' and ad.: K' -, T is an isomorphism (see Lemma 8.1), [k, z] = 0 which was what we wished to prove. Conversely, let s c- Ker(adK) n K'. We need to prove that s r= Z = ada ' Z, i.e. ada s e Z, i.e. [k, ada s] = 0, [k, [a, s]] = 0. Since s c- K' and [k, s] = 0, from the Jacobi identity and because [k, a] = 0, we obtain

[k, [a, s]] + [s, [k, all + [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(adK) n K'. If Z E Z, then [a, z] a K' and

[k, [a, z]] = - [z, [k, a]] - [a, [z, k]] = 0 since [k, a] = 0, [z, k] = 0, z E Z = Ker(adk) n T'. Thus, ada Z c Z. Since dim Z = dim Z, ada Z = Z. Therefore, ada R = It which completes the proof. REMARK The two mappings ada : Z - Z and ada ' : Z -+ 2 differ from each other by the transformation ad':;? - Z which is a non-degenerate linear self-mapping of Z. In order to construct the form F, 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 V by using the restriction of the standard Kirillov form on V Indeed, B(X, n]) = 0 for any X, , n E V since rl] E H and H is orthogonal to V. Of course, we could consider V which is a linear perturbation of the

plane V (produced e.g. by the parallel translation of the plane by the vector h e H, h j4 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 dFc = 0, X Q FC = 0. Since ads,: V - H, we need to construct the operator C: V - H using operators of the form ad,,, 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. 0 = (fj x fl/15 = fj. Since the mappings ad,,: V - H are not identical for the case fj = (.5 x fl/.5, the action of the operators ad, needs to be compensated by operators of the form ad-. We should combine the operators of both forms, i.e. ad and

104

A. T. FOMENKO AND V. V. TROFIMOV

ad -', because, as we mentioned above, the space V can be mapped in H

with the use of both ad and ad-' (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 + + 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. + ada ' ad,, + D: ada.

C=

0 0

0

0

ad, 'adb 0 0

,

D

a, a' c- 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, . ada' adb: R ada.: Z - Z. Since Z = Ker(adK) n V, Ker(adb: V - V) 2 Z (if b is in general position in K, then Ker(adb: V - V) = Z) and adb: V --p R; ada' : R -p 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. Let us define the form F, on the space V (and on the orbits 0 = O(X) (-- V) by the formula Fc(X, , r7) = B(CX, q]), the DEFINITION 8.2

operator C being as defined above C = C(a, b, a', D); X,, E V; B(X, Y) being the scalar product. It is clear that the 2-form Fc is skew-symmetric on V (with respect to ry). Following the general method, we would have to consider the operation [c', rl] instead of rl] ( = [X, s'], n = [X, q']). However, since we consider the semi-simple case only, the forms B(CX, rj) and B(CX, [ ', q']) 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(d,b,d',D);Q:R+Z+T.i +2+K;Q =ada. +ada'adb+D. It is clear that the field XQ is tangent to the orbits 0 of the coadjoint action of Sj on V

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 8.2

105

Let f be a smooth function on V and the vector field

s grad f e V be such that F,(s 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 f].

Proof We have Y(f) = B(Y, grad f ). Hence B(Y, grad f) = B(CX, [s grad f, Y]) = B([CX, s grad f], Y) ;

B(grad f - [CX, s grad f], Y) = 0. Since Y e V is arbitrary we obtain [CX, s grad f] - grad f = 0 which completes the proof. If the form Fc 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 Fc(s grad f, Y) = Y(f ).

9. THE CASE OF THE GROUP , =!6 = (, x -5)/Sj (SYMMETRIC SPACES OF TYPE II) Let us consider the semi-simple group .5 as a symmetric space. Then, as

we know (see e.g. [48]), this symmetric space can be written as (S5 x 6)/Sa, the involution a: S x Sa -+ .5 x Sa being defined by a(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 (D H being denoted by the same letter), a V = - V,

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

LEMMA 9.1

Proof Here T = {(t, -t)}, t e T', T' being a Cartan subalgebra in H and K = {(t, t)}, t c- T', so that clearly T and K are isomorphic to F. Since V and H correspond to the same group, the coadjoint action adH 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,, X. Hence, to find Z it is sufficient to find the centralizer T' in H. Because we consider the semi-simple case, we have

Z = 0, i.e. the centralizer coincides with T. 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 + R and H = K + R are isomorphic to the

106

A. T. FOMENKO AND V. V. TROFIMOV

Cartan decompositions (linear for V). Note that although 3 is diffeomorphic to !5, v is not embedded in 6 = , x , as a subgroup. We

can assume that H = {(X,X)} and V = {(X, -X)} are identified by using the natural mapping a: (X,X) = (X, -X). In particular, the orbits 0 c V coincide with the orbits of the standard coadjoint action of

b on H. The form Fc(X ; , q), C(a, a, 0, 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 0 of the coadjoint action of Sa on V. PROPOSITION 9.1

Proof Since V = T + R, X = rzX + Y, nX a T, Y e R and

CX =DirX +ad,'adbY=xX +Y=XeH. Thus, Fc(X, , q) = B(X,

q]). 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, q]) and B(X, q']) are invariant on V with respect to Ads. Therefore, it is sufficient to compare them at just one point in the general position X0 in V; = adXo ', q = adXo q'; B(X0, [adXo ', adXo q']) and B(X D, [c', q']) differ from each other by the non-degenerate linear transformation adXo which maps the tangent space TXOO into itself. Hence the form Fc in general is not closed on V (in this example, Fc is closed on the orbits only). It is in this example ('B = t') that we can see the

role of the operator ad, ' adb which enables us to identify R and R in a natural way: the action adb is compensated by the action ada ' from which we can obtain the identity operator E assuming a = b. Using only

one operator adb (or ada) we would not obtain the operator E as a particular case of the sectional operator C (because on the space T the set of the roots is a redundant basis and hence, the system of equations

a(t) = 1, a 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'adb:R-,I. By using the identification of V and H, we find that the field XQ on V (on H) is given as follows . = [X, cpa,b,D(X)], the operator cpa,b,D: V - H (i.e. H -+ H) being identical with the multi-dimensional rigid body operator

107

INTEGRABLE SYSTEMS ON LIE ALGEBRA

introduced in [92]. In our case, D: T - T, ada 1 adb: T' - T, T' being the subspace generated by the eigenvectors of the operators ad,, t e T, H = T + T' being the Cartan decomposition. As follows from [89], all flows of the form cpa,b,o (a, b e T are elements of the general position in T), being Hamiltonian ones with respect to the form Fc (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 Fc, XQ.

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

Let us consider the space 6/Sj with rank (i.e. dim T) equal to the rank of ($

(i.e. the dimension of the Cartan subalgebra of G). Example:

V = SU(n)/SO(n); the embedding of SO(n) in SU(n) being the standard one. If dim T = rk G, then T V is a Cartan subalgebra of G (not only of V), i.e. K = 0 (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 = 0 (the annihilator K = 0 in V coincides with V). The following lemma holds.

If the space' = fui/.5 is of maximal rank, then the 2-form Fc on V is generated by the curvature tensor of the symmetric space v: 4B(a', R(X, ), r1) = Fc(X, , r1), R being the curvature tensor, a' e T being a fixed vector. LEMMA 10.1

Proof Since K=R=R=0, b=0, X=nX+X', nXeT, X'eZ; CX = C(nnX + X') = DxX + ada. X' (a' e T). Since K = 0, D = 0, i.e. CX = ada. X',

Fc(X; , i) = B(CX,

B([a', X],

B(a', [X, [x,11]]) = 4B(a', R(X, )U),

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

Since K = 0, 0 = .5, i.e. the orbits of maximal dimension in V are diffeomorphic to the group .5. Thus, the 2-form Fc on 0 =

is obtained

by the restriction of the 2-form B(a', R(X, )q) to .5 c V, the field

108

A. T. FOMENKO AND V. V. TROFIMOV

R(X, )q being tangent to the orbits 0 = S5. The flow 9. = [X, QX] is of the form [X, [X, a]] = adX(a),

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 Tin V (i.e. R = 0). Let T J be the n x n matrix on n, with the only non-zero element being equal to one

and occupying the location (ij) (i is the row, j the column), Ii, j I = - p
a=iik=IakTkkc T, then ad. Y = Y_ (aq - ap)xpq(Tpq - Tqp) e so(n); p
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 Y xpgl p, ql + i Y_ xpTpp p
P=1

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

JCpq = Y (2aJ - ap - aq)xPJxjq + Y (2aJ - ap - aq)xpjxgj j>q

P<j
+ E (2aJ - ap - aq)xigxjp + (aq - gp)(xq - xP)xPg, J
)E;(aJ-a;)xj,

p
and C is the seasonal operator defined above for the symmetric space SU(n)/SO(n). Proof Let us consider the decomposition of the 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 = 0. Therefore, the element

109

INTEGRABLE SYSTEMS ON LIE ALGEBRA

0

11

)+(

0

ix,,q

=h+Y

X= 0

0

ixPq

ixnn

is mapped into ada 0

by the sectional operator C. Since [C(h + Y), h + Y] = [ada Y, h + Y] = [ada Y, h] + [ada Y, Y],

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)/SO(3), coincides with the Euler equations of the motion of rigid body with a fixed point and an arbitrary inertia tensor in I83. PROPOSITION 10.1

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

dp dt

= (B - C)qr,

B

dq

dt =

(C - A)rp,

C

dr dt

_

- (A - B)Pq

In our case X13 = (2a2 - a1 - a3)x12x23,

x12 = (2a3 - a1 - a2)x13x23,

x23 = (2a1 - a2 - a3)x13x12

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

C-A=(202-a1-a3)B,

B-C=(2a3-a1-a2)A,

A-B=(2a1 - a2 - a3)C always has a solution, i.e. we can obtain an arbitrary inertia tensor. Thus, the statement is proved. PROPOSITION 10.2

Let C be the sectional operator for the symmetric

space SU(n)/SO(n) which is given by the element a e T. Then the external 2-form Fcis closed on V and hence it is closed on the orbits O: dFc = 0.

Proof Let us remember Fc(X;

,

il) = B(CX, [, q]).

A. T. FOMENKO AND V. V. TROFIMOV

110

Let us define the matrix-valued differential form d = Il dx, j 11 on the

space V. We can take the exterior product of this matrix with itself a=dA If , q e V, then (d A d)(c, >l) is the matrix with the location (i, j) occupied by the element Cij = [,k (cikllkj - Ilk kj) On the other hand, let us consider Evidently, i.e. rl) = rl]. Therefore, we can write down the differential (d A form FC using the Cartan-Killing metric explicitly: Fc(X) _ tr[C(X)(d A dc)]. Let the matrix elements d A do be of the form n

CUij = Y dx,k A dxkj. k-1

Then,

d A dg = > Y_ dxik n dxkj Tip

.

k

The operator C is of the form

C(X) = Y (aj - a,)x,j(Tij - Ti) E so(n). i<j

If X = iY_ xpq(TPq+ Tg9)eV, P.q

then

Fc _ Y C(X)pgUUgp

and

C(X)pq = (aq - ap)x

q

P.g

provided that x pq = Xqp. Thus, dFc =

I (aq

- ap) dxiq n dxpq n dxgi

P.q.i

=

Y_ (ap - a,) dxip n dxpq n dxgi

3

p,q,i

+ Y (ai - aq) dxgi n dxip n dxpq P.q.i

+ Y_ (aq - ap) dxpq A dxgi n dxip P.g.i

I (ap-a,+a, - aq+aq -a,)dxip n dxpg A dXgi=0. P.q.i

This completes the proof.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

111

10.2. The symmetric space Sn-1 = 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 '4 and symplectic structures Fc. Let us consider the sphere Sn-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 = TQS"-1. 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 E.j (the location (ij) is occupied by + 1; the location (ji) by -1; the rest of the elements are zero). Here i denotes row number and j column number. For simplicity, let us denote E;, 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, a), (a, q)] =

(i, q). A basis for V is the set (1, k), 2 < k < n, i.e. if X e V, X = y_k = 2

xk(1, k), xk being real numbers (see Figure 26). Then

H= so(n-1) 0 0

V

0

0 T= A(12)

K = so(s-2)

KI

TI

Fig. 26.

112

A. T. FOMENKO AND V. V. TROFIMOV

H = so(n - 1) =

Y

h,fl(afl);

25a<#Sn n

F= I xk(lk);

T = 2(12);

k=3

K = so(n - 2) _

Y_

K' = Y_ 4k(2k).

y2p(a/3);

k=3

352
Thus K = so(n - 2) and hence Z = 0

(it is

easy to check that

T=AnnKr) V). Therefore V=T+R and H=K+1 (2=0).

Hence, the operator C: V - H is of the form CX = C(nX + Y) = DnX + ad, ' adi, Y,

D: T - K.

Since dim T = 1, any element a e T is of the form 2(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. The sectional operator for the symmetric space SO(n)/SO(n - 1) is determined by the two elements b, d e K = so(n - 2). LEMMA 10.3

LEMMA 10.4

Let a = (12), b c K = so(n - 2) be an arbitrary element,

d = D(a) = D(12) E K. Let b = 11 b,,, 11, d = II d ij 11. Then, the operator

C: V - H, C(a, b, 0, D) = C is of the form CX = x2

Y

35i<j5n

dij(ij) - I (.="3 Y_ xabak(2k).

/

k=3

Proof Since X = rzX + Y, YET'; rrX E T;

ad,, X = ad,, Y = 35i
bikxk [(ik), (1k)] +

Y_

35k<jSn

bkjxk [(kj), (1k)]

X (Y x,,bak(lk). k=3 2=3

' (R Evidently adil 2)(2k) = (1k), i.e. ad( i) : (1k) - (2k), i.e. ad-'(R -ada(R - 1). Therefore

ado' ad,, Y = -

k=3 Ca=3

x,b,k)(2k).

Since X = E1n=3 xk(lk), trX = x2(12), DrrX = x2 Y_35i<j
proves the lemma.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 10.5

113

The form Fc(X ; , q), C = C(a, b, 0, D), is given on the

space V = TS"-' by the formula a=n

Fc = -x2

E

3,i
d,j dx, n dxj + yk= 3

xabak) dx2 n dxk

Y3

(see Figure 27). The orbits 0 of the adjoint action SO(n - 1) = fj on V = TQS"-' are the spheres S"-2 defined by the equation n

- xk = R2

(R = const).

k=2

The restriction fc of the form Fc to the orbit SR-' is of the form

fc =

C

y-

wij dxi n dxj,

0

0

0

0

0

SV

10 -8y

-6, CIE K=so(n-2) YER Fig. 27.

A. T. FOMENKO AND V. V. TROFIMOV

114

(Oij = -x2 dij -

1 X2

Xz

[(

X,(b,jxi - b,ixj) ) + (X? + xj )bij], a #(i. j)

= R2 - -k = 3 Xk ; x3, ... , xn being coordinates on the sphere SR- 2.

Proof Since the embedding K c H c so(n - 1) is the standard one (see Figure 26),

z:Ixk=R2, n

O=SR

k=2

being Cartesian coordinates in _k=2 Sk(1k), n = Y-k=2 nk(1k). Then X2, ...zz,

x"

definition Fc(X; , n) = B(CX,

V = 118"-'. Let

-bin; +;ni

= By

n]). Using Lemma 10.4 we obtain

Fc = trace(DirX + ad, ' ad,,

x,b,l dx2 n dxk.

dijdxi n dxj +

Y

k

(.=n3

Let us obtain the restriction fc to the form Fc on O. Since on 0: xz = forSETTSn 2; c_-(1x2)Y-k=3SkXkLet us assume

fi =B(DnX,n]),

fc=fi +f2;

f2 = B(ada ' adb Y, [ , n]) Then

I n

f2 =

an YY

\\11

n2bk) =3 xx

(S ink

-

fi`

xx

- nibk) (x, Y- Xabak - Xk L xabai a=3

a=3

3,i
Y

x2 3-< i
C( 2, (i,k) 3-<2

xa(xibk - Xkbai))

n

+ (x + X2 Let us calculate the form fl on O. We have

-

115

INTEGRABLE SYSTEMS ON LIE ALGEBRA

fi = BI x2

-x2

Y

3<-i<j,,-n

dij(Ij), 2£i<j£n - y (bi'j -

E

3_< i<JEn

The form Fc (C = C(a, b, 0, D), a = (12), b e K =

THEOREM 10.1

so(n - 2)) is closed on V = T¢S" -1 (in particular, on the orbits 0 = SR- 2 as well) if and only if d + 2b = 0. In this case, the form Fc is given by dl, the 1-form I being defined by the vector field 1(X) = (0, x2b(Y)), Y = (x3, ... , xn), b c- so(n - 2). The form Fc is not invariant on V or on 0 with respect to the action of the groups K = SO(n - 2) and S = SO(n - 1). If the orbit 0 = SR- 2 is even-dimensional and the skew-

symmetric matrix b E K = so(n - 2) is non-degenerate (e.g. b is in general position in K), then the form fc = Fcl o (i.e. the restriction of the

form Fc 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 fc defines a symplectic structure (not invariant under the action

of the group) on the open everywhere dense subset of 0 = S"'. Proof Let us prove that Fc is closed on V provided that d + 2b = 0. Indeed, since

y dijdxi n dxj + - ( - xab k Idx2 A dxk, k=3 a=3

dFc = -

dij dx2 n dxi n dxj +

J

bak 3

y

y

1

dij dx2 n dxi n dxj - 2

36i<j£n

dxal n dx2 n dxk

/

bak dx2 n dxa n dxk

3F2
(dij + 2bij) dx2 A dxi n dxj.

3£i<jFn

Thus dFc = 0 if and only if dij + 2bij = 0 for all i, j. Evidently (see Lemma 10.5 and the explicit expression for the forms), the forms Fc and fc are not invariant. Let us prove that the form fc is non-degenerate on S" - 2 almost everywhere. Let us consider the point X0 E SR- 2, X0 = (R, 0, ... , 0), i.e. x2 = R, xk = 0, 3 < k < n. At the point X 0 on TXOSR- 2 the form is given by the skew-symmetric matrix II2Rbij II . This follows

from the fact that the 2-form

A. T. FOMENKO AND V. V. TROFIMOV

116

1 x2

as(iJ) R 3<5Sn

x,(b,jxi - b,ix,) I + (x; + xj )bijJ

vanishes at the point (R, 0,. . . , 0). Since R # 0 the non-degeneracy of

the form ff is equivalent to the non-degeneracy of b = II bii II Since

b e K = so(n - 2) and dim TSR- 2 = n - 2, provided that n is even, we obtain the proof (in a small neighborhood of the point X0 on the sphere Sn-2 the form is non-degenerate and the coefficients of the form are algebraic functions of x3, ... , x"). The form F, is non-degenerate on S"-2, however it is degenerate on V= TS"-'. Moreover, if n is odd, then Fc is degenerate on V" - I . Since the form F, is closed on V, Fc is exact. The 1-form I is such that dl = F, is of the form 1 = x2 y-3
0X2-axis is equal to zero; in particular, I is tangent to the spheres Sx2 3 c S2 = 0 (the latter are given by the equation x2 = i = const; see Figure 27). The field 1 on S=2 3 is given by the standard action of the

operator b e SO(n - 2) on Sn-2 c R" - 2(x3, ... , x"). We still need to check the expression

Fc=dl; n

dl = Y_

(=n

Z b,kx,)dx2

k=3

A dxk

3

+ x2 Y b,k dx, n dxk + Y b,k dx, n dxk) = Fc a
a>k

because d + 2b = 0. This completes the proof. We can explain the fact that the symplectic structure fc on SR_ 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 D. The operator C: V - H is defined by the parameters a', b', D' (see above). Since dim T = 1, a' = ),'(12). Therefore we can assume that a' = a = (12). The

element b' c- K = so(n - 2) is arbitrary; the operator D': T -9 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(x2, ... , x") as follows

117

INTEGRABLE SYSTEMS ON LIE ALGEBRA n

x2=0,

-d;),

4 =x2 y- xa(b a=3

3
This means that the flow ) preserves the decomposition of the sphere _2 into the spheres SX, 3 = {x2 = const} (see Figure 27). If O = SR X = (x2, Y), Y = (x3, ... , X,,) E Rn 2, then the flow A' on the sphere S." 3 coincides with the flow Y = x2 pY, x2 = const, p = b' - d' c- K = so(n - 2), pY being the image of the vector Y under the standard action

of so(n - 2) on R(x...... xn). Proof From Lemma 10.4, n

r

n

n

x2dij(ij) - y (E xabsk)(2k), E

X=[

k=3 a=3

3<-i<j-
n

x2d;jx[(ij),(1a)] -

k3 =

3-
Y,

xp(Ip)

p=2

( y-

k=3 a-3

xabak)[(2k),xk(1k)] (or=" 3

xabak)[(2k),x2(12)]

n

x2dij[(ij), (12)])

x2xad1' [(if), (10)] + 0 = 3-
3ici<j
1

y- E x2xxb'(1k) -E I y- xabak/ Ixk(12) +k=3a=3 k=3 a=3

y

d,'x2xa[(aj), (la)] +

Y-

3
x2xad;a[(ir. ), (la)]

n

+ I Y- x2xab' (1k) k=3a=3

_ (since

3
y

xaxkb' (12) = 0)

/

x2xxdja(11) +

1

j=3

3

xzxad;a(1J) +4 E (x2 E4

3
k=3 \

n

xad;+ E1k=3 (X2 yJ a3 xabak)(1k)

zk(lk). Y- xx(b;k - d')1(1k) = X = _ k=3 E [x2 a=3 k=2

a=3

118

A. T. FOMENKO AND V. V. TROFIMOV

The lemma is proved.

Note one obvious integral of the flow X: Xkxk = x2

k=3

L

X.Xkpak = 0,

35a,k,
i.e. the function El = 3 xk = const along the flow .9 (we noted this before since X = [C'X, X] preserves S"-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: x2gY, q c so(n - 2), Y e R(x3,... , x"), x2 = const on the sphere Sz22 = SR 2 n (x2 = const). We obtain the form F, on V as dl, the 1-form I corresponding to the field 1. 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 x2 = const of the flow X is trivial, it is

enough to consider the restriction of the flow X to the sphere S"-' = (x2 = const). Since p e so(n - 2), p` = -P; x2 pX is tangent to S'-'. Let us consider the linear operator I:R "-2(X" ..,x") --+ R" z- (x3,...,xn) and construct the function Q,(X) = B(X, IX) on Rn-2

The function Q1(X) is an integral of the flow X = pX provided that [I, p] = 0 and Q1(X) j6 0 on S"-'. LEMMA 10.7

Proof

Q,(X) = B(9, IX) + B(X, IX) _ -B(X, x2 pIX) + B(X, Ix2 pX) = 0 if [I, P] = 0. The condition B(X, IX) 0 0 on S,-3 means that the vector IX is not tangent to the sphere S"- 3 at the point X. PROPOSITION 10.3 Let p c- 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 c- S(p), dim S(p) = r = [(n - 2)/2]. Let p = tl, t2, ... , t, be linearly independent vectors in S(p) (e.g. an orthonormal basis). Then the functions Q;(X) = B(t;X, pX), 1 < i < r

are functionally independent integrals of the flow X = x2 pX = points in general position on the linear space ... , Xn) = {x2 = const}, these integrals being in involution. R" - 2(X3, (0, x2 pY) at

119

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Also, the functions Q 1, ... , Q, are functionally independent integrals of

the flow X on the sphere S" - 2 = 0 which has the coordinates (x3, ... , x") (if the projection is orthogonal, then S" 2 -, V"- 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 fc is non-degenerate on So-2 = 0 (see Theorem 10.1). Then the flow X is completely Liouville-

integrable; the number of the integrals Q1,... , Q, being equal to r = (the dimension of 0 divided by two). If p c 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 T'). If n = 2r + 3 is odd, then the number of the integrals Q 1, ... , Q. is equal to '(dim O - 1).

Proof Since the operators t; commute with p, from Lemma 10.7 it follows that the functions Q; are integrals of the flow X. Since the Q; are quadratic functions, grad Q;(X) = (t; p)X. Since all elements p = t 1, t2, .... t, belong to the same subalgebra S(p), we can assume that all these operators are simultaneously reduced to the canonical form in some basis cpl , ... , c2, (provided that n - 2 = 2r). Let the operator t; be

defined by a set of the numbers t,, ... , 4 (Wl = ticp2, t;92 = -ti( Pi, etc.). If E = 1 C; t`Pkxk = 0 were a linear dependence relation (for all k), then 1 C; t;` = 0. Since the t; are independent on S(p), we find C; = 0, 1 < i < r. Thus, the independence ofQ;(X) is proved (almost everywhere on R" -2) . Let us consider in R" - 2(x3, ... , x") the sphere S", 3,

n = 2r + 2. Restricting the linear space L of integrals (where we take linear combinations with constant coefficients) to Sn-3, we obtain a (r - 1)-dimensional space, i.e. one of the integrals is constant on the sphere S". 3. Indeed, the system of linear equations C; tkPk, I S k < r has a unique non-trivial solution {C;} (since the system is in general position). Geometrically, the solution can be interpreted as a 1

vector in L which is orthogonal to the sphere S" `.This vector generates

Ker n where Ir: L --* TSn-3 is the orthogonal projection. Since the projection of the gradient of the function x2 (defined on Sn-2) onto the plane (x3, ... , x") coincides with this vector in Ker it, we can regard the function x2 as an integral. Geometrically, the motion of the trajectories of the flow X on S2r = Sn-2 can be interpreted very simply. The flow X

preserves Sn-3 = S2r-1 c R2r(X3,...,X2r+2). In the standard basis 91, , 92, in V2r (see above) the flow X = (0, x2 pY) is defined explicitly as follows: X(t) = Yj=1 (rotations within each of the 2-dimensional planes of the standard basis). Evidently, in the general

120

A. T. FOMENKO AND V. V. TROFIMOV

position case, the closure of the trajectory X(t) in S"3 = S2r-1 is the

torus T' which is embedded in S2,-1 c S2r. Thus, the tori of the commutative Liouville theorem (T' in M2' = S2') are generated by the standard embeddings of the standard tori into S2r. This completes the proof. 10.3. Hamiltonian flows XQ, symplectic structures F, and the equations of motion of analogs of a multi-dimensional rigid body

For simplicity, let us consider the compact symmetric space B = (fi/Sa, where 6, 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 (fi(X) 6(X) n V.

The intersection of the orbit 6(X) with the plane V coincides with the orbit 0(X) c 5(X) c V LEMMA 10.8

Proof Evidently, O(X) c ((X) since S) c 6. Since K = Ann T, dim TO = dim K' = dim 5/K. Because [V, V] cH,0(X) c6(X)nV. This completes the proof.

Let a, b e G be two commuting elements in general position. Let us consider a Cartan subalgebra S(a, b) c G which contains a, b (we know that such S(a, b) exists). Let us construct the operators (pa,b,D: G - G

which were defined and studied above (see [89], [90]):

qpa b.D =

ado 1 adh + D. Then, the flow X = [X, cpa,b,D(X)] describes the motion of the analogs of a multi-dimensional rigid body. The flow is tangent to the orbits (fi(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: S(a, b) -. S(a, b)), the flow X varies on O(X) as well. Let K 0 0, K c H, K = Ann T, let the elements a, b e G (which were initially elements in general position in G) approach V and H and eventually (becoming, in general, singular) come to T and K, a E T c V, b e K c H.

Then, the flow X = [X, (pa,b,D(X)] reverses on the orbit 6(X) and, approaching a "limit," preserves the intersection 6(X) n V = 0 = 5(X). On 0 this extremal "rigid body" flow can be regarded as coinciding with the flow XQ, Q = ada. + ado 1 adb + 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 a we have a(a) = 0,

i.e. the operator ada annihilates some plane E. in G and the operator

INTEGRABLE SYSTEMS ON LIE ALGEBRA

121

ada ' is not defined on E,. Since the flow X Q is the limit of the orthogonal projections of the rigid body flow on the plane which is orthogonal to Es,

it is clear that this plane coincides with V. Since the rigid body flow is completely integrable with the use of the integrals f (X + )a), f (X) being constant on the orbit functions, we can obtain the integrals of the flow X"Q by restricting the functions f (X + Aa) from the orbit 6(X) to the orbit ,(X) = 0 c 6(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 S"-'. 10.4. The symmetric space S"-' = SO(n)/SO(n - 1) (the complex case)

Let us consider the sphere S" -' = 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 i1121 = i(T1 2 + T21), i Z = -1 as the element a; the matrix b = i -+3 Sn bPgl pql, b = ib, bpg = bqP as the element F. Evidently, a, E e u(n), [a, b] = 0. We can assume that the

elements a and 6 belong to the same Cartan subalgebra in u(n). For b to belong to su(n) it is enough to require trace(b) = i Y-k = 3 bkk = 0. Also, it is possible to conduct all the calculations for the case b = ib, b being any real operator (not necessarily symmetric). However, for simplicity let us assume b` = b. LEMMA 10.9

The operator ad.-' ads maps the plane V into itself. We

have explicitly

ada' adsX = i Let a=i1121,

k=3 x=3

x. b,* (2k).

bpq=bqp, D:T-iK=so(n-2)c

H = so(n - 1); d = d,# 11 = D(12). Then, the operator C: V -. H, C = ado ' ads + D = C(a, b, 0, D) is of the form

A. T. FOMENKO AND V. V. TROFIMOV

122

dij(ij) -

CX = X2 3
xabk(2k). k=3 (="

3

/

Proof Evidently, ada(2k) = i I lkI, i.e. ada 1I lkl = -i(2k). Let X c V, X = yk=2 xk(lk);

adsX = [b,X] _ -i Y_

("

x=bak')Ilkl;

E 3

k n

ad-' adsX

x,b,kl(2k). k=3 a=3 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 11 b,, 11 by the symmetric one. These calculations are

similar to the ones in Lemma 10.4. This completes the proof. The form Fe, C =ad a 1 ads + D (the elements a and b being as given above) is given on the space V = TQS"- 1 by the formula: LEMMA 10.10

Fc = F1 + F2 = -x2

dij dxi n dxj

Y_

3
+Y

x, bak I dx 2 A dxk ;

/

k=3 a=1

bzk = bk.z.

The form F2 is exact on the space V: F2 = dl, the 1-form I is given by the formula x2 Y-'(z.k) = 3 b,k xx dxk. The orbits 0 of the coadjoint action are

the spheres SR- 2. The restriction fc of the form Fe to 0 is of the form

fc=fi

Adxj

= -x2

Y

dij dxi n dxj

3
1

if

E x,(xib,j - xjbai) dxi n dxj,

x2 3
2-

1

xk

k=3

We can write this in another way:

(Oij = -x2dij -

1.

r

E (x.(xib,,j - xjb,2i) + xixj(bjj - bii) + bij(x? - x;)

X2 L 2=3

2 #UP

;

INTEGRABLE SYSTEMS ON LIE ALGEBRA

123

X3, ... , xn are coordinates on the sphere SR- 2.

Proof Evidently, the restriction of the form

-x2

fl = B(D7rX;

y3
dijdxi n dxj

to the orbit 0 is of the form

E

-x2(x3, ... , xn)

dij dxi n dxj.

3
Then

f2 = B(ada ' ad6 X,

_

I

k=3 a=3

rl]) = (repeat the proof of Lemma 10.5)

x,b,kl dx2 A dxk.

I

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

E

fC = -x2

dij dxi n dxj

3
--

Y- x,(xib,j - xjb,;)1dxi n dxj.

X2 3
R ewriting the form f2, we obtain x.z(xibxj - xjb.zi) a=3 Y=#(i,r) z #(i,j)

x3(xib,j - xjbai) + xi(xibij - xjbii) + xj(xibjj - xjbji) (xib,j - Xjbai) + xixj(bj! - bii) + bij(x - x ).

Let us prove that F2 = dl on V Let us consider dl = Y_ b,kx, dx2 A dxk + x2 Y_ b,k dx, n dxk a,k

a,k

_ > b,kx, dx2 A dxk + x2 y b,k(dx, n dxk + dxk n dx,) 2,k

a
y b,,kx, dx2 A dxk = F2, (2,k)=3

The lemma is proved.

124

A. T. FOMENKO AND V. V. TROFIMOV

REMARK Similarly to the real case (see 10.2), the 1-form 1 is generated by the vector field I = (0, x2 bX), b = II bij II : R(x3, ... , x") R(x3, ... , x") being a symmetric operator (rather than a skewsymmetric operator in the real case). In contrast to the real case, the field

I on

Rn-2 is

not tangent to Sn-3(x2 = const); I on Vn-':1 eT, TO.

THEOREM 10.2

The form Fc, C = ado ' ad6 + D (the elements a and 6

being as described above, d = D(12) e K = so(n - 2)), is closed on V" -' = TS"-' and, in particular, on the orbits 0 = SR- 2 if and only if d = 0. In this case, the form Fe on V" ' is exact and can be written as dl, l = x2 Y-k= 3 (Y-n.= 3 bAxa) dxk. On V^ and on 0, the form

Fc=dl= Y

(Y_

xabldx2 ndxk

k-3 a=3

is not invariant under the action of the groups K = SO(n - 2) and Sn-2 is even-dimensional (n = 2r + 2, .5 = SO(n - 1). If the orbit 0 = r = rk SO(2r)) and the symmetric operator b = IIbPgII is in general position (being the element II ibP9 lI , bpq = bqp of the Lie algebra u(n - 2)),

then the form fc = Fel o is non-degenerate almost everywhere on 0. Thus, if d = 0 (n is even and the operator b(b' = b) is in general position),

then the form fc defines a non-degenerate symplectic structure (not invariant under the action of the group) on open everywhere dense

subset of the sphere 0 = S". 2. Proof Let us obtain a necessary and sufficient condition for the form

Fe to be closed on

V.

Since F, = F1 + F2, dFc = dF, because

dF2 = d 21 = 0 (see above). Y-i < j wij dxi n dxj, then

Let

us

calculate

dF1.

If

F, =

dFi = E (aiwjk + aj(oki + akw,j) dxi n dxj n dxk. i<j
In our case,

dF, =

1I

(xidjk - xjd,k + xkdij) dx, A dxj n dxk

X2 i<j
because xz = R2 - a= 3 xa. Thus, dFr = dF1 = 0 if and only if xidjk - xjdki + xkdij = 0 for all i, j, k, i.e. d = 0. As follows from explicit formulae, the forms Fe and ff are not invariant. Let us prove that if n is

even and b is in general position, then fc is non-degenerate on 0. As follows from Lemma 10.10, on the sphere Sn-2 the form ff with the

125

INTEGRABLE SYSTEMS ON LIE ALGEBRA

coordinates x3, ... , x" can be regarded as a form on the linear space R"-2

fu =

1

"

- x2 2=3 xa(x;b,; - x;b,;) + x;x;(b;; - b;;) + b;;(x? - x;

It is enough to prove that for some operator b, the matrix II f; II is non-

degenerate. Let b be close to a diagonal matrix, i.e. b;; - 0 for i : j. Then, the matrix 11 f;11 becomes the matrix f; = (b;; - b;;)(x;x;/x2). Let

us assume b;; = b;. Then, it is enough to prove that the matrix 11(b; - b;)(x;x;/x2)11 is non-degenerate provided that the vector b = (b,,. . . , b") is in general position. Thus it is enough to prove that the matrix 11(b; - b;)Il is non-degenerate (when n is even) considering in = x" = 1. Rn-2(x3, ... , x") the point with the coordinates x3 = Using the condition that the vector (b3, ... , b") is in general position, let

us assume b,, = b3, b # 0. Then

lb;

0

1-b

1-b2

b-1

0

1-b

1-b2r1b2r-

- b;l b2r-1

-1

b2r-2

-1

0

As b approaches zero, we find that it is enough to check that the matrix L; <; (i, j) is non-degenerate. Considering the corresponding system of linear equations, we obtain 0

+ X2 + X3 + ... + X2r = 0

-X1 + 0 + X3 +

+ X2r = 0

-x1 - X2 - X3 - ... + 0 = 0

x2+x3=0, x3+x4=0, ...,

Hence

xk + xk+1 = 0, ... , x2,-1 + x2r = 0, i.e. (x2s+x2s+1)+x2r=0.

s=1

Therefore x2r = 0 and consequently the system has only the trivial solution. This completes the proof. Evidently, provided that for some pair (ij): b. = b;, the form II (b, - b,)(x;x;/x2)11 is degenerate on R"-2. Let

us calculate the flow go = [X, (X] on 0. The operator Q: V -+ H is of

A. T. FOMENKO AND V. V. TROFIMOV

126

the form ad," ad,, + D'. Since we suppose a' = Ai1121 = Ad, let us assume a' = a, b' = c = i[.,3sp<j<"Cpjl pjl; 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 fc(d = 0). Therefore, we denote d' by d when constructing X'Q.

Let the elements a, c(c` = c), d e K = so(n - 2) be

LEMMA 10.11

as described above. Then the flow Q = ad, ' ad, + D, is given on the space V(x2, .... x") by

chosen

[QX, X],

n

X2 = 35a,k-
Xk = X2 I xa(ck - dak), a=3

3 < k < n. In this case, c` = c, d` = -d. Having x2 + + x2 as an integral, the flow XQ is tangent to the orbits 0 = SR-2 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 = Zk=2 xk(1k); 'rX = x2(12). From Lemma 10.9 we obtain

r

n

(nY-

n

xac)(2k),

x2dij(ij) - Y

k=3 a=3

L 3-
p=2

_ (see the proof of Lemma 10.6)

(Xacak)Xk(12)

=k=3

xp(1,P)

+ " [x2 a=3 Xa(ck - dk)J(lk) k=3

=3

= X = z2(12) + YR zk(lk). k=3

Hence, n

x2

=-

I =3 C a=3

k=

x2Cak)Xk;

There is one obvious integral

xk = x2 a=3

xa(ck - dak).

x2+...+X2 of the flow 9.7 because

n

X2-i2 +

k=3

XkXk = - X2 E X,,XkCak a,k

+X2EXkXlCak -X2LX,,XkdA =0, a,k

a,k

INTEGRABLE SYSTEMS ON LIE ALGEBRA

127

since c` = c, d` = -d. This completes the proof. It is useful to write the first expression as n

cakxaxk = -B(CY,Y).

Y_ caaxa -2

x2

a=3

a
Let us consider the problem whether the flow XQ is Hamiltonian with

respect to the symplectic structure fc on the orbit 0 = Sn-2. THEOREM 10.3

Let the closed 2-form fc = Fe I o be given on the orbit 0

ado ' ad6 = c(a, b, 0, 0), the elements a, b being as by the operator described above; b = ib, b` = b. Let the flow XQ on 0 be given by the operator Q = ad, ' ad, + D, c = ic, c` = c, d = D(12) E K. Let us consider together with the symmetric operators c, b (which map Rn-2(x3, ... , x4) into itself) the symmetric operator h = sb + bs`;

s=c-d, i.e. h= (bc +cb)+ [b,d]; h`=h, h:Rn-2-pRn-2. Let us consider on R" 2 four vector fields cX = X, hX, bX, cX and the corresponding 1-forms n

EX =

x2 dxa; a=3

HX = Y (hX)a dxa; a=3 n

BX = j (bX)a dxa; a=3

CX = E (cX)a dxa. a=3

Then on the orbits 0 the form fc is invariant under the flow Xq (i.e. X¢ ff = 0) if and only if the 1-forms E, H, B, C satisfy the relationship E A H = 2B A C. The symbol A denotes exterior product. If n = 2r + 2 is even and the operator b is in general position, then fc is a non-degenerate symplectic structure almost everywhere on S2r and the condition E A H = 2B A C describes all such Hamiltonian flows XQ with respect to fc. In particular, if the flow XQ is fixed, varying b and din

such way that E A H = 2B A C, we obtain a series of different symplectic forms for which the flow XQ is Hamiltonian. If the operators

b and c are both diagonal, b = Y-i= 3 b. T;, C = E7- 3 c; 7i;, then the condition E A H = 2B A C becomes [b, d] = 0, [b, c] = 0, c = II c,i II , c;j = c; + cj. This means that if bk bp, then ck + c,, = 0 and dpk = 0; if bk = bp, then Ck + cp and dpk are arbitrary.

REMARK The vector fields eX, hX, bX, cX on Rn-2 can be identified with the gradients of the following quadratic functions fe = B(X, X),

f = ,B(hX, X), fb = B(bX, X), J = B(cX, X). Then a necessary and

A. T. FOMENKO AND V. V. TROFIMOV

128

sufficient condition for the flow A'Q to be Hamiltonian (which was formulated in Theorem 10.3) is

dfe ndfk=2dfb Adf;

E A (BC + CB + [B,d]) = 2B AC.

Thus we can vary the flow X'Q so as to preserve the form fc (for a fixed fe)

since the factorization of the form w = E A H = 2B A C in the Lie algebra of exterior forms on Rn-2(x3, ... , x") is not unique. The more such factorizations we have, the more flows IQ exist which are Hamiltonian with respect to the form fc. Proof of Theorem 10.3

As follows from Theorem 10.2, the form Fe on

V"-I is k=3 a=3

xabldx2

A dxk.

Let us assume t = c - d. Let us calculate X QFc = E

(E:i,,bk

dx2 A dxk + Y-( x,bak dx 2 A dxk

+E(Y- x,bA dx2 n dik =

I (E

dx2 A dxk + > (E x, b,k l d( - y xs xq csg) A dxk

b,k (Y- x2xs

/

lI//

4

+Ex,b,kldx2 A d(x2Exstsk) (Z b, (y xatsa1 I dx2 A dxk + x2 E (Y- xab,k) E t,k dx2 A

= x2

rk

(

x,b,k I

(xgc,q dx, + xsc,q dxq) A dxk

J s,9

ks

x2Ly f(y bA( x,t,a +

a

- 2 > (Y- x,b,kl(

/

XgC,q 9

\

)

J

J

l

(Y xab I tk, } dx2 n dxk] dx,) A dxk

dxs)

129

INTEGRABLE SYSTEMS ON LIE ALGEBRA

= X2 z [ X3(bak tsa + bsa tka) J dx2 A dxk a's

k

-2y- [(Y- xabald(Exgcsg)]dxs s,k

n dxk.

at

This calculation was for V' '. Evidently, on V we have J.

0, i.e. in

general the flow X'o is not Hamiltonian with respect to Fc. Let us consider the orbit 0. Let us restrict the derivative Fc to 0. Using the relationship n 1

dx2 = - X2

Y- xkdxk

k=3

we obtain

FC1 =

-y11 t XsCgs"J dXq A dxk [Y- XsXq(bktsa + bsatka) + 2(E E xabakl( k,q a,s a

- yq
t {XsXq(baktsa + bsztka) - xsxk(bzgtsa + bsatga)} J(

[y-

a,s

+2 (y- xabak)(xscqs)

-2

l

xscks)] dxq A dxk

xxbaq/

\a

s

- [.

s

s

\1

Xs(Xghks - Xkhgs) +

ly-

- 2 \Yabga

2( bkaXa)C Cgsxs)

11

(-S c. xs)] dxq n dxk . xa/

We are also using

h = btT + tb,

hks

= > (bkz(tas)T + tkabas); a

Y bkaxa = (bX)k ; a

Hence,

i0 =

jj

=

3

[ [xq( <

Y- Cqs X., = (CX )q

\11

\

hksxSJ - Xk(> hgsxs I

- 2{(bX)q(CX)k - (CX)q(bX)k}J dxq n dxk

A. T. FOMENKO AND V. V. TROFIMOV

130

21 [(Xq(hX)k - Xk(hX)q) - 2(dfb n df)gk] dxq n dxk q
I Y- (dfe n dfk)gk dxq n dxk - 2 E (dfb A dfe)gk dXq n dxk } q
q
=-dfeAdfb+2dfbAdf. Thus fr = -dfe n djb + 2dfb n df , i.e. fe = 0 if and only if E A H = 2B n C. This completes the proof of the first part of the theorem. Let the

operator b be diagonal and consider the equations for the operators c and d. Since b,j = S,jb,, the above system becomes Y- Xs[xghk, - Xkhqs + 2bkXkCq, - 2XgbgCks] = 0, s

EXS[Xq(hks - 2becks) - xk(hgs - 2bkCgs)] = 0, s

XkXq(hkk - 2bgCkk - hqq + 2bkCqq) +X2 (hkq - 2b qCkg) - Xk (hqk - 2bkCqk)

+ Y xs[xq(hks - 2becks) - xk(hq, - 2bkCgs)] = 0. s # (k,q)

Thus we obtain hkk - hqq - 2bgCkk + 2bkCgq = 0,

hkq - 2bqCkq = 0;

If s

hqk - 2bkCqk = 0.

(k, q), then hks - 2bq Cks = 0; hqs - 2bkcgs = 0. If the operator c is

diagonal, i.e. c,j = 6,,c,, then hkk - hqq - 2Ckbgq + 2Cqbkk = 0; hkq + 2Cqbqk = 0;

hqk + 2bkgCk = 0

and if s (k,q), then hks + 2b,kcq = 0; hqs + 2bsgCk = 0. If both operators b and c are diagonal, then hkk = 2bkck; and if k : s hks = bk t,k + tk, b, = - bk dks + dks bs =

(bs - bk)dks

Thus hkk - hqq + 2bqCk + 2bkCq = (Ck + Cq)(bk - bq).

bk, = Cks = 0 for k # s, hks = 0 for k # s. Therefore (b, - bk)dk, = 0. Finally if b, c are diagonal then we obtain the equations

Since

131

INTEGRABLE SYSTEMS ON LIE ALGEBRA

(c, + cj)(b, - bb) = 0, (b, - bj)d,; = 0 for b, c, d. Thus if b; 0 b; then d,; = 0; c; = -c1. This means [b, d] = 0 and [b, c'] = 0, c,j = c; + c3. Thus the theorem is proved completely. REMARK The equations for b, c, d can be derived from the condition XQI = 0 (Fe = dl on V) as well. Let us consider examples of the forms Fc and the flows go obtained above. Some of the flows we already know.

10.5. Examples of flows X¢ on S" - I (the complex case) EXAMPLE 1

Let us consider XQ,

ad, 1 ad, + D. Let c = 0,

d EK = so(n - 2). Then 2 = 0, Xk = -X2 D=3 xdk, i.e. the flow go coincides with the flow of the form X, (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 So assuming d = 0, c,, = b,,c; (i.e. c is

diagonal). Then, z2 = - D" = 3 ck xk; xk = X2Ck Xk, 3 < k < n. This flow is integrable explicitly on the sphere S"-I. The complete set of integrals (obtained by A. V. Brailov) is very simple: f,,j(x3, ... , x") = (x;')/(xjCi);

<j < n; f = xZ + x3 + + x.2. Calculating directly, we can check that X f, = 0. Evidently the integrals f34, fa5, .... functionally independent at the points in general position on the orbit 0 = S"-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 3

f"are

yk = In Xk, 3 < k < n the flow X Q can be reduced to z2 = - Ek = 3 ck e2yk; Yk = CkX2; 3 < k < n.

In Examples 1 and 2 we pointed out two "extremal" flows; the flow go 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). EXAMPLE 3

Let n = 4; 23 = S2 = SO(4)/SO(3); we shall define the flow

XQ using the operator

ada I ad, + D, c = ( 0

0 ), c = is e su(2), a

132

i.e.

A. T. FOMENKO AND V. V. TROFIMOV

C3 + c4 = 0. Let d e K = so(n - 2) = so(2), d = ( -0

A).

We

assume x2 = x, x3 = y, x4 = z. Then the flow X on V3 = TS2 is given by x = -cy2 + cz2, y = x(cy + Az), i = x(-cz - Ay). Assuming c = 1

we obtain z = -y2 + z2, y = x(y + Az), i = x(-z - ply). 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)/ f = u, (z - y)// = v. Then x = (z - Y)(z + y); (z + Y) = x(2 - 1)(z - Y); (z - Y) = -x(2 + 1)(z + y), i.e. we obtain x = 2uv, u = (A - 1)xv, = -(A + 1)xu in the variables x, u, v. Let us recall the general form of the equation of motion of a classical three-dimensional rigid body X = uv(113 - 223); li = Xv(123 - 212); v = x4(212 - A13), 2:; = (b; - bj)/(ai - aj) = a; + aj since b = a2 (see 7.2). Therefore z = uv(a1 - a2), zi = xv(a3 - a1), u = xu(a2 - a3); a1, a2, a3 being arbitrary. Comparing this with our system, we derive the

conditions a1 - a2 = 2, a2 - a1 = A - 1; a2 - a3 = A - 1. Therefore, the relationships a1 = (p, a2 = - 2, a3 = A + q0 - 1 are satisfied. Since the parameters a1, a2, a3 are determined up to a constant factor,

we may assume a1 = 1 to obtain a2 = 1 - (2/9), a3 = 1 + (A - 0191 i.e. (a1, a2i a3) = (1, a, ft), a, fl being independent and arbitrary (given the arbitrary choice of c = c4 = -c3, A = d34, i.e. given a permissible variation of the operator which defines XQ). Thus for n = 4 the flow XQ coincides with the flow which corresponds to the dynamics of an arbitrary three-dimensional rigid body with a fixed point. The flow has the following trivial integrals: x2 + u2 + v; (a1 + a2)x2 + (a1 + a3)u2 + (a2 - a3)v2. It is well known that this flow is Hamiltonian with respect to the classical symplectic structure on 0 = S2 which in our case coincides with the invariant 2-form giving the standard volume on

the sphere S2 c R3 = V. However, as proved above, the flow .Q preserves the form fc = Fcl o uniquely defined on 0 = S2 by the operator C = ada 1 adr (see above). The problem is: if the flow XQ (defined by c and d) is fixed, is it possible to choose a non-degenerate form in the class of the forms fe which are preserved by XQ. In other words, is the flow of rigid body X'Q Hamiltonian with respect to some form fc? The answer is PROPOSITION 10.4

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

133

INTEGRABLE SYSTEMS ON LIE ALGEBRA

operator Q =ada ' ad s + D, 6=i1121, c = is E su(2), c=

/'

(c30

C4 0

c3 + c4 = 0 coincides with the equations of motion of a threedimensional rigid body with a fixed point (al, a2, a3 being arbitrary). Moreover, this flow X is Hamiltonian with respect to the nondegenerate symplectic structure fe on 0 = S2 defined by the operator c = ad.-' adb (with an appropriate choice of the operator b = ib) according to the general construction given in 10.4. Proof Above we already have proved the first part of the theorem. We need to choose the closed non-degenerate form fc which is invariant

under g¢. For this, let us consider the form Fe = dl given by the arbitrary operator b= ib, b = (b33 b3al (not necessarily symmetric) `b43

boa

on V = R3. Then the explicit expression for Fc is

Fc = y I ± xab, k=3 \a=3

I dx2 A dxk.

/

The restriction fe to S2 is given by fe = f,, dx; A dxi,

f; = - 1 x2

La#c,.;I

xa(xibj - x;bat) + x,x!(b;; - b1i) + xi big - x; bj;l. JJ

In our case, all f; = 0 except

{

1

J34 = -

X2

Cx3x4(b44 - b33) + x 3b34 - x 4b43]

Since Fc = dl, this form is closed on V and 0 for any b. Since f34 # 0, i.e. 1

b#µ 0

0

1), E R, the form f,. is nondegenerate on S2(x3, x4). Thus it

is enough to verify that the equation '? fc = 0 (regarded as the equation

in the operators b, d) has a solution such that d # 0 (i.e. , # 0), bOu

1

0). Indeed, since c is diagonal, the equation .Y'¢ fc = 0 is

equivalent to the following system (see Theorem 10.3) h33 - h44 - 2433 - 2c3b44 = 0 h34 + 2b34C3 = 0

134

A. T. FOMENKO AND V. V. TROFIMOV h43 + 2b43C4 = 0

7

h=btT+tb=be+cb+[b,a]. Hence,

h=

2c3b33 - 2(b34 + b43)

b34(C3 + c4) + )(b33 - b44)

b43(C3 + C4) + 2(b33 - b44)

2x4b44 + A(b34 + b43)

1 )

-A(b34 + b43) + (C3 + C4)(b33 - b44) = 0 b34(3c3 + c4) + )(b33 - b44) = 0 b43(C3 + 3c4) + A(b33 - b44) = 0.

Since c3 + c4 = 0, from the system we obtain A

b34 = - b43 = 2C

(b33 - b44),

2#0, c=c4.

Thus,

b=

p

q

-q p -2cq

detb:0.

A

Since 2

0 and p, q are arbitrary real numbers, the theorem is proved.

If we are looking for a solution of the equation X¢fc = 0 in the class of symmetric operators b, from Theorem 10.3 we obtain REMARK

(d = 0, (c3 + c4)(b3 - 64) = 0, i.e. b = b03

b0a)

fc is closed and non-

degenerate on S2. In this case, due to the condition d = 0 (i.e. A = 0) it is guaranteed that such a rigid body flow is Hamiltonian only if p = 3, i.e. aI = 3, a2 = 1, a3 = 2, i.e. the inertia ellipsoid is of the form (1, 2, 3); the inertia ellipsoid is not arbitrary (though the axes are of different lengths).

The statement formulated in Proposition 10.4 can be proved not only for the rigid body operators (i.e. is c su(2), i.e. c3 + c4 = 0) but also for any diagonal c, c3 + c4 = 0. In this case, we have the relationship 22 = (C3 + C4)(3c3 + C4)(C3 + 3C4)I2(C4 - C3)

between A, c3, c4. Therefore, the equation X¢ fc = 0 can be solved.

135

INTEGRABLE SYSTEMS ON LIE ALGEBRA

EXAMPLE 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 g : x2 = -cx3, z3 = cx2x3. Hence, x = 2cxz,

z = cx2 + A; let c > 0. Then, if A > 0, the solutions are given by the functions l CA

arctg (

if A < 0, by the functions 1

Jcx)=t,

IxI
arcth(x)=t.

IxI > J .

arcth(-

The image in the phase space is given by T = 2(i)2 - Zy2; x (C2X, do = + 22x2 + µl ; U=

-f

Jl

o

z = y,

J(x) = 2c2x3 + 2cAx (see Figure 28).

REMARK A. V. Brailov suggested the following interpretation of some forms of the type F, in the case of the symmetric space Sn-1 in V. Let us consider the vector b e K c H and the parallel translation of the plane V along vector b. We obtain the plane Vb = 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 Fc (at least, in the real case on Sn-1).

U00

u

'\ <0

A>O

A> O

u

N Fig. 28.

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

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 method 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 + )a), A e R, 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 R '(x, y) expressed as a function of f(x,y) = ex. Then fa(x, y) = e"+2a = ea1ex,

i.e. fa(x,y) is expressed functionally by the initial function f(x, y). Taking a polynomial in general position as f, we obtain (translating fa(x) with a fixed) N new functions (N = deg f is the degree of the polynomial f): '(X) {{

f (X

=

N

+ Aa) _

pi(x, a)A'

i-0 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) = axe + bxy + cy2. Let us consider the

translation of the function f : 136

INTEGRABLE SYSTEMS ON LIE ALGEBRA

(f)(x) =

d

137

f(x + via). A=O

Then, f and (f) are functionally dependent if and only if

0=b2-4ac=0. The proof follows from the explicit expression of the Jacobi matrix J

of the functions f = axe + bxy + cy2; (f) = 2aAx + bBx + bAy + 2cBy, the translation vector being of the form a = (A, B) 0 0. 2ax + by

J _ - (2aA

bx + 2cy

+ bB bA + 2cB

The determinant J is equal to zero if and only if (4ac - b2)B = 0 and (b2 - 4ac)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.

THEOREM 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 0(t) of the coadjoint representation (invariants of coadjoint representation of the Lie group M- which corresponds to the Lie algebra K), a e K* being a fixed covector, A, u e 68 being arbitrary fixed numbers. We claim that the functions

FA(t) = F(t + Au) and 6,(t) = G(t + pa) are in involution with respect to the standard symplectic Kirillov structure on all the orbits 0(t) (for any A, It e R).

Proof Since s grad, F = adjF (t), we need to prove that
0,

<x, y> means the value of the linear functional x on the vector y. Indeed,

{F, G} = -w(s grad F, s grad G) = - o (adap(t), ad jc (t))

_

Let A96 j and t=a(t+via) + fl(t+pa), fl =.1/(,1-µ),a=u/(µ- A).

138

A. T. FOMENKO AND V. V. TROFIMOV

Then
a<(t + .ia), [dF(t + 2a), dG,,(t)]> + 14<(t + µa), [dF. (t), dG(t + pa)] > = a(adaF(r +AQ)(t + .la), dc,(t)>

- (t + µa), di 2(t)> = 0

because the functions F and G are constant on the orbits Ad*. if A = µ, 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 [74] (S. V. Manakov) for the case Lie algebra so(n) and was then developed by A. T. Fomenko and A. S. Mishchenko in [89] 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 + .la) I A E C, f E 1(G)} by SI. The equality dim SI = ?(dim G + ind G) holds if, and only if, the element a + Ax is regular for any complex number A E C.

COROLLARY The translation of the invariants of the coadjoint representation is the complete commutative set if, and only if, the inequality dim M511, > 1 holds, where M,;ng 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 all 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)

dimW
INTEGRABLE SYSTEMS ON LIE ALGEBRA

139

and a set of linear functionals c' (f) on G, i.e.

cf: f(Ada x) = c*(f) EG*:

d c*(f)() =

dt

dcf( ),

cf(Exp t=o

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

Under all the above assumptions, we claim that (s grad f)(x) = c;(f )fi (x) + ... + c*(f)f,(x) PROPOSITION 11.1

Proof Let us differentiate the equation (with respect to t for t = 0) f(Ad'exP« x) = c1(Exp tc)fi(x) +

+ c'(Exp )f(x): cI ()fi (x) + ... + cs*()f,(x)

-

* f(Ade"Pt4 x) = f.s

dt

=o

d

\dt

t=o

AdExPt4 x

x) = (ad* x)(f*.x) = x([ , f*,x])

=

x([f*,x, cj)


)

In the above expression, f* x c (G*)* but G** = G. Therefore f* x E G.

Thus we have obtained adf, x(x) = -c' fI (x) -

- c* f,(x). Let us

check that the definition of the vector field s grad f will be satisfied if we assumes grad f = c fI + + c* f,. Since the symplectic structure w is non-degenerate, this will complete the proof. We have

w(c' f(x), a(x, )) = w(-a(x, f*,x), a(x, cf))

_ <x, - [f*.x, fl> = <x, R, f*A = a(x, )(f*.x) = -

Let X. = c xk 8/ax; be differential operators in the space c'(G*), x, being coordinates in the basis e' c- G* adjoint to the basis e; E G, c be the structure tensor G in the basis e;. Then, PROPOSITION 11.2

a) if W is an invariant subspace in A(G*), then for any f c W, X; rr E W. Let f q Wand fI , f, be the basis in W. Then X. f = c{ f ; b) c*(f)_ -c; e'.

Proof' From the equality f(AdeXPt{ x) = c'(Exp

(x)

A. T. FOMENKO AND V. V. TROFIMOV

140

it follows that d dt e=0

(x),

f(AdExp:4 x) =

but dt f(Ad*

,

x) = df (ad* x) = (ad* x)(df

= <x, [ , dfx]> = -

-[(Xjf)e`]O. Therefore (Xi f )e' = - f (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 dt

f (AdEXPrc X) E W.

t=o

The validity of claim (b) follows from the equation

(-f c*) = [X;(f )]e` = c; f e` LEMMA 11.2 Let f, g be smooth functions on G*. Then, If, g} - 0 on all orbits Ad* if and only if c;jxk of/8x; 7g/8xj = 0; e; being a basis in G,

G being the structure tensor for G in the basis e,, e' being the adjoint basis in G*, x; being the coordinates in the basis e'.

Proof We have the expression s grad fx = add f(x) for the skew gradient. Therefore, If, g}x = co(s grad f , s grad gx) =

= = [ad* (x)](dfx) _ <x, [dgx, dfx])

ag of = ax; axe dgx

09

= ax.

(x)e',

k

ag of ax; axi

f

dfx = axj O

(x)et.

THEOREM 11.3 (Trofimov, V. V.) Let W be a finite-dimensional subspace of A(G*) invariant under the coadjoint representation of the

group 6 which corresponds to the Lie algebra G. Let fl, ... , f, be a

141

INTEGRABLE SYSTEMS ON LIE ALGEBRA

, h, E W such that

basis in W. Let us define in W the functions hl, . 1 < i, j < p,

cj*(dhi.s) = 0,

1 < k < s,

cj* = c*(h with respect to the basis fl, ... , f,. We assert that (a) {hi, h;} = 0, 1 < i, j < p on all orbits of the coadjoint representation of the Lie group 6; (b) the translates of h; are in involution on all orbits of

the coadjoint representation Ad*(6): {hi(x + .a), h;(x + µa)} _- 0,

1
(1)

Let us denote x + Aa = y. Then, from the left-hand side of (1) A = ck,;Yk

ah, ayi

ahz (Y)

ay;

(Y + va) -

k

Acakah,

ayi

(y)

ahz (Y + va), ay;

v = u - A. From the equality ah c (Yk + vak) ayfl (y + va) = ca*if,(Y + va)

(cQ*i is the i-th coordinate of the covector c''l*, 1 < y < p, 1 follows that ah

f

ah

1

jakay,(y+va)=V cT*if,(Y+va)-c Ykoyo(Y+va) Therefore, ah,

ahz A=ci;yk(Y) AA (Y+va) ayi ay;

C

[ca*iff(Y + va) - c'.;, A V

1+

I) c,; Yk ahz V

1+

_

aho

-ayi(Y) ay;

'1 cQ*iha(Y) ah v

ayi

ah' (y

+ va)

ayj

0h, &_i

(Y)

ah,

(Y + va) - - ce*I f'(Y + va) - (Y) v

ay,

a

ahz

v

aY,

(Y + va) - cQ*ify(Y + va)

(Y)

-(1 + )c(dh(Y + va))h6(Y) - v ca*(dhs(Y))fy(Y + va) =_ 0.

s) it

142

A. T. FOMENKO AND V. V. TROFIMOV

If we consider expression (1) as a limit, the proof follows for

V=µ-A=0.

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 hl , ... , hp e W, an S-representation of the group (5 ((fi being associated

with G) provided that the following two conditions are satisfied:

a) W is invariant with respect to the coadjoint representation Ad*((5);

b) - 0 on G*. REMARK 1 Evidently, the condition - 0 does not depend on the choice of the basis fl, ... , f, in the space W

REMARK 2 We have proved that if (h1,. .. , h,,; W) is an Srepresentation of the group 6, then h;(x + Aa), hj(x + µa) are in involution on all orbits Ad*(0) with respect to the Kirillov form. REMARK 3 A special case of Theorem 11.3 is the method of translation

of semi-invariants (see [10]). Note that in this case we can omit the condition 0. This is an important fact which follows from representation theory (see [127], [89], [12], [26], [27], [116]).

We can combine the operation of argument translation with the operation of the restriction of functions to subspaces of G*. We shall illustrate this for the case of invariants (the results were obtained by A. V. Brailov); however, it can be generalized for any representation in a trivial way. Let V be a vector subspace, a be an involutive automorphism of V, Vo be the eigensubspace of a which corresponds to the eigenvalue 1, V, be

the eigensubspace which corresponds to the eigenvalue (- 1). Then V = Vo Q V, is the decomposition of V with respect to the involutive automorphism u. Let a*: V* -+ V* be the automorphism dual to a. Evidently, a* is an involutive automorphism. Let V* = (V*)o Q (V*),

be the corresponding decomposition of

V*.

Restriction of the

functionals x e V* to the subspace Vo c V gives a natural isomorphism (V*)o -+ Vp , V* being dual to V0. Similarly, the restriction of the

functionals x e V* to V, c V allows the identification of (V*), and V* (the latter being the space dual to V,).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

143

Let G be a Lie algebra, a be an involutive automorphism of G, G = Go p G1 be the expansion of G with respect to a, G* = G* p G; be the expansion of G* with respect to a*. Since a is an automorphism of G, [G;, G;] c G, +j; i, j = 0, 1 being taken modulo 2. Let f be any smooth function on G*, we denote the restriction of the function by f LEMMA 11.3 Let a be an involutive automorphism of the Lie algebra G, G = Gap G1 and G* = Go p Gi be the corresponding

decomposition of G and G*, a E G* 1, A, p e R; f, g be invariants of G. are in Then the restrictions of the translates of the invariants fza and

involution on all orbits of the coadjoint representation of G* (the invariants are meant to be invariants of the coadjoint representation). Proof We have dja(x) = Z(dfa(x) + dJ a(x)), x e G** for the translate of

any smooth function f by the element a e G*, J(x) = f (a*(x)). If f eI(G), then feI(G) (I(G) being the invariants of the coadjoint representation of G). Therefore

4<x, [dl.(x) + dl-1.a(x),dgµa(x) + d9-,.(x)]>

=

a<x,

[dfxa(x), dgpa(x)]> +

a<x, [dfxa(x), d9-µa(x)]>

+ a<x,

a<x, [df-xa(x),dg-,,,.(x)1>

{fxa, gµa}(x) +( 4{f. ,, 4,1W W) + 4{J-ia,gp:}(x) + 4{ f_xa,g-pa}(X) = 0. The above expression follows from Theorem 11.1. Since f, g, 7,4 are invariants of the coadjoint representation Ad*((5).

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 e1, ... , e" be a basis of the Lie algebra G; e', ... , e" be the conjugate basis of G*; X "-- . , x" and x,,. . . , X. 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*.

Let f and g be functions on H* which are in involution on all orbits of Ad*(,). Let us consider the extension of these functions LEMMA 12.1

A. T. FOMENKO AND V. V. TROFIMOV

144

to G*. They remain in involution on all orbits of the coadjoint representation Ad* of the Lie group 6 which corresponds to the Lie algebra G.

Proof Let e1, ... , e, be a basis for H. Let us extend it to a basis for

el,...,e e,+l,.... e in

G.

Let i=l,...,s; a=s+1,...,n;

a = 1, ... , n. We shall use these notations throughout this section. In this case, the lifting of functions from H* to G* does not depend on the coordinates xa. Let us check the involutivity using Lemma 11.2. We have of ag CayX,

__ of ag

aXa aXb -

Cs(tXe

ax" OXp

+ C2LXC

= C1j X,

of ag + C12Xc

-of ag

Ox" ax;

ax1 axe

+ C xc

'

of ag tX1 aXj

ax; ag

= CXk of ag ax; ax,

= 0.

Considering ag/ax, = 0 and [e;, e;] E H, we obtain c j = 0. Finally, we obtain zero because we have the condition that { f, g} _ C Xk Of/ax; ag/ax; - 0 on all the orbits of Ad*(.5). LEMMA 12.2 Let G D H be a chain of subalgebras. If f is an invariant of the coadjoint representation Ad*(6), and g is the lift of a function on

H*, then f and g are in involution on all orbits of the coadjoint representation of the group (5 which corresponds to G. The proof follows from the equality Of ag k

' Xk aX a

X..

Of

k

09

- (c ij aza ) Xk

= 0.

If H (-_ G', G' being the derived algebra of the Lie algebra, f being a semi-invariant of Ad*(f), ag/ax8 = 0, then { f, g} - 0 on all the orbits of the coadjoint representation Ad*((5). LEMMA 12.3

Proof The function f is a semi-invariant of Ad* if and only if Ck;Xk Of/ax; = A; f. Therefore

INTEGRABLE SYSTEMS ON LIE ALGEBRA

of ag _ (° bc

CX°

aXb O

x-l

of 1 ag _

J aXh OXc

145

ag (Xf) ax,

a9 f=(A1 but ,; = 0, since the character on the derived algebra equals zero. H, such Thus, for each chain of subalgebras G H1 H2 we can DHs_1DH, GD G'=) H1=) H1:DH2DH2::) that construct functions in involution on G* using a large number of the functions in involution on HS . The latter functions may be obtained by

e.g. translation of invariants, with the subsequent addition of semiinvariants of the coadjoint representation of the group .51 which corresponds to the Lie algebra H;. We can add not only semi-invariants but also functions from some representations.

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 c V defines covectors c* E G* when we choose a basis in V. If the extension of a function g 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 t . LEMMA 12.4

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 Bore] subalgebras in the semi-simple Lie algebras (see [126], [127]).

The simplest example of the construction described above can be obtained if Fl, is a maximal Abelian subalgebra 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 H1 H2 such that Hi H2, H2 = 0, [H1,H2] = 0, then any function G G' H1

f E A(H?) and elements H2 (regarded as functions on Hz) are in involution on all orbits of AdT(G*).

146

A. T. FOMENKO AND V. V. TROFIMOV

Proof As follows from the condition [H1,H2] = 0, the operators Xd do not involve derivatives with respect to the coordinates conjugate to the coordinates in H1 and the function f does not depend on the rest of the coordinates. Therefore,

of c X. of ax -° =c,,xa - 5, aXb

aXb axe

of x,-=Xdf =0. h

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(' i), 13 being the Lie group which corresponds to the Lie algebra S, then the functions .I and j are in

involution on all orbits of Adm; B being the Lie group which corresponds to the Lie algebra V; 7 = f - it, g = g o 7r, n: V* -+ S* being the restriction mapping. If f is an invariant of the coadjoint

representation of the group V and g is the lifting of g e A(S*), then 0 on all the orbits of Ad**. 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 !B. 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 G1 c G2 c

c: G; dim G,/G; -I = 1. Invariants of the representation Ad* of the ideals G; (i = 1,... , dim G) were lifted to G*

by means of the natural projection G* -+ G* 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 G° = {X e G: (ad X)a* = 0} 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 e 1(G) be the invariant of G; g be a smooth function on (Ga)*. Then the function fo(x) = f (x + a) and g

(regarded as a function on G*) are in involution on all orbits of the

INTEGRABLE SYSTEMS ON LIE ALGEBRA

coadjoint representation of the group (

(t

147

corresponds to the Lie

algebra G) on the space G*.

Proof We have { fa, g}(x) = <x, [dfa(x), dg(x)] >

= -

= - = 0 because dg(x) e G*. The lemma is proved. LEMMA 12.7

Let a- be an involutive automorphism of the Lie algebra

G, G = Go Q G1 and G* = Go Q Gi being the corresponding decompositions of G and G* (see Section 11); a e Gi, f e I(G), g be a smooth function on (Go)* which is the space dual to the annihilator of a in G. Then, the function (the restriction to G* of the translate of the invariant f by the element a) and the function g (regarded as a function on G**) are in involution on G*.

Proof We have { fa,g}(x) = <x, [d3(x),dg(x)]>

=

z<x, [dfa(x) + d]` a(x), dg(x)] >

= -i
d]` a(x), dg(x)] >

= J<(ad dg(x))a, dfa(x) - df_a(x)> = 0, because dg(x) E G. 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 G' over the field k. We denote the ring of polynomials in the variables xl , ... , x over the

Let fI,...,f,Ek[x1,...,x,,] be such that field k by f,(0) _ = f,(0) = 0. Let 06 be the quotient-ring IK = k[x1, ... , xa] (f1., , f ); (fi, f,) being the ideal in k [xl , . . . , xa] generated by the be the natural VS polynomials f1,... , f,; let n: k[xl, ... ,

A. T. FOMENKO AND V. V. TROFIMOV

148

projection. We denote the image of x( under the mapping n by e,. Then, 6G is multiplicatively generated by the elements el,... , e" e K over the field k. Let us extend the ring of scalars of the Lie algebra G to 1K, i.e. consider the Lie algebra G'K = G'(& 6S over the ring K. The field k is contained in the ring K. Therefore, we can consider the Lie algebra GN over the field k. We denote this Lie algebra by Of (G), f = (fl, ... , f,) being a polynomial mapping f : R" -+ R' such that f (0) = 0. We claim that Of(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 Y=

e" (at.....2,)

'

'

en"xal.....a" ;

xal.....a" E G.

Let us set g(y) = x0,....0. 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 semi-

direct 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 ® 1K generated by the subspaces e1 G,... , E. G. Then Of (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. on G. Let us consider an arbitrary polynomial mapping K = 118[x]/(x2) in the construction described above. Then, we obtain the Lie algebra O (G) for which the algorithm for calculating the invariants REMARK

of the coadjoint representation (given the invariants of the coadjoint representation of the Lie algebra G) is given in [132].

Let us consider an arbitrary polynomial representation

REMARK

f:

ll

-' l omitting the condition f (O) = 0. Then, we obtain a Lie

algebra Of (G) which contains G as a subalgebra. Let us consider the Lie algebras Of (G) where

f = (zlm,..., x.,

We denote Of (G) by 52,,,E

)E(k[xl,...,zn])n

m"(G).

be a basis of the Lie algebra G. Then the vectors e "e;; 1 <, j - r; 0 < a; S mi; 1 5 i < n forma basis of the Lie

Let e1,

.

. .

el, algebra S2",t

, e,.

","(G). Let us denote the space dual to the space G by G*. If

e e G (i = 1, ... , r) is a basis of the space G, then we denote the dual

149

INTEGRABLE SYSTEMS ON LIE ALGEBRA

basis in G* by ei; it is defined by the relationship <e`, eJ> = 6'; < f, x>

being the value of f e G* on x e G. Let us denote coordinates in m,,...,m,lG)* in the basis dual to the basis ei' 1 < j < r, 0 <, a; < m,, 1 < i < n by x(al, ... , an)j. We denote coordinates in G* in the basis dual to e, (1 < i <, r) by xi (i = 1, ... , r). n,(G) new variables Let us introduce in Z, =

E

S" ... en"x(Y1, .

.

. ,

/Q

I'n)i

(1)

J = l .....n

Description of the algorithm (91). Let F(xl, ... , x,) be an analytic (e.g.

polynomial) function on the space G*. Let us consider the function F(zl, ... , z,) on the space S2ml m(G)* with the values in the ring K. Expanding F(xl,... , x,) in a Taylor series and substituting zi from (1), we obtain a finite sum because the higher powers of e, are equal to zero. Thus, F(zl, ... , z,) is a well-defined function on 52,,,E

in K. We can express this function as , ... F(Zl,... , Zr) 0<_1,' j

el

mp(G)* with values

,

en Fal.... x,(x(I'1, ... I'n)i)

We shall call the functions F, ,: 52,, n,(G)* - k the homogeneous components of the function F(z1,... , z,). By definition, the algorithm

(21) transforms the function F on the space G* into the set of homogeneous components 21(F) = {F2,,...,,j of the function F. The algorithm (91) has the following properties (see [129]):

Let F,,... , FN be functionally independent invariants of the coadjoint representation of the simply-connected Lie group 6 which is associated with the Lie algebra G. Then functions of the set THEOREM 13.1

{91(F1)} u u {91(FN)} are invariants of the coadjoint representation of the simply-connected Lie group SZmi m(6) associated with the Lie

algebra 52,,,1 m,(G); the invariants {91(F1)} u ... u {21(FN)} being functionally independent on the space S2n,l m^(G)*.

Let the functions F1(x),... , FN(x) defined on G* be in involution with respect to the Kirillov form on all orbits of the coadjoint THEOREM 13.2

representation of the Lie group 6 which is associated with the Lie algebra G. Then, after the algorithm (21) is applied, all functions {21(F1)} u

. u {21(FN)} are in involution with respect to the Kirillov

form on all orbits of the coadjoint representation of the Lie group S2,nl

n,(6) which is associated with the Lie algebra 52m1

,

(G). In

150

A. T. FOMENKO AND V. V. TROFIMOV

addition, if F1,... , FN are functionally independent on G*, then all functions of the set UN 191(F,) are functionally independent on We shall prove this theorem in the following section.

Let F,,... , F,v be a complete set of functionally independent invariants of the coadjoint representation of the Lie DEFINITION 13.1

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 6 (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 Q....... dimk I ind(G). In this case, the algorithm (91) transforms the complete set of the invariants of Ad*((5) into the complete set of the invariants of Ad*(5)

,

REMARK The algorithm from [132] is a special case of the algorithm (21). The following statement is a corollary of Theorems 13.2, 13.3 and [10], [89], [126], [127]. THEOREM 13.4

Let the invariants of the Lie algebra G separate the

orbits. Then, the algorithm (91) transforms a complete involutive set of functions on G* into a complete involutive set of functions on the space m(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 of functions on the space dual to the Lie algebra (1ii(2).....ia21(2)' ... a

may be constructed explicitly by means of the algorithm (91). Let us give explicit expressions for functions F from algorithm 21 and

some rings l = EXAMPLE 1

In [132] invariants of a Lie algebra (which we denote by

151

INTEGRABLE SYSTEMS ON LIE ALGEBRA

S21(G)) are constructed. This Lie algebra corresponds, evidently, to the ring l( = 18[x]/(x2). Let us consider the ring 18[x]/(x3). We have the coordinates x(0),,x(1)1,x(2)1 and the variables z, = x(0)1£2 + x(l)1E + x(2)1 (E3 = 0) in the space 02(G)*. Using the Taylor formula, we obtain

F(zl,... , z,) = Fo(z) + F1(z)E + F2(z)E2, Fo(z) = F(x(2)1) ;

F1(z)= E

aF(x(2)i)x(1).; ax(2)i

j=1

F2(z) =

a2F(x(2)r)

1

2 ,,q= 1 aX(2), ax(2)q

x(1) x(1) +I 9 P

aF(x(2)')

k.1

ax(2)k

x(0)k

Let us consider the Lie algebra which corresponds to the ring K = 18[x1,x2]/(xl,xz). Then K 21,1(G) = G + e1G + E2G + E1E2G. EXAMPLE 2

We denote the coordinates in i21,1(G)* by x(0,0),, x(1,0);, x(0, 1),, x(1, 1), in accordance with the decomposition of SZ1,1(G) as a direct sum

of linear subspaces. In this case, the variables are z, = E1E2x(0,0), + E1x(0, 1), + £2x(1,0), + x(1, 1),. Using the Taylor formula, we obtain F(z,, ... , z,) = F0,0 + E1F1,0 + s2F0,1 + E1E2F1,1, Fo(z) = F(x(1, 1),);

Fl

aF(x(1, .o(Z) =

k=1

1),)

ax(1, 1)k

x(0, 1)k;

aF(x(1, 1),)

Fo,1(z) = E

k=1

F1,1

02F(x(1,

'

(2) =

a( x1

P.9=1

1)k

1).)

X(110)k;

x(1,0)Px(0, 1)9

ax(1 1)P I ax(1, 1)q 8F(x(1, 1),)

+ k=1

ax( 1, 1)k

x(0 , 0)k

-

13.2. The proof of the general theorem

In what follows, we shall use the tensor notation, e.g. a`b, means summation with respect to i. Let c j be the structure tensor of the Lie algebra G in the basis e1, ... , e,; let ekj be the structure tensor of the

152

A. T. FOMENKO AND V. V. TROFIMOV

m^(G) in the basis c ' e'^e;. In the space c°°(G*) of smooth functions on the differential operators G* Xi = X(e;) = ck xk alaxj are defined. We shall call them operators of infinitesimal translation of the Lie algebra G in the basis e1, ... , e,. algebra SZm

The operators of infinitesimal translation of the Lie algebra S2m .....m^(G) in the basis el' e,'-e;, 0 < aj < mj, 1 j <, n, 1 < i < r (e; being the basis of G) are given by the formulas PROPOSITION 13.1

a

X(el' ... en-e,) _ 0
4X(a1 + fI, ... , an + Yn)k

p=1, .,n

p Fl1, ... , FNn)j ax((

The proof follows directly from the definition of the commutator in the Lie algebra °im,,....m^(G).

Let F be an invariant of the coadjoint representation of the Lie group 6 which corresponds to the Lie algebra PROPOSITION 13.2

G. Then, any function f e 21(F) is an invariant of the coadjoint representation of the group S2m ,...,m^(6) associated with 52,,,,

m^(G).

Proof The function f is an invariant of the coadjoint representation if and only if X(el' .e'ej)f = 0 for all ai,j (see 2.3). Since the operators of infinitesimal translation are linear, to check if all f e 21(F) are invariant it is enough to check the condition X(el, ... e''ej)F(zj,

ej), z,) = 0

in the ring 68[x1, ... , xn]/(x,'+', ... e^+1). We have X(el' ... k

ctjx(a1 + NI, ... , an + Yn)k 0
aF(zl,...,zn) ax(fi1,

.

p Yn)j

,

(2)

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

= em,

B, ... em^

B^ a azj

Using (3) we transform (2) X(el, ... en^ei)F(z)

(3)

INTEGRABLE SYSTEMS ON LIE ALGEBRA

153

CkijX(al + N1, ... , an + Nn)kEl 1 -Pi y04a +# <mo 1Sp1n

-Ck

OF

P.

azi

...En^

yazj OSa +f Sm

OF(z)

t3.

I Sp

Note that since /3; + a, mi, m; - II; > a,. Therefore it is possible to i.e. "divide" by el' X(E1' ... En.ei)F(z) = E1, ... En ,c1 OF azi // x(al + N1, ... , an + fn)kEl

X

-a, ... En -ft.

0-
1 SpZn

Denoting a; + i, by y, we obtain X(e' ... En"ei)F(z) = E1a, ...

Ea^Cki n

i

OF iaZj Y_

X(Y1, ... ' Yn)kEl,

Y, ...

Em^-Y.

(4)

ao
Let us add the summands X(71....

En^-Y^ (y; < ai for at °-Y^. least one subscript i) to the sum Y- x(y1, ... , Yn)kET' Y` En These

summands do not affect the result since from the condition ),; < a, it follows that a; + (m; - y;) > a; + m, - a; = m;, i.e. after multiplication by ell E'^ each new summand gives zero (i.e. has eai with S; > mi which is Tern) X(01 ... En^et)F(z) = E1, ... En^C OF x(y1, ... , yn)kEl 1 -Y1 ... En^ aZj OSy9Sm, 1
= E1, ... En^zkCk OF j azi

the latter expression is zero since F is an invariant of Ad*(0) and therefore C J-Zk aF(z)/azi = 0 (see 2.3). Proposition 13.2 is proved. Let F1,. .. , FN, be functionally independent functions on the space G* which is dual to the Lie algebra G. Let 52,,,, m,(G) be an extension of G constructed by means of the ring PROPOSITION 13.3

,xn^+1)

Then all functions from the set

154 {21(F1)} m^(G)*.

A. T. FOMENKO AND V. V. TROFIMOV

u {21(FN)} are functionally independent on the space

S2m

Proof Let us consider the function F(zl , ... , z,) on the space the z;'s being given by (1). Let us express z; as follows:

X 2'+2 +a #0

Eli ... En"x(Fil,

... , Nn)i + x(m1,....

n)i

1-
Let us apply the Taylor formula to F(z; + x(ml,... , mn);). Then, evidently, we obtain

F(z1,...,z,) = F(x(m1,.,mn)1,...,x(m1,...,mn)r) p

N

X

= (i,....1, p. ax ml, ... , mni, ... , aX(ml, ... , mn)i, p1 ... En-x(fi1,Q... , Yn)t,) X .. . C(Eli

Lx

(EEi'...En'x(l'...,fin)

(5)

The summation symbol means summation over all a;, /3, > 0 such that + a # 0. Let us separate the variables a; + (t; = mi, 1 < i < n, ai + into groups in such a way that within each group f + + fn = const. Let us arrange the groups according to increasing order of N1 + + fn 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(al , . . , a triangular matrix with diagonal elements .

A.3 = 8F;(x (ml , ... , mn) p)/Ox(m l , ... , inn),: A::

J=

0 A.

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 Fl , ... , F, be a complete set of functionally

P r o o f o f Theorem 13.3

independent invariants of the coadjoint representation of the simplyconnected Lie group 0 which corresponds to the Lie algebra G. Then,

dim OG = n - s; 0., being an orbit of maximal dimension of the coadjoint representation Ad* i of the Lie group 6. The algorithm (21) gives s dim, III functionally independent invariants of the Lie algebra C1

.,.....

m,(G): jW(Fl)} u ... u {%(Fjj.

This may not be a complete set of invariants, i.e. the number of the .,(G) is not less than s dim K. Therefore, invariants dim 0...... n.(G) <, dim G dim, K - s dim, K = dim, II((dim G - s) = dim, l( dim OG,

(6)

because the invariants G separate the orbits. In the above expression,

the orbit of maximal dimension of the coadjoint

is

representation for the extension SZm me(G). On the other hand, from m(G) we the explicit form of the operation of commutation in obtain

x(al,... , an),El' ... En-e')

dim Om

Bt,

dim, 1( rk

= rk 0

Btu

= dim, K dim OG(x(m l , ... ,me'),

OG(x) being the orbit corresponding to the point x e G*; B. _ c;`j x(m, , ... , m ),. Let us choose y = x(ml, ... , m )t e` such that the dimension of O,(y) is maximal. Then, dim O

,

...,m,(G) >, dim Om,,...,m,(G)(z) = dim, IK dim OG

because of the element y e

Z-

y

a(al, ..., an),E1 ... En-et

OSaDS1m 1

a(al, ... 4+

+aI#O

, an)tEal

... en"-e' + y.

156

A. T. FOMENKO AND V. V. TROFIMOV

Comparing this with (6) we obtain dim Oml from which our assertion follows.

m^(G) = dim,, U( dim O"

To prove Theorem 13.2 we need some preliminary considerations. Let F, H be two functions with their values in l 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 of the coadjoint representation Ad*((5) of the Lie group 6 which corresponds to the Lie algebra G if and only if c jx,, aF/ax; aG/ax; = 0 (see Lemma 11.2). Let us consider on the functions F, H with values in the ring space 12 (G)*, m = (ml, ... ,

0(_

Let us define a new bracket

{F, H} ® on such functions F,H. The function {F, H} ® is defined on a,,,(G)*, m = (ml,. . , and has the values in the IIB-algebra .

K ®RK = R[x...... ORR[xl,..

,xa]/(xi`+1,...,xn^+1

By definition, we set

DEFINITION 13.2

{F, H}

®_

c

w

k

aF(z) OH(z) a II( ® II(, aw; aw;

c , being the structure tensor of Slm(G), m = (m1, ... , Mn) in the basis running through those elements of 41 ... E'"e; and w = ( x ( 1 , . .. , Slm(G)*, m = (m1, ... , Mn) comprising the basis dual to Eil ... E,a,"e;; z; = z;(w;) being the function of the variables w; defined by equation (1). We assume aF(w) 0a);

E II(®{ 1} c K (E) K

and

OH(w) aw;

e{1}®a(-1(®II(.

Let us denote E; O 1 by C; E K ® K; 1 O E; by , e K ® K. Note that the bracket {F,H}® is linear with respect to each argument but is not skew-symmetric. Let us consider in the ring K (& K the element

A=` PROPOSITION 13.4

99-4°m. 1

11

52

2)...(

U

-

The following equality holds in the ring K ® K

157

INTEGRABLE SYSTEMS ON LIE ALGEBRA

AI

E

C I ...bn"x(N11.... Nn)k

a;+Ri=m1

1 i4n

Sil ... n"x(Fil, ... ,

=A

Nn)k)

1

i.e. A(1 ®zk) = A(z® ® 1) for any k.

Proof Let us consider the sum x(al+#11...Ian+/'n)kbml-a1

Y_

b

062,+po'm,

S

F

yS

(7)

1_
Let us denote aJ + f; by y, and mj - a; by ti. Then, fl, = yj + t; - mj and the sum (7) becomes

x(al +

Y1, ... , an + Yn)kb1 1

-a, ... Yn

-P1 ... Sn S

SS

a;.91

x(Y1,

Yn)krrrl ..

r

xnm.-7.

7;.r; K

x(Y1,...,Yn)kK11-Y1...

C'.

S

bb

Ynr1

m1-Y1

m"-Y.

(yi Denoting a; Q+ f; by yj and YYmj - f, by t,, we obtain for formula (7) n.-0. x(a1

+hi,...,an+Nn)kS11-a1...C

ai,p j

Yi.rl x(Y1,...>yn)kbl,-Y1...cm. Y.A

\\ Y.

l

Comparing this with the previous expression for the sum (7), we obtain the proof of Proposition 13.4.

Let F be a real-valued analytic function of the variables X1 , ... , x,. Let us assume COROLLARY

vi=2(xi®1+ 1®xi)El ®aS.

158

A. T. FOMENKO AND V. V. TROFIMOV

Then

AF(v...... v,)=AF(x1 ®1,...,x,(D 1)=AF(1®x,,...,1(& x,). Proof From Proposition 13.4 we obtain that A(2(x; (& 1 + 1 (& x1))" = A(xi (9 1)p = A(1(9 xi)p.

(8)

Expanding F as a power series, we obtain the proof from (8). Let F, H be analytic functions defined on the space G*. Let us consider F(z,, ... , z,), H(z1,... , z,) which are functions on S2m1 m(G)* with values in the ring K. Let IF, H}G be the Poisson bracket on the space G* arising from the Kirillov form. We state that PROPOSITION 13.5

{F, H}® = A{F, H} G(v, , ... , v,), A E 1K ® 1K being the element defined

above and v;=2cz,®1+1(9 z;). Proof From the definition of the commutator in the Lie algebra 12,,,

m(G) it follows that

F, H)

®_

aH

OF

c'j;x(al + fl, ... , an + Nn)k 0-M'+-<m,

aF ax(a1,..

001

1
OH

®ax(/I'1,...,fl((

);,

(OF/Ow;) ® (aH/aco;) means the product in the ring 1K (& 11< of the elements (aF/awt) ® 1, 10 (OH/6w;). Using equation (3) we obtain

{F,H}®_

aF

aH

`; az. ®az. x 0
x

+fl1,...,a.+Nn)kbi,-a, ...

S

So.-a.

1,-a1... c. -9.

(q)

Using Proposition 13.4 we obtain

x(a1+a"+Yn)kSj' 21...yn. 1
nn-dn

159

1NTEGRABLE SYSTEMS ON LIE ALGEBRA

yyal ..,

Z

A(

ai+fli-m, 1
=A(

(

,

/'n)k

aj+01=mr

.

(10)

I < i
In the light of (10) we can rewrite (9) as {F, H} ® =

A 2

ci,(zk ®1 + I ©zk)

aF(z ®1) aH(1 ®z) az

az

(we use the obvious relationship) OF(z)

azi

01=

aF(z (3 1) a(z ® 1)i

1®

3H(z)

3H(1 (9 z)

azi

all ® z),

Applying the corollary, we obtain

{F,H}®= Ac

aF(v) OH vk

avi

av; '

Vk = z(zk ® 1 + 1 (9 zk). Thus, Proposition 13.5 is proved.

Proof of Theorem 13.2 In Proposition 13.3 we proved that the functions in the set a = {2t(Fl)} u .. u {91(FN)} are functionally

independent. Let us prove that all functions in the set I are in involution. Let F, H E If, then Fa,,...,a"El` ... en",

F(zi) =

a

B,. .9A

Because the bracket {F, H} ® is linear, we obtain F, H } _

pa

J

{Fa,.....z"'FIRi,...,9.}S1

y

bn bl

z

Sn

From Proposition 13.5 we obtain {F, H} ® = A{F, H} G(2(zi ® 1 + 10 zj) = 0

since according to our condition {F, H} G = 0 and therefore all the brackets prove.

are equal to zero which was what we had to

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, x, } = 0 for all coordinate functions. The algorithm (91) transforms the coordinates in G* into the coordinates in Sam REMARK The algorithm (91) is also applicable to functions defined on some open set U c G*. Then the functions from 21(F) are defined on some open subset U c 52,,,, mq(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 (21) to the construction of S-representations

The algorithm (91) allows us to construct S-representations (see Definition 11.1) systematically from the simplest S-representations.

Let (h1, ... , hp; W) be an S-representation of the Lie group 6. Then, (91(h,),. . , 91(hp); 21(W)); 91(W) = {21(b), b e W} is an S-representation of the Lie group SZa(6) associated with the Lie algebra THEOREM 13.5

.

Qa(G).

If h1,. .. , hp are semi-invariants of the coadjoint representation Ad*(21), W = Q?=1 Rh;, then (hl,. . . , hp; W) is always an S-representation. This follows from [127], [89], [12], [26], [27], [116]. If we apply the algorithm (21) to this S-representation, then we REMARK

obtain the S-representation of the group SZa(6) which, in general, cannot be reduced to semi-invariants. A complete set of semi-invariants for some classes of Lie groups is described in [10], [126], [118], [132]. PROPOSITION 13.6

Let F e A(G*); then

X(E1' ...

En"ei)(F,...... A.) =

-at.....pFv

(X(es)F)p,

E 21(F) being the homogeneous component of the function F.

Proof From the definition of the commutator in 0.(G) it follows immediately that X(El, ... E2

)_

Q(

c x(CCl + N1, ... , an + AX l

a

3x(I'1, ... ,Yn)j

p£n

C k being the structure tensor of the Lie algebra Gin the basis e;. It is easy

to obtain from this equation that

INTEGRABLE SYSTEMS ON LIE ALGEBRA

X(E1, ... E,a,"ei)F(Z1, ...

161

, Zr) =Eli ... En (X (ei)F)(zj, ... , z,);

x(ei) being the operator in c°°(G*) which corresponds to the basis vector

ei E G and zi being substituted by its expression (1). Applying to the function F(z1,.. , z,) = C' Elt ... En"Fai.....a"(x(N1 ... a o am, 1sj
El-e) and using the equation

the operator X(Ei'

X(Ef, ... En"ei)F = El, ... En"(x(ei)F)

the proposition is clearly proved.

If W is an invariant subspace in A(G*) with respect to Ad* (0), then 91(W) = {21(b), b e W} is an invariant subspace with COROLLARY

respect to Ad*(Qa((5)). PROPOSITION 13.7

If W is a finite-dimensional invariant subspace in

A(G*), then 21(W) is a finite-dimensional invariant subspace in A(S2a(G)*).

Proof Let us prove that if fl, ...

is a basis in W, then {91(f1)} u u {9(f,)} is a basis in 21(W). This follows from the explicit form for the

homogeneous component of f

al

which is easy to obtain by

a"

expanding f (z1, ... , z,) in a Taylor series (see (5)). Let (h1, ... , hp; W) be an S-representation of the Lie group t and X(ei)hk = c;,(hk)fj (see [126]). Then since 91(W) is an invariant finitedimensional subspace,

X-( ... Ea"e h ,) rt 1

= CI.Q,..-..Q" h k.Y,.....Y"

( k,y,..... Y)i.al.....a" fjrQ 1.....Q" r

hk.y,,._.,y" e 2I(hk) is the homogeneous component hk E W. PROPOSITION 13.8

The following equality holds

5 being the Kronecker symbol.

Proof From Proposition 13.6 it follows that X(E11 ...E n"ei)(hk.Y,.....Y")

_ [(X(ei)(hk)]y,-a,.....Y"-a" J

(

c (hk)iJj.y,-a.....Y.-a".

( 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 91(W) is invariant. Let us check the condition (b) of Definition 11.1. We rewrite this condition using the coordinates


ax. '

c*i(h;) = c'*(h;)(e;) being the coordinates of a covector c*(h;) in the basis e; a G. Multiplying z

a

C*c l" "y:Qn(hj+ ,, ..fin) aXD

by Ej' ... e.2- " and adding, we find that it is enough to prove that the expression

c* 9,.Y9,(hi, ..J

ah(z)

is equal to zero (or, obviously, to the expression Oh(z)

azi

From Proposition 13.8 it follows that the last expression is CM,

S4'-a.c*(h;)i

ah(z) azi

which is obviously zero because, as is given, ck(h)i ah(z)/azi = 0. This completes the proof (see also [130]). 13A. Algebras with Poincare duality

Let A be a graded commutative algebra, A= AO p .

Q A,,,

A.A; c Ai+; (i +j < n). We assume that dim A. = 1. Let n be a linear functional on A which is identically zero on Ai if i < n - 1 and is not zero on A. Let fl(a, b) = St(ab) be a symmetric bilinear form on A. If fi 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 A0 = k. Let el = 1. Let us choose the arbitrary basis £2, ... , Eli in A1, then choose the arbitrary basis e;, +1, ... , e;, in A2, etc. up to degree n/2. In we choose a basis such that the matrix of the form fi is diagonal. In spaces Ai, i > n/2 we choose

the bases conjugate with respect to fi to the ones already chosen in A012) - ;. As a result we obtain el, ... , CN, a homogeneous basis of the algebra A self-conjugate with respect to fi; fl(e*i, ej) = 5 ,, * being a permutation. Let e1,.. . , e,,, be a basis in G*; x1, ... , x,,, 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 ®e, of the Lie algebra GA = G ® A (regarded as a Lie algebra over k) can be regarded

as linear coordinate functions on G. Let x = xi ®Ej be coordinates on

G. For a polynomial function P(x) on G* co

Y>

P(xl,... ,

Pi1,...,i,xi, ... Xik = P1x1

(12)

k =0 (i,,...,ik)

with 1 <,j < N let us define a polynomial function P(')(x) on GA: 00

PU)(X...... XN)

k=0 (1,.....it) U,....lk)

it(Ei, Si .

eJ;)X#It ... X*lt .

. .

.

/

= P1(Ei, EJ)x*',

(13)

... e;, on the basis vector e,. Let x e G*, Ann(x) = {g e G, (ada)*x = 0} be the (E;, ej ' ... ' E,,) being the projection of the element Ej,

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 c- G*}. Let F be a set of polynomial functions on G*. The set of polynomial functions on GA* = (G (9 A)* of the form PU), P E F, 1 j S N we denote by FA. (i) If F is an involutive set on G*, then FA is an involutive set on GA = (G (9 A)*. (ii) If F is a complete involutive set and the number of independent

THEOREM 13.6 (A. V. Brailov [19])

polynomial invariants of the Lie algebra G is equal to the index of the Lie algebra, then FA is a complete involutive set on GA* _ (G (9 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.

Let P1,. .. , P, be a set of independent polynomial invariants of the Lie algebra G with index r. Then, (i) ind(GA) = dim A ind(G); (ii) Plll), , Pr") is a complete set of independent THEOREM 13.7

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 *e; = e*;, *2 = 1. Let (X, Y) be the scalar product corresponding to the basis El,. .. , 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, bc) = (*b, ac), a, b, c e A. Let A e A ®A, A = L, (*E;) ® E;. In what follows, we omit the summation symbol for

repeated indices. The element A is an exact analog of diagonal cohomology class (see [69]). LEMMA 13.2 We have A(1 (& a) = A(a ® 1), a e A.

Proof The following equations hold A (1 ® a) = *E, ® Eja = (*ej, Eja)*E® ® *Ej = (*E,, Era)*E, ® ej

=Era®*Ej

Let P e k[xl, ... , x ]. We define a polynomial P E A[xl, ... , x,,,], P(x) = P,e,x*', 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('), Q(»} = (Ei (& E;, A{P, Q} (9 1).

Proof Since {PW, Q,;)} (W ) _ E, 0 E;,

a3x;) ®a x:

{xr,

x9})

q

it is enough to prove that OP(X) ®a(x) ax.,

axp

{x;, xp

A

P,

x) ®1.

(14)

The left-hand side we denote by {P,Q}®. The operation {P,Q}®

165

INTEGRABLE SYSTEMS ON LIE ALGEBRA

introduced by us is bilinear and if one argument is fixed, it becomes a derivation of the other one. This means {P1P2,Q}®= P1 ® 1{P2,Q}® + P2 ® 1{P1,Q}®

{P,Q1Q2}®= 1 ®Q1{P,Q2}®+ 1

®Q2{P,Q1}®.

(15) (16)

Other properties of the left-hand side are Q} ®1)

(17)

Q2}®1= 1®01(A-{P,Q2}®1) +

(18)

The property (17) follows from the fact that the operation P - P is multiplicative and from the properties of the standard bracket { f, 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) for linear

P and Q. {Xr,xq}®_ *Es ® *Ep{x;,xQ} _ (Ek, Es ' Ep)*Es ® *EpCrq Xt

_ (*Ek, Es Ep)*Es (9 *EpC'gXSk t *k _ (*Es, Ek Ep)*Es ®EPCrgXt t

*k

= EkEp ®*EpCrgXt

= A ' Ek ® 1 CrgX*k = A {xr, Xq} ®1 .

Let P1, .... Pr E k[x1, ... , XM] be polynomials which are independent at the point y e G*, y = (yl, ... , Then, the polynomials P11), , P(" are independent at any point x e G,*, such that the highest coordinates x' = y., 1 '< i '< M. LEMMA 13.4

Proof Let us suppose that the polynomials P1, ... , Pr are independent with respect to the first r coordinates,

det NM # 0' z

Let us consider the matrices M,q(x), 1 < s, q < N

166

A. T. FOMENKO AND V. V. TROFIMOV

(M.,(x))1j =

0P(g)(x) , ax*i

1 < i, j 5 r.

,

Ifs > q, then (Eq, a E,) = 0. Hence, if (Eq, 61) # 0 then ax*'/ax*' = 0. Therefore, when s > q the matrix M,q(x) consists of zeros. When s = q, we obtain aPjg)(x) 041,

= (Eq, EJ)PI

ax*' ax*8 *9

ax; *1 .. x;*1 ax*s x9,., t,r

t

1 = P!(y)

., -

xt*

a yi

Let us consider the matrix M,q;j(x) constructed with the matrices M,q(x).

This matrix is block-triangular, the diagonal being occupied by nondegenerate matrices aP,(y)/ay;. Therefore, it is non-degenerate and hence, the polynomials P(,) __P(" 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(0,x*'} = {P('),x, I = {E; ® e.,A {P,x,} ® 1} = 0. Hence the Pro`m's are invariants of the algebra GA = G ® A. If P1, ... , P,

are independent invariants, then from Lemma 13.4 it follows that rN copies of the invariants P1, , P;"I of the Lie algebra GA = G (9,,A are independent. Therefore, ind GA 3 rN. The converse inequality can be obtained as follows. Let L,q(x) be a square matrix (L,q(x));j = {x; , x q }(x). If s > q, then L. = 0. If s = q, then (L,q(x))ij = xk c . If y e G* (y, = x', t = 1, ... , m) is a regular point, then the rank of the matrix b. = xkc is m - ind G. Therefore,

ind GA < mN - rank L,q(x) = mN - (m - ind G)N = rN. Proof of Theorem 13.6

(i) From Lemma 13.3 it follows that FA is involutive; (ii) Let P1,. .. , Pb E F be independent and form a complete commutative set. In this case, k = Z(m + r), r = ind G. From Lemma 13.4 it follows that kN copies of the polynomials P11j, ... , P') (which

are in involution) are independent. For this set to be complete it is enough that kN = Z(dim GA - ind GA) (this follows from Theorem 13.7: Z(dim GA + ind GA) = z(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) = q;(x;)Z, q, j4 0 (1 < i <, m). Then QN(x) = q;xx*' is also nondegenerate quadratic form. It is easy to see that in the real case this form is always indefinite. REMARK The results of 13.1 in the polynomial case can be obtained

from 13.4 if we take l [xl, ... , xn]/(xj'+x ^+') as the algebra A. For the non-commutative case of Theorem 13.6 see [200].

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 of the algebra H in the space V.

If a basis x1, ... , x is fixed in V, this means that there is an (n x n)matrix W of 1-forms on H. Let us denote the value of this matrix on a vector h c- H by W. (thus Wh is the matrix of the transformation adh: V -+ V in the basis x1, ... , xn). We denote the n-tuple (xl,... , xn) which consists of the vectors in the basis for the space Vby X. Then

[h,X] = XWh,

(1)

[h, X] stands for the line ([h, xl], ... , [h, xn]), h e H. The elements wj of the matrix Ware 1-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 adh w;j = [h, w;j]. Then,

[h, W] = [W, W ]

Here on the right-hand side is the matrix commutator. Indeed, for any vector h' EH: B(adh wijh') = B(w,;, [h', h]) giving (adh W)h.

= W

Wh., W ] ,

(2)

168

A. T. FOMENKO AND V. V. TROFIMOV

B(X, Y) being the Cartan-Killing form. This is true for any h'. Therefore

adh W=[h,W]=[W,WW]. From (1), (2) it follows that [h, X W'] = [X K' ] . Let E be an r-form invariant under the action of the algebra H on ntuples (h: Y -+ YWh). Such forms exist because H is a semi-simple algebra. Then, E(X We1, ... , X We') is an H-invariant, i.e. [h, E(X We,, ... , X We')] = 0 if we consider E(X Wel, ... , X We,) 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.

Functions of the form E(X We,,... , X We,) on G* are in involution with the linear functions wi;, 1 < i, j n. PROPOSITION 14.1

It is possible that the H-module G has several submodules V', V"'. . , V("I 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(X We1, X'We2, ... , X("`I involution with all linear functions w;j; 1 < i, j <, n.

are in

Let us call functions of the form E(X.Wpi, X'We2,... , X(') W',) canonical H-invariants. DEFINITION 14.1

14.2. Involutivity of similar functions

According to the classical Cayley-Hamilton theorem,

"-I W. = Y_ a,(W)W`.

(3)

i=o

In the form of the canonical H-invariants E(XWe,, X'We2, ... , X(m)We,) the powers of the matrix W are given DEFINITION 14.2

explicitly. We shall call two canonical H-invariants similar if one of them can be obtained from the other by the replacement of the matrix W" by W'; the powers i being such that the function a1(W) in the expansion (3) is not identically zero.

Let E be the matrix of a bilinear form E. Then, E(X'W e1, X W e2) = X'We1E(X We2) T = (- 1)e2X'W We, +e2EX T

INTEGRABLE SYSTEMS ON LIE ALGEBRA

169

because WE + EWT = 0. 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 G) Vi m p VR is an expansion of the H-modules Gin irreducible submoduli. It is required that

a) H be a simple Lie subalgebra in G of the classical type A, B, C, D,; b) All Vi 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,W'EXT is canonical H-invariant. Then the Poisson bracket of the similar functions {Fi, F.} (X can be identical with X') is zero. The proof of this theorem will be given below. Let {wij,x,} = x,A,tij. We shall prove several auxiliary statements. LEMMA 14.1 We have the equality Astij = Aij,,.

Proof From the equation {wi, x,} = (X W.), = x,(w,,, wij)

it follows that A.,;j = Aij,,. Let us denote Moreover {h, W} = [W W]. LEMMA 14.2

Ws, = A,tij Wi.

Then,

[W, W] = 0.

Proof Let U = ?Sy(W2) be the invariant of the algebra Lie H. Then {h,{U,x,}} = {U,III, x,}} = {U,x,}(W),,. Moreover {U,x,} =

Hence

x,A,kij Wi(W,)k, = xk(Wh)k.,Astij W i - x5Astik(W )kj W i + xsAstki ij(W,,)jk

Since this equality is valid for all x h, we can omit x, and substitute W for Wh. Lemma 14.2 is proved.

170

A. T. FOMENKO AND V. V. TROFIMOV

Proof of Theorem 14.1 First, we prove the theorem for the special case when {x;p), 0 for any i, j, p, q. We have

0 = {F", F"} _

{F>ai(W)F}

a;(W){F", F;}

.

The following equation follows from the fact that (H, H) satisfies the Mcondition Y'"'X(n)WI(p.9)_'XT.X(q)wn+i-s-.(p.e)E'XT_

{F",F;} =

(4)

p.g

Hence we have the expression for {F", F.}. Clearly {F, F; } = 0, since otherwise I a;(W){F", F;}

0.

We will now consider the general case. Equation (4) follows from the M-

property. We have {F",Fp} = X'W"E*;{x;,x;}W*EXT - X'WpE*;{x;,xj}Wj*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 E;j{x;, commutes with W-or, which is equivalent, N(h) commutes with W;, for any h e H. Note that {h, N} = [N, Wh]. If {x;, xj} e H, then {h, N}(h) = {B([h, M], h) = 0. Thus, it follows that [N, Wh](h) = 0, which we wanted. Since {F", Fp} = Fp} - {Fp, F"}), z({F",

fXT =Z(XW"E*;{x',xj}W*EXT - XWpE*;{x;,xj}W*EXT) = 0.

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 (x1, ... , x", x'1, ... , x',) = Y form a basis for V + V. The actions of Hon the basis 2n-tuple Y and the basis 2n-tuple Y' = ( A x , ,. .. , Ax", x1, ... , x") are isomorphic. ThL.efore we can construct H-invariants for the module V + V. If the representation of H

INTEGRABLE SYSTEMS ON LIE ALGEBRA

in V is given by the matrix of 1-forms W, then in V + V corresponding matrix W is of the form

(O

W).

171 the

Then,

y'W'gyT = AXW`EXT + X'W`EXT. Using this example, it is easy to obtain a corollary. COROLLARY

Linear combinations of similar canonical H-invariants of

the form FP = CCiJkX(`)WpEJX(k),

aijk E R

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, co) 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 Co.

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 E be a group which acts on M by symplectic diffeomorphisms, H be a function which is constant on the orbits of E. Then, we shall say that

the group E is a group of symmetries of the Hamiltonian system (M, co, H).

Let E be a compact group of symmetries of the Hamiltonian system (M, co, H), a be a 1-invariant algebra of integrals, a, be a E-fixed subalgebra, N be a manifold of fixed points, a be the set of restrictions of the integrals in a to N. Then (a) PROPOSITION 15.1 (Restriction theorem)

(N, Co) is a symplectic manifold, Co = wl N; (b) a is closed with respect to

the Poisson bracket { f g}N; the restriction mapping is an epimorphism of Lie algebras (RE, { f, g} N) -. (R, {k, e} N); (c) ! is the algebra of integrals of the Hamiltonian system (N, Co, H); H being the restriction of

HtoN.

172

A. T. FOMENKO AND V. V. TROFIMOV

Let (M, co) be a symplectic vector space; p: E - End(W) be a completely reducible symplectic representation of the group E, Wo be a fixed subspace. Then, wl wo is a non-degenerate form. LEMMA 15.1

Proof Let i; E Wo and l be its orthogonal complement. Since

E Wo,

' is invariant under E. Hence, there exists an invariant onedimensional subspace R such that W = l O Rq. From the definition of n it follows that # 0. Given that l is 1-invariant, we deduce 7EWp. LEMMA 15.2

If F e aE, then s grad,,, F = s grads F.

Proof Let q e N. Because F is invariant, we obtain sgrad,,, F(q) e TqN. Therefore, to obtain the proof of the lemma it is enough to check the equality w(s grads F, X) = w(s gradN F, X)

(1)

for any vector X on M. Both sides of (1) are equal to X(F) which proves the lemma.

Proof of Proposition 15.1 Let q e N be a fixed point. Then, the correspondence a -+ dqa: TIN - TN is a symplectic representation of E on the space T. N. Since, under the hypothesis of the proposition, E 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 aE is a homomorphism and the image of aE 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 E:

Foadp(a), which evidently belongs to a, Thus we have proved assertion (b) of Proposition 15.1. Let us prove assertion (c). Indeed, let F E a be the integral (M, w, H). Then {H, F},,, = 0. Applying (b) we obtain {H, F}N = 0. 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 E be a compact group which acts by

173

INTEGRABLE SYSTEMS ON LIE ALGEBRA

automorphisms on the Lie algebra G, G,. be a fixed subalgebra, x e G* c G*, OG(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 of G.. Then, (a) E acts by symplectic diffeomorphisms on OG(x); (b) OG,,(x) is open in the manifold of E-fixed points in the orbit 0,(x); (c) if co, co,, are the Kirillov forms of the Lie algebras G, G. respectively, then co = C0I G..

First let us prove two lemmas. Let 1,0,(x) be as in Proposition 15.2; ac, Q*: G* -+G* be the linear mapping conjugate to a. Then a*(OG(x)) - OG(x) LEMMA 15.3

Proof Let x e G*. Let us define the neighborhood Vx, in 0,(x') Vs. _ {Exp(Ad9*)(x'), g e G}.

(2)

Applying a* to (2), we obtain Q*(Vx,) = {Exp(ad*

g E G}.

(3)

Hence, Q*(Vz) C OG(x')

(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 OG(x) is a connected set and OG(X) n a* -' (OG(o *(x)) is both open and closed as follows from (4). LEMMA 15.4

Let x e G*, , n e TxOG(x), co being the Kirillov form.

Then wx( , n) =

Q*n)

Proof By definition of the vectors and q, there are vectors ', n' E G such that i; = ads (x),

n = ad* (x),

then, by the definition of the Kirillov form,

(5)

<x, [g', n']>.

Applying a* to (5), we obtain

Q* = ad*-i.(a*x),

a*n = ad;-1,,'(Q*x)

(6)

Using (6))x it is easy to calculate the Kirillov form at the point u*x: Y a- ln]> = = cox(S' w a*(x)(c*S, Q*n) =
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 '-t(9))(x) = Exp(ad*)(x),

a E E.

(7)

From (7) we obtain ad: (x) E T. , N .

adB (x) = ad* )(x),

a c- 1.

(8)

Let Ann(x) _ u e G ad, (x) = 0}, g c- G, g = g' + g", g' a G. + Ann(x), g" E W, W being a E-invariant complement to G. + Ann(x) in G. Then, ad*, ,(x) = ad*., (x). Hence g" = 0. Indeed, if g" # 0, there exists a E E such that g" # ag", therefore, 0 # g" - ag" E W which contradicts the fact that g" - ag" E Ann(x). Hence we can strengthen (8): ad*(x) e TxN ca g" = 0. This means that TxN = TXOG,(x).

(9)

Since evidently OG,(x) c N, 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 7L2-graded Lie algebras

Let G be a Lie algebra. Let us suppose that G = H ® V, [H, H] c H, [H, V] c V, [V, V] c H. Then G is called a 712-graded Lie algebra. Let

H*(V*) be the subspace of all co-vectors which annihilate V (respectively H) in G*. Then G* = H* Q V*. Below, for any x e G* and g c- G, xH and xv stand for the H* and V* components of x; gE, and gv stand for the H and V components of g. Let us define a new commutator [x, y];, on G:

[9,9']x = 19H, 9A + 19v,9v] + [9H, 9v] + )19v.9'v]

(10)

The commutator [x, y],, satisfies the Jacobi identity. We denote the Lie algebra obtained by GA. DEFINITION 15.1

Let G be a 712-graded Lie algebra, G,. be the series of

Lie algebras constructed above corresponding to the commutators [x, y] A. We shall call the Lie algebra Go a contraction of the 712graded Lie algebra G. DEFINITION 15.2

Let G be a 712-graded Lie algebra. Let us denote the

175

INTEGRABLE SYSTEMS ON LIE ALGEBRA

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 7L2-graded Lie algebra, Go the

THEOREM 15.1

corresponding contraction. Then the index of Go is equal to the rank of the Lie algebra G (= index of G).

Let G be a 1L2-graded Lie algebra, x e G*, g e G, G = Hp V, then LEMMA 15.5

{x,g}o = {xH,gH} + {xv,gv} + {xv,g1}.

(11)

Proof If g' e G, then <{x,9}A,9'> = <x,[9,9lA>

= <x, [9H, 9H] + [9v, 9H] + [9H, 9v] + /Z[9v, 9v]> = <XH, [9H, 9H]> + <xv, [9v, 9H] + [9H, 9v]> + "<xH, [9v, 9v]>

= <{x., 9H} + {xv,9v} + {xv,9H} + ){xH,9v},9') Assuming A. = 0, we obtain (11).

Let G be a semi-simple Lie algebra, B(X, Y) be the Cartan-Killing form on G, g e G. Let us denote the covector dual to the vector g with respect to B(X, Y) by g*. DEFINITION 15.3

If g E GA, then g* a G* is defined by the formula

g*(u) = B(g, u), u e GA, the form B(X, Y) being on GA the same as on G (remember that G and GA are identical as vector spaces). If W c G is a subspace, then W* = {g* I g c- W}.

We know that in G we can choose a Cartan subalgebra T such that the

following conditions are satisfied: (a) T = TH p Tv, TH = T n H, Tv = T n V; (b) if g E V and [g, Tv] = 0, then g 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 (a)-(d) are satisfied; let T* c G* be the subspace of Go dual to T in the sense of Definition 15.3.

Then, for x c- T* (in general position) Anno(x) = T

Proof Let

x = xH + xv,

xH EH*,

xv E V*

and

u e Anno(x),

176

A. T. FOMENKO AND V. V. TROFIMOV

u = uH + uv, uH EH, uv E V. From (11) it follows that ({xH,uf} + {xv,uv} = 0 (12)

1{xv,uH} = 0.

By definition, x = g* for some g E T, g = gH + 9v, gH E H, gv E V. Since H is orthogonal to V. gH = xH, g*, = xv. Therefore the equations (12) can

be rewritten as 51911, UH] + 19v, uv] = 0

(13)

[9v,uH] = 0. For gv (in general position) from the second equation (13) it follows that

UH c K since 9H E TH c K and [g,, uH] E K. Therefore, taking the bracket of the first equality (13) with gv, we obtain (14)

[9v, [9v, uv]] = 0;

because Tv is reductive in G, from (14) it follows that [gv, uv] = 0. From

this equation and the first equation of (13) we obtain [gH, uH] = 0. Hence, using the relationship [gv, uv] = 0 we deduce for gH and g, (in general position) that uv E Tv, uH E TH. As a result, u E T. Thus we have

proved that Anno(x) c T. The inverse inclusion can be easily obtained from (13). If G is an arbitrary 7L2-graded Lie algebra, and Go is its contraction, then ind(G) < ind(G0). LEMMA 15.7

Proof Let Gx denote one of the Lie algebras introduced in the definition of contraction. The correspondence a.,,

9H+9v -' 9H+

9v

is an isomorphism Gx., -> GA if 2', 2" 0. Therefore, ind(G,i) = const(2) if 2 0. Let us prove that if 2 = 0, then the index can only increase. Indeed, the index is equal to the rank of the matrix l c (A)xk 11, c;`;(2) being the structure tensor of the Lie algebra GA; xk being the coordinates of a

covector in general position. If 2 varies slightly, then the rank of the matrix Ilci;(A)xk11 can only increase. Therefore, the index of GA can only

decrease. Hence, ind Go > ind Gx,

Proof of Theorem 15.1

A -A 0.

From Lemma 15.6 we obtain

INTEGRABLE SYSTEMS ON LIE ALGEBRA

177

ind(G°) < 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°) >, ind(G1) _ ind(G). Therefore ind(G°) = ind(G). This completes the proof. Let G = H ® V be as before a 712-graded Lie algebra. Similarly to the

above definition of the contraction of the commutator [x, y] -, [x, y]0 we can define an analog of the contraction for certain functions F on G*.

Let us suppose that the expansion of F in exterior powers on V terminates DEFINITION 15.4

F°(xH) + F'(xH, xV) + ... + F"(xH, xV);

Fk(xH,xv) being a k-form xV for fixed xH. Let us define FA(x) = 2n12F(xH, 2-'I2xv). Note that FA(x) is defined for A = 0: F0(x) = F"(xH, xV) as well.

Let G = H Q V be a 7L2-graded Lie algebra, F, F' be functions on G* such that (a) {F, F'} = 0; (b)F,1 and Fx are defined (see Definition 15.4; e.g. if F and F' are polynomials). Then, we assert that {F,1, F'} = 0 in particular, {F", F" 1 0 = 0; F", F'm being the terms of highest degree of the expansion into homogeneous components on V. PROPOSITION 15.3

1129v is a homoProof The correspondence gH + gV - gH + _ morphism of Lie algebras G - G,, if A 0. Under this F maps into

2"JZFA. Therefore IF,, F' }z = 0 if A : 0. 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 G. Let us show how we can use the proposition to determine the invariants of the contracted Lie algebra G0. Let G = H Q V be a 7L2-graded Lie algebra, F be a polynomial on G*, invariant under the coadjoint representation; let F0 be the contraction of F (see Definition 15.4), G0 be the contraction of G. Then F is an invariant of the coadjoint representation of G0. PROPOSITION 15.4

Proof Let u E G0, then u can be regarded as a linear function on G. Evidently, here the contraction u - u0 is equal to uV if u V # 0 and equal to uH in the opposite case. We can express the fact that F is Ginvariant as follows

178

A. T. FOMENKO AND V. V. TROFIMOV

{F, uH} - 0

{F, u,,} -0,

uHEH,

u,,E V.

(15)

From the remark made above it follows that {F0, uH} - 0, {F0, u,,}0 - 0. These equations imply that the contraction FO is Goinvariant. EXAMPLE The algebra so(4) is defined by the equations [eij, efk] = eik, i, j, k taking values from 1 to 4; e,j = - ej,. The subalgebra H = so(3) is

generated by the vectors ell, e13, e23; let V be the linear subspace (e14, e24, e34). It is easy to check that the decomposition so(4) = H $ V gives a 1L2-grading, the corresponding contraction Go being a semidirect sum so(3) $ i83 (in other words, the Lie algebra of the group of

motions of the Euclidean space R3). We know that the functions F= xi, and F' = x12x34 - x13x24 + x23x14 are invariants of so(4). Contracting F and F' we obtain the invariants of the Lie algebra

so (3)$l :Fo=x14+x24+x34,Fo= F'.

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 of functions which are constant on orbits in general position. In outline, the construction of the integrals can be described as follows. Let f be some invariant of the 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 of the function f (x), i.e. consider the function f (x + Aa), A e C or R. Since in our cases the functions f are

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 + Aa) = Y-k 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 [74] 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 [10], [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 of the form i = [x, px]; the operators cp = TP.,D define the Hamiltonians of the complex series (see Section 7). Let us consider the coadjoint action

of the complex semi-simple Lie group (! on the corresponding Lie algebra G (we assume G = G*). The group partitions the algebra G into orbits. We set

Adgx=gxg-1,

gE6.

LEMMA 16.1 Any smooth function f (x), x c 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 i = [x, p x]; gyp: G - G being an arbitrary self-conjugate operator.

The proof follows directly from Tx0 = {[x, y] }, they vector running over the whole algebra G. Note that in the complex case there exist the elements which do not belong to the orbit 0(t), t e H; H being a fixed Cartan subalgebra. Let us consider the set of all complex vectors grad f(x), f e IG, IG 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 c- Reg G, then H(x) = g0H(a, b)go 1 for some go e 6, 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 eIG}. 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 c- G. We denote by grad f (x) the value of the field grad fat the point x c- G.

Proof Remembering that Tx0 = {[x,f]},

running over the Lie

algebra G, we obtain (grad f(x), [x, f]) = 0 for any

because [x, f] f (x) = 0. Since the operator adx is skew-symmetric, <[grad f(x), x], > = 0 and because the Killing form is non-degenerate

this means that [grad f (x), x] = 0. The converse assertion can be checked similarly. REMARK The equation [x, grad f (x)] = 0 is obtained from c;`ixk of/8x; = 0 (see 2.3) by using the isomorphism G* = G given by the

Cartan-Killing form. PROPOSITION 16.1

Let f E IG, i.e. the function is invariant and

constant on the orbits. Then the complex functions h,A(x) = f (x + .la) are (for any ),) integrals of the equation z = [x, q bp(x)], W.,, being the

operator of the complex series. The function F(x) = <x, cpx> is an integral as well.

Proof Let us check the identity 0 = (d/dT)hx(x), T being the parameter along the trajectories of the flow z. This is equivalent to checking that = 0. We have

Fig. 29.

182

A. T. FOMENKO AND V. V. TROFIMOV

=

= - A

= <[grad f (x + 1a), x + 2a], cpx> - A = < [grad f (x + ;a), x + 2 a], cpx>

- 2 - 7 . We have used the definition of cp from Section 7; t e H(a, b), x' e V. The first summand is equal to zero because of Lemma 16.2 (applied to the point x + Aa). The third summand is equal to zero because D(t) E H(a, b). The second summand we compute as follows

A = . = - A,< [grad f (x + .la), x + Aa], b> = 0

taking into account Lemma 16.2 and the identity [b, t + ,.a] = 0. Thus (d/dt)h,A(x) = 0 along z. Further d F(x) dt

(px> + <x, cpz> = 2<[x, cpx], cpx> = 0

because

cp

is

symmetric and the operator ad is skew-symmetric. Let us consider the model example sl(n, C). Evidently, the standard 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 + 2a)' = [x + 2a, apx + ).b]. Indeed, expanding the

commutator and making trivial calculations, we obtain the initial

equation z = [x, px]. We have used the fact that [a, b] = 0 and [x, b] + [a, px] = 0 because of the definition of cp,bp. Thus the equation has not changed but we notice a new series of integrals: the symmetric polynomials of the eigenvalues of the matrix x + via. These integrals can be presented in two ways: (1) Given the expansion of the polynomial det(x + Aa -,uE) = E,,O P,,,A",uO in powers of 2 and u, all polynomials

P,O(x, a) are integrals of the equation; (2) Using the functions Sk = trac(x + Aa)" and their expansions in powers of A: Sk = Ya Qak'(x, a).l°. The link between the Newton polynomials and the symmetric polynomials a' also interrelates the Qakl and P.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

183

Let us construct the integrals of the compact series. Let Gu be a compact form of the Lie algebra G. Let X E G,,, a, b c- H,,, x - ,la c- G if .l

is real. Let us consider the action of 6 on G.. Unlike the complex

case, the union of the orbits defined by a Cartan subalgebra H. = H (a, b) coincides with G.. Let tp : G - G,, be an operator of the complex series.

Any smooth function f invariant under the coadjoint action of (5 (i.e. constant on the orbits) is an integral of the Euler [x, cpx], 'p: G. - G. being an arbitrary self-adjoint equation LEMMA 16.3

operator. The proof is trivial. Let IG. be a ring of invariant polynomials on G. Let us present explicitly the multiplicative structure of the ring 1G.. Let N be the normalizer of H. in G.; then, N/S = 0 is the Weyl group. Let

t E H., then the orbit 0(t) is orthogonal to the algebra H.. This orbit returns to H. intersecting H in a finite number of points which are the images of the element t under the action of the Weyl group. The ring IG,,

is identical with a ring of polynomials in H which are invariant with respect to the action of the Weyl group. This ring can be described

simply: if 6. is connected then the ring IG is a free algebra on r generators where r = the rank of G. among which it is possible to choose

homogeneous, algebraically independent polynomials Pk1'

, Pk,

where k. = deg Pk. For simple Lie algebras of number k., the degrees of the polynomials Pk; 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;

G2:2,6;

F,:2,6,8,12; E6: 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 Pk, can be given explicitly. Consider the linear representation of the algebra G. of minimal dimension with matrices of

184

A. T. FOMENKO AND V. V. TROFIMOV

the size (m x m), let A,,.

.

.

,

A,,, be the weights of the representation, i.e.

the linear functionals in H. corresponding to the eigenvectors of the operators in H. on the representation space. The coordinates A1,... , A, in H. can be linearly dependent. The polynomials Pk have the form: n+1

An: I A;'; j=1

Bn: Y j=1 n

Cn: Yj=1 n

D. : > j=1

k;=2,4,6,...,2n-2; P;,=A1A2...An A;'. It is clear If G. is a particularly simple algebra then Pk, 1 that all the rings IG are subrings of the ring of symmetric polynomials S(A...... A,,,). All of the indicated functions are of the form Trace xki = A;'; except for the series D. in which a further polynomial D-et x is added. PROPOSITION 16.2

Let f c IG,,, i.e. the function is constant on the

orbits of the algebra Gn. Then the functions hx(x) = f (x + )a) are (for any A) integrals of the equation x = [x, cpx] where cp is the operator of the compact series x + .la E G,,, a. E R. The function F(x) = <x, (px> 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 G. to Gn. The operators cpab: G. --> G. are generated by the vectors a, b c H,,, in particular a, b 0 G. and hence x + .la 0 G,,, if

xEG.,AcD. Let f e IG,,, i.e. the function f is constant on the orbits of the algebra Gn. Consider the functions gjx) where A E f8, PROPOSITION 16.3

INTEGRABLE SYSTEMS ON LIE ALGEBRA

185

x c- G. c G. which are the restriction of the functions h1(x) = f (x + Aa)

to G. c G. Then the functions gz are integrals of the equation z = [x, (pabx] where gpob is the operator of the normal series. The function F(x) = <x, cpx> is also an integral.

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. = 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 G. of the algebra G = sl(n, C); then su(2) coincides with the stationary points of the involution ox = It is

clear that G. decomposes into the sum of three one-dimensional subspaces generated by vectors of the following form:

E- = i(E2 - E-s), Eo

E+ = Ex + E-_, where E0 E H. = iHo,

Eo=(i

Oil,

E+=I-0 O),

E__(0 O).

The operator of the compact series q : su(2)l- su(2) acts in the following way: a(b)

coE+ =

a(a)

a(b)

E- = a(a) E_

E+

,

coEo = ) 0Eo,

0 is an arbitrary real number, b = A+a, A.+ j4 0, a(a) : 0, i.e. A+ = a(b)/a(a). Finally, cpE+ = ),+E+, cpE_ _ A+E_, cpEo = )LoEo and differs from zero. For cp in general position A+ A0. In the case where G. = su(2) the operators cp of the compact series form a twoparameter family (A+, 2o). If where A0

x=

(-

iz

x+iy

x + iyl

E su(2) -iz )

then

V(x) _

i)lo

+(x + iy)

2+(-x + iy)

-iAoz

)

186

A. T. FOMENKO AND V. V. TROFIMOV

It is clear that <x, i> = 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 E0, see Figure 30. The following functions should be the integrals of the flow: Trace(x + Aa)k. We have

i(z+qA)

x+iy

-x+iy -i(z+qA))' where a = qEo, q 96 0. Hence S1 = 0, S2 = -2(x2 + y2 + z2 + 2zqA + q2 .2). The integrals are the coefficients of each power of A,, i.e. Q1(x,a) = x2 + y2 + z2,

Q2(x,a) = zq,

Q3(x,a) = q2,

i.e. in reality the functions z = const, x2 + 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 Q1(x,a) = Q1(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 11,12, I3 are connected by the relationship 11 = 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, C), G. = su(3), G. = so(3), ax = _T, TX = z, G is the manifold of stationary points of the involutions a and T. The subalgebra

G. = so(3) is generated by the three vectors E,j = E,, + E,

Fig. 30.

187

INTEGRABLE SYSTEMS ON LIE ALGEBRA

0

E12=

1

0

- 0 0.

E13=

1

0

0

-

0

0

0

0

0 0,

1

0

1

0

Let a, b e H. = (diagonal purely imaginary matrices 3 x 3 with zero trace). Then the operators cpab: G. - G take the form: gE12 =

bl - b2 E12, a1 - a2

pE13 =

bl - b3

E13

pE23 =

a1 - a3

b2 - b3 E23 a2 - a3

The manifold {cpab} 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 = (bi - bj)/(ai - aj) then x - vP(213 - )23)E12 + ay(A23 - A12)E13 + a$(212 - )13)E23

where x = aE12 + YE13 + yE23. It is clear that <x,. > = 0, i.e. the vectors z are tangent to the spheres with centers at point 0. Let us recall that for so(3) = su(2) the Killing form coincides with a Euclidean scalar product. We have

fi .a1

x+2a=

a

fl

-a

i2a2

y

-fl

-y

iAa3

The integrals are defined by the functions Trace(x + ).a)", 1 < k < 3.

Fig. 31.

188

A. T. FOMENKO AND V. V. TROFIMOV

Computation gives P(x) = a2 + Q2 + y2, Q(x, a) = a2(a, + a2) + #2(a, + a3) + y2(a2 + 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 M2 = const. The Euler equation is fully integrable for any a, b e H. 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 Ax2 + B y2 + Cz2 = 1

I x2+y2+z2=a2 then it can be proved that at all points on this curve 4

(ABC)3/4

PIP2 PI + P2

ABCa2 - BC - CA - AB -

const,

where p, , 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

G,, = so(4) c G c su(4) c G = sl(4, C).

The algebra so(4) can be represented in G as the skew-symmetric matrices spanned by the vectors in standard form E;3 = E, + E_,. Let us write x c so(4) in the form x = aEI2 + PEI3 + yEI4 + SE23 + pE24 + EE34 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 H4 c su(4) then = a b, (panx

b2

EI2 + fi b, - b3

a, - a2 +S

b2 - b3

a2 - a3

a, - a3 E23 + p

+ y b,

EI3

b2 - b4

a2 - a4

b4 EI4 a, -- a4

E24 + E

b3 - b4

a3 - a4

E34.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

189

For every pair a, b in general position we obtain a flow .z on S2 X S2. The integrals will be the functions Trace(x + 2a)", 1 S k < 4 where

x+2a=

Aa 1

a

fi

-at

.ia 2

6

-Ii -6 Aa3 -y

-p

-E

y

E

Aa

Computations give four integrals: h1 = Trace x2, h2 = Trace x4, h3 = Trace x2a, h4 = 2 Trace x2a2 + trace xaxa. The integrals h1 and h2 are constant on the orbits and have the form h1 = a2 + #2 + y2 + 62 + p2 + E2 , h2 = h; + 4(Jibyp - a&ye + aple) - 2(a2E2 + $2p2 + y262).

In fact h2 is the square of the second degree integral q (after computing

the function hi from h2) where q = aE - fip + O. Thus, the two quadratic integrals h1 and q are the generators of the ring I so(4), i.e. any polynomial which is constant on the orbits can be decomposed in terms

of h1 and q. It is easy to prove that h1 and q are independent. The equations h1 = 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 :

h3 = a2(a1 + a2) + #2(a1 + a3) + y2(a1 + a4) + 62(a2 + a3) + p2(a2 + a4) + E2(a3 + a4),

h4 = a2(a2 + ata2 + a2) + $2(ai + a1a3 + a3) + y2(al + ala4 + a4) + O2(a2 + a2a3 + a3) + p2(a2 + a2a4 + a4) + E2(a3 + a3a4 + a4). It is easy to prove that the integrals hl, q, h3, 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 z = [x, rp.,,(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 hx(x, a) = f (x + .la) are integrals of the flow x for any Z. Any two integrals hA(x, a)

and gµ(x, a) arising from the functions f, g E IG are in involution in orbits. The Hamiltonian F = <x, Tx> of the flow x also commutes with all integrals of the form hx(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 of the form hx(x, a). (2) Let

G. be a compact real form of a semi-simple Lie algebra and let z = [x, Tx] be a Hamiltonian system determined by a compact series

operator gyp. Then the set of the functions of the form f (x + .la) where f EIG forms a complete commutative set on orbits in general

position in the Lie algebra G, (3) Let G be a compact normal subalgebra in the compact algebra G. and let z = [x, cpx] be a Hamiltonian system of the normal series. Then the set of functions of the form f (x + %a) where f e I G,, 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 f of G satisfies the identity

s grad f (x) _ [grad f (x), x].

Proof Let

be a vector in TO, then w(s grad f,

f f (x) _

; on determining the form co we obtain w(s grad f, ) _ = [x, y], and hence _ <s grad f, y> where <s grad f, y>, i.e. <[grad f, x], y> = <s grad f, y>. As this identity is true

for any y then s grad f = [grad f, x]. If F = <x, (px> then cpx = grad f (x) and hence - )i = [qpx, x] _ s grad f which proves that is Hamiltonian. Thus, if f and g are two functions of G then { f, g} _ <[x, grad f], grad g). Finally, (f, g} _ <x, [grad f, grad g]>. We have proved the following assertion. Any smooth functions f and g on G fulfil the identity If, g} = (x, [grad f, grad g] >.

LEMMA 16.5

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 16.6

191

Let f and g be smooth functions on G which are constant

on each orbit. Then [grad f, grad g] = 0. Proof Let x c- 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. 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 hA(x, a) =

f (x + .1a), dx(x, a) = g(x + µa) where a c- H(a, b). Then the integrals hx and dN commute. Moreover, {F, hA, } = 0 for any f c IG.

Proof Let us recall that the functions hx and dµ by Proposition 16.1 are integrals of the flow [x, cpx]. By Lemma 16.5 it is sufficient to prove that <x, [grad hz, grad d'] > = 0,

<x, [grad f (x + 1a), grad g(x + µa)]) = 0.

Let us put Y = x + %a then x= Y - .1a, x + pa = Y + va where v = p - .1. Let us suppose initially that v

0.

z = = <[Y, grad f (Y)], grad g(Y + va)> - <[Aa, grad g(Y + Aa), grad f (Y)]> . As f e I G then [Y, grad f (Y )] = 0 by Lemma 16.2. As g e I G then by the same lemma

[Y + va, grad g(Y + va)] = 0. Hence [Y, grad g(Y + va)]

v[a, grad g(Y + va)].

Substituting in z we obtain z= - < [Y, grad g (Y + va )] , grad f (Y) V

_ = 0 V

as f EIG. The assertion is proved for A 0- p. If A = p then

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A. T. FOMENKO AND V. V. TROFIMOV

0 = <x, grad f (x + .la), grad g(x + .la)] > by Lemma 16.6. It remains to be proved that IF, hx} = 0, i.e. to compute L = <x, [(px, grad f (x + tia)] >

as grad F(x) = (px. Let us put Y = x + )a then L = <[Y - 2a, q,Y - Acpa], grad f (Y)> _ <[Y, cpY], grad f(Y)> - )<[Y, cpa], grad f (Y)> - )<[a, qpY], grad f (Y)> + X1.2<[a, cpa], grad f (Y)> = 0

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, 9a] = 0, in the third [a, pY] = adb Y = [b, Y] and again [Y, grad f(Y)] = 0. The involutivity of the integrals of the compact series can be proved in the same way. PROPOSITION 16.5

Let f; g e IG., let us put h,(x, a) = f (x + da),

dµ(x, a) = g(x + µa) where a, b E H (a, b). Then the integrals h,, and d,, commute and IF, h2 } = 0 for any f e IG,,.

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 fl , ... _f, e I G

be the complete set of invariants of the algebra. At the point x there arises a set of complex vectors grad h2,k where hA,k(x, a) = fk(x + .la). Let V(x, a) be the subspace of G generated by vectors grad hx.k(x, a). We

must find a lower bound for dim V(x, a). Let us consider the p'2` where qk + 1 = deg fk. Let fk increase with increasing degree. Let N + 1 = q, + 1= deg f, be the highest degree among the generators fk. All the polynomials hx,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 h.,,k = El,, o U'1' where Uk(x, a) = grad pk(x, a) are polynomials of degree i in x and a. It is clear that UZk(a) does not depend on x as plk is linear in x. The vectors UZk generate the Cartan subalgebra H independent of the choice of x. It is decomposition h.,,k

clear that V(x,a) is generated by the vectors U.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

193

For each k the following recurrence relationships on the vectors Uk are satisfied: LEMMA 16.7

[ Uk , x] = 0

[Uk,x] + [Uko,a] = 0

[Uk,x] + [Uk I,a] = 0

[Uk,x] + [U'-', a] = 0 [Uk,a] = 0. Proof By Lemma 16.2 [x, grad fk(x)] = 0. Applying this identity to the functions hx.k we obtain [x + Aa, grad hA,k(x, a)] = 0, i.e. N

x+1a, I U.l' =0. =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) O+ V(x) then

grad hk,k(x,a)EH(x)Q V+(x), i.e. Uk EH(x) $ V+(x).

Further, let us consider that a e H(x) $ V+(x). Let us consider the simple roots al, ... , a, then every positive root can be given in the form

a = Yi-l m;a; where m; > 0 and m; are whole numbers. The whole number k = k(a) = Y = 1 m; is called the order or the height of the root a. We shall denote the subspace of V+(x) generated by the vectors xa for which k(a) = k by Vk+(x). Then, it is evident that V '(x) = VI+ $ . $ V , + and V I+ is generated by x,,, ... , xa,, i.e. by simple roots. Let us specify a choice of a a c- H(x) $ V + (x). Let a E VI+ and a = Y; = I v; xa, where all v; A 0, 1 < i < r. Then [ Vk+, a] c Vk+1, [H(x), a] c VI+ LEMMA 16.9

Let x, a be chosen as shown above. Then VI+ = [H(x), a]

and ada: H(x) - VI+ is an isomorphism.

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A. T. FOMENKO AND V. V. TROFIMOV

LEMMA 16.10

[l

,

Let x, a be the vectors shown above. Then V k' i =

a] i.a. ado: 1 k+ --* V +I is an epimorphism.

The following relations: U,° E H(x), Y+ Q ... (D Vi+ are valid for 1 < k <, r.

LEMMA 16.11 LEMMA 16.12

U k E H(x) @

Vectors U, j < k generate the entire subspace H(x) O+ Vi+ @ ... ® v+ .

LEMMA 16.13

Let a be an element in general position. Then

dim, V(x, a) >, dim H(x) ® V '(x) = 2(dim G + rank G), where V(x) is a complex subspace generated by all the vectors grad hz,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 enough to note that the elements x and a have been used symmetrically in all the previous statements, as

f

a polynomial of degree q. The proof of the completeness of the commutative sets, constructed above, in the cases of compact and normal series can be obtained from the mentioned scheme, taking into account the involutions defining those two series. For details see [89], [90]. 17. CASES OF COMPLETE INTEGRABILITY OF THE EQUATIONS OF INERTIAL MOTION OF A MULTI-DIMENSIONAL RIGID BODY IN AN IDEAL FLUID

Let us consider the embedding of the type of system mentioned in the

title, in the non-compact Lie algebra of the group of motions of Euclidean space (this has been done in the authors' works [123], [124],

[125]). It turns out that in this case too the method of argument translation makes it possible to construct a complete commutative set of integrals on the orbits in general position.

Let f be an invariant of the coadjoint representation of the group of motions of the Euclidean space Vi". Then the functions LEMMA 17.1

INTEGRABLE SYSTEMS ON LIE ALGEBRA

195

f(x + Aa) for any real A are the integrals of the Euler equations z = adQx 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, > is the value of the functional x on the vector . Obviously: A =

_ - - A.

As f is invariant, the first term is zero. Using the definition of the sectional operator Q(a, b, D) we obtain

-AA =

1 ad*, x1,a),df(x +)ia)> + ,

where x1 E K*I, x2 E K*. The second term is zero, as Dx2 E K, a E K*. The first term is equal to . As x2, b c- Ann(a), we have = = 0

because f is invariant. This concludes the proof of the lemma. THEOREM 17.1 (A. T. Fomenko, V. V. Trofimov) (1) The differential equations x = adQx 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 hx(x) = fix + Aa) are motion integrals any number A. Any two integrals hx and gu are in involution on all the orbits of the representation Ad* of the Lie group '(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 = [(n + 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 + .la) comprise a complete commutative set on the orbits in general position.

The statement codim 0 = [(n + 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:

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A. T. FOMENKO AND V. V. TROFIMOV

0

E(n)* =

I

SO(n) 0

\yl...y.

0

Let us call the minor of a matrix x at the intersection of rows i1,...

, i,

and the columns 11, where 1 < i1 < ... < is ,j: M'1 1 < jl < < js 5 n. Then the functions

n,

t

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 (f) be the full set of polynomial invariants, then N,

f (x + Aa) _

pis(x, a)A', s=0

df e E(n)** = E(n). Suppose df (x + ;.a) =

ui,(x, a)A', s=0

where ui, a E(n). LEMMA 17.2

The following recurrence relations hold: a(x, uio) = 0 a(x, ui1) + a(a, uio) = 0;

a(x, uiN) + a(a, ui,N,-1) = 0 a(a, ui,N) = 0.

Let n = 2s + 1. Consider the complexifcation CE(n). The Lie algebra so(n, C) is simple. Let

so(n,(C)=H®YG; ®EG; i;1

i31

where the subspaces Gt are spanned the root vectors ea with the weight

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

of the root a equal to ± i and H = graded subspace of E(n).

Q

E(n) + = (H (D Ce") $ (Gi $ B1)

I CE 2k + l,zk+ z Consider the

Q... $ (G; $ B3) $ E Gk k3s+1

= Iao QH where Bs+1 -j = C(ez j_ I + ie2j) c C", 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 G; + B1 such that all the components in decomposition upon the root basis in G' and the component of the base vector e" _ z + ie" _ I E C" are nontrivial. LEMMA 17.3

Let a E Gi ED B1 c E(n)* be the element mentioned E(n) we have a(a, Hj) c Hi+1 c E(n)* for i > 0.

above. Then, for H;

LEMMA 17.4 Let x, a c- E(n)* be chosen as indicated above, then the mapping H. - Hi,, c CE(n)* defined by y - a(a, y), y c- Hi c CE(n) is an epimorphism. LEMMA 17.5

The relations (a) ujo E Ho, (b) ujk E Hk hold for any j.

LEMMA 17.6

The vectors ujk generate the entire subspace Hk. We

conclude that the dimension of the subspace, generated by df(x + .la) is at least dim E(n)' = sz + 2s + 1; but for complete integrability we need

codim 0 + z(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 fl(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 KI(G)* with the "complex" series' sectional operators, constructed earlier.

Let G be a complex semi-simple Lie algebra and R(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 + .la), A E C, a E G* fixed covector. Applying algorithm (91) from Theorem 13.1 to the functions h(x) ER(G), we can construct functions h(y), h(x, y) on f )(G)* and obtain a set of functions R(f2(G)) on space fl(G)*. We remind the

reader that the construction of Section 13, applied to the Lie algebra Q(G) = G + EG, EZ = 0 enables us to construct functions F1(x, y), ... , F,(x, y) E Coo(S1(G*)), with F;(x, y) = (8F;(y)/8yj)xj (x, coordinates in G*

and y; in EG*), using functions F1(x), ... , F,(x) E C'(G)*. If F; is in involution on all the orbits of the coadjoint representation of the Lie group (! associated with Lie algebra G, then the functions (y), F,(y), P, (x, y)..... F,(x, y) are in involution on all the orbits F1(y), of the coadjoint representation of the Lie group D (O) associated with Lie algebra K )(G). If F; are functionally independent on fl(G)*, then Fl(y), Fj(x, y) are functionally independent on f)(G)* too. (See Section 13, Theorem 13.2).

Let a function h be functionally dependent on the family of functions a(f (G)), then the Euler equations z = a(x, dhx), x e Q(G)* are a completely integrable Hamiltonian system on all the THEOREM 18.1

orbits in general position of the coadjoint representation Ad* of the Lie group KI(T)) associated with Q (G).

INTEGRABLE SYSTEMS ON LIE ALGEBRA

199

Proof The involutivity of the given functions follows from Theorem 13.2. Let FI (x), ... , FN(x) be a complete set of involutive functions on G* (constructed above), then FI (y), ... , FN(y), FI (x, y), ... , FN(x, y) are functionally independent. In order to achieve complete integrability it is necessary to have s integrals, where s = Z(dim S2(G) + ind S2(G)) = Z(2 dim G + 2 ind G)

= 2['I(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 KI(G)* for x e O(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 0((1i), associated with fl(G). Or, more precisely, let F(x) be any smooth function on i2(G)*, invariant with respect to the coadjoint representation of the group f)(65 ), then all the functions F(x + .la), A e C are first integrals of the Euler equations for any A e C. Any two of those integrals F(x + 2a), H(x + pa), A, p c- C are

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 S2(6).

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 . = a(x, C(a, b, D)(x)), x e S2(G)*. To do that it is enough to check that = 0, where <J, x> is the value of the functional x on the vector f We have:

= (dF(x + .la), a(x + .la, C(x))> - = - .

The first term is equal to zero, as F is an invariant, thus

2

_ -1,

as DheKer4a, where x=x'+h, x'eV+eV, heH+eH, H is a

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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 = 0. We used here the facts that h, 2a e Ker p(b) and that, as F is an invariant, the expression a(x + Aa, dF(x + Ad)) = 0 is zero. We have, thus, proved that the shifts of the invariants are integrals of the Euler equations. We

shall prove now the functional independence of the shifts of the invariants. Let F1(x + Ad), ... , FN(x + Ad) be a complete set of functions in involution for G. Then, using these, a complete set of functions in involution on (2(G)* can be constructed: OF, (y + Ad) Ad),

.

x;, ...

ClYr

(y

x; . (1)

Let us examine now the invariants Ad* for S2(G): F1(Y), ... , FN(Y),

ay,Y)

x;, ... , ay Y) x;

(2)

we should take shifts of these invariants along a vector a where a is taken

from eG*, 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 of functions R(S2(G,,)), than the Euler equations

x = a(x, dhs), x E )(G )* are a completely integrable Hamiltonian

system on all the orbits

in

general position of the coadjoint

representation of the Lie group f2(6.), associated with S2(Gu). Let G,, be a compact form of a complex semi-simple Lie with f2(Gu). Then this system compact series operators C(a, b, D): S2(G )* is completely integrable in the Liouville sense on all the orbits in general THEOREM 18.4

algebra, z = a(x, C(a, b, D)(x)) the Euler equations on

position of the coadjoint representation Ad* of the Lie group S2(0 ),

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

associated with the Lie algebra SZ(GJ. Or, more precisely, let F(x) be any then smooth function on SQ(G.)* invariant with respect to

all the functions F(x + Aa) are integrals of the Euler equation for any 2 E R. Any two such integrals F(x + Aa) and H(x + µa) 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 We shall examine now the construction of integrals of the Euler equation with the "normal" operator series. Let f (x) be a function, invariant with respect to the Consider functions hs(x) = f (x + 2a)I Then the functions h2 are, for any A E U8, first integrals of the Euler equation z = a(x, C(a, b)(x)) on where C(a, h) is a "normal" series operator, a e iHo, h e iHo + EiH'. PROPOSITION 18.1

coadjoint representation of the Lie group

Proof Suppose that we have a differential df(x + Aa) in G,,, then df I G.(x + .la) is an orthogonal projection df (x + Aa) on G In the case

G + eG we obtain df (x + a ,a) = vl + Eve, vi e G (i = 1, 2) and df I n(,.)(x + Aa) = n(vI) + eir(v2)

where n

is

an orthogonal

projection on G. with respect to the Cartan-Killing form. Then = (df I II(G.)(x + .la), [Cx, x])

= (7C(vI) + En(v2), [Cx, x])

= (v1 + eve, [Cx,x]); as [Cx, x] e G it is possible, therefore, to add any term to the first factor, orthogonal to G. We have thus: = (df(x + via), [Cx, x])

= = 0,

as has been proved already in the case of the "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* of the Lie group with respect to the Kirillov form. Moreover, the number of functionally

independent integrals, given by Proposition 18.1 is equal to half the

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A. T. FOMENKO AND V. V. TROFIMOV

dimension of an orbit in general position of the coadjoint representation Ad* of the Lie group associated with Lie algebra S2(G.).

Proof We reduce all cases to that of a semi-simple "normal" series of operators. We have G. c G. Let f (x) be an invariant of G,,, then on G.

the complete set of functions in involution (see [89], [90]) can be obtained as the restriction to G. of functions f(x + .la). Applying to these functions the (21) algorithm, one finds a complete set of functions in involution on These functions have the form fl Y + ).a)/G (as

the operations of restriction and substitution are commutative) and W G. (Y + .la)/?yj)x;. Let us check that those functions are identical to the functions mentioned in the proposition. According to Theorem 13.1 the S2(Gu) invariants have the form f (y) and (Of (y)/8y;)x;. It is clear that the shifts of these functions along a suitable a after restriction to S2(G.) will give the necessary results. Thus, the algorithm (91) leads to the necessary set of functions, which proves the proposition.

The results are summarized in the following theorem.

Let G be a normal compact subalgebra in a complex semi-simple Lie algebra G; z = a(x, C(a, b)(c)). The Euler equations on with the operators of the "normal" series C(a, b):

THEOREM 18.5

S2(G,,)* --

are a completely integrable system in the Liouville

sense on all the orbits in general position of the coadjoint representation Or, more of the Lie group S2(6j, associated with the Lie algebra f precisely, let F(x) be any smooth function on Q (G.)*, invariant with respect to Ad*(S2((6.)), then all the functions F(x + Aa)I ((;,) are integrals of the Euler equations for any a. E R. Any two such integrals F(x + .la), H(x + µa) are in involution on all the orbits Ad* 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* In a way similar to the complex case the following theorem can be proved.

Let function h be functionally dependent on the functions of the family, then the Euler equations )i= a(x, dhx), x e f2(G.)* are a completely integrable Hamiltonian THEOREM 18.6

system on all orbits in general position of the coadjoint representation Ad* of the Lie group C)(6 ), associated with Lie algebra S2(G ).

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

18.2. Complete integrability of a geodesic flow on T*f2(6)

Let G be a complex semi-simple Lie algebra, with compact form, G. and

G a normal compact subalgebra in G. Let f2(6), f2(6 ), f2((%) be Lie groups corresponding to the Lie algebras f2(G), f2(Gu), f2(G ). Consider

the cotangent bundles T*f)(6),

Let quadratic

forms <, x e T* be given on the spaces T* = T*f2(6), T*f2((5.), T*f1(6.) (e is the group identity element), cp being an operator of the "complex," "compact" or the "normal" series correspondingly. Extend

these quadratic forms in a left-invariant way to the entire cotangent bundle space to obtain a function H. On the cotangent bundle T*971 there is a canonical symplectic structure w E f2Z(T*m) (see Section 1) given any smooth manifold SDt. THEOREM 18.7

Let f2(6), f2(6u) and f)((%) be the Lie groups as

defined above and let H be a left-invariant form defined on T*fl((6), which corresponds to the operators of the T*f2(0 ), "complex," "compact" or "normal" series; then the Hamiltonian flow

on T*0(6), T*f2(6.), T*f ((h) corresponding to H is completely Liouville-integrable in the classical sense.

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*1((5), T*0((6.),

corresponding to a quadratic form H, is completely Liouville-integrable in the non-commutative sense.

or f2(6 ). Then, since S acts Proof Let Sj be 0(6), symplectically on T? = T*Sj, a finite-dimensional Lie algebra of integrals V, exists on T*Sj which is isomorphic to T¢5. Above we pointed out that the commutative Lie algebra of integrals Vo (dim Vo = 2(dim T, !b - ind T,5) exists for the operators C(a, b, D) on the orbits of Ad*(,). Extending these functions to left-invariant functions on T*S in

a manner similar to 3.2 we obtain a commutative Lie algebra of functions Vo on T *5, which commutes with V, . The proof of the lemma

can now be obtained by verifying that dim T* 6 = dim(Vo + V,) + 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 + Y,)*. Above we have constructed a complete

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A. T. FOMENKO AND V. V. TROFIMOV

involutive set of functions on V*; hence, such a set exists on (V0 + V1)* as well. Since Vo is an Abelian Lie algebra, we can select any basis in Vo as a complete involutive set of functions on V* (any element of Vo is a linear function on Vo ).

18.3. Extensions of 1(G) for low-dimensional Lie algebras

In this section we shall consider Lie algebras over the field J 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 f20(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 coadjoint

representations is given in [106] (we shall use the notation given in [106]).

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 18.3

The proof can be obtained by a calculation of the rank of the matrix IlC.xk1l with the use of the tables given in [106]. PROPOSITION 18.4

Let G be a low-dimensional Lie algebra; then it

satisfies Condition (FJ) from 3.6.

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: A3,4, A3,6, A3,8, A3.9, A4.1, A4,8, A4,10, A5.1, A5,2, A5,4, A5,10, A5,40, A6,1, A6,2, A6,3, A6,16, A6,20, A6,22

We do not have enough shifts of invariants for the rest of Lie algebras. For those Lie algebras A6,r (1 < 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 semi-

invariants. We omit minor details of the calculations because of their complexity. This completes the proof.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

THEOREM 18.8

205

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 (21) of Theorem 13.2 to the functions on G* which were given in Proposition 18.4. Then on the space

S2a(G)* there exists a complete involutive set of functions, giving 83 infinite series of Lie algebras for which the Euler equations x = a(x, dfx), x E (1.(G)* are completely integrable in the classical sense on all orbits in

general position of the representation Ad* Qa(6). 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

If we take A": (su(n), su(n - 2));

B,,: (so(2n + 1), so(2n - 1));

C":(sp(2n),sp(2n - 2)) D": (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 z = s grad V, V = Y_ aij(W"X(i), X(j))kk are completely integrable.

For pairs of Lie algebras (so(n) p Ili;, so(n - 1), (so(n + 1), so(n - 1)), (su(n) Q C", su(n - 1)), (su(n + 1), su(n - 1)), the following more general result holds, namely the functions REMARK

Y_

[a..(W2pX(i) X(i)) + p..(W2p-IX(i),X(j')]

i,j=1,2

form an involute set.

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A. T. FOMENKO AND V. V. TROFIMOV

REMARK 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.

The series D", B. We consider a pair (so(n), so(n - 2)). The adjoint action of the algebra so(n - 2) on the algebra so(n) splits as a direct sum of invariant subspaces: so(n - 2), Rn-2, 68"-2, R1, 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 X'W2e+1Xt (the same matrix notation XW2eXt, X'W2eX't, invariants can be obtained using the form P = L; {xi,x;}; in what follows it will become clear why it is necessary to use this form). Note XW2eXt,

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 W" 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'W2eXt and X'W2e+1 Xt) is not equal to zero. Thus, the following set of functions is an involutive one:

Vo=a1YXr +0(2EXj2+a3YX,x,+a4Y{x;,x.}, V2p = a1XW2pXt + a2X'W2pX't + a3X'W2pXt + a4X'W2p-1Xt

(similarly with Y_ {x;, xt}, the functions are of the form X'W'P -'X' because Fo = E {x;, x;} = X'PXr and hence, F2p = X'W2pPXt =

XW2v-1

{W*t,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 co;3, 1 < i, j < n - 2) is [(n - 1)/2]. The proof of the f(lnctional 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

-

207

INTEGRABLE SYSTEMS ON LIE ALGEBRA

RI(n-1)12] is n - 2. According to Section 3, direct sum of so(n - 2) and we have an integrabler case of a Hamiltonian system since

dim so(n) -_ L2] = dim so(n - 2) +

Cn

2

11

+n-2

(the right-hand side is the sum of dimension and index of the algebra of integrals).

Series A", C". 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: x1

1

W Xn_2

-X1

-xn-2

-x1

... -Xn-2

xn_2 Iesu(n).

(1)

z

--

Provided that (1) holds, the irreducible invariant subspaces (apart from su(n - 2)) are V= {x1, ... , Xn-2}, V' = {X1, ... , xn-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" = >; a1 W', a.-, = 0 (from which it follows that a priori it is unknown whether, for example, {X W"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 a" W" _

w"-2 -

a; W'. From similar integrals which are functionally

independent on the orbit, we can select n - 1 (see Theorem 19.4) which, in turn, are functionally independent with integrals w,j (1

i,j
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

Let

F

be

a

canonical

invariant,

then

{z+z",F} = 0. 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

n2 - n; the dimension of the algebra of integrals is

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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 C 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. According to 3.6, for Liouville integrability to be attained, it is enough to show that the algebra of integrals satisfies Condition (FJ). REMARK

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, D) 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: Let (G1, H1) and (G2, H2) be pairs of real Lie algebras, such that G1 ® C and G2 ® C are isomorphic and that the isomorphism THEOREM 19.2

induces an isomorphism H1 ® C = H2 ® C. Then, a complete involutive set of one pair (G;, H.) 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)* in terms of the Lie algebra

209

INTEGRABLE SYSTEMS ON LIE ALGEBRA

G. ® C but from this point of view G1 ® C and G2 ® C would be indistinguishable. Let us consider as an example the pair (sl(n, R), sl(n - 2, l)) 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, J) into sl(n - 2, l )-submodules can be expressed by the following matrix: xl

x'

xn-2

x;,-2

W Y1

yl

...

Y,.-2

(2)

Z Z r"

yn-2

Comparing (1) and (2) we obtain involutive sets YWeX, Y'WeX', YWQX', Y'WeX. If we did not consider the "compact" case, it would be difficult to get these, because the bilinear transformation

B: (X, Y) --+ YX is not a bilinear form on the invariant subspace generated by the vectors x1,... , x" _ 2, y , , . .. , yn _ 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'WCX, YWeX' follows from X'WeX, XWeX'. Volume form

Let H = sl(n,1l ), V, V', ... , V("') be H-submodules; isomorphic

to each other and corresponding to a minimal representation of sl(n, 68) in 68", G = sl(n, R) ®(V + V' + + V(m)). Then, the functions det(X Wil, ... , X'Wi., ... , X"W',) are H-invariants. det(X W`l, ... , X'W'-, ... , X"W',, . . ) . Let us write W W (k times) .

instead of the matrices K' in the expression for the function F1, enumerating all the resulting copies of W. We then enumerate all row vectors X('") in the expression for F2. Let F1 be XWY(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,..., W) is a column; F2 = X(m)Z(fl), X(m) being the row vector with the number fi; Z(fl) = det(X, X', ... , X(k) , .. , W) is a column. Then

{Fl, F2} _

XtYi(a)A,,jjXsm)Zi6).

a.B

a.0

210

A. T. FOMENKO AND V. V. TROFIMOV

Direct computation gives A811J = B,Ai. Thus, {F,, Fz} _ > XZ(fl)X(m)Y(a). a,p

Now it is easy to verify that Statements 19.2, 19.3 and 19.4 are valid. The functions det(X, X W, ... , X W;,, X', X'W,... , X'We1) and det(X, X W,... , X W'`2, X', ... , X'We2) are in involution on PROPOSITION 19.2

H ® (V + V') provided that k, < k2 <(n - 1)/s]. A similar statement can be easily obtained for H ® (V + V' + V"). PROPOSITION 19.3

Functions FP = det(X, XW, ..., XWP, X', X'W,

... , X W" - P - 2), p = 0, ... , [(n - 2)/2] together with linear functions wig, x;, 1 < i, j <, n form a complete non-commutative set on

HQ(V+V'). Functions det(X, ..., XW'°, ..., X', ..., X'W'1, ... , X(')W'-) are invariants of the algebra H Q (V + + V(m)) PROPOSITION 19.4

_.rovided that i, = [n/(m + 1)] + (1 ± 1)/2, Y, i, + m + 1 = n. 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) + l". Let us take as coordinates on (so(n) + R n)* linear forms w;,, x; which are bases of so(n) and I?' respectively. Let us choose these linear forms so

that {wi;, w;i} = wii, {w1 , xj} = xj. Then the Hamiltonian in terms of these variables is V= 1E

w`' + > rixi 4i,j (ai+I)

(3)

i

The vector field s grad V is then given by

uiij_(ai-aj)

waw:J

(ai + ai)(ai +

+x;r.-xri

wii xi

If + Ml

The Lagrange case follows from (3) provided that r; = 0, If = C,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

211

1 < i < n - 1, a" = kc, r" 0. The corresponding Hamiltonian system involves a set of linear integrals co;,, 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 = Al Tr(W2) + A2 Y x? + A3z, where x, = co,,, x', = z and the factors A, depend on C, k and r". Evidently, the Hamiltonian is a linear combination of canonical so(n - 1)-invariants (z = 1/(n - 1) Ei {x;, x,}), and hence we can write down an involutive series of the first integrals FP = AI Tr(W2p+2) + A2XW2pX' + A3X'W2p-1X'.

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) Q R")* as a quadratic Hamiltonian. The following canonical invariants are available: XW2eX1, X'W2eX", X'W2eX', X'W2e+1X`

and hence, for the family of Hamiltonians V4,1 = AI Tr(W2) + A2 >2 x? + A3 > x;2 + A4

x,x; we can obtain involutive sets of first

integrals. Formally we can extend this family

Vxp=2 Tr(W2)+12

x?+A3>x;2+A4Yxix;+25z2

since Yl-1 X,2 + z2 is an invariant of the coadjoint representation. The proof of the completeness of this set is the same as in the previous case.

REMARK We can obtain the integrable case for the series given in [103] assuming A4 = 0. 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) (D l8", 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

212

A. T. FOMENKO AND V. V. TROFIMOV

(so(n + 1), so(n - 1)) -> (so(n) ® 1fi", so(n - 1))

is practicable for other Lie algebras as well, e.g. (su(n + 1), su(n - 1)) - (su(n) (D C", su(n - 1)) (u(n + 1), su(n - 1)) -, (u(n) (D C", su(n - 1))

(sl(n + 1, li), sl(n - 1, R) - (sl(n, li) (@ i", 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 -+ G. be the contraction of the algebra G. If we take (so(n) ® R", so(n - 1)), (su(n) ® C", su(n - 1)),

(u(n) (9 C", su(n - 1)), or (sl(n, R) ® li", sl(n - 1, 118)) as a pair (GB, H),

then the Hamiltonian systems are given by z = s grad B* V, the Hamiltonian

V = E (a;,(X(`)W", X1 )kk + fl;;P(X11)W(P , XU)

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, l8), sl(n - 1, li)).

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 ¢ = {F;, i = 1, ... , rk W, f e (H*)*, 1 < i < dim H}, where the Ft 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 0; or (b) if this cannot be done, for obvious reasons, by proving that the skew gradients of the set of functions 0 are linearly independent almost everywhere. The semi-simple case

In 19.2 examples of Hamiltonian systems on semi-simple algebras are

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 0, when the F; are canonical H-

invariants. It is clear that then the similar linear combinations of invariants included in the set 0 as the functions Fi will comprise a functionally independent set. THEOREM 19.4

The following sets of functions 0 are functionally

independent on orbits in general

position of the

coadjoint

representation of the corresponding Lie algebras: (1) so(2n): 0 = {elements so(n - 2) viewed as linear forms on so(n - 2)* and either (a) XW2pX`, or (b) X'W2pX", (c) X'W2pX" or (d) 0 5 p <, n - 1}; (2) so(2n + 1): 0 = {elements of so(2n - 1) viewed as linear forms on so(2n - 1)* and either (a) XWPX' or (b) X'W2pX", (c) X'W2pX`, 0 <, p < n or (d) Z,X'W2p-1X`, 0 < p < n - 1}; (3) su(n): 0 = {elements of su(n - 2) viewed as linear forms on su(n - 2)* and either (a) Z, XWPX or (b) Z",X'W"X', (c) Z' + Z', XWPX' + X'WPX, 0 < p < n - 3}; (4) sp(2n): 0 = {elements of sp(2n - 2) viewed as linear forms on sp(2n - 2)* and either (a) XW2X or (b) XW2i+1X}. X'W2P-1

Let U;: H* -> I (1 < i < rkH) be functionally independent invariants of the coadjoint representation of the group Exp H. Then the projections of the gradients grad U;, I <, i < rk H = r on the subspace W = {w1,. .. , w, j} c H are linearly independent at points in general position. LEMMA 19.1

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 0. 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 s grad Fp = {Fp, X} = Ei=O(P+2) f X'W, 0 <, p 5 [(n - 4)/2]. Vectors X, X W, ... , X W` are, in general, linearly independent, therefore the s grad FP are linearly independent too. It is not difficult to see also that the FP are functionally independent on an orbit in general position in

214

A. T. FOMENKO AND V. V. TROFIMOV

sl(n, R) Q (68" p If8") with linear integrals o , x;, 1 S i, j < n. And, finally,

dim(sl(n, R) p (68" + R n)) -

C1

r + n - 2[ ]) =dim(sl(n,68)QR1)+rn

2

21 + 1+Ln 2 21

(the left-hand side of teh equation is the dimension of an orbit in general position in sl(n, U8) (@ (118" + R") and the right-hand side is the sum of the

dimension and the index of the algebra of integrals). Note that the number of linearly independent skew gradients gives a lower bound the dimension of an orbit of the coadjoint representation of the algebra sl(n, V8) ® (118" + - + If8") and Proposition 19.4 gives an upper bound for this dimension. 20. THE INTEGRABILITY OF CERTAIN HANIILTONIAN SYSTEMS ON LIE ALGEBRAS Let G be a Lie algebra, G* its dual space, l(G*) the set of functions on G*

invariant under Ad* (invariants of Lie algebra G). For any element a c- G* we shall define a set of functions I. = Ia(G*) _ { fx, fA(x) = f(x + ,ia), f e I(G*)}. As we know, I. is an involutive set with respect to the standard bracket { f, g} on G* (see Section 11): { f, g}(x) = <x, [dx f, dx g]>. In the case of a reductive Lie algebra there is

an invariant identification of G with G*. Thus the notation Ia = Ia(G)

makes sense for a c- G as well. Let 0 c G* be an orbit of the representation Ad* of the group 6 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 i dim 0 independent functions. It was proved in [89], [90] that for the compact form G. of a complex semi-simple Lie algebra G the set Ia is completely involutive when a c- G. is an element in general position. In addition, if G. is a normal compact

subalgebra in Ga, the set T. for an element a c- G" with a 1 G. is completely involutive for an element a in general position, where I. = I. I G. The sets I. and Ia are called the compact and normal series integrals. The Euler equations x = add f(x)

(1)

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

are completely Liouville integrable for Hamiltonians f functionally dependent on the sets Ia or T. as these sets are completely involutive. It will be shown below (Theorem 20.2) that, in the case of G. = su(m)

for any element a the sets la and L. 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 Ia(G*) to complete involutive sets. We prove, as an auxiliary result, that the restriction of the set Ia(su(m)) to any singular orbit 0 gives 1 dim 0 independent functions for a certain fixed element a e su(m) (independent of the orbit). This result is similar to the result of Dao Chong Thi [21] in whose proof, however, there are some inaccuracies. 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 (Tj) = 6,bj,, be homogeneous polynomial functions, defined uniquely by the following identity: (

)5 Tr(x + R)s =

(- 1)nPa.q(x), n+q=s

deg Pa.q = n. Let su(m) = {x Egl(m, C), x = -x`, tr x = 0}; u(m) _ {x E gl(m, C), x = -z`}; so(m) = {x E gl(m, C), x = z = -x`}. THEOREM 20.1

Let a e so(m),

a = E'-I' a(71+1 - T + I,i),

(1 < i < m - 1). Then the restrictions of polynomials in su(m) give i dim 0 independent functions on it.

a;

0

to any orbit 0

Proof It is possible to select x E 0 for any orbit 0 in such a way that x1 s x = diag( x1,..., where Let xm. xm), G9 c gl(m, C) be the subspace, generated by the vectors

Ei.i+q -

()4+1(T

i+q ± (-

1)qT +q.i),

Gq

G+

G9 .

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, WW = W r G. Let also

G± = $q,o Gt, Gq = Q+ o
216

A. T. FOMENKO AND V. V. TROFIMOV

W.' = W n G9 . Let T= Tx TO be the tangent space to the orbit 0. The direct calculation of the differential of the polynomial at a point x as a function of (su(m))* leads to: m-q

d., Pa" = X CI(a) U^ - l,j dx j 9,j

+ RQ,9

j=1

where xj is the linear function on (su(m))* arising from the element E E u(m); C9 (a) = (q + 1)aj ... aj+q+1; UQ = Eko+ . +k4=, x.o ... xJq, R" ,q e G. _ 1. We shall show that dxP",9 e su(m)q . First let x be a point in

general position. Then by considering the sequence of identities [dxPa 9 + 1, x] _ [dxPa q, x] we obtain d,, 9 E su(m)q . For singular x we obtain dxPa q e su(m)' from continuity. As a consequence

9

be the projection of dxPa,9 on T. We shall generate T+ for q >, 1. Taking into account show below that the the projection of the E su(m)9 we obtain that d,c that onto T, generates T +. As dim T + = i dim 0, the differentials theorem is proved. R" ,q Esu(m)7+_1. Let

generate Tq

Let us prove in addition that the differentials (q >, 1). As

E su(m)q 1, m-q dz pa.9 =

CJI(a)Un

C, 9(a)

0.

j=1

It is enough therefore to show that the matrix Vq, l' = U7 is nonsingular, where n(1) < n(2) < < n(mq) is a maximal sequence of q (it follows from the definition of the indices such that xn(j) < sequence n(j) that mq = dim Ti). LEMMA 20.1 Let y1, ... , yN be indeterminates, for 1 <j, p < N, j + p < N, let W?(y) be the homogeneous component of i in the formal series Wp: W"(y) = (1 - y)-' - (1 - yj+p)-1. Let WP be the

(N - p) x (N - p) matrix of these polynomials 0

i < N - p - 1,

1<j
det W"(y) = (Yp+2 - Yi)(Yp+3 - Y2) . . (YN - YN-p-1)det WP +'(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 of terms (from top to bottom)

217

INTEGRABLE SYSTEMS ON LIE ALGEBRA

of the series W +1 - Wp, WP.,, = {term of degree i of the series W + 1 - Wp},j> 1. As(W01, Wo2,..., WO,N-p-1)= (1,0,0,..., 0) and we have Wj+1 - Wp = (Y;+p+l det Wp=detWp= (yp+2 Yj)Wjp+l,

-YN-p-1)detWp+1.

Let x1 < xm; 1 < q < m; n(1) < maximal sequence of indices, such that Xn(j) < LEMMA 20.2

.

< n(mq) be a

and r(1) < ... < r(m4) be a maximal sequence of subscripts amongst sequences satisfying the condition x,(j-) = x,u+1) _ = x,(j+s) s <, q. Then x,,(j)+s = x,u+s) where j = 1, ... , mq, s = 0,... , q.

Proof Let k(0) = 0, x1 = Xk(1)+1 = ... = Xk(2) _

= Xk(1) = A1,

2,

.

, xk(,-1)+1 = .. = Xk(t)=m =

i = 1,... , t;

!C(i) = min(q, k(i) - k(i - 1)),

k(0) = 0,

k(i) _ Y_ K(i) S=1

where the subscript p has been determined from the inequality k(p - 1) < j + s <, k(p). On the other hand if 0 <, s < q, X,(j+s). The lemma is tip. Whence We can check directly that X,u+s)

proved. Now everything is ready for calculating the determinant of the matrix V9. Let yj = x,(j), then V,? = Ug (j) = {the component of degree i of the

formal series (1 -

(1 - X"U) +q)-'I = {the component of

(1 - x,(j+q)) -' degree i of the formal series (1 - x,(j)) -' Wj (y). As det Wm 1 = 1, applying induction, we obtain det V9 = det W4 # 0. The theorem is proved. (i) For any element x c su(m) the set Ix(su(m)) can be extended to a completely involutive set. (ii) For any element x E su(m) THEOREM 20.2

orthogonal to so(m) the set Ix(so(m)) can also be extended to a completely involutive one.

Proof (i) Let G. = su(m), G. = so(m), Gx

g c G,,; [g, x] = 01 the

centralizer of x in G,,, in the same way let G; be the centralizer of x in G,,. As x is a semi-simple element, Gv is a reductive subalgebra in G. and the

rank of Gx is equal to the rank of G. We may take x = diag(

x1,...,

xm),

X1 < ... <' Xm.

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A. T. FOMENKO AND V. V. TROFIMOV

As G' is a reductive Lie algebra, there can be found on Gw an involutive f1,. .. , fk comprising k ='(dim GU + rank Gu) independent functions. Extend the functions f on G,, by taking them to be constant set

along vectors orthogonal to G'. These extended functions we shall denote by f too. LEMMA 20.3

The set Ix is in involution with the set {fi, ... , fk}.

Proof Let f (y) = J(y +' x), J E I(G j). Then {f,1;}(y) =

= -A<x,[df,dyf]> =2<[x,d,..T],dyf>=0 as dyf eGu.

It is known that dxI4 = d,lx (for any set of functions a, by definition dx a = {linear span dx f, f e tR}). 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 xT(dXI4) = i dim 0 and therefore

dim nT(d,I,) _' dim 0 ='(dim G. - dim G;) too. Therefore we can choose from the set Ix functions fk + i , , fk +s where s ='(dim G,, - dim Gx), so that TT(da f) (i = k + 1, ... , k + s) generate T +. As T + c T = Ts01 Gv and d, f E Gv (i = 1, ... , k) all the functions fi , ... , fk +, are also independent. Their number is equal to Therefore this is a completely involutive set on G. Statement (i) of the theorem is proved. (ii) Wodd = O G2n+1 Wed = W o G+ d.

For

any

subspace

WcG

let

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 dxla on T is nT(dxl,) = T+. As dxl, = dalx nT(dalx) = T. As dalx = nT

(dalx) = nTpdd ° nc,(dalx) = nc. ° nTo(daIx) = n,q(T+dd) =

Todd

219

INTEGRABLE SYSTEMS ON LIE ALGEBRA

LEMMA 20.4 We claim that !(dim G' + rg

dim(Gx)o+da

Proof As Gx = so(k1) ®... (D so (k,), GU = s(u(kI) (G4)

odd

®... (D u(kt)),

= (su(kl) ®... ® Su(kt))

dl

it is enough to show that z(dim so(k) + rg so(k)) = dim(su(k))'d but this was proved in [90]. The lemma is proved. It is possible to construct on Gx an involutive set

consisting

of k = !(dim G' + rg Q independent functions. In the same way as in (i) we shall extend the functions f to G. taking them to be constant along vectors orthogonal to G. As in (i) we have {Ix, j } = 0. And indeed,

{f,}(y) = =

_ -A<x, [!(d, - dyf-z),dyfi]> _ = 0 as

dyjEG'. It has been shown above that it is possible to select from set 1

functions fk+I,

.

, fk+s

in such a way that fk+s}) = T+odd.

fk+, are independent. Their dd. But, as was number is k + s = dim T+dd + dim(Gx)odd = noted above, dim(G) °+dd = !(dim G + rank G.). Therefore the set 11, ,fk+S} form a ,fk+s together with the set Ix u {jk+I,

As T+dd 1 G all the functions A

,

completely involutive set. The theorem is proved. 20.2. Completely involutive sets of functions on affine Lie algebras

Let G be a Lie algebra and let G = H ® V where H is a subalgebra and V is a commutative ideal. Let p = ad,, 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.

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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 of the action of the 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 = Ann(W, x) =

{g a W, adg x = 0} c W. If W is a subalgebra in G, then Wx is a subalgebra too. We shall need, when calculating the index of an affine Lie algebra. Let G be an affine Lie algebra which is the split extension of Lie algebra H determined by a representation p of the

THEOREM 20.3 (see [118])

algebra H on V. Then for an element x e G* in general position the equality ind G = ind Hx + ind p* holds, where p* is the representation of H on V*, dual to p. Proof Let x e G*, x = xH + xv, xH e H*, xv e V be an element such that the following conditions are satisfied: (a) ind p* = dim H dim Hx; (b) ind Hxv = inf find HY, y e V*); (c) dim Ann(Hxv, xIH=,,) =

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 Hx + ind p* follows from (a)-(c). Let g = gH + gv a Gx, then ada (xH) + ad* (xv) = 0

(2)

ad* (xv) = 0.

(3)

It follows from (3) that gH a Hx°. Consider the restriction of equation (2)

to H'v: = - = 0;

Hxv> = <xH, ad* (Hxv)) = <XHI H=v, ad9H(Hxv)>

= ;

so, restricting (2) to Hxv we find (ada*H)(xIH=v) = 0. Let nH be the projection from G onto H along V. We have proved that 7CH(Gx) c Ann(Hxv, xIH=,,). On the other hand, let gH c Ann(Hxv, xIH=, ). Then

INTEGRABLE SYSTEMS ON LIE ALGEBRA

221

equation (3) holds automatically. We shall show that it is possible to choose g,, e V in such a way that equation (2) holds too. LEMMA 20.5

If y e H* and yl H=y = 0, then there is g,, e V such that

y = adg,,(x,,).

Proof We have shown above that (ad,,)* xvl H=v = 0, therefore it is sufficient to check that dim(ad,,)* x,, = dim{y EH*, yIH= = 0}. Clearly = 0} = dim H/HXV. On the other hand,

dim{y E H*, yIH'v.

dim(adv)* x,, = dim V/Vxv. Let L(x, y) = <x,,, [x, y]) be a bilinear skew-symmetric form on G. It is easy to check that H and V are isotropic

subspaces under the form L and HXV $ V'v = Ker L, from which it follows that dim V/Vxv. The lemma is proved. As xH)I H=v = 0 we can choose, on the strength of the lemma, g,, in such a way that (ad,,,)* XH + (ad9,,)* x,, = 0. Hence it follows that

gHE7CH(Gs) and nH(Gx) = Ann(Hxf,xIH=). We now evaluate the dimension of a fiber of the projection nH: Gx - nH(Gx). It follows from equation (2) that dim(nH)-'(gH) = dim Vxv. From which dim Gx = dim Ann(Hxv, xiH-) + dim Vxv.

Note now that dim Vxv = dim V - dim V/Vxv. It was proved above that dim V/VXI = dim H/Hxv, therefore

dim Vv = dim V - dim H + dim H''' = 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 = 0. As V is an ideal, V c Ga. As V is a commutative ideal, it follows from Lemma 20.3 that Ia(G*) 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* is completely involutive. Then for an in general position the set

element a e Hx in general position the set 10(G*) u V is completely involutive.

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A. T. FOMENKO AND V. V. TROFIMOV

Proof We

shall show further that dim(dxla(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 dxla(G*).

Let G be any Lie algebra satisfying condition (a) of the theorem. Let I = I (G*), x, a e G*, I. = Ia(G*) and (Px,a: G --' G be the partial many-valued operator (px a = (bx ' 0a, where ca(x) = ad* a. Then for a sufficiently large N and for x e G* in general position, LEMMA 20.6

dxla = ((Px.a)N(G" )

Proof Let f be a polynomial invariant. We shall define a polynomial

.f by equation acg f

f q(x + a) = E fq(x),

degfq=degf - q.

q =0

For the polynomials f q = f .q the identity (ad d x f q +')* x = (ad dx f q)*a holds and, therefore, dx fq+' e Tx,a(dx fq). Whence dxfq E ((Px,a)q(Gx) and dxla c ((Px a)N(Gx) for N > max{deg fl, ... , deg f2}, where

fl . ... . f, is any complete set of polynomial invariants. The reverse dxla follows from the fact that for x E G* in general position, dxl = Gx and that the ambiguity of the operator (Px,a coincides with Gx and, consequently, with dxl too. The lemma is proved. inclusion (cpx a)"'(Gx)

LEMMA 20.7

Let x E G*, a E H*, x' = XI,.,, a' = al,.,. Then 7rH(dxla(G*)) = dx,la,((Hxv)*).

It is Proof Let I. = la(G*), IQ. = 1a,((Hx'')*), (P = (Px.a, (P' = enough to show, on the strength of Lemma 20.6, that for k > 0: nH((Pk(Gx)) = ((P')k(Ann(Hxv, x')).

(4)

By virtue of Lemma 20.6 it is enough to show that for k > 0: nH((Pk(Gx)) = ((,')k(Ann(Hxv, x')).

(5)

It has been shown above that nH(Gx) = Ann(Hxv, x'), hence the equation (5) is true for k = 0. Let us suppose that we have proved (5) already for

some k>0and gk E (Pk(Gx) _ ((P')k(Ann(Hx'', x')) O V,

gk = gH + gv, g' E (9')k(Ann(Hxv, x')), g,, E V. We want to prove that

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223

nH((p(g)) = cp'(gH) a consequence of which is the equation (5) fork + 1. According to the definition of the operators cp and (p' we obtain pp(gk) = {gk+1, (ad

(p'(gH) =

g+1)* x = (ad gk)* a} ,

{gH+l, (adH gH+1)* x' = (adH gk)* a'}.

Let gk+' E cp(gk), which means that the following equations hold:

(adgH+l)*xH + (adgV+l)*xv = (adgH)*a,

(6)

(ad gH+')* xv = 0.

(7)

We obtain from

(7) gH+ 1 E Hxv, from which it follows that ((ad gH+ )* xH) I H=v = (ad gk,+' )* x'. As for gH, gH E H"' so that ((ad gH)* a)I H=v = (ad gk )* a'. In addition, ((ad gV+')* xv)I H=v = 0,

therefore restricting elements of G*, which appears in (6) to Hxv, we get gk+ 1 E gp(gk) if (ad gk,+' )* x' = (ad g1)* a'. Thus, then nH(gk+1) = gH+1 E rp(g'). On the other hand, let gH+' a (p'(gk), then it is possible to choose gk + 1 E (p(gk) in such a way that 7CH(gk + 1) = gH+' And indeed, it follows from gH+I e gp'(gk) that

((ad g')* a - (ad

gH+ 1)*

xH) I H=v = 0.

Therefore (Lemma 20.5) there is an element gkv+' such that (ad g ,+ l) * x v = (ad gH)* a - (ad gH+' )* xH . k+1 and n1(gk+1)=g'+'. Thus E co(gk) + nH((p(gk)) = 4'(gH) and equation (5) has been proved for k + 1. The

Hence

gk+1 =

gH+1

lemma is proved.

We obtain from Lemma 20.7 dxI,(G*) + V = dx,Ia,((Hxv)*) O V,

therefore

dim(d., I, + V) = dim dx,I', + dim V = '(dim Hxv + ind Hxv) + dim V

='(ind Hxv + dim G + dim Hxv + dim V - dim H). Note now that ind p* = dim V - dim H/Hxv, therefore

dim(dxI, + V) ='(ind Hxv + dim G + ind p*) ='(ind G + dim G)

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A. T. FOMENKO AND V. V. TROFIMOV

in accordance with Theorem 20.3. That means that I. + V is a completely involutive set on G*. The theorem is proved. COROLLARY

Let G be the extension of a compact Lie algebra H

determined by a representation p: H -p gl(V), Then for an element a e H* in general position the set 1,(G*) u V is a completely involutive set.

Proof For any y E V* the subalgebra HY is reductive. Therefore, Ia.(H'')

is a completely involutive set for an element a' c- (HY)* in general position. That polynomial invariants exist follows from the fact that [G, G] = G. REMARK The stabilizers Hy of elements yin 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. 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 % be 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 = g x f, g e (fi, fEG*, ad* f = x f, E G, fEG*. The number r = ind G = dim G* - sup fEG* dim 0(f) is called the index of Lie algebra, where 0(f) is the orbit of the coadjoint representation passing through f E G*,

r= ind G = inf dim G(f) where G(f) = { E G ad* f =c x f =01. The point f e G* is called a point in general position if dim 0(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 6o and .5 be the connected Lie subgroups of (fi 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*, Go = { f E G* I f I H ° 0} and H* is isomorphic to H* _ {h E G* I hlGo = 0} c G*. We can therefore consider G0* and H* as subspaces of G*. The representations Ad*:60 - GL(G0*) and ad*: Go End(Go*) are defined. We introduce the notations Ada f =

g Of if g e 60, f c -Go* and ad* f = ®f if E Go, f c G0*. Thus, if

fEG**cG*, ge60c6, EGo CG then x fEG*, ®fEGo*, g x f E G*, g® f c- Go and, generally, x f and ® f as well as g x f and g ® f do not coincide in G*. Let n0 be the projection of G* onto Go along H* (n1 the projection of G* onto H* along Go), then we obtain the following simple relations

no( x f) = Of, EGo, LEMMA 21.1

na(g x f) = g Ox .f,

.fEGo,

gE60.

(1)

Let h e H*, fEG*, then O(f + h) = 0(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 We have g x (f +h)= g x f+ g x h, g E 6, f E G*, h E H. As H is a commutative subalgebra and Go an ideal in G, g x h = h for all

g c 6, i.e. 0(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 H* in G* be obtained by translating space

H* along the vector f, i.e. f' E H* if and only if no(f') = f. Then 0(f) n H* is a subgroup with respect to addition in H* . Indeed, if f + h; E Off ), h; E H*, i = 1, 2 then it follows from the

lemma that f + hI + h2 and f - hl belong to 0(f). COROLLARY 2 It follows from the lemma that f c- G* is a point in general position if and only if n0(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*o such that for any f e Wo, f is, at the same time, a point in general position both in G* and G*o. Let f c- W0 c Go, i.e. f is a point in general position both for G* and

Go, and let 00(f) be an orbit of the coadjoint representation of 60 on Go, i.e. 00(f) = {6 ® f I g e 60} = 60 ® f. Then the tangent space to ®f I E G0} = Go ®f. 00(f) at the point f is the space Tf 0(f) On the strength of equation (1) we have

226

A. T. FOMENKO AND V. V. TROFIMOV

00(f) = no(u(f)),

(2)

where

u(f)={gx Go ®f = Tf00(f) = 7C0({c x f,

E GO})

LEMMA 21.2 Let f c- W0 c G**, h e H*. Then if x f = f + th. for all the t e E8: (Exp

7ro(Go x f).

x f = h, e G0 then

Proof If h = 0, then our assertion follows from the definition; we can thus assume h 0.{ 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* = Gi + R h (i.e. by considering a new decomposition of G as the direct sum of an ideal G1 and a one-dimensional subalgebra). Let T(t) = Exp(tf) x f be a curve in G*. We have to prove that y(t) = f + th. It follows from the condition x f = h, e G0 and equation (2) that ® f = ic0(h) = 0, therefore (Exp t t a 60. But n0[(Ex0 x f ] _ (Exp ®f, therefore y(t) _ (Exp x f E H f. The latter means that y(t) = f + a(t)h, a(t) E R. Let t1, t2 e ER, then

y(t1 + t2) = [Exp(t1 + t2) ] x f = (Exp t1) x [(Exp i2) x f] = (Exp t1) x If + a(t2)h] = f + a(t1)h + a(t2)h because

(Exp t1) x h = h. Therefore a(t1 + t2) = a(r1) + a(t2). 0 = h or i(t)j, 1, therefore a(t) = t, which was to be

Moreover y(t)1

proved. COROLLARY

Let f c- W 0 c Go, then 60 x f

{ f+ (G0 x f n H*)}

where (j0 is the Lie group corresponding to G0. There is a subspace Vo c H* such that Vo = G0 x f n H*, f e W1 where W1 is open and everywhere dense in G. In particular, Vo c O(f) n H f for any f E W1. LEMMA 21.3

Proof We shall prove the lemma using the method of induction. Let G

be decomposable as a direct sum of an ideal G0 and an Abelian subalgebra H, dim H = 1: G = G0 + H. Consider the restriction of the coadjoint representation on G* to 60, then G* is partitioned into the orbits of this action60 x f, f e G*. Similarly to Corollary 2 of Lemma

21.2 we have: f E G* is in general position (60 x f has maximal dimension) if and only if 7r0(f) is in general position for this action. This means that there is in Go an open everywhere dense set W' of points in

INTEGRABLE SYSTEMS ON LIE ALGEBRA

227

general position for the action of 60 on G*. Let Wl = Wo n W'. It is obvious that W1 is open and everywhere dense in Go also. Let us take any two vectors fl, fl e W, then fl, f2 are in general position both for G* and G*, i.e. dim(G0 x fl) = dim(G0 (9 f2) and dim(G0 x fl) = dim(G0 x f2). Suppose that G0 x f1 n H* A 0 thus G1 x fl n H* = H* as dim H* = 1. According to (2) we have: implying n0(Go x fl) = G0 ® f dim G0 0 f1 = dim G0 x f - 1, therefore

dim G0 ®f2 = dim G0 ®f1 = dim G0 x f1 - 1 = dim G0 x f2 - 1, but G0 ® f2 = n0(G0 x f2), thus G0 x f2 n H* = G0 x fl n H* = H*, which means that there is a subspace Vo c H* (here either Vo = 0, or Vo =H*) such that G0x fnH*= l' for all the feW1. Suppose we have proved the lemma for the case dim H = m. Consider the decomposition G = G0 + H, G0 an ideal, H an Abelian subalgebra, dim H = m + 1 . Let us choose in G a basis ell e2, ... , ejo,

e0+1..... ejo+m+I for H. We denote the conjugate basis by G* as eI , ejo+m+I where e'(ej) = 5 , i, j = I.... , j0 + m + 1. Consider G1 = Go + <ejo+I, .. , ejo+m) = G0 + H1 where <ejo+1, ...., e,,+,> is the subspace of H with basis ejo+I, ... , ejo+m It is obvious that G1 is a Lie algebra which can be decomposed as the direct sum of an ideal G0

and an Abelian subalgebra H1, dim H1 = m. In accordance with the

inductive hypothesis, there is V0* in H* = <ejo+I, .... ejo+m> such that of/the G0 x (1) f n Hi = V0* where i; x (1) f denotes the action vector e Go on the vector f c- Wl c Go under the representation of G1 in Gi, where Wl is an open and everywhere dense set in G. According to the construction, we have a decomposition

G=G1+<en>=G1+Re,, =G1+H2, where G1 is an ideal in G, H2 an Abelian subalgebra, dim H2 = 1. In the same way as when dim H = 1, we find in Go an open and everywhere dense subset W' such that dim(G0 ® fl) = dim(Go ®f2),

dim(G0 x fl) = dim(G0 x f2) for all the fl, f2 E W', but G0 ®f = n0(G0 x fl), i=1,2, therefore dim(G0 x f1 nH*)=dim(G0 x f2 r) H*) for all fl, f2 e W c G*0. Write W1 = W' n W1, then W1 is the open and everywhere dense set in G. Let TII be a projection of G* on Gi along H2, then from relation (2) we obtain :

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A. T. FOMENKO AND V. V. TROFIMOV

Go x(1)f = nt(Go x f),

(Go x(1)f)nHip=n1[(Go x f)nH*],

i= 1,2, f1,f2EW1.

We have the following equations:

dim[(Go x fl) n H*] = dim[(Go x f2) n H*]

[(Go x(1)f) nHi] = n1[(Go x f) r) H*],

i = 1, 2;

dim[Go x (1) fl) n Hi] = dim[(Go x (1) f2 n Hi] = dim Vo for any f1, f2 E W1. But H* = Hi + H2* and dim HZ = 1, therefore it follows from these equations that there is a subspace Vo of H* such that Go x f n H* = Vo for all f E W1, which was what we had to prove. We introduce the notation k = dim Vo , where Vo = Go x f n H*, f e W1. It follows from Lemma 21.3 and Corollary 1 of Lemma 21.1 that for f e W1 there is a basis e1, ... , ego, ego+1, . . . , e,, of G such that Go = <e1, e2, ... , e,0>, H = <eio+1, ... , e.) and REMARK

k

60 x f n H* = f + > Re* +1 + i=1

s(f)

7Le*+k+s

s=1

where k = dim V* does not depend on f and Ek= 1 Re*+ 1 = V*. Let, as before, G be expanded as the direct sum of an ideal Go and an Abelian

subalgebra H: G = Go + H, let 6, 60,.5 be the connected Lie groups corresponding to these Lie algebras. Consider the restriction of the coadjoint representation 6 to G0*, so that 6 acts naturally on G0*. We denote by 6 ® f for f E Go the orbit of the point f under this action.

Obviously, 60 ®f contains the orbit 0,(f)=000f of the coadjoint representation of 60 on G. According to equation (2) we have 6 (& f = no(6 ® f). There is in G*0 an open and everywhere dense set W2 of points in general position for the action of (5 on G. We denote its intersection with W1 by W: W = W1 n W2. Then W is also an open and

everywhere dense set in G. In a way similar to Lemma 21.3, the following lemma can be proved. LEMMA 21.4

There is in the space H* a subspace V*(V*

Vo ),

dim V* = k + I such that for all f E W we have:

6x fnH*=V*.

(3)

It follows, obviously, from Lemma 21.4 that V* may be written as a direct sum of two subspaces V* = Vo + Vl*, where Vo is as constructed

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

in Lemma 21.3, dim V1* = 1.

Let G be any Lie algebra, n = dim G, r = ind G. Let m smooth functions be defined on G*:F1(f),...,Fm(f), F.(f)eC°°(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 F1, . . , F. are mutually in involution on all orbits, i.e. (Fi(f ), F1(f )} - 0, 1 <, i, j 5 m, rk(dF1 (f), . . . , dFm(f) 3 (n + 2)/2 on all orbits in general position. Let G = Go + H, and let functions F(f) be defined on Go* c G*, then F(f) .

can be extended to functions F(f) on G* by setting F(f + h) = F(f), where f e Go, h e H* (see Section 12). The following theorem is the main theorem of this Section.

Let G be a Lie algebra decomposable as a direct sum of

THEOREM 21.1

an ideal

Go and an Abelian subalgebra H: G = Go + H. Let

Fl (f ), ... F.(f) be a completely involutive set of functions on Go. These functions Fl , ... , Fm can be naturally extended to functions F1, ... , Fm on G* (see Section 12). Then we claim that the class of functions F,,. . , F. 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 * VV,dim VV =k, dimV*=k+1=s(V*= Vo +V*,

dimV*=1), Gx fnH*=V*, Gox fnH*=V* for any feW, where W is an open and everywhere dense subset in G. As the functions

Fl, ... , Fm are mutually in involution, their extensions F1, ... , F. to G*, as proved in [127] (see also Section 12) are in involution on G*, i.e. 0 = {F;(f ), F;(f )} for all i,j. Obviously, df;(f) = dF;(f) E Go for any f e Go, 1 , (jo + ro)/2 = Mo, where jo = dim Go and ro = ind Go. In order to prove the theorem it is necessary to check the inequality

rk(dF1(f),... ,dF.(f)) = q %

n

2

2

- (n -jo - s)

n + r - 2n + 2jo + 2s 2

-

2jo+2s+r-n 2

for which it is enough to show that Mo

=Lo

+ro 2

2jo+2s+r-n 2

(4)

We shall prove the inequality (4) using induction. Let us prove it first for

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A. T. FOMENKO AND V. V. TROFIMOV

s=Oands= I. Lets=0,i.e.G x fnH*=0 foranyfeW.AsfcW, f is in general position both for G*o and G*. For linear subspaces Go ® f and Go x f we obtain from equation (2): Go ® f = 7ro(Go x f) (Go ® f is the tangent space to the orbit 00(f) of the coadjoint representation of

(50 in G*o). As Go x f n H* = 0, we have dim Go ® f = dim Go x f and dim Go ® f = dim Go x f < dim G x f = n - r. Therefore

jo+r 2j0-dim Go®f 2jo+r-n Mo

2

-

2

2

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 = 0. In the first case dim(G0 x f n H*) = 1, but Go Of = 7r0(Go x f), therefore

dim(G0 ®f) = dim Go x f - 1 < n - r - 1. In the second case dim(G x f n H*) = 1, dim(Go x f n H*) = 0, therefore dim Go x f < dim G x f - 1. And, as a consequence, dim Go ® f = dim Go x f
Mo = z(2jo - dim Go ®f) >(2jo + r + 1 - n), but Z(n + r) is an integer, therefore, z(n - r) is an integer too, hence

M0> (2jo+r+2-n)=z(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 3 2) and let us prove it now for s = so. Let so = k0 + l0; consider the two cases: (a) k0 0, (b)

k0=0,10=s0

0.

The first case. We choose for G a basis e,, e2, ... , ejo, ejo+, , ... , e Go = <e1, ... , ejo>. Let

such that G = Go + H, H = (ejo+I, ... ,

H'=(eo+i, -e.>, e,*eGox f r) H*. Write G°(el,...,ejo,ejo+ H, = G = G1 + H1 then G1 may be decomposed as a direct sum of an ideal Go and an Abelian subalgebra H2 = G1=Go+Hz. Let G1x(1) f be the coadjoint representation of Lie group 6, in G'. In accordance with equation (2): Go x (l) f* = 7rG,(G0 x f), where ire, is the projection of G* on G' along H' and Go x(1) f nHz = n6l(Go x f n H*). The construction is such

that Hi =

c Go x f n H*, therefore

dim(Go x (1) f n HZ) = dim(G0 x fnH*) - 1 = k0 - 1.

INTEGRABLE SYSTEMS ON LIE ALGEBRA

231

Similarly, from the fact that G1 x (1) f = it,,(G1 x f) it follows that dim[(G1 x (1) f) n H*] = 1o + ko - 1 = so - 1. Therefore, according to the inductive assumption, we have Mo Z(2j0 + 2(s0 - 1) + r1 - n1), n1 = dim G1, r1 = ind G1 but -dim(G1 x(1) f) 3 -dim(G x f) + 1, therefore nl = n - 1;

r1 = nl - dim G1 x(1) f > r. Thus, Mo%i(2jo+2(so- 1)+r-n)+Z.

i(2j0 + 2(so - 1) + r - n) are integers we Having noticed that M0 and obtain Mo ?(2j0 + 2s + r - n). Likewise, for the second case, when ko = 0, l0 = so: Hi = <e*> = G x f n H*, dim(G0 x(l) f n HZ) _ k0 = 0 and dim(G1 x (1) f n HZ) = 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 = G0 + 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 F1(f ), ... , Fm(f ) (m = '(n0 + r0)); the corollary follows from this assertion and from our theorem.

Let G be any Lie algebra of dimension n, r = ind G, and let I1, ... J, be a complete set of invariants for G (i.e. I. is invariant under the coadjoint representation). It is obvious that for any F(f) E C`°(G*):

{F(f),1;(f } - 0, i = 1, ... , r. This means that if a set of smooth functions F,(f ), ... , F(f) on G* is complete and commutative, then the set F1,... , Fm, Ii,. .. ,1, is commutative. From Theorem 21.1 immediately follows the theorem:

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 I1(f),. . . , I,(f) be a complete set of invariants for G(r = ind G). Let F1(f ), ... , Fr(f) be a completely involutive set of smooth functions on THEOREM 21.2

G. Then the set F1(f ), ... , Fm(f ), 11(f),. .. , I,(f) is a completely involutive set on G, where F1(f ), ... , Fm(f) are the liftings of functions F1 , ... , Fm to G*. COROLLARY 1

Let G be decomposable as the direct sum of a nilpotent

232

A. T. FOMENKO AND V. V. TROFIMOV

ideal and an Abelian subalgebra : G = Go + H. If there is a complete set of invariants for G, then a completely involutive set of functions on it exists.

Let G = Go + H, Go a nilpotent ideal, H an Abelian subalgebra and r = ind G = 0, then a completely involutive set of

COROLLARY 2

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 of functions on G for Borel subalgebras BG.

Let G be the semisimple complex Lie algebra, BG = Q+; Rh; ED Y,,,, Re. be a Bore] subalgebra in G, coo be the element of the group Weyl of the maximal height (see [11]). If 0 be the orbit of the maximal dimension of the representation, coadjoint then codim 0 = i card A, where A = {a EA I (-coo)a a}; A be the set of the simple roots of the Lie THEOREM 21.3 (Trofimov, V. V., [126], [127]).

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(AdB f) = X(g)F(f) for any g E 05, f E G* where X(g) is a character of the group 05 and Ad* is the coadjoint representation of the group (5 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 0(f) is in involution with a semi-invariant F(f) if and only if = 0 for all f c- G* (doff) E G). Let the semiinvariants F(f) and t(f) of the algebra G be in involution

{F(f ), 0(f)) = 0, f e G*, then for any h c- G* and any 2, p e R the functions Fz,,,(f) = F(f + Ah), 0,,,ti(f) = Off + ph) involution {FA,h(f ), 4,,,h(f )}

0.

are also in

233

INTEGRABLE SYSTEMS ON LIE ALGEBRA

Let {el , ... , en } be a basis in G; let G - be the structure tensor of the Lie algebra G with respect to this basis. Denote by (f1 ,f2, ... , fn) the system

of -coordinates in G* relative to

(e1, e2, . . , e") where e'(ej) = S;, i , j = 1 , ... , n (el, ... , e") is the conjugate basis in G*. Then for any .

vector f c- G* we have dim O(f) = rk Ckj fk II , where 0(f) is the orbit of the coadjoint representation passing through f, and f = Y_h- 1 fie' gives f with respect to the basis (e1, e2, ... , e"). Let M(n, l) be the space of all matrices with n rows and n columns. Define the matrices Toj, by Tjo = (olio bjj.), i, j = 1, 2, ... , n. Then the T j, i, j = 1, 2, ... , n form a basis for the space M(n, 68):

M(n, R) _ Y7 J_1 RTij. Let T. be the space of all upper-triangular matrices T. = [_1,i,j," RTij. T. is a Lie algebra. Using the scalar product (x, y) = T,(x, y), x, y c M(n, l) it is possible to identify the space T,* with the space of all lower-triangular matrices Y_ 1 <j
11

i2'

-

,i'

(il'j2.....)p/

be the minor at the intersection of the rows

numbered i1,. .. , ip and the columns numbered jl, ... , jp in the matrix X. Consider the following functions on Tn* SP,k(x) =

XIl'...,ip,n-k+ 1,...,n 1' 2'...' k, ll,.

k
.

.

, lp

p36,k_>0,p+2k_
s0,0(x) = 1, x e T,* then SP.k(x), p = 0, 1, 0 < k

[(n - p)/2] are semi-

invariants of the coadjoint representation and are mutually in involution (see [10]). Moreover, if A E T"* is a point in general position (in particular A = x0 = T",1 + Tn_1,2 + + T--E,/21-1J./21) then the coefficients of the terms in ) of the polynomials Sp,k(X + AA), p = 0, 1,

0 < k < [(n - p)/2] form completely involutive set of functions (see [10]).

Consider Lie algebras of the following form: L = V + Y-1,i<j,n lTj, where V is any subspace of the n-dimensional space of diagonal matrices V c_ 1 RTi. It is obvious that L is a Lie subalgebra of T. and L* = V + E1 sj
j

n be a system of coordinates on the space T"* with respect to the basis T j (1 < j < i < n). Then there is a set of polynomials P = { pi(x,,#), i = 1, 2,... , Z(dim T" + ind Tn), 1 jl S a < n) on T"* such that (a) P is a completely involutive set of LEMMA 21.5

Let (x1 ), 1

i

234

A. T. FOMENKO AND V. V. TROFIMOV

functions; (b) p;(x,p) = x,, + x;j, i = 1, 2,... , [(n - 1)12],j = n + 1 - i, pk(x,0), k > [(n + 1)/2] does not depend on the variables x,,,

a= 1,2,...n.

Proof By properties of the Poisson bracket, we have {F1, F2F3} = {F,, F2}F3 + {F,, F3}F2 for any three functions F,, F2, F3. It follows from this that if F1, F2, ... , F. is a completely involutive set of functions

on Tn* then the set of functions F,......

m where Fi = F. for any i # io, i = 1,... , m and either Fia = F,0 + f (,)-(1.2.....m) F, or

Fi0 = F10 - f](,}-{1, 2,....m) F. will satisfy the same condition. Let us take xo = T,,,, + + T. _(n,'2]+ 1,[n/2] The coefficients of terms in A of the polynomials Sp,k(x + Axo), p = 0, 1, 0 < k S [(n - p)/2] have the desired properties. The lemma is proved.

THEOREM 21.4

Let L be a Lie algebra of the form L = V +

L < <.,n 68T j, where V is a subspace of the space of diagonal matrices V c + _ , RT,,. Then there is on L * a set of polynomials q, , q2, ... , q,, m = Z(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*.

Proof According to Lemma 21.5 we have a set of polynomials P = { p;(x,,,)} on Tn* , such that p; = x,, + x11, i = 1, 2, ... , [(n + 1)/2],

j = n + 1 - i and pk(x,fl), k > [(n + 1)/2] does not depend on the variables x,,, a = 1, 2, ... , n. In addition, {p;, pj} - 0 for all i, j = 1, 2, ... , M = J(dim T. + ind Tn). Consider in the subalgebra L of the algebra T. the element xo = Tn, I + + T. _ [,,/21+1,[./2] E L*. As this element xo E Tn* is in general position in this space, ind Tn = are the structure constants of T,, with where respect to (Tj), 1 < i < j < n; letting (x° V) denote the coordinates of xo in Tn* we have the following formulas: C30V,19 = (SeyS,& - S,BSySfl)(1 - 60,6,)

C)(1 - aflybz9 uv

0

n+1

Cae,rexno = (bflya2+e -

(5)

6,06n+1 n+1

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 + Y-,
T..

I ,;Ti,a= 1, 2,... , k form a basis in V. As xoeL* one may

235

INTEGRABLE SYSTEMS ON LIE ALGEBRA

consider the matrix II C"v,,,ex° II , where C"'.,,9 are the structure constants

of L with respect to the basis (Tj, 1 < i < j < n, T,,, a = 1, ... , k), dim 0(xo), where O(xo) is an orbit of the coadjoint representation of the Lie algebra L. It follows from (5) that cork 11 Caa,ve xuV 11 = cork I I C,p,ye II = cork A,

where the matrix A has the form (n = 2m): 1-1

T1,r

1-1

T.,.

0

0

T2,2

0

T.,.

A=T.+iA.+1 T2.,i

1-1 1-1

0

0

0

It is obvious that cork A= ind T = m. In addition cork 1 Cie,rex°V II = cork 11C,R,eII = cork B where the matrix B has the form:

Ti.I /0

C

T.,%

B=

Tm+I.m -C C

0

andC=(C,,6y),1
each row 0, of the matrix C is a linear combination of rows

of the matrix D, where A = set { , ; } ,

( fro) with coefficients taken from the

1 < a< k, 1 < i < n. Let B, , ... , 0, (1= rk C) be the l linearly

236

A. T. FOMENKO AND V. V. TROFIMOV

independent rows of this matrix (1 < s1,... , si <

k). Let ((z,o),

1 < ft < a n, (z°a), a = 1, 2, ... , k) be a system of coordinates in L* and for 1 s < k, s # si, 1 i l suppose 0, = =1 ai 0,,. By dint of the construction of the matrices D and C we have:

ji

i i

jo) = bsio - l;,Jo,

i=1

io + ja = n + 1.

(6)

For each 0, consider on L* the first degree polynomial qe, = Yi=1 aiz,;, - z then it follows from (6) that q0, _

i=1

fi(xii + xn+1

+1

Yi E R

(7)

As the polynomials pk(x°,5) fork > [(n + 1)/2] = m on Tn* do not depend on the variables x,a, 1 a n, one may take these polynomials to be defined on L*: pk(x°p) = qk(i ). Thus we have defined on L* a set of {g9,(x11' .. xnn), polynomials s # si, i = 1, 2, .. , 1; q&4),

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 Z(dim L + codim O(xo)) of them. But ?(dim L + codim O(x0)) , 2(dim L + ind L).

Thus the theorem is proved.

22. INTEGRABILITY OF EULER'S EQUATIONS ON SINGULAR ORBITS OF SENII-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

tHR, teC

Let G be a semi-simple Lie algebra. An element a E G is called semisimple if the endomorphism ad° is semi-simple. In this case G = [G, G] Q G°, where G° is the centralizer of a in G. The restriction ad°I Ic,c] is invertible. Thus the mapping ada 1: [a, G] - [a, G] is well

INTEGRABLE SYSTEMS ON LIE ALGEBRA

237

defined. Let b e G°, then (adb)([a, G]) [a, G] and therefore the mapping ada 1 adb: [a, G] - [a, G] is defined. Let D: G° - G' be an operator symmetric with respect to a non-singular Ad-invariant form Q. We define operators PabD by the matrix

/,

_

`YabD -

ada 1 adb

0

0

D)

(see Section 7) with respect to the decomposition G = [a, G] ® Ga. PROPOSITION 22.1

The decomposition G = [a, G] @ G° 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 G° of a is also a non-singular form. PROPOSITION 22.3

Suppose that the hypotheses of Proposition 22.2

are satisfied, and that b E G° and D : G° -> G° is an operator symmetric with respect to Q, then the operator cpahD: G -> G is symmetric with respect to Q also. Let G be a Lie algebra, a e G a semi-simple element, Q an invariant symmetric bilinear form on G, G° the centralizer of a, Cent G° its center, D: Ga --+ G° an operator symmetric with respect to Q, Q the restriction of Q to G°, and f1(Y),... , f,(Y) integrals (in involution) of the equation of motion Y = [Y, D(Y )], Y E Ga. Then (a) if b c- Ga, Euler's equations of motion x = [x, PabD(x)], x c- G have integrals J(x + .la) f (Y), where J e I(G), A e R; (b) if b E Cent G°, the functions where Y is the projection of x onto G° along [a, G], are integrals of the LEMMA 22.1

equation of motion; (c) all the above-mentioned integrals of the equation of motion x = [x, (pabD(x)], x c G are in involution with respect to the Kirillov bracket transferred to the Lie algebra G using Q. Let a be an involutive automorphism of the Lie algebra G, G* the space dual to G, a* the automorphism of G and G* arising from a and a* (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 a and, using <X, Y> to identify G with G*, G** is identified with Go, Gi

with G1, where G1 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), a an involutive automorphism of G, G = Go p G1 the decomposition of G arising from

238

A. T. FOMENKO AND V. V. TROFIMOV

a; let f , g be invariants of G, a E GI, A, µ e ll ; let fa(x) = f (x + .la), g,,,a(x) = g(x + µa) be their translates, fx a, g a the restrictions of fl,a, g,,,a to Go, Ga the centralizer of a, Cent Ga its center, Go = Ga n Go,

D : Ga - G a, an operator symmetric with respect to the restriction <X, Y> of (X, Y> to Go, f1(Y),... , fk(Y) integrals in involution (under the Kirillov form transferred to Go by <X, Y>) of the equation of motion

3' = [Y, D(Y)], Y E G. Then we claim that (a) if b e Ga n GI, then functions J(x + Aa), J E 1(G), 2 E l are integrals of the equations of motion z = [x, (pabD(x)], x c Go; (b) if b c- Ga n G1, the functions f(x) = f,(Y), where Y is the projection of x onto G a along [a, G1] are also integrals of x = [x, (pabD(x)], x e Go; (c) all the above-mentioned integrals of the equation x = [x, (pabD(x)] : J(x + Aa) and f(x) (for b c (Cent(Ga) n G1)) are in involution on Go under the Poisson bracket, which corresponds to the restriction of <X, Y> to Go.

The proof of these statements is standard and we omit it. We note a further property of the center of the centralizer Ga. Let G be a real or complex semi-simple Lie algebra, a c G semi-simple element, Ga the centralizer of a, Cent Ga its center, <X, Y> LEMMA 22.3

an invariant non-singular symmetric bilinear form which we use to

identify G with G*, fl, ... , fk generators of I(G), the algebra of invariants. Then the differentials dfl(a), ... , dfk(a) generate Cent Ga.

Proof (1) The complex case. Let H be a Cartan subalgebra of G. Then, as is well known, the restriction mapping j: 1(G) - S(H) where S(H) is the algebra of polynomial functions on H, is an embedding j(I(G)) = S(H)W where S(H)W is the subalgebra of S(H) comprising the elements invariant under the Weyl group W. Let b c- Cent Ga, Wa the stabilizer of a in W, Wb the stabilizer of b in W. We have then Wa c Wb. Let {a1, a2, ... , aa} be an orbit of a under the Weyl group. We choose a positive function g on H in such a way that dg(a) = b and dg(a;) = 0 for a;

a. Let g = n ZW E W g - co. Then d4(a) = b and g c- S(H) W. Therefore,

f = j -1(g) is an invariant of G such that df(a) = b. Thus we have shown

that if fl, , f are generators of the invariants of the algebra G, f = P(f1, ... , fk) for a suitable polynomial P. Therefore, . . .

b = E;`= I OP/8f d f (a), which was what we had to prove.

(2) The real case. Let G be a real Lie algebra. Consider the complexification G, of G. Then G is a real form of Gc; let a be the

conjugation on G. Let r = rk G and fl, ... , f, be generators of

INTEGRABLE SYSTEMS ON LIE ALGEBRA

algebra

I(G).

Let

239

9,+I=

( , / -l ) _'(fi - J ° Q),

92, = (.,/- l)-'(f2 - f, ° u), where the line

denotes complex conjugation. Then g, , ... , 92r c- I (G,) and all of these are real on G, therefore their restrictions to G, which we also denote by 91, ... 1 92,, are invariants of G. Let g be an invariant of G. Let gc be the complex extension of g to Gc. Then gc is an invariant of Gc and gc = f2) for a suitable polynomial P. As f, = g, + 9r+I,

f, = 9, + 1/ -192x, gc and g are polynomials in g,,... 1 92,. Let G' be the centralizer of a in Gc and Cent Go its center. Then Cent GaC _ Cent G° + /_-1 Cent G° and any element b e Cent G° is a linear combinations of differentials df, (a), ... , df,(a) with complex coefficients

by dint of (1), therefore, b is a linear combination of differentials dg, (a), ... , dg2r(a) with real coefficients which was what we had to prove.

Let G be a semi-simple Lie algebra over the field k = l or C; H a splitting Cartan subalgebra of G; R = R(G, H) a root system of the split Lie algebra (G, H). For any root a c- R let G° = {x c- G: [h, x] = a(h)x for all the h e H}; the dimension of each of the vector spaces G', [G', G-'] is equal to one. For any root a c- R the space [G', G -'] is contained in H an element H° E [G1, G-'] is uniquely defined by the condition a(H2) = 2. We define the real subspace HR in H in the following way HR = LER RH°. Note that in case of k = R we have H;, = H. Let (G, H) be a split semi-simple Lie algebra over the field k = l or C; 0 an orbit of G, intersecting the set tHR where t c- k; let a c G be a semi-simple element, G° its centralizer, b c- G°, Q a nonTHEOREM 22.1

singular invariant symmetric bilinear form on G, D: G° -* G° a symmetric operator with respect to Q. Then Euler.s equation of motion X = [X, (pObp(X)] ,

XE0

(1)

has integrals in involution J(x + Ia), J E I(G), 2 E R, 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 [26]). 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 at c- B. Suppose h = E°ER a2H°. For any root a e B denote by b,, and c° scalars such that

240

A. T. FOMENKO AND V. V. TROFIMOV

b,c, = a, and let x, a G", x _, e G where [x x _,] = H8, U = Y_,EB b,x, v = Y_8EB cax_s, S = ku + kh + kv. We claim that (a) [h, u] = 2u, [h, v] = -2v, [u, v] = h, with dim G" = dim G" = r; (b) consider G as an S-module under the adjoint representation. Let G = Al Q . p A. be some decomposition of this module as a direct sum of simple S-modules of dimensions v1 + 1, ... , v" + 1, where v1 < < v". Then n = r; (c) let J1, ... , J, be homogeneous algebraically independent generators of the algebra of invariants 1(G) of degrees

m1 + 1, ... , m, + 1, where m1 < < m,. Then vi = 2m, for any 1 < i 5 r; (d) differentials of functions J1, . . , J, are linearly independent at any point in the set u + G For the element h of this lemma all the eigenvalues of the .

endomorphism ad,, are even. For an integer n let G" be an eigenspace of ad,, for the eigenvalue 2n. This subspace is called the n-th diagonal of the Lie algebra G (with respect to basis B). We have [G`, G']

G'+'

(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 = Y-eEB mfl, where mp are integers such that a = E,,, ma - P. From the definition of the diagonals of the Lie algebra G it follows that for any integer n : 0 we have G" = " G8. 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:

R°(x) = {a a R: a(x) = 0} ,

B°(x)=BnR°(x), R° (x, B) = R + (B) n R°(x),

R'(x) = R - R°(x), B'(x) = B n R'(x), R' (x, B) = R+(B) n R'(x).

Let C = {x e HR: a(x) 3 0 for all a E B}, the closure of positive Weyl

chamber, t e k and x e tC. Then any root a in R° = R° (x, B) is an integer linear combination of roots in B° = B°(x) R'+ = R'+ (x, B) we have an embedding

(R'++B)nRuR'+.

and for (3)

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 of the positive Weyl chamber, x e tC, 0 LEMMA 22.5

an orbit in G passing through x, T = Ts0 the tangent space, T" = T n G" the intersection of T with G", the n-th diagonal of Lie

INTEGRABLE SYSTEMS ON LIE ALGEBRA

241

algebra G. Then: (a) T = 0"E, T"; (b) ads: T" -+ T" is an isomorphism; (c) (ada)(T") c Tn+1 for a = LE, xa and elements xa as in Lemma 22.4.

Proof (a) The equality T = ED. c, T" follows from the fact that T = [x, G], x e H = G° and from formula (2). (b) ads(T") c T" as a consequence of formula (2). As the endomorphism ads is semi-simple, ads: T -+ T is an isomorphism. It follows from this that ads: T" -+ T" is also an isomorphism. (c) This follows from formula (3). The lemma is proved.

Let J1, ... , J, be algebraically independent generators of the algebra of invariants I(G), m1 + 1, ... , m, + 1 their degrees, m1 < . < m,. The numbers m1, . . , m, 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,...,mt + 1): m" 1

Jt(x + .la) = Y A'J,a(x).

(4)

1=0

As J1, ... , Jr are invariants

[x + 2a, grad Jt(x + 2a)] = 0.

(5)

We obtain from (4) and (5) m;+1

Y t'([x, uf] + [a, u; -']) = 0,

(6)

where u = grad Jj,a(x) (i = 1 , ... , r; j = 1, ... , mt + 1), u; 1 = 0 (1 < i < r). As J"'Q+ 1(x) does not depend on x, u;"; + 1 = 0 and we obtain,

as a result, the following chain of equalities (see [89], [90]):

[x, u°] = 0 [x, us] + [a, u,°] = 0 (7)

[x,

[a, um; -1 ]

=0

[a, u""] = 0. LEMMA 22.6 Let (G, H) be a split semi-simple Lie algebra, R its root system, B a basis for R, G' the first diagonal of the Lie algebra G, x E H, a e G1; J1,.. . , J, homogeneous algebraically independent generators of the algebra of invariants 1(G) of degrees m1 + 1,. . ,m, + 1, where .

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A. T. FOMENKO AND V. V. TROFIMOV

<, m,. are the indices of the Lie algebra G, J,° homogeneous

m1

polynomials from the decomposition Ji(x + .la) _ ! of 'J a(x) r); u = grad their gradients, Vp = Vp(x, a) the linear span of the u such that i = 1, ... , r, j = 0,... , p. Then Vp c G° + + G".

Proof Suppose, first, that x e H is a regular element of G. In this case the centralizer Gx of the element x is equal to H. As u° = grad Ji(x) E Gx,

V° c Gx = H = G° and for p = 0 the lemma is proved. We proceed by induction. Suppose that VP -I G° + + GP-'. As a consequence of + GP, consequence of formulas (7) Lemma 22.5(c) [V,- 1, a] c G1 + and Lemma 22.5(b) we obtain -

As x is regular, by hypothesis, Vp c G° + + G". 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 Vp c G" which is not, in general, true (see [21]). LEMMA 22.7

Under the hypotheses of Lemma 22.6 suppose that

a = Y-aEB x° where for any a E B the element xa 0 0 and xa e G. Let + GP, G° be the centralizer of a, Gp = Gp n G°. Then Gp = G° + Gp

Vp.

Proof Note that u" = grad Jmo(x) = grad Ji(a). Let Wp be the linear span of those ui for which m; < p. As a consequence of Lemma 22.4(a)(d) we have dim G° = r and the gradients grad Ji(x) generate G°, therefore

the gradients u"" = grad Ji(a) (1 < i < r) are linearly independent. Therefore dim Wp/Wp _ 1 = m(p) where m(p) is the number of indices mt

of the Lie algebra G equal to p. Let A,_., A, be simple modules as in Lemma 22.4, where S = ku + kh + kv and u = a, AP = Al n GP, ...,

A; = A, n G". As GP is an eigenspace of the endomorphism ad,,, (AF n G°). Note that GP = AP p ... Q A; and GP n G° 1 i A,° n G° 76 0 only if m, = p and then dim(A? n G°) = 1. Hence dim(G' n G°) = m(p). Therefore, dim Wp = dim G. By dint of Lemma 22.6 Wp c G ,, whence Wp = G. Finally, Gp = Wp c Vp. The lemma is proved. LEMMA 22.8

Let us assume the notations and hypotheses of the

previous Lemmas 22.5, 22.6, 22.7. Then T" c Vp.

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243

Proof If p is greater than all the indices ml, ... , m, then Tp = 0 and the

lemma is proved. Let at be a root of height p and a(x) 0 0. As a consequence of Lemma 22.5(a) for any xa E G° we have [x°, a] e

Tp+1.

As a consequence of Lemma 22.5(b) there is a u e Tpsuch that [u, x] + [x°, a] = 0. Suppose, by induction that Tp+' Vp+1, then u = Y; =1 Y°+i c; u , where c are scalars depending on u. Using formulas (7) we obtain r

p+1

[u, X]+LY_ Y_ Ciu;-',a =0. rr

i=1 j=1

Therefore, x° - Y; y +1 c u -' e G. As x° E Tp and u/' e G. for all the i = 1,... , r and j = 1,... , p + 1 (by dint of Lemma 22.6) then also

-Y!=, y_=1 j + u -'i E G. As a consequence of Lemma 22.7: Gc c Vp. Hence x° E Vp. Therefore Tp c Vp. The lemma is proved. xa

Proof of Theorem 22.1 Let B be a basis of the root system R = R(G, H), (xz)ZEB a set of non-zero elements x,, e G°, a = E°EB x2, x e tHR n 0. As the Weyl group acts transitively on the Weyl chambers, we can assume, without loss of generality, that x E tC where C is the closure of the positive (with respect to B) Weyl chamber. Let J1, ... , Jr be homogeneous algebraically independent generators of the algebra of

invariants of G of degrees m1 + 1, ... , m2 + 1, J,°(x) the functions found in the decomposition J.(x + Aa) = Y; _o' 2jJ °(x) for i = 1.... , r and j = 0, ... , m;; let V(x, a) be the linear span of their gradients grad J,°(x) (1 i < r, 0 < j m;), T = TX TO the tangent space; for any integer p: Tp = T n GP, T+ = Qp,o T". 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 + a), where J is any invariant of G, A E R. 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 of the functions J(x + Aa) on the parameter a. The rest of the theorem follows from Lemma 22.5(a). The theorem is proved. Let G. be a compact semi-simple Lie algebra, Q a nonsingular bilinear symmetric invariant form on G°, element a e G°, Ga its centralizer, b c- G', D: Ga -+ Gy an operator symmetric with respect to Q, THEOREM 22.2

0 an orbit in G. Then Euler's equations X = [X, co ,,a(X)],

XE0

(8)

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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 + .la),

where J E I(G ), A e R. For a general position element a in G. it is possible to choose among these integrals independent functions, equal in number to half the dimension of the orbit 0. Proof Let G, be the complexification of the Lie algebra G., H a Cartan subalgebra of G,, R = R(G,, H) the root system, (x,),ER a Weyl basis for

G, modulo H. As all the compact real forms G, are isomorphic, we assume, without loss of generality, that

G. = v - iHR O I

¢ER,

R(xa + x -z)) O (

¢ER,

68(x, - x _ 2)) .

The space H. = V -1H, is a Cartan subalgebra of G. Therefore, the orbit 0 intersects H. Let O, be the orbit of Int(G,), containing 0. As 0

intersects H. Or intersects H. =HR. According to Lemma 22.1 we can choose z dim, 0, = i dim, 0 independent functions among the complex-value polynomial functions J(x + Aa), where J E 1(G), 2 e C for any element a E G, from a Zariski-open non-empty subset of G. Since

manifold 0 x G. is a real form of the manifold O, x G, (more exactly: 0 x G. is a connected component of the set of fixed points in O, x G, under conjugation, defined by the real form G,, of G,) it is also possible to choose among the functions J(x + .la), where J e I (G ), A e 18, i dimR 0

independent on 0 functions for any element a e G. in a non-empty subset of G. open in the Zariski topology. The assertions that J(x + Ia) where J E I(G ), 2 E R are motion integrals of equation (8) and that they are in involution follow from Lemma 22.1. The theorem is proved. REMARK

Equations (1) and (8) are Hamiltonian with the Hamilton

function equal to habD(x) = Q(x, (pabD(x)). The quadratic form habD(x) in

Theorem 22.1 in the case of the field I is neither positive nor negative definite if b 0, and for regular semi-simple b there are always at least Z(dim G - rg G) "minuses" and as many "pluses." In Theorem 22.2 the form habD(x) may be positive definite. 22.2. Integrability of Eider's equations x = [x, q abD(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

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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 '(dim G + rk G). THEOREM 22.3

Let G be a compact semi-simple Lie algebra, Q a non-

singular invariant symmetric bilinear form on G; a e G, G° the centralizer of a in G, Cent G° its center; D: Ga - G' an operator symmetric with respect to Q, f, (y), ... , fk(y) an involutive (under the restriction of Q to G°) set of motion integrals of the equation Y = [y, D(y)],

yEG

(9)

a) For any b e G° the functions J(x + .la), where J E I (G), A E 9 are motion integrals of Euler's equation z = [x,(P.bD(x)],

x e G.

(10)

b) For any b E Cent G°, 1 < i < k the functions g;(x) = f (y), where y is the projection of x onto G° 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 f,, . . , fk 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 p G° (because of Proposition 22.1). LEMMA 22.9 (see [89], [90])

Let x c G; let V(x, a) be the linear span of

gradients grad J(x + La), where J is any invariant of G, A E R an arbitrary number. We have then V(x, a) = V(a, x). Proof As the algebra of invariants 1(G) is generated by homogeneous invariants, we may assume that J is a homogeneous invariant of degree n. We have

grad, J(x + La) _ s grad. J(x + aa) = A" grade J(a + .? -'x),

where for the function 4(x, a) in two variables x and a (x, a c- G), grad, O(x, a) (respectively grad. O(x, a)) is the gradient of the function

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A. T. FOMENKO AND V. V. TROFIMOV

t(x, a) viewed as a function of x (respectively a) alone. Hence V(x, a) = V(a, 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 V(a, x) onto T along G° has dimension dim YT(a, x) =' dim T. By Lemma 22.9 V(a, x) = V(x, a), therefore, for a general position point x E G in G it is possible to select

' dim T independent functions on the set x + T among the functions J E I(G), 2 E R. Let these be the functions 9k + where s =' dim T. As the gradients of functions

9k + 1,

g1,

- -

, gk

at any point G belong to G° and G° is Q-orthogonal to T, the

functions g1,.. ,gk+, are functionally independent on G. Let us compute their total number k + s. As f1, ... , fk are independent on G° we have, according to the definition of a completely involutive set,

k ='(dim G° + rk G°), As any Cartan subalgebra H, containing the element a is a Cartan subalgebra of G°, rk G° = dim H = rk G. Therefore

k+s='(dimG°+rkG°) +' dimT='(dimG°+rkG) +'dimT ='(dim G° + dim T + rk G) =' (dim G + rk G). We have therefore proved that g1,. .. on G. The theorem is proved.

, 9k +, is

a completely involutive set

Let G be a reductive Lie algebra, G = Cent(G) $ 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 COROLLARY 1

there is a completely involutive set of polynomial functions on G.

Proof Let Q bean invariant form on G, x1 , .. , xk a basis for Cent(G), y) (i = 1, ... , k). Let f1,. .. , ff linear forms on G, /(y) = involutive set on S, a completely be fk + 1, . , fk +, s ='(rk S + dim S). As rkG = rk S +k and dim G = dim S + k, then '(rkG + dim G) = k + s. Therefore f1,. .. , fk +, is a completely involutive set on G. -

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INTEGRABLE SYSTEMS ON LIE ALGEBRA

22.3. Integrability of Euler's equations x = [z, (PabD(x)] on the subalgebra G. fixed under the canonical involutive automorphism a: 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; Ga, H H. as in 22.1. Let (x,),ER be a Weyl basis for G modulo H. We shall define Go = HR E

G = HR O O+

68(x, + x

( zR 68x, _.))C

o (O+

x

2ER, OR, where R+ is a set of positive roots in some basis B of the system R, Ga. We have Go = G Q V the Cartan G = Go n G,,, V = Go n c V, [a, V] c G. and for the decomposition. Let a E V. Then [a, centralizer Go of a in Go we have Go = G Q Va, where G,, = Go n G, V. = V n G. Whence we obtain the following decompositions

Go =GnO[G,,,a]ED VaO[ V,a], G. = [a,

(11)

O Ga .

Let Q be a non-singular symmetric bilinear form on Go, invariant under all automorphisms of Go. Then V is 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,, and Ga is used to determine the involutivity of the functions on G,, and Ga.

Let Go be the above-mentioned real semi-simple Lie algebra, H, a Cartan subalgebra, Q an invariant form on Go, a E H, Go THEOREM 22.4

the centralizer of a in Go, Cent(Go) its center, b e Cent(Go) n V, D: Ga - C' an operator symmetric with respect to Q, fl(Y), ... , ;(Y) a completely involutive set of independent motion integrals of Euler.s equation

Y= [y,D(y)], Then Euler's equations on G

yEGn.

(12)

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A. T. FOMENKO AND V. V. TROFIMOV

xeG,,

(13)

have motion integrals J(x + Aa), where J E 1(G), A E Q8 and motion integrals gi(x) = fi(y) where y is the projection of x onto G' along [a, V]. These integrals comprise a completely involutive set.

Let B be a basis of the root system R = R(G°, HR), a E HR; x± = 1LEB X+ a, x = x + + x _ ; y, = x, ± (-1)H"Hx _,, for any root a E R of height Jai. For any integer p >,O we define Fp Rys , FO=O, F°_ = HR, FP = Ft p... Q Fr . I et J...... J, be homogeneous algebraically independent generators of the algebra of invariants of G° of degrees m, + 1, ... , m, + 1, m, LEMMA 22.10

+_+_=

m2; functions J j,,, (1 < i < r and 0 < j ` mi) are defined by the expansion J,(a + Ax) = Y, , AjJ{x(a) for any integer p > 0; let VP = Vp(a, x) be the linear span of the gradients grad J ,x(a) such that

I
Proof Suppose, first, that a e H, is a regular element of the Lie algebra G°. For such a we have V° c: Go = HR = Fo F. We proceed by induction. Suppose that Vp _, FP _ 1. As x e F1 it follows that [x, VP _ ,] c Fp . As

a EHR = F°_, we obtain Y. c FP using formulas (7) from 22.1. For regular a the lemma is proved. For singular elements a c HR the lemma follows from the continuous dependence of gradients grad J x(a) on a. The lemma is proved.

Let (G, H) be a split Lie algebra, R = R(G, H) the root system, B a basis of R, for any integer n let G" be the n-th diagonal of G. yi, where y, e G` is called the diagonal component of Let y c- G, v = E y of degree i. If yi : 0 and yj = 0 for all j > i (respectively < i), then y, is called the maximal (respectively minimal) diagonal component of y and the number i the maximal (respectively minimal) diagonal degree of y. We suppose further that all the hypotheses of Lemma 22.10 are satisfied. Let DP = grad J°, (a) - grad J?xt(a). Then either the maximal LEMMA 22.11

(respectively minimal) diagonal degree of D+ (respectively D?) is no

more (respectively no less) than p - 1 (respectively - (p - 1)) or DP = 0. Proof Let us prove the lemma for D. The proof for D°_ is the same as for D+ after changing at appropriate places (+) for (-) and vice versa. By formulas (7) from 22.1 we have [grad JP, +1(a), a] + [grad JX(a), x] = 0,

INTEGRABLE SYSTEMS ON LIE ALGEBRA

249

[grad JP+ 1(a), a] + [grad Js (a), x+] = 0. Subtracting one equation from the other we obtain [D++ +1,a]

= -[D°+,x+] - [grad Jx(a),x-]

(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 JP(a)

does not exceed p, therefore the maximal diagonal degree of [grad Jx(a), x _] 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++', a]. Suppose that aeHR is a regular element of Go, then Ker ad, = HR and ado(G') = G' (for j # 0), therefore, D++' has maximal diagonal degree no greater than p. By continuity, we deduce that the same is true for any a E H. By induction, we see that the lemma is true

for anyp>0. LEMMA 22.12 We keep the notations of Lemmas 22.10 and 22.11. Let

T = [a, bo], for any integer p ,>O let Vp be the projection of VP = VD(a, x) onto T along G. We have then T n FP = VDT.

Proof For p = 0 we have T n FP = 0 and VDT = 0 and the lemma is proved. Suppose, inductively, that FP -I n T = 1'T 1. By Lemma 22.8 T n GP c VP(a, x +). Therefore, the maximal diagonal components (of degree p) of the gradients grad J°, (a), where 1 < i < r generate T n GP. By Lemma 22.11 the same is also true of the gradients J?s(a), where 1 <, i < r. Therefore, for any root a of height jai = p with a(a) # 0 we have x, = E'= 1 C; grad J?x(a) + M., where M. has maximal diagonal

degree no greater than p - 1. As a consequence of Lemma 22.10

therefore the minimal diagonal degree of the sum C grad J°x(a) is equal to (- p) and the minimal diagonal Y; component of this sum (of degree (-p)) is equal to -(-1)"x_. J?s(a) e FP ,

Therefore the minimal diagonal degree of M, is also equal to - p and the

minimal diagonal component of M, (of degree -p) is equal to (-1)Px_,. Recall that ya = x, - (-1)Px_,. It follows from the above that yS = i= C; grad Jps(a) + N, where N. e Fp_ 1. Projecting all the components of this equation onto T along Go, we obtain yz = Y-1 -1 CQ gradT J?x(a) + Na , where grad T J?x(a) (respectively N=) is the

250

A. T. FOMENKO AND V. V. TROFIMOV

projection of grad Jf (a) (respectively of N2) on T along Go. As F_1 = (Fp_1 nT)$(F;_1 nGo) and N2eFp_1, NQ eFp_1 nT By induction T n Fp_ 1 = VpT 1(a, x), therefore NT E VpT 1(a, x) and yQ E VPT(a, x). As we supposed at the beginning that a was any root of height p such that a(a) 0 0, it follows that T n FP 1 c VVT(a, x). As a consequence of Lemma 22.10 Vp(a, x) c FP 1 therefore T n FP = VPT (a, x). By induction the lemma is proved for any integer p > 0. LEMMA 22.13

Let Go and G. be Lie algebras as in Theorem 22.4,

r = rk G0, rk G be the rank of let m1, ... , m, be the indices of the Lie algebra Go. Then the rank of G is equal to the number of odd indices among m1, ... , m,. 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 A1, B, (r 3 2), C, (r > 2), D, (r is even and r 3 4), E.,, E8, F4 or G2.

Then all the indices m1, ... , m, are odd (see [11]). Therefore an automorphism of Hs equal to (-1) belongs to the Weyl group [11]. Therefore the canonical automorphism a: Go -. Go equal to (- 1) on HR and mapping xa into x for any root a is an inner automorphism. As G coincides with the set of fixed points of a, it follows that rk G = rk Go = r. As all the indices ml,. . . , 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 3 2). Indices: 1, 2, ... , r; G. = 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 indices is equal to (r + 1)/2 and rk so(r + 1) = 2(r + 1). The series of roots of D, (r is odd, r 3 3). Indices: 1,3,5,.. . , 2r - 3 and r - 1. The number of odd indices is equal to r - 1,

G = so(r) p so(r) and rk G = r - 1. The series of roots E6. Indices: 1, 4, 5, 7, 8, 11; G. = sp(4) number of odd indices is equal to 4 and rk G. = 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, HR) is a real split semi-simple Lie algebra. Let R = R(G0, HR) be the

root system, B its basis, R+ the system of positive roots, R°dd the set of

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251

roots of odd height, Rodd = R n R°dd. For any finite set M let Card M be the number of its elements. Then

J(rk G. + dim G") = Card(R'd).

Proof We have G = Q+.,R, R(xa + x_8) and dim G. = Card R+. Therefore it is enough to prove that

rk G = 2 Card(R°dd) - dim G = 2 Card(R°+dd) - Card(R+)

= Card(R°dd) - Card(R+°)

_

m(2i + 1), i30

where R+ is the set of roots of even height and m(2i - 1) the number of indices of the Lie algebra G° equal to 2i + 1. This is exactly what was proved in Lemma 22.13. The lemma is proved. Again under the hypotheses of Theorem 22.4 we remind the reader that a is an element of HR, G0 its centralizer, G" = G" n G. Let R = R(G°, HR) be the root system of the split Lie algebra (G0, HR) and B a basis such that for any root a c- B we have a(a) > 0. Then LEMMA 22.15

i(rk GQ + dim

Card(R° r) Rod +d),

where R° is the set of roots a e R, equal to zero on a and R°+dd the set of roots at e R positive (with respect to B) and of odd height.

Proof As the element a is semi-simple in G° (see [26]) Ga is a reductive Lie algebra. Let Cent Go be its center, S = [Go, Go] is a semi-simple

ideal in G. Then Go = (Cent Ga) Q S. We have HR c Go and the normalizer of HR in Ga coincides with H. Therefore Cent G0 c HR and HS = HR n S is a split Cartan subalgebra in S. Let R = R(S, Hs) be the

root system of the split Lie algebra (S, Hs), R' the set of roots a e R

not equal to zero on a. We have Go = HR ® ($ ER° Ga) and Ga). Hence it follows that associating with each root S = Hs ® a c R° its restriction to Hs we obtain a one-to-one mapping of the set R°

onto R. Let B° = B n R° _ (a°, ... , ak), a e R°. Then for suitable integers n l , ... , nk we have a = D-1 ni a°. Let y c -A and y = a/Hs. Then y = Y+=1 ni fiiQ where fii = a°/Hs for every 1 < i < k. Therefore

B = (f1, . .

.

, Yk) is a

basis of R and the height of root a e R° with respect

252

A. T. FOMENKO AND V. V. TROFIMOV

to B is equal to the height of the root y = a/HS e R with respect to B. Let

°+, where S. = S n G. By Lemma 22.14 z(rk S + dim S,) = Card R°d R+d is the set of roots fi e R of odd height and positive with respect to B. It follows from these considerations that Card(R°+dd) = Card(R° n R°+dd).

Note that S. = QaERo R(xa + x _a) = G., where R° is the set of positive The roots a in R°. Therefore J(rk G.' + dim G.a) = Card(R° n lemma is proved. R°+dd).

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 + .la) where J E 1(G°), .? e R and the functions J(x + la) are viewed as functions of

the variable x E G°, let V(x, a) be the linear span of the gradients grad, J(x + Ia) where J e 1(G°), 1 e B and the functions J(x + la) are viewed as functions of the variable x e G. Then for any x c- 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, G°] along Go and VT (a, x) = VT (x, a) the projection of V(x, a) = V(a, x) onto T n G. = [a, V] along G', F = Fp , where the space FD is defined in Lemma 22.10 and p is greater than all the indices of the Lie algebra G°. As a consequence of Lemma 22.12 we have VT (a, x) = T n F - for x = Y-1EB (xa + x -J, given the basis B such that

for any a e B the value a(a) > 0. Therefore, VT(x, a) = VT(a, x) _ T n F°dd, where F°dd = JaER.dd ICY: (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 F°dd = Card(R' n Rod +d) where R' is the set of roots a e R such that a(a) # 0. As (fl_ .. , fk) is a completely involutive set on then k = Z(rk Ga + dim G"). Since grad g;(x) E G: (1 <, i < k) we have

dim( V(x, a) Q I Q B grad g;(x) J \i-1 = Card(R' n R+d) +'(rk Ga + dim

=Card(R'nRodd) +Card(R°nR odd) = Card(R+d) = Z(dim R , + 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 x1, ... , x" be the coordinates in a rigid system of coordinates connected with it rigidly; let Tj denote the elementary n x n matrix with zeros everywhere except for a one at the intersection of the ith row and j-th column; let E, denote the elementary skew-symmetric matrix E, = T j - T;. Let us suppose that a rigid body revolves with an

angular velocity S2 in a moving system of coordinates which has a corresponding elementary skew-symmetric matrix E.3. In this case the kinetic energy T is given by the formula T = J . . . I p(x)(x? + xj )d"x, z 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, = J . J p(x)x?d"x (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 f2 = y_;<; a)ijE;j equals 1 Y w j(ai + a). Hence we find an expression for the kinetic moment M = A(Q), A(0) = IS2 + QI, where I = diag(a,,... , a"). The Euler equations have the following form, M = [M, f2], where M = A(S2). The

matrix I is called the inertia tensor of the rigid body. Let Q be an invariant symmetric bilinear form on so(n) with respect to which the basis (Eij)1,i<;,n has been made orthonormal, THEOREM 22.5

let I = diag(a...... a") be the inertia tensor of the rigid body, and M = A(S2) be the kinetic moment of the body in a moving system of

coordinates where A(S2) = I0 + SDI, a = I2, b = I, so(n)" is the centralizer of a in so(n). Then,

a) for an appropriate operator symmetric with respect to Q, D: so(n)s -+ so(n)s we have A-'(M) = cpabo(M) b) the functions tr(M + AI2)k (k = 2, ... , n, A E 68) are the integrals of

motion of the Euler equation M = [M, A-'(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-'(M)] by including polynomial functions which depend only on the Q-orthogonal projection of M onto so(n)a.

Proof Without loss of generality, we assume that a, > a2 a" > 0. Let 7 1, ... , y, be numbers such that

254

A. T. FOMENKO AND V. V. TROFIMOV

al = ... = aqi = YI > aq,+1 = ' = a4, = Y2 > ... =aq.=Y >aqi_1+I =

Yl >...>Y3

Then so(n)° = so(P1) O ®so(P,) where pl =9j, P2=q2 - ql, ..., p, = q3 - R,- 1. Let the operator D: so(n)° _+ so(n)" be multiplication by y; on so(p;) (i = 1, ... , s). Then it is easily verified that A-l(S2) = W°bD(Q) Let G = sl(n, Vi), G = so(n) in Theorem 22.4. The functions tr M'` (2 < k <, n) are algebraically independent homogeneous generators of the algebra of invariants of sl(n, 68). Thus, if we give a completely involutive set of integrals of the equation

Y = [Y,D(Y)],

Yeso(n)°,

(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)' is a set of

integrals of motion of equation (15). Therefore we can take any completely involutive set on so(n)" 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 SPACES

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 cP°b (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 dsabD on symmetric spaces

Let the Lie group 6 with the Lie algebra G act smoothly on a manifold M. In this case to each element g e G corresponds vector field g on M. The action of G on M induces a symplectic action of 6 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

on the left-hand side denotes the pairing of

G* and G, while on the right-hand side it denotes the pairing of

and TM. As was noted in [88] every moment mapping P has the following property which is very important to us: for any two smooth functions f and g on G* we have the equality

{f°P,g°P}={f,g}°P

(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 (i 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

P = PQ: T*M - G,

Q(P(x),g) =

where g c G, x e T. *M. It is clear that for the moment mapping P = PQ

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 m e M the moment mapping P is linear along x c- T*M. Therefore, if the operator cp,,,p has been defined on G (a and b

may be elements of the Lie algebra which contains G), then HabD(x) = Q(P(x), cpabD(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 dsobD 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. Let M be a globally symmetric Riemannian manifold, rk M its rank, (5 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 TD is a compact semi-simple Lie group. THEOREM 23.1

a) If rk M = rk G, cpabD, is an arbitrary positive operator on G (a and b may be elements of a larger Lie algebra), and if fI (x),. . . , fk(x) is the completely involutive set of independent integrals of Euler's equation

256

A. T. FOMENKO AND V. V. TROFIMOV

[x, cp°bD(x)] (x e G), k = Z(rk G + dim G) then k = dim M and the functions Ji (P(x)), ... , fk(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, b c G, cpabD is a positive operator, then the geodesic flow on T*M of the Riemannian metric dsobD of the moment mapping PQ: T*M -+ G has an involutive set of integrals of motion J(P(x) + Aa), J E I(G), A e R 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 m e M. An isometry of M which leaves point m fixed and which moves the geodesics passing through m, defines an involutive automorphism of (si (see [48]). This automorphism of (fi defines an automorphism of the Lie algebra G, which we denote by a. Let G = H Q V be the decomposition of G such that the automorphism a equals 1 on H and (- 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 6 in G intersects V. As is well known, the action of i 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)), ... , fk(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 = z(dim G + rk G). Let a be an arbitrary element of V, where G° is the centralizer of a in G, H° = H n G°, V° = V n G. Since

[H,H] cH, [VV] cH, [H,V] (-- V and aEV, G°=H°+V°. We define on G a skew-symmetric form L°(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 + H°, V' = H + V° where H1 and V' are skew-orthogonal complements to H and V in G with respect to L°. Consequently the quotient L. on G/G° = H/H° Q V/V° is non-degenerate and realizes a pairing of the spaces H/H° and V/V°. Consequently, dim H/H° = dim V/V°. Let K be a Cartan subalgebra in

G such that K c V and a E K is an element regular in G. Then G°= V°=K, H°=0. We have:

257

INTEGRABLE SYSTEMS ON LIE ALGEBRA

z(dim G + rk G) = Z(dim G + dim G°)

= z(dim H + dim V + dim V) = J(dim H - dim H° + dim V - dim V°) + dim V° = dim V/Va + dim Va = dim V = dim M.

b) Let K be a Cartan subalgebra in G such that K,, = K n V is the maximal commutative subspace in V. In this case dim K,, = rk M. For an element a E K,, in general position in K. we have Va = K, therefore for the orbit 0 of the group Oti in G which passes through point a, we have

1 dim O =

i dim G/Ga = Z(dim H/H° + dim V/V°) = dim V/Va

.

It follows from Theorem 22.1 that the set of functions (J(x + 1a)), J E I (G), A e R contains # dim 0 functions independent on 0. Since the

invariants of G are constant on 0 but are not constant on K, then by using Lemma 22.3 on the image of the mapping P, i.e. on P(T*M), we shall get i dim 0 + dim Kv independent functions in the set J;, a, J e 1(G), 2 E R. Consequently, there are at least z dim 0 + dim K,, functions independent on T*M among the functions J(P(x) + 2a). Further: j dim 0 + dim K,, = dim V/ Va + dim Va = dim V= dim M

.

The functions J 1,a(x) are integrals of motion in involution of the Euler equation z = [x, cp°,,D(x)], which is Hamiltonian on the orbits of (ci in G with the Hamiltonian habD(x) = ZQ(x,(pabD(x)). In view of formula (1) we find that the functions J(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 dso D, and since the set of integrals of motion J(P(x) + 2a) contains functions independent on T*M equal in number to the dimension of M, the theorem is proved.

23.2. The metrics ds;,, on a sphere S"

Let S"

y e 118i+1: Y2

+

2 + yn+1 = 1} be a sphere; let SO(n + 1) be

the special orthogonal Lie group consisting of (n + 1) x (n + 1) matrices g such that gg` = E and det g = 1 where g' denotes the transposed matrix, E = diag(1, ... , 1); let so(n + 1) be the Lie algebra of

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 = -x`. In the standard way the group SO(n + 1) acts on the sphere S": the element g c- SO(n + 1) sends the point y E S" to the point (g)t)', where gy` is the matrix product of matrix g and of the matrix y` which is the transpose of the row matrix y = (y,, .. , y"+1). THEOREM 23.2

Let

a, b Egl(n + 1, B ),

a = diag(a1, ... an+1),

b=diag(bl,...,b"+1), a1>...>an+1>0, b1>...>b"+1; where Q(X, Y) = - tr(XY) is the invariant symmetric non-degenerate bi-

linear form oni gl(n + 1, R), positive definite on so(n + 1) c gl(n + 1); let Pab = ado ' adb be symmetric with respect to Q; the group SO(n + 1) acts in a standard way on the sphere S" = {yi + + yz,+1 = 1}, P = PQ: T*S" - so(n + 1) is the corresponding moment mapping; dsab is the Riemannian metric on S" that corresponds (under the Legendre

transofrmation) to the quadratic Hamiltonian Hab, where Hab(q) _ ZQ(P(q),1Pab(P(q))) for q c- T*S Then:

a) the geodesic flow on T*S" of the Riemannian metric ds;b has n

independent quadratic integrals in involution H1, ... , H", where Hk(q) = iQ(P(q),1Pa.at(P(q))), a' = diag(al, ... , an+l) and q E T*S"; b) if -b = diag(a1 ' , ... , a-+ 1), then with the substitution y; = x,/ aj (i = 1,... , n + 1) the metric dsab becomes a metric z

z

[+...+i]'(dx+...+dx+1) n+I

1

that is conformally equivalent to the standard metric dxi + of the ellipsoid

xlz + ... +

x"+1

a1

an+1

+ dxn+1

z

(pac(X)) for Proof (a) Let c = diag(c1, ... , cn+1), ha<(X) = X egl(n + 1, 01). As follows from the results of [89], the quadratic 1Q(x,

function ha,(X) is a linear combination of the functions J(X + .i.a) where J is the invariant of gl(n + 1, R), .l e R. As a corollary of Lemma 11.3, the

259

INTEGRABLE SYSTEMS ON LIE ALGEBRA

quadratic functions h, where h,, is the restriction of h,, to so(n + 1) c

gl(n + 1, l) are pairwise in involution for any diagonal matrix c e gl(n + 1, R). As an involutive automorphism Q, whose set of fixed points is so(n + 1), we may take the automorphism Q(X) = -X`. As a corollary of the fundamental property of the moment mapping (formula (1) from 23.1), the functions H,,. . . , H. are pairwise in involution on T*S". In order to prove the independence of these functions we shall need to

have their explicit calculation in local coordinates. Let S+ be the hemisphere given by the inequality yn+1 > 0. The functions y1,

are local coordinates on S+, while yn+l = ' +

, Y"

+ y,2. Let

z1,. .. , zn be the corresponding impulse variables on T*S+. Then , y") is a system of coordinates on T*S"+ and the (z1, , zn, Y l, standard symplectic structure is co=y"=1 dz, A dy,. The matrix X = I x. j II E so (n + 1) has a corresponding vector field on S" which in the

local coordinates yl, ... , yn equals n-1

(

"

a

a

)+

11

y E xr,ly;a-Y;Yja+ Y,E x,.n+lY"+1 aY, . ,=1 j i+1 i=1 Thus, P(z, y) = 11 P,j(z, y) II , where P,j(z, y) = z, yj - Zj Y, provided that i, j < n, and P,," + 1(z, Y) = zi Yn + 1 provided that 1 <, i < n. Hence we

find that

1 [nil Hac(z, Y) = 2

if

E

c -C

j (z, yj - zj Yd'

i=l j=,+l a, - aj

+

'.

Let yo = (0, ... , 0, 1). Then n

Hk(z,Yo) _

From this we find the Jacobian

1

,

k

an +l

2,=1 a, - an+1

C,-Cn+1

,=l ai - an+l

z,z.

z

2

z,Yn+l

A. T. FOMENKO AND V. V. TROFIMOV

260

aHk(z, yo) 3zi

+ an+1

a2 z

z

2

al + alan+l + an+1

ai-1an+1

al +

2

a2 + a2an+1 + an+l

+ ... + a"+1

az +

a"+1 1

n nn an + an + 1

z

2 an + anon+l + an+

... an + a"-'a,+,

-}- an+1

-i-

To calculate the determinant of this matrix we make the following (n - 1) elementary transformations. The first transform. tion consists of subtracting the (n - 1)-th row, multiplied by from the n-th row. The second transformation consists of subtracting the (n - 2)-th row,

multiplied by an+l, from the (n - 1)-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

al

a2

a

alft

a2n

1

...

u,.

whose determinant, the well-known Vandermond determinant, is nonzero, in view of the fact that a1 > > an. Thus, the involutivity and

independence of the functions H1, ... , H has been proved. These functions are integrals of the geodesic flow of the metric dsan, since the

Hamiltonian Hnl, which corresponds to this metric

a linear

is

combination of H1, ... , H for any diagonal matrix b. b) Let -b = diag(a1 1, ... , an +l). Then l

H0n(z, l')

2

n-1 i=1 J=i+1

a-1 - a '

ai - aj

1

(zi y. - z;1'i)2 n

+ [_, i=1

a, + 1 - ai

ai - an+1

1

2

2

Zi yn + 1

INTEGRABLE SYSTEMS ON LIE ALGEBRA

-

1 r n-1 2

261

n

i=1j=i+1

(aiaj)-1(z y, + zz y? - 2zizjyiy;) n

122 1 zi yn+1 J

+y

i=1

= 2[ini a lz?(y =

j#i

1

ifj

J

(aiaj) 1ziz;yiYj

-1

n

z

i=1 1

=2

n

i=1

n

a, 1z

ai 1 zi2

a-1yj2 - ` a; lz2a` 'y2

'=1

i=1 n

Vin`

(aia;)-1Zi

[r (

i=1j=1 1

2

'

+an+l yn+l y- ai

n

zj yi yj +

i=1

(aiai)-lziziyiyi

2j

1

Zi

i=1

2

[Ct

a1z); 1ajlyj)

-

n

ai 1Ziyi)C;

n

aj 1zJyj

i

= ZZ(A - Y)Z`,

where Z = (z1, ... , is a row matrix, A = diag(al 'c, ... , a 'c), C = ;+I 11, Yi; _ (aia;) 'yi y,. Whence we obtain the a; 1 y; , Y Y Y; expression for the Lagrangian '(y, y) corresponding to the Hamiltonian Hab: 2'(y, y) = Zy(A - Y) 'y` where y = (yl, ... , But this is, obviously, the Lagrangian of a free particle on a sphere with metric dsab = (dy)(A - Y)-'(dy)`, where (dy) _ (dy1,... , The only

expression left to calculate is (A - Y)-'. We have

(A -

- A -'Y))-' = (E - A lY) lA = A-' + A 'YA-' + A-'(YA ')YA '

Y)-1 = (A(E

+ A 'YA 'YA 'YA ' + Note that

.

1

A. T. FOMENKO AND V. V. TROFIMOV

262

(YA-1Y)ij =

(atak)-lytykakc 1(akaj)-lYkY; k=1

a,-lyic-la, lyjak lykakak lYk

k=1

_

(aiaj)-1Yiyjc-1

ak 1yk = LYij, k=0

where L = c 1 yk =1 ak lyk . From which we obtain

(A - Y)-1 = A-1 + A-1YA-1 + LA-1YA-1 + L2A-1YA-1 + = A-1 +

1

1

L A-1YA-1.

Proceeding further: n+1

1

1

1 -L

1-

-1 2

j=1 aj Yj

nk=1ak-1Yk2 =

-1 2 a+1Yn+1

n+l -1 2 j=1 aj Yj

Hence we obtain n

+az+l

ds b = c-1

ai(dyi)2

n

Yn+l i.j=1

=1

YtYj dYi dYj

As

(dyn+1) = d

=Yldy1+...+yndyn

yi+...+yn

2 +...+

Yl

2

yn

we obtain an+1(dyn+1)

2 = an+l

E Yiyjdyidyj.

2

Yn+1 i,;=1

Therefore 1 n+1

n+1

dsab = c-1 i=1

ai(dyf)2 = r a. 1Yi j=1

The change of variables xi =

ai yi

(i = 1, ... , n + 1) leads to the

metric ds;b taking the form n+1

1 n+1

j=1

i=1

a-2x2 Y - i

af(dy1)2.

i=1

INTEGRABLE SYSTEMS ON LIE ALGEBRA

conformally equivalent to the metric (dx1)z +

+

263

on an

ellipsoid X' {+...+ -=i}. a1 an xz

+1

23.3. Applications to non-commutative integrability

Let (M, w) be a symplectic manifold, 6 a Lie group with a Poissonian action on M (see, for instance [1]); let P: M -+ G* be the corresponding moment mapping. Let E G*, and let t i stabilizer of under the coadjoint representation; denote by M4 = level surface. As the moment mapping P maps the Poissonian action of 6 on M onto the coadjoint action on G*, and the stabilizer 64 leaves the level surface M4 invariant. If the level's surface MS is a smooth manifold and the factor set

N4 = M4/64 has the structure of a smooth manifold such that the canonical projection n: M4 - N4 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 N4. The symplectic manifold (N4, w4) is called a reduced symplectic manifold. Let H be a Hamiltonian on M, invariant, under 6, and let G be the algebra of the Hamiltonian system (M, w, H). Let X, be a Hamiltonian vector field, related to the Hamiltonian H, i.e. X = s grad H; let X /M, denote the restriction of this vector field to M4 (note that X is tangent to W. As the vector field is invariant with respect to 6V, it is projected onto a uniquely defined vector field X on N4 which, as one can easily check, is Hamiltonian on N4 with Hamiltonian H4(y) = H(x), where y c- N4 is the projection of x. Let 640 be the subgroup 6' leaving each point x c- M4 fixed. (f is, obviously, a normal subgroup in W. Let 64R = 64164 be the effective stabilizer of . It can be shown that for a general position point E P(M) in P(M) the connected component of the unit of the group 6 is commutative (for compact Lie algebras G the entire stabilizer 64 and, therefore, (5 also are connected Lie groups). From which it follows that the reduced Hamiltonian system (N4, w4, H4) is quadrature equivalent to the initial Hamiltonian system (M, (0, H) for

any point

e P(M) in general position in P(M) (the mapping it is

considered to be given by known functions). Suppose that the stabilizer 64 acts locally transitively on a surface M4 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 (N4,104, H,) is trivially integrable. The initial system, therefore, (M, co, 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 non-

commutatively 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, co, H) has motion integrals in involution of the form J(P(x) + Aa), J E I (G), A E 18 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 c G.

Proof We shall show that the codimension of a general position orbit 0 in P(M) is equal to the dimension of M,, c e 0. Let TT(P(M))1 be the intersection of the kernels of all functionals rl e TT(P(M)); let TT 01 be the intersection of the kernels of all functionals rl E TT 0. The codimension of

0 in P(M) is equal to dim TT01/TT(P(M))1. On the other hand, TT0' is the Lie algebra of 64, and T,(P(M))1 is the Lie algebra of 60. Since 64 acts locally transitively on M, we have

r = dim TT01/TT(P(M))1 = dim 64 = dim M. By Theorem 22.2 the set of functions J(x + .1a), J e 1(G), A E 1 for a

general position element a e G in G contains 1 dim 0 independent

functions on 0. Adding r more invariants G (the existence of r independent invariants on P(M) follows from Lemma 22.3) we obtain

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r + dim 0 independent functions in the set J(x + .la), J e I (G), A, a 18 z on P(M). Hence we have the same number of independent functions in

the set J(P(x) + 2a), J e I (G), A E R. 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, c), 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 M fgcok, where k =' dim M). THEOREM 23.4

Let (M, co, H) be a Hamiltonian system, G an integral

Lie algebra. Suppose that for any h the iso-energetic surface M,, _ {x c- M: H(x) = h} is compact. Then G is a compact Lie algebra. Proof Let g e G, and let x9 be the corresponding Hamiltonian vector field on M. The vector field xa is tangent to M,, 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 6 belonging to the Lie algebra G on the manifold M is defined. This action is Poissonian as, by the definition of vector field x9, 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, w, H), the iso-energetic surfaces are invariant under the action of 6 and their images P(Mh) 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 ad9 is a nilpotent endomorphism of G* and the mapping t - Exp(a4,,) x is polynomial for any x e G*. Since P(Mh) is invariant for x e P(M,,) this mapping is a mapping from R to P(Mh). As P(M,,) is compact and the mapping t - Exp(ad a) x polynomial, it is constant, Exp(ad to*) x = x for

all t. Therefore, for any x e P(M) we have ada* x = 0. 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,

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[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 o 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 M2 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 H1(Q, 7L) of one-dimensional integral homologies (or the data on the fundamental group), using the fixed three-dimensional surface Q 3 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 W. 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), 248-26 1). At the same time

there is also a natural development of some of the important ideas of

S. P. Novikov [96], V. V. Kozlov [59], 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 : 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 = 0},

then {H, f } = AH, where A = const. This equation is more common than {H, f } = 0. 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 zf is non-degenerate in the subspaces that are normal to these submanifolds. DEFINITION 24.1

The class of these integrals is wider than the class of analytic integrals. Accumulated experience of investigating specific mechanical systems shows that most of the integrals which have already been discovered are Morse-type. Suppose y is a closed integral trajectory of the system v on Q3 (i.e. a periodic solution). We will say that y is stable if some of its DEFINITION 24.2

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A. T. FOMENKO AND V. V. TROFIMOV

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 H1(Q,7L) or the fundamental group 7r1(Q). 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 nonTHEOREM 24.1 (A.

singular compact three-dimensional surface of the level Q of the Hamiltonian H using the Morse-type integral f on Q. Then, if the group

of homologies H1(Q,1) is finite cyclic, v has no less than one stable periodic solution on Q; and if H1(Q,7L) 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 H1(Q,7L) is usually easy. In specific examples the surfaces Q of constant energy (or

their reduction) are often diffeomorphic either to the sphere S3, the projective space RP3, 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 = S' 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 = S' x S2. If the Hamiltonian H has an isolated minimum or maximum point in M°, 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 18P3, or to S' X Sz. Then the system v has

at least two stable periodic solutions on S3 and at least one stable periodic solution on l8P3, S' x SZ. 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 3 Q= Suppose R = rank 7C1(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. If the rank of the group H1(Q,7L) > 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 = T3, the rank H1(Q, 7L) = 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. 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 COROLLARY 1

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?

Not every three-dimensional smooth compact closed orientable manifold can play the role of a surface of constant energy of COROLLARY 2

a Hamiltonian system, integrable using the Morse integral (on this surface).

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A. T. FOMENKO AND V. V. TROFIMOV

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 P(T) 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 f 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 V. (1) The complete torus S' x D2. (2) The cylinders T2 x D'. (3) The direct product (we shall call it the

oriented saddle) N2 x S', where N2 is a disk with two holes. (4) Consider the non-trivial fibration A3 -. N2 S' with the base S' and the fiber N2. The boundary of the manifold A3 is the two tori T2. It is clear that A3 (we will call it a non-oriented saddle) is implemented in l3 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 -°' K2 with the base K2 = Klein bottle and the fiber D' = 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 f reaches a local minimum or maximum (then they are stable). Then Q = m(S' x D2) + p(T 2 x D') + q(N2 x S') + s(A3) + r(K3), 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 f 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 Men and F: Men -+ l is a mapping of the moment, i.e. F(x) = (fl (x),. . . , f"(x), where f are commuting smooth integrals and fl = H. The point x c- M

is regular if the rank dF(x) = n and it is critical otherwise. Suppose

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271

N c M is a set of critical points and E = F(N) is a set of critical values

(bifurcation diagram). If a e R"\E, then the compact fiber B. = F-1(a) c Men consists of Liouville tori. For the deformation a outside E, the fiber B. is transformed by means of the diffeomorphisms. If the curve y, along which a moves, meets E, then the fiber BQ 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 E < n - 1, then all the fibers B where a e R"\E are diffeomorphic. The basic problem is when dim E = n - 1. Consider five types of (n + 1)-dimensional manifolds. (1) We shall call the direct product D2 x T" -1 a dissipative complete torus. Its boundary is the torus Tn. (2) We will call the product T" x D1 a cylinder. Its boundary is the two tori T". (3) Suppose N2 is a two-dimensional disk with two holes. We shall call the direct product N2 x T" -1 an oriented torus saddle. Its boundary is the three tori Tn. (4) Consider all the nonequivalent fibration A,, _ N2 T" -1 with base the torus Tn-1, with a fiber

N2. They are classified by the elements aeH1(Tn-1,7L2) = 7L2-1 N2 x T" -1 when a = 0 is a special case. If a # 0, the fibration A. is nontrivial and all manifolds A,, are diffeomorphic. The manifolds Aa when

a # 0 will be called non-orientable torus saddles. They have a boundary-the two tori T". (5) Let us consider the manifolds K" = T"/Gs, where T" is the torus, a = 0, 1, and G. is the group of the transformations defined as follows (this action was introduced by A. V. Brailov and V. T. Fomenko): R (a) =

i

(-Q1, Qz + 2, a3, (a2,a1,a3 +2 ,a4,...,a.),

a = 0, a = 1,

where a = (a1,.. . , an) e 118"/7L" = T. Here n > 2 in case a = 0 and n > 3 in case a = 1. Then Ka = Ko x Tn-2, K1 = Ki x Tn-3 and Ki = K2 x S'. Let us consider the two-fold covering p: T" - K" and let Kn+1 = Kp1 is the cylinder of the map p. It is clear that 8K,+1 = T° We will describe five types of rearrangement of the torus T. (1) The torus T", implemented like the boundary of the dissipative complete torus D2 x Tn-1, contracts to its "axis," the torus Tn-1 (we will put T" -> T" - 1 -+ 0). (2) The two tori Ti" and T20--the boundaries of the cylinder T" x D1 moves in opposite directions and merge into one torus T" (i.e. 2Tn - Tn - 0). (3) The torus T"-the lower boundary or the oriented torus saddle N2 x T" -1 rises upward and, in accordance with the topology N2 x T"', splits into two tori Ti and Tz (i.e. T" -+ 2T").

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A. T. FOMENKO AND V. V. TROFIMOV

(4) The torus T"-one of the boundaries A,, rises with respect to A. and is rearranged in its "middle," becoming once more a single (twice wound) torus (i.e. T" - T"). These rearrangements are parametrized by the nonzero elements a e H1 (T"-1,1L2) = 7L2-1 (5) Let us realize the torus T" as the boundary of Kr I . Let us deform T" in K" + 1 and collapse T" on K.". We obtain p: T" -+ K. We shall fix the values of the last n - 1 integrals

f2,. .. , f" and shall consider the resulting (n + 1)-dimensional surface Xn+1 Limiting in it f1 = H, we obtain the smooth function fin X"'. We will say that the rearrangement of the Liouville tori, which generate the non-singular fiber B. (assumed compact), is a rearrangement of the common location if, in the neighborhood of the rearrangement the torus T", the surface X" +I is compact, non-singular and the restriction f of the energy f1 = H on Xn+1 is a Morse function in the sense of Section 1 in this neighborhood. In terms of the diagram E, this means that the path y along which a moves, transversally intersects E at the point C, whose neighborhood in E is a smooth (n - 1)-dimensional submanifold in IR", and the last n - 1 integrals f2,. . ., f" are independent on Xn+1 in the neighborhood of the torus T". THEOREM 24.4 (A. T. Fomenko) (Theorem of the classification of the rearrangements of Liouville tori) (1) If dim E < n - 1, then all the non-singular fibers B. are diffeomorphic. (2) Suppose dim E = n - 1.

Suppose the non-degenerate Liouville torus T" moves along the common non-singular (n + 1)-dimensional surface of the level of the integrals f2,... , f", which is entrapped by the change in value of the energy integral f1 = H. This is equivalent to the fact that the point a = F(T") E li" moves along the path yin the direction of E. Suppose the torus T" undergoes rearrangement. This occurs when and only when T" meets the critical points N of the mapping of the moment F (i.e. the path y at the point C transversally pierces the (n - 1)-dimensional sheet E). Then all the possible types of rearrangement of the common location are

exhausted by the compositions of the above five canonical rearrangements 1, 2, 3, 4, 5. In case 1 (the rearrangement T" - T" -1 -+ 0) as the energy H increases the torus T" becomes a degenerate torus T" -1,

after which it disappears from the surface of the constant energy H = const (the limiting degeneration). In case 2 (the rearrangement 2T" -+ T" - 0) as the energy H increases the two tori T1 and Ti" merge

into one torus T", after which they disappear from the surface H = const. In case 3 (i.e. T" -+a 2T") as H increases the torus "penetrates" the critical energy level and splits into two tori T, and T2"

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on the surface H = const. In case 4 (i.e. T" -a T") as H increases the torus T" "penetrates" the critical energy level and once more becomes the torus T" (a non-trivial transformation of a double coil). In case 5 the torus T" merge into the manifold K; and disappears from the surface H = const. Changing the direction of the motion of the torus T", we obtain five inverse processes: (1) the production of the torus T" from the torus T", (2) the trivial production of the two tori Tl and Tz from one torus T", (3) the non-trivial merging of the two tori Ti" and Tz into one torus T", (4) the non-trivial transformation of the torus T" into the torus T" (double coil), (5) the transformation of Ka into the torus T. 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 T" drifts along the surface X"" of the level of the integrals f2,. . , f". It can happen that T" contracts to 1

-

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 H and causes, consequently, a slow evolution (drift) of the Liouville tori along

Xn+1. An answer to the question-What kind of topology is the topology of the surfaces Xn+1?-is given by the following theorem. THEOREM 24.5

Suppose M2 is a smooth symplectic manifold and the

system v = s grad H is integrable using the smooth independent f2,.. ., f ". Suppose Xn+1 is any fixed noncommuting integrals H = singular compact common surface of the level of the last n - 1 integrals. Suppose the restriction H on X"+1 is a Morse function. Then

X"+i

m(D2 x

T"-1) +

p(T" x D1) + q(N2 x

T"-1)

+ Y sa(A j) +

raK" 1,

a#o

i.e. a splice of boundary tori (using some diffeomorphisms) of the following "elementary bricks" is obtained: m dissipative complete tori, p

cylinders, q torus oriented saddles, s = aeo sa torus non-oriented saddles and r = ro + r1 non-oriented cylinders. The number m equals

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the number of limiting degenerations of the system v in X"', in which H

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 Men, such that the rank G + dim G = dim Men. Then the trajectories of the system move with respect to the tori T', where r = rank G. When proving the above results we use the following statements.

Suppose in the singular fiber B, there is exactly one critical saddle torus Tn-1. (1) Suppose the integral f is orientable in Xn+1 and a < c < b, where a and b are close to c. Then Cb = (f < b) is LEMMA 24.1

homotopically equivalent to C. = (f < a), to which the manifold P-1 x D1 is attracted with respect to the two non-intersecting tori T;,- I and T2a 1. (2) Suppose the integral f is non-orientable. Then Cb is homotopically equivalent to Ca, to the boundary B. of which, using the

torus T"', is attracted the n-dimensional manifold Y" which has the

a fibration y",, 'D' Tn- 1, which corresponds to the nonzero element aE71z-1 =H,(Tn-1,7L2). (3)

boundary P-1 and which

is

Further, each of the tori Tl a 1, T2, a 1, T" 1 always realizes one of the generatrices in the group of homologies Hn

(T.", Z) = 7L" -1. If any of

these (n - 1)-dimensional tori are attached to one and the same Liouville torus T", they do not intersect and they realize one and the same generatrix of the group of homologies H. -I (Ta , 71), and therefore

they are always isotopic in the torus T".

We will provide one more description of the three-dimensional surfaces Q of constant energy of the integrable (using an oriented Morse

integral) systems on W. Let us suppose that all critical manifolds of integral f are 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 Ma of the genus g, where g > 1 (i.e. a sphere with g handles) and take the product Mg x S'. We shall separate an arbitrary finite set of non-intersecting and self-non-intersecting smooth circles a;

in Ma , among which there are exactly m contractible circles (the remainder are non-contractible in MB ). In Mp X S2 the circles 01i determine the tori Ti' = at x S1. We will cut out Ma x S' 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.

Find an explicit convenient corepresentation of the group ir1(Q), where Q' is the surface from Theorem 24.3. Give an explicit PROBLEM

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 Q', 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. 243. 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 surface Q' = (H = const); f : Q - R is a second independent Bottian integral on Q. The critical submanifolds of f are isoenergetic

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 -'(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, f ), 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 f 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:ZP h':Z'-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 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 f, then fQ is a union of a finite number of Liouville's tori. Let us represent these tori by the points in R3 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 f, be a connected component of a critical level surface of the integral. We denote by N, the set of critical points of the integral f on f c.

Let us consider two cases: (a) N, = f, (b) Nc c f c. In Section 24.2 A. T. Fomenko gives the complete description of all cases and the topological structure of f,. (See [149], [150].) Let us consider case (a). Here only three types of critical sets are possible.

The "minimax circle" type. Here Nc = f and this set is homeomorphic to a circle S'. The integral f has a minimum or maximum on S'. The circle S' 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 N, = fc. This set is homeomorphic to a two-

dimensional 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 D1. We represent this minimax torus by the white point (the vertex of the graph). Two edges of

the graph Z enter this vertex or emerge from it. The "Klein bottle" type. Here N, = f . 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 KZ is homeomorphic to the skew product of K2 and the interval D'. We represent this minimax Klein bottle 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 f and N, # f. Here N, is a union of non-intersecting critical circles in f,. Each of these circles is a saddle circle for f. We shall call the corresponding connected component f a saddle component. Each saddle component f, is represented by a flat horizontal square in R' 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 T. 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 S'. 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 fc. 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 f . Then the trajectory meets

some torus on the upper non-singular level fc+,. We obtain a certain correspondence (homeomorphism) between regular domains on the

levels f _ and f,+,. 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 T . All edges of the graph T 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 T, are identified with some ends of the edges of the graph A. Graph T 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 f, with a nonorientable separatrix diagram of some critical circles on f. 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 T. Finally, we define the graph Z as the union Z = A + E, T, 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) + E, P(T) (here {c} are the critical values off) where P(A) and P(T) are two-dimensional surfaces with boundary. Here P(A) = (A x S') + > D2 + Y U2 + Z S1 x V. Here A = Int A, > D2 denotes the non-intersecting 2-discs corresponding to the vertices of the graph A, which have a "minimax circle" type; 2]p2 denotes the non-intersecting MObius bands, corresponding to the vertices of the graph A, which have a "Klein bottle" type; Y S' x D1 denotes the non-intersecting cylinders, corresponding to the vertices of the graph A, which have a "torus type".

The corresponding boundary circles of A x S' are identified with the boundary circles of D2, t2, S' x D' by some homeomorphisms. Let us construct the surfaces P(T,). Let us consider the orientable case. Fomenko proves (see Section 24.2 and [149], [150]) that in this case the surface f is homeomorphic to direct product K, x S1, where K, is some

graph. The graph K, is constructed from several circles, which are tangent in some points. Such circles can be realized as a cycle on the

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torus contained in f.. This cycle intersects with a critical circle on f only in one point. The surface f 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 f,. They cut f into the sum of flat rings. Consequently, the circle y (non-homologous to zero) is uniquely defined on a critical level surface

f. We can choose the circle a which is a generator on the torus contained

in f. 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 K, is the union of all ovals. The surface P(TA) can be realized as "normal section" of a small neighborhood of a critical level surface f in Q. The intersection of P(T)

with f is the graph K,. To realize the surface P(T,) in Q, we must consider the small intervals on the integral trajectories of the field w = grad f, which intersect the graph K, This definition is correct in all non-critical points on K, Let us consider the vertices of the graph K, i.e. the critical points of the integral f on f . Then we consider the small squares orthogonal to the critical circles on f. 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(7 ). The graph K, is embedded in the surface P(T). We obtain some graph K as the union of all graphs K, and all boundary circles, described above.

THEOREM 24.7 (A. T. Fomenko) The graph Z(Q, f) is conjugate to

the graph K(Q,f) in the surface P(Q, f ). 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 f be two arbitrary Bottian integrals of a

system v. Then the homeomorphism h : Z(Q, f) -+Z(Q, f') (see

Theorem 24.6) transforms the subgraphs TT 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, f'). 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, f '). 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, f'). 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, f'). This event corresponds (from the analytical point of view) to the operation f -+ f 2 (square of function) or, conversely, to the operation f (square root). 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.

DEFINITION 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 Hamiltonian. 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.

Bibliography

[1] Arnol'd V. I., Mathematical Methods in Classical Mechanics, Nauka, Moscow, 1974 (English translation, Springer-Verlag, Berlin and New York, 1978). [2] Adler M., van Moerbeke P., Completely integrable systems, Euclidean Lie algebras and curves, Adv. in Math., 38, no. 3 (1980), 267-313.

[3] Adler M., van Moerbeke P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math., 38, no. 3 (1980), 318-379. [4] Avez A., Lichnerowich A., Diaz-Miranda A., Sur l'algi bre des automorphises infinitesimaux d'une varii ti 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-de Vries type equations, Invent. Math., 50 (1979),219-248. [6] Adler M., Some finite dimensional systems and their scattering behaviour, Comm. Math. Phys., 55 (1977). [7] Adler M., On a trace functional for formal pseudo-differential operators and the Hamiltonian structure of Korteweg-de 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] Airault H., McKean H. P., Moser J., Rational and elliptic solutions of the Korteweg-de Vries equation and related many body problem, Comm. Pure Appl. Math., 30 (1977), 95-148. [10] Arkhangel'sky A. A., Completely integrable Hamiltonian systems on a group of triangular matrices, Mat. Sb., 108, no. 1 (1979), 134-142 (in Russian). [11] Hourbaki N., Groupes et Algebres de Lie, Chapters IV-VI, Hermann, Paris VI, 1968.

[12] Borho W., Gabriel P., Rentschler R., Primideal in Einhiillenden Auflbsbaren LieAlgebren, Lecture Notes in Math., no. 357 (1973), Springer-Verlag, Berlin. [13] Belyayev A. V., On the motion of a 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. Yestnik, 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., Sur la 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), 51-62 (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] Fomenko 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

283

Institute of Mathematics of Ukrainian SSR Academy of Sciences, pp. 3-19 (in Russian). [36] (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. 1108 (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. [37] Flaschka H., On the Toda lattice I, Phys. Rev. B, 9 (1974), 1924-1926. [38] Flaschka H., Toda lattice II, Progr. Theor. Phys., 51 (1974), 703-716. [39] Gorr G. V., Kudryashova L. V., Stepanova L. A., Classical Problems of Rigid Body Dynamics. Development and Contemporary State, Scientific Thought, Kiev, 1978 (in Russian).

[40] 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).

[41] 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). [42] Gardner C., Korteweg-de Vries equation and generalization IV. The Korteweg-

de Vries equation as a Hamiltonian system, J. Math. Phys., 13, no. 8 (1971), 1548-1551. [43] Guillemin V., Quillen D., Sternberg S., The integrability of characteristics, CPAM, 23 (1970), 39-77. [44] Guillemin V., Sternberg S., Geometric asymptotics, Math. Surveys, no. 14 (1977), American Mathematical Society, Providence, RI.

[45] Gardner C. S., Geen J. M., Kruskall M. D., Miura R. M., Korteweg-de Vries equation and generalization VI. Methods for exact solutions, Comm. Pure Appl. Math., 27 (1974), 97-133.

[46] Godbillon C., Geombtrie differentielle et mecanique analytique, Collection Methodes, Hermann, Paris, 1969. [47] Goto M., Grosshans F. D., Semisimple Lie algebras, Lecture Notes in Pure and Appl. Math., 38 (1978), Marcel Dekker, New York and Basel. [48] Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962. [49] Jacob A., Stemberg S., Coadjoint structures, solutions and integrability, Lecture Notes in Physics, 120 (1980), Springer-Verlag, Berlin. [50] Jacobson N., Lie Algebras, Interscience (John Wiley and Sons), New York and London.

[51] Kochin N. E., Kibel' I. A., Roze N. V., Theoretical Hydromechanics, pt.

1,

Fizmatgiz, Moscow, 1963 (in Russian).

[52] Krichiver I. M., Methods of algebraic geometry in the theory of non-linear equations, UMN, 33, pt. 6 (1977), 183-208. [53] Kostant B., Quantization and unitary representations: Part 1, Prequantization.

284

A. T. FOMENKO AND V. V. TROFIMOV

Lectures in Modem Analysis and Applications, Lecture Notes in Math., no. 170 (1970), Springer-Verlag, Berlin. [54] Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34, no. 3 (1980), 195-338. [55] Kulish P. P., Reymann A. G., The hierarchy of symplectic forms for Schriidinger and Dirac equations on a straight line, Records Sci. Seminars LOMI, 77 (1978), 134-147 (in Russian).

[56] 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.

[57] Krasil'shchik I. S., Hamiltonian cohomologies of canonical systems, Reports USSR Acad. Sci., 251, no. 6 (1980), 1306-1309 (in Russian). [58] Knorrer H., Geodesics on the ellipsoid, Invent. Math., 59, no. 2 (1980), 119-144. [59] Kozlov V. V., Qualitative Analysis Methods in Rigid Body Dynamics, MGU, Moscow, 1980 (in Russian). [60] Kozlov V. V., Onishcherlko D. A., The non-integrability of the Kirchoff equations, Reports USSR Acad. Sci., 266, no. 6 (1982), 1298-1300 (in Russian). [61] Kupershmidt B. A., Wilson G., Conservation laws and symmetries of generalized Sine-Gordon equations, Comm. Math. Phys., 81 (1981), 189-202.

[62] Kupershmidt B. A., Wilson G., Modifying Lax equations and the second Hamiltonian structure, Invent. Math., 62 (1981), 403-436. [63] Kirillov A. A., Unitary representations of nilpotent Lie groups, UMN, 17, pt. 4 (1962), 57-110 (in Russian). [64] Landau L. D., Lifshits E. M., Continuous Medium Electrodynamics, Fizmatgiz, Moscow, 1959 (in Russian). [65] Langlois M., Contribution a 1'etude du mouvement du corps rigide a N dimensions autour d'un point fixe, in These presentee d la faculte des sciences de l'universite de Besancon, Besangon, 1971. [66] Lax P. D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21, no. 5 (1968), 467-490.

[67] 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). [68] Loos 0., Symmetric Spaces, vol. 1, 2, Benjamin, New York and Amsterdam, 1969. [69] Milnor J., Stasheff J., Characteristic Classes, Princeton University Press, NJ, 1974. [70] McKean H., Integrable systems and algebraic curves, in Global Analysis, Lecture Notes in Math., no. 755 (1979), 83-200, Springer-Verlag, Berlin. [71] 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. [72] van Moerbeke P., Mumford D., The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154.

[73] 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). [74] Manakov S. V., Note on the integration of Euler equations of the dynamics of a ndimensional rigid body, Functional Anal. App!., 10, pt. 4 (1976),93-94 (in Russian).

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 Anal. Appl., 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.

[86] 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. App!., 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. App!., 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 LustemikSchnirelman-Morse (LSM) theory. I, Functional Anal. Appl., 15, pt. 3 (1981, 54-66; 11, Functional Anal. App!., 15, pt. 4 (1982), 3-49 (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. App!., 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 of the classical generalized Toda models, Invent. Math., 56 (1976), 261-269. [101] Olshanetsky M. A., Perelomov A. M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math., 37 (1976), 93-109. [102] Poincarb H., Methodes Nouvelles, 1, Chapter 5, 1892. [103] Perelomov A. M., Some observations on the integration of the equations of motion of a 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 of the 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., Winternitz 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. 1., 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 le 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] Rals M., Indices of semi-direct product Exp (5, 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 le theorime d'existence des algebres de Lie semi-simples, Publ. Math. Paris, 31 (1966), 21-58. [123] Trofimov 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 A. 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] Trofimov 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] Trofunov V. V., Finite-dimensional representations of Lie algebras and completely integrable systems, Mat. Sb., 3, pt. 4 (1980), 610-621 (in Russian). [127] Trofimov 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] Trofunov V. V., Euler equations on finite-dimensional soluble Lie groups, Izv. Akad. Nauk SSSR (Math. Ser.), 44, no. 5 (1980), 1191-1199 (in Russian). [129] Trofunov, V. V., Completely integrable geodesic flows of left-invariant metrics on

Lie groups, connected with commutative graduated algebras with Poincar% duality, Dokl. Akad. Nauk SSSR, 263, no. 4 (1982), 812-816 (in Russian). [130] Trofimov V. V., Methods for constructing S-representations, MGU "Vestnik" Ser. Math. Mekhanika, no. 1 (1984), 3-9 (in Russian). [131] Trofimov 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 sur 1'algebre symmbtrique d'une algebre de Lie nilpotente, Bull. Soc. Math. France, 100 (1972), 301-335.

[135] Vesclov 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 formalism?, 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). [138] Wilson G., Commuting flow and conservation laws for Lax equations, Math. Proc. Cambridge Philos. Soc., 86 (1979), 131. [139] 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, 1960. [141] Wintner A., Celestial Mechanics, Princeton University Press, Princeton, 1947. [142] Zakharov V. E., Faddeyev L. D., The Korteweg-de Vries equation-a completely integrable Hamiltonian system, Functional Anal. Appl., 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., % (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, 90A, 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 Diff. 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 of the Kelvin-Stuart cat's eyes flow, Proc. AMS. SIAM Summer Seminar, Santa Fe, July, 1984, 1-16.

[162] Holm D., Kupershmidt B., Levermore D., Hamiltonian differencing of fluid dynamics, 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-difference 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 at 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-de Vries equations associated to super extensions of the Virasoro algebra, Physics Letters, 109A, 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), 133-150. [179] Marsden J. 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 delta Accademia della Scienza di Torino, 117 (1983),289-340. [182] McDuff D., Examples of simply-connected symplectic non-Kiihlerian manifolds, J. Di/j 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 diJerentielle, 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. D(fl 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 l'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 automorphisms 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 Differential 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). [192] 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). [193] Weinstein A., Symplectic manifolds and their Lagrangian submanifolds, Adv. in Math., 6 (1971), 329-346. [194] Weinstein A., Symplectic geometry, Bull. Amer. Math. Soc., 5 (1981), 1-13. [195] Weinstein A., The local structure of Poisson manifolds, J. of Diff. 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.), 50, no. 4 (1986),661-674 (in Russian). [199] 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

l8-the set of all real numbers. C-the set of all complex numbers. 7L-the set of all integers. Q---the set of all rational numbers. l8"-n-dimensional real linear space. Cmt A (02 -the exterior product of differential forms co, and W2.

X`, X'-the matrix transpose of a matrix X. T; the elementary matrix: (Tj)Pq = (Sjp 5Jq).

t

E1 = T1 - T; the elementary skew-symmetric matrix. Ii, j I = Tj + T the elementary symmetric matrix, ad, (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. B(X, Y)--the Cartan-Killing form. <X, Y>-pairing between the space V and the space V* that is dual to it. C°°(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 -+ 6 of the Lie algebra G into the Lie group 6. 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. W(M)-the space of the differential k-forms on the manifold M.

Index

Adjoint representation, 18 Affine Lie algebras, 219 Algebra with Poincarb duality, 162 Algebraic variety, 8 Argument translation, 137

Functions in involution, 30

Bifurcation diagram, 271 Bounded domain, 8

Index of the Lie algebra, 33 Integral, 12 Invariant, 27

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 f-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

Geodesic flow, 10

Hamiltonian field, 8

Kahler 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