Integrable Systems in the Realm of Algebraic Geometry (Lecture Notes in Mathematics, 1638) 3540423370, 9783540423379

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1638

Springer

Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Pol Vanhaecke

Integrable Systems in the realm of Algebraic Geometry

SecondEdition

~ Springer

Author Pol Vanhaecke Drpartement de Math~matiques UFR Sciences SP2MI Universit6 de Poitiers Trlrport 2 Boulevard Marie et Pierre Curie BP 30179 86962 Futuroscope Chasseneuil Cedex, France E-mail: Pol.Vanhaecke @mathlabo.univ-poitiers.fr

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Vanhaecke, Pol: Integrable systems in the realm of algebraic geometry / Pol Vanhaecke. - 2. ed.. - Berlin ; Heidelberg ; New York ; Barcelona ; I-IongKong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 2001 (Lecture notes in mathematics ; 1638) ISBN 3-540-42337-0 Mathematics Subject Classification (2000): 14K20, 14H70, 17B63, 37J35 ISSN 0075- 8434 ISBN 3-540-42337-0 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-61886-4 (lst edition)Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 1996, 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10844943 41/3142/LK - 543210 Printed on acid-free paper

P r e f a c e to the s e c o n d e d i t i o n

The present edition of this book, five years after the first edition, has been spiced with several recent results which fit naturally in the point of view that had been adapted in the original text and with some new examples and constructions that will help the reader to appreciate better our approach to integrable systems. On this occasion I wish to thank my collaborators from the last five years, to wit Christina Birkenhake, Peter Bneken, t~ui Fernandes, Masoto Kimura, Vadim Kuznetsov, Marco Pedroni, Michael Penkava, Luis Piovan and Claude Roger for a fruitful interaction and for their warm friendship. Most of the results that have been added are taken from, or are inspired by, joint work with some of them; I acknowledge their permission to add these, sometimes unpublished, results. The colleagues at my newest working environment, the University of Poitiers (France), created for me a pleasant and stimulating working environment. I wish to acknowledge the support of all of them. Special thanks go to Marc van Leeuwen, Claude Quitt4 and Patrice Tanvel for sharing their insights with me, which usually led to a real improvement of parts of the text. Last but not least, Yvette Kosmann-Schwarzbach, who was not acknowledged in the first version of this book - - most probably because my gratitude to her was too big and too obvious! - - is thanked here in all possible superlatives, for her constant support and for her sincere friendship. Merci Yvette!

Table of Contents

I. I n t r o d u c t i o n

. . . . . . . . . . . . . . . . . . . . . . . . . . .

II. Integrable HamUtonian

1.

systems on aftlne Poisson varieties . . . . . .

17.

1. I n t r o d u c t i o n

17.

2. Affine Poisson varieties a n d their morphisms 2.1. Affine Poisson varieties

. . . . . . . . . . . . . . . .

19.

. . . . . . . . . . . . . . . . . . . . . . .

2.2. M o r p h i s m s of affine Poisson varieties 2.3. Constructions of affme Poisson varieties

19.

. . . . . . . . . . . . . . . . .

26.

. . . . . . . . . . . . . . . .

28.

2.4. Decompositions a n d invariants of affine Poisson varieties . . . . . . . . . 3. Integrable H a m i l t o n i a n systems a n d their morphisms

. . . . . . . . . . . .

3.1. Integrable H a m i l t o n i a n systems on affine Poisson varieties 3.2. Morphisms of integrable H a m i l t o n i a n systems

37. 47.

. . . . . . . .

47.

. . . . . . . . . . . . .

54.

. . . . . . . . . . . .

57.

3.4. Compatible a n d multi-Hmniltonian integrable systems . . . . . . . . . .

62.

3.3. Constructions of integrable H a m i l t o n i a n systems

4. Integrable H a m i l t o n i a n systems o n other spaces . . . . . . . . . . . . . . . 4.1. Poisson spaces

65.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

65.

4.2. Integrable H a m i l t o n i a n systems on Poisson spaces . . . . . . . . . . . . III. Integrable Hamiltonian 1. I n t r o d u c t i o n

systems and symmetric

products

of curves .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. T h e systems a n d their integrability

. . . . . . . . . . . . . . . . . . . .

2.1. N o t a t i o n 2.2. T h e compatible Poisson structures {-,-}~ . . . . . . . . . . . . . . . 2.3. Polynomials in involution for {., .}~ . . . . . . . . . . . . . . . . . . 2.4. T h e hypereUiptic case . . . . . . . . . . . . . . . . . . . . . . . . 3. T h e geometry of the level manifolds . . . . . . . . . . . . . . . . . . . . 3.1. T h e real a n d complex level sets . . . . . . . . . . . . . . . . . . . . 3.2. T h e s t r u c t u r e of t h e complex level manifolds . . . . . . . . . . . . . . 3.3. T h e s t r u c t u r e of the real level manifolds . . . . . . . . . . . . . . . . 3.4. Compactification of t h e complex level manifolds . . . . . . . . . . . . 3.5. T h e significance of the Poisson structures {-, -}~ viii

. . . . . . . . . . . .

69. .

71. 71. 73. 73. 73. 78. 83. 85. 85. 87. 89. 93. 95.

IV. Interludium: the geometry 1. I n t r o d u c t i o n

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. D i v i s o r s a n d line b u n d l e s 2.1. D i v i s o r s

of Abelian varieties

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2. L i n e b u n d l e s

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3. S e c t i o n s of line b u n d l e s

. . . . . . . . . . . . . . . . . . . . . . .

2.4. T h e R i e m a n n - R o c h T h e o r e m

. . . . . . . . . . . . . . . . . . . . .

2.5. L i n e b u n d l e s a n d e m b e d d i n g s in p r o j e c t i v e s p a c e

. . . . . . . . . . . .

2.6. H y p e r e l l i p t i c c u r v e s . . . . . . . . . . . . . . . . . . . . . . . . . 3. A b e l i a n varieties

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1. C o m p l e x tori a n d A b e l i a n varieties 3.2. L i n e b u n d l e s o n A b e l i a n varieties 3.3. A b e l i a n s u r f a c e s

97. 97. 99. 99. 100. 101. 103. 105. 106. 108.

. . . . . . . . . . . . . . . . . .

108.

. . . . . . . . . . . . . . . . . . .

109.

. . . . . . . . . . . . . . . . . . . . . . . . . .

111.

4. J a c o b i varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114.

4.1. T h e a l g e b r a i c J a c o b i a n

. . . . . . . . . . . . . . . . . . . . . . .

114.

4.2. T h e a n a l y t i c / t r a n s c e n d e n t a l J a c o b i a n

. . . . . . . . . . . . . . . . .

114.

4.3. A b e l ' s T h e o r e m a n d J a c o b i i n v e r s i o n

. . . . . . . . . . . . . . . . .

119.

4.4. J a c o b i a n d K u m m e r s u r f a c e s . . . . . . . . . . . . . . . . . . . . .

121.

5. A b e l i a n s u r f a c e s o f t y p e (1,4) . . . . . . . . . . . . . . . . . . . . . . .

123.

5.1. T h e g e n e r i c c a s e

. . . . . . . . . . . . . . . . . . . . . . . . . .

5.2. T h e n o n - g e n e r i c case V. Algebraic

completely

1. I n t r o d u c t i o n 2. A.c.i. s y s t e m s

. . . . . . . . . . . . . . . . . . . . . . . .

integrable

Hamiltonian systems

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI. The Mumford systems

2. G e n e s i s

127. 127.

4. T h e l i n e a r i z a t i o n o f t w o - d k m e n s i o n a l a.e.i, s y s t e m s

1. I n t r o d u c t i o n

124.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. P a i n l e v ~ a n a l y s i s for a.c.i, s y s t e m s

5. L a x e q u a t i o n s

. . . . . . . .

123.

. . . . . . . . . . . . . . . . . . . . .

135. 138. 140.

143.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145.

2.1. T h e a l g e b r a o f p s e u d o - d i f f e r e n t i a l o p e r a t o r s

. . . . . . . . . . . . . .

2.2. T h e m a t r i x a s s o c i a t e d to two c o m m u t i n g o p e r a t o r s

. . . . . . . . . . .

145. 146.

2.3. T h e i n v e r s e c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . . . . . .

150.

2.4. T h e K P v e c t o r fields

152.

. . . . . . . . . . . . . . . . . . . . . . . .

ix

3. M u l t i - H a m i l t o n i a n s t r u c t u r e a n d symmetries . . . . . . . . . . . . . . . . 3.1. T h e loop algebra 9(q

. . . . . . . . . . . . . . . . . . . . . . . .

3.2. R e d u c i n g t h e R-brackets a n d the vector field ~

. . . . . . . . . . . . .

4. T h e odd a n d t h e even Mumford systems . . . . . . . . . . . . . . . . . . 4.1. T h e (odd) M u m f o r d system . . . . . . . . . . . . . . . . . . . . .

155. 155. 157. 161. 161.

4.2. T h e even M u m f o r d system . . . . . . . . . . . . . . . . . . . . . .

163.

4.3. Algebraic complete integrability a n d Laurent solutions . . . . . . . . . .

164.

5. T h e general case

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII. T w o - d i m e n s i o n a l a.c.i, s y s t e m s a n d a p p l i c a t i o n s 1. I n t r o d u c t i o n

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. T h e genus two M u m f o r d systems

. . . . . . . . . . . . . . . . . . . . .

168.

175. 175. 177.

. . . . . . . . . . . . . . . .

177. 179. 181.

3. Application: generalized K u m m e r surfaces . . . . . . . . . . . . . . . . .

185.

2.1. T h e genus two odd Mumford system

. . . . . . . . . . . . . . . . .

2.2. T h e genus two even Mumford system

. . . . . . . . . . . . . . . . .

2.3. T h e Bechlivanidis-van Moerbeke system

3.1. Genus two curves with a n a u t o m o r p h i s m of order three . . . . . . . . . 3.2. T h e 94 configuration on t h e J a c o b i a n of I ~ . . . . . . . . . . . . . . .

185. 186.

3.3. A projective e m b e d d i n g of the generalised K u m m e r surface . . . . . . . .

190.

4. T h e Gaxnier potential . . . . . . . . . . . . . . . . . . . . . . . . . .

196.

4.1. T h e Garnier potential a n d its integrability . . . . . . . . . . . . . . .

196.

4.2. Some moduli spaces of Abelian surfaces of type (t,4) . . . . . . . . . . 4.3. T h e precise relation with the canonical J a c o b i a n . . . . . . . . . . . .

202. 206.

4.4. T h e relation w i t h t h e canonical J a c o b i a n m a d e explicit . . . . . . . . . 4.5. T h e central G a r n i e r potentials . . . . . . . . . . . . . . . . . . . .

211. 216.

5. A n integrable geodesic flow on SO(4)

. . . . . . . . . . . . . . . . . . .

220.

5.1. T h e geodesic flow on SO(4) for metric II . . . . . . . . . . . . . . . .

220.

5.2. Linearizing variables . . . . . . . . . . . . . . . . . . . . . . . . .

222.

5.3. T h e m a p A,t --+ M s . . . . . . . . . . . . . . . . . . . . . . . . .

226.

6. T h e H~non-Heiles hierarchy

. . . . . . . . . . . . . . . . . . . . . . .

230.

6.1. T h e cubic H~non-Heiles potential . . . . . . . . . . . . . . . . . . .

230.

6.2. T h e quartic H~non-Heiles potential . . . . . . . . . . . . . . . . . . 6.3. T h e H~non-Heiles hierarchy . . . . . . . . . . . . . . . . . . . . .

232. 233.

7. T h e T o d a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Different forms of t h e Toda lattice . . . . . . . . . . . . . . . . . .

235. 235.

7.2. A m o r p h i s m to t h e genus 2 even Mumford system . . . . . . . . . . . .

237.

7.3. Toda a n d A b e l i a n surfaces of type (1,3)

240.

References Index

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

243.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253.

Chapter

I

Introduction

Integrable systems first appeared as mechanical systems for which the equations of motion are solvable by quadratures, i.e., by a sequence of operations which included only algebraic operations, integration and application of the inverse function theorem. Apart from some non-trivial examples which were constructed before, the first main result (due to Liouville, but essentially an application of a result due to Hamilton) was that if a mechanical system with n degrees of freedom of the form dqi dt

(H

8H

dpi

M

iqpvi

Wt

o9qi

any function in the coordinates qj,

which is H, then it can be solved involution if their Poisson bracket

pi) has n independent functions in involution, one of two functions f and g are said to be in quadratures; by

n

gg (,gf ;, 9qi

If'g1

vanishes, If, gj

=

0 and

f

is called

9pi

a

first

integral

19f '9g 9pi 9%

of the system if

f

and H

are

in involution.

Mechanical systems which

satisfy the conditions of Liouville's Theorem are called Liouin the sense of Liouville. A quite short but important list integrable integrable of (non-trivial) examples of Liouville integrable systems were found during the last century: a few integrable tops (the Euler top, the Lagrange top, Kowalevski's top, the GoryachevChaplygin top), free motion of a particle on an ellipsoid (Jacobi), motion of a rigid body in an ideal fluid (Clebsch, Kirchhoff and Steklov case), motion in the field of a central potential (Newton) and a few others. Both finding these systems (i.e., showing that enough first integrals in involution exist, which was done by constructing them) and solving them explicitly ville

or

P. Vanhaecke: LNM 1638, pp. 1 - 16, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001

-

-

Chapter

1. Introduction

by quadratures required a lot of ingenuity and often quite long calculations. In the more complicated cases the solution was written down in terms of two-dimensional theta functions by a non-trivial use of the rich analytical properties of these functions. In turn it motivated the reseaxch in theta functions and Abelian varieties, which originated in the beginning of that century in the works of Riemann and Abel. A rich interaction between integrable systems and complex analysis' was about to develop from it. But it did not happen. There were two reasons for this. The first one is that Poincax6 showed at the beginning of this century that a general mechanical system of the above Hamiltonian form is not Liouville integrable. In particular he also showed that the famous three body problem is not

integrable.

This declined the interest in

mathematicians second

reason

(as

far

integrable systems

for

physicists,

astronomers and

distinction between these three groups could be made). is that complex analysis and algebraic geometry started to develop in as a

pletely different directions and the theory

The com-

of theta functions and Abelian

varieties, which is to be situated on their intersection where analytic and algebraic objects come together, also faded away from the picture. For about 60 years there was neither progress nor interest in 2 integrable systems. The renewed interest

motivated

by the discovery of a new approach to solve nonGardner, Greene, Kruskal and Miura in 1967. This method has been known as the inverse scattering method and was originally designed for studying evolution equations such as the Korteweg-de Vries equation. It became apparent that these "integrable" evolution equations possessed a Hamiltonian formulation, having an infinite munber of independent first integrals in involution and that they could therefore be interpreted as integrable systems with an infinite number of degrees of freedom; the inverse scattering method provided the first integrals and led to explicit solutions. Later the method was succesfully applied in the context of classical (finite-dimensional) integrable systems, first by Flaschka, (see [Flal] and [Fla,2]), Manakov (see [Man]) and Moser (see [Mosl] and [Mos2]) and later by many others. In a short period of time a connection was discovered with many branches of mathematics, especially Lie theory, representation theory, algebraic and differential geometry. The revival of the interest in integrable systems was as much present in physics as in mathematics- many physical interesting systems were found to be well-enough described by integrable systems and (infinite-dimensional) integrable systems play nowadays linear evolution

a

was

equations, due

to

dominant role in modern field theories.

Traditionally integrable systems are considered as differential geometric objects. The phase space is a smooth (or analytic) manifold, equipped with a symplectic structure and the functions in involution are smooth (or analytic) functions. Many important constructions relate to the existence of certain coordinates

or

transformations and

are

of

a

transcendental

we think here of the construction of Darboux coordinates, canonical transformations, separating variables, generating functions, etc. In most examples however the different elements which constitute the integrable system are algebraic: the phase space has the structure of (the real part of) an affine algebraic variety, the Poisson bracket of two regular functions is a regular function and the functions in involution are also regular functions. This suggests that algebraic geometric tools may be helpful in studying and solving integrable systems. For example the n-dimensional manifolds which are obtained by fixing the values of the n

nature;

1

At that time the

theory of theta functions

analysis. 2 Basically everything that

was

known about

tained in Whittaker's classical book

[Whil.

was

considered

as

a

chapter

in

complex

integrable systems before the revival

is

con-

1. Affine Poisson varieties

integrals are n-dimensional affine algebraic varieties whose precise nature influences the possible types of flows: if such a fixed level manifold admits a compactification to which the n integrable vector fields extend in a 4olomorphic way and remain independent then the level manifold must be a complex torus and the flows of these vector fields are linear on it. first

Algebraic-geometric

tools

were

already used

in the

study of integrable systems

at the

end of the 19-th century, but it was Adler and van Moerbeke (see [AM2]-[AM9] and [MM]) who clarified the meaning of the integrability of most classical systems by introducing the

algebraic completely integrable system (a.c.i. system) and developed new tools to analyse and solve such systems. The basic conditions which they impose for an integrable system to be a.c.i. is that the general fiber of the momentum map is an affine part of an Abelian variety and that the flows of the integrable vector fields are linear on them. The main tool they developed for studying a.c.i. systems is the asymptotic analysis of integrable concept of

an

systems and goes back to KowaJevski. The results which were obtained from it include explicit embeddings of Abelian varieties (and related varieties such as Kummer varieties) in projective space, a detailed analysis of divisors and their singularities on Abelian varieties, a classification of integrable flows on SO(4), the construction of Lax equations, a theorem which allows one to conclude from the asymptotic analysis that a given integrable system is a.c.i., some intricate relations to Lie theory, ...

The present work originated and was largely influenced by the work of Adler and van Moerbeke. It was presented as a "Habilitation A diriger des recherches" at the Universit6 de Lille (France) and contains some of the author's work QBV2], [Van3] and [Van5]; part of [Van6] and the main result of

[Van4]

are

sketched).

did the effort to rewrite the relevant

In order to

present

our

theory (and the above papers)

coherent way we completely in the language work in

a

algebraic geometry. Although trivial at many points, extra work was often needed to do this. In the rest of this introduction we explain in more detail the different notions which are of

introduced and the main results which

are

established.

1. Affine Poisson varieties

Chapter II by introducing the concept of an affine Poisson variety. An affine Poisson variety (M, I-, J) is an affine variety M (defined over the field of complex numbers) on its algebra O(M) of regular functions, i.e., O(M) with a Poisson algebra structure has a Lie algebra structure I-, J We start

O(M)

X

O(M)

(f,g) which is

a

-4

O(M)

-+

ff,gl

derivation in each of its arguments. The last condition implies that there is f e O(M) a vector field Xf (called its Hamiltonian vector field)

associated to each function

At every point the Poisson bracket has a rank which is constant on a Zariski open subset of M; this constant, which is always even in view of the skew symmetry of the Poisson bracket, is called the rank of the affine Poisson variety and is denoted by RkJ-, .1.

defined

by Xf

=

I-, f I.

regular function whose associated Hamiltonian vector field is zero is called a Casimir. The Casimirs form a subalgebra Cas(M) of O(M) and lead to an important decomposition of M, the Casimir decomposition, which is given by the fibers of the morphism

A

7r :

M -+

Spec Cas(M),

Chapter induced

the inclusion

by

Cas(M)

O(M);

1. Introduction

when

Cas(M)

is

finitely generated, Spec Cas(M) morphism. Picking a fiber may be interpreted as fixing some of the values of the Casimirs; the fibers over closed points are the ones which correspond to fixing all values of all Casimirs, while the fibers over other points in Spec Cas(M) correspond to fixing only the values of the constants of the Casimirs which belong to some subalgebra. A general point in the spectrum being by definition a closed point which does not belong to a certain divisor, a general fiber corresponds to picking "generic" values for all Casimirs. Said differently the fibers of -7r axe just the level sets of the Casimirs. is

an

c

E

affine

algebraic variety

C

and

is

-7r

a

We will show that the Poisson bracket

Spec Cas(M) and that all

vector fields

on

M restrictS3 to the fiber

are

tangent

to these

restriction of the Poisson structure to each fiber is less than

Poisson structure, with that

equality for

a

over any point the rank of the

fibers; equal to the

or

rank of the

general fiber (Proposition 11.2.38). Moreover

dimCas(M)

:. _

dim(M)

where dim Cas(M) is the Krull dimension of

-

we

have

we

have

Rk(M),

Cas(M) (Proposition 11-2.40).

When

equality in this equation we will say that the Poisson bracket is maximal; maximality is preserved by restriction to the general fiber which implies that the general fiber has a rank equal to its dimension. Examples will be given which show that not all fibers (over closed points) need to have the same dimension or rank and that the algebra of Casimirs of the restricted Poisson structure needs not be maximal.

Restricting the Poisson structure duce

new

to

a

affine Poisson varieties from old

level set of the Casimirs is ones.

To

give

a

complete

obvious way to prodescription of some other one

constructions it is useful to introduce first the concept of a morphism between affine Poisson varieties (MI, 1., -11) and (M2, J'i *12): it is a morphism 0 : M, -+ M2 which preserves the

i.e., the following diagram is commutative.

Poisson structure,

0(w 0.

X

M1

morphism

-+

{','12

0(w

X0*I

0A) A Poisson

0(w

X

10. 0A) T-

M2 does

not

0(mi)

necessarily

map the

algebra

of Casimirs of M2

one of MI. Conditions for this to happen will be given. Also the image of M, by a morphism needs not be an affine Poisson variety since the image needs not even be an affine variety. When the image is an affine subvariety of M2 then it inherits a Poisson bracket from

in the

M2 and the

map to this

image

is Poisson

(Proposition 11.2.16).

An

important property of a morphism 0: A, 1-, -11) -+ (Ai J* 1 *12) Of affine Poisson varieties is that the rank at a point of M, is always higher than the rank at the image of this

point, Vm E M, 3

Poisson

Rk.,J., .11 :: 'Rko(x&

SpecCas(M) gives (by restriction) each

The fibers of

Poisson structure

:

ir

:

M -+

are

algebraic

i

but need not be

irreducible; the

irreducible component the structure of

variety. 4

an

affine

2.

In

particular

one

Integrable Hamiltonian systems

has equality for an isomorphism. This leads to an invariant polynomial as follows. The definition of the rank at a point gives a second

for affine Poisson varieties

decomposition of M into algebraic varieties which we call the rank decomposition. Each element of this decomposition (more precisely its closure) is given by natural

Mi

==

Jp

I Rkpl-, -1

E M


0

bilinear

a

(M)

fp+,-i

fi,...,

following,

at the

by Der' (M).

map

F -1

gets

one

the vector

n-derivations

subspace of skew-symmetric

and its

Hamiltonian

upside down

the above definition

Turning

3. tion

Integrable

Il.

E

O(M) by

fp+q-1)

I

o,ESq,p-i

1:

+

i

...

fa(p+q-1))

aESp,q_i

shuffles the set of (p,q) a of (permutations 11,...'p is the sign of a). < a(p + q); c(a) a(p) and a(p + 1) < P E DerP (M) and Q E Derq (M) then [P, Q] E Derp+q-1 (M). Thus

Sp,

where

o-(1)




system Let

Lemma3.2

context

of

Let

3.1

f f, Al

choice

Hamiltonian

involutive

called one

fixed

f

c-

systems

-1)

be

0;

we

an

Poisson A

Poisson

say that The triple

A.

affine

on

affine

we

it

variety. complete if

is

(M,

A)

varieties

subalgebra

moreover

called

is

a

for

A of any

(complete)

f

O(M) E

is

O(M)

involutive

-

(M,

A)

be

an

involutive

If A is complete then A is integrally The integral closure of A in O(M) A is finitely generated.

Hamiltonian closed

is also

system.

O(M);

in

and is

involutive

finitely

generated

when

Proof The

proof of (i.)

goes in

exactly

the

the

proof of Proposition 2.46, replacing that if A is finitely generated then its integral in closure of all elements 0 of O(M) for which there as the set exists with coefficients in A, which has 0 as a root) a monic is also a finitely polynomial generated algebra (see e.g., [AD] Ch. 5). To check that it is involutive, check that we first element of the closure of in is A involution with all elements of integral A. every Thus, let 0 be an element of O(M) for which there exists a polynomial

Cas(M) by

A and g

E

O(M) by g Ei O(M) (defined

p(X) for

which

P(0)

=

of minimal

degree.

Proposition checked by

2.46 a

all

f

E

that

similar

=

0 and with For

any

10, f I argument

=

0,

that

same

way

as

A. It is well-known

Xn +

ai

a1Xn-1

any

-

-

-

+ an

belonging to A; we assume that the polynomial 0 implies the proof as in equality f P(o), f J the P. of it can now using minimality Using this, in the integral two functions closure are in involution.1

A the upon

+

=

47

is

of

be

Chapter

algebra not unique.

Every involutive latter

is in

general

of

closure

integral

an

involutive

is contained

in the

is contained This

in

(M, 1-, .1, A) be an involutive of A. field of fractions

Let

Lemma3.3

Hamiltonian

Integrable

II.

systems

algebra following

which is

by A the

system and denote

Hamiltonian

but the

complete,

lemma.

the

(3.) The subalgebra An o(m) of O(M) is also involutive; A; (2) If A is complete then A n O(M) involutive contained in is an A subalgebra B of O(M) (3) =

if dim B

=

which is

complete;

it is

unique

dim A.

Proof Recall

O(M) in

A,

for

(e.

g.,

[AD]

from

which there

which has

0

Ch.

exists

a

if

root.

as a

A n o (m) can be identified polynomial (which is not necessarily 0 E A n O(M) and

5)

P(X) is

a

follows

at

aXn-I

aoXn

+

(with

coefficients

degree

+

-

-

-

the set of elements

monic)

0 of

coefficients

with

+ an

ai

in

A)

for

which

P(O)

=

in the

of P

the

=

=

Proposition

JP(O), O'l

=

as

0, then proof of 0 implies minimality (again as 0, upon using 10, Al 2.46). In turn this implies that if 0' is another element of An 0 (M) the equality 0. Thus A n O(M) is involutive, 0 leads to 10, O'l showing (i.); from it (2) of minimal

polynomial

JP(O), Al

that

=

=

once.

complete we pass to AO A n O(M); if the latter is complete it is the unique involutive subalgebra of O(M) which contains A and is complete. If not, we to 0 and repeat the above construction element add ail f E O(M) \ AO for which If, AOI of because number of finite after done 1 a dim AO + we are steps; obtain A,. Since dim A, is not unique in general examples (interesting the choice of f the algebra which is obtained but not

If A is involutive

=

=

==

of this

are

given below).

0

in involutive algebras of the maximal possible only be interested Weknow from Lemma3.3 that such an algebra A dimension, given by the next proposition. will denote which A I if A we has a unique completion, by Compl(A) (or by Complf fl, is generated A 1) by If,, In

this

text

will

we

.

Proposition

3.4

.

.

Let

-

,

(M,

A)

be

an

Hamiltonian

involutive 1

dim A : ' , dim M

-

2

Rkj-,

system.

Then

(3.1)

.1.

Proof

general fiber.F A C O(M). By Proposition map Consider

a

also

involutivity

which

is

induced

by

the

inclusion

2.37, dim.F

dim.F

SpecA

of the map M-+

=

dim M

-

of O(Y) equals the number of independent derivations constructed be derivations such can that of A implies 48

(3.2)

dim A. at

general point using functions a

of F and from

A.

3.

To

see

the

97 is

mE

latter, arbitrary

Integrable

recall

Hamiltonian

the ideal

that

and

but fixed

f

X.

nested

-

Cas(M) independent

dim

Aj+j

dim

independent

point) It

morphisms

by the functions

A. For any g

over

If, gj

=

A

E

dim

=

fields

vector

obviously following

then

gives the

derivations,

ni

A,

+

lower

1,

C

(2.40),

(3.2) dimA

Wefinally

get

C

A2

:5 bound

(3.3)

1,

ni +

-

X'-"(f)

where

have

we

0,

=

a

lower

Rkj

=

A, (i.e.,

no

=

to

the

Y and

bound for

construct

we can

elements

of A lead

diM.F.

of

=

to

Consider

a

-

-

,

O(M),

1.

If

ni

denotes

having independent

0 and n,

=

vectors

follows

It

r.

number of

the

that

at

ni

general

a

i for

=

dim Cas (M).

-

all

i.

(3.3)

find

M+ dim Cas (M))

to the definition

A,

c

...

r

dim A

>

we

(dim

C

particular

in

ni+l

and


1 with topological algebraic group. Then Hom(7r, (E), G) is if p : 7r, (F,) --+ G and g E G more precisely variety on which G acts by conjugation, is the homomorphism ir,(E) G defined by -+

fundamental an

from

apart

-

is

decomposed (in just a three-holed

be

2 trinions consist of 2g sphere and such a (in the case of genus two there exist precisely two such decompostions) Each trinion bounded being by three curves (which two by two) one gets 3g are identified 3 curves on E and what is important here is that they are non-intersecting. Calling these curves C, Gg-3 we find from Goldman's formula (3.6) that the functions thus one obtains are in involution; fo ...... an involutive algebra -

-

I

...

53

I

Chapter

A

Compllfc

=

fc,,, -j

......

bracket

the Poisson

3g dim M

for

i.e.,

by

one

(2g

-

dim

relation,

2)

A will

3

-

=

Since

6.

-

integrable

dim G and A is

for

dimG

integrable fields corresponding 3.2.

Since

3.

=

G

for

=

Morphisms

SL(2); to all

with our parallel morphisms of integrable In

then the

2)

-

G,

to

G

of

Hamiltonian

-

Since the rank

3.

of

only if dimM, bound

are

dimension

we

find

that

A is

the Hamiltonian

that

pictures

only

vector

systems of affine

morphisms

SL(n)

=

super-integrable.

actually

Hamiltonian

discussion

3g

which 2g generators, has M hence dim G, 1)

of -

dim

above

the

are

2

(2g

ourselves

fc,

functions

=

to be

if

(2g

=

1

-1

system

a

only

from

Poisson

varieties

we now

turn

to

systems.

(M2&,'j2,A2) -+ (M2ij*i'j2iA2)

and

Let

Definition3.12

systems, M2 with

clear

integrable

of

2

restricted

we

it is

Rkj-,

-

has

6

-

if and

dimension

if and

6g i.e.,

1

dimM

systems

computed

is

integrable

be

iri(E) Hom(iriL (E), G) has

6g

=

dimension

and its

maximal,

is

Hamiltonian

Integrable

11.

morphism 0: (Mj,j-,-jj,Aj) properties following a

be two is

Hamiltonian integrable a morphism 0: M,

(j-) 0 is a Poisson morphism; (2) 0* CaS(M2) C Ca$(MI); (3) O*A2 CAI -

Schematically,

(2)

of the map and

regularity

Cas(M2)

is in

Ikom the

phism (hence

very we

-+ we

it

definition have

a

represented

be

-

Al

-

such

clear

category).

It

a

-

map an

the

that is

follows:

(3.7)

0*

(W J'i *12, A2)

call

is

-------

as

O(M2)

0*

Cas(MI) morphism 0: (M17j*7"j1iA1) a morphism: automatically the diagram to be bijective).

can

A2

-

0.

A

(3)

and

also

O(Mi)

has an inverse biregular all inclusion forces isomorphism (it

which is

immediate

morphisms any biregular

of two

composition that

for

is

which maps

a mor-

0 : unique

map

Hamiltonian system (MI, 1- 7 -11, A,) there exists a M2 and for any integrable and a bracket on algebra A2 C O(M2) such that M2 Poisson unique integrable 1. -12 is A, and A2 an isomorphism; explicitly 0: (Ml 7 J* i'll IAI) -+ (M2 I i j 2 1 A2)

M,

-+

*

i

Ifi 912

(0-1)*

10*f7 0*911 54

Vig

G

O(m2)-

Integrable

3.

(i.)

Conditions

(2)

and

axe

3.13

Hamiltonian

conditions

algebras.

the level of the integrable phism of the corresponding proposition. on

Proposition

Hamiltonian

at the

spaces

resp.

1

following

-+

a

be

0 induces a morfollowing

is shown in the

morphism of integrable

a

diagram commutative, M2

I7rC-(M2) Spec Cas(M2)

morphism Spec A,

:

which

*121 A2)

that

than

Spec Cas(M2)

Spec Cas(MI)

as

as

rather

Then

7rc-(Ml)I well

structures,

(3) implies

base spaces,

M,

as

morphisms

of the Poisson

0: (MI, 1-, -11, A,) -+ (M2i I' 0 induces a morphism

Let

systems.

makes the

(2)

resp.

0: Spec Cas(MI) which

level

Condition

paxameter

and their

systems

makes the

diagram

following

Spec A2

commutative.

M,

M2

IrAjI

I-A2

Spec A, If 0* Cas(M2)

=

Cas(MI)

(resp.

O*A2

=

Spec A2

Aj

(resp.

then

)

is

injective.

Proof

The first of

0* implies Said

mapped into

into

following

are

a

level

level

condition set sets

(3)

A2; condition

and relations

from

in

Of A2 and if of

meaning examples and propositions. the

diagram (3.7) by taking of the corresponding spectra.

immediate

at the level

injectivity

differently,

different

illustrate

assertions

Definition

O*A2

(2)

3.12

implies

that

spectra;

also

0

each level

level sets of A, A, then different be given a similax interpretation.

can

between

the three

55

conditions

surjectivity

in

Definition

set are

A, is mapped

of

Wefurther

3.12

in

the

Chapter

Example

Let

3.14

Consider

C4 (with

8ij,

I

as

Jqi,

qj Casimir.

show that

us

coordinates

I

fpi,

pj We look

=

Integrable

11.

Hamiltonian

Definition

in

P2) C-3 (with coordinates

q1, q2 7 P1 i

0, and this C3

=

(2)

neither

3.12

with

systems

the canonical q1, q21

nor

Poisson

PI)

with

(3)

follow

Jqi,

structure

Jq1,

p,

(1).

from

I

pj I I and q2

=

=

in C4 and denote by 0 the projection qlq2PI-plane however morphism, O*q2 is not a Casimir of C4 showing Notice that in this case 0 does not induce a map 0 as in (2). 3.13. Taking two different functions the algebras generated by them) on C2 (i.e., does not imply W(i.) at

map along P2. Then 0 is that (3.) does not imply

Proposition shows that

a

the

as

Poisson

,

3.12 large class of morphisms for which condition (2) in Definition that of closed these include the (i.), namely universally morphisms; proper the finite morphisms and, in particular, morphisms (see [Har] pp. 95-105). Weprove this in the following however we restrict ourselves to the case of finite proposition, morphisms, since in result the this however use case we will verbatim to the case of only (the proof generalizes closed universally morphisms).

There is however

follows

a

from

(MI,

-11)

be two affine Poisson varieties and (M2, J*)'12) a finite M2 0 M, morphism (for example a (possibly suppose ramified) covering map). If 0 is a Poisson morphism then 0* Cas(M2) C Cas(MI); if 0 is moreover dominant then Cas(Mi) closure is the integral of 0* Cas(M2) in O(Mi).

Proposition

Let

3.15

that

and

is

-4

:

Proof

show that

Let

us

elements

of

morphisms

0

if

is finite

then

for

any

f

Cas(M2),

E

O*f

is in involution

The main property which is used about finite O(MI). is that if 0: M, -+ M2 is such a morphism then O(MI) is

where P' denotes as

desired.

the derivative

of the

we

take

an

O*O(M2). Cas(MI)

in

proving that be arbitrary,

element

g E

Cas(MI)

We show that is the

integral

P has

closure

=

10*f P(g)11

=

for

all

polynomial

we

f

E

degree)

with

gI of P we find

JO*f gJ ,

=

0

Cas(MI).

and call

polynomial

P its

actually of 0* Cas(M2).

coefficients

its

in

To do

this,

as

above,

with

coef-

0* Cas(M2), thereby let O*f E O*O(M2)

JO*f, g'

,

+

O*alg'-'

find

0* If,

that

O(M2),

has its

so

that

ai

ai

coefficients

I

=

E

0 for

all

Cas(M2)

+

i.

for

O*O(M2) Since i

=

56

1,

0 -

-

0 *a.11

10*f, O*a.11

O*Ifi

+'*'+ in

+

+... +

O*Jf,a1J2gn-1 degree,

,

By minimality

P.

C

10*f, O*alllg'-' this

P,(g)lo*f

minimal

then 0

Since

=

polynomial

0* Cas(M2)

We have shown that

Next

ficients

10*f, PWI

=

all

(or universally closed) over O*O(M2). integral

Thus any element g E O(MI) is a root of a monic polynomial P (of in O*O(M2). As in the proof of Proposition coefficients 2.46 we find 0

with

anJ2-

and since is dominant .'

n.

P

it

was

supposed of minimal

follows

that

If,

ai

0 0

Integrable

3.

It

be

can

in

seen

Hamiltonian

way that

similar

a

and their

systems

0

if

:

morphisms

(MI, {-, -11, A,)

Hamiltonian systems which morphism of integrable Of O*A2 in O(Mi) closure (for a proof, use completeness integral corollary. following Let (MI, I 3.16 j 1, A,) -+ (M2 Corollary whose image is an affine subvariety of M2. Then 0 morphism. su7jective 1

21

A2)

A2)

(M21

-+

and dominant

is finite

A,).

of

A,

then

leads

It

is

a

is the to

the

and be a morphism which is finite and a of an injective composition

is the

Proof 2.16 that, Proposition 0 s o . Define

Weknow from

1-, -1)

(O(Mi),

say

f,g

A

E

*f

Then for an

*Jf,gj

have

we

If, A}

If

involutive.

=

O(MI) is integrable

the

same as

the

E

J *f,

I *f

O(O(Mi))

J *f, *g}

=

0 then

Hamiltonian

If

=

A,}

=

0

can

be

decomposed via

for

one

Ai}

of * we see that A is 0; by injectivity closure of O*A in O(MI). A, is the integral also complete. Finally the dimension count

0

M, since

Clearly

system.

E

=

0 since

of A, and A is

A, by completeness

E

morphism,

Poisson

=

A For

a

as

%

axe

follows

It

is finite.

and

of

1-, -1, A)

(O(MI),

that

morphisms

is

Hamiltonian

integrable

systems.

3

Example

then

dominant

If

3.17

Poisson

a

Cas(M.1)

morphism 0

may be

example for (M2, J* '}2) ample 2.54), for M, the plane x

larger

map. Then

Cas(M2)

=

C[xl

the

than

the Lie-Poisson

Take for

=

0* Cas(M2)

hence

-+

integral structure

the trivial

0 with

(W 1'7 *12) is finite but not of 0* Cas(M2) in O(MI). closure for the Heisenberg algebra (Ex-

(MI, l'I'll)

:

=

Poisson

and for

structure

C, while Cas(MI)

=

0

the inclusion

O(MI).

7,11) -+ (W J*,*12) is finite and morphism 0: (Mill* Take for example on C3 the from 0* Cas(M2). dominant then Cas(MI) may be different from Example 3.14 and consider the finite Poisson structure covering map 0 : C3 -+ C' given however the Casimir q2 is Poisson this a morphism; (qj, pl, q22). Obviously by O(ql, pl, q2) C -+ C is in this that Notice F for function E O(C3). is not of the form O*F any 2 to condition remark applies (3) in case not injective, being given by (q2) q2 A similar Even if

Example3.18

a

Poisson

=

=

In Section

Using

ones.

systems fiber

on

2.3

these them.

we

ducible

gave several

give

we now

Wefirst

of the parameter

Proposition

integrable

of

Constructions

3.3.

.

3.12.

Definition

3.19

Hamiltonian to build

constructions

the

show that

corresponding an integrable

new

Hamiltonian

A)

(M, general

Let

is

an

integrable

of

any

Poisson

for

from old

varieties

Hamiltonian

integrable

system restricts

Hamiltonian

Then level of the Casimirs. component of a is the inclusion and a morphism. map System

levels

affine

constructions

to

a

general

map.

Hamiltonian

general

systems

subalgebra

of

the

Casimirs.

57

system and

(,F, f -, JI.F, AI.F) The property

is

also

an

T

an

irre-

integrable for the

holds

Chapter

II.

Integrable

Hamiltonian

systems

Proof B be any subalgebra of Cas(M) and let Y be of M-+ Spec B. Weknow already from Proposition

Let

fiber

and from

structure

If

we

is

complete

and Y is

dimY This

Proposition

A to Y then

restrict

-

3.19

dimM

Ay is

an

suffices

dim B

-

algebra.

integrable

is called

One may think

of

subsystem

trivial

a

of Casimirs

algebra

A

of this

2

the inclusion

obtained

from

2 a

RkJ-, -I.F. morphism.

(M,

by fixing

A

since

A,77,

of

map is

general Poisson

is maximal.

structure

RkJ-, .1

=

system obtained

being

a

induced

an

Ap which is complete

I

B)

dim

-

Clearly

as

Y has

compute the dimension

to

(dim

-

component of

2.38 that

involutive

an

Any integrable Hamiltonian a trivial subsystem.

Definition3.20 sition

Thus it

algebra

the

get again

general.

dim A

shows that

2.42 that

we

irreducible

an

2

A) by Propo-

the values

of

of

some

the Casimirs.

Example

examples

In the

3.21

(i.e.,

fiber.F

in the choice

check that

one

of values

has however

assigned

sense general enough The dimension general fiber. rank may be lower; then

those

with

in the

F is of

dim.F

AI.F)

(F,

so

integrable that

is

Proposition -7ri

trivial, 3.22

integrable

an

system

Al,

while

.1y

11, 21

when

Casimirs).

both the dimension

of

a

>

dhnA

on

the fiber

let

projection

(MI

special

fiber

picking a particular Namely one has

and rank

F may be

to

of Y coincide

and/or

higher

its

dimAly,

x

system. Reconsider e.g. Example 2.54: C' for this Poisson structure will lead to an

=

0,

since

X

M27 f'i

(Mi, I., Ji, A,) map

M,

x

be

the

induced

an

integrable

Poisson

structure

on

Hamiltonian

system

M2 -+ Mi Then

-1m, xm2,-7r,*Al

Hamiltonian integrable system and the projection Hamiltonian of the integrable system is a product of level Set Of (W J* i'}27 A2)-

is

the

Hamiltonian

systems

on

be careful

54 0(.F)

r

For i E

the natural

denote

Rkf-,

-

Hamiltonian

Hamiltonian

fiber

and let

is not

integrable

of the

none

that

a

to

(some of)

to

an

set

0

*7r2*A2)

maps 7ri a

level

(3.8) Each level morphisms. of (MI, f -, -11, A,) and a

are

set

Proof The

Poisson-part

of

this

proposition

already

was

given

in

Proposition

2.21.

involutivity,

firi Ai

(2)

7r2* A2 7r,*1 Al ,

(9

7r;2 A2 I

mi

.

m,

58

-"::::

7ri 1*J&

A111

+

1r*2JA2, A212 2

0-

As for

Integrable

3.

Wecount

Hamiltonian

morphisms

dimensions:

-7r,*Al

dim

7r2*A2

0

=

dim A, + dim A2

=

dim Mi

?r,*Al

?r2*A2

1

complete

RkJ-, -11

-

2

dim(Mi

=

Since

and their

systems

X

M2)

Rk

2

and involutive

+ dim

with

M2

1

Itkf*

-

2

1'12

Q-, JM1 xM2) the

this product bracket, earch of the projection 7r,*Ai (8),7r2*A2 computation integrable. maps iri these projection The fibers of the C 7r1*A1 0 7rM2, one has -7ri*Ai maps are morphisms. of M, x M2 -+ Spec(7r,*Al that is, of the momentum map are given by the fibers 0 lr2*A2), product map M, x M2 -+ Spec A, x Spec A2 hence all fibers are products of level sets of A, and A2. I (8)

is

shows that

It

is easy to show in addition

if

Ham(Ai)

field

(or Ham(A2))

Definition3.23

Wecall which

A construction which will

be used several

product

the

related

to

(but

on

of

7r2* A2)

contains

(M1,J-,J1,A1)

a

and

from)

different

chapters,

in the next

times

systems which depend

Hamiltonian

(9

super-integrable

vector

does.

(3-8) is

Ham(-7r,* A,

that

to

respect

Since for

is

the

is obtained

By this

parameters.

(M2,J',*}2,A2)-

product

when

we mean

construction

dealing that

with

we

and

integrable

have

an

affine

of parameters we have an c of a set possible (M, I J) variety algebra A, on it. This set of parameters is assumed here to be the points on an integrable N and we assume that A, (i.e., its elements) affine variety on c. Then we depends regularly which contains all the integrable Poisson variety Hamiltonian a big affine can build systems proposition.8 subsystems. This is given by the following (M, 1., .1, A,) as trivial Poisson

and for

-

,

values

all

Hamiltoaffine variety and for each c r= N an integrable Poisson variety nian system (M, I., Jm, A,.), on c is given on an affine depending regularly (M x N, I-, J) and of an affine Poisson variety (M, 1-, .1) then M x N has a structure subalgebra A such that each (M, I-, Jm, A,) is isomorphic O(M x N) contains an integrable x to a trivial subsystem of (M N, 1-, -1, A) via the inclusion maps

Proposition

If

3.24

N is

an

0,:

M-+ Mx N:

mi-+

(m,c).

Proof For N one takes a

Poisson

manifold.

the trivial The

is maximal

and

Cas(M

that

there

exists

means

of the 8

projection

:

Mx

so

of Casimirs

that on

Cas(N) this

=

product

O(N)

which makes Mx N into

is maximal

since

the

:

P -+

-7r(-)

general

N is

(n)

n; both

corresponds

to

one on

M

on c N) Cas(M) (9 O(N). The fact that A, depends regularly to A, on the fiber over c subalgebra A of O(M x N) which restricts N -+ N. Clearly its dimension is given by dim A dim A, +dim N =

=

in Example 2.24, namely when considered to the situation generalizes a dominant morphism, for each n E N, I-, -In is a Poisson bracket on the and An is an involutive for subaJgebra of 0 (-7r(- 1) (n)) which is integrable N. and 3.24 to G axe on n An supposed depend regularly Proposition I-, Jn

The proposition

fiber

ir

p,

algebra

x a

structure

the

special

case

P

=

Mx N considered

59

at

the end of

Example

2.24.

Chapter so

dim A

that

O(N)

Since

=

is

a

the restriction which is

subalgebra

isomorphism

construction

so

-

(one

1

Rkf Cas(M

corresponds

when restricted

discuss

to such

is that

of

Hamiltoniau

setup which

the

to

a

and involutive a

level

Mvia

one on

is

it

integrable.

of the Casimirs

set

the

and

mor Phism.

fiber.

taking

quotient.

a

This

is of

interest,

because

systems possess discrete or continuous symmetry here has the virtue to allow to pass easily to the

we use

does not need to worry

systems

A is complete the fiber over p is

N)

x

structure

integrable

Hamiltonian

since

2

of

we

many of the classical The algebraic groups.

quotient

N)

x

of the Poisson

an

The next

dim(M

Integrable

II.

about

the action

being free,

picking

on).

regular

values

and

Proposition

3.25 Let G be a finite reductive or a Poisson action group and consider is A M, where (M, 1-, -1) is an affine Poisson variety. involutive an If algebra such that for each g (=- G the biregular map X, : M-+ Mdefined by X, (m) X(g, m) leaves A invariant, C A, then (MIG, j.'.10, i.e., X*A AG) is an involutive Hamiltonian system 9 and the quotient Here 1., -10 is the quotient bracket on MIG given by map -7r is a morphism. 2.25. G then is is Proposition finite If (MIG, f.,.}O, AG) integrable.

X: G x M-+

=

Proof

of AG is immediate from Proposition Involutivity completeness of A implies completeness of A n

Then a

finite

group

we

O(M)G

=

dim.A

=

dimM

=

one

An

we

used in the first

shows that

O(M)G

equality

algebra integrable;

that

dim

O(M)G.

As for

G is finite.

that

now

dimensions,

since

G is

2

M/G

I -

2

-1

-,

Rkf

-,

-jo,

dim O(M) and A c O(M). Similarly being given by Cas(M) n O(M)G. Thus O(M)G) C A, hence the quotient map is a =

maximal,

is

-7r*(A

obviously

1Rkf

-

O(M)'

dim

of Casimirs

the

is

Suppose

have

dimAn

where

2.25.

n

morphism.

0

We will

Example

encounter

A

3.26

O(M)G). Namely, the level

sets

A similar

of

-

,

for

action

correspond

to

j

o

leaves the

5

quotient interesting.

A (&

diagonal

occurs

each level

A)

Here

sets

are

when A C set

(M, f

of

precisely

are

the level

The look

examples later.

case

case

(M, I-, -}, A) and consider M x M by interchanging and the

of

special

(MIG, I

Example 3.27 systems

lot

in this

applies

result

which

a

the

-,

leads

One may e.g.

J, A)

quotients

to

first

observations.

O(M)G (which implies

of the Casimirs

construction

some

a

with

is stable

for

of the level

in

case

lot

of

Cas(MIG)

the action sets

of

Cas(MIG) new

integrable

C

(M, f

c

of G and -

-

,

O(M)G.

1, A).

Hamiltonian

an integrable Hamiltonian system M, I-, -Imxm, A (9 A). The group Z2 acts on the factors in the product. Obviously this is a Poisson action A invariant, to The level sets which a quotient. thereby leading are symmetric level sets. products of the original

its

square

(M

x

60

start

3.

the group G in For future

that

Notice

Integrable

of M.

phism. group quasi-automorphism.

Hamiltonian

systems

Proposition

3.25

use

we

be

can

introduce

morphisms

and their

also

slightly

(A I-, J,A) bean integrable Hamiltonian an isomorphism (M, I-, -}, A) -+ (M, I-, -}, A). More generally, an isomorphism Poisson brackets on Mthen (M, -.11, A) quasi-automorphism. Definition3.28

Let

is

The final

Proposition be a function

construction 3.29

which

is to

(M, 1-, -1, A)

Let

is

remove

not

a

be

divisor

an

integrable

Then there

constant.

phase

from

subgroup of the automornotion of a more general

seen as a

the

J* *12 1., -11 (A {-, '12, A) is

-+

AN)

and

1

are

two

called

a

space.

system and let f

integrable

an

E

Hamiltonian

morphism (N, J* 7'IN7 AN) -+ (M, 1-, -1, A) which the complement (in M) of the zero locus of f as image.

(N, f"i'lN,

automorphism

and

Hamiltonian exists

An

system. if

is

a

O(M) system having

dominant,

Proof

Most

of the of that

notation

proof (the proposition.

Poisson

part)

We start

was

with

given

the

Proposition

in

f

case

E

A.

If

2.35 we

and

we use

AN

define

dimension since 7r is a Poisson morpbism and it has the right then AN is involutive then Let EE Weneed to verify to be integrable. fiti O(N) completeness. Ein-0 j-

IN

n

fit',

in order

n

AN

i=O

the

7r*A[t]

:--

0

:>

Effi,

7r*A[t]lNti

0

i=O n

Effii

lr*AlNfn-i

0

i=O n

1:1& Alfn-i

0

i=O

E ffn-i,

A

0

i=O n

E fjn-i

E

A

i=O n

1:

ffn-itn

G AN

i=O n

1:

fit'

CE

AN-

i=O

Since desired

the

AN is involutive equivalence.

last

line

also

implies

the

first

line,

so

we

have established

the

if (M, I available Of AN is still J,A) satisfies description explicit also N of In that case the fibers 3.7. of Proposition the conditions satisfy the Spec 7r*A In general 7r*A. one has 3.7 hence -7r*A is complete and AN of Proposition conditions available. is not and a more description explicit AN Compl(-7r*A) If

f

A then

an

-

,

=

61

Chapter

Compatible

3.4.

Integrable

11.

Hamiltonian

and multi-Hamiltonian

We now introduce

a

few concepts

systems

integrable

which

relate

to

systems

compatible

integrable

Hamiltonian

systems. ion

Definit

3.30

Let

i

1,

be

(linearly

Poisson independent) compatible Hamiltonian for each integrable system i these systems axe called n then 1, Hamiltonian compatible integrable systems. Any field Y on Mwhich is integrable vector non-zero with respect to (in particular Hamiltonian) all Poisson structures i.e., for which there exist fl, f,, E A such that

brackets =

on an

.

.

affine

variety

=-=

n

(M, I-, ji,

M. If

n

A)

is

an

,

.

.

Y

is called

an

ways;

Remark 3.31

system that satisfied

in

any of the

...

if

integrable

demand in

We do not

vector integrable Examples 3.33 and 3.34

all

=

n

=

1', Aln,

=

2)

Hamiltonian

system (bi-Hamiltonian

multi-Hamiltonian

integrable

f., fill

(bi-Hamiltonian

multi-Hamiltonian

a

many different

=

the

the

fields it

.

.

,

field,

vector

systems when

definition

of

an

be multi-Hamiltonian.

is far

too

restrictive

in

n

since

it is Hamiltonian

(M, I-, ji, 2).

A)

is then

in

called

=

integrable

multi-Hamiltonian

Although

this

condition

is

general.

All

and basic constructions propositions given above are easily adapted to the case of or multi-Hamiltonian but this will not be made explicit here. compatible Just structures, of a reductive which one example: an action is Poisson action with a both group respect to of two compatible Poisson structures Hamiltonian integrable systems yields on the quotient two compatible Hamiltonian which are specific integrable systems. Here are some properties to compatible Hamiltonian integrable systems. 3.32 Proposition (1) Compatible

(2)

integrable

The Poisson

rank,

Hamiltonian

systems have the same level sets; Hamiltonian of compatible integrable systems have the same equals the rank of a general linear combination of these Poisson

brackets

also

which

structures

If (M, I., -1j, A)

(3)

linear

integrable

compatible

are

I-, +x of

combination

Hamiltonian

integrable

the Poisson

Hamiltonian structures

then

system

the system

for

(M,

general A) is an

a

system.

Proof The

proof of (l.)

is obvious

since the level

sets

are

determined

the rank ofall

structures

is

by A only. Since Rkf ji I., ji equal. To determine the rank of a linear combination of these structures Poisson matrix (with one looks at the corresponding of O(M)) which is given by the same linear combination of respect to a system of generators the Poisson matrices of the structures I-, ji. Now a general linear combination of invertible matrices is invertible, which applied to a non-singular minor of size Rkj-, ji leads to (2). 2 dimM-2 dimA

For

a

dimA

linear =

we

find

combination dimM

showing W-

-

1L 2

Rkj-,

that

I-, .1,\ ji,

of

(maximal) rank Rkj-, jj one has (M, I-, +\, A) is an integrable

hence

62

that

JA, A},\

Hamiltonian

=

0 and

system,

Integrable

3.

We will

examples Example qj,

integrable

q2, p,

P2)

For A c

O(C4)

structures

are

are

all

Example

compatible Here

the Poisson

1-, -11

structures

1' J2

and

1

C4 (with

on

by

0

0

1

0

0

0

0

1

0

0

0

1

0

0

1

0

-1

0

0

0

0

-1

0

0

0

-1

0

0

-1

0

0

0

take those

the Poisson

which

independent of q, and q2. Then both vector fields are of the form integrable

their

f

are

a

C9 + g

9ql

coordinates

matrices

and

functions

and since

compatible

integrable simple

two

are

systems.

1 f,g

9q2

E

Poisson

A

bi-Hamiltonian. from

Recall

3.34

Example 2.11 that

U( defines

of

systems.

defined

I they

morphisms

examples

integrable

bi-Hamiltonian

Consider

3.33

and

and their

systems

many (non-trivial) multi-Hamiltonian

text

and of

systems of

this

in

encounter

Ha,miltonian

Hamiltonian

any u and F in O(C') to obtain a non-triviaJ

for

5

OF

j','ju,F+G

G}

=

OF Ox

Poisson

Poisson

defines

the roles

0

OY

0

TX_

G is any other non-constant hence 1' 1 *}u,F and J* , ju,G are

dependent, A ComplIF, However, by interchanging

-OF

-OF

;9__V

structure

element

of F and G.

us

denote

O(C3)

of

then

this

Hamiltonian we

find

I-, Ju,F

system A also

that

non-constant

Poisson

and, assuming that

compatible

integrable

an

C3 F is assumed

on

Let

structure.

If

1

OF Oz

OF

a

here in order

by J* juF.

0

the matrix

structure

l'i"ju,G

+

__"

F and G are inon

(C3, J* ju,F),

defines

an

integrable

Hamilsystem on (C3, J* , ju,G) I hence leading to a pair of compatible integrable Since moreover the Hamiltonian fields with respect to both Poisson vector systems.

Hamiltonian tonian

structures

are

given by

fuoVF we

conclude

Closely

that

A defines

related

its

use.

the

Let

sequence of functions

integrable of

concept

hierarchy.

multi-Hamiltonian

explain

to

an

Let

1-, -11 jfj I i

and E

us

The

following

property

is

essentially

integrable

an

this

be two

is called

I-, fiJ2

VG 10

--::

c

Al

bi-Hamiltonian

define

J",'}2

ZI

x

I'

a

i

in the

system

case

of

compatible

(i

due to Lenaxd and

63

C3.

multi-Hamiltonian a

hierarchy E

Z).

Magri.

system

bi-Hamiltonian

Poisson

bi-Hamiltonian

fi+111i

on

brackets if

is that

hierarchy on

M.

of

a

and

Then

a

Chapter

Proposition If

one

of

All

3.35

with

involution

respect with

in involution

functions

fi

of

to both Poisson

functions

these

Integrable

11.

is

a

(for

of

systems

bi-Hamiltonian

a

(hence

brackets

Casimir

the elements

Hamiltonian

I

jfj

hierarchy

with

Z} are combination).

i

E

to any linear

respect

then all these fi of the structures) bi-Hamiltonian hierarchy.

either

any other

are

in

also

Proof If

jfj

I

i E

ZI

forms

a

hierarchy,

then for

any i