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 Second Edition
Y,Vkl
Springer "841
Geometry
Author Pol Vanhaecke
D6parternent de Math6matiques UFR Sciences SP2MI
Universit6 de Poitiers
T616port
2
Boulevard Marie et Pierre Curie BP 30179
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Futuroscope
E-mail:
Chasseneuil Cedex, France
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CIP-Einheitsaufnahme
Vanhaecke, Pol: Integrable systems
in the realm of algebraic geometry / Pol Vanhaecke. 2. Berlin ; Heidelberg New York ; Barcelona ; Hong Kong ; London Milan ; Paris ; Singapore Tokyo : Springer, 2001
ed..
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(Lecture notes in mathematics ; ISBN 3-540-42337-0 Mathematics
1638)
Subject Classification (2000): 14K20, 14H70, 17B63,
37J35
ISSN 0075- 8434 ISBN 3-540-42337-0 ISBN 3-540-61886-4
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Preface to the second edition
book, five years after the first edition, has been spiced with 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. The present edition of this
several recent results which fit
On this occasion I wish to thank my collaborators from the last five years, to wit Christina
Birkenhake, Peter Bueken, Rui 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 axe taken from, or are inspired by, joint work with some of them; I acknowledge their permission to add these, sometimes unpublished, results.
colleagues at my newest working environment, the University of Poitiers (aance), me a pleasant and stimulating working enviromnent. I wish to acknowledge the support of all of them. Special thanks go to Marc van Leeuwen, Claude Quitt6 and Patrice Tauvel for sharing their insights with me, which usually led to a real improvement of parts The
created for
of the text.
least, Yvette Kosmann-Schwambach, who was not acknowledged in the most probably because my gratitude to her was too big and too is thanked here in all possible superlatives, for her constant support and for her obvious! sincere friendship. Merci Yvette! Last but not
first version of this book -
-
Acknowledgments
indispensable for establishing and presenting the results Not enough credit can be given to those who created at home, at the Max-Planck-Institut in Bonn, at the University of Lille and finally at the University of California at Davis a pleasant and stimulating atmosphere. Even some people I don't know by name should be thanked here. The
which
help
are
of many
people
was
contained in this work.
Special thanks
are
due to Mark Adler and Pierre
van
Moerbeke, whose fundamental work
a.c.i. systems was the starting point for the research contained in this book. Stimulating discussions with them have led to an improvement of many of the results and to a better on
understanding of the subject. Also Michble Audin deserves a special plarce here for sharing insights with me through long discussions and letters. Extremely helpful for a thorough understanding were several algebraic-geometric explanations by Laurent Gruson. her
I wish to thank my collaborators Jos6 Bertin and Marco Pedroni for
a
fruitful interac-
tion. I have also benefited from discussions with my colleagues at Lille, in particular Jean d'Almeida, Robert Gergondey, Johannes Huebschmann, Rapha6l Freitas, Armando Treibich,
Gijs Taymnan and Alberto Verjowski and at UC Davis, in particular Josef Mattes, Mulase, Michael Penkava, Albert Schwarz and Craig Tracy.
Motohico,
I also acknowledge my other friends scattered around the globe, to wit, Christina Birkenhake, Robert Brouzet, Peter Bueken, Jan Denef, Paul Dhooghe, Jean Fastr6, Ljubomir Gavrilov, Luc Haine, Horst Knbrrer, Franco Magri, Askold Perelomov, Luis Piovan, Elisa Prato and Taka Shiota for their interest in my work and helpful related discussions. For useful comments
on
the manuscript I
referee and several students in my Last but not
this adventure.
graduate
least, special thanks
am
indebted to Mich6le
course
in UC Davis
Audin,
an
anonymous
(Spring 1996).
to my wife Lieve for her constant assistance
through
Table of Contents
1. Introduction IT.
.
.
.
.
.
.
.
Hamiltonian systems
Integrable
1. Introduction
.
.
.
.
.
.
.
.
2.1. Affine Poisson varieties 2.2.
.
.
.
2. Affine Poisson varieties and their
.
.
.
.
.
.
.
.
morphisms
.
.
.
.
.
3.
Morphisms of affine Poisson varieties
Decompositions
Integrable
Hamiltonian systems and their Hamiltonian systems
.
Integrable
.
.
on
.
.
systems
2.2. The
.
.
.
.
and their .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
case
.
.
3.1. The real and
.
complex level
3.2. The structure of the
.
.
.
.
.
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.
.
.
.
.
.
Compactification
significance
.
.
.
.
.
sets
1.
17. 17.
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19.
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19.
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26.
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28. 37.
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47. 47.
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54.
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57.
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62. 65.
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65.
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69.
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.
curves
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71.
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71.
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73.
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73.
-jwd
73. 78.
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.
.
.
complex level manifolds
of the
.
.
.
3.3. The structure of the real level manifolds 3.4.
.
I., -1d'
3. The geometry of the level manifolds
3.5. The
.
.
Poisson spaces
on
.
.
morphisms
structures
in involution for
hyperelliptic
.
.
integrability
compatible Poisson
PolynomiaJs
2.4. The
.
.
Hamiltonian systems and symmetric products of
2. 1. Notation
2.3.
.
.
.
other spaces
Integrable Hamiltonian systems
1. Introduction 2. The
.
.
.
Compatible and multi-Hamiltonian integrable systems
.
.
.
affine Poisson varieties
on
.
.
integrable Hamiltonian systems
Hamiltonian systems
.
.
Morphisms of integrable Hamiltonian systems
Integrable
.
.
.
Integrable
4.2.
.
.
3.2.
4.1. Poisson spaces
111.
.
3.1.
3.4.
.
and invariants of affine Poisson varieties
3.3. Constructions of
4.
.
.
2.3. Constructions of affine Poisson varieties
2.4.
.
affine Poisson varieties
on
.
.
.
.
complex level manifolds
of the Poisson structures viii
-j'Pd
.
.
83.
.
.
.
85.
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.
.
85.
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87.
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89.
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93.
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95.
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.
IV. Interludium: the
1. Introduction
.
.
geometry of Abelian varieties
.
.
.
.
2. Divisors and line bundles
2.1. Divisors
.
.
.
2.2. Line bundles
.
.
.
.
.
.
.
Hyperelliptic
3. Abelian varieties 3.1.
Complex
.
.
.
on
4. Jacobi varieties
.
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.
97.
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99.
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99.
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100.
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101.
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103.
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.
105.
in
projective
.
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106.
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108.
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108.
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109.
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111.
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114.
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.
algebraic
4.2. The
analytic/transcendental
.
.
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.
114.
Jacobian
.
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.
114.
4.3. Abel's Theorem and Jacobi inversion
4.4. Jacobi and Kummer surfaces
V.
generic
5.2. The
non-generic
(1,4)
.
.
case
.
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119.
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121.
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123.
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123.
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.
124.
Algebraic completely integrable
1. Introduction 2. A.c.i.
.
systems
3. Painlev6
Hamiltonian systems
.
.
.
.
127.
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127.
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129.
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135.
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138.
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.
140.
.
VI. The Mumford
.
.
analysis for a.c.i. systems
equations
.
.
.
4. The linearization of two-dimensional a.c.i.
5. Lax
space
.
Jacobian
case
97.
.
.
4.1. The
5.1. The
.
.
.
.
5. Abelian surfaces of type
.
.
Abelian varieties
3.3. Abelian surfaces
.
.
tori and Abelian varieties
3.2. Line bundles
.
.
embeddings
curves
.
.
2.4. The Riemann-Roch Theorem
2.6.
.
.
.
2.3. Sections of line bundles
2.5. Line bundles and
.
.
.
.
.
systems
.
.
.
.
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.
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.
.
.
.
systems .
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143.
.
1. Introduction
.
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143.
2. Genesis
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145.
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145.
2.1. The
.
.
algebra
of
pseudo-differential operators
.
.
.
.
.
.
.
.
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.
.
146.
2.3. The inverse construction
.
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.
150.
2.4. The KP vector fields
.
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.
152.
2.2. The matrix associated to two
.
commuting operators
ix
3. Multi-Hamiltonian structure and
3.1. The 3.2.
loop algebra
4. The odd and the
4.2. The
4.3.
(odd) even
.
.
.
.
general
Mumford system
case
.
.
.
.
VII. Two-dimensional a.c.i. 1. Introduction
.
.
Mumford systems
even
Mumford system
.
.
.
.
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.
.
2.2. The genus two
even
Application: generalized
.
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155.
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157.
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161.
potential
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161.
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163.
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164.
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.
168.
and Laurent solutions .
.
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.
.
Linearizing variables
5.3. The map M -+ M .
6. The H6non-Heiles
.
.
hierarchy
.
.
6.1. The cubic H6non-Heiles 6.2. The
.
.
.
.
7. The Toda lattice
.
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175.
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177.
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177.
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179.
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181.
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185.
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185.
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186. 190.
.
of order three
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196.
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196.
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202.
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206.
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211.
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216. 220.
.
.
(1,4)
to the genus 2
even
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220.
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222.
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226.
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230.
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230.
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232.
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233.
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235.
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235.
.
.
.
.
.
.
.
.
.
.
.
.
237.
.
.
.
.
.
.
.
.
.
.
.
.
240.
Mumford system
7.3. Toda and Abelian surfaces of type
References
.
explicit
.
7.1. Different forms of the Toda lattice
morphism,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
II
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
175.
.
potential
hierarchy
.
.
quartic H6non-Heiles potential
6.3. The H6non-Heiles
.
.
.
.
.
.
on SO(4) SO(4) for metric
.
.
.
potentials
on
.
.
integrable geodesic flow geodesic flow
.
.
4.4. The relation with the canonical Jacobian made
5.1. The
.
.
potential and its integrability
4.5. The central Garnier
.
.
system
.
.
.
automorphism
.
.
.
Kummer surfaces
.
.
.
Mumford system
an
.
.
4.3. The precise relation with the canonical Jacobian
.
.
.
.
.
4.2. Some moduli spaces of Abelian surfaces of type
Index
.
.
.
.
4.1. The Garnier
7.2. A
.
.
configuration on the Jacobian of r projective embedding of the generalised Kummer surface
4. The Gaxnier
5.2.
.
.
.
.
3.2. The 94
5. An
.
.
systems
with
curves
.
.
2.3. The Bechlivanidis-van Moerbeke system
3.1. Genus two
155.
.
.
.
.
2.1. The genus two odd Mumford,
3.3. A
.
.
systems and applications
.
2. The genus two Mumford.
3.
.
.
.
.
Algebraic complete integrability
5. The
.
.
.
the R-brackets and the vector field V
Pteducing
4.1. The
91,
symmetries
(1,3) .
.
x
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
243.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
253.
Chapter
II
Integrable Hamiltonian systems affine Poisson varieties
on
1. Introduction
In this
chapter
give the basic definitions and properties of integrable Hamiltonian morphisms. In Section 2 we define the notion of a Poisson bracket (or Poisson structure) on an affine algebraic variety. The Poisson bracket is precisely what is needed to define Hamiltonian mechanics on a space, as is well-known from the theory of symplectic and Poisson manifolds. We shortly describe the simplest Poisson structures (i.e., constant, linear, affine and quadratic Poisson structures; also general Poisson structures on C2 and C') and describe two natural decompositions of affine Poisson varieties, one is given by the algebra of Casimirs, the other comes from the notion of rank of a Poisson systems
on
we
affine Poisson varieties and their
structure
(at
from old
ones.
a
point).
We also describe several ways to build
new
affine Poisson varieties
Morphisms of affine Poisson variety are regular maps which preserve the Poisson bracket. Isomorphisms preserve the rank at each point, leading to a polynomial invariant for affine Poisson varieties. This invariant permits us on the one hand to distinguish many different affine Poisson varieties, on the other hand it allows us to display in a structured way the basic characteristics of the Poisson structure. It will be computed for many different examples and a
refinement of this invariant is also discussed. In Section 3
we turn to integrable Hamiltonian systems. We motivate our definition by propositions and (counter-) examples. The notions of super-integrability, compatibility and integrable multi-Hamiltonian systems fit very well into the picture and most of our propositions are easily adapted to the case that the integrable Hamiltonian systems under
several
discussion have
one
of these extra structures.
decomposition of the variety,
as
the
one
The notion of momentum map leads to it is much finer).
given by the Casimirs (however
17
P. Vanhaecke: LNM 1638, pp. 17 - 70, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
a
Chapter We also define
11.
Integrable Hamiltonian systems
morphisms of integrable Hamiltonian systems; they
are
Poisson
mor-
algebra of functions in involution. It allows one to state precisely the relation between different integrable Hamiltonian systems, for example between new systems and the old ones from which they were constructed. Our discussion is parallel to the one of affine Poisson varieties (up to some modifications). Some really interesting examples of integrable Hamiltonian systems will be given in later chapters. phisms which
preserve the
The final section
(Section 4)
is devoted to
a
generalization
of
our
definitions to the
of other spaces. We draw special attention to the case of real Poisson manifolds. The main difference is that on the one hand the algebras we work with in the case of an affine case
variety are in general not finitely generated so that many constructions do not apply polynomial invariant), on the other hand many local constructions (e.g., Darboux coordinates, action-angle variables) which cannot be performed for affine Poisson varieties, play a dominant role in the study of some other Poisson spaces, including Poisson manifolds. Poisson
(e.g.,
the
Apart from Section
4
we
will in this
chapter always work
numbers.
18
over
the field of
complex
2. Affine Poisson varieties and their
2. Affine Poisson varieties and their
morphisms
morphisms
2.1. Affine Poisson varieties Phase space will closed subset of C'
always
(closed
be
an
affine
vaxiety
for the Zariski
in the
topology).
sense
Such
a
of
[Har], i.e.,
variety
an
irreducible
M C CI is the
zero
prime ideal Im of C[xi.... Xn], and its ring (or C-algebra) of regular functions denoted, resp. defined by
locus of
a
1
is
C[Xi'...' Xn]
O(M)
=
IM
integral domain (it has no zero divisors) and it is finitely generated; M can be reconstructed, up to isomorphism, from O(M) as SpecmO(M), the set of closed points in
O(M)
is
an
SpecO(M). The extra structure which Poisson bracket
on
a
Lie
algebra
algebra
Let M be
Definition 2.1 is
its
structure
we use
of
an
I-, j
to describe Hamiltonian
systems
on
M is
given by
a
fanctions.
regular
affine variety. A Poisson bracket or Poisson structure on M O(M), which is a bi-derivation, i.e., for any f G O(M) the
on
C-linear map
Xf:O(M)-+O(M) -+Ig,fl
g
is
a
derivation
(satisfies
the Leibniz
rule),
Xf (gh)
=
(2.1)
(Xf g) h + gXf h
for all g, h E O(M). The derivation Xf is called the Hamiltonian derivation associated to the Hamiltonian f and we write Ham (M, f -, -1) for the (vector) space
Ixf
=
I., f I I f
of Hamiltonian derivations. A function
Xf
=
0,
is called
a
Casimir
function
Cas
f
or a
(M, 1., -1)
E
E
O(M)
OMI whose Hamiltonian vector field is zero, we denote
Casimir and
=
If
O(M) I Xf
G
=
01
(vector) space of Casimirs; it is the center of the Lie bracket I-, j hence it is a Lie (O(M), I-, J). When no confusion can arise, either argument in Ham (M, I-, J) and Cas (M, f J) is omitted.
for the
ideal of
-
,
Remarks 2.2 1.
Xf being a
derivation may be refrased in
TM,
reason we
usually call
the elements
a
geometric
way
by saying
that it is
a
global
HO(M, Tm) (for the definition of the sheaf algebraic variety see [Hax] Section 11.8). For this Xf of Ham (M, 1., -1) Hamiltonian vector fields.
the tangent sheaf to M, i.e., of differentials and the tangent sheaf to an section of
Xf
E
Using the above mentionned correspondence between an affine variety and its algebra regular functions we have that affine Poisson varieties correspond to finitely generated Poisson algebras without zero divisors. 2.
of
19
Chapter 3.
Turning
upside down
the above definition
a
and its
subspace
one
gets
at the
following, equivalent definin
denote the vector space Hom(A 0 (M), 0 (M)) by C' (M) of skew-symmetric n-derivations by Der' (M). For every p, q > 0 a bilinear
Poisson bracket. Let
tion of
Hamiltonian systems
Integrable
Il.
us
map
F -1
:
,
is defined for P E
[Pj Q] (fl
CP(M), Q I... I
CP M
and for
Cq(M)
E
C, (M)
X
-+
CP+"- I (M)
fi,..., fp+,-i
E
O(M) by
fp+q-1)
o,ESq,p-i
1:
+
...
i
fa(p+q-1))
aESp,q_i where
o-(1)
Sp,
j. (3) For any r < d/2, there exists an affine Poisson variety whose invariant is RrSd. =
=
=
Proof M we have Prd Since M is irreducible and Md 1, all other Mij have by definition being given as the intersection of hypersurfaces in M they also have lower =
=
lower rank and
dimension. This shows
(2)
As for
(iL).
rely on the symplectic foliation, described in Section 4 below; an algebraic proof which would allow to remove the assumption about M being non-singular is still missing (in view of Proposition 2.18 it would suffice to show that the irreducible components of the Mi are affine Poisson subvarieties of M). Through every point of M passes a leaf which inherits a symplectic structure from the Poisson structure, so on the one hand all Hamiltonian vector fields at this point (which span a subspace of dimension equal to the rank 2r of the Poisson structure at this point) are tangent to such a leaf, on the other hand such a leaf is entirely contained in the subset M2,; thus every irreducible component of M2, has dimension at least 2r showing (2). For
Before
(3)
we
we
need to
take the canonical Poisson structure of rank 2r
give
a
refinement of the
invariant, let
us
consider
on
C2d (Example 2.7).
some
first
0
examples.
Example 2.50 An affine Poisson variety is regular if and only if its invariant polynomial is a monomial, i.e., is of the form R'S', where 2r is the rank and s the dimension of the variety. In particular the invariant polynomial of the trivial structure on an afline Poisson variety of dimension s is S'.
Example 2.51 For the Poisson structures on C2, which axe defined by a single polynomial jx, yj, with W:A 0 we have p RS2 + kS, where k is the number of components of W(x, y) the plane curve defined by W(x, y) 0. Its invariant matrix is thus given by =
=
=
( It follows in
0
k
0
0
0). 1
particular that the polynomial invariant is not a complete invariant: all nonpolynomials W(x, y) lead to a Poisson structure on C2 with invariant
constant irreducible p
=
RS2
+ S.
Example
2.52
The
Sklyanin brackets and
their
generalizations (see Example 2.9) lead for
the various values of the parameters to a lot of different invariant polynomials, giving an easy proof that many of these Poisson structures are different. We give the different polynomials -
which
are
easily computed
-
in the
following
table
(the integers i, j, k
and range from 1 to 3; a dash means that the values of the parameters the relation alb, a2b2 + a3b3 0)-
:--
42
are
are
taken different
incompatible with
P
all b
=
0
2. Alfine Poisson varieties and their
morphisms
all
ak
a
=
0
s4
=
bj
=
0
RS4
+
2S3
bi
=
bk
=
0
RS4
+
2S3
bi
=
all b
0
RS4
2S3
+
RS4 + S3
RS4
+
S3 +,52
0
RS4
+
S3 + S2
=k
=
aj
0
=
S3 +
+
all
S2
RS4
a
+
: S'
0
+ S
3S2
S2
+
RS4 +,52 RS4 + S3 + S
RS4
RS4 +3S2
0
+
RS4
0
=
=
RS4
bi
bk
ai
RS4
+
+
RS4
2S2
s2 +2S
+
RS4
+ 2S
S2 + 2S + 4S
Table I
A
above there
precise description of
more
polynomial corresponds are
Spec Cas(M) by
E
affine Poisson
affine Poisson vaxieties. Then
components for each
c
an
variety
can
be
given by combining the
invariant with Proposition 2.38. We know from that proposition that to each point of the affine variety Spec Cas(M) a fiber whose irreducible
P,-(M)
=
P
we
may define
a
polynomial
invariant
p,(M)
(-7r-I (M) (C) Cas
assumption that the fiber over c is irreducible; if not then the right hand side in just replaced by the sum over all irreducible components. Thus we label each point of Spec Cas(M) by the invariant polynomial of the corresponding fiber over it and obtain in this way a more sensitive invariant for affine Poisson varieties. In the examples which follow we will only consider the fibers over closed points c.
under the
this definition is
Example
2.53
the dual of
a
of this space
The
simplest non-trivial example is given by the Lie-Poisson structure on semi-simple Lie algebra (see Example 2.8). A basis Ix, y, zJ be chosen such that the corresponding Poisson matrix takes the form
three-dimensional can
( The
algebra
zero,
we
-Z
Y
Z
0
X
-Y
-X
0
(2.22)
.
y2 Z2] hence Spec Cas(M) can be clearly given by C[X2 2 Z 2; we denote the corresponding by evaluation on the element X2 Y Since (2.22) has only rank zero at the origin, which lies in the fiber over
of Casimirs is
identified with C
coordinate
0
by
u.
conclude that P
=
,
RS3 + 1 and
Pc
RS2 RS2 +1
43
if if
U(c) :;:A 0, U(C) 0. =
Chapter It may also be
depictured
as
11.
Integrable Hamiltonian systems
follows.
0 X
RS2+1
U
RS2
2.54 For the Heisenberg algebra the Lie-Poisson structure can be written as x. As above one finds that the algebra of Casimirs is given 0, ly, zj jx, zj jxj yj by C[x], and again its spectrum can be identified with C (with coordinate u) by evaluation
Example =
on
=
the Casimir
=
The Poisson structure has
x.
entire level of the Casimirs
level
sets).
(showing
It follows that p
RS3
=
that
is
case
depictured
as
in
zero on the plane x Proposition 2.38 needs
=
0 which is
an
not hold for all
S2 and
+
f RS2
PC
This
rank
now
equaJity
(c) U(c)
if
S2
0,
U
if
0.
follows. 0 X
S2
Example
An
2.55
interesting example is found by taking the Lie-Poisson structure on following basis
Consider the
gf(2)*.
1
X
=
0
0
( 0)
for g and let x,
.
.
.
,
t be the
to
x
T=
0
generators of 0 (Z),
1
T. The
X,
(0 0), corresponding Poisson
0
Y
-Y
0
we
-
X
-Y
-
t
Y
0
-Z
Z
0
=
+ t and xt
the points
t
0
-Z X
Cas(q*) C[x+t, xt-yz]. It follows that Spec Cas(g*) is in this case isomorphic pick the isomorphism. such that the standard coordinates u and v on C2 correspond
have
we
C2 ;
0
0 0),
given by
0
to
Z=
0
Z
and
0
(0 1),
Y=
,
0
matrix is
U
RS2
on
-
yz
(in
the line y
that =
z
order). =
pe
0,
x
Since the rank of the Poisson structure is two except for RS4 + S and t, we find that in this case p
=
RS2 RS2 +I
=
if
U2(e)
if U2 (C)
44
4v(c), 4V (C).
2. Affine Poisson varieties and their
Example
2.56
(Section VII.7).
The
following example will
structure determined
by the
up later when
come
t6l
In terms of coordinates
morphisms
for C'
we
studying the
Toda lattice
consider the Lie-Poisson
Poisson matrix
0
-t2
tj
0
t3 -t3
-t1
t2
0
tT
0
(-T )
with T
0
(2.23)
C[t1t2t3j t4 + t5 + t6], so that (in Paragraph VII.7.1) that CaS(C6) C2, with coordinates u and v, corresponding to t1t2t3 and t4 + t5 + t6 (in that order). By computing a few determinants one sees that,the rank is zero 0 (1 < i < j ! 3) on the three-plane tj t2 0, two on the three four-planes ti t3 tj We will show later
Spec Cas (C6)
can
=
be identified with
=
=
=
=
and four elsewhere. From it
one
p=R2,56 PC
It is
==
=
easily obtains the following invariant polynomials:
f3R
+
3R84
+
S3,
R2S4 2S4 + 3RS3 + S2
if if
U(c) U(C)
0, 0.
represented by the following diagram.
;3+S2 u
Proposition 2.57 Let (M, I., .1m) and (N, I* JN) be two affine their product M x N be equipped with the product bracket. Then
p(M In
x
N)
Poisson varieties and let
p(M)p(N).
=
particular, if the invariant polynomial of an affine Poisson variety variety is not a product (with the product bracket).
is irreducible then this
Poisson
Proof We use as above Mi, Nj and (M x N)i as notation for the determinantal varieties associated to M, N and M x N respectively. The coefficients of the invariant polynomials and By Proposition 2.21, we have p(M), p(N) and p(M x N) are written as pi'.,
pi2j
(M
x
N)i
U k+l=i
45
pi'j.
Mk
x
N1.
Chapter
11.
Integrable
Hamiltonian systems
Using the fact that the irreducible components irreducible components,
pixj
we
#j-dim. irred.
E
#j-dim.
of
a
product
are
precisely
the
products of
find comp. of
(M
x
irred. comp. of
N)j Mk
x
N,
k+l=i
E 1:
(#m-dim.
irred. comp. of
Mk) (#n-dim.
irred. comp. of
NI)
k+l=i m+n=j
1: 1: PklrnPin
-
k+l=i m+n=j
This shows that
p(M
Remark 2.58
It would be interesting to determine the invariant(s) of the Lie-Poisson arbitrary semi-simple Lie algebra and to relate it to the theory of (co-)
structure of
an
x
N)
=
p(M)p(N).
adjoint orbits.
46
3.
3.
Integrable Hamiltonian systems and their morphisms
Integrable Hamiltonian systems
and their
morphisms
In the
study of semi-simple Lie algebras the notion of a Cartan subaJgebra plays a corresponding object for affine Poisson spaces is an integrable algebra: a maximal commutative (in this context called involutive) subaJgebra. An affine Poisson variety with a fixed choice of integrable algebra is what we call an integrable Hamiltonian system. The study of integrable Hamiltonian systems can be seen as a chapter in Poisson geometry; for example we will see that all propositions which we proved for affine Poisson varieties have their equivalents for integrable Hamiltonian systems. Our definition is an adaption of the classical definition of an integrable system on a symplectic manifold (see e.g., [AMI]) to the case of an affine Poisson variety. Notice that we do not ask that the rank of the Poisson variety be maximal (or constant). Another difference is that the classical definition demands for having the right number of independent functions in involution, while we ask for having a complete algebra (of the right dimension) of functions in involution, completeness meaning here that this algebra contains every function which is in involution with all the elements of this algebra. On the one hand this adaption is very natural, it is even inevitable if one wants to discuss morphisms and isomorphisms of integrable Hamiltonian systems. On the other hand it is not easy to verify completeness of an involutive algebra, e.g., the (polynomial) algebra generated by a maximal number of functions in involution needs not be complete. Accordingly we will also prove some propositions in this section which will be useful for describing and determining explicitly the integrable algebra in the case of concrete examples. dominant role. The
3.1.
Integrable Hamiltonian systems
Definition 3.1
one
has
f f, Al
(M, JA, Al
-1)
Let
called involutive if =
0 -#>
Hamiltonian system
f
0; c-
A.
be we
an
on
affine Poisson
say that it is
The
affine Poisson varieties
variety. A subalgebra A of O(M) is complete if moreover for any f E O(M)
triple (M,
A)
is called
a
(complete)
involutive
-
Lemma 3.2
Let (M, A) be an involutive Hamiltonian system. (i.) If A is complete then A is integrally closed in O(M); (2) The integral closure of A in O(M) is also involutive and is finitely generated
when
A is finitely generated. Proof The
proof of (i.)
goes in
exactly the same way as the proof of Proposition 2.46, replacing O(M) by g Ei A. It is well-known that if A is finitely generated then its integral closure in O(M) (defined as the set of all elements 0 of O(M) for which there exists a monic polynomial with coefficients in A, which has 0 as a root) is also a finitely generated algebra (see e.g., [AD] Ch. 5). To check that it is involutive, we first check that
Cas(M) by
A and g
E
every element of the integral closure of A is in involution with all elements of A. be an element of O(M) for which there exists a polynomial
p(X) for which
P(0)
=
Xn +
a1Xn-1
+
-
-
-
Thus,
let
0
+ an
0 and with all ai belonging to A; we For any f E A the equality f P(o), f J
that the polynomial is implies as in the proof of Proposition 2.46 that 10, f I 0, upon using the minimality of P. Using this, it can now be checked by a similar argument that any two functions in the integral closure are in involution.1 of minimal
=
degree.
=
47
assume =
0
Chapter II. Integrable Hamiltonian systems
Every involutive algebra is contained in an involutive algebra which is complete, but the general not unique. This is contained in the following lemma.
latter is in
integral
(3.) (2) (3)
(M, 1-, .1, A)
Let
Lemma 3.3
be
an
involutive Hamiltonian system and denote
of the field of fractions of A. The subalgebra An o(m) of O(M) is also involutive; A; If A is complete then A n O(M) A is contained in an involutive subalgebra B of O(M) which dim A. if dim B
by A the
closure
=
is
complete;
it is
unique
=
Proof
(e. g., from [AD] Ch. 5) that A n o (m) can be identified as the set of elements 0 of for which there exists a polynomial (which is not necessarily monic) with coefficients
Recall
O(M) in
A,
which has
0
root. if
as a
0
P(X)
E
A n O(M) and
=
aoXn
+
aXn-I
+
-
-
+ an
-
0, then polynomial of minimal degree (with coefficients ai in A) for which P(O) of in the P of the as proof 0 minimality (again using 0, upon implies 10, Al JP(O), Al Proposition 2.46). In turn this implies that if 0' is another element of An 0 (M) the equality JP(O), O'l 0 leads to 10, O'l 0. Thus A n O(M) is involutive, showing (i.); from it (2)
is
=
a
=
=
=
=
follows at
once.
A n O(M); if the latter is complete complete we pass to AO unique involutive subalgebra of O(M) which contains A and is complete. If not, we 0 and repeat the above construction to add ail element f E O(M) \ AO for which If, AO I dim AO + 1 we are done after a finite number of steps; because of obtain A,. Since dim A, the choice of f the algebra which is obtained is not unique in general (interesting examples 0 of this are given below). If A is involutive but not
=
it is the
=
==
only be interested in involutive algebras of the maximal possible proposition. We know from Lemma 3.3 that such an algebra A dimension, given by A I if A has a unique completion, which we will denote by Compl(A) (or by Complf fl, is generated by If,, A 1) In this text
will
we
the next
.
Proposition
3.4
.
.
-
,
Let
(M,
A)
be
an
involutive Hamiltonian system. Then 1
dim A ::' , dim M
-
2
(3.1)
Rkj-, .1.
Proof Consider map A C
a
general fiber.F of
the map M
-+
SpecA
which is induced
by
the inclusion
O(M). By Proposition 2.37, dim.F
=
dim M
-
dim A.
(3.2)
equals the number of independent derivations of O(Y) at a general point of F and involutivity of A implies that such derivations can be constructed using functions from A.
dim.F also
48
Integrable Hamiltonian systems and their morphisms
3.
To m
see
the
latter, recall that the ideal of F is generated by the functions f arbitrary but fixed and f ranges over A. For any g E A we have
E 97 is
Xg(f hence
X.
-
X-M)
is tangent to the locus defined
If, gj
=
=
by the ideal of F, i.e.,
to Y and
O(Y) using elements of A. Next we show that the dim Cas(M) independent derivations, giving a lower bound
-
nested sequence of
where
construct
we can
elements of A lead to for diM.F. Consider
a
subalgebras Cas
where dim Aj+j
X'-"(f)
0,
derivations of dim A
-
=
Ao
C
Ai
C
A2
C
c
...
A,
=
O(M),
dim A, + 1, in particular r Rkj 1. If ni denotes the number of independent vector fields on M coming from A, (i.e., having independent vectors at a general point) then obviously ni < ni+l :5 ni + 1, no 0 and n, r. It follows that ni i for all i. It gives the following lower bound =
=
-
-
,
=
dim.F > dim A
Combining (2.40), (3.2)
and
(3.3)
we
=
-
=
dim Cas (M).
(3.3)
find I
dimA
We
finally get
1 with
fundamental group 7r, (E) and let G be a reductive algebraic group. Then Hom (7r, (E), G) is an affine variety on which G acts by conjugation, more precisely if p : 7r, (F,) --+ G and g E G then g-p is the homomorphism ir,(E) -+ G defined by g-P
for C E
?r,(E).
It turns out
(see [Gol]) M
(which
is
G
SL (n)
fc a
well-defined
O(M).
It
g(P(O)g
that the quotient
=
Hom(7r, (F,), G) IG
affine
explicitly
is
=
variety since G is reductive) has a natural Poisson structure which can be described for the classical groups. For simplicity let us consider the case in the standard representation. For a curve C G 7ri (E) the function
an
very =
M
was
:
M -+ G
:
p j-+
regular function on M and it by Goldman (see [Gol]) given by
shown
maximal rank is
I fc, fc, I
T ace(p(C))
can
that
(p; C, C)
be shown that these functions generate on such functions a Poisson bracket of
fc, C',
fc fc,
-
(3-6)
n
PEC#C1
The
sum runs over
intersect curves on
the intersection
transversally)
and
C and C' intersect at p, at p which is obtained
E, based A
points of C and C' (one may suppose that the curves is a sign which is determined by the way the (oriented) upon using the orientation of E. Finally, CpCp' is the curve
e(p; C, C')
large
involutive
by
first
following
C and then
for this bracket is obtained
following C'.
as follows. E can be decomposed (in algebra trinions; a trinion, also called a pair of pants, is just a three-holed sphere and such a decomposition will consist of 2g 2 trinions (in the case of genus two there exist precisely two such decompostions) Each trinion being bounded by three curves (which are identified two by two) one gets 3g 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 fo in thus one obtains an involutive algebra are involution; ......
several
ways)
into so-called
-
-
I
...
53
I
Chapter 11. Integrable Hamiltonian systems
A
=
and its dimension is computed to be 3g maximal, A will be integrable if and only if
Compllfc...... fc,,, -j
the Poisson bracket is
-
3
=
dimM
3. Since the rank of
1
1
3g
-
Rkj-, -1
-
2
=
2
dimM,
6. Since iri(E) has a system of 2g generators, which are bound 6g i.e., for dim M 1) dim G, hence M has dimension by one relation, dim Hom(iriL (E), G) has dimension (2g (2g 2) dim G and A is integrable if and only if =
-
-
-
6g i.e., for dimG
=
Since
3.
we
-
6
(2g
=
-
2) dim G,
restricted ourselves to G
=
SL(n)
we
SL(2); it is clear from the above pictures that the integrable for G fields corresponding to all functions fc, are actually super-integrable. =
3.2.
find that A is
only
Hamiltonian vector
Morphisms of integrable Hamiltonian systems
In parallel with our discussion of morphisms morphisms of integrable Hamiltonian systems.
Definition3.12
we now
turn to
(M2&,'j2,A2) be two integrable Hamiltonian -+ (M2ij*i'j2iA2) is a morphism 0: M,
and
Let
of affine Poisson varieties
systems, then a morphism 0: (Mj,j-,-jj,Aj) M2 with the following properties
(j-) 0 is a Poisson morphism; (2) 0* CaS(M2) C Ca$(MI); (3) O*A2 CAI -
the map and
Schematically, regularity of
Cas(M2)
(2)
-
and
(3)
can
be
Cas(MI)
-
--------
Al
as
follows:
O(M2)
A2
(3.7)
0*
0*
0.
represented
-
O(Mi)
morphism 0: (M17j*7"j1iA1) -+ (W J'i *12, A2) which is biregular has an inverse which is automatically a morphism: we call such a map an isomorphism (it forces all inclusion maps in the diagram to be bijective).
A
Ikom the very definition it is clear that the composition of two morphisms is a morIt is also immediate that for any biregular map 0 : we have a category). Hamiltonian for and system (MI, 1- 7 -11, A,) there exists a unique -+ integrable M2 M, any Poisson bracket 1. -12 on M2 and a unique integrable algebra A2 C O(M2) such that
phism (hence
0: (Ml J* i'll IAI) 7
-+
(M2 I i
Ifi 912
*
i
j 2 A2) 1
is
an
isomorphism; explicitly A2
(0-1)* 10*f7 0*911 54
Vig
G
O(m2)-
A, and
3.
Conditions on
(i.)
Integrable Hamiltonian systems
and
(2)
axe
and their morphisms
conditions at the level of the Poisson structures, rather than
integrable algebras. Condition (2) resp. (3) implies that 0 induces a morcorresponding paxameter spaces resp. base spaces, as is shown in the following
the level of the
phism of the proposition. Proposition
3.13
Let
0: (MI, 1-, -11, A,) -+ (M2i I' *121 A2) 0 induces a morphism 1
be
a
morphism of integrable
Hamiltonian systems. Then
0: Spec Cas(MI) which makes the
-+
Spec Cas(M2)
following diagram commutative, M,
M2
7rc-(Ml)I
I7rC-(M2) Spec Cas(M2)
Spec Cas(MI)
as
well
as a
morphism :
which makes the
following diagram
Spec A,
Spec A2
commutative.
M,
M2
IrAjI
I-A2 Spec A2
Spec A,
If 0* Cas(M2)
=
Cas(MI) (resp. O*A2
=
Aj
then
(resp. )
is
injective.
Proof
The first assertions of
0* implies injectivity
are
diagram (3.7) by taking spectra; corresponding spectra.
immediate from
at the level of the
also
surjectivity 0
differently, condition (3) in Definition 3.12 implies that each level set of A, is a level set Of A2 and if O*A2 A, then different level sets of A, are mapped into different level sets of A2; condition (2) can be given a similax interpretation. We further illustrate the meaning and relations between the three conditions in Definition 3.12 in the following examples and propositions. Said
mapped
into
55
Chapter 11. Integrable Hamiltonian systems
Example 3.14 Let us show that in Definition 3.12 neither (2) nor (3) follow from (1). Consider C4 (with coordinates q1, q2 P1 P2) with the canonical Poisson structure Jqi, pj I I and q2 0, and C-3 (with coordinates q1, q21 PI) with Jq1, p, I fpi, pj I 8ij, Jqi, qj I as Casimir. We look at this C3 as the qlq2PI-plane in C4 and denote by 0 the projection map along P2. Then 0 is a Poisson morphism, however O*q2 is not a Casimir of C4 showing that (3.) does not imply (2). Notice that in this case 0 does not induce a map 0 as in Proposition 3.13. Taking two different functions on C2 (i.e., the algebras generated by them) shows that (i.) does not imply W7
=
=
i
=
=
,
of morphisms for which condition (2) in Definition 3.12 universally closed morphisms; these include the proper morphisms and, in particular, the finite morphisms (see [Har] pp. 95-105). We prove this in the following proposition, however we restrict ourselves to the case of finite morphisms, since we will only use the result in this case (the proof however generalizes verbatim to the case of universally closed morphisms). There is however
follows from
a
large class
(i.), namely
that of
Proposition 3.15 Let (MI, -11) and (M2, J*)'12) be two affine Poisson varieties and suppose that 0 : M, -4 M2 is a finite morphism (for example a (possibly ramified) covering map). If 0 is a Poisson morphism then 0* Cas(M2) C Cas(MI); if 0 is moreover dominant then Cas(Mi) is the integral closure of 0* Cas(M2) in O(Mi). Proof Let
show that if
us
elements of
morphisms
0
is finite then for any
f
E
Cas(M2), O*f
is in involution with all
The main property which is used about finite is that if 0: M, -+ M2 is such a morphism then O(MI) is
O(MI).
(or universally closed) integral over O*O(M2).
Thus any element g E O(MI) is a root of a monic polynomial P (of minimal coefficients in O*O(M2). As in the proof of Proposition 2.46 we find 0
10*f, PWI
=
=
desired. We have shown that
with
P,(g)lo*f gI ,
where P' denotes the derivative of the as
degree)
polynomial P. By minimality of P we find JO*f gJ 0* Cas(M2) C Cas(MI). ,
=
0
take an element g E Cas(MI) and call P its polynomial as above, with coefO*O(M2). We show that P has actually its coefficients in 0* Cas(M2), thereby proving that Cas(MI) is the integral closure of 0* Cas(M2). To do this, let O*f E O*O(M2) Next
we
ficients in be
arbitrary, then 0
=
10*f P(g)11
=
JO*f, g'
,
+
O*alg'-'
10*f, O*alllg'-'
O*Jf,a1J2gn-1
+... +
+
+
+'*'+
0 *a.11
10*f, O*a.11
O*Ifi anJ2-
polynomial has its coefficients in O*O(M2) and since P was supposed of minimal 0 for all i. Since 0 is dominant it follows that If, ai 0 degree, we find that 0* If, ai I for all f E O(M2), so that ai E Cas(M2) for i n. 0 1, Since this
=
=
-
56
-
.'
Integrable Hamiltonian systems and
3.
their
morphisms
It can be seen in a similar way that if 0 : (MI, {-, -11, A,) -+ (M21 A2) is a morphism of integrable Hamiltonian systems which is finite and dominant then A, is the integral closure Of O*A2 in O(Mi) (for a proof, use completeness of A,). It leads to the following corollary.
Corollary 3.16 Let (MI, I j 1, A,) -+ (M2 whose image is an affine subvariety of M2. Then 0 su7jective morphism. 1
21
A2)
is the
be a morphism which is finite and composition of an injective and a
Proof
Proposition 2.16 that, 0 s o . Define
We know from
(O(Mi), 1-, -1)
say
A For
f,g
A
E
involutive. If
*f
Then for an
E
as a
Poisson
morphism, 0
can
be
decomposed
via
=
have
we
If, A}
=
If
=
E
O(O(Mi)) I *f
E
Ai}
0; by injectivity of * we see that A is J *f, *g} *Jf,gj J *f, A,} 0 since A, is the integral closure of O*A in O(MI). =
=
0 then
=
A, by completeness of A, and A is also complete. Finally the dimension
O(MI) is the same as the one for M, since 0 is integrable Hamiltonian system. Clearly % and
finite. It follows that axe
is
integrable Hamiltonian
of
morphisms
count
(O(MI), 1-, -1, A)
3
systems.
If a Poisson morphism 0 : (MI, l'I'll) -+ (W 1'7 *12) is finite but not Cas(M.1) may be larger than the integral closure of 0* Cas(M2) in O(MI). Take for example for (M2, J* '}2) the Lie-Poisson structure for the Heisenberg algebra (Ex0 with the trivial Poisson structure and for 0 the inclusion ample 2.54), for M, the plane x C[xl hence 0* Cas(M2) C, while Cas(MI) O(MI). map. Then Cas(M2)
Example
3.17
dominant then
=
Even if
Example3.18
=
=
=
a
Poisson
morphism 0: (Mill* 7,11)
-+
(W J*,*12)
is finite and
Cas(MI) may be different from 0* Cas(M2). Take for example on C3 the Poisson structure from Example 3.14 and consider the finite covering map 0 : C3 -+ C' given (qj, pl, q22). Obviously this is a Poisson morphism; however the Casimir q2 by O(ql, pl, q2) dominant then
=
O*F for any function injective, being given by (q2)
is not of the form case
not
=
O(C3).
F E 2
q2
C
Notice that
A similar remark
.
applies
-+
C is in this
to condition
(3)
in
Definition 3.12.
3.3. Constructions of In Section 2.3 ones.
Using these
systems
on
we
integrable
Hamiltonian systems
gave several constructions to build new affine Poisson varieties from old give the corresponding constructions for integrable Hamiltonian
we now
them. We first show that
an
integrable
Hamiltonian system restricts to
a
general
fiber of the parameter map.
Proposition
3.19
Let
(M,
A)
is
an
integrable Hamiltonian system
AI.F)
and T
an
irre-
is an integrable ducible component of a general level of the Casimirs. Then (,F, f -, JI.F, Hamiltonian System and the inclusion map is a morphism. The property also holds for the
general levels of
any
subalgebra of
the Casimirs.
57
Chapter II. Integrable Hamiltonian systems Proof Let B be any
subalgebra of Cas(M) and let Y be an irreducible component of a general Spec B. We know already from Proposition 2.38 that Y has an induced Poisson structure and from Proposition 2.42 that the algebra of Casimirs of this structure is maximal. If we restrict A to Y then we get again an involutive algebra Ap which is complete since A is complete and Y is general. Thus it suffices to compute the dimension of A,77, fiber of M
-+
I
dimY
-
dim A
This shows that
dimM
Ay is
Definition3.20 sition 3.19 is called
dim B
-
-
(dim A
-
dim B)
=
2
RkJ-, .1
integrable algebra. Clearly the inclusion map
an
RkJ-, -I.F.
2
is
a
morphism.
Any integrable Hamiltonian system obtained from (M, a trivial subsystem.
One may think of
trivial
a
subsystem
being
as
obtained
2
A) by Propo-
by fixing the values
of
some
of
the Casimirs.
Example fiber.F
3.21
(i.e.,
In the
examples
in the choice of values
one
has however to be careful when
assigned
(some of)
to
the
picking
particular
a
Casimirs). Namely
one
has to
check that F is
general enough in the sense that both the dimension and rank of Y coincide with those of a general fiber. The dimension of a special fiber F may be higher and/or its rank may be lower; then dim.F
(F,
so
none
AI.F)
of the
integrable
is not
integrable
trivial, while
Proposition
3.22
>
dhnA
dimAly,
integrable Hamiltonian system.
an
Hamiltonian systems Hamiltonian system on the fiber
that fiber is
on x
Reconsider e.g.
Example
C' for this Poisson structure will lead =
0,
2.54: to
an
since the induced Poisson structure
on
Al, 54 0(.F) r
For i E
and let -7ri denote the natural
11, 21
let
projection
(MI is
Rkf-, .1y
-
X
(Mi, I., Ji, A,) map
M,
x
M2
be -+
M27 f'i -1m, xm2,-7r,*Al
an integrable Hamiltonian system Mi Then
0
*7r2*A2)
(3.8)
integrable Hamiltonian system and the projection maps 7ri are morphisms. Each level of the integrable Hamiltonian system is a product of a level set of (MI, f -, -11, A,) and a level Set Of (W J* i'}27 A2)an
set
Proof The
Poisson-part of this proposition
was
already 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
3.
Integrable Hamiltonian systems and their morphisms
We count dimensions: dim -7r,*Al 0
7r2*A2
=
dim A, + dim A2
=
dim Mi
1 2
dim(Mi
=
1
-
RkJ-, -11
M2)
X
+ dim M2
2
Rk 2
-
Itkf* 1'12
Q -, JM1 xM2)
Since ?r,*Al (8) ?r2*A2 is complete and involutive with respect to the product bracket, this computation shows that 7r,*Ai (8),7r2*A2 is integrable. Since for earch of the projection maps iri The fibers of the one has -7ri*Ai C 7r1*A1 0 7rM2, these projection maps are morphisms. momentum map are given by the fibers of M, x M2 -+ Spec(7r,*Al 0 lr2*A2), that is, of the product map M, x M2 -+ Spec A, x Spec A2 hence all fibers are products of level sets of A, and A2. I It is easy to show in addition that Ham(Ai) (or Ham(A2)) does.
Ham(-7r,* A, (9 7r2* A2)
contains
a
super-integrable vector
field if
We call
Definition3.23
(3-8)
the
product of (M1,J-,J1,A1) and (M2,J',*}2,A2)-
A construction which is related to
(but
which will be used several times in the next
different
chapters,
the product construction and dealing with integrable
from)
is obtained when
Hamiltonian systems which depend on parameters. By this we mean that we have an affine Poisson variety (M, I , J) and for all possible values c of a set of parameters we have an -
integrable algebra A, on it. This set of parameters is assumed here to be the points on an affine variety N and we assume that A, (i.e., its elements) depends regularly on c. Then we can build a big affine Poisson variety which contains all the integrable Hamiltonian systems (M, 1., .1, A,) as trivial subsystems. This is given by the following proposition.8 for each c r= N an integrable Hamiltois given on an affine Poisson variety (M, 1-, .1) then M x N has a structure of an affine Poisson variety (M x N, I-, J) and O(M x N) contains an integrable subalgebra A such that each (M, I-, Jm, A,) is isomorphic to a trivial subsystem of (M x N, 1-, -1, A) via the inclusion maps
Proposition nian
3.24
system (M,
If
N is
an
affine variety
and
I., Jm, A,.), depending regularly
0,:
M
-+
M
x
N:
on c
m i-+
(m,c).
Proof For N a
one
takes the trivial structure
Poisson manifold. The
is maximal and
so
algebra of Casimirs
that on
Cas(N)
this
=
O(N)
product
which makes M
is maximal since the
x
N into
one on
M
N) Cas(M) (9 O(N). The fact that A, depends regularly on c means that there exists a subalgebra A of O(M x N) which restricts to A, on the fiber over c of the projection p, : M x N -+ N. Clearly its dimension is given by dim A dim A, +dim N Cas(M
x
=
=
8
generalizes to the situation considered in Example 2.24, namely when morphism, for each n E N, I-, -In is a Poisson bracket on the fiber -7r(-) (n) and An is an involutive subaJgebra of 0 (-7r(- 1) (n)) which is integrable for general n; both I-, Jn and An axe supposed to depend regularly on n G N. Proposition 3.24 ir
:
The proposition
P -+ N is
corresponds
a
dominant
to the
special
case
P
=
M
x
N considered at the end of
59
Example
2.24.
Chapter so
that dim A
Since
O(N)
is
dim(M x N)
=
a
subalgebra
Integrable Hamiltonian systems
1
2 Rkf Cas(M
-
of
II.
since A is x
N)
the fiber
complete and involutive p is
over
the restriction of the Poisson structure which is
corresponds to the isomorphism when restricted to such a fiber.
an
a
it is
integrable.
level set of the Casimirs and
one on
M via the
morP hism.
The next construction we discuss is that of taking a quotient. This is of interest, because many of the classical integrable Hamiltoniau systems possess discrete or continuous symmetry groups. The algebraic setup which we use here has the virtue to allow to pass easily to the
quotient (one does so
not need to worry about the action
being free, picking regular
values and
on).
3.25 Let G be a finite or reductive group and consider a Poisson action M, where (M, 1-, -1) is an affine Poisson variety. If A is an involutive algebra such that for each g (=- G the biregular map X, : M -+ M defined by X(g, m) leaves X, (m) A invariant, i.e., X*A C A, then (MIG, j.'.10, AG) is an involutive Hamiltonian system 9 and the quotient map -7r is a morphism. Here 1., -10 is the quotient bracket on MIG given by Proposition 2.25. If G is finite then (MIG, f.,.}O, AG) is integrable.
Proposition X: G
x
M
-+
=
Proof
Involutivity of AG is immediate from Proposition 2.25. Suppose now that G is finite. completeness of A implies completeness of A n O(M)G. As for dimensions, since G is
Then a
finite group
we
have
dimAn
O(M)G
=
dim.A
=
dimM
1Rkf
-
2
-,
-1
I =
where one
A
n
we
dim M/G
-
2
Rkf -, -jo,
dim O(M) and A c O(M). Similarly equality that dim O(M)' algebra of Casimirs is maximal, being given by Cas(M) n O(M)G. Thus integrable; obviously -7r*(A n O(M)G) C A, hence the quotient map is a
used in the first
=
shows that the
O(M)G
is
morphism.
0
We will encounter
Example
A
3.26
O(M)G). Namely,
a
lot of examples later. Here
special
in this
case
case
occurs
the level sets of
(MIG, I
A similar result
applies for the level
Example 3.27
The
-
,
when A C
each level set of
j o A) 5
quotient
are
precisely
are some
O(M)G (which implies Cas(MIG)
(M, f -, J, A) the
construction leads to
a
(M
c
is stable for the action of G and
quotients of the level
sets of the Casimirs in
systems which look interesting. One may e.g. start with
(M, I-, -}, A) and consider its x M by interchanging the
first observations.
case
sets of
Cas(MIG)
C
(M, f
-
-
,
1, A).
O(M)G.
lot of an
new integrable Hamiltonian integrable Hamiltonian system
M, I-, -Imxm, A (9 A). The group Z2 acts on product. Obviously this is a Poisson action and the action leaves A (& A invariant, thereby leading to a quotient. The level sets which correspond to the diagonal are symmetric products of the original level sets.
M
square
x
factors in the
60
3.
Integrable Hamiltonian systems and their morphisms
Notice that the group G in
phism. group of M. For future quasi-automorphism.
Proposition 3.25 use we
be
can
seen as a
introduce also the
slightly
subgroup of the automormore general notion of a
(A I-, J,A) bean integrable Hamiltonian system. An automorphism -+ (M, I-, -}, A). More generally, if 1., -11 and J* *12 are two Poisson brackets on M then an isomorphism (M, -.11, A) -+ (A {-, '12, A) is called a quasi-automorphism. Definition3.28 is
Let
isomorphism (M, I-, -}, A)
an
The final construction is to
1
remove a
divisor from
phase
space.
Proposition 3.29 Let (M, 1-, -1, A) be an integrable Hamiltonian system and let f E O(M) be a function which is not constant. Then there exists an integrable Hamiltonian system (N, f"i'lN, AN) and a morphism (N, J* 7'IN7 AN) -+ (M, 1-, -1, A) which is dominant, having the complement (in M) of the zero locus of f as image. Proof
proof (the Poisson part) was given in Proposition 2.35 and we proposition. We start with the case f E A. If we define AN then AN is involutive since 7r is a Poisson morpbism and it has the right dimension to be integrable. We need to verify completeness. Let Ein-0 fiti EE O(N) then Most of the
notation of that
use
:--
the
7r*A[t]
in order
j-
n
fit', AN i=O
IN
n
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 AN is involutive the last line also desired
implies the first line,
so we
have established the
equivalence.
an explicit description Of AN is still available if (M, I J,A) satisfies Spec 7r*A also satisfy the Proposition 3.7. In that case the fibers of N 7r*A. In general one has conditions of Proposition 3.7 hence -7r*A is complete and AN AN Compl(-7r*A) and a more explicit description is not available.
If
f
A then
-
,
the conditions of
=
61
Chapter
Compatible
3.4.
We
now
11.
Integrable Hamiltonian systems
and multi-Hamiltonian
introduce
a
integrable systems
few concepts which relate to
compatible integrable Hamiltonian
systems. Definit ion 3.30
brackets
Let
affine
i
variety M.
=-=
If
1,
n
be
(linearly independent) compatible
n
(M, I-, ji, A)
is
Poisson
integrable Hamiltonian system for each i n then these systems axe called compatible integrable Hamiltonian 1, systems. Any non-zero vector field Y on M which is integrable (in particular Hamiltonian) with respect to all Poisson structures i.e., for which there exist fl, f,, E A such that on an
an
=
.
.
.
,
.
Y
is called
a
multi-Hamiltonian
f., fill
=
= ...
(bi-Hamiltonian
many different ways; any of the an
=
.
.
,
1', Aln,
if n 2) vector field, since it is Hamiltonian in integrable Hamiltonian systems (M, I-, ji, A) is then called =
integrable multi-Hamiltonian system (bi-Hamiltonian when
Remark 3.31
We do not demand in the definition of
system that all the integrable satisfied in
an
=
2).
integrable multi-Hamiltonian
vector fields be multi-Hamiltonian.
3.33 and 3.34 it is far too restrictive in
Examples
n
Although
this condition is
general.
All propositions and basic constructions given above are easily adapted to the case of compatible or multi-Hamiltonian structures, but this will not be made explicit here. Just one example: an action of a reductive group which is a Poisson action with respect to both Poisson structures of two compatible integrable Hamiltonian systems yields on the quotient two compatible integrable Hamiltonian systems. Here are some properties which are specific to compatible integrable Hamiltonian systems.
Proposition 3.32 (1) Compatible integrable Hamiltonian systems have the same level sets; (2) The Poisson brackets of compatible integrable Hamiltonian systems have the same rank, which also equals the rank of a general linear combination of these Poisson structures
(3) If (M, I., -1j, A)
are
linear combination
integrable
compatible integrable Hamiltonian system then for
I-, +x of
the Poisson structures the system
(M,
general A) is an
a
Hamiltonian system.
Proof The
proof of (l.) is obvious since the level sets
determined
by A only. Since Rkf ji equal. To determine the rank of a linear combination of these structures one looks at the corresponding Poisson matrix (with respect to a system of generators of O(M)) which is given by the same linear combination of 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 combination =
dimM
-
1L 2
are
find that the rank ofall structures
we
I., ji
is
0 and I-, .1,\ of (maximal) rank Rkj-, jj one has that JA, A},\ Rkj-, ji, hence (M, I-, +\, A) is an integrable Hamiltonian system, =
showing W-
62
Integrable Hamiltonian systems
3.
We will encounter in this text many
and their
morphisms
(non-trivial) examples
of
compatible integrable Here are two simple
Ha,miltonian systems and of integrable multi-Hamiltonian systems. examples of integrable bi-Hamiltonian systems.
Example
Consider the Poisson structures
3.33
qj, q2, p, and
P2)
defined
by the
1-, -11
and
1' J2 1
on
C4 (with coordinates
Poisson matrices
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
and
O(C4)
For A c
structures
are
take those functions which
compatible and
since their
I they
are
are independent of q, and q2. Then both integrable vector fields are of the form
a
f
Poisson
C9 + g
9ql
1 f,g
9q2
A
E
all bi-Hamiltonian.
Example
Recall from
3.34
Example 2.11 that the matrix OF Oz
-OF
0
OF Ox
OF
-OF
;9__V
TX_
0
0
U(
OF
5
OY
defines for any u and F in O(C') a Poisson structure on C3 F is assumed non-constant here in order to obtain a non-triviaJ Poisson structure. Let us denote this Poisson structure
by J* juF. 1
j','ju,F+G
If G is any other non-constant element of O(C3) then I-, Ju,F + l'i"ju,G 1' 1 *}u,F and J* , ju,G are compatible and, assuming that F and G are in__"
hence
dependent, A ComplIF, G} defines an integrable Hamiltonian system on (C3, J* ju,F)However, by interchanging the roles of F and G. we find that A also defines an integrable Hamiltonian system on (C3, J* ju,G) hence leading to a pair of compatible integrable Hamil=
,
I
,
tonian systems. Since structures
are
moreover
the Hamiltonian vector fields with respect to both Poisson
given by
fuoVF we
conclude that A defines
Closely
an
integrable
x
VG
10
c
Al
bi-Hamiltonian system
on
C3.
related to the concept of an integrable multi-Hamiltonian system is that of a hierarchy. Let us define this in the case of a bi-Hamiltonian hierarchy and
multi-Hamiltonian
explain
its
use.
Let
sequence of functions
1-, -11 and J",'}2 be jfj I i E ZI is called I-, fiJ2
The
following property
is
--::
two a
compatible
I' fi+111i
essentially due
Poisson brackets
bi-Hamiltonian
i
(i
to Lenaxd and
63
E
hierarchy
Z).
Magri.
if
on
M.
Then
a
Chapter
11.
Integrable Hamiltonian systems
All functions fi of a bi-Hamiltonian hierarchy jfj I i E Z} are in 3.35 involution with respect to both Poisson brackets (hence with respect to any linear combination). If one of these functions is a Casimir (for either of the structures) then all these fi are also
Proposition
in involution with the elements
of
any other bi-Hamiltonian
hierarchy.
Proof If
jfj I
i E
ZI
forms
a
hierarchy,
then for any i