ńņš. 1 |

Noncommutative Algebras

Ana Cannas da Silva1

Alan Weinstein2

University of California at Berkeley

December 1, 1998

1

acannas@math.berkeley.edu, acannas@math.ist.utl.pt

2

alanw@math.berkeley.edu

Contents

Preface xi

Introduction xiii

I Universal Enveloping Algebras 1

1 Algebraic Constructions 1

1.1 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . 1

1.2 Lie Algebra Deformations . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 The Graded Algebra of U(g) . . . . . . . . . . . . . . . . . . . . . . . 3

2 The PoincarĀ“-Birkhoļ¬-Witt Theorem

e 5

Almost Commutativity of U(g) . . . . . . . .

2.1 . . . . . . . . . . . . . 5

Poisson Bracket on Gr U(g) . . . . . . . . . .

2.2 . . . . . . . . . . . . . 5

2.3 The Role of the Jacobi Identity . . . . . . . . . . . . . . . . . . . . . 7

2.4 Actions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Proof of the PoincarĀ“-Birkhoļ¬-Witt Theorem

e . . . . . . . . . . . . . 9

II Poisson Geometry 11

3 Poisson Structures 11

3.1 Lie-Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Almost Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Structure Functions and Canonical Coordinates . . . . . . . . . . . . 13

3.5 Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 14

3.6 Poisson Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Normal Forms 17

4.1 Lieā™s Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 A Faithful Representation of g . . . . . . . . . . . . . . . . . . . . . 17

4.3 The Splitting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.4 Special Cases of the Splitting Theorem . . . . . . . . . . . . . . . . . 20

4.5 Almost Symplectic Structures . . . . . . . . . . . . . . . . . . . . . . 20

4.6 Incarnations of the Jacobi Identity . . . . . . . . . . . . . . . . . . . 21

5 Local Poisson Geometry 23

5.1 Symplectic Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Transverse Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 The Linearization Problem . . . . . . . . . . . . . . . . . . . . . . . 25

5.4 The Cases of su(2) and sl(2; R) . . . . . . . . . . . . . . . . . . . . . 27

III Poisson Category 29

v

vi CONTENTS

6 Poisson Maps 29

6.1 Characterization of Poisson Maps . . . . . . . . . . . . . . . . . . . . 29

6.2 Complete Poisson Maps . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3 Symplectic Realizations . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.4 Coisotropic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.5 Poisson Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.6 Poisson Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7 Hamiltonian Actions 39

7.1 Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.2 First Obstruction for Momentum Maps . . . . . . . . . . . . . . . . 40

7.3 Second Obstruction for Momentum Maps . . . . . . . . . . . . . . . 41

7.4 Killing the Second Obstruction . . . . . . . . . . . . . . . . . . . . . 42

7.5 Obstructions Summarized . . . . . . . . . . . . . . . . . . . . . . . . 43

7.6 Flat Connections for Poisson Maps with Symplectic Target . . . . . 44

IV Dual Pairs 47

8 Operator Algebras 47

8.1 Norm Topology and C ā— -Algebras . . . . . . . . . . . . . . . . . . . . 47

8.2 Strong and Weak Topologies . . . . . . . . . . . . . . . . . . . . . . 48

8.3 Commutants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.4 Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Dual Pairs in Poisson Geometry 51

9.1 Commutants in Poisson Geometry . . . . . . . . . . . . . . . . . . . 51

9.2 Pairs of Symplectically Complete Foliations . . . . . . . . . . . . . . 52

9.3 Symplectic Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 53

9.4 Morita Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

9.5 Representation Equivalence . . . . . . . . . . . . . . . . . . . . . . . 55

9.6 Topological Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . 56

10 Examples of Symplectic Realizations 59

10.1 Injective Realizations of T3 . . . . . . . . . . . . . . . . . . . . . . . 59

10.2 Submersive Realizations of T3 . . . . . . . . . . . . . . . . . . . . . . 60

10.3 Complex Coordinates in Symplectic Geometry . . . . . . . . . . . . 62

10.4 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 63

10.5 A Dual Pair from Complex Geometry . . . . . . . . . . . . . . . . . 65

V Generalized Functions 69

11 Group Algebras 69

11.1 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

11.2 Commutative and Noncommutative Hopf Algebras . . . . . . . . . . 72

11.3 Algebras of Measures on Groups . . . . . . . . . . . . . . . . . . . . 73

11.4 Convolution of Functions . . . . . . . . . . . . . . . . . . . . . . . . 74

11.5 Distribution Group Algebras . . . . . . . . . . . . . . . . . . . . . . 76

CONTENTS vii

12 Densities 77

12.1 Densities . . . . . . . . ...... . . . . . . . . . . . . . . . . . . . . 77

12.2 Intrinsic Lp Spaces . . ...... . . . . . . . . . . . . . . . . . . . . 78

12.3 Generalized Sections . ...... . . . . . . . . . . . . . . . . . . . . 79

12.4 PoincarĀ“-Birkhoļ¬-Witt

e Revisited . . . . . . . . . . . . . . . . . . . . 81

VI Groupoids 85

13 Groupoids 85

13.1 Deļ¬nitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 85

13.2 Subgroupoids and Orbits . . . . . . . . . . . . . . . . . . . . . . . . 88

13.3 Examples of Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . 89

13.4 Groupoids with Structure . . . . . . . . . . . . . . . . . . . . . . . . 92

13.5 The Holonomy Groupoid of a Foliation . . . . . . . . . . . . . . . . . 93

14 Groupoid Algebras 97

14.1 First Examples . . . . . . . ..... . . . . . . . . . . . . . . . . . . 97

14.2 Groupoid Algebras via Haar Systems . . . . . . . . . . . . . . . . . . 98

14.3 Intrinsic Groupoid Algebras ..... . . . . . . . . . . . . . . . . . . 99

14.4 Groupoid Actions . . . . . . ..... . . . . . . . . . . . . . . . . . . 101

14.5 Groupoid Algebra Actions . ..... . . . . . . . . . . . . . . . . . . 103

15 Extended Groupoid Algebras 105

15.1 Generalized Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

15.2 Bisections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

15.3 Actions of Bisections on Groupoids . . . . . . . . . . . . . . . . . . . 107

15.4 Sections of the Normal Bundle . . . . . . . . . . . . . . . . . . . . . 109

15.5 Left Invariant Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 110

VII Algebroids 113

16 Lie Algebroids 113

16.1 Deļ¬nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

16.2 First Examples of Lie Algebroids . . . . . . . . . . . . . . . . . . . . 114

16.3 Bundles of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 116

16.4 Integrability and Non-Integrability . . . . . . . . . . . . . . . . . . . 117

16.5 The Dual of a Lie Algebroid . . . . . . . . . . . . . . . . . . . . . . . 119

16.6 Complex Lie Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . 120

17 Examples of Lie Algebroids 123

17.1 Atiyah Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

17.2 Connections on Transitive Lie Algebroids . . . . . . . . . . . . . . . 124

17.3 The Lie Algebroid of a Poisson Manifold . . . . . . . . . . . . . . . . 125

17.4 Vector Fields Tangent to a Hypersurface . . . . . . . . . . . . . . . . 127

17.5 Vector Fields Tangent to the Boundary . . . . . . . . . . . . . . . . 128

viii CONTENTS

18 Diļ¬erential Geometry for Lie Algebroids 131

18.1 The Exterior Diļ¬erential Algebra of a Lie Algebroid . . . . . . . . . 131

18.2 The Gerstenhaber Algebra of a Lie Algebroid . . . . . . . . . . . . . 132

18.3 Poisson Structures on Lie Algebroids . . . . . . . . . . . . . . . . . . 134

18.4 Poisson Cohomology on Lie Algebroids . . . . . . . . . . . . . . . . . 136

18.5 Inļ¬nitesimal Deformations of Poisson Structures . . . . . . . . . . . 137

18.6 Obstructions to Formal Deformations . . . . . . . . . . . . . . . . . 138

VIII Deformations of Algebras of Functions 141

19 Algebraic Deformation Theory 141

19.1 The Gerstenhaber Bracket . . . . . . . . . . . . . . . . . . . . . . . . 141

19.2 Hochschild Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 142

19.3 Case of Functions on a Manifold . . . . . . . . . . . . . . . . . . . . 144

19.4 Deformations of Associative Products . . . . . . . . . . . . . . . . . 144

19.5 Deformations of the Product of Functions . . . . . . . . . . . . . . . 146

20 Weyl Algebras 149

20.1 The Moyal-Weyl Product . . . . . . . . . . . . . . . . . . . . . . . . 149

20.2 The Moyal-Weyl Product as an Operator Product . . . . . . . . . . 151

20.3 Aļ¬ne Invariance of the Weyl Product . . . . . . . . . . . . . . . . . 152

20.4 Derivations of Formal Weyl Algebras . . . . . . . . . . . . . . . . . . 152

20.5 Weyl Algebra Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 153

21 Deformation Quantization 155

21.1 Fedosovā™s Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

21.2 Preparing the Connection . . . . . . . . . . . . . . . . . . . . . . . . 156

21.3 A Derivation and Filtration of the Weyl Algebra . . . . . . . . . . . 158

21.4 Flattening the Connection . . . . . . . . . . . . . . . . . . . . . . . . 160

21.5 Classiļ¬cation of Deformation Quantizations . . . . . . . . . . . . . . 161

References 163

Index 175

Preface

Noncommutative geometry is the study of noncommutative algebras as if they were

algebras of functions on spaces, like the commutative algebras associated to aļ¬ne

algebraic varieties, diļ¬erentiable manifolds, topological spaces, and measure spaces.

In this book, we discuss several types of geometric objects (in the usual sense of

sets with structure) which are closely related to noncommutative algebras.

Central to the discussion are symplectic and Poisson manifolds, which arise

when noncommutative algebras are obtained by deforming commutative algebras.

We also make a detailed study of groupoids, whose role in noncommutative geom-

etry has been stressed by Connes, as well as of Lie algebroids, the inļ¬nitesimal

approximations to diļ¬erentiable groupoids.

These notes are based on a topics course, āGeometric Models for Noncommuta-

tive Algebras,ā which one of us (A.W.) taught at Berkeley in the Spring of 1997.

We would like to express our appreciation to Kevin Hartshorn for his partic-

ipation in the early stages of the project ā“ producing typed notes for many of

the lectures. Henrique Bursztyn, who read preliminary versions of the notes, has

provided us with innumerable suggestions of great value. We are also indebted

to Johannes Huebschmann, Kirill Mackenzie, Daniel Markiewicz, Elisa Prato and

Olga Radko for several useful commentaries or references.

Finally, we would like to dedicate these notes to the memory of four friends and

colleagues who, sadly, passed away in 1998: MoshĀ“ Flato, K. Guruprasad, AndrĀ“

e e

Lichnerowicz, and Stanislaw Zakrzewski.

Ana Cannas da Silva

Alan Weinstein

xi

Introduction

We will emphasize an approach to algebra and geometry based on a metaphor (see

Lakoļ¬ and NuĖez [100]):

n

An algebra (over R or C) is the set of (R- or C-valued) functions on a space.

Strictly speaking, this statement only holds for commutative algebras. We would

like to pretend that this statement still describes noncommutative algebras.

Furthermore, diļ¬erent restrictions on the functions reveal diļ¬erent structures

on the space. Examples of distinct algebras of functions which can be associated

to a space are:

ā¢ polynomial functions,

ā¢ real analytic functions,

ā¢ smooth functions,

ā¢ C k , or just continuous (C 0 ) functions,

ā¢ Lā , or the set of bounded, measurable functions modulo the set of functions

vanishing outside a set of measure 0.

So we can actually say,

An algebra (over R or C) is the set of good (R- or C-valued) functions on a space

with structure.

Reciprocally, we would like to be able to recover the space with structure from

the given algebra. In algebraic geometry that is achieved by considering homomor-

phisms from the algebra to a ļ¬eld or integral domain.

Examples.

1. Take the algebra C[x] of complex polynomials in one complex variable. All

homomorphisms from C[x] to C are given by evaluation at a complex number.

We recover C as the space of homomorphisms.

2. Take the quotient algebra of C[x] by the ideal generated by xk+1

xk+1 = {a0 + a1 x + . . . + ak xk | ai ā C} .

C[x]

The coeļ¬cients a0 , . . . , ak may be thought of as values of a complex-valued

function plus its ļ¬rst, second, ..., kth derivatives at the origin. The corre-

sponding āspaceā is the so-called kth inļ¬nitesimal neighborhood of the

point 0. Each of these āspacesā has just one point: evaluation at 0. The limit

as k gets large is the space of power series in x.

3. The algebra C[x1 , . . . , xn ] of polynomials in n variables can be interpreted as

the algebra Pol(V ) of āgoodā (i.e. polynomial) functions on an n-dimensional

complex vector space V for which (x1 , . . . , xn ) is a dual basis. If we denote

the tensor algebra of the dual vector space V ā— by

ā—k

T (V ā— ) = C ā• V ā— ā• (V ā— ā— V ā— ) ā• . . . ā• (V ā— ) ā• ... ,

xiii

xiv INTRODUCTION

ā—k

where (V ā— ) is spanned by {xi1 ā— . . . ā— xik | 1 ā¤ i1 , . . . , ik ā¤ n}, then we

realize the symmetric algebra S(V ā— ) = Pol(V ) as

S(V ā— ) = T (V ā— )/C ,

where C is the ideal generated by {Ī± ā— Ī² ā’ Ī² ā— Ī± | Ī±, Ī² ā V ā— }.

There are several ways to recover V and its structure from the algebra Pol(V ):

ā¢ Linear homomorphisms from Pol(V ) to C correspond to points of V . We

thus recover the set V .

ā¢ Algebra endomorphisms of Pol(V ) correspond to polynomial endomor-

phisms of V : An algebra endomorphism

f : Pol(V ) ā’ā’ Pol(V )

is determined by the f (x1 ), . . . , f (xn )). Since Pol(V ) is freely generated

by the xi ā™s, we can choose any f (xi ) ā Pol(V ). For example, if n = 2, f

could be deļ¬ned by:

x1 ā’ā’ x1

x2 ā’ā’ x2 + x2 1

which would even be invertible. We are thus recovering a polynomial

structure in V .

ā¢ Graded algebra automorphisms of Pol(V ) correspond to linear isomor-

phisms of V : As a graded algebra

ā

Polk (V ) ,

Pol(V ) =

k=0

where Polk (V ) is the set of homogeneous polynomials of degree k, i.e.

symmetric tensors in (V ā— )ā—k . A graded automorphism takes each xi to

an element of degree one, that is, a linear homogeneous expression in the

xi ā™s. Hence, by using the graded algebra structure of Pol(V ), we obtain

a linear structure in V .

4. For a noncommutative structure, let V be a vector space (over R or C) and

deļ¬ne

Īā¢ (V ā— ) = T (V ā— )/A ,

where A is the ideal generated by {Ī± ā— Ī² + Ī² ā— Ī± | Ī±, Ī² ā V ā— }. We can view

this as a graded algebra,

ā

ā¢ ā—

Īk (V ā— ) ,

Ī (V ) =

k=0

whose automorphisms give us the linear structure on V . Therefore, as a

graded algebra, Īā¢ (V ā— ) still ārepresentsā the vector space structure in V .

The algebra Īā¢ (V ā— ) is not commutative, but is instead super-commutative,

i.e. for elements a ā Īk (V ā— ), b ā Ī (V ā— ), we have

ab = (ā’1)k ba .

INTRODUCTION xv

Super-commutativity is associated to a Z2 -grading:1

Īā¢ (V ā— ) Ī[0] (V ā— ) ā• Ī[1] (V ā— ) ,

=

where

Ī[0] (V ā— ) = Īeven (V ā— ) Īk (V ā— ) ,

:= and

k even

Ī[1] (V ā— ) = Īodd (V ā— ) Īk (V ā— ) .

:=

k odd

Therefore, V is not just a vector space, but is called an odd superspace;

āoddā because all nonzero vectors in V have odd(= 1) degree. The Z2 -grading

allows for more automorphisms, as opposed to the Z-grading. For instance,

ā’ā’

x1 x1

ā’ā’

x2 x2 + x1 x2 x3

ā’ā’

x3 x3

is legal; this preserves the relations since both objects and images anti-

commute. Although there is more ļ¬‚exibility, we are still not completely free

to map generators, since we need to preserve the Z2 -grading. Homomor-

phisms of the Z2 -graded algebra Īā¢ (V ā— ) correspond to āfunctionsā on the

(odd) superspace V . We may view the construction above as a deļ¬nition: a

superspace is an object on which the functions form a supercommutative

Z2 -graded algebra. Repeated use should convince one of the value of this type

of terminology!

5. The algebra ā„¦ā¢ (M ) of diļ¬erential forms on a manifold M can be regarded as

a Z2 -graded algebra by

ā„¦ā¢ (M ) = ā„¦even (M ) ā• ā„¦odd (M ) .

We may thus think of forms on M as functions on a superspace. Locally, the

tangent bundle T M has coordinates {xi } and {dxi }, where each xi commutes

with everything and the dxi anticommute with each other. (The coordinates

{dxi } measure the components of tangent vectors.) In this way, ā„¦ā¢ (M ) is the

ā—¦ ā—¦

algebra of functions on the odd tangent bundle T M ; the indicates that

here we regard the ļ¬bers of T M as odd superspaces.

The exterior derivative

d : ā„¦ā¢ (M ) ā’ā’ ā„¦ā¢ (M )

has the property that for f, g ā ā„¦ā¢ (M ),

d(f g) = (df )g + (ā’1)deg f f (dg) .

Hence, d is a derivation of a superalgebra. It exchanges the subspaces of even

ā—¦

and odd degree. We call d an odd vector ļ¬eld on T M .

6. Consider the algebra of complex valued functions on a āphase spaceā R2 ,

with coordinates (q, p) interpreted as position and momentum for a one-

dimensional physical system. We wish to impose the standard equation from

quantum mechanics

qp ā’ pq = i ,

1 The term āsuperā is generally used in connection with Z2 -gradings.

xvi INTRODUCTION

which encodes the uncertainty principle. In order to formalize this condition,

we take the algebra freely generated by q and p modulo the ideal generated by

qp ā’ pq ā’ i . As approaches 0, we recover the commutative algebra Pol(R2 ).

Studying examples like this naturally leads us toward the universal envelop-

ing algebra of a Lie algebra (here the Lie algebra is the Heisenberg algebra,

where is considered as a variable like q and p), and towards symplectic

geometry (here we concentrate on the phase space with coordinates q and

p).

ā™¦

Each of these latter aspects will lead us into the study of Poisson algebras,

and the interplay between Poisson geometry and noncommutative algebras, in par-

ticular, connections with representation theory and operator algebras.

In these notes we will be also looking at groupoids, Lie groupoids and groupoid

algebras. Brieļ¬‚y, a groupoid is similar to a group, but we can only multiply certain

pairs of elements. One can think of a groupoid as a category (possibly with more

than one object) where all morphisms are invertible, whereas a group is a category

with only one object such that all morphisms have inverses. Lie algebroids are

the inļ¬nitesimal counterparts of Lie groupoids, and are very close to Poisson and

symplectic geometry.

Finally, we will discuss Fedosovā™s work in deformation quantization of arbitrary

symplectic manifolds.

All of these topics give nice geometric models for noncommutative algebras!

Of course, we could go on, but we had to stop somewhere. In particular, these

notes contain almost no discussion of Poisson Lie groups or symplectic groupoids,

both of which are special cases of Poisson groupoids. Ample material on Poisson

groups can be found in [25], while symplectic groupoids are discussed in [162] as

well as the original sources [34, 89, 181]. The theory of Poisson groupoids [168] is

evolving rapidly thanks to new examples found in conjunction with solutions of the

classical dynamical Yang-Baxter equation [136].

The time should not be long before a sequel to these notes is due.

Part I

Universal Enveloping Algebras

1 Algebraic Constructions

Let g be a Lie algebra with Lie bracket [Ā·, Ā·]. We will assume that g is a ļ¬nite

dimensional algebra over R or C, but much of the following also holds for inļ¬nite

dimensional Lie algebras, as well as for Lie algebras over arbitrary ļ¬elds or rings.

1.1 Universal Enveloping Algebras

Regarding g just as a vector space, we may form the tensor algebra,

ā

gā—k ,

T (g) =

k=0

which is the free associative algebra over g. There is a natural inclusion j : g ā’ T (g)

taking g to gā—1 such that, for any linear map f : g ā’ A to an associative algebra

A, the assignment g(v1 ā— . . . ā— vk ) ā’ f (v1 ) . . . f (vk ) determines the unique algebra

homomorphism g making the following diagram commute.

jE

T (g)

g

d

g

d

d

fd

Ā‚

d c

A

Therefore, there is a natural one-to-one correspondence

HomAssoc (T (g), A) ,

HomLinear (g, Linear(A))

where Linear(A) is the algebra A viewed just as a vector space, HomLinear de-

notes linear homomorphisms and HomAssoc denotes homomorphisms of associative

algebras.

The universal enveloping algebra of g is the quotient

U(g) = T (g)/I ,

where I is the (two-sided) ideal generated by the set

{j(x) ā— j(y) ā’ j(y) ā— j(x) ā’ j([x, y]) | x, y ā g} .

If the Lie bracket is trivial, i.e. [Ā·, Ā·] ā” 0 on g, then U(g) = S(g) is the symmet-

ric algebra on g, that is, the free commutative associative algebra over g. (When g

is ļ¬nite dimensional, S(g) coincides with the algebra of polynomials in gā— .) S(g) is

the universal commutative enveloping algebra of g because it satisļ¬es the universal

property above if we restrict to commutative algebras; i.e. for any commutative

associative algebra A, there is a one-to-one correspondence

HomCommut (S(g), A) .

HomLinear (g, Linear(A))

1

2 1 ALGEBRAIC CONSTRUCTIONS

The universal property for U(g) is expressed as follows. Let i : g ā’ U(g) be

the composition of the inclusion j : g ā’ T (g) with the natural projection T (g) ā’

U(g). Given any associative algebra A, let Lie(A) be the algebra A equipped with

the bracket [a, b]A = ab ā’ ba, and hence regarded as a Lie algebra. Then, for

any Lie algebra homomorphism f : g ā’ A, there is a unique associative algebra

homomorphism g : U(g) ā’ A making the following diagram commute.

iE

U(g)

g

d

g

d

d

fd

Ā‚

d c

A

In other words, there is a natural one-to-one correspondence

HomAssoc (U(g), A) .

HomLie (g, Lie(A))

In the language of categories [114] the functor U(Ā·) from Lie algebras to associative

algebras is the left adjoint of the functor Lie(Ā·).

Exercise 1

What are the adjoint functors of T and S?

1.2 Lie Algebra Deformations

The PoincarĀ“-Birkhoļ¬-Witt theorem, whose proof we give in Sections 2.5 and 4.2,

e

says roughly that U(g) has the same size as S(g). For now, we want to check that,

even if g has non-zero bracket [Ā·, Ā·], then U(g) will still be approximately isomorphic

to S(g). One way to express this approximation is to throw in a parameter Īµ

multiplying the bracket; i.e. we look at the Lie algebra deformation gĪµ = (g, Īµ[Ā·, Ā·]).

As Īµ tends to 0, gĪµ approaches an abelian Lie algebra. The family gĪµ describes a

path in the space of Lie algebra structures on the vector space g, passing through

the point corresponding to the zero bracket.

From gĪµ we obtain a one-parameter family of associative algebras U(gĪµ ), passing

through S(g) at Īµ = 0. Here we are taking the quotients of T (g) by a family of

ideals generated by

{j(x) ā— j(y) ā’ j(y) ā— j(x) ā’ j(Īµ[x, y]) | x, y ā g} ,

so there is no obvious isomorphism as vector spaces between the U(gĪµ ) for diļ¬erent

values of Īµ. We do have, however:

Claim. U(g) U(gĪµ ) for all Īµ = 0.

Proof. For a homomorphism of Lie algebras f : g ā’ h, the functoriality of U(Ā·)

and the universality of U(g) give the commuting diagram

f

g Eh

d

d

ih ā—¦ f

ig ih

d

c ā!g d c

Ā‚

U(g) E U(h)

1.3 Symmetrization 3

In particular, if g h, then U(g) U(h) by universality.

Since we have the Lie algebra isomorphism

m1/Īµ

gĪµ ,

g' E

mĪµ

1

and Īµ, we conclude that U(g) U(gĪµ ) for Īµ = 0.

given by multiplication by 2

Īµ

In Section 2.1, we will continue this family of isomorphisms to a vector space

isomorphism

U(g) U(g0 ) S(g) .

The family U(gĪµ ) may then be considered as a path in the space of associative

multiplications on S(g), passing through the subspace of commutative multiplica-

tions. The ļ¬rst derivative with respect to Īµ of the path U(gĪµ ) turns out to be an

anti-symmetric operation called the Poisson bracket (see Section 2.2).

1.3 Symmetrization

Let Sn be the symmetric group in n letters, i.e. the group of permutations of

{1, 2, . . . , n}. The linear map

1

s : x1 ā— . . . ā— xn ā’ā’ xĻ(1) ā— . . . ā— xĻ(n)

n!

ĻāSn

extends to a well-deļ¬ned symmetrization endomorphism s : T (g) ā’ T (g) with

the property that s2 = s. The image of s consists of the symmetric tensors and

is a vector space complement to the ideal I generated by {j(x) ā— j(y) ā’ j(y) ā— j(x) |

x, y ā g}. We identify the symmetric algebra S(g) = T (g)/I with the symmetric

tensors by the quotient map, and hence regard symmetrization as a projection

s : T (g) ā’ā’ S(g) .

The linear section

S(g) ā’ā’ T (g)

Ļ„:

ā’ā’ s(x1 ā— . . . ā— xn )

x1 . . . xn

is a linear map, but not an algebra homomorphism, as the product of two symmetric

tensors is generally not a symmetric tensor.

The Graded Algebra of U(g)

1.4

Although U(g) is not a graded algebra, we can still grade it as a vector space.

We start with the natural grading on T (g):

ā

T k (g) = gā—k .

T k (g) ,

T (g) = where

k=0

Unfortunately, projection of T (g) to U(g) does not induce a grading, since the

relations deļ¬ning U(g) are not homogeneous unless [Ā·, Ā·]g = 0. (On the other hand,

symmetrization s : T (g) ā’ S(g) does preserve the grading.)

4 1 ALGEBRAIC CONSTRUCTIONS

The grading of T (g) has associated ļ¬ltration

k

(k)

T j (g) ,

T (g) =

j=0

such that

T (0) ā T (1) ā T (2) ā . . . T (i) ā— T (j) ā T (i+j) .

and

We can recover T k by T (k) /T (kā’1) T k .

What happens to this ļ¬ltration when we project to U(g)?

Remark. Let i : g ā’ U(g) be the natural map (as in Section 1.1). If we take

x, y ā g, then i(x)i(y) and i(y)i(x) each āhas length 2,ā but their diļ¬erence

i(y)i(x) ā’ i(x)i(y) = i([y, x])

has length 1. Therefore, exact length is not respected by algebraic operations on

U(g). ā™¦

Let U (k) (g) be the image of T (k) (g) under the projection map.

Exercise 2

Show that U (k) (g) is linearly spanned by products of length ā¤ k of elements

of U (1) (g) = i(g).

We do have the relation

U (k) Ā· U ( ) ā U (k+ )

,

so that the universal enveloping algebra of g has a natural ļ¬ltration, natural in the

sense that, for any map g ā’ h, the diagram

g Eh

c c

U(g) E U(h)

preserves the ļ¬ltration.

In order to construct a graded algebra, we deļ¬ne

U k (g) = U (k) (g)/U (kā’1) (g) .

There are well-deļ¬ned product operations

U k (g) ā— U (g) ā’ā’ U k+ (g)

[Ī±] ā— [Ī²] ā’ā’ [Ī±Ī²]

forming an associative multiplication on what is called the graded algebra asso-

ciated to U(g):

ā

U k (g) =: Gr U(g) .

j=0

Remark. The constructions above are purely algebraic in nature; we can form

Gr A for any ļ¬ltered algebra A. The functor Gr will usually simplify the algebra

ā™¦

in the sense that multiplication forgets about lower order terms.

2 The PoincarĀ“-Birkhoļ¬-Witt Theorem

e

Let g be a ļ¬nite dimensional Lie algebra with Lie bracket [Ā·, Ā·]g .

Almost Commutativity of U(g)

2.1

Claim. Gr U(g) is commutative.

Proof. Since U(g) is generated by U (1) (g), Gr U(g) is generated by U 1 (g). Thus

it suļ¬ces to show that multiplication

U 1 (g) ā— U 1 (g) ā’ā’ U 2 (g)

is commutative. Because U (1) (g) is generated by i(g), any Ī± ā U 1 (g) is of the form

Ī± = [i(x)] for some x ā g. Pick any two elements x, y ā g. Then [i(x)], [i(y)] ā

U 1 (g), and

[i(x)][i(y)] ā’ [i(y)][i(x)] = [i(x)i(y) ā’ i(y)i(x)]

= [i([x, y]g )] .

As i([x, y]g ) sits in U (1) (g), we see that [i([x, y]g )] = 0 in U 2 (g). 2

When looking at symmetrization s : T (g) ā’ S(g) in Section 1.3, we constructed

a linear section Ļ„ : S(g) ā’ T (g). We formulate the PoincarĀ“-Birkhoļ¬-Witt theorem

e

using this linear section.

Theorem 2.1 (PoincarĀ“-Birkhoļ¬-Witt) There is a graded (commutative) al-

e

gebra isomorphism

Ī» : S(g) ā’ā’ Gr U(g)

given by the natural maps:

Ļ„

S k (g) E T k (g) E U (k) (g) E U k (g) ā‚ Gr U(g)

ā‚ E

E1 vĻ(1) ā— . . . ā— vĻ(k)

v1 . . . vk E [v1 . . . vk ] .

k!

ĻāSk

For each degree k, we follow the embedding Ļ„ k : S k (g) ā’ T k (g) by a map

to U (k) (g) and then by the projection onto U k . Although the composition Ī» :

S(g) ā’ Gr U(g) is a graded algebra homomorphism, the maps S(g) ā’ T (g) and

T (g) ā’ U (g) are not.

We shall prove Theorem 2.1 (for ļ¬nite dimensional Lie algebras over R or C)

using Poisson geometry. The sections most relevant to the proof are 2.5 and 4.2.

For purely algebraic proofs, see Dixmier [46] or Serre [150], who show that the

theorem actually holds for free modules g over rings.

Poisson Bracket on Gr U(g)

2.2

In this section, we denote U(g) simply by U, since the arguments apply to any

ļ¬ltered algebra U,

U (0) ā U (1) ā U (2) ā . . . ,

5

Ā“

6 2 THE POINCARE-BIRKHOFF-WITT THEOREM

for which the associated graded algebra

ā

Uj U j = U (j) U (jā’1) .

Gr U := where

j=0

is commutative. Such an algebra U is often called almost commutative.

For x ā U (k) and y ā U ( ) , deļ¬ne

ā’1 ā’1) ā’2)

{[x], [y]} = [xy ā’ yx] ā U k+ = U (k+ /U (k+

so that

ā’1

{U k , U } ā U k+ .

This collection of degree ā’1 bilinear maps combine to form the Poisson bracket on

Gr U. So, besides the associative product on Gr U (inherited from the associative

product on U; see Section 1.4), we also get a bracket operation {Ā·, Ā·} with the

following properties:

1. {Ā·, Ā·} is anti-commutative (not super-commutative) and satisļ¬es the Jacobi

identity

{{u, v}, w} = {{u, w}, v} + {u, {v, w}} .

That is, {Ā·, Ā·} is a Lie bracket and Gr U is a Lie algebra;

2. the Leibniz identity holds:

{uv, w} = {u, w}v + u{v, w} .

Exercise 3

Prove the Jacobi and Leibniz identities for {Ā·, Ā·} on Gr U.

Remark. The Leibniz identity says that {Ā·, w} is a derivation of the associative

algebra structure; it is a compatibility property between the Lie algebra and the

associative algebra structures. Similarly, the Jacobi identity says that {Ā·, w} is a

ā™¦

derivation of the Lie algebra structure.

A commutative associative algebra with a Lie algebra structure satisfying the

Leibniz identity is called a Poisson algebra. As we will see (Chapters 3, 4 and 5),

the existence of such a structure on the algebra corresponds to the existence of a

certain diļ¬erential-geometric structure on an underlying space.

Remark. Given a Lie algebra g, we may deļ¬ne new Lie algebras gĪµ where the

bracket operation is [Ā·, Ā·]gĪµ = Īµ[Ā·, Ā·]g . For each Īµ, the PoincarĀ“-Birkhoļ¬-Witt theorem

e

will give a vector space isomorphism

U(gĪµ ) S(g) .

Multiplication on U(gĪµ ) induces a family of multiplications on S(g), denoted ā—Īµ ,

which satisfy

1

Īµk Bk (f, g) + . . .

f ā—Īµ g = f g + Īµ{f, g} +

2

kā„2

for some bilinear operators Bk . This family is called a deformation quantization

of Pol(gā— ) in the direction of the Poisson bracket; see Chapters 20 and 21. ā™¦

2.3 The Role of the Jacobi Identity 7

2.3 The Role of the Jacobi Identity

Choose a basis v1 , . . . , vn for g. Let j : g ā’ T (g) be the inclusion map. The algebra

T (g) is linearly generated by all monomials

j(vĪ±1 ) ā— . . . ā— j(vĪ±k ) .

If i : g ā’ U(g) is the natural map (as in Section 1.1), it is easy to see, via the

relation i(x) ā— i(y) ā’ i(y) ā— i(x) = i([x, y]) in U(g), that the universal enveloping

algebra is generated by monomials of the form

i(vĪ±1 ) ā— . . . ā— i(vĪ±k ) , Ī±1 ā¤ . . . ā¤ Ī± k .

However, it is not as trivial to show that there are no linear relations between these

generating monomials. Any proof of the independence of these generators must use

the Jacobi identity. The Jacobi identity is crucial since U(g) was deļ¬ned to be an

universal object relative to the category of Lie algebras.

Forget for a moment about the Jacobi identity. We deļ¬ne an almost Lie

algebra g to be the same as a Lie algebra except that the bracket operation does not

necessarily satisfy the Jacobi identity. It is not diļ¬cult to see that the constructions

for the universal enveloping algebra still hold true in this category. We will test the

independence of the generating monomials of U(g) in this case. Let x, y, z ā g for

some almost Lie algebra g. The jacobiator is the trilinear map J : g Ć— g Ć— g ā’ g

deļ¬ned by

J(x, y, z) = [x, [y, z]] + [y, [z, x]] + [z, [x, y]] .

Clearly, on a Lie algebra, the jacobiator vanishes; in general, it measures the ob-

struction to the Jacobi identity. Since J is antisymmetric in the three entries, we

can view it as a map g ā§ g ā§ g ā’ g, which we will still denote by J.

Claim. i : g ā’ U(g) vanishes on the image of J.

This implies that we need J ā” 0 for i to be an injection and the PoincarĀ“-

e

Birkhoļ¬-Witt theorem to hold.

Proof. Take x, y, z ā g, and look at

i (J(x, y, z)) = i ([[x, y, ], z] + c.p.) .

Here, c.p. indicates that the succeeding terms are given by applying circular per-

mutations to the x, y, z of the ļ¬rst term. Because i is linear and commutes with

the bracket operation, we see that

i (J(x, y, z)) = [[i(x), i(y)]U (g) , i(z)]U (g) + c.p. .

But the bracket in the associative algebra always satisļ¬es the Jacobi identity, and

so i(J) ā” 0. 2

Exercise 4

1. Is the image of J the entire kernel of i?

2. Is the image of J an ideal in g? If this is true, then we can form the

āmaximal Lie algebraā quotient by forming g/Im(J). This would then

lead to a reļ¬nement of PoincarĀ“-Birkhoļ¬-Witt to almost Lie algebras.

e

Ā“

8 2 THE POINCARE-BIRKHOFF-WITT THEOREM

Remark. The answers to the exercise above (which we do not know!) should

involve the calculus of multilinear operators. There are two versions of this theory:

ā¢ skew-symmetric operators ā“ from the work of FrĀØlicher and Nijenhuis [61];

o

ā¢ arbitrary multilinear operators ā“ looking at the associativity of algebras, as

in the work of Gerstenhaber [67, 68].

ā™¦

2.4 Actions of Lie Algebras

Much of this section traces back to the work of Lie around the end of the 19th

century on the existence of a Lie group G whose Lie algebra is a given Lie algebra

g.

Our proof of the PoincarĀ“-Birkhoļ¬-Witt theorem will only require local existence

e

of G ā“ a neighborhood of the identity element in the group. What we shall construct

is a manifold M with a Lie algebra homomorphism from g to vector ļ¬elds on M ,

Ļ : g ā’ Ļ(M ), such that a basis of vectors on g goes to a pointwise linearly

independent set of vector ļ¬elds on M . Such a map Ļ is called a pointwise faithful

representation, or free action of g on M .

Example. Let M = G be a Lie group with Lie algebra g. Then the map

taking elements of g to left invariant vector ļ¬elds on G (the generators of the right

ā™¦

translations) is a free action.

The Lie algebra homomorphism Ļ : g ā’ Ļ(M ) is called a right action of the

Lie algebra g on M . (For left actions, Ļ would have to be an anti-homomorphism.)

Such actions Ļ can be obtained by diļ¬erentiating right actions of the Lie group G.

One of Lieā™s theorems shows that any homomorphism Ļ can be integrated to a local

action of the group G on M .

Let v1 , . . . , vn be a basis of g, and V1 = Ļ(v1 ), . . . , Vn = Ļ(vn ) the corresponding

vector ļ¬elds on M . Assume that the Vj are pointwise linearly independent. Since

Ļ is a Lie algebra homomorphism, we have relations

[Vi , Vj ] = cijk Vk ,

k

where the constants cijk are the structure constants of the Lie algebra, deļ¬ned

cijk vk . In other words, {V1 , . . . , Vn } is a set of vector

by the relations [vi , vj ] =

ļ¬elds on M whose bracket has the same relations as the bracket on g. These

relations show in particular that the span of V1 , . . . , Vn is an involutive subbundle

of T M . By the Frobenius theorem, we can integrate it. Let N ā M be a leaf of the

corresponding foliation. There is a map ĻN : g ā’ Ļ(N ) such that the Vj = ĻN (vj )ā™s

form a pointwise basis of vector ļ¬elds on N .

Although we will not need this fact for the PoincarĀ“-Birkhoļ¬-Witt theorem,

e

we note that the leaf N is, in a sense, locally the Lie group with Lie algebra g:

Pick some point in N and label it e. There is a unique local group structure on

a neighborhood of e such that e is the identity element and V1 , . . . , Vn are left

invariant vector ļ¬elds. The group structure comes from deļ¬ning the ļ¬‚ows of the

vector ļ¬elds to be right translations. The hard part of this construction is showing

that the multiplication deļ¬ned in this way is associative.

2.5 Proof of the PoincarĀ“-Birkhoļ¬-Witt Theorem

e 9

All of this is part of Lieā™s third theorem that any Lie algebra is the Lie algebra

of a local Lie group. Existence of a global Lie group was proven by Cartan in [23].

Claim. The injectivity of any single action Ļ : g ā’ Ļ(M ) of the Lie algebra g on

a manifold M is enough to imply that i : g ā’ U (g) is injective.

Proof. Look at the algebraic embedding of vector ļ¬elds into all vector space

endomorphisms of C ā (M ):

Ļ(M ) ā‚ EndVect (C ā (M )) .

The bracket on Ļ(M ) is the commutator bracket of vector ļ¬elds. If we consider

Ļ(M ) and EndVect (C ā (M )) as purely algebraic objects (using the topology of M

only to deļ¬ne C ā (M )), then we use the universality of U(g) to see

ĻE E EndVect (C ā (M ))

Ļ(M )

g ā‚ ā‚

ĀØ

B

ĀØĀØ

ā!Ė ĀØ

ĀØ

Ļ

i ĀØ

ĀØĀØ

cĀØĀØĀØ

U(g)

Thus, if Ļ is injective for some manifold M , then i must also be an injection. 2

The next section shows that, in fact, any pointwise faithful Ļ gives rise to a

faithful representation Ļ of U(g) as diļ¬erential operators on C ā (M ).

Ė

2.5 Proof of the PoincarĀ“-Birkhoļ¬-Witt Theorem

e

In Section 4.2, we shall actually ļ¬nd a manifold M with a free action Ļ : g ā’ Ļ(M ).

Assume now that we have g, Ļ, M, N and Ļ : U(g) ā’ EndVect (C ā (M )) as described

Ė

in the previous section.

Choose coordinates x1 , . . . , xn centered at the āidentityā e ā N such that the

images of the basis elements v1 , . . . , vn of g are the vector ļ¬elds

ā‚

Vi = + O(x) .

ā‚xi

The term O(x) is some vector ļ¬eld vanishing at e which we can write as

ā‚

O(x) = xj aijk (x) .

ā‚xk

j,k

We regard the vector ļ¬elds V1 , . . . , Vn as a set of linearly independent ļ¬rst-order

diļ¬erential operators via the embedding Ļ(M ) ā‚ EndVect (C ā (M )).

Lemma 2.2 The monomials Vi1 Ā· Ā· Ā· Vik with i1 ā¤ . . . ā¤ ik are linearly independent

diļ¬erential operators.

This will show that the monomials i(vi1 ) Ā· Ā· Ā· i(vik ) must be linearly independent

in U(g) since Ļ(i(vi1 ) Ā· Ā· Ā· i(vik )) = Vi1 Ā· Ā· Ā· Vik , which would conclude the proof of the

Ė

PoincarĀ“-Birkhoļ¬-Witt theorem.

e

Ā“

10 2 THE POINCARE-BIRKHOFF-WITT THEOREM

Proof. We show linear independence by testing the monomials against certain

j

functions. Given i1 ā¤ . . . ā¤ ik and j1 ā¤ . . . ā¤ j , we deļ¬ne numbers Ki as follows:

j

(Vi1 Ā· Ā· Ā· Vik ) (xj1 Ā· Ā· Ā· xj ) (e)

Ki :=

ā‚ ā‚

+ O(x) Ā· Ā· Ā· + O(x) (xj1 Ā· Ā· Ā· xj ) (e)

= ā‚xi1 ā‚xik

1. If k < , then any term in the expression will take only k derivatives. But

j

xj1 Ā· Ā· Ā· xj vanishes to order at e, and hence Ki = 0.

2. If k = , then there is only one way to get a non-zero result, namely when

the jā™s match with the iā™s. In this case, we get

0 i=j

j

Ki =

cj > 0 i=j .

i

3. If k > , then the computation is rather complicated, but fortunately this

case is not relevant.

Assume that we had a dependence relation on the Vi ā™s of the form

bi1 ,...,ik Vi1 Ā· Ā· Ā· Vik = 0 .

R=

i1 ,...,ik

kā¤r

Apply R to the functions of the form xj1 Ā· Ā· Ā· xjr and evaluate at e. All the terms

of R with degree less than r will contribute nothing, and there will be at most one

monomial Vi1 Ā· Ā· Ā· Vir of R which is non-zero on xj1 Ā· Ā· Ā· xjr . We see that bi1 ,...,ir = 0

for each multi-index i1 , . . . , ir of order r. By induction on the order of the multi-

indices, we conclude that all bi = 0. 2

To complete the proof of Theorem 2.1, it remains to ļ¬nd a pointwise faithful

representation Ļ for g. To construct the appropriate manifold M , we turn to Poisson

geometry.

Part II

Poisson Geometry

3 Poisson Structures

Let g be a ļ¬nite dimensional Lie algebra with Lie bracket [Ā·, Ā·]g . In Section 2.2, we

deļ¬ned a Poisson bracket {Ā·, Ā·} on Gr U(g) using the commutator bracket in U(g)

and noted that {Ā·, Ā·} satisļ¬es the Leibniz identity. The PoincarĀ“-Birkhoļ¬-Witt

e

ā—

theorem (in Section 2.1) states that Gr U(g) S(g) = Pol(g ). This isomorphism

induces a Poisson bracket on Pol(gā— ).

In this chapter, we will construct a Poisson bracket directly on all of C ā (gā— ),

restricting to the previous bracket on polynomial functions, and we will discuss

general facts about Poisson brackets which will be used in Section 4.2 to conclude

the proof of the PoincarĀ“-Birkhoļ¬-Witt theorem.

e

3.1 Lie-Poisson Bracket

Given functions f, g ā C ā (gā— ), the 1-forms df, dg may be interpreted as maps

Df, Dg : gā— ā’ gā—ā— . When g is ļ¬nite dimensional, we have gā—ā— g, so that Df

ā—

and Dg take values in g. Each Āµ ā g is a function on g. The new function

{f, g} ā C ā (gā— ) evaluated at Āµ is

{f, g}(Āµ) = Āµ [Df (Āµ), Dg(Āµ)]g .

Equivalently, we can deļ¬ne this bracket using coordinates. Let v1 , . . . , vn be a basis

for g and let Āµ1 , . . . , Āµn be the corresponding coordinate functions on gā— . Introduce

the structure constants cijk satisfying [vi , vj ] = cijk vk . Then set

ā‚f ā‚g

{f, g} = cijk Āµk .

ā‚Āµi ā‚Āµj

i,j,k

Exercise 5

Verify that the deļ¬nitions above are equivalent.

The bracket {Ā·, Ā·} is skew-symmetric and takes pairs of smooth functions to

smooth functions. Using the product rule for derivatives, one can also check the

Leibniz identity: {f g, h} = {f, h}g + f {g, h}.

The bracket {Ā·, Ā·} on C ā (gā— ) is called the Lie-Poisson bracket. The pair

(gā— , {Ā·, Ā·}) is often called a Lie-Poisson manifold. (A good reference for the Lie-

Poisson structures is Marsden and Ratiuā™s book on mechanics [116].)

Remark. The coordinate functions Āµ1 , . . . , Āµn satisfy {Āµi , Āµj } = cijk Āµk . This

ā—

implies that the linear functions on g are closed under the bracket operation.

Furthermore, the bracket {Ā·, Ā·} on the linear functions of gā— is exactly the same as

the Lie bracket [Ā·, Ā·] on the elements of g. We thus see that there is an embedding

of Lie algebras g ā’ C ā (gā— ). ā™¦

11

12 3 POISSON STRUCTURES

Exercise 6

As a commutative, associative algebra, Pol(gā— ) is generated by the linear func-

tions. Using induction on the degree of polynomials, prove that, if the Leibniz

identity is satisļ¬ed throughout the algebra and if the Jacobi identity holds on

the generators, then the Jacobi identity holds on the whole algebra.

In Section 3.3, we show that the bracket on C ā (gā— ) satisļ¬es the Jacobi identity.

Knowing that the Jacobi identity holds on Pol(gā— ), we could try to extend to

C ā (gā— ) by continuity, but instead we shall provide a more geometric argument.

3.2 Almost Poisson Manifolds

A pair (M, {Ā·, Ā·}) is called an almost Poisson manifold when {Ā·, Ā·} is an almost

Lie algebra structure (deļ¬ned in Section 2.3) on C ā (M ) satisfying the Leibniz

identity. The bracket {Ā·, Ā·} is then called an almost Poisson structure.

Thanks to the Leibniz identity, {f, g} depends only on the ļ¬rst derivatives of f

and g, thus we can write it as

{f, g} = Ī (df, dg) ,

where Ī is a ļ¬eld of skew-symmetric bilinear forms on T ā— M . We say that Ī ā

Ī“((T ā— M ā§ T ā— M )ā— ) = Ī“(T M ā§ T M ) = Ī“(ā§2 T M ) is a bivector ļ¬eld.

Conversely, any bivector ļ¬eld Ī deļ¬nes a bilinear antisymmetric multiplication

{Ā·, Ā·}Ī on C ā (M ) by the formula {f, g}Ī = Ī (df, dg). Such a multiplication sat-

isļ¬es the Leibniz identity because each Xh := {Ā·, h}Ī is a derivation of C ā (M ).

Hence, {Ā·, Ā·}Ī is an almost Poisson structure on M .

Remark. The diļ¬erential forms ā„¦ā¢ (M ) on a manifold M are the sections of

ā§ā¢ T ā— M := ā• ā§k T ā— M .

There are two well-known operations on ā„¦ā¢ (M ): the wedge product ā§ and the

diļ¬erential d.

The analogous structures on sections of

ā§ā¢ T M := ā• ā§k T M

are less commonly used in diļ¬erential geometry: there is a wedge product, and there

is a bracket operation dual to the diļ¬erential on sections of ā§ā¢ T ā— M . The sections of

ā§k T M are called k-vector ļ¬elds (or multivector ļ¬elds for unspeciļ¬ed k) on M .

The space of such sections is denoted by Ļk (M ) = Ī“(ā§k T M ). There is a natural

commutator bracket on the direct sum of Ļ0 (M ) = C ā (M ) and Ļ1 (M ) = Ļ(M ).

In Section 18.3, we shall extend this bracket to an operation on Ļk (M ), called the

ā™¦

Schouten-Nijenhuis bracket [116, 162].

3.3 Poisson Manifolds

An almost Poisson structure {Ā·, Ā·}Ī on a manifold M is called a Poisson structure

if it satisļ¬es the Jacobi identity. A Poisson manifold (M, {Ā·, Ā·}) is a manifold M

equipped with a Poisson structure {Ā·, Ā·}. The corresponding bivector ļ¬eld Ī is then

called a Poisson tensor. The name āPoisson structureā sometimes refers to the

bracket {Ā·, Ā·} and sometimes to the Poisson tensor Ī .

3.4 Structure Functions and Canonical Coordinates 13

Given an almost Poisson structure, we deļ¬ne the jacobiator on C ā (M ) by:

J(f, g, h) = {{f, g}, h} + {{g, h}, f } + {{h, f }, g} .

Exercise 7

Show that the jacobiator is

(a) skew-symmetric, and

(b) a derivation in each argument.

By the exercise above, the operator J on C ā (M ) corresponds to a trivector

ļ¬eld J ā Ļ3 (M ) such that J (df, dg, dh) = J(f, g, h). In coordinates, we write

ā‚f ā‚g ā‚h

J(f, g, h) = Jijk (x) ,

ā‚xi ā‚xj ā‚xk

i,j,k

where Jijk (x) = J(xi , xj , xk ).

Consequently, the Jacobi identity holds on C ā (M ) if and only if it holds for

the coordinate functions.

Example. When M = gā— is a Lie-Poisson manifold, the Jacobi identity holds

on the coordinate linear functions, because it holds on the Lie algebra g (see Sec-

tion 3.1). Hence, the Jacobi identity holds on C ā (gā— ). ā™¦

Up to a constant factor, J = [Ī , Ī ], where [Ā·, Ā·] is the Schouten-

Remark.

Nijenhuis bracket (see Section 18.3 and the last remark of Section 3.2). Therefore,

the Jacobi identity for the bracket {Ā·, Ā·} is equivalent to the equation [Ī , Ī ] = 0.

ā™¦

We will not use this until Section 18.3.

3.4 Structure Functions and Canonical Coordinates

Let Ī be the bivector ļ¬eld on an almost Poisson manifold (M, {Ā·, Ā·}Ī ). Choosing

local coordinates x1 , . . . , xn on M , we ļ¬nd structure functions

Ļij (x) = {xi , xj }Ī

of the almost Poisson structure. In coordinate notation, the bracket of functions

f, g ā C ā (M ) is

ā‚f ā‚g

{f, g}Ī = Ļij (x) .

ā‚xi ā‚xj

Equivalently, we have

1 ā‚ ā‚

ā§

Ī = Ļij (x) .

2 ā‚xi ā‚xj

Exercise 8

Write the jacobiator Jijk in terms of the structure functions Ļij . It is a homo-

geneous quadratic expression in the Ļij ā™s and their ļ¬rst partial derivatives.

14 3 POISSON STRUCTURES

Examples.

1. When Ļij (x) = cijk xk , the Poisson structure is a linear Poisson struc-

ture. Clearly the Jacobi identity holds if and only if the cijk are the structure

constants of a Lie algebra g. When this is the case, the x1 , . . . , xn are coordi-

nates on gā— . We had already seen that for the Lie-Poisson structure deļ¬ned

on gā— , the functions Ļij were linear.

2. Suppose that the Ļij (x) are constant. In this case, the Jacobi identity is

trivially satisļ¬ed ā“ each term in the jacobiator of coordinate functions is zero.

By a linear change of coordinates, we can put the constant antisymmetric

matrix (Ļij ) into the normal form:

ļ£« ļ£¶

0 Ik

0ļ£ø

ļ£ ā’Ik 0

0 0

where Ik is the k Ć— k identity matrix and 0 is the Ć— zero matrix. If

we call the new coordinates q1 , . . . , qk , p1 , . . . , pk , c1 , . . . , c , the bivector ļ¬eld

becomes

ā‚ ā‚

ā§

Ī = .

ā‚qi ā‚pi

i

In terms of the bracket, we can write

ā‚f ā‚g ā‚f ā‚g

{f, g} = ā’ ,

ā‚qi ā‚pi ā‚pi ā‚qi

i

which is actually the original form due to Poisson in [138]. The ci ā™s do not

enter in the bracket, and hence behave as parameters. The following relations,

called canonical Poisson relations, hold:

ā¢ {qi , pj } = Ī“ij

ā¢ {qi , qj } = {pi , pj } = 0

ā¢ {Ī±, ci } = 0 for any coordinate function Ī±.

The coordinates ci are said to be in the center of the Poisson algebra; such

functions are called Casimir functions. If = 0, i.e. if there is no center,

then the structure is said to be non-degenerate or symplectic. In any

case, qi , pi are called canonical coordinates. Theorem 4.2 will show that

this example is quite general.

ā™¦

3.5 Hamiltonian Vector Fields

Let (M, {Ā·, Ā·}) be an almost Poisson manifold. Given h ā C ā (M ), deļ¬ne the linear

map

Xh : C ā (M ) ā’ā’ C ā (M ) Xh (f ) = {f, h} .

by

The correspondence h ā’ Xh resembles an āadjoint representationā of C ā (M ). By

the Leibniz identity, Xh is a derivation and thus corresponds to a vector ļ¬eld, called

the hamiltonian vector ļ¬eld of the function h.

3.6 Poisson Cohomology 15

Lemma 3.1 On a Poisson manifold, hamiltonian vector ļ¬elds satisfy

[Xf , Xg ] = ā’X{f,g} .

Proof. We can see this by applying [Xf , Xg ] + X{f,g} to an arbitrary function

h ā C ā (M ).

Xf Xg h ā’ Xg Xf h + X{f,g} h

[Xf , Xg ] + X{f,g} h =

Xf {h, g} ā’ Xg {h, f } + {h, {f, g}}

=

{{h, g}, f } + {{f, h}, g} + {{g, f }, h} .

=

The statement of the lemma is thus equivalent to the Jacobi identity for the Poisson

bracket. 2

Historical Remark. This lemma gives another formulation of the integrability

condition for Ī , which, in fact, was the original version of the identity as formulated

by Jacobi around 1838. (See Jacobiā™s collected works [86].) Poisson [138] had

introduced the bracket {Ā·, Ā·} in order to simplify calculations in celestial mechanics.

He proved around 1808, through long and tedious computations, that

{f, h} = 0 and {g, h} = 0 {{f, g}, h} = 0 .

=ā’

This means that, if two functions f, g are constant along integral curves of Xh ,

then one can form a third function also constant along Xh , namely {f, g}. When

Jacobi later stated the identity in Lemma 3.1, he gave a much shorter proof of a

ā™¦

yet stronger result.

3.6 Poisson Cohomology

A Poisson vector ļ¬eld, is a vector ļ¬eld X on a Poisson manifold (M, Ī ) such

that LX Ī = 0, where LX is the Lie derivative along X. The Poisson vector ļ¬elds,

also characterized by

X{f, g} = {Xf, g} + {f, Xg} ,

are those whose local ļ¬‚ow preserves the bracket operation. These are also the

derivations (with respect to both operations) of the Poisson algebra.

Among the Poisson vector ļ¬elds, the hamiltonian vector ļ¬elds Xh = {Ā·, h} form

the subalgebra of inner derivations of C ā (M ). (Of course, they are āinnerā only

for the bracket.)

Exercise 9

Show that the hamiltonian vector ļ¬elds form an ideal in the Lie algebra of

Poisson vector ļ¬elds.

Remark. The quotient of the Lie algebra of Poisson vector ļ¬elds by the ideal of

hamiltonian vector ļ¬elds is a Lie algebra, called the Lie algebra of outer deriva-

tions. Several questions naturally arise.

16 3 POISSON STRUCTURES

ā¢ Is there a group corresponding to the Lie algebra of outer derivations?

ā¢ What is the group that corresponds to the hamiltonian vector ļ¬elds?

In Section 18.4 we will describe these āgroupsā in the context of Lie algebroids.

ā™¦

We can form the sequence:

C ā (M ) Ļ2 (M )

0 Ļ(M )

E E E

ā’ā’

h Xh

ā’ā’ LX Ī

X

where the composition of two maps is 0. Hence, we have a complex. At Ļ(M ), the

homology group is

Poisson vector ļ¬elds

1

HĪ (M ) := .

hamiltonian vector ļ¬elds

This is called the ļ¬rst Poisson cohomology.

The homology at Ļ0 (M ) = C ā (M ) is called 0-th Poisson cohomology

0

HĪ (M ), and consists of the Casimir functions, i.e. the functions f such that

{f, h} = 0, for all h ā C ā (M ). (For the trivial Poisson structure {Ā·, Ā·} = 0, this is

all of C ā (M ).)

See Section 5.1 for a geometric description of these cohomology spaces. See Sec-

tion 4.5 for their interpretation in the symplectic case. Higher Poisson cohomology

groups will be deļ¬ned in Section 18.4.

4 Normal Forms

Throughout this and the next chapter, our goal is to understand what Poisson

manifolds look like geometrically.

4.1 Lieā™s Normal Form

We will prove the following result in Section 4.3.

Theorem 4.1 (Lie [106]) If Ī is a Poisson structure on M whose matrix of

structure functions, Ļij (x), has constant rank, then each point of M is contained

in a local coordinate system with respect to which (Ļij ) is constant.

Remarks.

1. The assumption above of constant rank was not stated by Lie, although it

was used implicitly in his proof.

2. Since Theorem 4.1 is a local result, we only need to require the matrix (Ļij )

to have locally constant rank. This is a reasonable condition to impose, as

the structure functions Ļij will always have locally constant rank on an open

dense set of M . To see this, notice that the set of points in M where (Ļij )

has maximal rank is open, and then proceed inductively on the complement

of the closure of this set (exercise!). Notice that the set of points where the

rank of (Ļij ) is maximal is not necessarily dense. For instance, consider R2

with {x1 , x2 } = Ļ•(x1 , x2 ) given by an arbitrary function Ļ•.

3. Points where (Ļij ) has locally constant rank are called regular. If all points

of M are regular, M is called a regular Poisson manifold. A Lie-Poisson

manifold gā— is not regular unless g is abelian, though the regular points of gā—

form, of course, an open dense subset.

ā™¦

4.2 A Faithful Representation of g

We will now use Theorem 4.1 to construct the pointwise faithful representation of

g needed to complete the proof of the PoincarĀ“-Birkhoļ¬-Witt theorem.

e

On any Poisson manifold M there is a vector bundle morphism Ī : T ā— M ā’ T M

deļ¬ned by

for any Ī±, Ī² ā T ā— M .

Ī±(Ī (Ī²)) = Ī (Ī±, Ī²) ,

We can write hamiltonian vector ļ¬elds in terms of Ī as Xf = Ī (df ). Notice that Ī

is an isomorphism exactly when rank Ī = dim M , i.e. when Ī deļ¬nes a symplectic

structure. If we express Ī by a matrix (Ļij ) with respect to some basis, then the

same matrix (Ļij ) represents the map Ī .

Let M = gā— have coordinates Āµ1 , . . . , Āµn and Poisson structure {Āµi , Āµj } =

cijk Āµk . If v1 , . . . , vn is the corresponding basis of vectors on g, then we ļ¬nd

a representation of g on gā— by mapping

vi ā’ā’ ā’XĀµi .

17

18 4 NORMAL FORMS

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