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Geometric Models for

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 Poincar´-Birkho¬-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 Poincar´-Birkho¬-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 Lie™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 — -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 Poincar´-Birkho¬-Witt
e Revisited . . . . . . . . . . . . . . . . . . . . 81


VI Groupoids 85

13 Groupoids 85
13.1 De¬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 De¬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 Di¬erential Geometry for Lie Algebroids 131
18.1 The Exterior Di¬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 In¬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 A¬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 Fedosov™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 Classi¬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 a¬ne
algebraic varieties, di¬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 in¬nitesimal
approximations to di¬erentiable groupoids.
These notes are based on a topics course, “Geometric Models for Noncommuta-
tive Algebras,” 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 “ 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: Mosh´ Flato, K. Guruprasad, Andr´
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
Lako¬ and Nu˜ez [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, di¬erent restrictions on the functions reveal di¬erent structures
on the space. Examples of distinct algebras of functions which can be associated
to a space are:

• polynomial functions,

• real analytic functions,

• smooth functions,

• C k , or just continuous (C 0 ) functions,

• L∞ , 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 ¬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 ∈ C} .
C[x]

The coe¬cients a0 , . . . , ak may be thought of as values of a complex-valued
function plus its ¬rst, second, ..., kth derivatives at the origin. The corre-
sponding “space” is the so-called kth in¬nitesimal neighborhood of the
point 0. Each of these “spaces” 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 “good” (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 — by
—k
T (V — ) = C • V — • (V — — V — ) • . . . • (V — ) • ... ,


xiii
xiv INTRODUCTION

—k
where (V — ) is spanned by {xi1 — . . . — xik | 1 ¤ i1 , . . . , ik ¤ n}, then we
realize the symmetric algebra S(V — ) = Pol(V ) as

S(V — ) = T (V — )/C ,

where C is the ideal generated by {± — β ’ β — ± | ±, β ∈ V — }.
There are several ways to recover V and its structure from the algebra Pol(V ):

• Linear homomorphisms from Pol(V ) to C correspond to points of V . We
thus recover the set V .
• Algebra endomorphisms of Pol(V ) correspond to polynomial endomor-
phisms of V : An algebra endomorphism

f : Pol(V ) ’’ Pol(V )

is determined by the f (x1 ), . . . , f (xn )). Since Pol(V ) is freely generated
by the xi ™s, we can choose any f (xi ) ∈ Pol(V ). For example, if n = 2, f
could be de¬ned by:
x1 ’’ x1
x2 ’’ x2 + x2 1

which would even be invertible. We are thus recovering a polynomial
structure in V .
• Graded algebra automorphisms of Pol(V ) correspond to linear isomor-
phisms of V : As a graded algebra

Polk (V ) ,
Pol(V ) =
k=0


where Polk (V ) is the set of homogeneous polynomials of degree k, i.e.
symmetric tensors in (V — )—k . A graded automorphism takes each xi to
an element of degree one, that is, a linear homogeneous expression in the
xi ™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
de¬ne
Λ• (V — ) = T (V — )/A ,
where A is the ideal generated by {± — β + β — ± | ±, β ∈ V — }. We can view
this as a graded algebra,

• —
Λk (V — ) ,
Λ (V ) =
k=0

whose automorphisms give us the linear structure on V . Therefore, as a
graded algebra, Λ• (V — ) still “represents” the vector space structure in V .
The algebra Λ• (V — ) is not commutative, but is instead super-commutative,
i.e. for elements a ∈ Λk (V — ), b ∈ Λ (V — ), we have

ab = (’1)k ba .
INTRODUCTION xv


Super-commutativity is associated to a Z2 -grading:1
Λ• (V — ) Λ[0] (V — ) • Λ[1] (V — ) ,
=
where
Λ[0] (V — ) = Λeven (V — ) Λk (V — ) ,
:= and
k even


Λ[1] (V — ) = Λodd (V — ) Λk (V — ) .
:=
k odd
Therefore, V is not just a vector space, but is called an odd superspace;
“odd” 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,
’’
x1 x1
’’
x2 x2 + x1 x2 x3
’’
x3 x3
is legal; this preserves the relations since both objects and images anti-
commute. Although there is more ¬‚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 Λ• (V — ) correspond to “functions” on the
(odd) superspace V . We may view the construction above as a de¬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 „¦• (M ) of di¬erential forms on a manifold M can be regarded as
a Z2 -graded algebra by
„¦• (M ) = „¦even (M ) • „¦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, „¦• (M ) is the
—¦ —¦
algebra of functions on the odd tangent bundle T M ; the indicates that
here we regard the ¬bers of T M as odd superspaces.
The exterior derivative
d : „¦• (M ) ’’ „¦• (M )
has the property that for f, g ∈ „¦• (M ),
d(f g) = (df )g + (’1)deg f f (dg) .
Hence, d is a derivation of a superalgebra. It exchanges the subspaces of even
—¦
and odd degree. We call d an odd vector ¬eld on T M .
6. Consider the algebra of complex valued functions on a “phase space” 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 ’ pq = i ,
1 The term “super” 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 ’ pq ’ 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).

™¦
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. Brie¬‚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 in¬nitesimal counterparts of Lie groupoids, and are very close to Poisson and
symplectic geometry.
Finally, we will discuss Fedosov™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 [·, ·]. We will assume that g is a ¬nite
dimensional algebra over R or C, but much of the following also holds for in¬nite
dimensional Lie algebras, as well as for Lie algebras over arbitrary ¬elds or rings.

1.1 Universal Enveloping Algebras
Regarding g just as a vector space, we may form the tensor algebra,

g—k ,
T (g) =
k=0

which is the free associative algebra over g. There is a natural inclusion j : g ’ T (g)
taking g to g—1 such that, for any linear map f : g ’ A to an associative algebra
A, the assignment g(v1 — . . . — vk ) ’ f (v1 ) . . . f (vk ) determines the unique algebra
homomorphism g making the following diagram commute.
jE
T (g)
g
d
g
d
d
fd

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

If the Lie bracket is trivial, i.e. [·, ·] ≡ 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 ¬nite dimensional, S(g) coincides with the algebra of polynomials in g— .) S(g) is
the universal commutative enveloping algebra of g because it satis¬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 ’ U(g) be
the composition of the inclusion j : g ’ T (g) with the natural projection T (g) ’
U(g). Given any associative algebra A, let Lie(A) be the algebra A equipped with
the bracket [a, b]A = ab ’ ba, and hence regarded as a Lie algebra. Then, for
any Lie algebra homomorphism f : g ’ A, there is a unique associative algebra
homomorphism g : U(g) ’ A making the following diagram commute.
iE
U(g)
g
d
g
d
d
fd

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(·) from Lie algebras to associative
algebras is the left adjoint of the functor Lie(·).
Exercise 1
What are the adjoint functors of T and S?



1.2 Lie Algebra Deformations
The Poincar´-Birkho¬-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 [·, ·], then U(g) will still be approximately isomorphic
to S(g). One way to express this approximation is to throw in a parameter µ
multiplying the bracket; i.e. we look at the Lie algebra deformation gµ = (g, µ[·, ·]).
As µ tends to 0, gµ approaches an abelian Lie algebra. The family gµ 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 gµ we obtain a one-parameter family of associative algebras U(gµ ), passing
through S(g) at µ = 0. Here we are taking the quotients of T (g) by a family of
ideals generated by
{j(x) — j(y) ’ j(y) — j(x) ’ j(µ[x, y]) | x, y ∈ g} ,
so there is no obvious isomorphism as vector spaces between the U(gµ ) for di¬erent
values of µ. We do have, however:
Claim. U(g) U(gµ ) for all µ = 0.

Proof. For a homomorphism of Lie algebras f : g ’ h, the functoriality of U(·)
and the universality of U(g) give the commuting diagram
f
g Eh
d
d
ih —¦ f
ig ih
d
c ∃!g d c

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/µ
gµ ,
g' E

1
and µ, we conclude that U(g) U(gµ ) for µ = 0.
given by multiplication by 2
µ

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(gµ ) 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 ¬rst derivative with respect to µ of the path U(gµ ) 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 — . . . — xn ’’ xσ(1) — . . . — xσ(n)
n!
σ∈Sn

extends to a well-de¬ned symmetrization endomorphism s : T (g) ’ 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) — j(y) ’ j(y) — j(x) |
x, y ∈ 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) ’’ S(g) .

The linear section
S(g) ’’ T (g)
„:
’’ s(x1 — . . . — 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):

T k (g) = g—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 de¬ning U(g) are not homogeneous unless [·, ·]g = 0. (On the other hand,
symmetrization s : T (g) ’ S(g) does preserve the grading.)
4 1 ALGEBRAIC CONSTRUCTIONS


The grading of T (g) has associated ¬ltration
k
(k)
T j (g) ,
T (g) =
j=0

such that
T (0) ⊆ T (1) ⊆ T (2) ⊆ . . . T (i) — T (j) ⊆ T (i+j) .
and
We can recover T k by T (k) /T (k’1) T k .
What happens to this ¬ltration when we project to U(g)?
Remark. Let i : g ’ U(g) be the natural map (as in Section 1.1). If we take
x, y ∈ g, then i(x)i(y) and i(y)i(x) each “has length 2,” but their di¬erence
i(y)i(x) ’ i(x)i(y) = i([y, x])
has length 1. Therefore, exact length is not respected by algebraic operations on
U(g). ™¦
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 ¤ k of elements
of U (1) (g) = i(g).

We do have the relation
U (k) · U ( ) ⊆ U (k+ )
,
so that the universal enveloping algebra of g has a natural ¬ltration, natural in the
sense that, for any map g ’ h, the diagram
g Eh




c c
U(g) E U(h)
preserves the ¬ltration.
In order to construct a graded algebra, we de¬ne
U k (g) = U (k) (g)/U (k’1) (g) .
There are well-de¬ned product operations
U k (g) — U (g) ’’ U k+ (g)
[±] — [β] ’’ [±β]
forming an associative multiplication on what is called the graded algebra asso-
ciated to U(g):

U k (g) =: Gr U(g) .
j=0

Remark. The constructions above are purely algebraic in nature; we can form
Gr A for any ¬ltered algebra A. The functor Gr will usually simplify the algebra
™¦
in the sense that multiplication forgets about lower order terms.
2 The Poincar´-Birkho¬-Witt Theorem
e
Let g be a ¬nite dimensional Lie algebra with Lie bracket [·, ·]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 su¬ces to show that multiplication

U 1 (g) — U 1 (g) ’’ U 2 (g)

is commutative. Because U (1) (g) is generated by i(g), any ± ∈ U 1 (g) is of the form
± = [i(x)] for some x ∈ g. Pick any two elements x, y ∈ g. Then [i(x)], [i(y)] ∈
U 1 (g), and
[i(x)][i(y)] ’ [i(y)][i(x)] = [i(x)i(y) ’ 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) ’ S(g) in Section 1.3, we constructed
a linear section „ : S(g) ’ T (g). We formulate the Poincar´-Birkho¬-Witt theorem
e
using this linear section.

Theorem 2.1 (Poincar´-Birkho¬-Witt) There is a graded (commutative) al-
e
gebra isomorphism
» : S(g) ’’ Gr U(g)
given by the natural maps:


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


E1 vσ(1) — . . . — vσ(k)
v1 . . . vk E [v1 . . . vk ] .
k!
σ∈Sk


For each degree k, we follow the embedding „ k : S k (g) ’ T k (g) by a map
to U (k) (g) and then by the projection onto U k . Although the composition » :
S(g) ’ Gr U(g) is a graded algebra homomorphism, the maps S(g) ’ T (g) and
T (g) ’ U (g) are not.
We shall prove Theorem 2.1 (for ¬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
¬ltered algebra U,
U (0) ⊆ U (1) ⊆ U (2) ⊆ . . . ,


5
´
6 2 THE POINCARE-BIRKHOFF-WITT THEOREM


for which the associated graded algebra

Uj U j = U (j) U (j’1) .
Gr U := where
j=0

is commutative. Such an algebra U is often called almost commutative.
For x ∈ U (k) and y ∈ U ( ) , de¬ne
’1 ’1) ’2)
{[x], [y]} = [xy ’ yx] ∈ U k+ = U (k+ /U (k+
so that
’1
{U k , U } ⊆ U k+ .
This collection of degree ’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 {·, ·} with the
following properties:
1. {·, ·} is anti-commutative (not super-commutative) and satis¬es the Jacobi
identity
{{u, v}, w} = {{u, w}, v} + {u, {v, w}} .
That is, {·, ·} 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 {·, ·} on Gr U.


Remark. The Leibniz identity says that {·, 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 {·, w} is a
™¦
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 di¬erential-geometric structure on an underlying space.
Remark. Given a Lie algebra g, we may de¬ne new Lie algebras gµ where the
bracket operation is [·, ·]gµ = µ[·, ·]g . For each µ, the Poincar´-Birkho¬-Witt theorem
e
will give a vector space isomorphism
U(gµ ) S(g) .
Multiplication on U(gµ ) induces a family of multiplications on S(g), denoted —µ ,
which satisfy
1
µk Bk (f, g) + . . .
f —µ g = f g + µ{f, g} +
2
k≥2

for some bilinear operators Bk . This family is called a deformation quantization
of Pol(g— ) in the direction of the Poisson bracket; see Chapters 20 and 21. ™¦
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 ’ T (g) be the inclusion map. The algebra
T (g) is linearly generated by all monomials

j(v±1 ) — . . . — j(v±k ) .

If i : g ’ U(g) is the natural map (as in Section 1.1), it is easy to see, via the
relation i(x) — i(y) ’ i(y) — i(x) = i([x, y]) in U(g), that the universal enveloping
algebra is generated by monomials of the form

i(v±1 ) — . . . — i(v±k ) , ±1 ¤ . . . ¤ ± 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 de¬ned to be an
universal object relative to the category of Lie algebras.
Forget for a moment about the Jacobi identity. We de¬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 di¬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 ∈ g for
some almost Lie algebra g. The jacobiator is the trilinear map J : g — g — g ’ g
de¬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 § g § g ’ g, which we will still denote by J.
Claim. i : g ’ U(g) vanishes on the image of J.

This implies that we need J ≡ 0 for i to be an injection and the Poincar´-
e
Birkho¬-Witt theorem to hold.
Proof. Take x, y, z ∈ 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 ¬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 satis¬es the Jacobi identity, and
so i(J) ≡ 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
“maximal Lie algebra” quotient by forming g/Im(J). This would then
lead to a re¬nement of Poincar´-Birkho¬-Witt to almost Lie algebras.
e
´
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:
• skew-symmetric operators “ from the work of Fr¨licher and Nijenhuis [61];
o
• arbitrary multilinear operators “ looking at the associativity of algebras, as
in the work of Gerstenhaber [67, 68].
™¦


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 Poincar´-Birkho¬-Witt theorem will only require local existence
e
of G “ 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 ¬elds on M ,
ρ : g ’ χ(M ), such that a basis of vectors on g goes to a pointwise linearly
independent set of vector ¬elds on M . Such a map ρ 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 ¬elds on G (the generators of the right
™¦
translations) is a free action.
The Lie algebra homomorphism ρ : g ’ χ(M ) is called a right action of the
Lie algebra g on M . (For left actions, ρ would have to be an anti-homomorphism.)
Such actions ρ can be obtained by di¬erentiating right actions of the Lie group G.
One of Lie™s theorems shows that any homomorphism ρ can be integrated to a local
action of the group G on M .
Let v1 , . . . , vn be a basis of g, and V1 = ρ(v1 ), . . . , Vn = ρ(vn ) the corresponding
vector ¬elds on M . Assume that the Vj are pointwise linearly independent. Since
ρ 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, de¬ned
cijk vk . In other words, {V1 , . . . , Vn } is a set of vector
by the relations [vi , vj ] =
¬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 ⊆ M be a leaf of the
corresponding foliation. There is a map ρN : g ’ χ(N ) such that the Vj = ρN (vj )™s
form a pointwise basis of vector ¬elds on N .
Although we will not need this fact for the Poincar´-Birkho¬-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 ¬elds. The group structure comes from de¬ning the ¬‚ows of the
vector ¬elds to be right translations. The hard part of this construction is showing
that the multiplication de¬ned in this way is associative.
2.5 Proof of the Poincar´-Birkho¬-Witt Theorem
e 9


All of this is part of Lie™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 ρ : g ’ χ(M ) of the Lie algebra g on
a manifold M is enough to imply that i : g ’ U (g) is injective.

Proof. Look at the algebraic embedding of vector ¬elds into all vector space
endomorphisms of C ∞ (M ):

χ(M ) ‚ EndVect (C ∞ (M )) .

The bracket on χ(M ) is the commutator bracket of vector ¬elds. If we consider
χ(M ) and EndVect (C ∞ (M )) as purely algebraic objects (using the topology of M
only to de¬ne C ∞ (M )), then we use the universality of U(g) to see

ρE E EndVect (C ∞ (M ))
χ(M )
g ‚ ‚

¨
B
¨¨
∃!˜ ¨
¨
ρ
i ¨
¨¨
c¨¨¨
U(g)

Thus, if ρ is injective for some manifold M , then i must also be an injection. 2
The next section shows that, in fact, any pointwise faithful ρ gives rise to a
faithful representation ρ of U(g) as di¬erential operators on C ∞ (M ).
˜

2.5 Proof of the Poincar´-Birkho¬-Witt Theorem
e

In Section 4.2, we shall actually ¬nd a manifold M with a free action ρ : g ’ χ(M ).
Assume now that we have g, ρ, M, N and ρ : U(g) ’ EndVect (C ∞ (M )) as described
˜
in the previous section.
Choose coordinates x1 , . . . , xn centered at the “identity” e ∈ N such that the
images of the basis elements v1 , . . . , vn of g are the vector ¬elds

Vi = + O(x) .
‚xi

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


O(x) = xj aijk (x) .
‚xk
j,k

We regard the vector ¬elds V1 , . . . , Vn as a set of linearly independent ¬rst-order
di¬erential operators via the embedding χ(M ) ‚ EndVect (C ∞ (M )).

Lemma 2.2 The monomials Vi1 · · · Vik with i1 ¤ . . . ¤ ik are linearly independent
di¬erential operators.

This will show that the monomials i(vi1 ) · · · i(vik ) must be linearly independent
in U(g) since ρ(i(vi1 ) · · · i(vik )) = Vi1 · · · Vik , which would conclude the proof of the
˜
Poincar´-Birkho¬-Witt theorem.
e
´
10 2 THE POINCARE-BIRKHOFF-WITT THEOREM


Proof. We show linear independence by testing the monomials against certain
j
functions. Given i1 ¤ . . . ¤ ik and j1 ¤ . . . ¤ j , we de¬ne numbers Ki as follows:
j
(Vi1 · · · Vik ) (xj1 · · · xj ) (e)
Ki :=
‚ ‚
+ O(x) · · · + O(x) (xj1 · · · xj ) (e)
= ‚xi1 ‚xik


1. If k < , then any term in the expression will take only k derivatives. But
j
xj1 · · · 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 j™s match with the i™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 ™s of the form

bi1 ,...,ik Vi1 · · · Vik = 0 .
R=
i1 ,...,ik
k¤r



Apply R to the functions of the form xj1 · · · 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 · · · Vir of R which is non-zero on xj1 · · · 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 ¬nd a pointwise faithful
representation ρ for g. To construct the appropriate manifold M , we turn to Poisson
geometry.
Part II
Poisson Geometry
3 Poisson Structures
Let g be a ¬nite dimensional Lie algebra with Lie bracket [·, ·]g . In Section 2.2, we
de¬ned a Poisson bracket {·, ·} on Gr U(g) using the commutator bracket in U(g)
and noted that {·, ·} satis¬es the Leibniz identity. The Poincar´-Birkho¬-Witt
e

theorem (in Section 2.1) states that Gr U(g) S(g) = Pol(g ). This isomorphism
induces a Poisson bracket on Pol(g— ).
In this chapter, we will construct a Poisson bracket directly on all of C ∞ (g— ),
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 Poincar´-Birkho¬-Witt theorem.
e

3.1 Lie-Poisson Bracket

Given functions f, g ∈ C ∞ (g— ), the 1-forms df, dg may be interpreted as maps
Df, Dg : g— ’ g—— . When g is ¬nite dimensional, we have g—— g, so that Df

and Dg take values in g. Each µ ∈ g is a function on g. The new function
{f, g} ∈ C ∞ (g— ) evaluated at µ is

{f, g}(µ) = µ [Df (µ), Dg(µ)]g .

Equivalently, we can de¬ne this bracket using coordinates. Let v1 , . . . , vn be a basis
for g and let µ1 , . . . , µn be the corresponding coordinate functions on g— . Introduce
the structure constants cijk satisfying [vi , vj ] = cijk vk . Then set

‚f ‚g
{f, g} = cijk µk .
‚µi ‚µj
i,j,k




Exercise 5
Verify that the de¬nitions above are equivalent.


The bracket {·, ·} 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 {·, ·} on C ∞ (g— ) is called the Lie-Poisson bracket. The pair
(g— , {·, ·}) is often called a Lie-Poisson manifold. (A good reference for the Lie-
Poisson structures is Marsden and Ratiu™s book on mechanics [116].)
Remark. The coordinate functions µ1 , . . . , µn satisfy {µi , µj } = cijk µk . This

implies that the linear functions on g are closed under the bracket operation.
Furthermore, the bracket {·, ·} on the linear functions of g— is exactly the same as
the Lie bracket [·, ·] on the elements of g. We thus see that there is an embedding
of Lie algebras g ’ C ∞ (g— ). ™¦



11
12 3 POISSON STRUCTURES

Exercise 6
As a commutative, associative algebra, Pol(g— ) is generated by the linear func-
tions. Using induction on the degree of polynomials, prove that, if the Leibniz
identity is satis¬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 ∞ (g— ) satis¬es the Jacobi identity.
Knowing that the Jacobi identity holds on Pol(g— ), we could try to extend to
C ∞ (g— ) by continuity, but instead we shall provide a more geometric argument.

3.2 Almost Poisson Manifolds

A pair (M, {·, ·}) is called an almost Poisson manifold when {·, ·} is an almost
Lie algebra structure (de¬ned in Section 2.3) on C ∞ (M ) satisfying the Leibniz
identity. The bracket {·, ·} is then called an almost Poisson structure.
Thanks to the Leibniz identity, {f, g} depends only on the ¬rst derivatives of f
and g, thus we can write it as

{f, g} = Π(df, dg) ,

where Π is a ¬eld of skew-symmetric bilinear forms on T — M . We say that Π ∈
“((T — M § T — M )— ) = “(T M § T M ) = “(§2 T M ) is a bivector ¬eld.
Conversely, any bivector ¬eld Π de¬nes a bilinear antisymmetric multiplication
{·, ·}Π on C ∞ (M ) by the formula {f, g}Π = Π(df, dg). Such a multiplication sat-
is¬es the Leibniz identity because each Xh := {·, h}Π is a derivation of C ∞ (M ).
Hence, {·, ·}Π is an almost Poisson structure on M .
Remark. The di¬erential forms „¦• (M ) on a manifold M are the sections of

§• T — M := • §k T — M .

There are two well-known operations on „¦• (M ): the wedge product § and the
di¬erential d.
The analogous structures on sections of

§• T M := • §k T M

are less commonly used in di¬erential geometry: there is a wedge product, and there
is a bracket operation dual to the di¬erential on sections of §• T — M . The sections of
§k T M are called k-vector ¬elds (or multivector ¬elds for unspeci¬ed k) on M .
The space of such sections is denoted by χk (M ) = “(§k T M ). There is a natural
commutator bracket on the direct sum of χ0 (M ) = C ∞ (M ) and χ1 (M ) = χ(M ).
In Section 18.3, we shall extend this bracket to an operation on χk (M ), called the
™¦
Schouten-Nijenhuis bracket [116, 162].


3.3 Poisson Manifolds

An almost Poisson structure {·, ·}Π on a manifold M is called a Poisson structure
if it satis¬es the Jacobi identity. A Poisson manifold (M, {·, ·}) is a manifold M
equipped with a Poisson structure {·, ·}. The corresponding bivector ¬eld Π is then
called a Poisson tensor. The name “Poisson structure” sometimes refers to the
bracket {·, ·} and sometimes to the Poisson tensor Π.
3.4 Structure Functions and Canonical Coordinates 13


Given an almost Poisson structure, we de¬ne the jacobiator on C ∞ (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 ∞ (M ) corresponds to a trivector
¬eld J ∈ χ3 (M ) such that J (df, dg, dh) = J(f, g, h). In coordinates, we write

‚f ‚g ‚h
J(f, g, h) = Jijk (x) ,
‚xi ‚xj ‚xk
i,j,k

where Jijk (x) = J(xi , xj , xk ).
Consequently, the Jacobi identity holds on C ∞ (M ) if and only if it holds for
the coordinate functions.
Example. When M = g— 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 ∞ (g— ). ™¦


Up to a constant factor, J = [Π, Π], where [·, ·] 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 {·, ·} is equivalent to the equation [Π, Π] = 0.
™¦
We will not use this until Section 18.3.


3.4 Structure Functions and Canonical Coordinates

Let Π be the bivector ¬eld on an almost Poisson manifold (M, {·, ·}Π ). Choosing
local coordinates x1 , . . . , xn on M , we ¬nd structure functions

πij (x) = {xi , xj }Π

of the almost Poisson structure. In coordinate notation, the bracket of functions
f, g ∈ C ∞ (M ) is
‚f ‚g
{f, g}Π = πij (x) .
‚xi ‚xj
Equivalently, we have
1 ‚ ‚
§
Π= πij (x) .
2 ‚xi ‚xj


Exercise 8
Write the jacobiator Jijk in terms of the structure functions πij . It is a homo-
geneous quadratic expression in the πij ™s and their ¬rst partial derivatives.
14 3 POISSON STRUCTURES


Examples.
1. When πij (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 g— . We had already seen that for the Lie-Poisson structure de¬ned
on g— , the functions πij were linear.
2. Suppose that the πij (x) are constant. In this case, the Jacobi identity is
trivially satis¬ed “ each term in the jacobiator of coordinate functions is zero.
By a linear change of coordinates, we can put the constant antisymmetric
matrix (πij ) into the normal form:
« 
0 Ik
0
 ’Ik 0
0 0

where Ik is the k — k identity matrix and 0 is the — zero matrix. If
we call the new coordinates q1 , . . . , qk , p1 , . . . , pk , c1 , . . . , c , the bivector ¬eld
becomes
‚ ‚
§
Π= .
‚qi ‚pi
i

In terms of the bracket, we can write
‚f ‚g ‚f ‚g
{f, g} = ’ ,
‚qi ‚pi ‚pi ‚qi
i

which is actually the original form due to Poisson in [138]. The ci ™s do not
enter in the bracket, and hence behave as parameters. The following relations,
called canonical Poisson relations, hold:
• {qi , pj } = δij
• {qi , qj } = {pi , pj } = 0
• {±, ci } = 0 for any coordinate function ±.
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.
™¦


3.5 Hamiltonian Vector Fields
Let (M, {·, ·}) be an almost Poisson manifold. Given h ∈ C ∞ (M ), de¬ne the linear
map
Xh : C ∞ (M ) ’’ C ∞ (M ) Xh (f ) = {f, h} .
by
The correspondence h ’ Xh resembles an “adjoint representation” of C ∞ (M ). By
the Leibniz identity, Xh is a derivation and thus corresponds to a vector ¬eld, called
the hamiltonian vector ¬eld of the function h.
3.6 Poisson Cohomology 15


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

[Xf , Xg ] = ’X{f,g} .

Proof. We can see this by applying [Xf , Xg ] + X{f,g} to an arbitrary function
h ∈ C ∞ (M ).

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

Xf {h, g} ’ 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 Π, which, in fact, was the original version of the identity as formulated
by Jacobi around 1838. (See Jacobi™s collected works [86].) Poisson [138] had
introduced the bracket {·, ·} 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 .
=’

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
™¦
yet stronger result.


3.6 Poisson Cohomology

A Poisson vector ¬eld, is a vector ¬eld X on a Poisson manifold (M, Π) such
that LX Π = 0, where LX is the Lie derivative along X. The Poisson vector ¬elds,
also characterized by
X{f, g} = {Xf, g} + {f, Xg} ,
are those whose local ¬‚ow preserves the bracket operation. These are also the
derivations (with respect to both operations) of the Poisson algebra.
Among the Poisson vector ¬elds, the hamiltonian vector ¬elds Xh = {·, h} form
the subalgebra of inner derivations of C ∞ (M ). (Of course, they are “inner” only
for the bracket.)

Exercise 9
Show that the hamiltonian vector ¬elds form an ideal in the Lie algebra of
Poisson vector ¬elds.


Remark. The quotient of the Lie algebra of Poisson vector ¬elds by the ideal of
hamiltonian vector ¬elds is a Lie algebra, called the Lie algebra of outer deriva-
tions. Several questions naturally arise.
16 3 POISSON STRUCTURES


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

• What is the group that corresponds to the hamiltonian vector ¬elds?
In Section 18.4 we will describe these “groups” in the context of Lie algebroids.
™¦
We can form the sequence:

C ∞ (M ) χ2 (M )
0 χ(M )
E E E
’’
h Xh
’’ LX Π
X

where the composition of two maps is 0. Hence, we have a complex. At χ(M ), the
homology group is
Poisson vector ¬elds
1
HΠ (M ) := .
hamiltonian vector ¬elds
This is called the ¬rst Poisson cohomology.
The homology at χ0 (M ) = C ∞ (M ) is called 0-th Poisson cohomology
0
HΠ (M ), and consists of the Casimir functions, i.e. the functions f such that
{f, h} = 0, for all h ∈ C ∞ (M ). (For the trivial Poisson structure {·, ·} = 0, this is
all of C ∞ (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 de¬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 Lie™s Normal Form

We will prove the following result in Section 4.3.

Theorem 4.1 (Lie [106]) If Π is a Poisson structure on M whose matrix of
structure functions, πij (x), has constant rank, then each point of M is contained
in a local coordinate system with respect to which (πij ) 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 (πij )
to have locally constant rank. This is a reasonable condition to impose, as
the structure functions πij 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 (πij )
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 (πij ) is maximal is not necessarily dense. For instance, consider R2
with {x1 , x2 } = •(x1 , x2 ) given by an arbitrary function •.
3. Points where (πij ) 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 g— is not regular unless g is abelian, though the regular points of g—
form, of course, an open dense subset.
™¦


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 Poincar´-Birkho¬-Witt theorem.
e
On any Poisson manifold M there is a vector bundle morphism Π : T — M ’ T M
de¬ned by
for any ±, β ∈ T — M .
±(Π(β)) = Π(±, β) ,
We can write hamiltonian vector ¬elds in terms of Π as Xf = Π(df ). Notice that Π
is an isomorphism exactly when rank Π = dim M , i.e. when Π de¬nes a symplectic
structure. If we express Π by a matrix (πij ) with respect to some basis, then the
same matrix (πij ) represents the map Π.
Let M = g— have coordinates µ1 , . . . , µn and Poisson structure {µi , µj } =
cijk µk . If v1 , . . . , vn is the corresponding basis of vectors on g, then we ¬nd
a representation of g on g— by mapping

vi ’’ ’Xµi .


17
18 4 NORMAL FORMS


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