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Page 1

Geometric Models for

Noncommutative Algebras

Ana Cannas da Silva1

Alan Weinstein2

University of California at Berkeley

December 1, 1998

1
[email protected], [email protected]

2
[email protected]

Page 2

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�e-Birkho�-Witt Theorem 5
2.1 Almost Commutativity of U(g) . . . . . . . . . . . . . . . . . . . . . 5
2.2 Poisson Bracket on Gr U(g) . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Role of the Jacobi Identity . . . . . . . . . . . . . . . . . . . . . 7
2.4 Actions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Proof of the Poincaré-Birkhoff-Witt Theorem . . . . . . . . . . . . . 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

Page 97

13.1 Definitions and Notation 87

in several ways” (see Section 13.2). We refer to Brown [19, 20], as well as [171], for
extensive general discussion of groupoids.

Examples.

1. A group is a groupoid over a set X with only one element.

2. The trivial groupoid over the set X is defined by G = X, and α = β =
identity.

3. Let G = X ×X, with the groupoid structure defined by

X ×X
π1 ↓↓ π2

α(x, y) := π1(x, y) = x , β(x, y) := π2(x, y) = y ,

(x, y)(y, z) = (x, z) ,
ε(x) = (x, x) ,

(x, y)−1 = (y, x) .

This is often called the pair groupoid, or the coarse groupoid, or the
Brandt groupoid after work of Brandt [17], who is generally credited with
introducing the groupoid concept.

X

X

ε(X)

π1

π2

?
















Remarks.

1. Given a groupoid G, choose some φ 6∈ G. The groupoid multiplication on G
extends to a multiplication on the set G ∪ {φ} by

gφ = φg = φ
gh = φ , if (g, h) ∈ (G×G) \G(2) .

The new element φ acts as a “receptacle” for any previously undefined prod-
uct. This endows G ∪ {φ} with a semigroup structure. A groupoid thus
becomes a special kind of semigroup as well.

2. There is a natural way to form the product of groupoids:

Page 98

88 13 GROUPOIDS

Exercise 40
If Gi is a groupoid over Xi for i = 1, 2, show that there is a naturally defined
direct product groupoid G1 ×G2 over X1 ×X2.

3. A disjoint union of groupoids is a groupoid.



13.2 Subgroupoids and Orbits

A subset H of a groupoid G over X is called a subgroupoid if it is closed under
multiplication (when defined) and inversion. Note that

h ∈ H ⇒ h−1 ∈ H ⇒ both ε(α(h)) ∈ H and ε(β(h)) ∈ H .

Therefore, the subgroupoid H is a groupoid over α(H) = β(H), which may or may
not be all of X. When α(H) = β(H) = X, H is called a wide subgroupoid.

Examples.

1. If G = X is the trivial groupoid, then any subset of G is a subgroupoid, and
the only wide subgroupoid is G itself.

2. If X is a one point set, so that G is a group, then the nonempty subgroupoids
are the subgroups of G, but the empty set is also a subgroupoid of G.

3. If G = X×X is the pair groupoid, then a subgroupoid H is a relation on X
which is symmetric and transitive. A wide subgroupoid H is an equivalence
relation. In general, H is an equivalence relation on the set α(H) = β(H) ⊆
X.



Given two groupoids G1 and G2 over sets X1 and X2 respectively, a morphism
of groupoids is a pair of maps G1 → G2 and X1 → X2 which commute with all
the structural functions of G1 and G2. We depict a morphism by the following
diagram.

G1 - G2

X1

α1

?

β1

?
- X2

α2

?

β2

?

If we consider a groupoid as a special type of category, then a morphism between
groupoids is simply a covariant functor between the categories.

For any groupoid G over a set X, there is a morphism

G
(α, β)
- X ×X

X

α

?

β

?
= X

π1

?

π2

?

Page 193

INDEX 183

de�nition of symplectic structure,
14

dual pair, 53
E-symplectic form, 135
E-symplectic structure, 135
foliation, 23
form, 20
groupoid, 127
leaf, 23
Lie algebroid of a symplectic man-

ifold, 125
manifold, 20
Poisson cohomology, 23
realization, 32, 59
symplectically complete foliation,

53

tangent bundle
as a Lie algebroid, 114
complexi�ed, 62

tensor algebra, 1
theorem

Darboux’s, 20, 21
double commutant, 50
Gel’fand-Naimark, 48
Lie’s, 17
splitting, 19
unique Haar measure, 74

topological groupoid, 92
topology

norm, 47
of convergence of matrix elements,

49
of pointwise convergence, 48
on bounded operators, 47, 48
strong, 48
weak, 49

torus
irrational foliation, 59
maximal, 91
quantum, 152

transformation
groupoid, 90
Lie algebroid, 114

transitive
groupoid, 89
Lie algebroid, 123, 124

translation maps, 76
transverse

Lie algebra, 24

Poisson structure, 24
structure function, 24

uncertainty principle, xvi
unimodular group, 75
unit or identity, 69
unital, 49
universal

algebra, 1
property, 1

universal enveloping algebra
almost commutativity, 5
de�nition, 1
grading, 3
Poisson bracket, 5

vector �eld
ϕ-related, 29
hamiltonian, 14, 20
left invariant, 111
odd, xv
Poisson, 15
set of hamiltonian vector �elds,

40
set of Poisson vector �elds, 40

vector �elds tangent to
the boundary, 128
a hypersurface, 127

von Neumann algebra, 49
von Neumann, J., 47, 50, 151

weak topology, 49
Weinstein, A., 19, 26, 33, 34, 126
Weyl algebra

a�ne invariance, 152
automorphism, 153
bundle, 153
de�nition, 149
derivation, 152
�ltration, 158

at connection, 154
formal, 150, 158
Moyal-Weyl product, 149, 150
Weyl product, 150

Weyl curvature, 160
Weyl group, 91
Weyl groupoid, 91
Weyl product, 150
Weyl symbol, 151

Xu, P., 56

Page 194

184 INDEX

Y -tangent bundle, 127
Yang-Baxter equation, 135

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