Silva A. da, Weinstein A. - Geometric Models for Noncummutative Algebras.pdf

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Geometric Models for
Noncommutative Algebras
Ana Cannas da Silva 1
Alan Weinstein 2
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 Poincare-Birkho-Witt Theorem 5
2.1 Almost Commutativity of U(g) . . . . . . . . . . . . . . . . . . . . . 5
2.2 Poisson Bracket onGrU(g) . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Role of the Jacobi Identity . . . . . . . . . . . . . . . . . . . . . 7
2.4 Actions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Proof of the Poincare-Birkho-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 ofg . . . . . . . . . . . . . . . . . . . . . 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 ofsu(2) andsl(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 T 3 . . . . . . . . . . . . . . . . . . . . . . . 59
10.2 Submersive Realizations of T 3 . . . . . . . . . . . . . . . . . . . . . . 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 L p Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
12.3 Generalized Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
12.4 Poincare-Birkho-Witt Revisited . . . . . . . . . . . . . . . . . . . . 81
VI Groupoids
85
13 Groupoids 85
13.1 Denitions 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 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Dierential Geometry for Lie Algebroids 131
18.1 The Exterior Dierential 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 Innitesimal 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 Ane 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 Classication of Deformation Quantizations . . . . . . . . . . . . . . 161
References
163
Index
175

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