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of an E-multivector ¬eld, 132
connection
density
Ehresmann, 45
±-density, 77
Fedosov™s, 155
de¬nition, 77
¬‚at, 44, 102
¬‚attening, 156, 160 generalized, 80
INDEX 177


derivation Lie algebroid of a Lie groupoid,
114
inner, 15, 157
Lie algebroid of a Poisson mani-
law for Dirac™s bracket, 152
fold, 125
of a Poisson algebra, 15
Lie algebroid of a symplectic man-
of a superalgebra, xv
ifold, 125
outer, 15
tangent bundle, 114
diagonal subgroupoid, 89
transformation Lie algebroid, 114
di¬eological space, 118
transitive Lie algebroid, 123
di¬erentiable groupoid, 93
Y -tangent bundle, 127, 128
Dirac™s quantum Poisson bracket, 152,
extended groupoid algebra, 105
157
Dirac, P., 151, 152, 157, 158
F -homotopic, 94
distribution
factor, 50
compactly supported, 79
Fedosov quantization, 161
group algebra, 76
Fedosov, B., 154
distributional section, 79
¬‚at connection, 45, 102
Douady, A., 118
foliation
dual of a Lie algebroid, 119
F -homotopy, 94
dual pair
graph, 94
from complex geometry, 65
holonomy groupoid, 93
in algebra, 50
irrational, 59
in Poisson geometry, 51
leaf, 93
symplectic, 53
one-sided holonomy, 95
Reeb foliation, 94
E-cohomology, 132
symplectic, 23
E-di¬erential form symplectically complete, 53
de¬nition, 131 formal adjoint, 80
degree, 131 formal Weyl algebra, 150, 158
E-k-form, 131 formality conjecture, 144
homogeneous, 131 Frobenius theorem, 8, 19, 45
properties of dE , 131 Fuchssteiner, B., 125
E-Gerstenhaber bracket, 133 function
E-Lie derivative, 113, 133, 137 Casimir, 14, 16, 136
E-multivector ¬eld, 132 collective, 66
E-Π-cohomology, 136 hamiltonian, 40
E-Poisson bivector ¬eld, 135 fundamental groupoid, 89, 94
E-symplectic form, 135
E-symplectic structure, 135 Gel™fand, I., 48
e¬ective curvature, 160 general linear groupoid, 102
Ehresmann, C., 45, 92 generalized
exact Poisson bivector ¬eld, 137 density, 80
examples section, 79, 105
of a groupoid algebra, 97 Gerstenhaber algebra
of groupoid, 89 E-Gerstenhaber algebra, 133
examples of Lie algebroids E-Gerstenhaber bracket, 133
Atiyah algebra, 123 de¬nition, 133
Atiyah sequence, 123 of a Lie algebroid, 133
bundle of Lie algebras, 114 Gerstenhaber, M., 134, 141
Lie algebra, 114 Ginzburg, V., 57
178 INDEX


grading Lie, 93
formal Weyl algebra, 158 measurable, 93
universal enveloping algebra, 3 morphism, 88
normal bundle of G(0) , 100
graph, 94
Grothendieck, A., 89 orbit, 89
group pair groupoid, 87, 94
de¬nition, xvi, 87 principal groupoid, 90
isotropy subgroup, 89 product, 85
group algebra product of groupoids, 87
distribution, 76 relation, 88
distribution group algebra, 76 representation, 102
group(-element)-like, 74 set of composable pairs, 85
list of structures, 73 source, 85
measure, 73 subgroupoid, 88
groupoid symplectic groupoid, 127
action, 90, 101 target, 85
action of a bisection, 107 topological, 92
as a category, xvi transformation groupoid, 90
Baer groupoid, 91 transitive, 89
Borel, 93 trivial, 87
Brandt groupoid, 87 Weyl groupoid, 91
bundle of groups, 93 wide subgroupoid, 88
bundle of Lie algebras, 116 with structure, 92
bundle of Lie groups, 116
Haar measure, 74
coarse groupoid, 87
Haar system, 92, 98
comparison with group, xvi
C — -algebra, 98 Hae¬‚iger, A., 95
hamiltonian
de¬nition, 85
action, 39, 44
diagonal subgroupoid, 89
function, 40
di¬erentiable, 93
set of hamiltonian vector ¬elds,
ergodic theory, 89
40
example of a groupoid algebra,
strongly, 44
97
vector ¬eld, 14, 17, 20
examples, 89
vector ¬eld on a Lie algebroid,
extended groupoid algebra, 105
136
fundamental, 89
weakly, 44
fundamental groupoid, 94
harmonic oscillator, 63
Galois theory, 89
Harrison cohomology, 142
general linear groupoid, 102
Heisenberg algebra, 150
groupoid algebra, 98
Hochschild cohomology
groupoid algebra action, 103
action of symmetric groups, 143
groupoid algebra with coe¬cients
algebraic structure, 143
in a vector bundle, 103
cup product, 143
Haar system, 98
decomposition, 143
holonomy of a foliation, 93
de¬nition, 142
identity section, 85
Gerstenhaber, 134
intrinsic groupoid algebra, 99
Gerstenhaber bracket, 143
inversion, 86
squaring map, 143
isotropy subgroupoid, 89
left invariant vector ¬eld, 111 Hochschild complex, 142
INDEX 179


Hodge decomposition, 156 Jacobi identity
de¬nition, 6
holonomy
deformation of products, 145, 146
de¬nition, 45, 93
for elements of A2 (V ), 142
description, 93
jacobiator, 7, 13, 145
equivalence relation, 93
Poisson structure, 12
¬‚at connection, 104
super-Jacobi identity, 133, 141,
groupoid, 115
142
groupoid of a foliation, 93, 94
Jacobi, C., 15
on a regular Poisson manifold, 24
jacobiator, 7, 13, 145
one-sided, 95
homogeneous
Karasev, M., 33
E-di¬erential form, 131
Keel, S., 89
E-multivector ¬eld, 132
Kirillov, A., 23
Hopf algebra
Kontsevich, M., 144
antipode, 69
Kostant, B., 42
associativity of multiplication, 70
co-unit or coidentity, 69
Lako¬, G., xiii
coassociativity of comultiplication,
Lazard, M., 118
70
leaf
commutative, 72
breaking the leaves, 27
comultiplication, 69
de¬nition, 93
de¬nition, 69
left invariant
examples, 69
measure, 74
multiplication, 69
vector ¬eld, 111
noncommutative, 72
Leibniz identity
Poisson, 72
de¬nition, 6
quantum group, 72
deformation of products, 146
relation to groups, 72
for abstract products, 145
unit or identity, 69
in the Weyl algebra, 152
Lie algebroid, 113
in¬nitesimal deformation Lie algebroid of a Poisson mani-
obstructions to continuing, 138 fold, 126
of a Poisson structure, 137 super-Leibniz identity, 133
trivial, 138 Lie algebra
in¬nitesimal neighborhood, xiii, 155 action, 8
inner derivation, 15, 157 almost, 7
integrability bundle of Lie algebras, 114
conditions, 21 cohomology, 132, 142
Jacobi identity, 15 deformation, 2
Newlander-Nirenberg theorem, 120 degenerate, 26
of Lie algebroids, 114, 117 non-degenerate, 26
intrinsic groupoid algebra, 99 representation, 17
intrinsic Lp spaces, 78 structure constant, 8
irrational foliation, 59 super-Lie algebra, 133
isotropic, 34 transverse, 24
isotropy Lie algebroid
algebroid, 113 as a supermanifold, 131
subgroup, 89 cohomology, 132
subgroupoid, 89 complex Lie algebroid, 120
180 INDEX


connection, 124 Lie-Poisson bracket
curvature, 124 de¬nition, 11
de¬nition, 113 dual of a Lie algebroid, 119
degree of an E-form, 131 Lie-Poisson manifold
di¬erential complex, 136 de¬nition, 11
di¬erential geometry, 131 hamiltonian action, 39
dual, 119 Jacobi identity, 13
E-di¬erential form, 131 Lie-Poisson bracket, 11
E-Gerstenhaber bracket, 133 normal form, 20
E-k-form, 131 rank, 17
E-Lie derivative, 113, 133, 137 linear Poisson structure, 14
E-Π-cohomology, 136 linearizable Poisson structure, 25
E-Poisson bivector ¬eld, 135 linearized Poisson structure, 24
E-symplectic form, 135 Liouville vector ¬eld, 137
E-symplectic structure, 135 local bisection, 107
examples, 114, 123 Lu, J.-H., 57
exterior di¬erential algebra, 131
M¨bius band, 94
o
Gerstenhaber algebra, 132, 133
Mackenzie, K., 118
hamiltonian vector ¬eld, 136
Mackey, G., 89
history, 115
maximal torus, 91
homogeneous E-form, 131
measurable groupoid, 93
integrability, 117
measure
Leibniz identity, 113
algebras of measures on groups,
Lie-Poisson bracket, 119
73
morphism, 120
class, 93
multivector ¬eld, 132
group algebra, 73
of a Lie groupoid, 114
Haar measure, 74
of a Poisson manifold, 125
left-invariant, 74
of a symplectic manifold, 125
quasi-invariant, 74
orbits, 113
Melrose, R., 127
Poisson bracket, 136
modular character, 75
Poisson cohomology, 136
modular function, 75
Poisson structure, 134
Molino, P., 118
Poisson vector ¬eld, 137
moment map
properties of dE , 131
groupoid action, 101
squaring map, 139
vs. momentum map, 101
Lie bracket, 6
momentum
Lie derivative
phase space, xv
Cartan™s magic formula, 21, 126,
momentum map
159
de¬nition, 39, 40
Lie algebroid, 113, 133, 137
equivariance, 42
Lie group
¬rst obstruction, 40, 43
modular character, 75
for a group action, 42
modular function, 75
second obstruction, 41“43
unimodular, 75
vs. moment map, 101
Lie groupoid
Mori, S., 89
de¬nition, 93
Morita equivalence, 55, 56
Lie algebroid of a, 114
morphism of groupoids, 88
Lie™s theorem, 17
Lie, S., 8, 9, 17, 40 Moyal-Weyl product, 149“151
INDEX 181


multilinear maps de¬nition, 29
group of Poisson automorphisms,
brackets, 142
29
symmetric, 142
Poisson bivector ¬eld
multivector ¬eld
de¬nition, 135
•-related, 30
E-Poisson bivector ¬eld, 135
de¬nition, 12
exact, 137
Lie algebroid, 132
on a Lie algebra, 135
Naimark, M., 48 Poisson bracket
Newlander-Nirenberg theorem, 120 di¬erential operators, 149
Newton™s method, 156 Lie algebroid, 136
non-degenerate Lie algebra, 26 universal enveloping algebra, 5
norm topology, 47 Poisson cohomology
Novikov, S., 95 ¬rst, 16
Nu˜ez, R., xiii
n on a Lie algebroid, 136
symplectic case, 23
obstruction 0-th, 16
to a holomorphic connection, 121 Poisson Hopf algebra, 72
to a momentum map, 40“43 Poisson Lie group
to deformation of a Poisson struc- de¬nition, 72
ture, 138 non-linearizability, 26
to the Jacobi identity, 7 Poisson manifold
odd almost symplectic, 20
di¬erential forms, 78 coisotropic, 34
vector ¬eld, xv de¬nition, 12
one-sided holonomy, 95 Lie algebroid of a, 125
operator regular, 17
bounded, 47 symplectic, 20
compact, 48 Poisson map
product, 151 complete, 31
orbit de¬nition, 29
coadjoint, 39 Poisson quotient, 34
groupoid, 89 Poisson relation, 34
of a Lie algebroid, 113 Poisson structure
almost, 12
outer derivation, 15
canonical coordinates, 13
pair groupoid, 87, 94 de¬nition, 12
Palais, R., 118 formal deformation, 137, 138
permutation group, 143 in¬nitesimal deformation, 137
phase space, xv Lie™s theorem, 17
•-related multivector ¬eld, 30 linear, 14
•-related vector ¬eld, 29 linearization, 25
Planck™s constant, 146 linearized, 24
Poincar´-Birkho¬-Witt theorem
e normal form, 17
and group algebras, 81 obstructions to deformation, 138
discussion, 7 on a Lie algebroid, 134
proof, 9 structure functions, 13
statement, 5 transverse, 24
Poisson algebra, 6 Poisson submanifold, 36
Poisson automorphism Poisson tensor, 12
182 INDEX


Poisson vector ¬eld section
de¬nition, 15 admissible section, 106
Lie algebroid, 137 bisection, 106
set of hamiltonian vector ¬elds, distributional, 79
40 generalized, 79, 105
Poisson™s theorem, 15, 19 of the normal bundle, 109
Poisson, S.-D., 14, 15 semigroup, 87, 106, 107
Poisson-algebra homomorphism, 29 Smale, S., 42
Pradines, J., 115 Souriau, J.-M., 42, 118
principal groupoid, 90 spectrum, 48
product splitting theorem, 19
coproduct, 69 squaring map, 139, 142, 143
of groupoids, 87 strong topology, 48
star, 151 structure constant, 8
von Neumann, 151 structure function
de¬nition, 13
quantization for a Lie algebroid, 119
classi¬cation, 161 transverse structure, 24
deformation, 155 subgroup
Fedosov, 161 isotropy, 89
patching from local, 155
subgroupoid
quantum
as a relation, 88
group, 72
de¬nition, 88
operator, 151
diagonal, 89
quasi-invariant measure, 74
isotropy, 89
wide, 88
rank
submanifold
of a Lie algebra, 18
Poisson, 36
of a Poisson structure, 18
super-
Poisson structure with constant
commutativity, xiv
rank, 17, 20
commutativity of cup product, 143
realization
derivation, xv
injective, 59
Jacobi identity, 133, 141, 142
submersive, 60
Leibniz identity, 133, 134
symplectic, 59
Lie algebra, 133
Reeb foliation, 94
manifold, 131
Reeb, G., 94
space, xv
regular equivalence relation, 34
vector ¬eld, 132
regular Poisson manifold
symmetric
de¬nition, 17
algebra, 1
holonomy, 24
group, 3, 143
relation, 88
tensor, 3
representation
symmetrization, 3
of a groupoid, 102
symplectic
pointwise faithful, 8
almost symplectic manifold, 20
representation equivalent, 56
canonical coordinates, 14
Rinehart, G., 115
canonical structure on a cotan-
gent bundle, 36, 119
Schouten-Nijenhuis bracket, 12, 135
Schr¨dinger, E., 151
o Darboux™s theorem, 20, 21
INDEX 183


de¬nition of symplectic structure, Poisson structure, 24
14 structure function, 24
dual pair, 53
uncertainty principle, xvi
E-symplectic form, 135
unimodular group, 75
E-symplectic structure, 135
unit or identity, 69
foliation, 23
unital, 49
form, 20
universal
groupoid, 127
algebra, 1
leaf, 23
property, 1
Lie algebroid of a symplectic man-
universal enveloping algebra
ifold, 125
almost commutativity, 5
manifold, 20
de¬nition, 1
Poisson cohomology, 23
grading, 3
realization, 32, 59
Poisson bracket, 5
symplectically complete foliation,
53
vector ¬eld
•-related, 29
tangent bundle
hamiltonian, 14, 20
as a Lie algebroid, 114
left invariant, 111
complexi¬ed, 62
odd, xv
tensor algebra, 1
Poisson, 15
theorem
set of hamiltonian vector ¬elds,
Darboux™s, 20, 21
40
double commutant, 50
set of Poisson vector ¬elds, 40
Gel™fand-Naimark, 48
vector ¬elds tangent to
Lie™s, 17
the boundary, 128
splitting, 19
a hypersurface, 127
unique Haar measure, 74
von Neumann algebra, 49
topological groupoid, 92
von Neumann, J., 47, 50, 151
topology
norm, 47 weak topology, 49
of convergence of matrix elements, Weinstein, A., 19, 26, 33, 34, 126
49 Weyl algebra
of pointwise convergence, 48 a¬ne invariance, 152
on bounded operators, 47, 48 automorphism, 153
strong, 48 bundle, 153
weak, 49 de¬nition, 149
torus derivation, 152
irrational foliation, 59 ¬ltration, 158
maximal, 91 ¬‚at connection, 154
quantum, 152 formal, 150, 158
transformation Moyal-Weyl product, 149, 150
groupoid, 90 Weyl product, 150
Lie algebroid, 114 Weyl curvature, 160
transitive Weyl group, 91
groupoid, 89 Weyl groupoid, 91
Lie algebroid, 123, 124 Weyl product, 150
translation maps, 76 Weyl symbol, 151
transverse
Lie algebra, 24 Xu, P., 56
184 INDEX


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

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. 8
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