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Tuynman G.M. — Supermanifolds and Supergroups: Basic Theory
Tuynman G.M. — Supermanifolds and Supergroups: Basic Theory

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Название: Supermanifolds and Supergroups: Basic Theory

Автор: Tuynman G.M.

Аннотация:

Supermanifolds and Supergroups explains the basic ingredients of super manifolds and super Lie groups. It starts with super linear algebra and follows with a treatment of super smooth functions and the basic definition of a super manifold. When discussing the tangent bundle, integration of vector fields is treated as well as the machinery of differential forms. For super Lie groups the standard results are shown, including the construction of a super Lie group for any super Lie algebra. The last chapter is entirely devoted to super connections. The book requires standard undergraduate knowledge on super differential geometry and super Lie groups. Some knowledge of ordinary differential geometry is helpful when reading the text.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 431

Добавлена в каталог: 17.11.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathcal{A}$-Lie algebra      270
$\mathcal{A}$-Lie algebra, associated to an $\mathcal{A}$-Lie group      271
$\mathcal{A}$-Lie group      142 266
$\mathcal{A}$-Lie subgroup      292
$\mathcal{A}$-Lie subgroup, normal      296
$\mathcal{A}$-Lie subgroup, proper      300
$\mathcal{A}$-manifold      128
$\mathcal{A}$-module      2
$\mathcal{A}$-vector space      85
$\mathfrak{A}$-graded $\mathcal{A}$-module      6
$\mathfrak{A}$-graded commutative ring      3
$\mathfrak{A}$-graded commutativity      3
$\mathfrak{A}$-graded ring      3
$\mathfrak{A}$-grading      3
Action, effective      143
Action, left/right      142 266
Adjoint representation      31 32 283 284 325
Affine connection      374
Algebra, $\mathfrak{A}$-graded commutative      28
Algebra, $\mathfrak{A}$-graded Lie      28
Algebra, $\mathfrak{A}$-graded Lie of parity $\alpha$      31
Algebra, associative      28
Algebra, associative of parity $\alpha$      32
Associated bundle      151 364
Atlas      124
Atlas, adapted to a subbundle      160
Atlas, trivializing      145
Automorphism of a module      11
Basis      46
Basis, dual      67
Basis, ordered      59
Basis, orthonormal      192
Batchelor's theorem      196
Berezinian      41 55 78
Bianchi identities      355 357
Bilinear map      8
Bimodule      2
Body of $\mathcal{A}$      57
Body of a (proto) $\mathcal{A}$-manifold      126
Body of a linear map      82
Body of a matrix      58
Body of a module      82
Body of a smooth function      105 129
Border point      228
Bracket of a Lie algebra      28
Bundle of morphisms      170
Bundle, associated      151 364
Bundle, fiber      145
Bundle, fiber, principal      155 301
Bundle, frame      361
Bundle, pull-back      151
Bundle, structure      361
Bundle, trivial      147
Bundle, vector      156
Cartan's structure equations      355 357—359
Center of a Lie algebra      325
Central extension of a Lie algebra      325
Central extension of an $\mathcal{A}$-Lie group      325
Chain rule      116
Chart of an $\mathcal{A}$-manifold      124
Christoffel symbols      370
Closed differential form      262
Cocycle of a Lie algebra      325 332
Cocycle of an $\mathcal{A}$-Lie group      325
Cohomology of a Lie algebra      325 332
Cohomology of an $\mathcal{A}$-Lie group      325 332
Cohomology, de Rham      263 327
Commutativity, $\mathfrak{A}$-graded      3
Commuting flows      236
Compatible chart of a fiber bundle      144
Compatible chart of an $\mathcal{A}$-manifold      124
Complex conjugation      188
Connection      342 345 351
Connection, 1-form      351
Connection, affine      374
Connection, Ehresmann      342
Connection, flat      342
Connection, FVF      345
Connection, integrable      342
Connection, linear      374
Connection, principal      351
Contraction, elementary      74 75 274
Contraction, operator      12
Convergence      286
Coordinate (even/odd)      104
Cotangent bundle      248
Covariant derivative      366
Covariant derivative, exterior      354
Covariant derivative, induced      368
Covering      311—312
Coxeter group      23
Curvature 2-form      355
Curvature tensor      394
De Rham cohomology      263 327
Decomposition      2
Degree-$\alpha$ part of a vector bundle      167
Derivation (right/left)      29
Determinant      41 61 77 80 111
Determinant, graded      41 55 78—80 111 118 285
DeWitt topology      93
Diffeomorphic      128
Diffeomorphism      128
Differential form      248
Differential form with values in a vector bundle      383
Differential form with values in an $\mathcal{A}$-vector space      323
Dimension of a (proto) $\mathcal{A}$-manifold      124
Dimension, differential      207
Dimension, even/graded/odd/total      61
Direct product of $\mathcal{A}$-manifolds      133
Direct product of bundles      153
Direct sum      7 14
Direct sum of bundles      164
Dual basis      67
Dual bundle      170
Dual of a module      9
Dual of a morphism      14
Effective action      143
Ehresmann connection      342
Elementary contraction      74 75 274
Embedding      214
Endomorphism      9
Enlarging the structure group      154
Equivalence of $\mathcal{A}$-Lie subgroups      293
Equivalence of bases      83
Equivariant map      305
Euler vector field      261
Evaluation operator      12
Even      3 4 56
Exact differential form      262
example      3 4 21 28 56 57 59 71 74 97 110 118 125 132 133 137 143 144 152 231 280 283 287 297 332 339 367 396 399
Example, counter      7 10 60 63 65 66 76 81 84 87 103 123 130 148 163 191 192 216 221 224 244 300 344
Exponential map      278
Exponential map of matrices      287
Exterior algebra      27
Exterior covariant derivative      354
Exterior derivative      249
Exterior power      25
Exterior power of a vector bundle      167
f.g.p      47—54 59 172 180 181 196 250
Family of $\mathcal{A}$-Lie group morphisms      282
Family of Lie algebra morphisms      282
Fiber, bundle      145
Fiber, bundle, map      147
Fiber, bundle, principal      155 301
Fiber, bundle, structure      145
Fiber, over a point      145
Fiber, typical      145
Finite dimensional      59
Finite type      46
Finitely generated      46 48 50 51 196
Flat connection      342
Flow      228
Flow, commuting      236
Flow, global      234
Flow, local      228
Foliation      243 341
Form, differential k-      248
Frame      361
Frame, bundle      361
Free $\mathfrak{A}$-graded $\mathcal{A}$-module      16
Frobenius' Theorem      243 244 342 355
Fundamental vector field      299
FVF connection      345
Gauge      351
Gauge, transformation      351
Generator      16 46
Graded      56
Graded, determinant      41 55 78—80 111 118 285
Graded, subspace      59 86
Graded, trace      75 76 80 285
Graded, transpose      71
Graph      134 149
Homogeneous      3 4
Homomorphism      9
Horizontal k-form      395
Horizontal lift      342
Horizontal map      342
Horizontal part of a tangent vector      342
Horizontal section      342
Horizontal submanifold      342
Horizontal tangent vector      342
Ideal      296
Identification      11
Immersion      214
Implicit function theorem      122
Independent elements      46
Initial condition      228
Integrable connection      342
Integrable subbundle      243 244 342
Integrable vector field      228
Integral manifold      244
Interchanging map      22
interval      228
Invariance of dimension      121
Invariant k-form      320
Invariant k-form, left/right      320
Invariant vector field      315
Inverse function theorem      121
Invertible homomorphism      11
Involutive subbundle      243
Isomorphic fiber bundles      147
Isomorphic modules      11
Isomorphism      11
Isomorphism of $\mathcal{A}$-Lie groups      267
Isomorphism of fiber bundles      147
Isomorphism of Lie algebras      270
Isomorphism of vector bundles      157
Isotropy subgroup      304
Jacobi identity      28 30—32 270—272 284 326 332 356
Jacobian      116
k-additive      2
k-form      248
k-form with values in a vector bundle      383
k-form with values in an $\mathcal{A}$-vector space      323
k-form, M-dependent      257
k-linear map (left/right)      8
Kronecker delta      58
Leaf      244 246 294
Left multiplication      2
Left translation      268
Left-invariant vector field      270
Lie derivative      253
Lift      377
Linear connection      374
Linear map (left/right)      8
Local flow      228
Locally finite      134
Maurer — Cartan 1-form      339 357
Metric on a free graded $\mathcal{A}$-module      192
Metric on a vector bundle      194
Modeled, an $\mathcal{A}$-manifold on an $\mathcal{A}$-vector space      124
Module      2—6
Momentum map      332
Morphism      9
Morphism of $\mathcal{A}$-Lie groups      142 267
Morphism of Lie algebras      30 270
Morphism of vector bundles      157
Nilpotent vector      81
Normal $\mathcal{A}$-Lie subgroup      296
Notational shorthand      119
Odd      56
Ordered basis      59
Orthogonal complement      193
Orthonormal basis      192
Parallel transport      344 364
Parity      3 4
Parity of a linear map      8
Parity of a section      158
Parity, reversal      102
Parity, shift operation      32 102
Partition of unity      95 128 135 136 159 167 175 194 195 197 199 263
Plateau function      136 160 176 205 250 327
Principal connection      351
Principal fiber bundle      155 301
Product of bundles      153
Projective      47
Proper $\mathcal{A}$-Lie subgroup      300
Proto $\mathcal{A}$-manifold      124
Pseudo effective action      143
Pseudo metric on a vector bundle      194
Pseudo metric, pseudo metric on a free graded $\mathcal{A}$-module      188
Pull-back of a differential form      255
Pull-back of a differential form, E-valued      324
Pull-back of a differential form, generalized      257 319
Pull-back of a section      183
Pull-back, bundle      151
Pull-back, map      183
Push forward of a section      178
Push forward of a vector field      218
Quotient      16
Quotient, bundle      164
Rank      73
Rank of a function      122
Rank of a matrix      58
Rank of a vector bundle      156
Reducing the structure group      154
Regular value      214
Related vector fields      218
Representation of a Lie algebra      30
Representation of an $\mathcal{A}$-Lie group      267
Restriction of a bundle to a submanifold      146
Right multiplication      2
Right translation      268
Right-invariant vector field      270
Ring, $\mathfrak{A}$-graded      3
Ring, commutative      3
Second countable topology      128 245 247
Section      148
Shorthand, notational      119
Signature      24
Simply connected      312
Skew-symmetric      24
Skew-symmetrization operator      40
Smooth $\mathcal{A}$-structure      124
Smooth functions      94—101
Smooth linear map      85
Smooth map between $\mathcal{A}$-manifolds      128
Smooth system      96
Smooth system, maximal      97
Smooth tree      96
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