Hall B.C. — Lie Groups, Lie Algebras, and Representations: An Elementary Understanding :: Электронная библиотека попечительского совета мехмата МГУ

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Hall B.C. — Lie Groups, Lie Algebras, and Representations: An Elementary Understanding

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Название: Lie Groups, Lie Algebras, and Representations: An Elementary Understanding

Автор: Hall B.C.

Аннотация:

This book addresses Lie groups, Lie algebras, and representation theory. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation.

Язык:

Рубрика: Математика/Алгебра/Теория представлений/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 351

Добавлена в каталог: 26.03.2005

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Предметный указатель

      47
      48
      131 244 256
      262
      183 270
,       220
      244 257
      262
      266 270
      262
      268 270
      265 270
, ,       271
      271
$GL(n;\mathbb{R})$ , $GL(n;\mathbb{C})$       see general linear group
      244
$H_{\alpha}$       see co-root
      99 209
$M_n(\mathbb{C})$       3
,       200 220
      see positive root
$SL(n;\mathbb{R})$ , $SL(n;\mathbb{C})$       see special linear groups
      9
$s^{\alpha}$       169
      see permutation group
$w_{\alpha}$       178 243
$\Delta$       see half-sum of positive roots
$\delta_{jk}$       5
$\mathbb{R}\mathbb{P}^3$       17
$\mathfrack{b}^+$ , $\mathfrak{b}^-$       220
$\mathfrak{g}_{\alpha}$       165
$\pi_1$       see fundamental group
$\varepsilon_{klm}$       325
Abelian      see commutative
Absolute convergence      28
Abstract root systems      243
Adjoint mapping      47
Adjoint of a linear operator      298
Adjoint of a matrix      6 293
Adjoint representation      96
Ado's Theorem      55
Algebraically integral elements      216
Analytic subgroups      see connected Lie subgroups
Analytically integral elements      215
Angles allowed in root systems      246
Averaging method      120
Baker — Campbell — Hausdorff formula      63
Base of a root system      181 249
Basepoint      331
Bilinear form, skew-symmetric      8
Bilinear form, symmetric      6 7
Borel subalgebra      220
Borel — Weil theory      218
Bracket      44
Campbell — Hausdorff formula      see Baker — Campbell — Hausdorff formula
Canonical commutation relations      122
Canonical form      see Jordan canonical form
Cartan subalgebra      143 162
Cauchy sequence      28
Center of a compact Lie group      273
Center of a group      114 284
Center of a Lie algebra      61 88 185
Center, discrete subgroup of      25 90
centralizer      187
Chamber, Weyl      see Weyl chamber
Change of coordinates      304
Character formula      see Weyl character formula
Character of a representation      211
Characteristic polynomial      291
Class function      211 213
Classical groups and Lie algebras      7 265
Classification of irreducible root systems      272
Classification of semisimple Lie algebras      272
Clebsch — Gordan theory      112 322
Clifford algebra      81
Closed subgroup      4
Closed, bounded sets      11
Co-root      167 172 174 248
Commutative group      279
Commutative group, representations of      114
Commutator      44
Compact group, complete reducibility of      120
Compact group, fundamental group of      333 336
Compact real form      159
Compactness      11
Compactness of       225 228
Complete reducibility      118 156
Complete vector field      308 311
Completeness      see convergence of Cauchy sequences
Complex Lie algebra      44 53
Complex Lie group      313
Complex manifold      309
Complex matrix Lie group      44 218
Complexification of a real Lie algebra      56
Component      12
Conjugacy class      211
Conjugate-linear      171 298
Connected Lie subgroups      82
Connectedness      12
Contragredient representation      see dual representation
Convergence of a sequence of matrices      3
Convergence of Cauchy sequences      28
Convergence, absolute      28
Convex hull      149 153 232
Coordinate system      see local coordinates
Covering group      see universal covering group
Covering space      see universal cover
Cross product      53 60
Decomposition, polar      see polar decomposition
Denominator, Weyl      see Weyl denominator
Derivation of $C^{\infty}(\mathcal{M})$       307
Derivation of a Lie algebra      61
Derivative of exponential mapping      70
Diagonalizable matrices, denseness of      35
Diagonalization      293
Differential forms      318
Differential of a Lie group homomorphism      311
Differential of a smooth mapping      305
Differential of exponential mapping      see derivative of exponential mapping
Dimension formula      see Weyl dimension formula
Direct product of groups      285
Direct product of matrix Lie groups      56 110
Direct sum of Lie algebras      56
Direct sum of root systems      245
Discrete subgroup      see center discrete
Double cover      81
Dual lattice      339 340
Dual representation      112 152 230
Dual root system      181 248 258
Dual space      112 299
Dynkin diagram      269
E(n)      see Euclidean group
Eigenspace      292
Eigenvalue      291
Eigenvector      291
Embedded submanifold      308
Equivalence class      287
Equivalence of representations      92
Equivalence of root systems      246
Euclidean group      9 42
Exceptional root systems      271
Exponential mapping      48
Exponential of a locally nilpotent operator      205
Exponential of a matrix      27
Faithful      91
Fiber bundle      334
Flow along a left-invariant vector field      311
Flow along a vector field      307
Freudenthal's formula      232
Fundamental domain      274

Fundamental group      16 331
Fundamental representations      139
Fundamental weights      147 255
Fundamental Weyl chamber      194 253
General linear group      3 39
Generalized eigenvector      294
Generalized orthogonal group      7 41
Group      279
Group versus Lie algebra homomorphisms      76
Haar measure      120 318
Half-sum of positive roots      212 233 235 236 264
Heisenberg group      8 41 63 100
Higher      132 197
Highest weight      133 197 201
Highest weight cyclic representation      135 197
Hilbert — Schmidt inner product      298
Hilbert — Schmidt norm      28
Holomorphic function      223
Holomorphic representation      219
Homomorphism of groups      285
Homomorphism of Lie algebras      54
Homomorphism of Lie groups      311
Homomorphism of matrix Lie groups      17
Homotopic      331
Homotopy group      333
Hyperplane      178 244 250
Ideal      61 157
Identity component      12
Identity element      279
Indecomposable Lie algebra      157
Indecomposable positive root      250
Infinite-dimensional representations      201
Infinitesimal elements      39
Inner product      297
Integers mod n      283
Integral element      130 192 254
Integration over group      see Haar measure
Intertwining map      92
Invariant inner product      120 121 146 163
Invariant subspace      91
Inverse      280
Inverse root      see co-root
Irreducible representation      91
Irreducible root system      245
Isomorphism of groups      285
Isomorphism of Lie algebras      54
Isomorphism of matrix Lie groups      17
Jacobi identity      53
Jacobian      237
Joint eigenvector      see simultaneous eigenvector
Jordan canonical form      296
Kernel of a homomorphism      285
Kernel of exponential mapping      337
Kostant's formula      232
Lattice      273
Left-invariant vector field      310
Left-translation      120 223
Length ratios allowed in root systems      246
Lie algebra homomorphism      see homomorphism of Lie algebras
Lie algebra of a matrix Lie group      38
Lie algebra, general      53
Lie group      20 309
Lie group homomorphism      see homomorphism of Lie groups
Lie product formula      35
Lie subgroup      see connected Lie subgroup
Linear functional      299
Linear Lie group      see matrix Lie group
Linear operator      292
Local coordinates      303
Locally nilpotent operator      205
Logarithm for a matrix Lie group      51
Logarithm of a matrix      32
Loop      331
Lorentz group      7 41
lower      132 197
Lowering operator      103
Manifold      21 303
Matrix Lie group      3
Maximal commutative subalgebra      162 163
Maximal compact subgroup      343
Maximal torus      187 211 217 240 336
Module      see representation
Morphism      see intertwining map
Multiplicity      130 150 194 233
Negative root      181 249
Nilpotent      31 60 190 295
Nilpotent, locally      see locally nilpotent operator
Nonmatrix Lie groups      314
Nontrivial ideal      157
Nontrivial invariant subspace      91
Norm of a matrix      27
Normal subgroup      286
normalizer      187
O(N)      see orthogonal group
O(n;k)      see generalized orthogonal group
One-parameter subgroup      37
Orbit      150 153
Orthogonal      266
Orthogonal complement      298
Orthogonal group      5 40 265
Orthogonal matrix      5
Orthonormal basis      293
Path-connected      12 22
Permutation group      184 283
Peter — Weyl theorem      210
Physicists' convention      39
Poincare group      10 43
Polar decomposition      19 342
Polynomials, action of SU(2) on      97
Polynomials, action of SU(3) on      152
Positive matrix      19 20
Positive root      181 249
Positive simple root      131 181 249 250
Product rule      44 59 305
Projective representation      330
Quotient group for Lie groups      312 317
Quotient group, general      287
Quotient representation      203
Raising operator      103
Rank      164 243
Rank three      262
Rank two      256
Real root      212 336
Real weight      212
Reductive Lie algebra      157
Representation      91
Right-translation      223
Root      129 130 164 171
Root beer      325
Root space      165
Root system      180 243
Root vector      165
Rotation group      see special orthogonal group
Schur's lemma      113
Schwarz inequality      28 58
Self-adjoint      293
Semisimple Lie algebra      157
Simple connectedness      15 76 216 221 339
Simple connectedness of      15 338
Simple Lie algebra      157
Simple root      see positive simple root
Simultaneous diagonalization      300
Simultaneous eigenvector      299
Skew self-adjoint      293
SN decomposition      32 295
SO(3), fundamental group of      16 335 338
SO(3), representations of      116
SO(N)      see special orthogonal group
Special linear group      5 40
Special orthogonal group      5 40

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