Brocker Th., Dieck T.T. — Representations of Compact Lie Groups :: Электронная библиотека попечительского совета мехмата МГУ

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Brocker Th., Dieck T.T. — Representations of Compact Lie Groups

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Название: Representations of Compact Lie Groups

Авторы: Brocker Th., Dieck T.T.

Аннотация:

This book is an introduction to the representation theory of compact Lie groups, following Hermann Weyl's original approach. Although the authors discuss all aspects of finite-dimensional Lie theory, the emphasis throughout the book is on the groups themselves. The presentation is consequently more geometric and analytic than algebraic in nature. The central results, culminating the Weyl character formula, are reached directly and quickly, and they appear in forms suitable for applications to physics and geometry.
This book is a good reference and a source of explicit computations, for physicists and mathematicians. Each section is supplemented by a wide range of exercices, and geometric ideas are illustrated with the help of 24 figures

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1995

Количество страниц: 313

Добавлена в каталог: 21.04.2008

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Предметный указатель

$Ad_{G/T}: T \rightarrow Aut L(G/T)$       49 160
$Alt^{k}V$ (alternating k-forms)      41
$A_{n}$ (Dynkin diagram)      212 216 218
$B_{n}$ (Dynkin diagram)      212 216 220
$c(\gamma+\varrho) \in Irr(G,T)$       241
$C^{0}(G,K)$ as a representation      123 ff
$C^{0}_{c}(X)$       40
$C_{n}$ (Dynkin diagram)      212 216 221
$D_{n}$ (Dynkin diagram)      212 216 219
$e(x)=e^{2\pi ix}$       240
$e_{+}$ , $e_{-}$ , $e^{\mathbb{C}}_{\mathbb{R}}$ , $e^{\mathbb{H}}_{\mathbb{C}}$ , $e^{\mathbb{H}}_{\mathbb{R}}$       94f
$E_{6}$ , $E_{7}$ , $E_{8}$ (Dynkin diagram)      212 216
$GL(n,\mathbb{C})$ , $GL(n,\mathbb{H})$ , $GL(n,\mathbb{R})$       3 7
$G_{2}$ (Dynkin diagram)      212 216 227
$Hom_{G}(U,V)$       67
$Irr(G,\mathbb{C})$ , $Irr(G,\mathbb{H})$ , $Irr(G,\mathbb{R})$       69 96
$i^{G}_{H}V$ : induced representation      143
$L^{2}{G)$ square-integrable functions      134
$l_{x}$ : left translation      14
$L_{\alpha n} = \alpha^{-1}{n}, \alpha \in R_{+}, n\in \mathbb{Z}$       226
$n\alpha\beta$ (Cartan number)      198 ff
$PGL(n,\mathbb{C})$ , $PGL(n,\mathbb{R})$       4 11
$r^{C}_{R}$ , $r^{H}_{C}$ , $r^{H}_{R}$ , $r_{+}$ , $r_{-}$       95
$sign(\sigma)$       41
$sl(2,\mathbb{C})$       115 ff 196 202
$SL(n,\mathbb{C})$ , $SL(n,\mathbb{R})$       4
$Sp(n,\mathbb{C})$       9 155
$Spin(4) \cong Spin(3) \times Spin(3)$       292
$Spin(6) \cong SU(4)$       292
$SU(4) \cong Spin(6)$       292
$S^{1}$ (circle)      3
$S^{1}$ , representations      125
$S^{i}(V)$ (symmetric power)      75
$S^{n}$ (sphere)      36
$s_{+}$ , $s_{-}$       94
$s_{\alpha}$ (reflection ass. to \alpha\in R)      192 197
$V(k,l)=V^{\bigotimes k}\bigotimes\overline{V}^{\bigotimes l}$       137
$V_{k}(\mathbb{C}^{n})$ , $V_{k}(\mathbb{R}^{n})$       37 f
$Z\langle I*\rangle$       257
$А(\lambda)$ (alternating sum)      240
$Т_{p}M$ , $T_{p}f$ : tangent space      11 ff
$\Gamma(Q)$ (Clifford group)      57
$\lambda$ -ring      104
$\mathbb{H}$       5
$\mathbb{R}P^{n}$ (projective space)      11
$\mathbb{Z}/n$ : cyclic group      10
$\mathcal{H} = ker\alpha, \alpha\in R$       192 198
$\mathcal{J}$ (real or qualernionic structure)      93
$\mathcal{S}_{p}$ (algebra of germs)      12 169
$\mathcal{T}(G,K)$ (representative functions)      125 ff
$\mathfrak{G}$ : harmonic polynomials      88
$\mathfrak{p}(\zeta)$ (decomposition into positive roots)      257
$\Omega^{k}(М)$ (alternating differential forms)      42
A(n) (alternating group)      10
Abelian Lie group      25 30 39
Abelian Lie group, representations      69 107
Abelian subalgebra, maximal      188
Abstract subgroup      28
Action of a group on a space      30 ff
Ad, ad: adjoint representation      18
Adams operation      104 ff
Adjoint representation      18 71 183 222
Affine variety      152
Affine Weyl group: extended Weyl group      226
Alcove      226
Algebra, associative      54
Algebra, Clifford      54
Algebra, Lie      14 ff 19 20
Algebraic group      156
Alternating      41 240 244
Alternating group      10
Alternating sum      240
Alternating tensor      77
Analytic structure on Lie groups      138
Angle of roots      199 205
Angular velocity      22
Anti-automorphism      6 55
Antipodal map      52
Ascoli      130
Atlas bundle      33
Atlas manifold      2 43
Aut(V)      3
Base space      32
Basis of root system      204
Borel — Weil — Bott theorem      256
Boundary of manifold      50
Bounded operator      130
Bundle      32 188
Bundle atlas      33
Bundle chart      32
Bundle chart, tangent      14
Bundle map      32
C(Q), $C_{n}$ : Clifford algebra      55
Canonical decomposition of representation      70 101
Canonical metric of root system      213 215f
Cartan composite      253 262
Cartan matrix      210
Cartan matrix, determinant      216
Cartan number      198
Cartan subgroup      176 ff 297
Casimir operator      122
Center      165 189 201 229 235
Center of $\mathbb{H}$       5 11
Central element      230
centralizer      165 169 189
Chamber      192 227
Change of coordinates      2
Character of a representation      80 ff 107
Character of a representation, formula      239 ff
Character of a representation, group      82 83 107
Character of a representation, ring: representation ring      102 ff
Chart domain      2
Chart of bundle      32
Chart, manifold      2
Class function      81 134 166
Classical groups      1 ff 169
Classical groups, complexification      155
Classical groups, definitions      4 ff
Classical groups, maximal torus      169 ff
Classical groups, representations      265 IT
Classical groups, root systems      216 ff
Classical groups, Weyl groups      169 ff
Clebsch — Gordan formula      87 92 260
Clifford algebra      7 54 283
Clifford algebra, classification      286 291
Clifford algebra, modules      287 f
Clifford group      57
Coassociative      147
Codimension      27
Coinverse      146
Compact group, structure      232 ff
Compact operator      130 ff
Compact scmisimplc Lie algebra      209
Compact-open topology      142
Complex root: root      185
complex type      97
Complexification      151 ff
Comultiplication      146
Con(G) (conjugation classes)      166
Conjugate elements in G and T      166 180
Conjugate homomorphisms      67
Conjugate representations      75
Conjugate tori      159 ff
Conjugation      95
Conjugation in Clifford algebra      57
Conjugation of quaternions      6
Conjugation, complex      75
Conjugation, theorem for tori      159 ff 177
Connected component of unit      10
Contained (a representation in another)      69

Conv(A)      250
Convolution      67 83 129 139
coordinates      2
Counit      146
Cover, covering      54 228 233
Coxeter graph      211
Cyclic group      10 129 180
Cyclic group, representations      108 129
Cyclic, topologically      38 40 177
C[G]: group ring      66
Decomposable      204
Decomposition into positive roots      257
deg(f) (mapping degree)      51
Derivation      12
Descending chain of manifolds      136 138
Determinant, Cartan matrix      216
Determinant, conjugation map      161 242
Diagonal action      82
Diffeomorphism      40
Differentiable      2 15 41
Differentiable bundle      33
Differentiable G-space      31
Differentiable manifold      2
Differentiable map of manifolds      2
Differential      12
Differential form      41 ff
Dimension of manifold      2
Dimension of representation      65 242 249
Dirac approximation      135 143
Dirac matrix      292
Direct product      3 82 105
Direct sum      69 72 74
Discrete normal subgroup      10 26
Divisible group      25 169
Division algebra      5
Domain of chart      2
Dominant weight      249 ff
Dual representation      75
Duality (Tannaka — Krein)      146
Dynkin diagram      209 ff
Dynkin diagram, classification      212
Elementary symmetric function      175
Embedding      28
Embedding, equivariant      138 143
End(V)      3 99
Equicontinuous      130
Equivalent representations      67
Equivariant      31
Euclidean scalar product      4
Euler angles      53
Euler operator      122
Exponential map      17 21
Exponential map, surjectivity      24 30 165
Extended Weyl group      226
Extension      95
Exterior derivative      50
Exterior power      75 103 175
Exterior product      42
Extremal point      256
Factorial      249
Faithful      66 136
Fiber      32
Fibered product      90 110
Filtration of Clifford algebra      62
Fixed point      31 77 181
Flow      16 39
Frame      37
Frobenius reciprocity      127 144 295
Frobenius theorems      9 234
Fubini      49
Fundamental group      54 61 223
Fundamental representation      254
Fundamental system of weights      254
Fundamental Weyl chamber      204
G-manifold      31
G-space      30 ff
General element of a Lie group      168 181
General position      231
Generator of a torus      38
Germ      12 169
Global root      185 189 195
Global weight: weight      108
Graded      56
Gram — Schmidt orthogonalization      11
Grassmann manifold      38
Group ring      66 83 129
H(G)      264 277
Half sum of positive roots      207 241
Half-space      206
Half-spin representation      279 ff
Half-spin representation, type      290
Harmonic polynomial      88 ff 117
Hermitian      4 21 153
Higher (order on LT*)      250
Hom(U,V)      75
Homogeneous polynomial      84 117
Homogeneous space      30 ff 35
Homomorphism of Lie groups      2 29
Homotopic      51 53 179
Homotopy group      36 f 187 223
Hopf algebra      147
Hopf deration      37 40
I: integral lattice      195
Immersion      27 40
Indecomposable      204 254
Index of self-conjugate irreducible representation      262
Induced representation      143 ff 256 292
Infinitesimal weight      112 ff
Initial value of integral curve      16
Inner product      4 8 9 79 245
Integral ( $\int$ )      40 ff 44 48 139
Integral ( $\int$ ) curve      16
Integral ( $\int$ ) formula, Weyl’s      163
Integral ( $\int$ ) on LT      245
Integral ( $\mathbb{Z}$ ) elements of LT      195
Integral ( $\mathbb{Z}$ ) form      195
Integral ( $\mathbb{Z}$ ) lattice      195 221 230
Intertwining operator      67
Invariant      15 31
Invariant inner product      68 92 114
Invariant integral      40 ff 139
Invariant integral of SO(3)      53
Invariant subspace      31
Invariant vector field      15
Inverse root      195 198 208 223
Irr(G,T)      242
Irreducible character      80
Irreducible module      72 287
Irreducible representation      68 141
Irreducible root system      211
Isomorphism of G-modules      67
Isomorphism of Lie groups      2 22
Isotropy group      31 35 138
Isotropy group in Weyl group      202
Isotypical      70 73 83 101 126 128
j (quaternion)      5 62
J (symplectic form)      9
Jacobi identity      19
Jacobian determinant      41
Jacobian matrix      14
k (quaternion)      6 62
Killing form      209 214 229
Kostant’s formula      258
Krein      146
Kronecker product      75
Kronecker theorem      38
L(G): Lie algebra      14
L(G/T) = LG/LT      160
Laplace operator      88
Largest root      257

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