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

Left invariant      11 ff 15 41 46 49
Left translation      14 181
Legendre function      121
Lie algebra      14 ff 19 20
Lie algebra, representation      111
Lie derivative      111
Lie group      1 ff
Lie group, analytic structure      138
Lie group, holomorphic      30
Lie group, homomorphism      2
Lie group, isomorphism      2
Lie product      18 ff
Linear groups      3 ff
Linear groups over $\mathbb{H}$       7
Linear groups, standard inclusions      10
Linked      40
Local coordinates      2
localization      249
Locally finite      128
Locally trivial      14 32
LOG      28 f
Long root      213
Lorentz group      9
lower      250
Mapping degree      51 179
Matrix algebra      284 287
Matrix representation      66
Maximal abelian Lie algebra      164
Maximal abelian subgroup      164 f
Maximal compact subgroup      90 153
Maximal torus; tonts, maximal      157
Minkowski space      292
Mod(R)      287
Module      67 72 112
Module over convolution algebra      67 83 140
Monotone (integral)      40
Morphism of G-modules      67
Multiplicity      69 70 257
Naturality of exp      23
Negative root      204
Norm, Clifford      57
Norm, operator      131
Norm, quaternion      6
Norm, symplectic      8
Normal subgroup      35
Normalized      41
Normalizer of maximal torus      158 ff
Normalizer of reflection group      196 201
O(3)-representations      84 ff
O(n): orthogonal group      4
One-parameter group      11 ff 17
Operation of a group on a space      30 ff
Operator      72
Orbit      31 35
Orbit of extended Weyl group      227 232
Orbit of Weyl group      202
Order on LT*      250
Orientation      43 51 52
Orientation, preserving      43
Orthogonal group      4 10 11 36 38 155
Orthogonal group, representation ring      292 297
Orthogonal group, representations      292 ff
Orthogonal representation      93
Orthogonality relation      79 81 83 101 142 245
Partition of unity      44 53
Pauli spin matrix      62
Peter — Weyl theorem      133 ff
Physicists’ convention      21 62
Pin(n)      60
Positive (in R(G))      103
Positive root      204 252
Positive root, decomposition into      257
Positive weight      257
Positive-definite      11 153
Precompact      130
primitive      116
Product of characters      80 254 259
Product of representative functions      125
Product, direct      3 82 105
Projection, principal bundle      32
Projection, tangent bundle      14
Projective group      4 11 90
Projective representation      90 ff 110
Projective space      11 37 52
Pure quaternion      6 61
Quadratic form      54
Quaternion      5 61
Quaternion group      7 61 84 91 115 185 292
Quaternion, pure      6
Quaternionic representation      93 ff 261
Quaternionic structure      93 102
Quaternionic type      97
Quotient group      35
R(G): representation ring      103
Rank      165 189 197
Rank one      185 199
Rank two, root systems      200 211 227
Real representation      93 ff 136
Real root: root      185
Real structure      93 101 102
Real type      97 261
Real weight      114
Reduced root system      197
Reducible      68
Reducible root system      211
Reflection      192 ff 202
Regular element in Lie group      168 224 226
Regular value      51 52 179
Rep(G, K)      94
Representation      65 ff
Representation group, quaternionic      263
Representation of Lie algebras      111 ff
Representation ring      102 ff 167 255
Representation ring is Noctherian      256
Representation ring, $R(G)=R(T)^{W}$       249
Representation ring, real      263 f
Representation ring, ring structure      255
Representation space      65
Representation, canonical decomposition      70 101
Representation, faithful      66 136
Representation, infinite-dimensional      127
Representative function      78 123
Res: restriction      71
Restriction      71 95 144
Riemann sphere      37 63
Riemannian metric      226
Right action      31
Right invariant      47
RO(G)      105
Root      185 ff 197
Root system      189 ff 197
Root system of classical groups      216 ff
Root system, canonical metric      213
Root system, irreducible      211
Rotation      22
S(n) (symmetric group)      10
Scalar product on function space      79 245
Scalar product, canonical of root system      213 215

Scalar product, Euclidean      4
Scalar product, Hermitian      4
Scalar product, symplectic      8 11
Schur’s lemma      69 72
Self-adjoint      21 131
Self-conjugate      98 261
Semisimple      71 ff 201 214 229 237 249 256
Separation of points      132
Short root      213
Similar      67
Simple module      72 287
Simple reflection      205
Simple root      204
Singular element in Lie group      168 189 224 226
Skew field      5 7 74
Smooth      2
SO(3)-representations      84 ff 115
SO(n): special orthogonal group      4
Sp(n): symplectic group      8
Special      168 181
Special $\lambda$ -ring      104
Special linear group      4
Special orthogonal group      4 36
Special orthogonal group, complextfication      155
Special orthogonal group, maximal torus      171
Special orthogonal group, representation ring of SO(2n)      282
Special orthogonal group, representations      272 ff
Special orthogonal group, root system      219 f 222
Special orthogonal group, Weyl group      171
Special unitary group      5 36
Special unitary group, complexification      155
Special unitary group, maximal torus      170
Special unitary group, representations      265
Special unitary group, root system      218
Special unitary group, Weyl group      170
Spectral theorem      132
Sphere      36
Spherical coordinates      120
Spherical harmonic      88 ff
Spin(n): spinor group      54 ff
Spinor group      54 ff 91
Spinor group, maximal torus      173 ff
Spinor group, representation ring      279 f
Spinor group, representations      278 ff
Spinor group, root system      222
Spinor group, types of representations      290
Spinor group, Weyl group      173 ff
Splitting principle      106
Standard basis of quaternions over $\mathbb{C}$       5
Standard basis of quaternions over $\mathbb{R}$       6
Standard isomorphism $\mathbb{C}^{2}\rightarrow\mathbb{H}$       6
Standard isomorphism $\mathbb{R}^{4}\rightarrow\mathbb{H}$       6
Standard representation      69 71
Star operator      274
Steinberg’s Formula      259
Stiefel diagram      223
Stiefel manifold      37
Stokes’ Theorem      50
Stone — Weierstrass      132
String of roots      201
Structure group      32
SU(2)-representations      84 ff
SU(n): special unitary group      5
Subgroup      27
Subgroup, closed      28
Submanifold      27 28
Submodule      68 72
Subreprescntation      68
Sum of characters      80
Sum of representations      69
Sum of representative functions      125
Sum of root systems      211
Supp(f): support      43
Support      43 257
Symmetric character      240
Symmetric group      10
Symmetric operator      131 ff
Symmetric power      75
Symmetric sum      252
Symmetric tensor      77
Symmetry      197
Symplectic      7 ff
Symplectic bilinear form J      9
Symplectic group      8 11 37 155
Symplectic group, complex      9 155
Symplectic group, maximal torus      173
Symplectic group, representations      269 ff
Symplectic group, root system      221
Symplectic group, Weyl group      173
Symplectic representation      93
System of generators      254
Tangent bundle      14
Tangent map      12
Tangent space      4 ff
Tannaka — Krein duality      146 ff
Tensor product      74 80 84 87 137
Tensor product, graded      56
Tensor product, multiplicities of weights      259
Topological group      29
Topologically cyclic      38 190
Torus      3 25 38
Torus, maximal      157 ff 223
Total space of tangent bundle      14
Tr(A), trace      76
Transformation of integrals      41 51
Transitive group action      31 35
Triangular matrix      11
Trigonometric polynomial      125 136
Trivial representation      69 137
Twisted      293
TYPE      93 ff 261
Type I, type II      293
U(2)-representations      84 ff
U(n): unitary group      4
Unitary group      4 11 36 153
Unitary group, maximal torus      170
Unitary group, representation ring      175
Unitary group, representations      267
Unitary group, root system      217 f
Unitary group, Weyl group      170
Unitary representation      68
Vector bundle      14
Vector field      11 ff 15
Vector product      11 22 118
Velocity vector      11
Virtual character: representation ring      103
Volume form      43 49
wall      192 202 226
Wedge product      42
Weight      108 ff 116 184 239
Weight space      108 112
Weight vector      114
Weyl chamber      192 202
Weyl character formula      239 ff
Weyl group      158 165 177 193 198
Weyl group, extended      226
Weyl integral formula      163
Z(H): centralizer      165

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