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| Varadarajan V.S. — Lie Groups, Lie Algebras, and Their Representations |
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| Предметный указатель |
Adjoint representation, of a Lie algebra 50
Adjoint representation, of a Lie algebra in its universal enveloping algebra 172
Adjoint representation, of a Lie group 104
Admissible topological group 62
Ado's Theorem 237
Algebraic group, complex 45
Algebraic group, real 45
Algebraic set 23
Analytic group 41
Analytic manifold, complex 21
Analytic manifold, real 20
Analytic structure, real 20
Analytic subgroup of a Lie group 42
Analytic system of tangent spaces 25
Baker — Campbell — Hausdorff formula 114
Bracket, Lie, in a Lie algebra 47
Bracket, Lie, of vector fields 6
Canonical coordinates, of the first kind 88
Canonical coordinates, of the second kind 89
Cartan matrix 290
Cartan subalgebra (CSA) 262
Cartan — Killing form 208
Casimir element, associated with a representation of a semisimple Lie algebra 218
Casimir element, of the universal enveloping algebra of a semisimple Lie algebra 217
Casimir polynomial 208
Center, of a Lie algebra 50
Central series, ascending and descending, of a nilpotent Lie algebra 193
Chamber, Weyl 282
Chamber, Weyl, associated with root systems 370 379
Chevalley's theorem, on conjugacy of CSA's of a complex Lie algebra 263
Chevalley's theorem, on invariant polynomials on a semisimple Lie algebra 335
Chevalley's theorem, on polynomials invariant with respect to a finite reflection group 383
Chevalley's theorem, on stabilizers in a finite reflection group 377
Classical groups, compact 46
Classical groups, complex 46
Compact form, of a complex semisimple Lie algebra 348
Compact form, of a complex semisimple Lie group 348
Complexification of a Lie algebra 47
Covering group 62
Covering homomorphism 62
Cyclic vector for a representation 154
Derivation, of an algebra 49
Derived algebra, of a Lie algebra 200
Diffeomorphism 15
Differentiable functions 2
Differentiable manifold 2
Differentiable manifold, dimension of 2
Differentiable mapping 14
Differentiable structure 1
Differentiable structure, of a homomorphism of one Lie group into another 70
Differentiable structure, of a mapping 14
Differential expression 4
Differential operator 6
Differential operator,  -related to another 14
Differential operator, expression of, at a point 6
Differential operator, formal adjoint of 14
Differential operator, left invariant, on a Lie group 53
Dominant linear function 314
Dynkin diagram 293
Engel's theorem 190
Enveloping algebra, of a Lie group 55
Exponential map 85
Exponential map, differential of 108
Exponential of a matrix 89
Exterior algebra 166
Exterior differential form 11
Exterior differentiation 11
Filtered algebra 176
Filting decomposition 152
Free action of a group 81
Free associative algebra 164
Free Lie algebra 174
Frobenius theorem, global 30
Frobenius theorem, local 28
Fundamental group 61
G-space 74
G-space, transitive 74
Galois extension 149
Germ of a function 3
Graded algebra 176
Graded algebra, associated with a filtered algebra 177
Harish — Chandra's theorem, on the center of the universal enveloping algebra of a semisimple Lie algebra 339
Hilbert's fifth problem 42
Homogeneous space 74
Homomorphism, of Lie algebras 47
Ideal, of a Lie algebra 47
Imbedding 16
Imbedding, quasiregular 17
Imbedding, regular 16
Immersion 16
Infinitesimal character 184 339
Integral linear function 314
Involutive analytic system, of tangent spaces 25
Involutive analytic system, of tangent spaces, canonical 26
Involutive analytic system, of tangent spaces, coordinates adapted to 28
Involutive analytic system, of tangent spaces, integrable 25
Involutive analytic system, of tangent spaces, integral manifold of 25
Jacobi identity 6 47
Jordan decomposition, of an element of a semisimple Lie algebra 215
Jordan decomposition, of an endomorphism of a vector space 151
Levi decomposition, of a Lie algebra 224
Levi decomposition, of an analytic group 245
Levi subalgebra, of a Lie algebra 224
Levi subgroup, of an analytic group 245
Lexicographic ordering, of a real vector space 285
Lie algebra 46
Lie algebra, abelian 48
Lie algebra, of a Lie group 51
Lie algebra, quotient of 48
| Lie algebra, structure constants of 47
Lie algebra, subalgebra of 47
Lie group 41
Lie group, analytic action of, on a manifold 75
Lie group, quotient of 80
Lie subalgebra defined by a Lie subgroup 57
Lie subgroup 42
Local homomorphism 63
Locally isomorphic groups 63
Maurer — Cartan forms 138
Maximal torus, of a compact semisimple analytic group 353
Measure, corresponding to an n-form 12
monodromy 63
Nil ideal, of a Lie algebra 206
Nil radical, of a Lie algebra 206
Nil radical, of an analytic group 244
Nil representation, of a Lie algebra 189
Nilpotent analytic group 195 243
Nilpotent component, of an element of a semisimple Lie algebra 215
Nilpotent element, of a semisimple Lie algebra 215
Nilpotent Lie algebra 189
Orbit 80
Orientable manifold 12
Orientation, of a manifold 12
Poincare series, of a graded vector space 381
Poincare — Birkhoff — Witt theorem 168
Positive system of roots 280 370 379
Product manifolds 19
Product, of Lie algebras 48
Product, of Lie groups 41
Quotient, of a manifold relative to a map 16
Radical, of a Lie algebra 204
Radical, of an analytic group 244
Rank, of a Lie algebra 260
Real form, of a complex Lie algebra 133
Reductive Lie algebra 232
Reflection group 369
Reflection, in a vector space 369
Regular element, of a Lie algebra 261
Regular point of an algebraic set 23
Replica, of an endomorphism of a vector space 157
Representation with weights 316
Representation, complex analytic, of a complex Lie group 102
Representation, irreducible 51
Representation, of associative algebra 152
Representation, of Lie algebra 49
Representation, of Lie group 102
Representation, quotient representation of 51 153
Representation, semisimple 153
Representation, subrepresentation of 51 153
Representations, of a Lie algebra, direct sum of 50
Representations, of a Lie algebra, equivalence of 51
Representations, of a Lie algebra, outer tensor product of 50
Representations, of a Lie algebra, tensor product of 50
Root 273 370
Root space decomposition, of a semisimple Lie algebra 274
Root subspace 273
Root system 370
Root system, extended integral 378
Root system, integral 377
Schur's lemma 153
Semidirect product, of analytic groups 228
Semidirect product, of Lie algebras 224
Semisimple analytic group 207
Semisimple component, of an element of a semisimple Lie algebra 215
Semisimple element, of a semisimple Lie algebra 215
Semisimple Lie algebra 204
Simple Lie algebra 211
Simple system of roots 280 370 379
Solvable analytic group 204 243
Solvable Lie algebra 201
Spin representation 329
Stability subgroup 74
Submanifold of a differentiable manifold 18
Submanifold of a differentiable manifold, quasi-regular 18
Submanifold of a differentiable manifold, regular 18
Submersion 15
Symmetric algebra 165
Symmetrizer map 180
Symplectic group 46
Tangent space 3
Tangent space, holomorphic 22
Tangent vector 3
Tangent vector, holomorphic 22
Taylor expansions, on a Lie group 94
Tensor algebra 163
Tensor, degree of 163
Tensor, skew-symmetric 165
Tensor, symmetric 165
Torus 43
Transformation group, global 123
Transformation group, infinitesimal 123
Transformation group, local 122
Unipotent representation, of an analytic group 245
Unipotent subgroup of GL(F) 196
Unitarian trick 349
Universal complexification, of a real analytic group 256
Universal covering group 62
Universal enveloping algebra, of a Lie algebra 167
Vector field 5
Weight, highest, of a representation 317
Weight, of a representation 193 316
Weight, subspace 193 316
Weyl's theorem, on finiteness of fundamental groups of compact semi-simple groups 345
Weyl's theorem, on semisimplicity of representations 222 348
Weyl, basis 280
Weyl, character formula of 366
Weyl, dimension formula of 327 368
Weyl, group 279
Weyl, integral formula of 363
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