 -space 7
Abelian group 12
Absolutely continuous measure 40
Absolutely simple Lie algebra 92
Abstract group 12
Action (of a group) on the left 13
Action (of a group) on the right 14
Adjoining a unit 25
Adjoint representation of a Lie algebra 97
Adjoint representation of a Lie group 97
Admissible chart 63
Admissible subalgebra 236
Ado’s theorem 88
Aleksandrov’s line 63
Algebra of type I 60
Algebra over K 32
Algebraic integer 157
Algebraic manifold 68
Algebraic number field 28
Algebraically irreducible representation 110
Almost everywhere 39
Alternating group 262
Amenable group 129
Analytic vector 155
Antirepresentation 14
Associativity 12
Atlas 62
Automorphism 11 13
Banach algebra 45
Banach space 37
Banach — Steinhaus theorem 42
Base 78
Base for a topology 6
Basis 31
Basis measure 54
Bicommutant 58
Bifunctor 6
Bigebra 178
Binary relation 2
Borel set 6
Borel — Weil — Bott theorem 140
Boundary of a manifold 77
Bounded set 41
Burnside’s Theorem 140
C*-algebra 49
C*-algebra of a group 143
Campbell — Hausdorff — Dynkin formula 108
Canonical embedding 4
Canonical mapping 2
Canonical matrix 92
Canonical parameter 85
Canonical projection 4
Cartan’s criterion 92
Cartan’s subalgebra 93
Category 3
Central extension 20
centralizer 17
Change of scalars 111
Character group 167
Character of a finite dimensional representation 156
Character of a infinite dimensional representation 161
Character of Banach algebra 46
Chart 62
Choquet’s theorem 42
Class  63
Classical simple algebra 92
Clifford algebra 34
Closed form 76
Closed operator 114
Closed set 6
Co-adjoint representation 226
Coboundary 10 21
Coboundary operator 10
Cochain 10
Cocycle 10 21
Cohomology group 21
Coinduced representation 187
Commutant 58
Commutation 86
Commutative algebra 87
Commutative group 12
Commutator of vector fields 72
Compact 7
Compact operator 42
Complete measure 39
Completely continuous operator 42
Completely irreducible representation 112
Completely reducible representation 110
Completion 9
Complex manifold 68
Complexification 111
Component of a section 79
Conjugate representation 109
Conjugate subgroup 15
Connected set 7
Continuous product of algebras 60
Continuous sum of spaces 58
Continuously diagonal operator 58
Contragredient representation 109
Contravariant functor 5
Contravariant tensor 33
Convolution 140
Convolution of generalized functions 148
Cotangent bundle 79
Countably additive measure 38
Countably normable space 37
Courant’s priniciple 56
Covariant functor 5
Covariant tensor 33
Cross norm 42
Cross section 79
Cyclic group 16
Cyclic representation 53
Cyclic vector 53
De Rham cohomology ring 76
Decomposable extension 20
Decomposable operator 58
Decomposable representation 110
Decreasing filtration 26
Derivative along a vector 69
Derivative of a family of mappings 71
Derivative of a mapping 70
Derivative of a measure 40
Determining (family of prenorms) 36
Diagonal operator 58
Diffeomorphic manifolds 64
Diffeomorphism 64
Differentiability along a vector field 73
Differential 74
Differential form 74
Differential form of type (0,r) 82
Differential of a function 70
Dimension of a minifold 62
Dimension of a vector space 31
Direct sum 31
Direct sum of bundles 79
Directed set 3
Direction 3
Disjoint measures 40
Disjoint representations 109
Distance 8
Division algebra 27
Dixmier — Glimm — Sakai theorem 162
Dolbeault’s theorem, generalized 82
Double coset 16
Dual category 4
Dual group 167
Dual module 30
Dual object of a group 113
Dynkin diagramm 94
| Engel’s theorem 91
Envelopping algebra 149
Epimorphism 13
Equivalent atlases 67
Equivalent extensions 19
Equivalent measures 40
Equivalent representations 48 109
Even permutation 262
Exact form 74
Exponential Lie group 103
Exponential mapping 103
Extension (of a group) 18
Exterior algebra 34
Exterior differentiation 74
Exterior product 34
Factor 62
Factor group 15
Factor Lie algebra 33
Factor module 29
Factor representation 110
Factor ring 25
Factor set 2
Factor space 7
Faithful representation 13 48
Fiber 78
Field 27
Filtering to the right 3
Filtration 26
Finite measure 39
Finite representation 115
Flag manifold 291
Formal dimension 139
Fourier transform 166
Free algebra 33
Free K-module 29
functor 6
Fundamental direction 9
Fundamental group 9
G-orbit 14
Garding space 153
Gel’fand theorem 49
Gel’fand transform 46
Gel’fand — Kirillov conjecture 152
Gel’fand — Mazur theorem 45
Gel’fand — Naimark theorem 83
Gel’fand — Raikov theorem 146
Generalized infinitesimal character 165
Generalized infinitesimal generating function 232
Generalized infinitesimal Gleason — Montgomery — Zippin theorem 83
Graded 26
Graph of an operator 121
Grassmann algebra 34
Grassmann manifold 65
Grothendieck group 110
Group 12
Group algebra 140
Group ring 139
Hahn — Banach theorem 42
Hamiltonian vector field 231
Hausdorff space 7
Heisenberg group 237 287
Hermitian element 48
Hilbert space 37
Hilbert tensor product 44
Hilbert — Schmidt operator 44
Holomorphic bundle 80
Holomorphic cross section 80
Holomorphicaly induced representation 204
Homogeneous algebra 60
Homogeneous G-space 15
Homogeneous symplectic manifold 233
Homomorphism 12
Homotopic mappings 9
Homotopy class 9
I-adic topology 27
Ideal of a Lie algebra 89
Idempotent 122
Identity 12
Image 13
Increasing filtration 26
Induced representation 182
Induced representation in the sense of Mackey 188
Inductive limit 5
Infinitesimal character 163
Infinitesimal Lie group 86
Inner automorphism 13 97
Integrable function 39
integral 39
Integral of a form 76
Integral orbit 239
Intertwining number 109
Intertwining operator 109
Invariant mean 129
Invariant subgroup 15
Inverse morphism 4
Inverse relation 2
Isomorphic Lie groups 99
Isomorphic objects 4
Isomorphism 4 13
Jacobson topology 147
Joint spectrum 47
k-irredicible 113
k-representation 226
Kazdan’s theorem 181
Kernel 13
Killing form 91
Kostant — Auslander theorem 241
Krein — Mil’man theorem 42
Lagrange distribution 246
Law of a composition 3
Left coset 15
Left fraction 152
Left G-space 13
Left Haar measure 130
Left ideal 25
Left K-module 28
Leray’s theorem 11
Lie algebra 86
Lie algebra, homomorphism 87
Lie group 83
Lie operator 73
Lie skew field 152
Lie’s theorem 90
Linear mapping 29
Linear operator 31
Linear representation 108
Linear representation of a Lie group 87
Linear space 31
Linearization of a projective representation 220
Linearly ordered set 2
Local coordinates 62
Locally convex space 36
Locally finite measure 39
Locally isomorphic Lie groups 99
Mackey’s criterion of inducibility 196
Mackey’s theorem 197
Manifold 62
Manifold with boundary 77
Maschke’s Theorem 140
Matrix element 135
Measurable function 39
Measurable set 38
Measurable vector function 57
Measure 38
Metric space 8
Minkowski functional 36
Mixed tensor 33
Modulus of a group 131
Monomial group 198
Monomial representation 182
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