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de Graaf W.A. — Lie Algebras: Theory and Algorithms
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Название: Lie Algebras: Theory and AlgorithmsАвтор: de Graaf W.A. Аннотация: he aim of the present work is two-fold. Firstly it aims at a giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras. Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras. These two subject areas are intimately related. First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., [42], [48], [77], [86]). Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincaré-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples). Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists). Secondly, the algorithms can be used to arrive at a better understanding of the theory. Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory of life.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2000Количество страниц: 393Добавлена в каталог: 04.12.2009Операции: Положить на полку |
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125 131 Abelian Lie algebra 15 Action of a Lie algebra on a vector space 23 Adjoint map 3 Adjoint representation 24 AdjointMatrix 14 Admissible configuration 143 Ado's Theorem 212 Algebra 1 Algebra, associative 1 Algebra, Lie 2 Alternating element 287 Appliance 221 243 245 ArelsomorphicSSLieAlgebras 182 Associative law 2 Automorphism of a Lie algebra 22 Bilinear form 4 BracketingOfHallWord 239 240 Branching rule 301 BranchingRule 302 Canonical basis 155 Canonical generators 153 CanonicalGenerators 154 Cartan matrix 136 Cartan's criterion for semisimplicity 77 Cartan's criterion for solvability 43 CartanMatrixToRootSystem 137 CartanSubalgebraBigField 58 CartanSubalgebraRestricted 59 Casimir operator 81 Casimir operator, universal 280 Central component 29 centralizer 14 15 Centralldempotents 315 Centre 15 Coefficient of a representation 205 CollectionInUAE 193 Commutative Lie algebra 15 Commutator 2 Completely reducible module 24 composition 190 247 Composition series 24 Composition, useful 249 Composition, useless 249 ConjugateDominantWeight 268 Coxeter diagram 143 DecomposeCharacterTensorProduct 300 Decomposing element 105 315 DecompositionBySuccessiveSubtractions 296 Deglex order 186 193 Degree compatible order 193 Degree of free magma elements 218 Derivation 19 Derivation, inner 20 Derivation, outer 20 Derivations 20 Derived series 17 Derived subalgebra 17 DerivedSeries 17 Descending chain condition 186 193 221 241 Diagram automorphism 164 DimensionPolynomial 292 Direct sum of L-modules 24 Direct sum of Lie algebras 21 Direct sum, root systems 129 DirectSumDecomposition 31 108 Dominant integral linear function 263 DominantWeightsOfHWModule 279 Dorroh extension 305 Dynkin diagram 141 Engel's theorem 35 Equivalence of Cartan matrices 136 Equivalence of L-modules 23 Even orthogonal Lie algebra 6 ExtendRepresentation 209 ExtensionSeriesNilpotentCase 203 ExtensionSeriesSolvableCase 203 Faithful representation 23 Finite presentation 220 Finitely presented Lie algebra 220 Fitting decomposition 54 Fitting-null component 54 Fitting-one component 54 FittingOneComponent 55 foliage 232 Formal character 287 Free algebra 218 Free associative algebra 186 Free Lie algebra 219 Free magma 218 Free monoid 186 Freudenthal's multiplicity formula 280 Fundamental dominant weights 266 Groebner basis in a free algebra 222 Groebner basis in a free associative algebra 188 Groebner basis in a free Lie algebra 246 Groebner basis in a universal enveloping algebra 195 Groebner basis, left 201 GroebnerBasis 198 250 Hall order 229 Hall set 229 Hall word 233 HallSet 229 Height of a root 136 Highest weight 122 263 Highest-weight module 263 Highest-weight vector 122 263 Hypercentre 18 Ideal 1 Idempotent 309 Idempotent, central 313 Idempotent, orthogonal 309 Idempotent, primitive 309 Inner automorphism 23 Inner automorphism group 23 Integral linear function 262 InterReduce 250 Irreducible character 295 Irreducible module 24 Irreducible rootsystem 129 IsHallElement 229 Isomorphism of algebras 1 Isomorphism of root systems 129 IsomorphismOfSSLieAlgebras 157 IsRestricted 27 Iwasawa's theorem 215 Jacobi identity 2 Jordan decomposition 312 Killing form 76 Klymik's formula 298 Kostant's formula 294 Largest nilpotency ideal 205 Las Vegas algorithm 12 Leading monomial 186 193 Left-normal form 201 LeftGroebnerBasis 201 LeftNormalForm 201 Leibniz formula 19 Length of a Weyl group element 274 letters 230 Level of a weight 278 Levi subalgebra 89 Levi's theorem 87 LeviSubalgebra 111 118 LeviSubalgebraByLCSeries 112 Lexicographical order 237 Lie bracket 2 Lie polynomial 233 Lie's theorem 41 Linear associative algebra 305 Linear lie algebra 4 Longest element in a Weyl group 275 Lower central series 17 LowerCentralSeries 17 Lyndon — Shirshov words 238 Module 23 Morphism of algebras 1 Morphism of L-modules 23 Multiplication 1 Multiplication table 8 Multiplicative order 186 193 221 241 Multiplicity of a weight 280 Negative root 134 Nilpotency class 17 Nilpotency ideal 205 Nilpotent, associative algebra 306 Nilpotent, element 45 Nilpotent, Lie algebra 17 Nilradical 36 Non-degenerate bilinear form 4 NonNilpotentElement 46 Normal monomials 222 246 Normal words 186 195 NormalForm 187 194 243 normalizer 15 16 Odd orthogonal Lie algebra 6 p-map 26 p-th power mapping 26 Poincare — Birkhoff — Witt theorem 191 Polynomial map 62 Positive root 134 Primary components 50 Primary decomposition 50 PrimaryDecompositionBigField 104 PrimaryDecompositionSmallField 105 Product space 16 ProductSpace 17 Projection matrix 302 Quasi-Hall set 229 Quotient algebra 3 Quotient module 24 QuotientAlgebra 13 Racah's formula 294 297 Radical 307 Radical of an associative algebra 306 Rank of a Lie algebra 57 Rank of a root system 128 Reduced expression (for a Weyl group element) 267 Reduction 187 195 Reduction order 241 Reflecting hyperplane 127 Reflection 127 Regular element 57 Representation 211 215 Representation of a Lie algebra 23 RepresentationNilpotentCase 210 RepresentationSolvableCase 211 Restricted Lie algebra 25 Reverse lexicographical order 239 RewriteMagmaElement 235 Root 60 Root lattice 160 Root order 132 Root order, lexicographical 134 Root space 60 Root space decomposition 60 Root system 127 Root, long 170 Root, short 170 S-element 195 Self-reduced set 188 222 247 Semidirect sum 21 Semisimple, associative algebra 307 Semisimple, Lie algebra 77 Semisimple, linear transformation 311 Serre's theorem 259 Shirshov's theorem 247 Sign of an element of the Weyl group 287 Simple, Lie algebra 77 Simple, reflection 140 Simple, root 134 Simple, system of roots 135 Simple, type 150 SimpleLieAlgebra 172 Simply-laced root system 160 Solvable Lie algebra 17 Solvable radical 38 SolvableRadical 45 Special linear Lie algebra 4 Split, Cartan subalgebra 59 Split, linear Lie algebra 40 Splitting element 102 314 SplittingElementDeterministic 103 SplittingElementRandom 103 Standard factorization 244 Standard monomial 192 Standard sequence 230 Steinberg's formula 297 Structural invariant 121 Structural invariant, complete 121 Structure constants 7 Subalgebra 1 Submodule 24 Symmetric element 287 Symplectic Lie algebra 5 Toral subalgebra 115 ToralSubalgebra 116 Trace form 76 TYPE 180 Type of a semisimple Lie algebra 175 Upper central series 18 UpperCentralSeries 18 Verma module 265 Weight 122 255 262 Weight lattice 266 Weight space 122 255 262 Weight vector 122 255 262 Weight-coordinate vectors 272 WeightsOfHWModule 278 Weyl group 139 Weyl vector 139 Weyl's character formula 291 Weyl's dimension formula 291 Weyl's theorem 83 WeylOrbit 277
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