Lie algebras: theory and algorithms
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2000
|
| Edition: | 1. ed. |
| Series: | North-Holland mathematical library
56 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 393 S. Ill. |
| ISBN: | 0444501169 |
Staff View
MARC
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| 245 | 1 | 0 | |a Lie algebras |b theory and algorithms |c Willem A. de Graaf |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2000 | |
| 300 | |a XII, 393 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Basic constructions 1
1.1 Algebras: associative and Lie 1
1.2 Linear Lie algebras 4
1.3 Structure constants 7
1.4 Lie algebras from p groups 10
1.5 On algorithms 13
1.6 Centralize™ and normalizers 17
1.7 Chains of ideals 19
1.8 Morphisms of Lie algebras 21
1.9 Derivations 22
1.10 (Semi)direct sums 24
1.11 Automorphisms of Lie algebras 26
1.12 Representations of Lie algebras 27
1.13 Restricted Lie algebras 29
1.14 Extension of the ground field 33
1.15 Finding a, direct sum decomposition 34
1.16 Notes 38
2 On nilpotency and solvability 39
2.1 Engel s theorem 39
2.2 The nilradical 42
2.3 The solvable radical 44
2.4 Lie s theorems 47
2.5 A criterion for solvability 49
2.6 A characterization of the solvable radical 51
2.7 Finding a non nilpotent element 54
2.8 Notes 56
X
3 Cartan subalgebras 57
3.1 Primary decompositions 58
3.2 Cartan subalgebras 64
3.3 The root space decomposition 69
3.4 Polynomial functions 72
3.5 Conjugacy of Cartan subalgebras 73
3.6 Conjugacy of Cartan subalgebras of solvable Lie algebras . . 76
3.7 Calculating the nilradical 79
3.8 Notes 86
4 Lie algebras with non degenerate Killing form 89
4.1 Trace forms and the Killing form 90
4.2 Semisimple Lie algebras 91
4.3 Direct sum decomposition 92
4.4 Complete reducibility of representations 95
4.5 All derivations are inner 99
4.6 The Jordan decomposition 101
4.7 Levi s theorem 104
4.8 Existence of a Cartan subalgebra 107
4.9 Facts on roots 108
4.10 Some proofs for modular fields 116
4.11 Splitting and decomposing elements 120
4.12 Direct sum decomposition 128
4.13 Computing a Levi subalgebra 130
4.14 A structure theorem of Cartan subalgebras 134
4.15 Using Cartan subalgebras to compute Levi subalgebras .... 136
4.16 Notes 141
5 The classification of the simple Lie algebras 143
5.1 Representations of s^F) 144
5.2 Some more root facts 146
5.3 Root systems 150
5.4 Root systems of rank two 154
5.5 Simple systems 157
5.6 Cartan matrices 161
5.7 Simple systems and the Weyl group 164
5.8 Dynkin diagrams 167
5.9 Classifying Dynkin diagrams 169
5.10 Constructing the root systems 176
5.11 Constructing isomorphisms 181
xi
5.12 Constructing the semisimple Lie algebras 187
5.13 The simply laced case 189
5.14 Diagram automorphisms 194
5.15 The non simply laced case 195
5.16 The classification theorem 206
5.17 Recognizing a semisimple Lie algebra 206
5.17.1 Identifying a semisimple Lie algebra 207
5.17.2 Isomorphism of semisimple Lie algebras 213
5.18 Notes 217
6 Universal enveloping algebras 219
6.1 Ideals in free associative algebras 220
6.2 Universal enveloping algebras 226
6.3 Grobner bases in universal enveloping algebras 229
6.4 Grobner bases of left ideals 238
6.5 Constructing a representation of a Lie algebra of character¬
istic 0 240
6.5.1 Calculating a series of extensions 240
6.5.2 The extension space 242
6.5.3 Extending a representation 245
6.5.4 Ado s theorem 249
6.6 The theorem of Iwasawa 253
6.7 Notes 255
7 Finitely presented Lie algebras 257
7.1 Free Lie algebras 258
7.2 Finitely presented Lie algebras 261
7.3 Grobner bases in free algebras 261
7.4 Constructing a basis of a finitely presented Lie algebra .... 266
7.5 Hall sets 271
7.6 Standard sequences 273
7.7 A Hall set provides a basis 276
7.8 Two examples of Hall orders 281
7.9 Reduction in L(X) 285
7.10 Grobner bases in free Lie algebras 292
7.11 Presentations of the simple Lie algebras of characteristic zero 300
7.12 Notes 308
xii
8 Representations of semisimple Lie algebras 311
8.1 The weights of a representation 312
8.2 Verma modules 315
8.3 Integral functions and the Weyl group 317
8.4 Finite dimensionality 321
8.5 On representing the weights 323
8.6 Computing orbits of the Weyl group 326
8.7 Calculating the weights 331
8.8 The multiplicity formula of Freudenthal 333
8.9 Modifying Freudenthal s formula 338
8.10 Weyl s formulas 341
8.11 The formulas of Kostant and Racah 349
8.12 Decomposing a tensor product 352
8.13 Branching rules 359
8.14 Notes 361
A On associative algebras 363
A.I Radical and semisimplicity 364
A.2 Algebras generated by a single element 367
A.3 Central idempotents 373
A.4 Notes 377
Bibliography 379
Index of Symbols 387
Index of Terminology 389
Index of Algorithms 393
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-09T18:42:57Z |
| institution | BVB |
| isbn | 0444501169 |
| language | English |
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| physical | XII, 393 S. Ill. |
| publishDate | 2000 |
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| publisher | Elsevier |
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| series | North-Holland mathematical library |
| series2 | North-Holland mathematical library |
| spelling | Graaf, Willem A. de Verfasser aut Lie algebras theory and algorithms Willem A. de Graaf 1. ed. Amsterdam [u.a.] Elsevier 2000 XII, 393 S. Ill. txt rdacontent n rdamedia nc rdacarrier North-Holland mathematical library 56 Algoritmen gtt Lie-algebra's gtt Lie algebras Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 s DE-604 North-Holland mathematical library 56 (DE-604)BV000005206 56 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009050772&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Graaf, Willem A. de Lie algebras theory and algorithms North-Holland mathematical library Algoritmen gtt Lie-algebra's gtt Lie algebras Lie-Algebra (DE-588)4130355-6 gnd |
| subject_GND | (DE-588)4130355-6 |
| title | Lie algebras theory and algorithms |
| title_auth | Lie algebras theory and algorithms |
| title_exact_search | Lie algebras theory and algorithms |
| title_full | Lie algebras theory and algorithms Willem A. de Graaf |
| title_fullStr | Lie algebras theory and algorithms Willem A. de Graaf |
| title_full_unstemmed | Lie algebras theory and algorithms Willem A. de Graaf |
| title_short | Lie algebras |
| title_sort | lie algebras theory and algorithms |
| title_sub | theory and algorithms |
| topic | Algoritmen gtt Lie-algebra's gtt Lie algebras Lie-Algebra (DE-588)4130355-6 gnd |
| topic_facet | Algoritmen Lie-algebra's Lie algebras Lie-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009050772&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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