Introduction to Lie algebras and representation theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u. a.]
Springer
[2003]
|
| Ausgabe: | 10.[Dr.] |
| Schriftenreihe: | Graduate texts in mathematics
9 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 172 S. |
| ISBN: | 0387900535 |
Internformat
MARC
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| 245 | 1 | 0 | |a Introduction to Lie algebras and representation theory |c James E. Humphreys |
| 250 | |a 10.[Dr.] | ||
| 264 | 1 | |a New York [u. a.] |b Springer |c [2003] | |
| 300 | |a XII, 172 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 9 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-020126239 | ||
Datensatz im Suchindex
| _version_ | 1804142692731453440 |
|---|---|
| adam_text | Table of Contente
PREFACE
...........
vii
I.
BASIC
CONCEPTS
.......... 1
1.
Definitions and first examples
....... 1
1.1
The notion of Lie algebra
...... 1
1.2
Linear Lie algebras
....... 2
1.3
Lie algebras of derivations
...... 4
1.4
Abstract Lie algebras
....... 4
2.
Ideals and homomorphisms
....... 6
2.1
Ideals
......... 6
2.2
Homomorphisms and representations
.... 7
2.3
Automorphisms
........ 8
3.
Solvable and
nilpotent
Lie algebras
...... 10
3.1
Solvability
......... 10
3.2
Nilpotency
......... 11
3.3
Proof of Engel s Theorem
...... 12
II.
SEMISIMPLE
LIE ALGEBRAS
...... 15
4.
Theorems of Lie and Cartan
....... 15
4.1
Lie s Theorem
........ 15
4.2
Jordan-Chevalley decomposition
..... 17
4.3
Cartan s Criterion
........ 19
5.
Killing form
.......... 21
5.1
Criterion for semisimplicity
...... 21
5.2
Simple ideals of
L
........ 22
5.3
Inner derivations
........ 23
5.4
Abstract Jordan decomposition
..... 24
6.
Complete reducibility of representations
..... 25
6.1
Modules
....... 25
6.2 Casimir
element of a representation
.... 27
6.3
Weyľs
Theorem
........ 28
6.4
Preservation of Jordan decomposition
.... 29
7.
Representations of si
(2,
F)
....... 31
7.1
Weights and maximal vectors
...... 31
7.2
Classification of irreducible modules
.... 32
8.
Root space decomposition
....... 35
8.1
Maximal
toral
subalgebras and roots
.... 35
8.2
Centralizer of
H
......... 36
8.3
Orthogonality properties
...... 37
8.4
Integrality properties
....... 38
8.5
Rationality properties. Summary
..... 39
ix
x
Table of Contents
III. ROOT SYSTEMS
......... 42
9.
Axiomatics
.......... 42
9.1
Reflections in a euclidean space
..... 42
9.2
Root systems
........ 42
9.3
Examples
......... 43
9.4
Pairs of roots
........ 44
10.
Simple roots and Weyl group
....... 47
10.1
Bases and Weyl chambers
...... 47
10.2
Lemmas on simple roots
......
SO
10.3
The Weyl group
........ 51
10.4
Irreducible root systems
...... 52
11.
Classification
......... 55
11.1
Cartan matrix of
Φ
....... 55
11.2
Coxeter graphs and Dynkin diagrams
.... 56
11.3
Irreducible components
....... 57
11.4
Classification theorem
....... 57
12.
Construction of root systems and automorphisms
... 63
12.1
Construction of types A-G
...... 63
12.2
Automorphisms of
Φ
....... 65
13.
Abstract theory of weights
....... 67
13.1
Weights
......... 67
13.2
Dominant weights
....... 68
13.3
The weight
б
........ 70
13.4
Saturated sets of weights
...... 70
IV. ISOMORPHISM AND CONJUGACY THEOREMS
. . 73
14.
Isomorphism theorem
........ 73
14.1
Reduction to the simple case
...... 73
14.2
Isomorphism theorem
....... 74
14.3
Automorphisms
........ 76
15.
Cartan subalgebras
........ 78
15.1
Decomposition of
L
relative to
ad x
.... 78
15.2 Engel
subalgebras
....... 79
15.3
Cartan subalgebras
....... 80
15.4
Functorial properties
....... 81
16.
Conjugacy theorems
........ 81
16.1
The group <f(L)
........ 82
16.2
Conjugacy of CSA s (solvable case)
.... 82
16.3
Borei
subalgebras
........ 83
16.4
Conjugacy of
Borei
subalgebras
..... 84
16.5
Automorphism groups
....... 87
V. EXISTENCE THEOREM
....... 89
17.
Universal enveloping algebras
....... 89
17.1
Tensor and symmetric algebras
..... 89
17.2
Construction of U(L)
....... 90
17.3
PBW Theorem and consequences
..... 91
17.4
Proof of PBW Theorem
...... 93
Table
of Contents
xi
17.5
Free Lie algebras
........ 94
18.
Generators and relations
....... 95
18.1
Relations satisfied by
L
....... 96
18.2
Consequences of (S1HS3)
...... 96
18.3
Serre s Theorem
........ 98
18.4
Application: Existence and uniqueness theorems
. . 101
19.
The simple algebras
........ 102
19.1
Criterion for semisimplicity
...... 102
19.2
The classical algebras
....... 102
19.3
The algebra G2
........ 103
VI. REPRESENTATION THEORY
...... 107
20.
Weights and maximal vectors
....... 107
20.1
Weight spaces
........ 107
20.2
Standard cyclic modules
...... 108
20.3
Existence and uniqueness theorems
.... 109
21.
Finite dimensional modules
....... 112
21.1
Necessary condition for finite dimension
. . . . 112
21.2
Sufficient condition for finite dimension
. . . .113
21.3
Weight strings and weight diagrams
. . . .114
21.4
Generators and relations for V( )
..... 115
22.
Multiplicity formula
........ 117
22.1
A universal
Casimir
element
...... 118
22.2
Traces on weight spaces
...... 119
22.3
Freudenthaľs
formula
....... 121
22.4
Examples
......... 123
22.5
Formal characters.
....... 124
23.
Characters
.......... 126
23.1
Invariant polynomial functions
..... 126
23.2
Standard cyclic modules and characters
. . . .128
23.3
Harish-Chandra s Theorem
...... 130
Appendix
......... 132
24.
Formulas of Weyl, Kostant, and Steinberg
. . . .135
24.1
Some functions on H*
....... 135
24.2
Kostant s multiplicity formula
..... 136
24.3
Weyl s formulas
........ 138
24.4
Steinberg s formula
....... 140
Appendix
143
VII.
CHEVALLEY ALGEBRAS AND GROUPS
.... 145
25.
Chevalley basis of
L
........
І45
25.1
Pairs of roots
........ 145
25.2
Existence of a Chevalley basis
..... 145
25.3
Uniqueness questions
....... 146
25.4
Reduction modulo a prime
...... 148
25.5
Construction of Chevalley groups (adjoint type)
. . 149
26.
Kostant s Theorem
........ 151
26.1
A combinatorial lemma
....... 152
Table
of Contents
26.2
Special case:
st
(2,
F)
....... 153
26.3
Lemmas on commutation
...... 154
26.4
Proof of Kostant s Theorem
...... 156
27.
Admissible lattices
......... 157
27.1
Existence of admissible lattices
..... 157
27.2
Stabilizer of an admissible lattice
..... 159
27.3
Variation of admissible lattice
..... 161
27.4
Passage to an arbitrary field
...... 162
27.5
Survey of related results
...... 163
References
........... 165
Afterword
(1994).......... 167
Index of Terminology
......... 169
Index of Symbols
......... 172
|
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| author | Humphreys, James E. 1939-2020 |
| author_GND | (DE-588)108120848 |
| author_facet | Humphreys, James E. 1939-2020 |
| author_role | aut |
| author_sort | Humphreys, James E. 1939-2020 |
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| bvnumber | BV025517123 |
| classification_rvk | SK 340 |
| classification_tum | MAT 173f |
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| discipline | Mathematik |
| edition | 10.[Dr.] |
| format | Book |
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| id | DE-604.BV025517123 |
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| indexdate | 2024-07-09T22:35:49Z |
| institution | BVB |
| isbn | 0387900535 |
| language | English |
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| physical | XII, 172 S. |
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| series | Graduate texts in mathematics |
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| spelling | Humphreys, James E. 1939-2020 Verfasser (DE-588)108120848 aut Introduction to Lie algebras and representation theory James E. Humphreys 10.[Dr.] New York [u. a.] Springer [2003] XII, 172 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 9 Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Darstellungstheorie (DE-588)4148816-7 s Lie-Algebra (DE-588)4130355-6 s DE-604 Graduate texts in mathematics 9 (DE-604)BV000000067 9 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020126239&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Humphreys, James E. 1939-2020 Introduction to Lie algebras and representation theory Graduate texts in mathematics Darstellungstheorie (DE-588)4148816-7 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| subject_GND | (DE-588)4148816-7 (DE-588)4130355-6 (DE-588)4151278-9 |
| title | Introduction to Lie algebras and representation theory |
| title_auth | Introduction to Lie algebras and representation theory |
| title_exact_search | Introduction to Lie algebras and representation theory |
| title_full | Introduction to Lie algebras and representation theory James E. Humphreys |
| title_fullStr | Introduction to Lie algebras and representation theory James E. Humphreys |
| title_full_unstemmed | Introduction to Lie algebras and representation theory James E. Humphreys |
| title_short | Introduction to Lie algebras and representation theory |
| title_sort | introduction to lie algebras and representation theory |
| topic | Darstellungstheorie (DE-588)4148816-7 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| topic_facet | Darstellungstheorie Lie-Algebra Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020126239&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT humphreysjamese introductiontoliealgebrasandrepresentationtheory |