Lie groups and Lie algebras: [2] Chapters 4 - 6
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Ausgabe: | 1. softcover printing of the 1. Engl. ed. of 2002 |
| Schriftenreihe: | Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics
[7,2] |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 300 S. |
| ISBN: | 9783540691716 |
Internformat
MARC
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| 008 | 101217s2008 |||| 00||| eng d | ||
| 020 | |a 9783540691716 |9 978-3-540-69171-6 | ||
| 035 | |a (OCoLC)637488638 | ||
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| 100 | 1 | |a Bourbaki, Nicolas |d ca. 20. Jh.- |e Verfasser |0 (DE-588)140993142 |4 aut | |
| 240 | 1 | 0 | |a Groupes et algèbres de Lie |
| 245 | 1 | 0 | |a Lie groups and Lie algebras |n [2] |p Chapters 4 - 6 |c Nicolas Bourbaki |
| 250 | |a 1. softcover printing of the 1. Engl. ed. of 2002 | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XI, 300 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics |v [7,2] | |
| 490 | 0 | |a Bourbaki, Nicolas: Elements of mathematics |v ... | |
| 773 | 0 | 8 | |w (DE-604)BV004042069 |g 2 |
| 830 | 0 | |a Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics |v [7,2] |w (DE-604)BV002373127 |9 7,2 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-020787093 | ||
Datensatz im Suchindex
| _version_ | 1804143576418877440 |
|---|---|
| adam_text | CONTENTS
INTRODUCTION TO CHAPTERS IV, V AND VI
........
V
CONTENTS
................................................
VII
CHAPTER IV COXETER GROUPS AND TITS SYSTEMS
§ 1.
Coxeter Groups
......................................... 1
1.
Length and reduced decompositions
..................... 1
2.
Dihedral groups
...................................... 2
3.
First properties of Coxeter groups
...................... 4
4.
Reduced decompositions in a Coxeter group
.............. 5
5.
The exchange condition
............................... 7
6.
Characterisation of Coxeter groups
...................... 10
7.
Families of partitions
.................................. 10
8.
Subgroups of Coxeter groups
........................... 12
9.
Coxeter matrices and Coxeter graphs
.................... 13
§ 2.
Tits Systems
............................................ 15
1.
Definitions and first properties
......................... 15
2.
An example
.......................................... 17
3.
Decomposition of
G
into double cosets
.................. 18
4.
Relations with Coxeter systems
......................... 19
5.
Subgroups of
G
containing
В
........................... 21
6.
Parabolic subgroups
................................... 22
7.
The simplicity theorem
................................ 23
Appendix. Graphs
.......................................... 27
1.
Definitions
........................................... 27
2.
The connected components of a graph
................... 27
3.
Forests and trees
..................................... 29
Exercises for
§ 1............................................... 31
Exercises for
§ 2............................................... 44
VIII CONTENTS
CHAPTER V GROUPS GENERATED BY REFLECTIONS
§ 1. Hyperplanes,
chambers and facets
...................... 61
1.
Notations
............................................ 61
2.
Facets
............................................... 62
3.
Chambers
........................................... 64
4.
Walls and faces
....................................... 65
5.
Intersecting
hyperplanes ............................... 67
6.
Simplicial cones and simplices
.......................... 68
§ 2.
Reflections
.............................................. 70
1.
Pseudo-reflections
..................................... 70
2.
Reflections
........................................... 71
3.
Orthogonal reflections
................................. 73
4.
Orthogonal reflections in a euclidean
affine
space
......... 73
5.
Complements on plane rotations
........................ 74
§ 3.
Groups of displacements generated by reflections
....... 76
1.
Preliminary results
.................................... 77
2.
Relation with Coxeter systems
......................... 78
3.
Fundamental domain, stabilisers
........................ 79
4.
Coxeter matrix and Coxeter graph of
W
................. 81
5.
Systems of vectors with negative scalar products
.......... 82
6.
Finiteness theorems
................................... 84
7.
Decomposition of the linear representation of
W
on
Τ
..... 86
8.
Product decomposition of the
affine
space
E
.............. 88
9.
The structure of chambers
............................. 89
10.
Special points
........................................ 91
§ 4.
The geometric representation of a Coxeter group
....... 94
1.
The form associated to a Coxeter group
................. 94
2.
The plane E^- and the group generated by aa and as>
___ 95
3.
The group and representation associated to a Coxeter
matrix
.....................................
gg
4.
The contragredient representation
...................... 97
5.
Proof of lemma
1 ........................... 99
6.
The fundamental domain of
W
in the union of the
chambers
...................................
доі
7.
Irreducibility of the geometric representation of a Coxeter
group
............................................... 102
8.
Finiteness criterion
................................... 103
9.
The case in which BM is positive and degenerate
.......... 105
CONTENTS
IX
§ 5.
Invariants
in
the symmetric algebra
..................... 108
1.
Poincaré
series of graded algebras
....................... 108
2.
Invariants of a finite linear group: modular properties
..... 110
3.
Invariants of a finite linear group: ring-theoretic properties
. 112
4.
Anti-invariant elements
................................ 117
5.
Complements
........................................ 119
§ 6.
The Coxeter transformation
............................ 121
1.
Definition of Coxeter transformations
................... 121
2.
Eigenvalues of a Coxeter transformation: exponents
....... 122
Appendix: Complements on linear representations
.......... 129
Exercises for
§ 2............................................... 133
Exercises for
§ 3............................................... 134
Exercises for
§ 4............................................... 137
Exercises for
§ 5............................................... 144
Exercises for
§ 6............................................... 150
CHAPTER VI ROOT SYSTEMS
§ 1.
Root systems
........................................... 155
1.
Definition of a root system
............................. 155
2.
Direct sum of root systems
............................. 159
3.
Relation between two roots
............................ 160
4.
Reduced root systems
................................. 164
5.
Chambers and bases of root systems
.................... 166
6.
Positive roots
........................................ 168
7.
Closed sets of roots
................................... 173
8.
Highest root
......................................... 178
9.
Weights, radical weights
............................... 179
10.
Fundamental weights, dominant weights
................. 180
11.
Coxeter transformations
............................... 182
12.
Canonical bilinear form
................................ 184
§ 2.
Affine Weyl
group
...................................... 186
1.
Affine Weyl
group
.................................... 186
2.
Weights and special weights
............................ 187
3.
The
normaliser
of Wa
................................. 188
4.
Application: order of the Weyl group
.................... 190
5.
Root systems and groups generated by reflections
......... 191
X
CONTENTS
§ 3.
Exponential invariants
.................................. 194
1.
The group algebra of a free abelian group
................ 194
2.
Case of the group of weights: maximal terms
............. 195
3.
Anti-invariant elements
................................ 196
4.
Invariant elements
.................................... 199
§ 4.
Classification of root systems
........................... 201
1.
Finite Coxeter groups
................................. 201
2.
Dynkin graphs
....................................... 207
3. Affine Weyl
group and completed Dynkin graph
.......... 210
4.
Preliminaries to the construction of root systems
......... 212
5.
Systems of type B( (I
> 2) ............................. 214
6.
Systems of type
C¡
(I
> 2) ............................. 216
7.
Systems of type
A¡
(I
> 1) ............................. 217
8.
Systems of type Dj (I
> 3) ............................. 220
9.
System of type F4
.................................... 223
10.
System of type E8
.................................... 225
11.
System of type E7
.................................... 227
12.
System of type E6
.................................... 229
13.
System of type G2
.................................... 231
14.
Irreducible non-reduced root systems
.................... 233
Exercises for
§ 1............................................... 235
Exercises for
§ 2............................................... 240
Exercises for
§ 3............................................... 241
Exercises for
§4............................................... 242
HISTORICAL NOTE (Chapters IV, V and VI)
............ 249
BIBLIOGRAPHY
.......................................... 255
INDEX OF NOTATION
.................................... 259
INDEX OF TERMINOLOGY
.............................. 261
PLATE I. Systems of type
A¿
(I
> 1) ........................ 265
PLATE II. Systems of type Bj (I
> 2) ....................... 267
PLATE III. Systems of type C,
(¿
> 2) ...................... 269
PLATE IV. Systems of type
D, (Í
> 3) ...................... 271
PLATE V. System of type Ee
............................... 275
PLATE VI. System of type E7
.............................. 279
PLATE VH. System of type E8
............................. 283
CONTENTS
XI
PLATE
VIII.
System
of type F4
............................ 287
PLATE
IX.
System
of type G2
............................. 289
PLATE
X. Irreducible systems of rank
2 .................... 291
Summary of the principal properties of root systems
....... 293
|
| any_adam_object | 1 |
| author | Bourbaki, Nicolas ca. 20. Jh.- |
| author_GND | (DE-588)140993142 |
| author_facet | Bourbaki, Nicolas ca. 20. Jh.- |
| author_role | aut |
| author_sort | Bourbaki, Nicolas ca. 20. Jh.- |
| author_variant | n b nb |
| building | Verbundindex |
| bvnumber | BV036871527 |
| ctrlnum | (OCoLC)637488638 (DE-599)BVBBV036871527 |
| dewey-full | 512.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.55 |
| dewey-search | 512.55 |
| dewey-sort | 3512.55 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. softcover printing of the 1. Engl. ed. of 2002 |
| format | Book |
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| id | DE-604.BV036871527 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:49:51Z |
| institution | BVB |
| isbn | 9783540691716 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020787093 |
| oclc_num | 637488638 |
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| owner_facet | DE-703 DE-355 DE-BY-UBR DE-19 DE-BY-UBM |
| physical | XI, 300 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics |
| series2 | Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics Bourbaki, Nicolas: Elements of mathematics |
| spelling | Bourbaki, Nicolas ca. 20. Jh.- Verfasser (DE-588)140993142 aut Groupes et algèbres de Lie Lie groups and Lie algebras [2] Chapters 4 - 6 Nicolas Bourbaki 1. softcover printing of the 1. Engl. ed. of 2002 Berlin [u.a.] Springer 2008 XI, 300 S. txt rdacontent n rdamedia nc rdacarrier Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics [7,2] Bourbaki, Nicolas: Elements of mathematics ... (DE-604)BV004042069 2 Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics [7,2] (DE-604)BV002373127 7,2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020787093&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bourbaki, Nicolas ca. 20. Jh.- Lie groups and Lie algebras Bourbaki, Nicolas, ca. 20. Jh.-: Elements of mathematics |
| title | Lie groups and Lie algebras |
| title_alt | Groupes et algèbres de Lie |
| title_auth | Lie groups and Lie algebras |
| title_exact_search | Lie groups and Lie algebras |
| title_full | Lie groups and Lie algebras [2] Chapters 4 - 6 Nicolas Bourbaki |
| title_fullStr | Lie groups and Lie algebras [2] Chapters 4 - 6 Nicolas Bourbaki |
| title_full_unstemmed | Lie groups and Lie algebras [2] Chapters 4 - 6 Nicolas Bourbaki |
| title_short | Lie groups and Lie algebras |
| title_sort | lie groups and lie algebras chapters 4 6 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020787093&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004042069 (DE-604)BV002373127 |
| work_keys_str_mv | AT bourbakinicolas groupesetalgebresdelie AT bourbakinicolas liegroupsandliealgebras2 |