Lie algebras and algebraic groups:
Gespeichert in:
Hauptverfasser: | , |
---|---|
Format: | Buch |
Sprache: | English |
Veröffentlicht: |
Berlin [u. a.]
Springer
2005
|
Schriftenreihe: | Springer monographs in mathematics
|
Schlagworte: | |
Online-Zugang: | Inhaltsverzeichnis |
Beschreibung: | Auch als Internetausgabe |
Beschreibung: | XVI, 653 S. |
ISBN: | 9783540241706 3540241701 |
Internformat
MARC
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015 | |a 05,A26,0796 |2 dnb | ||
016 | 7 | |a 973670207 |2 DE-101 | |
020 | |a 9783540241706 |c Pp. : EUR 74.85 (freier Pr.), sfr 123.50 (freier Pr.) |9 978-3-540-24170-6 | ||
020 | |a 3540241701 |c Pp. : EUR 74.85 (freier Pr.), sfr 123.50 (freier Pr.) |9 3-540-24170-1 | ||
024 | 3 | |a 9783540241706 | |
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035 | |a (DE-599)BVBBV020002674 | ||
040 | |a DE-604 |b ger |e rakddb | ||
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044 | |a gw |c XA-DE-BE | ||
049 | |a DE-703 |a DE-91G |a DE-29T |a DE-384 |a DE-634 |a DE-20 |a DE-355 |a DE-19 |a DE-11 |a DE-83 | ||
050 | 0 | |a QA252.3 | |
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084 | |a SK 340 |0 (DE-625)143232: |2 rvk | ||
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084 | |a MAT 173f |2 stub | ||
084 | |a 510 |2 sdnb | ||
100 | 1 | |a Tauvel, Patrice |e Verfasser |4 aut | |
245 | 1 | 0 | |a Lie algebras and algebraic groups |c Patrice Tauvel ; Rupert W. T. Yu |
264 | 1 | |a Berlin [u. a.] |b Springer |c 2005 | |
300 | |a XVI, 653 S. | ||
336 | |b txt |2 rdacontent | ||
337 | |b n |2 rdamedia | ||
338 | |b nc |2 rdacarrier | ||
490 | 0 | |a Springer monographs in mathematics | |
500 | |a Auch als Internetausgabe | ||
650 | 4 | |a Algebraische Gruppe | |
650 | 4 | |a Lie-Algebra | |
650 | 4 | |a Lie algebras | |
650 | 4 | |a Linear algebraic groups | |
650 | 0 | 7 | |a Lie-Algebra |0 (DE-588)4130355-6 |2 gnd |9 rswk-swf |
650 | 0 | 7 | |a Algebraische Gruppe |0 (DE-588)4001164-1 |2 gnd |9 rswk-swf |
689 | 0 | 0 | |a Lie-Algebra |0 (DE-588)4130355-6 |D s |
689 | 0 | |5 DE-604 | |
689 | 1 | 0 | |a Algebraische Gruppe |0 (DE-588)4001164-1 |D s |
689 | 1 | |5 DE-604 | |
700 | 1 | |a Yu, Rupert W. T. |e Verfasser |4 aut | |
856 | 4 | 2 | |m Digitalisierung UB Augsburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013324371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
999 | |a oai:aleph.bib-bvb.de:BVB01-013324371 |
Datensatz im Suchindex
_version_ | 1804133554545754112 |
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adam_text | Contents
1
Results on topological spaces
.............................. 1
1.1
Irreducible sets and spaces
............................... 1
1.2
Dimension
............................................. 4
1.3
Noetherian spaces
....................................... 5
1.4
Constructible
sets
....................................... 6
1.5
Gluing topological spaces
................................ 8
2
Rings and modules
........................................ 11
2.1
Ideals
................................................. 11
2.2
Prime and maximal ideals
................................ 12
2.3
Rings of fractions and localization
......................... 13
2.4
Localizations of modules
................................. 17
2.5
Radical of an ideal
...................................... 18
2.6
Local rings
............................................. 19
2.7
Noetherian rings and modules
............................ 21
2.8
Derivations
............................................ 24
2.9
Module of differentials
................................... 25
3
Integral extensions
........................................ 31
3.1
Integral dependence
..................................... 31
3.2
Integrally closed domains
................................ 33
3.3
Extensions of prime ideals
............................... 35
4
Factorial rings
............................................. 39
4.1
Generalities
............................................ 39
4.2
Unique factorization
..................................... 41
4.3
Principal ideal domains and Euclidean domains
............. 43
4.4
Polynomials and factorial rings
........................... 45
4.5
Symmetric polynomials
.................................. 48
4.6
Resultant and discriminant
............................... 50
χ
Contents
5
Field extensions
........................................... 55
5.1
Extensions
............................................. 55
5.2
Algebraic and transcendental elements
..................... 56
5.3
Algebraic extensions
.................................... 56
5.4
Transcendence basis
..................................... 58
5.5
Norm and trace
......................................... 60
5.6
Theorem of the primitive element
......................... 62
5.7
Going Down Theorem
................................... 64
5.8
Fields and derivations
................................... 67
5.9
Conductor
............................................. 70
6
Finitely generated algebras
................................ 75
6.1
Dimension
............................................. 75
6.2
Noether s Normalization Theorem
......................... 76
6.3
Krull s Principal Ideal Theorem
........................... 81
6.4
Maximal ideals
......................................... 82
6.5
Zariski topology
........................................ 84
7
Gradings and nitrations
................................... 87
7.1
Graded rings and graded modules
......................... 87
7.2
Graded
submodules
..................................... 88
7.3
Applications
........................................... 90
7.4
Filtrations
............................................. 91
7.5
Grading associated to a filtration
......................... 92
8
Inductive limits
............................................ 95
8.1
Generalities
............................................ 95
8.2
Inductive systems of maps
............................... 96
8.3
Inductive systems of magmas, groups and rings
............. 98
8.4
An example
............................................100
8.5
Inductive systems of algebras
.............................100
9
Sheaves of functions
.......................................103
9.1
Sheaves
................................................103
9.2
Morphisms
.............................................104
9.3
Sheaf associated to a presheaf
............................106
9.4
Gluing
................................................109
9.5
Ringed space
...........................................110
10
Jordan decomposition and some basic results on groups
... 113
10.1
Jordan decomposition
...................................113
10.2
Generalities on groups
...................................117
10.3
Commutators
..........................................118
10.4
Solvable groups
.........................................120
10.5 Nilpotent
groups
........................................121
Contents
XI
10.6
Group actions
..........................................122
10.7
Generalities on representations
...........................123
10.8
Examples
..............................................126
11
Algebraic sets
.............................................131
11.1 Affine
algebraic sets
.....................................131
11.2
Zariski topology
........................................132
11.3
Regular functions
.......................................133
11.4
Morphisms
.............................................134
11.5
Examples of morphisms
..................................136
11.6
Abstract algebraic sets
..................................138
11.7
Principal open subsets
...................................140
11.8
Products of algebraic sets
................................142
12
Prevarieties and varieties
..................................147
12.1
Structure sheaf
.........................................147
12.2
Algebraic prevarieties
...................................149
12.3
Morphisms of prevarieties
................................151
12.4
Products of prevarieties
..................................152
12.5
Algebraic varieties
......................................155
12.6
Gluing
................................................158
12.7
Rational functions
......................................159
12.8
Local rings of a variety
..................................162
13
Projective
varieties
........................................167
13.1
Projective
spaces
.......................................167
13.2
Projective
spaces and varieties
............................168
13.3
Cones and
projective
varieties
............................171
13.4
Complete varieties
......................................176
13.5
Products
...............................................178
13.6
Grassmannian variety
...................................180
14
Dimension
.................................................183
14.1
Dimension of varieties
...................................183
14.2
Dimension and the number of equations
...................185
14.3
System of parameters
...................................187
14.4
Counterexamples
.......................................190
15
Morphisms and dimension
.................................191
15.1
Criterion of affmeness
...................................191
15.2 Affine
morphisms
.......................................193
15.3
Finite morphisms
.......................................194
15.4
Factorization and applications
............................197
15.5
Dimension of fibres of a morphism
........................199
15.6
An example
............................................203
XII Contents
16
Tangent
spaces
............................................205
16.1
A first approach
........................................205
16.2
Zariski tangent space
....................................207
16.3
Differential of a morphism
...............................209
16.4
Some lemmas
..........................................213
16.5
Smooth points
..........................................215
17
Normal varieties
...........................................219
17.1
Normal varieties
........................................219
17.2
Normalization
..........................................221
17.3
Products of normal varieties
..............................223
17.4
Properties of normal varieties
............................225
18
Root systems
..............................................233
18.1
Reflections
............................................ . 233
18.2
Root systems
...........................................235
18.3
Root systems and bilinear forms
..........................238
18.4
Passage to the field of real numbers
.......................239
18.5
Relations between two roots
..............................240
18.6
Examples of root systems
................................243
18.7
Base of a root system
...................................244
18.8
Weyl chambers
.........................................247
18.9
Highest root
...........................................250
18.10
Closed subsets of roots
..................................250
18.11
Weights
...............................................253
18.12
Graphs
................................................255
18.13
Dynkin diagrams
.......................................256
18.14
Classification of root systems
.............................259
19
Lie algebras
...............................................277
19.1
Generalities on Lie algebras
..............................277
19.2
Representations
.........................................279
19.3 Nilpotent
Lie algebras
...................................282
19.4
Solvable Lie algebras
....................................286
19.5
Radical and the largest
nilpotent
ideal
.....................289
19.6
Nilpotent radical
........................................291
19.7
Regular linear forms
.....................................292
19.8
Cartari subalgebras
.....................................294
20 Semisimple
and reductive Lie algebras
.....................299
20.1 Semisimple
Lie algebras
.................................299
20.2
Examples
..............................................301
20.3
Semisimplicity of representations
..........................302
20.4 Semisimple
and nilpotent elements
........................305
20.5
Reductive Lie algebras
...................................307
Contents
XIII
20.6
Results on the structure of
semisimple
Lie algebras
..........310
20.7
Subalgebras of
semisimple
Lie algebras
....................313
20.8
Parabolic subalgebras
...................................316
21
Algebraic groups
..........................................319
21.1
Generalities
............................................319
21.2
Subgroups and morphisms
...............................321
21.3
Connectedness
..........................................322
21.4
Actions of an algebraic group
.............................325
21.5
Modules
...............................................326
21.6
Group closure
..........................................327
22 Affine
algebraic groups
....................................331
22.1
Translations of functions
.................................331
22.2
Jordan decomposition
...................................333
22.3
Unipotent groups
.......................................335
22.4
Characters and weights
..................................338
22.5
Tori and diagonalizable groups
...........................340
22.6
Groups of dimension one
.................................345
23
Lie algebra of an algebraic group
..........................347
23.1
An associative algebra
...................................347
23.2
Lie algebras
............................................348
23.3
Examples
..............................................352
23.4
Computing differentials
..................................354
23.5
Adjoint representation
...................................359
23.6
Jordan decomposition
...................................362
24
Correspondence between groups and Lie algebras
..........365
24.1
Notations
..............................................365
24.2
An algebraic subgroup
...................................365
24.3
Invariants
..............................................368
24.4
Functorial properties
....................................372
24.5
Algebraic Lie subalgebras
................................375
24.6
A particular case
.......................................380
24.7
Examples
..............................................383
24.8
Algebraic adjoint group
..................................383
25
Homogeneous spaces and quotients
........................387
25.1
Homogeneous spaces
....................................387
25.2
Some remarks
..........................................389
25.3
Geometric quotients
.....................................391
25.4
Quotient by a subgroup
..................................393
25.5
The case of finite groups
.................................397
XIV Contents
26
Solvable groups
............................................401
26.1
Conjugacy classes
.......................................401
26.2
Actions of diagonalizable groups
..........................405
26.3
Fixed points
...........................................406
26.4
Properties of solvable groups
.............................407
26.5
Structure of solvable groups
..............................409
27
Reductive groups
..........................................413
27.1
Radical and unipotent radical
............................413
27.2 Semisimple
and reductive groups
..........................415
27.3
Representations
.........................................416
27.4
Fimteness properties
....................................420
27.5
Algebraic quotients
.....................................422
27.6
Characters
.............................................424
28
Borei
subgroups, parabolic subgroups,
Cartari
subgroups
. . 429
28.1
Borei
subgroups
........................................429
28.2
Theorems of density
.....................................432
28.3
Centralizers and tori
....................................434
28.4
Properties of parabolic subgroups
.........................435
28.5
Cartan subgroups
.......................................437
29
Cartan subalgebras,
Borei subalgebras
and parabolic
subalgebras
................................................441
29.1
Generalities
............................................441
29.2
Cartari
subalgebras
.....................................443
29.3
Applications to
semisimple
Lie algebras
....................446
29.4
Borei
subalgebras
.......................................447
29.5
Properties of parabolic subalgebras
........................450
29.6
More on reductive Lie algebras
...........................453
29.7
Other applications
......................................454
29.8
Maximal subalgebras
....................................456
30
Representations of
semisimple
Lie algebras
................459
30.1
Enveloping algebra
......................................459
30.2
Weights and primitive elements
...........................461
30.3
Finite-dimensional modules
..............................463
30.4
Verma modules
.........................................464
30.5
Results on existence and uniqueness
.......................467
30.6
A property of the Weyl group
............................469
31
Symmetric invariants
......................................471
31.1
Invariants of finite groups
................................471
31.2
Invariant polynomial functions
............................475
31.3
A free module
..........................................478
Contents XV
32
S-triples
...................................................481
32.1
Jacobson-Morosov Theorem
..............................481
32.2
Some lemmas
..........................................484
32.3
Conjugation of S-triples
.................................487
32.4
Characteristic
..........................................488
32.5
Regular and principal elements
...........................489
33
Polarizations
.......................................·......493
33.1
Definition of polarizations
................................493
33.2
Polarizations in the
semisimple
case
.......................494
33.3
A non-polarizable element
...............................497
33.4
Polarizable elements
.....................................499
33.5
Richardson s Theorem
...................................502
34
Results on orbits
..........................................507
34.1
Notations
..............................................507
34.2
Some lemmas
..........................................508
34.3
Generalities on orbits
....................................509
34.4
Minimal
nilpotent
orbit
.....................·............511
34.5
Subregular
nilpotent
orbit
...............................513
34.6
Dimension of
nilpotent
orbits
.............................517
34.7
Prehomogeneous spaces of parabolic type
..................518
35
Centralizers
...............................................521
35.1
Distinguished elements
..................................521
35.2
Distinguished parabolic subalgebras
,......................523
35.3
Double centralizers
......................................525
35.4
Normalizes
............................................528
35.5
A
semisimple
Lie
subalgebra
.............................530
35.6
Centralizers and regular elements
.........................533
36
er-root
systems
.............................................537
36.1
Definition
..............................................537
36.2
Restricted root systems
..................................539
36.3
Restriction of a root
.....................................544
37
Symmetric Lie algebras
....................................549
37.1
Primary subspaces
......................................549
37.2
Definition of symmetric Lie algebras
.......................553
37.3
Natural subalgebras
.....................................554
37.4
Cartan subspaces
.......................................555
37.5
The case of reductive Lie algebras
.........................557
37.6
Linear forms
...........................................559
XVI Contents
38 Semisimple
symmetric Lie algebras
........................561
38.1
Notations
..............................................561
38.2
Iwasawa decomposition
..................................562
38.3
Coroots
................................................565
38.4
Centralizers
............................................568
38.5
S-triples
...............................................570
38.6
Orbits
.................................................573
38.7
Symmetric invariants
....................................579
38.8
Double centralizers
......................................584
38.9
Normalizers
............................................588
38.10
Distinguished elements
..................................589
39
Sheets of Lie algebras
......................................593
39.1
Jordan classes
..........................................593
39.2
Topology of Jordan classes
...............................596
39.3
Sheets
.................................................601
39.4
Dixmier sheets
.........................................603
39.5
Jordan classes in the symmetric case
......................605
39.6
Sheets in the symmetric case
.............................608
40
Index and linear forms
.....................................611
40.1
Stable linear forms
......................................611
40.2
Index of a representation
................................615
40.3
Some useful inequalities
.................................616
40.4
Index and semi-direct products
...........................618
40.5 Heisenberg
algebras in
semisimple
Lie algebras
.............621
40.6
Index of Lie subalgebras of
Borei subalgebras
...............625
40.7
Seaweed Lie algebras
....................................629
40.8
An upper bound for the index
............................630
40.9
Cases where the bound is exact
...........................635
40.10
On the index of parabolic subalgebras
.....................638
References
.....................................................641
List of notations
...............................................645
Index
..........................................................647
|
any_adam_object | 1 |
author | Tauvel, Patrice Yu, Rupert W. T. |
author_facet | Tauvel, Patrice Yu, Rupert W. T. |
author_role | aut aut |
author_sort | Tauvel, Patrice |
author_variant | p t pt r w t y rwt rwty |
building | Verbundindex |
bvnumber | BV020002674 |
callnumber-first | Q - Science |
callnumber-label | QA252 |
callnumber-raw | QA252.3 |
callnumber-search | QA252.3 |
callnumber-sort | QA 3252.3 |
callnumber-subject | QA - Mathematics |
classification_rvk | SK 340 |
classification_tum | MAT 173f |
ctrlnum | (OCoLC)254250067 (DE-599)BVBBV020002674 |
dewey-full | 512.482 |
dewey-hundreds | 500 - Natural sciences and mathematics |
dewey-ones | 512 - Algebra |
dewey-raw | 512.482 |
dewey-search | 512.482 |
dewey-sort | 3512.482 |
dewey-tens | 510 - Mathematics |
discipline | Mathematik |
format | Book |
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id | DE-604.BV020002674 |
illustrated | Not Illustrated |
indexdate | 2024-07-09T20:10:34Z |
institution | BVB |
isbn | 9783540241706 3540241701 |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013324371 |
oclc_num | 254250067 |
open_access_boolean | |
owner | DE-703 DE-91G DE-BY-TUM DE-29T DE-384 DE-634 DE-20 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-83 |
owner_facet | DE-703 DE-91G DE-BY-TUM DE-29T DE-384 DE-634 DE-20 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-83 |
physical | XVI, 653 S. |
publishDate | 2005 |
publishDateSearch | 2005 |
publishDateSort | 2005 |
publisher | Springer |
record_format | marc |
series2 | Springer monographs in mathematics |
spelling | Tauvel, Patrice Verfasser aut Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu Berlin [u. a.] Springer 2005 XVI, 653 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Auch als Internetausgabe Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 s DE-604 Algebraische Gruppe (DE-588)4001164-1 s Yu, Rupert W. T. Verfasser aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013324371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
spellingShingle | Tauvel, Patrice Yu, Rupert W. T. Lie algebras and algebraic groups Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups Lie-Algebra (DE-588)4130355-6 gnd Algebraische Gruppe (DE-588)4001164-1 gnd |
subject_GND | (DE-588)4130355-6 (DE-588)4001164-1 |
title | Lie algebras and algebraic groups |
title_auth | Lie algebras and algebraic groups |
title_exact_search | Lie algebras and algebraic groups |
title_full | Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu |
title_fullStr | Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu |
title_full_unstemmed | Lie algebras and algebraic groups Patrice Tauvel ; Rupert W. T. Yu |
title_short | Lie algebras and algebraic groups |
title_sort | lie algebras and algebraic groups |
topic | Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups Lie-Algebra (DE-588)4130355-6 gnd Algebraische Gruppe (DE-588)4001164-1 gnd |
topic_facet | Algebraische Gruppe Lie-Algebra Lie algebras Linear algebraic groups |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013324371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
work_keys_str_mv | AT tauvelpatrice liealgebrasandalgebraicgroups AT yurupertwt liealgebrasandalgebraicgroups |