Linear algebraic groups:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2009
|
| Edition: | Repr. of the 1998 second ed. |
| Series: | Modern Birkhäuser classics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Originally published as Vol. 9 in the series 'Progress in Mathematics' |
| Physical Description: | XII, 334 S. |
| ISBN: | 9780817648398 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804138555083063296 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
.............................................xiii
1.
Some Algebraic Geometry
...............................................1
1.1.
The Zariski topology
....................................................1
1.2.
Irreducibility of topological spaces
.......................................2
1.3. Affine
algebras
.........................................................4
1.4.
Regular functions, ringed spaces
.........................................6
1.5.
Products
.....................................!........................10
1.6.
Prevarieties and varieties
...............................................11
1.7.
Projective
varieties
............,........................................14
1.8.
Dimension
.............................................................16
1.9.
Some results on morphisms
.............................................17
Notes
.....................................................................20
2.
Linear Algebraic Groups, First Properties
...............................21
2.1.
Algebraic groups
......................................................21
2.2.
Some basic results
.....................................................25
2.3.
G-spaces
.............................................................28
2.4.
Jordan decomposition
..................................................31
2.5.
Recovering a group from its representations
..............................37
Notes
.....................................................................41
3.
Commutative Algebraic Groups
.........................................42
3.1.
Structure of commutative algebraic groups
...............................42
3.2.
Diagonalizable groups and tori
..........................................43
viii Contents
3.3.
Additive functions
......................................................
3.4.
Elementary unipotent groups
............................................
51
Notes
.....................................................................
56
4.
Derivations, Differentials, Lie Algebras
..................................57
4.1.
Derivations and tangent spaces
..........................................57
4.2.
Differentials, separability
...............................................
60
4.3.
Simple points
.........................................................66
4.4.
The Lie algebra of
alinear
algebraic group
...............................69
Notes
.....................................................................
77
5.
Topological Properties of Morphisras, Applications
.......................78
5.1.
Topological properties of morphisms
....................................78
5.2.
Finite morphisms, normality
.............................................82
5.3.
Homogeneous spaces
..................................................86
5.4.
Semi-simple automorphisms
............................................88
5.5.
Quotients
.............................................................91
Notes
.....................................................................97
6.
Parabolic Subgroups,
Borei
Subgroups, Solvable Groups
.................98
6.1.
Complete varieties
.....................................................98
6.2.
Parabolic subgroups and
Borei
subgroups
...............................101
6.3.
Connected solvable groups
.............................................104
6.4.
Maximal tori, further properties of
Borei
groups
........, 108
Notes
.......................................................
ИЗ
Contents ix
7. Weyl Group,
Roots, Root
Datum.......................................114
7.1.
The Weyl group
......................................................114
7.2.
Semi-simple groups of rank one
........................................117
7.3.
Reductive groups of semi-simple rank one
..............................120
7.4.
Root data
............................................................124
7.5.
Two roots
............................................................128
7.6.
The unipotent radical
.................................................130
Notes
....................................................................131
8.
Reductive Groups
......................................................132
8.1.
Structural properties of a reductive group
...............................132
8.2.
Borei
subgroups and systems of positive roots
...........................137
8.3.
The Bruhat decomposition
.............................................142
8.4.
Parabolic subgroups
..................................................146
8.5.
Geometric questions related to the Bruhat decomposition
.................149
Notes
....................................................................153
9.
The Isomorphism Theorem
.............................................154
9.1.
Two dimensional root systems
.........................................154
9.2.
The structure constants
................................................156
9.3.
The elements na
......................................................162
9.4.
A presentation of
G
...................................................164
9.5.
Uniqueness of structure constants
.......................................168
9.6.
The isomorphism theorem
.............................................170
Notes
....................................................................174
x
Contents
10.
The Existence Theorem
...............................................
I75
10.1.
Statement of the theorem, reduction
....................................175
10.2.
Simply laced root systems
............................................
I77
10.3.
Automorphisms, end of the proof of
10.1.1.............................181
Notes
....................................................................
184
11.
More Algebraic Geometry
............................................185
11.1.
F-structures on vector spaces
___......................................185
11.2.
F-varieties: density, criteria for ground fields
..........................191
11.3.
Forms
..............................................................196
11.4.
Restriction of the ground field
........................................198
Notes
....................................................................207
12.
F-groups: General Results
............................................208
12.1.
Field of definition of subgroups
.......................................208
12.2.
Complements on quotients
...........................................212
12.3.
Galois cohomology
..................................................216
12.4.
Restriction of the ground field
.........................................220
Notes
................................................................-.... 222
13.
F-tori
................................................................223
13.1.
Diagonalizabic groups over
F
.........................................223
13.2.
F-tori
...................................................... 225
13.3.
Tori in F-groups
....................................................227
13.4.
The groups
Ρ(λ)
....................................................233
....................................................................
236
Contents xi
14.
Solvable
F-groups....................................................237
14.1.
Generalities
.........................................................237
14.2.
Action of Ga on an
affine
variety, applications
..........................239
14.3.
F-split solvable groups
..............................................243
14.4.
Structural properties of solvable groups
................................248
Notes
....................................................................251
15.
F-reductive Groups
...................................................252
15.1.
Pseudo-parabolic F-subgroups
........................................252
15.2.
A fixed point theorem
................................................254
15.3.
The root datum of an F-reductive group
................................256
15.4.
The groups Uw
......................................................262
15.5.
The index
...........................................................265
Notes
....................................................................268
16.
Reductive F-groups
..................................................269
16.1.
Parabolic subgroups
.................................................269
16.2.
Indexed root data
....................................................271
16.3.
F-split groups
......................................................274
16.4.
The isomorphism theorem
............................................278
16.5.
Existence
...........................................................281
Notes
....................................................................284
17.
Classification
.........................................................285
17.1.
Type
Л„_і
...........................................................285
xii
Contents
17.2.
Types Bn and Cn
..... .................................!..............289
17.3.
Type Dn
............................................................293
17.4.
Exceptional groups, type
ог
..........................................300
17.5.
Indices for types
Fą
and Eg
...........................................302
17.6.
Descriptions for type
Fą
..............................................305
17.7.
Type £6
.............................................................310
17.8.
Type
Εη
............................................................312
17.9.
Trialitarian type
Dą
..................................................315
17.10.
Special fields
....................................................... 317
Notes
....................................................................319
Table of Indices
..........................................................320
Bibliography
.............................................................323
Index
....................................................................331
|
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| author_facet | Springer, Tonny A. 1926- |
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| callnumber-first | Q - Science |
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| callnumber-search | QA179.S67 1998 |
| callnumber-sort | QA 3179 S67 41998 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 220 SK 240 SK 260 |
| classification_tum | MAT 140f MAT 204f |
| ctrlnum | (OCoLC)316248063 (DE-599)BVBBV035268534 |
| dewey-full | 512/.55 512/.5521 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512/.55 512/.55 21 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Repr. of the 1998 second ed. |
| format | Book |
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| institution | BVB |
| isbn | 9780817648398 |
| language | English |
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| physical | XII, 334 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
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| publisher | Birkhäuser |
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| series2 | Modern Birkhäuser classics |
| spelling | Springer, Tonny A. 1926- Verfasser (DE-588)115799729 aut Linear algebraic groups T. A. Springer Repr. of the 1998 second ed. Boston [u.a.] Birkhäuser 2009 XII, 334 S. txt rdacontent n rdamedia nc rdacarrier Modern Birkhäuser classics Originally published as Vol. 9 in the series 'Progress in Mathematics' Linear algebraic groups Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 Algebraische Gruppe (DE-588)4001164-1 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017073949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Springer, Tonny A. 1926- Linear algebraic groups Linear algebraic groups Lineare algebraische Gruppe (DE-588)4295326-1 gnd Algebraische Gruppe (DE-588)4001164-1 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4295326-1 (DE-588)4001164-1 (DE-588)4001161-6 |
| title | Linear algebraic groups |
| title_auth | Linear algebraic groups |
| title_exact_search | Linear algebraic groups |
| title_full | Linear algebraic groups T. A. Springer |
| title_fullStr | Linear algebraic groups T. A. Springer |
| title_full_unstemmed | Linear algebraic groups T. A. Springer |
| title_short | Linear algebraic groups |
| title_sort | linear algebraic groups |
| topic | Linear algebraic groups Lineare algebraische Gruppe (DE-588)4295326-1 gnd Algebraische Gruppe (DE-588)4001164-1 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Linear algebraic groups Lineare algebraische Gruppe Algebraische Gruppe Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017073949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT springertonnya linearalgebraicgroups |