Structure and geometry of Lie groups:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; Dordrecht ; Heidelberg ; London
Springer
[2012]
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | x, 744 Seiten |
| ISBN: | 9780387847931 9781489990068 |
Staff View
MARC
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| 100 | 1 | |a Hilgert, Joachim |d 1958- |e Verfasser |0 (DE-588)1017959528 |4 aut | |
| 245 | 1 | 0 | |a Structure and geometry of Lie groups |c Joachim Hilgert, Karl-Hermann Neeb |
| 264 | 1 | |a New York ; Dordrecht ; Heidelberg ; London |b Springer |c [2012] | |
| 264 | 4 | |c © 2012 | |
| 300 | |a x, 744 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics | |
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Record in the Search Index
| _version_ | 1804148749866369024 |
|---|---|
| adam_text | Table
of
Contents
1
Introduction
.............................................. 1
1.1
Teaching Suggestions
................................... 6
1.2
Fundamental Notation
.................................. 6
Part I Matrix Groups
2
Concrete Matrix Groups
.................................. 9
2.1
The General Linear Group
............................... 9
2.2
Groups and Geometry
.................................. 21
2.3
Quatemionic Matrix Groups
............................. 33
3
The Matrix Exponential Function
........................ 39
3.1
Smooth Functions Defined by Power Series
................ 39
3.2
Elementary Properties of the Exponential Function
......... 45
3.3
The Logarithm Function
................................ 51
3.4
The Baker-Campbell-Dynkin-Hausdorff Formula
........... 54
4
Linear Lie Groups
........................................ 61
4.1
The Lie Algebra of a Linear Lie Group
.................... 61
4.2
Calculating Lie Algebras of Linear Lie Groups
............. 69
4.3
Polar Decomposition of Certain Algebraic Lie Groups
....... 72
Part II Lie Algebras
5
Elementary Structure Theory of Lie Algebras
............. 79
5.1
Basic Concepts
......................................... 79
5.2 Nilpotent
Lie Algebras
.................................. 91
5.3
The Jordan Decomposition
.............................. 95
5.4
Solvable Lie Algebras
...................................103
5.5 Semisimple
Lie Algebras
.................................112
5.6
The Theorems of
Levi
and Malcev
........................122
5.7
Reductive Lie Algebras
..................................129
viii Table of
Contents
6
Root Decomposition
......................................133
6.1
Cartan Subalgebras
..................................... 133
6.2
The Classification of
Simple
sl2(K)-Modułes
............... 144
6.3
Root Decompositions
of
Semisimple
Lie Algebras
........... 150
6.4
Abstract Root Systems and Their Weyl Groups
............ 158
7
Representation Theory of Lie Algebras
...................167
7.1
The Universal Enveloping Algebra
........................168
7.2
Generators and Relations for
Semisimple
Lie Algebras
.......173
7.3
Highest Weight Representations
..........................181
7.4
Ado s Theorem
.........................................189
7.5
Lie Algebra Cohomology
................................194
7.6
General Extensions of Lie Algebras
.......................214
Part III Manifolds and Lie
Groupe
8
Smooth Manifolds
........................................229
8.1
Smooth Maps in Several Variables
........................230
8.2
Smooth Manifolds and Smooth Maps
.....................234
8.3
The Tangent Bundle
....................................250
8.4
Vector Fields
..........................................261
8.5
Integral Curves and Local Flows
..........................271
8.6
Submanifolds
..........................................280
9
Basic Lie Theory
.........................................285
9.1
Lie Groups and Their Lie Algebras
.......................285
9.2
The Exponential Function of a Lie Group
.................296
9.3
Closed Subgroups of Lie Groups and Their Lie Algebras
.....317
9.4
Constructing Lie Group Structures on Groups
..............326
9.5
Covering Theory for Lie Groups
..........................335
9.6
Arcwise Connected Subgroups and Initial Subgroups
........347
10
Smooth Actions of Lie Groups
............................359
10.1
Homogeneous Spaces
....................................359
10.2
Frame Bundles
......................................... 370
10.3
Integration on Manifolds
................................ 388
10.4
Invariant Integration
....................................407
10.5
Integrating Lie Algebras of Vector Fields
.................. 419
Part IV Structure Theory of Lie Groups
11
Normal Subgroups,
Nilpotent
and Solvable Lie Groups
... 429
11.1
Normalizers, Normal Subgroups, and Semidirect Products
... 430
Table
of
Contents
ix
11.2
Commutators,
Nilpotent
and Solvable Lie Groups
..........442
11.3
The Automorphism Group of a Lie Group
.................452
12
Compact Lie Groups
.....................................459
12.1
Lie Groups with Compact Lie Algebra
....................459
12.2
Maximal Tori in Compact Lie Groups
.....................471
12.3
Linearity of Compact Lie Groups
.........................483
12.4
Topological Properties
..................................488
13 Semisimple
Lie Groups
...................................505
13.1
Cartan Decompositions
.................................505
13.2
Compact Real Forms
...................................512
13.3
The Iwasawa Decomposition
.............................522
14
General Structure Theory
................................529
14.1
Maximal Compact Subgroups
............................529
14.2
The Center of a Connected Lie Group
.....................533
14.3
The Manifold Splitting Theorem
.........................539
14.4
The Exponential Function of Solvable Groups
..............547
14.5
Dense Integral Subgroups
................................552
14.6
Appendix: Finitely Generated Abelian Groups
.............559
15
Complex Lie Groups
......................................565
15.1
The Universal Complexification
..........................566
15.2
Linearly Complex Reductive Lie Groups
...................571
15.3
Complex Abelian Lie Groups
............................579
15.4
The Automorphism Group of a Complex Lie Group
.........586
16
Linearity of Lie Groups
...................................587
16.1
Linearly Real Reductive Lie Groups
......................587
16.2
The Existence of Faithful Finite-Dimensional Representations
591
16.3
Linearity of Complex Lie Groups
.........................598
17
Classical Lie Groups
......................................605
17.1
Compact Classical Groups
...............................605
17.2
Noncompact Classical Groups
............................610
17.3
More Spin Groups
......................................616
17.4
Conformai
Groups
......................................622
18
Nonconnected Lie Groups
................................629
18.1
Extensions of Discrete Groups by Lie Groups
..............629
18.2
Coverings of Nonconnected Lie Groups
....................642
18.3
Appendix: Group Cohomology
...........................645
Table
of Contents
Part V Appendices
A Basic Covering Theory
.....................................653
A.I The Fundamental Group
................................653
A.
2
Coverings
.............................................657
В
Some Multilinear Algebra
..................................665
B.I Tensor Products and Tensor Algebra
......................665
B.2 Symmetric and Exterior Products
........................671
B.3 Clifford Algebras, Pin and Spin Groups
...................686
С
Some Functional Analysis
..................................701
C.I Bounded Operators
.....................................701
C.2 Hubert Spaces
.........................................703
C.3 Compact Symmetric Operators on Hubert Spaces
...........704
D
Hints to Exercises
..........................................707
References
....................................................717
Index
.........................................................723
|
| any_adam_object | 1 |
| author | Hilgert, Joachim 1958- Neeb, Karl-Hermann 1964- |
| author_GND | (DE-588)1017959528 (DE-588)112163920 |
| author_facet | Hilgert, Joachim 1958- Neeb, Karl-Hermann 1964- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.482 |
| dewey-search | 512/.482 |
| dewey-sort | 3512 3482 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T00:12:03Z |
| institution | BVB |
| isbn | 9780387847931 9781489990068 |
| language | English |
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| physical | x, 744 Seiten |
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| spelling | Hilgert, Joachim 1958- Verfasser (DE-588)1017959528 aut Structure and geometry of Lie groups Joachim Hilgert, Karl-Hermann Neeb New York ; Dordrecht ; Heidelberg ; London Springer [2012] © 2012 x, 744 Seiten txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s DE-604 Neeb, Karl-Hermann 1964- Verfasser (DE-588)112163920 aut Erscheint auch als Online-Ausgabe 978-0-387-84794-8 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024676001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hilgert, Joachim 1958- Neeb, Karl-Hermann 1964- Structure and geometry of Lie groups Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4035695-4 |
| title | Structure and geometry of Lie groups |
| title_auth | Structure and geometry of Lie groups |
| title_exact_search | Structure and geometry of Lie groups |
| title_full | Structure and geometry of Lie groups Joachim Hilgert, Karl-Hermann Neeb |
| title_fullStr | Structure and geometry of Lie groups Joachim Hilgert, Karl-Hermann Neeb |
| title_full_unstemmed | Structure and geometry of Lie groups Joachim Hilgert, Karl-Hermann Neeb |
| title_short | Structure and geometry of Lie groups |
| title_sort | structure and geometry of lie groups |
| topic | Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Lie-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024676001&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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