Basic Lie theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2007
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 427 S. graph. Darst. |
| ISBN: | 9812706984 9812706992 9789812706980 9789812706997 |
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MARC
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| 245 | 1 | 0 | |a Basic Lie theory |c Hossein Abbaspour ; Martin Moskowitz |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2007 | |
| 300 | |a XV, 427 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804136561136107520 |
|---|---|
| adam_text | Contents
Preface
and Acknowledgments
xi
Notations
xv
0
Lie Groups and Lie Algebras; Introduction
1
0.1
Topological Groups
.................... . 1
0.2
Lie Groups
......................... 6
0.3
Covering Maps and Groups
................ 10
0.4
Group Actions and Homogeneous Spaces
......... 15
0.5
Lie Algebras
......................... 25
1
Lie Groups
31
1.1
Elementary Properties of a Lie Group
.......... 31
1.2
Taylor s Theorem and the Coefficients of expX
exp
Y .
. 39
1.3
Correspondence between Lie Subgroups and Subalgebras
45
1.4
The Functorial Relationship
................ 48
1.5
The Topology of Compact Classical Groups
....... 60
1.6
The Iwasawa Decompositions for GL(n,R) and GL(n,C)
67
1.7
The Baker-Campbell-Hausdorff Formula
......... 69
2 Haar
Measure and its Applications
89
2.1 Haar
Measure on a Locally Compact Group
....... 89
2.2
Properties of the Modular Function
............ 100
2.3
Invariant Measures on Homogeneous Spaces
....... 101
2.4
Compact or Finite Volume Quotients
........... 106
2.5
Applications
......................... 112
2.6
Compact
Linear
Groups and Hubert s 14th Problem
. . 121
3
Elements of the Theory of Lie Algebras
127
3.1
Basics of Lie Algebras
................... 127
3.1.1
Ideals and Related Concepts
........... 127
3.1.2 Semisimple
Lie Algebras
.............. 138
3.1.3
Complete Lie Algebras
............... 139
3.1.4
Lie Algebra Representations
............ 140
3.1.5
The Irreducible Representations of sl(2,k)
.... 142
3.1.6
Invariant Forms
................... 145
3.1.7
Complex, Real and Rational Lie Algebras
.... 147
3.2 Engel
and Lie s Theorems
................. 150
3.2.1
Engel s Theorem
.................. 150
3.2.2
Lie s Theorem
................... 153
3.3
Cartan s Criterion and
Semisimple
Lie Algebras
..... 157
3.3.1
Some Algebra
.................... 157
3.3.2
Cartan s Solvability Criterion
........... 162
3.3.3
Explicit Computations of Killing Form
...... 166
3.3.4
Further Results on Jordan Decomposition
.... 170
3.4
Weyl s Theorem on Complete Reducibility
....... 173
3.5
Levi-Malcev Decomposition
................ 179
3.6
Reductive Lie Algebras
................... 187
3.7
The Jacobson-Morozov Theorem
............. 192
3.8
Low Dimensional Lie Algebras over
R
and
С
...... 198
3.9
Real Lie Algebras of Compact Type
........... 202
4
The Structure of Compact Connected Lie Groups
207
4.1
Introduction
......................... 207
4.2
Maximal Tori in Compact Lie Groups
.......... 208
4.3
Maximal Tori in Compact Connected Lie Groups
.... 210
4.4
The Weyl Group
...................... 217
4.5
What Goes Wrong If
G
is Not Compact
......... 221
5
Representations of Compact Lie Groups
223
5.1
Introduction
......................... 224
5.2
The
Schur
Orthogonality Relations
............ 226
5.3
Compact Integral Operators on a Hubert Space
..... 228
5.4
The Peter-Weyl Theorem and its Consequences
..... 234
5.5
Characters and Central Functions
............. 243
5.6
Induced Representations
.................. 250
5.7
Some Consequences of Frobenius Reciprocity
...... 255
6
Symmetric Spaces of Non-compact Type
261
6.1
Introduction
......................... 261
6.2
The Polar Decomposition
................. 264
6.3
The Cartan Decomposition
................ 267
6.4
The Case of Hyperbolic Space and the
Lorentz
Group
. 274
6.5
The G-invariant Metric Geometry of
F
.......... 278
6.6
The Conjugacy of Maximal Compact Subgroups
.... 289
6.7
The Rank and Two-Point Homogeneous Spaces
..... 294
6.8
The Disk Model for Spaces of Rank
1 .......... 299
6.9
Exponentiality of Certain Rank
1
Groups
........ 304
7 Semisimple
Lie Algebras and Lie Groups
313
7.1
Root and Weight Space Decompositions
......... 313
7.2
Cartan Subalgebras
..................... 316
7.3
Roots of Complex
Semisimple
Lie Algebras
....... 323
7.4
Real Forms of Complex
Semisimple
Lie Algebras
.... 337
7.5
The Iwasawa Decomposition
................ 343
8
Lattices in Lie Groups
355
8.1
Lattices in Euclidean Space
................ 355
8.2
GL(n,R)/GL(n,Z) and SL(n,R)/SL(n,Z)
....... 360
8.3
Lattices in More General Groups
............. 371
8.4
Fundamental Domains
................... 374
9
Density Results for Cofinite Volume Subgroups
377
9.1
Introduction
......................... 377
9.2
A Density Theorem for Cofinite Volume Subgroups
. . . 379
9.3
Consequences and Extensions of the Density Theorem
389
A Vector Fields
397
В
The
Kronecker
Approximation Theorem
403
С
Properly Discontinuous Actions
407
D
The Analyticity of Smooth Lie Groups
411
Bibliography
413
Index
421
|
| adam_txt |
Contents
Preface
and Acknowledgments
xi
Notations
xv
0
Lie Groups and Lie Algebras; Introduction
1
0.1
Topological Groups
. . 1
0.2
Lie Groups
. 6
0.3
Covering Maps and Groups
. 10
0.4
Group Actions and Homogeneous Spaces
. 15
0.5
Lie Algebras
. 25
1
Lie Groups
31
1.1
Elementary Properties of a Lie Group
. 31
1.2
Taylor's Theorem and the Coefficients of expX
exp
Y .
. 39
1.3
Correspondence between Lie Subgroups and Subalgebras
45
1.4
The Functorial Relationship
. 48
1.5
The Topology of Compact Classical Groups
. 60
1.6
The Iwasawa Decompositions for GL(n,R) and GL(n,C)
67
1.7
The Baker-Campbell-Hausdorff Formula
. 69
2 Haar
Measure and its Applications
89
2.1 Haar
Measure on a Locally Compact Group
. 89
2.2
Properties of the Modular Function
. 100
2.3
Invariant Measures on Homogeneous Spaces
. 101
2.4
Compact or Finite Volume Quotients
. 106
2.5
Applications
. 112
2.6
Compact
Linear
Groups and Hubert's 14th Problem
. . 121
3
Elements of the Theory of Lie Algebras
127
3.1
Basics of Lie Algebras
. 127
3.1.1
Ideals and Related Concepts
. 127
3.1.2 Semisimple
Lie Algebras
. 138
3.1.3
Complete Lie Algebras
. 139
3.1.4
Lie Algebra Representations
. 140
3.1.5
The Irreducible Representations of sl(2,k)
. 142
3.1.6
Invariant Forms
. 145
3.1.7
Complex, Real and Rational Lie Algebras
. 147
3.2 Engel
and Lie's Theorems
. 150
3.2.1
Engel's Theorem
. 150
3.2.2
Lie's Theorem
. 153
3.3
Cartan's Criterion and
Semisimple
Lie Algebras
. 157
3.3.1
Some Algebra
. 157
3.3.2
Cartan's Solvability Criterion
. 162
3.3.3
Explicit Computations of Killing Form
. 166
3.3.4
Further Results on Jordan Decomposition
. 170
3.4
Weyl's Theorem on Complete Reducibility
. 173
3.5
Levi-Malcev Decomposition
. 179
3.6
Reductive Lie Algebras
. 187
3.7
The Jacobson-Morozov Theorem
. 192
3.8
Low Dimensional Lie Algebras over
R
and
С
. 198
3.9
Real Lie Algebras of Compact Type
. 202
4
The Structure of Compact Connected Lie Groups
207
4.1
Introduction
. 207
4.2
Maximal Tori in Compact Lie Groups
. 208
4.3
Maximal Tori in Compact Connected Lie Groups
. 210
4.4
The Weyl Group
. 217
4.5
What Goes Wrong If
G
is Not Compact
. 221
5
Representations of Compact Lie Groups
223
5.1
Introduction
. 224
5.2
The
Schur
Orthogonality Relations
. 226
5.3
Compact Integral Operators on a Hubert Space
. 228
5.4
The Peter-Weyl Theorem and its Consequences
. 234
5.5
Characters and Central Functions
. 243
5.6
Induced Representations
. 250
5.7
Some Consequences of Frobenius Reciprocity
. 255
6
Symmetric Spaces of Non-compact Type
261
6.1
Introduction
. 261
6.2
The Polar Decomposition
. 264
6.3
The Cartan Decomposition
. 267
6.4
The Case of Hyperbolic Space and the
Lorentz
Group
. 274
6.5
The G-invariant Metric Geometry of
F
. 278
6.6
The Conjugacy of Maximal Compact Subgroups
. 289
6.7
The Rank and Two-Point Homogeneous Spaces
. 294
6.8
The Disk Model for Spaces of Rank
1 . 299
6.9
Exponentiality of Certain Rank
1
Groups
. 304
7 Semisimple
Lie Algebras and Lie Groups
313
7.1
Root and Weight Space Decompositions
. 313
7.2
Cartan Subalgebras
. 316
7.3
Roots of Complex
Semisimple
Lie Algebras
. 323
7.4
Real Forms of Complex
Semisimple
Lie Algebras
. 337
7.5
The Iwasawa Decomposition
. 343
8
Lattices in Lie Groups
355
8.1
Lattices in Euclidean Space
. 355
8.2
GL(n,R)/GL(n,Z) and SL(n,R)/SL(n,Z)
. 360
8.3
Lattices in More General Groups
. 371
8.4
Fundamental Domains
. 374
9
Density Results for Cofinite Volume Subgroups
377
9.1
Introduction
. 377
9.2
A Density Theorem for Cofinite Volume Subgroups
. . . 379
9.3
Consequences and Extensions of the Density Theorem
389
A Vector Fields
397
В
The
Kronecker
Approximation Theorem
403
С
Properly Discontinuous Actions
407
D
The Analyticity of Smooth Lie Groups
411
Bibliography
413
Index
421 |
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| id | DE-604.BV022473094 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:45:24Z |
| indexdate | 2024-07-09T20:58:21Z |
| institution | BVB |
| isbn | 9812706984 9812706992 9789812706980 9789812706997 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015680546 |
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| owner | DE-384 DE-703 DE-824 DE-20 |
| owner_facet | DE-384 DE-703 DE-824 DE-20 |
| physical | XV, 427 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Abbaspour, Hossein Verfasser (DE-588)140995684 aut Basic Lie theory Hossein Abbaspour ; Martin Moskowitz Singapore [u.a.] World Scientific 2007 XV, 427 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lie groups Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s Lie-Algebra (DE-588)4130355-6 s DE-604 Moskowitz, Martin A. Verfasser (DE-588)114556628 aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015680546&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Abbaspour, Hossein Moskowitz, Martin A. Basic Lie theory Lie groups Lie-Algebra (DE-588)4130355-6 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4130355-6 (DE-588)4035695-4 |
| title | Basic Lie theory |
| title_auth | Basic Lie theory |
| title_exact_search | Basic Lie theory |
| title_exact_search_txtP | Basic Lie theory |
| title_full | Basic Lie theory Hossein Abbaspour ; Martin Moskowitz |
| title_fullStr | Basic Lie theory Hossein Abbaspour ; Martin Moskowitz |
| title_full_unstemmed | Basic Lie theory Hossein Abbaspour ; Martin Moskowitz |
| title_short | Basic Lie theory |
| title_sort | basic lie theory |
| topic | Lie groups Lie-Algebra (DE-588)4130355-6 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Lie groups Lie-Algebra Lie-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015680546&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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