Representations of linear groups: an introduction based on examples from physics and number theory
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Wiesbaden
Vieweg
2007
|
| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Auch als Internetausgabe |
| Physical Description: | VIII, 270 S. graph. Darst. |
| ISBN: | 9783834803191 3834803197 |
Staff View
MARC
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| 245 | 1 | 0 | |a Representations of linear groups |b an introduction based on examples from physics and number theory |c Rolf Berndt |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Wiesbaden |b Vieweg |c 2007 | |
| 300 | |a VIII, 270 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Auch als Internetausgabe | ||
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| 650 | 4 | |a Representations of groups | |
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Contents
Introduction
ix
0
Prologue:
Some Groups and their Actions
1
0.1
Several Matrix Groups
. 1
0.2
Group Actions
. 3
0.3
The Symmetric Group
. 5
1
Basic Algebraic Concepts
7
1.1
Linear Representations
. 7
1.2
Equivalent Representations
. 9
1.3
First Examples
. . 10
1.4
Basic Construction Principles
. 14
1.4.1
Sum of Representations
. 14
1.4.2
Tensor Product of Representations
. 14
1.4.3
The Contragredient Representation
. 15
1.4.4
The Factor Representation
. 16
1.5
Decompositions
. 16
1.6
Characters
. 21
2
Representations of Finite Groups
23
2.1
Characters as Orthonormal Systems
. 23
2.2
The Regular Representation
. 27
2.3
Characters as Orthonormal Bases
. 28
3
Continuous Representations
31
3.1
Topological and Linear Groups
. 31
3.2
The Continuity Condition
. 33
3.3
Invariant Measures
. 38
3.4
Examples
. 40
4
Representations of Compact Groups
43
4.1
Basic Facts
. 43
4.2
The Example
G
=
SU{2)
. 48
4.3
The Example
G
=
SO(3)
. 52
5
Representations of Abelian Groups
59
5.1
Characters and the Pont
r j
agin Dual
. 59
5.2
Continuous Decompositions
. 60
6
The Infinitesimal Method
63
6.1
Lie Algebras and their Representations
. 63
6.2
The Lie Algebra of a Linear Group
. 67
6.3
Derived Representations
. 70
6.4
Unitarily
Integrable
Representations of sl(2,R)
. 73
6.5
The Examples su(2) and heis(R)
.-. 82
6.6
Some Structure Theory
. 84
6.6.1
Specifications of Groups and Lie Algebras
. 85
6.6.2
Structure Theory for Complex
Semisimple
Lie Algebras
. 89
6.6.3
Structure Theory for Compact Real Lie Algebras
. 93
6.6.4
Structure Theory for Noncompact Real Lie Algebras
. 95
6.6.5
Representations of Highest Weight
. 97
6.7
The Example su(3)
. 104
7
Induced Representations
117
7.1
The Principle of Induction
. 117
7.1.1
Preliminary Approach
. 118
7.1.2
Mackey's Approach
. 120
7.1.3
Final Approach
. 125
7.1.4
Some Questions and two Easy Examples
. 126
7.2
Unitary Representations of SL(2,R)
. . 130
7.3
Unitary Representations of SL(2,C) and of the
Lorentz
Group
. 143
7.4
Unitary Representations of Semidirect Products
. 147
7.5
Unitary Representations of the
Poincaré
Group
. 154
7.6
Induced Representations and Vector Bundles
. 161
8
Geometric Quantization and the Orbit Method
173
8.1
The Hamiltonian Formalism and its Quantization
. 173
8.2
Coadjoint Orbits and Representations
. 178
8.2.1
Prequantization
. 178
8.2.2
Example: Construction of Line Bundles over
M
=
Ρ
l(C)
. 181
8.2.3
Quantization
. 184
8.2.4
Coadjoint Orbits and Hamiltonian G-spaces
. 186
8.2.5
Construction of an Irreducible Unitary Representation by an Orbit
196
8.3
The Examples SU(2) and SL(2, R)
. 197
8.4
The Example Heis(R)
. 202
8.5
Some Hints Concerning the Jacobi Group
. 209
9
Epilogue: Outlook to Number Theory
215
9.1
Theta Functions and the
Heisenberg
Group
. 216
9.2
Modular Forms and SL(2,R)
. 221
9.3
Theta Functions and the Jacobi Group
. 236
9.4
Hecke's Theory of L-Functions Associated to Modular Forms
. 239
9.5
Elements of Algebraic Number Theory and
Hecke
L-Functions
. 246
9.6
Arithmetic L-Functions
. 250
9.7
Summary and Final Reflections
. 256
Bibliography
261
Index
266 |
| adam_txt |
Contents
Introduction
ix
0
Prologue:
Some Groups and their Actions
1
0.1
Several Matrix Groups
. 1
0.2
Group Actions
. 3
0.3
The Symmetric Group
. 5
1
Basic Algebraic Concepts
7
1.1
Linear Representations
. 7
1.2
Equivalent Representations
. 9
1.3
First Examples
. . 10
1.4
Basic Construction Principles
. 14
1.4.1
Sum of Representations
. 14
1.4.2
Tensor Product of Representations
. 14
1.4.3
The Contragredient Representation
. 15
1.4.4
The Factor Representation
. 16
1.5
Decompositions
. 16
1.6
Characters
. 21
2
Representations of Finite Groups
23
2.1
Characters as Orthonormal Systems
. 23
2.2
The Regular Representation
. 27
2.3
Characters as Orthonormal Bases
. 28
3
Continuous Representations
31
3.1
Topological and Linear Groups
. 31
3.2
The Continuity Condition
. 33
3.3
Invariant Measures
. 38
3.4
Examples
. 40
4
Representations of Compact Groups
43
4.1
Basic Facts
. 43
4.2
The Example
G
=
SU{2)
. 48
4.3
The Example
G
=
SO(3)
. 52
5
Representations of Abelian Groups
59
5.1
Characters and the Pont
r j
agin Dual
. 59
5.2
Continuous Decompositions
. 60
6
The Infinitesimal Method
63
6.1
Lie Algebras and their Representations
. 63
6.2
The Lie Algebra of a Linear Group
. 67
6.3
Derived Representations
. 70
6.4
Unitarily
Integrable
Representations of sl(2,R)
. 73
6.5
The Examples su(2) and heis(R)
.-. 82
6.6
Some Structure Theory
. 84
6.6.1
Specifications of Groups and Lie Algebras
. 85
6.6.2
Structure Theory for Complex
Semisimple
Lie Algebras
. 89
6.6.3
Structure Theory for Compact Real Lie Algebras
. 93
6.6.4
Structure Theory for Noncompact Real Lie Algebras
. 95
6.6.5
Representations of Highest Weight
. 97
6.7
The Example su(3)
. 104
7
Induced Representations
117
7.1
The Principle of Induction
. 117
7.1.1
Preliminary Approach
. 118
7.1.2
Mackey's Approach
. 120
7.1.3
Final Approach
. 125
7.1.4
Some Questions and two Easy Examples
. 126
7.2
Unitary Representations of SL(2,R)
. . 130
7.3
Unitary Representations of SL(2,C) and of the
Lorentz
Group
. 143
7.4
Unitary Representations of Semidirect Products
. 147
7.5
Unitary Representations of the
Poincaré
Group
. 154
7.6
Induced Representations and Vector Bundles
. 161
8
Geometric Quantization and the Orbit Method
173
8.1
The Hamiltonian Formalism and its Quantization
. 173
8.2
Coadjoint Orbits and Representations
. 178
8.2.1
Prequantization
. 178
8.2.2
Example: Construction of Line Bundles over
M
=
Ρ
l(C)
. 181
8.2.3
Quantization
. 184
8.2.4
Coadjoint Orbits and Hamiltonian G-spaces
. 186
8.2.5
Construction of an Irreducible Unitary Representation by an Orbit
196
8.3
The Examples SU(2) and SL(2, R)
. 197
8.4
The Example Heis(R)
. 202
8.5
Some Hints Concerning the Jacobi Group
. 209
9
Epilogue: Outlook to Number Theory
215
9.1
Theta Functions and the
Heisenberg
Group
. 216
9.2
Modular Forms and SL(2,R)
. 221
9.3
Theta Functions and the Jacobi Group
. 236
9.4
Hecke's Theory of L-Functions Associated to Modular Forms
. 239
9.5
Elements of Algebraic Number Theory and
Hecke
L-Functions
. 246
9.6
Arithmetic L-Functions
. 250
9.7
Summary and Final Reflections
. 256
Bibliography
261
Index
266 |
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| genre_facet | Lehrbuch |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:18:12Z |
| indexdate | 2024-07-20T09:21:55Z |
| institution | BVB |
| isbn | 9783834803191 3834803197 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015822212 |
| oclc_num | 180750215 |
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| physical | VIII, 270 S. graph. Darst. |
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| spelling | Berndt, Rolf 1940- Verfasser (DE-588)140923950 aut Representations of linear groups an introduction based on examples from physics and number theory Rolf Berndt 1. ed. Wiesbaden Vieweg 2007 VIII, 270 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Auch als Internetausgabe Linear algebraic groups Matrix groups Representations of groups Lineare Gruppe (DE-588)4138778-8 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Lineare Gruppe (DE-588)4138778-8 s Darstellungstheorie (DE-588)4148816-7 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2980273&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015822212&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berndt, Rolf 1940- Representations of linear groups an introduction based on examples from physics and number theory Linear algebraic groups Matrix groups Representations of groups Lineare Gruppe (DE-588)4138778-8 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4138778-8 (DE-588)4148816-7 (DE-588)4123623-3 |
| title | Representations of linear groups an introduction based on examples from physics and number theory |
| title_auth | Representations of linear groups an introduction based on examples from physics and number theory |
| title_exact_search | Representations of linear groups an introduction based on examples from physics and number theory |
| title_exact_search_txtP | Representations of linear groups an introduction based on examples from physics and number theory |
| title_full | Representations of linear groups an introduction based on examples from physics and number theory Rolf Berndt |
| title_fullStr | Representations of linear groups an introduction based on examples from physics and number theory Rolf Berndt |
| title_full_unstemmed | Representations of linear groups an introduction based on examples from physics and number theory Rolf Berndt |
| title_short | Representations of linear groups |
| title_sort | representations of linear groups an introduction based on examples from physics and number theory |
| title_sub | an introduction based on examples from physics and number theory |
| topic | Linear algebraic groups Matrix groups Representations of groups Lineare Gruppe (DE-588)4138778-8 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Linear algebraic groups Matrix groups Representations of groups Lineare Gruppe Darstellungstheorie Lehrbuch |
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