Dirac operators in representation theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2006
|
| Series: | Mathematics: theory and applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 199 S. |
| ISBN: | 9780817632182 0817632182 |
Staff View
MARC
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| 245 | 1 | 0 | |a Dirac operators in representation theory |c Jing-Song Huang ; Pavle Pandžić |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2006 | |
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| 490 | 0 | |a Mathematics: theory and applications | |
| 650 | 4 | |a Differential operators | |
| 650 | 4 | |a Dirac equation | |
| 650 | 4 | |a Representations of groups | |
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Record in the Search Index
| _version_ | 1804135737406259200 |
|---|---|
| adam_text | Contents
Preface
.........................................................
v
1
Lie Groups, Lie Algebras and Representations
.................... 1
1.1
Lie groups and algebras
..................................... 1
1.2
Finite-dimensional representations
............................ 9
1.3
Infinite-dimensional representations
........................... 21
1.4
Infinitesimal characters
...................................... 26
1.5
Tensor products of representations
............................ 30
2
Clifford Algebras and Spinors
.................................. 33
2.1
Real Clifford algebras
....................................... 33
2.2
Complex Clifford algebras and spin modules
.................... 40
2.3
Spin representations of Lie groups and algebras
................. 45
3
Dirac Operators in the Algebraic Setting
......................... 57
3.1
Dirac operators
............................................ 57
3.2
Dirac cohomology and Vogan s conjecture
..................... 61
3.3
A differential on (U(g)®C(p))K
............................. 65
3.4
The homomorpnism
ζ
....................................... 70
3.5
An extension of Parthasarathy s Dirac inequality
................ 71
4
A Generalized Bott-Borel-Weil Theorem
........................ 73
4.1
Kostant cubic Dirac operators
................................ 73
4.2
Dirac cohomology of finite-dimensional representations
.......... 76
4.3
Characters
................................................ 78
4.4
A generalized Weyl character formula
......................... 81
4.5
A generalized Bott-Borel-Weil theorem
....................... 82
5
Cohomological Induction
...................................... 85
5.1
Overview
................................................. 85
5.2
Some generalities about adjoint functors
....................... 89
χ
Contents
5.3 Homological
algebra
of Harish-Chandra modules
................ 95
5.4
Zuckerman functors
........................................ 101
5.5
Bernstein functors
.......................................... 106
6
Properties of Cohomologically Induced Modules
..................115
6.1
Duality theorems
........................................... 115
6.2
Infinitesimal character, A -types and vanishing
.................. 121
6.3
Irreducibility and unitarity
................................... 127
6.4
Αα(λ)
modules
............................................. 129
6.5
Unitary modules with strongly regular infinitesimal character
...... 132
7
Discrete Series
...............................................133
7.1
L2-index theorem
.......................................... 134
7.2
Existence of discrete series
................................... 137
7.3
Global characters
........................................... 138
7.4
Exhaustion of discrete series
................................. 140
8
Dimensions of Spaces of Automorphic Forms
.....................145
8.1
Hirzebruch proportionality principle
........................... 145
8.2
Dimensions of spaces of automorphic forms
.................... 147
8.3
Dirac cohomology and (q, /f)-cohomology
.................... 148
8.4
Cohomology of discrete subgroups
............................ 150
9
Dirac Operators and
Nilpotent
Lie Algebra Cohomology
...........153
9.1
u-homology and u-cohomology differentials
.................... 154
9.2
Hodge decomposition in the finite-dimensional case
............. 158
9.3
Hodge decomposition for p~
-
cohomology in the unitary case
..... 160
9.4
Calculating Dirac cohomology in stages
........................ 162
9.5
Hodge decomposition for
п
-cohomology
in the unitary case
....... 168
9.6
Homological properties of Dirac cohomology
................... 172
10
Dirac Cohomology for Lie Superalgebras
........................177
10.1
Lie superalgebras of Riemannian type
......................... 177
10.2
Dirac operator for (g, go)
.................................... 183
10.3
Analog of Vogan s conjecture
................................ 185
10.4
Dirac cohomology for Lie superalgebras
....................... 187
References
......................................................193
Index
...........................................................197
|
| adam_txt |
Contents
Preface
.
v
1
Lie Groups, Lie Algebras and Representations
. 1
1.1
Lie groups and algebras
. 1
1.2
Finite-dimensional representations
. 9
1.3
Infinite-dimensional representations
. 21
1.4
Infinitesimal characters
. 26
1.5
Tensor products of representations
. 30
2
Clifford Algebras and Spinors
. 33
2.1
Real Clifford algebras
. 33
2.2
Complex Clifford algebras and spin modules
. 40
2.3
Spin representations of Lie groups and algebras
. 45
3
Dirac Operators in the Algebraic Setting
. 57
3.1
Dirac operators
. 57
3.2
Dirac cohomology and Vogan's conjecture
. 61
3.3
A differential on (U(g)®C(p))K
. 65
3.4
The homomorpnism
ζ
. 70
3.5
An extension of Parthasarathy's Dirac inequality
. 71
4
A Generalized Bott-Borel-Weil Theorem
. 73
4.1
Kostant cubic Dirac operators
. 73
4.2
Dirac cohomology of finite-dimensional representations
. 76
4.3
Characters
. 78
4.4
A generalized Weyl character formula
. 81
4.5
A generalized Bott-Borel-Weil theorem
. 82
5
Cohomological Induction
. 85
5.1
Overview
. 85
5.2
Some generalities about adjoint functors
. 89
χ
Contents
5.3 Homological
algebra
of Harish-Chandra modules
. 95
5.4
Zuckerman functors
. 101
5.5
Bernstein functors
. 106
6
Properties of Cohomologically Induced Modules
.115
6.1
Duality theorems
. 115
6.2
Infinitesimal character, A"-types and vanishing
. 121
6.3
Irreducibility and unitarity
. 127
6.4
Αα(λ)
modules
. 129
6.5
Unitary modules with strongly regular infinitesimal character
. 132
7
Discrete Series
.133
7.1
L2-index theorem
. 134
7.2
Existence of discrete series
. 137
7.3
Global characters
. 138
7.4
Exhaustion of discrete series
. 140
8
Dimensions of Spaces of Automorphic Forms
.145
8.1
Hirzebruch proportionality principle
. 145
8.2
Dimensions of spaces of automorphic forms
. 147
8.3
Dirac cohomology and (q, /f)-cohomology
. 148
8.4
Cohomology of discrete subgroups
. 150
9
Dirac Operators and
Nilpotent
Lie Algebra Cohomology
.153
9.1
u-homology and u-cohomology differentials
. 154
9.2
Hodge decomposition in the finite-dimensional case
. 158
9.3
Hodge decomposition for p~
-
cohomology in the unitary case
. 160
9.4
Calculating Dirac cohomology in stages
. 162
9.5
Hodge decomposition for
п
-cohomology
in the unitary case
. 168
9.6
Homological properties of Dirac cohomology
. 172
10
Dirac Cohomology for Lie Superalgebras
.177
10.1
Lie superalgebras of Riemannian type
. 177
10.2
Dirac operator for (g, go)
. 183
10.3
Analog of Vogan's conjecture
. 185
10.4
Dirac cohomology for Lie superalgebras
. 187
References
.193
Index
.197 |
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| index_date | 2024-07-02T15:52:07Z |
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| isbn | 9780817632182 0817632182 |
| language | English |
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| spelling | Huang, Jing-Song Verfasser aut Dirac operators in representation theory Jing-Song Huang ; Pavle Pandžić Boston [u.a.] Birkhäuser 2006 X, 199 S. txt rdacontent n rdamedia nc rdacarrier Mathematics: theory and applications Differential operators Dirac equation Representations of groups Kohomologie (DE-588)4031700-6 gnd rswk-swf Dirac-Operator (DE-588)4150118-4 gnd rswk-swf Lie-Superalgebra (DE-588)4304027-5 gnd rswk-swf Automorphe Form (DE-588)4003972-9 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 s Dirac-Operator (DE-588)4150118-4 s Automorphe Form (DE-588)4003972-9 s Kohomologie (DE-588)4031700-6 s Lie-Superalgebra (DE-588)4304027-5 s DE-604 Pandžić, Pavle Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015027076&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Huang, Jing-Song Pandžić, Pavle Dirac operators in representation theory Differential operators Dirac equation Representations of groups Kohomologie (DE-588)4031700-6 gnd Dirac-Operator (DE-588)4150118-4 gnd Lie-Superalgebra (DE-588)4304027-5 gnd Automorphe Form (DE-588)4003972-9 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4031700-6 (DE-588)4150118-4 (DE-588)4304027-5 (DE-588)4003972-9 (DE-588)4148816-7 |
| title | Dirac operators in representation theory |
| title_auth | Dirac operators in representation theory |
| title_exact_search | Dirac operators in representation theory |
| title_exact_search_txtP | Dirac operators in representation theory |
| title_full | Dirac operators in representation theory Jing-Song Huang ; Pavle Pandžić |
| title_fullStr | Dirac operators in representation theory Jing-Song Huang ; Pavle Pandžić |
| title_full_unstemmed | Dirac operators in representation theory Jing-Song Huang ; Pavle Pandžić |
| title_short | Dirac operators in representation theory |
| title_sort | dirac operators in representation theory |
| topic | Differential operators Dirac equation Representations of groups Kohomologie (DE-588)4031700-6 gnd Dirac-Operator (DE-588)4150118-4 gnd Lie-Superalgebra (DE-588)4304027-5 gnd Automorphe Form (DE-588)4003972-9 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Differential operators Dirac equation Representations of groups Kohomologie Dirac-Operator Lie-Superalgebra Automorphe Form Darstellungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015027076&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT huangjingsong diracoperatorsinrepresentationtheory AT pandžicpavle diracoperatorsinrepresentationtheory |