Clifford algebras and Lie theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin
Springer
2013
|
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; 58 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XX, 321 Seiten 235 mm x 155 mm |
| ISBN: | 9783642362156 364236215X 9783642544668 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1807955061744074752 |
|---|---|
| adam_text |
IMAGE 1
CONTENTS
PREFACE VII
CONVENTION XI
LIST OF SYMBOLS XIX
1 SYMMETRIC BILINEAR FORMS 1
1.1 QUADRATIC VECTOR SPACES 1
1.2 ISOTROPIC SUBSPACES 3
1.3 SPLIT BILINEAR FORMS 5
1.4 E. CARTAN-DIEUDONNE THEOREM 7
1.5 WITT'S THEOREM 11
1.6 ORTHOGONAL GROUPS FOR K = K, C 12
1.7 LAGRANGIAN GRASSMANNIANS 18
2 CLIFFORD ALGEBRAS 23
2.1 EXTERIOR ALGEBRAS 23
2.1.1 DEFINITION 23
2.1.2 UNIVERSAL PROPERTY, FUNCTORIALITY 24
2.1.3 DERIVATIONS 25
2.1.4 TRANSPOSITION 26
2.1.5 DUALITY PAIRINGS 26
2.2 CLIFFORD ALGEBRAS 27
2.2.1 DEFINITION AND FIRST PROPERTIES 27
2.2.2 UNIVERSAL PROPERTY, FUNCTORIALITY 29
2.2.3 THE CLIFFORD ALGEBRAS CI ( N , M ) 30
2.2.4 THE CLIFFORD ALGEBRAS C/() 31
2.2.5 SYMBOL MAP AND QUANTIZATION MAP 32
2.2.6 TRANSPOSITION 34
2.2.7 CHIRALITY ELEMENT 35
2.2.8 THE TRACE AND THE SUPER-TRACE 36
2.2.9 LIE DERIVATIVES AND CONTRACTIONS 37
X I I I
HTTP://D-NB.INFO/1030083630
IMAGE 2
X I V CONTENTS
2.2.10 THE LIE ALGEBRA Q (A 2 (V)) 39
2.2.11 A FORMULA FOR THE CLIFFORD PRODUCT 41
2.3 THE CLIFFORD ALGEBRA AS A QUANTIZATION 42
2.3.1 DIFFERENTIAL OPERATORS 42
2.3.2 GRADED POISSON ALGEBRAS 44
2.3.3 GRADED SUPER POISSON ALGEBRAS 45
2.3.4 POISSON STRUCTURES ON A(V) 46
3 THE SPIN REPRESENTATION 49
3.1 THE CLIFFORD GROUP AND THE SPIN GROUP 49
3.1.1 THE CLIFFORD GROUP 49
3.1.2 THE GROUPS PIN(V) AND SPIN(V) 51
3.2 CLIFFORD MODULES 54
3.2.1 BASIC CONSTRUCTIONS 54
3.2.2 THE SPINOR MODULE S/R 56
3.2.3 THE DUAL SPINOR MODULE S F 58
3.2.4 IRREDUCIBILITY OF THE SPINOR MODULE 59
3.2.5 ABSTRACT SPINOR MODULES 60
3.3 PURE SPINORS 62
3.4 THE CANONICAL BILINEAR PAIRING ON SPINORS 65
3.5 THE CHARACTER X : R(V)P K X 69
3.6 CARTAN'S TRIALITY PRINCIPLE 70
3.7 THE CLIFFORD ALGEBRA C L ( V ) 74
3.7.1 THE CLIFFORD ALGEBRA C L ( V ) 74
3.7.2 THE GROUPS SPIN C (V) AND PIN C (V) 75
3.7.3 SPINOR MODULES OVER C L ( V ) 77
3.7.4 CLASSIFICATION OF IRREDUCIBLE C/(V)-MODULES 79
3.7.5 SPIN REPRESENTATION 80
3.7.6 APPLICATIONS TO COMPACT LIE GROUPS 83
4 COVARIANT AND CONTRAVARIANT SPINORS 87
4.1 PULL-BACKS AND PUSH-FORWARDS OF SPINORS 87
4.2 FACTORIZATIONS 90
4.2.1 THE LIE ALGEBRA O(V* V) 90
4.2.2 THE GROUP SO(V* V ) 91
4.2.3 THE GROUP SPIN(V* V ) 92
4.3 THE QUANTIZATION MAP REVISITED 94
4.3.1 THE SYMBOL MAP IN TERMS OF THE SPINOR MODULE 94
4.3.2 THE SYMBOL OF ELEMENTS IN THE SPIN GROUP 95
4.3.3 ANOTHER FACTORIZATION 97
4.3.4 THE SYMBOL OF ELEMENTS EXP(Y (A)) 99
4.3.5 CLIFFORD EXPONENTIALS VERSUS EXTERIOR ALGEBRA EXPONENTIALS . 99
4.3.6 THE SYMBOL OF ELEMENTS EXP(Y ( A ) - E/R 1 ) 101
4.3.7 THE FUNCTION A I- - 5?{A) 103
4.4 VOLUME FORMS ON CONJUGACY CLASSES 103
IMAGE 3
CONTENTS XV
5 ENVELOPING ALGEBRAS 109
5.1 THE UNIVERSAL ENVELOPING ALGEBRA 109
5.1.1 CONSTRUCTION 109
5.1.2 UNIVERSAL PROPERTY 110
5.1.3 AUGMENTATION MAP, ANTI-AUTOMORPHISM 110
5.1.4 DERIVATIONS ILL
5.1.5 MODULES OVER U ( Q ) ILL
5.1.6 UNITARY REPRESENTATIONS ILL
5.1.7 GRADED OR FILTERED LIE ALGEBRAS AND SUPER LIE ALGEBRAS . 112
5.1.8 FURTHER REMARKS 112
5.2 THE POINCARE-BIRKHOFF-WITT THEOREM 113
5.3 T/(G) AS LEFT-INVARIANT DIFFERENTIAL OPERATORS 116
5.4 THE ENVELOPING ALGEBRA AS A HOPF ALGEBRA 118
5.4.1 HOPF ALGEBRAS 118
5.4.2 HOPF ALGEBRA STRUCTURE ON S ( E ) 120
5.4.3 HOPF ALGEBRA STRUCTURE ON U (G) 121
5.4.4 PRIMITIVE ELEMENTS 123
5.4.5 CODERIVATIONS 124
5.4.6 CODERIVATIONS OF S ( E ) 125
5.5 PETRACCI'S PROOF OF THE POINCARE-BIRKHOFF-WITT THEOREM 126
5.5.1 A G-REPRESENTATION BY CODERIVATIONS 127
5.5.2 THE FORMAL VECTOR FIELDS X 1 * (( ) 128
5.5.3 PROOF OF PETRACCI'S THEOREM 130
5.6 THE CENTER OF THE ENVELOPING ALGEBRA 131
6 WEIL ALGEBRAS 135
6.1 DIFFERENTIAL SPACES 135
6.2 SYMMETRIC AND TENSOR ALGEBRA OVER DIFFERENTIAL SPACES 137
6.3 HOMOTOPIES 137
6.4 KOSZUL ALGEBRAS 140
6.5 SYMMETRIZATION 141
6.6 G-DIFFERENTIAL SPACES 143
6.7 THE G -DIFFERENTIAL ALGEBRA AG* 145
6.8 G-HOMOTOPIES 148
6.9 THE WEIL ALGEBRA 148
6.10 CHERN-WEIL HOMOMORPHISMS 151
6.11 THE NON-COMMUTATIVE WEIL ALGEBRA W G 153
6.12 EQUIVARIANT COHOMOLOGY OF G-DIFFERENTIAL SPACES 156
6.13 TRANSGRESSION IN THE WEIL ALGEBRA 158
7 QUANTUM WEIL ALGEBRAS 163
7.1 THE G-DIFFERENTIAL ALGEBRA CL(G) 163
7.2 THE QUANTUM WEIL ALGEBRA 167
7.2.1 POISSON STRUCTURE ON THE WEIL ALGEBRA 167
7.2.2 DEFINITION OF THE QUANTUM WEIL ALGEBRA 169
7.2.3 THE CUBIC DIRAC OPERATOR 171
IMAGE 4
XVI CONTENTS
7.2.4 W(G) AS A LEVEL 1 ENVELOPING ALGEBRA 172
7.2.5 CONJUGATION 173
7.3 APPLICATION: DUFLO'S THEOREM 174
7.4 RELATIVE DIRAC OPERATORS 176
7.5 HARISH-CHANDRA PROJECTIONS 182
7.5.1 ENVELOPING ALGEBRAS 182
7.5.2 CLIFFORD ALGEBRAS 184
7.5.3 QUANTUM WEIL ALGEBRAS 188
8 APPLICATIONS TO REDUCTIVE LIE ALGEBRAS 191
8.1 NOTATION 191
8.2 HARISH-CHANDRA PROJECTIONS 192
8.2.1 HARISH-CHANDRA PROJECTION FOR T/(G) 192
8.2.2 HARISH-CHANDRA PROJECTION OF THE QUADRATIC CASIMIR . 194 8.2.3
HARISH-CHANDRA PROJECTION FOR CL(G) 195
8.3 EQUAL RANK SUBALGEBRAS 197
8.4 THE KERNEL OF SFY 203
8.5 Q -DIMENSIONS 206
8.6 THE SHIFTED DIRAC OPERATOR 208
8.7 DIRAC INDUCTION 209
8.7.1 CENTRAL EXTENSIONS OF COMPACT LIE GROUPS 209
8.7.2 TWISTED REPRESENTATIONS 211
8.7.3 THE P-REPRESENTATION OF G AS A TWISTED REPRESENTATION OF G . 212
8.7.4 DEFINITION OF THE INDUCTION MAP 213
8.7.5 THE KERNEL OF 215
9 @(G, T) AS A GEOMETRIC DIRAC OPERATOR 219
9.1 DIFFERENTIAL OPERATORS ON HOMOGENEOUS SPACES 219
9.2 DIRAC OPERATORS ON MANIFOLDS 222
9.2.1 LINEAR CONNECTIONS 222
9.2.2 PRINCIPAL CONNECTIONS 223
9.2.3 DIRAC OPERATORS 225
9.3 DIRAC OPERATORS ON HOMOGENEOUS SPACES 227
10 THE HOPF-KOSZUL-SAMELSON THEOREM 231
10.1 LIE ALGEBRA COHOMOLOGY 231
10.2 LIE ALGEBRA HOMOLOGY 233
10.2.1 DEFINITION AND BASIC PROPERTIES 233
10.2.2 SCHOUTEN BRACKET 235
10.3 LIE ALGEBRA HOMOLOGY FOR REDUCTIVE LIE ALGEBRAS 238
10.3.1 HOPF ALGEBRA STRUCTURE ON (AG) 0 240
10.4 PRIMITIVE ELEMENTS 241
10.5 HOPF-KOSZUL-SAMELSON THEOREM 242
10.6 CONSEQUENCES OF THE HOPF-KOSZUL-SAMELSON THEOREM 244
10.7 TRANSGRESSION THEOREM 245
IMAGE 5
CONTENTS XVII
11 THE CLIFFORD ALGEBRA OF A REDUCTIVE LIE ALGEBRA 249
11.1 CL(G) AND THE P-REPRESENTATION 249
11.2 RELATION WITH EXTREMAL PROJECTORS 255
11.3 THE ISOMORPHISM (CLG) 0 = CL(P(G)) 260
11.4 THE P-DECOMPOSITION OF ELEMENTS § E G C CLG 262
11.4.1 THE SPACE HOM 0 (G, A(SG)) 262
11.4.2 THE SPACE HOM 0 (G, Y(/G)) 265
11.5 THE HARISH-CHANDRA PROJECTION OF Q ( P ( Q ) ) C CLG 269
11.6 RELATION WITH THE PRINCIPAL TDS 271
APPENDIX A GRADED AND FILTERED SUPER SPACES 275
A.L SUPER VECTOR SPACES 275
A.2 GRADED SUPER VECTOR SPACES 277
A.3 FILTERED SUPER VECTOR SPACES 279
APPENDIX B REDUCTIVE LIE ALGEBRAS 281
B.L DEFINITIONS AND BASIC PROPERTIES 281
B.2 CARTAN SUBALGEBRAS 282
B.3 REPRESENTATION THEORY OF S 1(2, C) 283
B.4 ROOTS 284
B.5 SIMPLE ROOTS 287
B.6 THE WEYL GROUP 288
B.7 WEYL CHAMBERS 291
B.8 WEIGHTS OF REPRESENTATIONS 293
B.9 HIGHEST WEIGHT REPRESENTATIONS 295
B. 10 EXTREMAL WEIGHTS 298
B.L 1 MULTIPLICITY COMPUTATIONS 299
APPENDIX C BACKGROUND ON LIE GROUPS 301
C.L PRELIMINARIES 301
C.2 GROUP ACTIONS ON MANIFOLDS 302
C.3 THE EXPONENTIAL MAP 303
C.4 THE VECTOR FIELD +$ R ) 306
C.5 MAURER-CARTAN FORMS 307
C.6 QUADRATIC LIE GROUPS 309
REFERENCES 311
INDEX 317 |
| any_adam_object | 1 |
| author | Meinrenken, Eckhard |
| author_GND | (DE-588)1034538942 |
| author_facet | Meinrenken, Eckhard |
| author_role | aut |
| author_sort | Meinrenken, Eckhard |
| author_variant | e m em |
| building | Verbundindex |
| bvnumber | BV040736193 |
| classification_rvk | SK 230 SK 340 |
| ctrlnum | (OCoLC)835293058 (DE-599)DNB1030083630 |
| dewey-full | 512.57 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.57 |
| dewey-search | 512.57 |
| dewey-sort | 3512.57 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-08-21T00:31:47Z |
| institution | BVB |
| isbn | 9783642362156 364236215X 9783642544668 |
| language | English |
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| physical | XX, 321 Seiten 235 mm x 155 mm |
| publishDate | 2013 |
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| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |
| spelling | Meinrenken, Eckhard Verfasser (DE-588)1034538942 aut Clifford algebras and Lie theory Eckhard Meinrenken Berlin Springer 2013 XX, 321 Seiten 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge 58 Hier auch später erschienene, unveränderte Nachdrucke Lie-Theorie (DE-588)4251836-2 gnd rswk-swf Clifford-Algebra (DE-588)4199958-7 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Clifford-Algebra (DE-588)4199958-7 s Lie-Theorie (DE-588)4251836-2 s DE-604 Lie-Algebra (DE-588)4130355-6 s Nachgedruckt als Meinrenken, Eckhard Clifford algebras and Lie theory Springer, 2014 978-3-642-54466-8 (DE-604)BV047925397 Erscheint auch als Online-Ausgabe 978-3-642-36216-3 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge ; 58 (DE-604)BV000899194 58 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4240471&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025716222&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Meinrenken, Eckhard Clifford algebras and Lie theory Ergebnisse der Mathematik und ihrer Grenzgebiete Lie-Theorie (DE-588)4251836-2 gnd Clifford-Algebra (DE-588)4199958-7 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| subject_GND | (DE-588)4251836-2 (DE-588)4199958-7 (DE-588)4130355-6 |
| title | Clifford algebras and Lie theory |
| title_auth | Clifford algebras and Lie theory |
| title_exact_search | Clifford algebras and Lie theory |
| title_full | Clifford algebras and Lie theory Eckhard Meinrenken |
| title_fullStr | Clifford algebras and Lie theory Eckhard Meinrenken |
| title_full_unstemmed | Clifford algebras and Lie theory Eckhard Meinrenken |
| title_short | Clifford algebras and Lie theory |
| title_sort | clifford algebras and lie theory |
| topic | Lie-Theorie (DE-588)4251836-2 gnd Clifford-Algebra (DE-588)4199958-7 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| topic_facet | Lie-Theorie Clifford-Algebra Lie-Algebra |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4240471&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025716222&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT meinrenkeneckhard cliffordalgebrasandlietheory |