Dualities and Representations of Lie Superalgebras

322 Pages·2012·1.682 MB·English

by  Shun-jen Cheng, Weiqiang Wang

Dualities and
Representations of
Lie Superalgebras
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(cid:23)(cid:4)(cid:6)(cid:24)(cid:6)(cid:8)(cid:9)(cid:22)(cid:10)(cid:23)(cid:8)(cid:9)(cid:22)
(cid:25)(cid:5)(cid:8)(cid:26)(cid:18)(cid:8)(cid:12)(cid:4)(cid:10)(cid:15)(cid:12)(cid:18)(cid:26)(cid:6)(cid:4)(cid:27)(cid:10)
(cid:6)(cid:9)(cid:10)(cid:11)(cid:8)(cid:12)(cid:13)(cid:4)(cid:3)(cid:8)(cid:12)(cid:6)(cid:7)(cid:27)
(cid:28)(cid:16)(cid:14)(cid:18)(cid:3)(cid:4)(cid:10)(cid:29)(cid:30)(cid:30)
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Dualities and
Representations of
Lie Superalgebras

Dualities and
Representations of
Lie Superalgebras
Shun-Jen Cheng
Weiqiang Wang
Graduate Studies
in Mathematics
Volume 144
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
David Cox (Chair)
Daniel S. Freed
Rafe Mazzeo
Gigliola Staffilani
2010 Mathematics Subject Classification. Primary 17B10, 17B20.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-144
Library of Congress Cataloging-in-Publication Data
Cheng,Shun-Jen,1963–
DualitiesandrepresentationsofLiesuperalgebras/Shun-JenCheng,WeiqiangWang.
pagescm. —(Graduatestudiesinmathematics;volume144)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-9118-6(alk.paper)
1.Liesuperalgebras. 2.Dualitytheory(Mathematics) I.Wang,Weiqiang,1970– II.Title.
QA252.3.C44 2013
512(cid:2).482—dc23
2012031989
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Contents
Preface xiii
Chapter1. LiesuperalgebraABC 1
§1.1. Liesuperalgebras: Definitionsandexamples 1
1.1.1. Basicdefinitions 2
1.1.2. ThegeneralandspeciallinearLiesuperalgebras 4
1.1.3. Theortho-symplecticLiesuperalgebras 6
1.1.4. ThequeerLiesuperalgebras 8
1.1.5. TheperiplecticandexceptionalLiesuperalgebras 9
1.1.6. TheCartanseries 10
1.1.7. Theclassificationtheorem 12
§1.2. StructuresofclassicalLiesuperalgebras 13
1.2.1. Abasicstructuretheorem 13
1.2.2. Invariantbilinearformsforglandosp 16
1.2.3. RootsystemandWeylgroupforgl(m|n) 16
1.2.4. RootsystemandWeylgroupforspo(2m|2n+1) 17
1.2.5. RootsystemandWeylgroupforspo(2m|2n) 17
1.2.6. Rootsystemandoddinvariantformforq(n) 18
§1.3. Non-conjugatepositivesystemsandoddreflections 19
1.3.1. Positivesystemsandfundamentalsystems 19
1.3.2. Positiveandfundamentalsystemsforgl(m|n) 21
1.3.3. Positiveandfundamentalsystemsforspo(2m|2n+1) 22
1.3.4. Positiveandfundamentalsystemsforspo(2m|2n) 23
1.3.5. Conjugacyclassesoffundamentalsystems 25
§1.4. Oddandrealreflections 26
1.4.1. Afundamentallemma 26
vii
viii Contents
1.4.2. Oddreflections 27
1.4.3. Realreflections 28
1.4.4. Reflectionsandfundamentalsystems 28
1.4.5. Examples 30
§1.5. Highestweighttheory 31
1.5.1. ThePoincare´-Birkhoff-Witt(PBW)Theorem 31
1.5.2. RepresentationsofsolvableLiesuperalgebras 32
1.5.3. HighestweighttheoryforbasicLiesuperalgebras 33
1.5.4. Highestweighttheoryforq(n) 35
§1.6. Exercises 37
Notes 40
Chapter2. Finite-dimensionalmodules 43
§2.1. Classificationoffinite-dimensionalsimplemodules 43
2.1.1. Finite-dimensionalsimplemodulesofgl(m|n) 43
2.1.2. Finite-dimensionalsimplemodulesofspo(2m|2) 45
2.1.3. Avirtualcharacterformula 45
2.1.4. Finite-dimensionalsimplemodulesofspo(2m|2n+1) 47
2.1.5. Finite-dimensionalsimplemodulesofspo(2m|2n) 50
2.1.6. Finite-dimensionalsimplemodulesofq(n) 53
§2.2. Harish-Chandrahomomorphismandlinkage 55
2.2.1. Supersymmetrization 55
2.2.2. Centralcharacters 56
2.2.3. Harish-ChandrahomomorphismforbasicLiesuperalgebras 57
2.2.4. Invariantpolynomialsforglandosp 59
2.2.5. ImageofHarish-Chandrahomomorphismforglandosp 62
2.2.6. Linkageforglandosp 65
2.2.7. Typicalfinite-dimensionalirreduciblecharacters 68
§2.3. Harish-Chandrahomomorphismandlinkageforq(n) 69
2.3.1. Centralcharactersforq(n) 70
2.3.2. Harish-Chandrahomomorphismforq(n) 70
2.3.3. Linkageforq(n) 74
2.3.4. Typicalfinite-dimensionalcharactersofq(n) 76
§2.4. Extremalweightsoffinite-dimensionalsimplemodules 77
2.4.1. Extremalweightsforgl(m|n) 77
2.4.2. Extremalweightsforspo(2m|2n+1) 80
2.4.3. Extremalweightsforspo(2m|2n) 82
§2.5. Exercises 85
Notes 89
Chapter3. Schurduality 91
Contents ix
§3.1. Generalitiesforassociativesuperalgebras 91
3.1.1. Classificationofsimplesuperalgebras 92
3.1.2. WedderburnTheoremandSchur’sLemma 94
3.1.3. Doublecentralizerpropertyforsuperalgebras 95
3.1.4. Splitconjugacyclassesinafinitesupergroup 96
§3.2. Schur-SergeevdualityoftypeA 98
3.2.1. Schur-Sergeevduality,I 98
3.2.2. Schur-Sergeevduality,II 100
3.2.3. Thecharacterformula 104
3.2.4. TheclassicalSchurduality 105
3.2.5. Degreeofatypicalityofλ(cid:2) 106
3.2.6. Categoryofpolynomialmodules 108
§3.3. RepresentationtheoryofthealgebraH 109
n
3.3.1. Adoublecover 110
(cid:2)
3.3.2. SplitconjugacyclassesinB 111
n
−
3.3.3. AringstructureonR 114
3.3.4. Thecharacteristicmap 116
3.3.5. Thebasicspinmodule 118
3.3.6. Theirreduciblecharacters 119
§3.4. Schur-Sergeevdualityforq(n) 121
3.4.1. Adoublecentralizerproperty 121
3.4.2. TheSergeevduality 123
3.4.3. Theirreduciblecharacterformula 125
§3.5. Exercises 125
Notes 128
Chapter4. Classicalinvarianttheory 131
§4.1. FFTforthegenerallinearLiegroup 131
4.1.1. Generalinvarianttheory 132
4.1.2. TensorandmultilinearFFTforGL(V) 133
4.1.3. FormulationofthepolynomialFFTforGL(V) 134
4.1.4. Polarizationandrestitution 135
§4.2. PolynomialFFTforclassicalgroups 137
4.2.1. AreductiontheoremofWeyl 137
4.2.2. Thesymplecticandorthogonalgroups 139
4.2.3. FormulationofthepolynomialFFT 140
4.2.4. FrombasictogeneralpolynomialFFT 141
4.2.5. Thebasiccase 142
§4.3. TensorandsupersymmetricFFTforclassicalgroups 145
4.3.1. TensorFFTforclassicalgroups 145
4.3.2. FromtensorFFTtosupersymmetricFFT 147