Yangians of Lie Superalgebras

98 Pages·2007·1.256 MB·English

by  Lucy Gow

Yangians of Lie Superalgebras
Lucy Gow
Athesissubmittedinfulfillment
oftherequirementsfor
thedegreeof DoctorofPhilosophy
SchoolofMathematicsandStatistics
TheUniversityofSydney
November 25, 2007
ii
Abstract
This thesis is concerned with extending some well-known results about the Yan-
giansY(gl )andY(sl )tothecaseofsuper-Yangians.
N N
FirstweproduceanewpresentationoftheYangianY(gl ),usingtheGauss
m|n
decomposition of a matrix with non-commuting entries. Then, by writing the
quantum Berezinian in terms of generators from the new presentation we prove
thatitscoefficientsgeneratethecentreZ ofY(gl ). WeshowthattheYangian
m|n m|n
Y(sl )isisomorphictoasubalgebraoftheYangianY(gl ),andinparticularif
m|n m|n
m 6= n,then
∼
Y(gl ) = Z ⊗Y(sl ).
m|n m|n m|n
Finally, we show that a Yangian Y(psl ) associated with the projective special
n|n
linearLiesuperalgebramaybeobtainedfromY(sl )byquotientingouttheideal
n|n
generatedbythecoefficientsofthequantumBerezinian.
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iv
Acknowledgements
IgratefullyacknowledgethehelpofmysupervisorAlexMolev,whoprovidedthe
original plan for this thesis project and has been helpful and supportive through-
outitscompletion. IalsoacknowledgethehelpofmyassociatesupervisorRuibin
Zhang,whomadehimselfavailabletoexplainsomemathematicstomeonanum-
berofoccasions.
The School of Mathematics and Statistics at the University of Sydney pro-
vided a friendly community in which to carry out this research. I would partic-
ularly like to thank my office-mates James Parkinson and Stephen Ward, as well
as fellow student Ben Wilson, for many interesting mathematical discussions and
adviceontheuseofLaTeX.DavidEasdownandAndrewMathas,aspostgraduate
coordinators,alsogaveveryusefuladvicethathelpedmetocompletethisthesis.
Thanks also to Mark Fisher for providing me with a space in his office while
I added the finishing touches in Melbourne, and to my brother Ian who read my
draft and corrected various typographical errors. Finally, I’d like to thank two
mathematicians from faraway places, Jon Brundan and Vladimir Stukopin, who
kindlyexplaineddetailsoftheirworktomeviaemail.
This thesis was supported financially by an Australian Postgraduate Award,
a supplementary top-up scholarship from the School of Mathematics and Statis-
tics, funds from the Postgraduate Student Support Scheme, and additional funds
forconferenceexpensesfromtheUniversityofSydneyAlgebraGroupandAMSI.
I declare this thesis to be wholly my own work, unless stated otherwise. No part
ofthisthesishasbeenusedinthefulfilmentofanyotherdegree.
30/06/2007
LucyGow
UniversityofSydney
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Contents
Abstract iii
Acknowledgements v
1 Introduction 1
1.1 YangiansofClassicalLieAlgebras . . . . . . . . . . . . . . . . . . . 1
1.2 YangiansofLiesuperalgebras . . . . . . . . . . . . . . . . . . . . . . 3
1.3 SummaryofThesisResults . . . . . . . . . . . . . . . . . . . . . . . 4
2 TheYangianofgl 7
m|n
2.1 DefinitionofY(gl ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
m|n
2.2 HopfSuperalgebraStructure . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 RelationshipwithU(gl ) . . . . . . . . . . . . . . . . . . . . . . . . 14
m|n
2.5 ThePoincare´-Birkhoff-WittTheorem . . . . . . . . . . . . . . . . . . 15
3 TheGaussDecomposition 19
3.1 Quasideterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 QuasideterminantsintheYangian . . . . . . . . . . . . . . . 20
3.2 GaussDecompositioninY(gl ) . . . . . . . . . . . . . . . . . . . . 20
m|n
3.3 Relationsbetweenquasideterminants . . . . . . . . . . . . . . . . . 22
3.3.1 TwomapsbetweenYangians . . . . . . . . . . . . . . . . . . 22
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3.3.2 RelationsintheYangianY(gl ) . . . . . . . . . . . . . . . . . 26
N
3.3.3 RelationsintheYangianY(gl ) . . . . . . . . . . . . . . . . 27
1|1
3.3.4 RelationsintheYangianY(gl ) . . . . . . . . . . . . . . . . 28
2|1
3.3.5 RelationsintheYangianY(gl ) . . . . . . . . . . . . . . . . 31
m|n
3.4 NewPresentationofY(gl ) . . . . . . . . . . . . . . . . . . . . . . 32
m|n
4 TheCentreofY(gl ) 39
m|n
4.1 TheQuantumBerezinian . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Stukopin’sPresentationofY(sl ) 45
m|n
5.1 QuantizationofSuperLieBialgebras . . . . . . . . . . . . . . . . . . 45
5.1.1 SuperLieBialgebras . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.2 Co-PoissonHopfSuperalgebras . . . . . . . . . . . . . . . . 46
5.1.3 Theh-adictopology . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.4 DefinitionofQuantization . . . . . . . . . . . . . . . . . . . . 48
5.2 Stukopin’sPresentation . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 NewPresentationofY(sl ) 53
m|n
6.1 NewpresentationofY(sl ) . . . . . . . . . . . . . . . . . . . . . . 53
m|n
6.2 IsomorphismBetweentheTwoPresentations . . . . . . . . . . . . . 55
6.3 TheYangianY(psl ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
n|n
6.3.1 TheHopfstructureonY(psl ) . . . . . . . . . . . . . . . . . 60
n|n
7 Conclusion 65
A Superalgebras 67
A.1 Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2 TheRuleofSigns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.3 TheLieSuperalgebragl . . . . . . . . . . . . . . . . . . . . . . . . 69
m|n
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A.4 TheLieSuperalgebrassl andpsl . . . . . . . . . . . . . . . . . 70
m|n n|n
A.5 RootSpaceDecomposition . . . . . . . . . . . . . . . . . . . . . . . . 71
A.6 CartanMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.7 TheKillingForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.8 UniversalEnvelopingAlgebras . . . . . . . . . . . . . . . . . . . . . 74
A.9 CasimirElements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.10 TheSymmetricGroupActsonCm|n ⊗...⊗Cm|n . . . . . . . . . . . 76
B Proofthatφisahomomorphism 77
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