Lectures on Algebraic Categorification

129 Pages·2012·1.118 MB·English

by  Volodymyr Mazorchuk

The QGM Master Class Series
Edited by Jørgen Ellegaard Andersen, Henning Haahr Andersen,
Nigel Hitchin, Maxim Kontsevich, Robert Penner and Nicolai Reshetikhin
The Centre for Quantum Geometry of Moduli Spaces (QGM) in Aarhus, Denmark,
focuses on collaborative cutting-edge research and training in the quantum
geometry of moduli spaces. This discipline lies at the interface between mathe-
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continuing the tradition initiated by the former Center for the Topology and
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This series contains lecture notes, textbooks and monographs arising from the
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study of geometrical aspects and mathematical foundations of quantum field
theory and string theory.
Previously published in this series:
Robert C. Penner, Decorated Teichmüller Theory
Volodymyr Mazorchuk
Lectures on
Algebraic
Categorification
Author:
Volodymyr Mazorchuk
Department of Mathematics
Uppsala University
P.O. Box 480
SE-75106 Uppsala
Sweden
Email: [email protected]
2010 Mathematics Subject Classification: 18-01, 18D05; 17B10, 17B55, 18A40, 18E30, 57M27
Key words: Categorification, 2-category, Lie algebra, module, category O, functor, knot, Jones polynomial,
Verma module, translation functor
ISBN 978-3-03719-108-8
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9 8 7 6 5 4 3 2 1
Preface
Themosttraditionalapproachtosolvingmathematicalproblemsis: startwithadifficult
problemandsimplifyituntilitbecomeseasyenoughtobesolved. However,developing
mathematicaltheoriesquiteoftengoesintheoppositedirection: startingwithan“easy”
theoryonetriesto“generalize”ittosomethingmorecomplicatedwhichcouldhopefully
describeamuchbiggerclassofphenomena.
An example of the latter is the development of what is now known as Khovanov
homology. TheJonespolynomialisaveryelementaryclassicalcombinatorialinvariant
appearinginlowdimensionaltopology. However,asmostotherknowntopologicalin-
variants,itisnotanabsoluteone(itdoesnotdistinguishallknots). Sometwelveyears
agoMikhailKhovanovdevelopedaveryadvanced“refinement”oftheJonespolyno-
mial which seriously increased the level of theoretical sophistication necessary to be
abletodefineandworkwithit. Insteadofelementarycombinatoricsandbasicalgebra,
Khovanov’sdefinitionwasbasedoncategorytheoryandhomologicalalgebraandvery
soon led to the study of higher categorical structures. This “categorification” of the
Jones polynomial created a new direction in topology and attracted a lot of attention
fromsomeotherpartsofmathematics,notablyalgebraandcategorytheory. Withina
fewyearscategorificationbecameanintensivelystudiedsubjectinseveralmathemat-
icalareas. Itcompletelychangedtheviewpointonmanylong-standingproblemsand
ledtoseveralspectacularresultsandapplications.
Thistextisawrite-upofthelecturesgivenbytheauthorduringtheMasterClass
“Categorification”atÅrhusUniversity,Denmark,inOctober2010. Itmostlyconcen-
tratesonalgebraicaspectsofthetheory,presentedinahistoricalperspective,butalso
containsseveraltopologicalapplications,inparticular,analgebraic(or,moreprecisely,
representationtheoretical)approachtocategorificationoftheJonespolynomialmen-
tionedabove. Thetextconsistsoffifteensectionscorrespondingtothefifteenone-hour
lectures given during the Master Class and fairly accurately describes the content of
theselectures. Therearesomeexercises(whichwereproposedtotheparticipantsof
theMasterClass)collectedattheendofthetextandaratherextensivelistofreferences.
Unfortunately,thetimeconstraintsontheMasterClassresultedinthefactthatseveral
recentdevelopmentsrelatedtocategorificationdidnotmakeitintothetext.
Thetextisaimedtobeanintroductoryoverviewofthesubjectratherthanafully
detailedmonograph. Theemphasisisondefinitions,examplesandformulationsofthe
results. Mostproofsareeitherbrieflyoutlinedoromitted; however, completeproofs
canbefoundbytrackingreferences. Itisassumedthatthereaderisfamiliarwithbasics
ofcategorytheory,representationtheory,topologyandLiealgebras.

Contents
Preface v
1 Basics: decategorificationandcategorification 1
1.1 Theideaofcategorification . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Grothendieckgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Decategorification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 (Pre)categorificationofanF-module . . . . . . . . . . . . . . . . . . 4
1.5 Gradedsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Someconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Basics: fromcategorificationoflinearmapsto2-categories 9
2.1 Categorificationoflinearmaps . . . . . . . . . . . . . . . . . . . . . 9
2.2 Naïvecategorification . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Weakcategorification . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 (Genuine)categorification . . . . . . . . . . . . . . . . . . . . . . . 14
3 Basics: 2-representationsoffinitary2-categories 17
3.1 2-representationsof2-categories . . . . . . . . . . . . . . . . . . . . 17
3.2 Fiat-categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Principal2-representationsoffiat-categories . . . . . . . . . . . . . . 19
3.4 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Cellsmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Homomorphismsfromacellmodule . . . . . . . . . . . . . . . . . . 22
3.7 Serresubcategoriesandquotients . . . . . . . . . . . . . . . . . . . . 22
3.8 Naturallycommutingfunctors . . . . . . . . . . . . . . . . . . . . . 22
4 CategoryO: definitions 25
4.1 DefinitionofcategoryO . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Vermamodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Blockdecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 BGGreciprocityandquasi-hereditarystructure . . . . . . . . . . . . 28
4.5 TiltingmodulesandRingelself-duality . . . . . . . . . . . . . . . . 29
4.6 ParaboliccategoryO . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.7 gl -example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2
viii Contents
5 CategoryO: projectiveandshufflingfunctors 33
5.1 Projectivefunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Translationsthroughwalls . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 DescriptionviaHarish-Chandrabimodules. . . . . . . . . . . . . . . 35
5.4 Shufflingfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5 Singularbraidmonoid . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.6 gl -example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2
6 CategoryO: twistingandcompletion 41
6.1 Zuckermanfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Twistingfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Completionfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 Alternativedescriptionvia(co)approximations . . . . . . . . . . . . . 45
6.5 Serrefunctor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 CategoryO: gradingandcombinatorics 47
7.1 Doublecentralizerproperty . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Endomorphismsoftheantidominantprojective . . . . . . . . . . . . 48
7.3 GradingonB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
(cid:2)
7.4 Heckealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.5 CategorificationoftherightregularH-module. . . . . . . . . . . . . 51
7.6 CombinatoricsofO . . . . . . . . . . . . . . . . . . . . . . . . . . 52
(cid:2)
8 S -categorification: Soergelbimodules,cellsandSpechtmodules 55
n
8.1 Soergelbimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.2 Kazhdan–Lusztigcells . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.3 Cellmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8.4 Categorificationoftheinducedsignmodule . . . . . . . . . . . . . . 58
8.5 CategorificationofSpechtmodules . . . . . . . . . . . . . . . . . . . 59
9 S -categorification: (induced)cellmodules 61
n
9.1 CategoriesORy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.2 Categorificationofcellmodules . . . . . . . . . . . . . . . . . . . . 61
9.3 Categorificationofpermutationmodules . . . . . . . . . . . . . . . . 62
9.4 ParabolicanaloguesofO . . . . . . . . . . . . . . . . . . . . . . . . 64
9.5 Categorificationofinducedcellmodules . . . . . . . . . . . . . . . . 65
10 CategoryO: Koszulduality 67
10.1 Quadraticdualofapositivelygradedalgebra . . . . . . . . . . . . . 67
10.2 Linearcomplexesofprojectives . . . . . . . . . . . . . . . . . . . . 68
10.3 Koszulduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.4 Koszuldualfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . 72
10.5 Alternativecategorificationofthepermutationmodule . . . . . . . . 72
Contents ix
11 sl -categorification: simplefinite-dimensionalmodules 75
2
11.1 ThealgebraU .sl / . . . . . . . . . . . . . . . . . . . . . . . . . . 75
v 2
11.2 FinitedimensionalrepresentationsofU .sl / . . . . . . . . . . . . . 76
v 2
11.3 CategorificationofV˝n . . . . . . . . . . . . . . . . . . . . . . . . . 77
1
11.4 CategorificationofV . . . . . . . . . . . . . . . . . . . . . . . . . . 79
n
11.5 Koszuldualpicture . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
12 Application: categorificationoftheJonespolynomial 81
12.1 KauffmanbracketandJonespolynomial . . . . . . . . . . . . . . . . 81
12.2 Khovanov’sideaforcategorificationofJ.L/ . . . . . . . . . . . . . . 82
12.3 Quantumsl -linkinvariants . . . . . . . . . . . . . . . . . . . . . . 82
2
12.4 Functorialquantumsl -linkinvariants . . . . . . . . . . . . . . . . . 84
2
13 sl -categorificationofChuangandRouquier 87
2
13.1 Genuinesl -categorification . . . . . . . . . . . . . . . . . . . . . . 87
2
13.2 AffineHeckealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 88
13.3 Morphismsofsl -categorifications . . . . . . . . . . . . . . . . . . . 89
2
13.4 Minimalsl -categorificationofsimplefinite-dimensionalmodules . . 90
2
13.5 sl -categorificationoncategoryO . . . . . . . . . . . . . . . . . . . 91
2
13.6 Categorificationofthesimplereflection . . . . . . . . . . . . . . . . 92
14 Application: blocksofFŒS (cid:2)andBroué’sconjecture 93
n
14.1 Jucys–Murphyelementsandformalcharacters . . . . . . . . . . . . . 93
14.2 Inductionandrestriction . . . . . . . . . . . . . . . . . . . . . . . . 94
14.3 Categorification of the basic representation of an affine Kac–Moody
algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.4 Broué’sconjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.5 Broué’sconjectureforS . . . . . . . . . . . . . . . . . . . . . . . . 96
n
14.6 Dividedpowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
15 ApplicationsofS -categorifications 99
n
15.1 WedderburnbasisforCŒS (cid:2) . . . . . . . . . . . . . . . . . . . . . . 99
n
15.2 Kostant’sproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15.3 Structureofinducedmodules . . . . . . . . . . . . . . . . . . . . . . 102
16 Exercises 105
Bibliography 109
Index 117