Frobenius-Schur indicators for a class of fusion categories

FUSIONCATEGORIES

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Abstract.Wegiveanexplicitdescription,uptogaugeequiva-lence,ofgroup-theoreticalquasi-Hopfalgebras.Weusethisde-scriptiontocomputetheFrobenius-Schurindicatorsforgroup-theoreticalfusioncategories.

1.Introduction

TherearestronganalogiesbetweenthetheoryoffinitegroupsandthetheoryofsemisimpleHopfalgebras;someofthem,however,stillremainconjectural.Inparticular,theproblemofclassifyingsemisim-pleHopfalgebras,sayoverthefieldofcomplexnumbers,seemstobeaconsiderablydifficultone,eveninlowdimensions.Perhapsthemostimportantfeatureoftheseobjects,whichrelatesthemtootherbranchesofmathematicsandphysics,isthattheircategoryofrepresentationsisaspecialcaseofasocalledfusioncategory.Thisfactleadstothecon-siderationoftheclassificationproblem,notonlymoduloHopfalgebraisomorphisms,butmodulogaugeequivalences:roughly,twofinitedi-mensional(quasi)-HopfalgebrasHandH′giverisetothesamefusioncategoryofrepresentationsifandonlyiftheyaregaugeequivalent,inthesensethatH=H′asalgebras,andthecomultiplicationofH′isobtainedby’twisting’thatofHbymeansof∆H′(h)=F∆(h)F−1,forsomegaugetransformationF∈(H⊗H)×.

Animportantclassofexamplesofsemisimplequasi-HopfalgebraswasintroducedbyOstrik[17]andstudiedlaterbyEtingof,NikshychandOstrik[7]:thesearecalledgrouptheoreticaland,bydefinition,theyareexactlythoseforwhichthecategoryofrepresentationsisagrouptheoreticalcategoryC(G,ω,F,α),whereGisafinitegroup,F⊆Gisasubgroup,ω:G×G×G→k×isanormalized3-cocycleandα:F×

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F→k×isanormalized2-cochain,suchthatω|F=dα.Moreprecisely,C(G,ω,F,α)isthecategoryofkαF-bimodulesinthetensorcategoryVecGωoffinitedimensionalG-gradedvectorspaces,withassociativityconstraintgivenbyω.AllconcreteknownexamplesofsemisimpleHopfalgebras(andthisdoesnotextendtothequasi-context),turnouttobegrouptheoretical;thiscanbeseenasaconsequenceofourresultsin[16].Thefollowingquestionhasbeenposedin[7].Ananswertothisquestion(evenanaffirmativeone)wouldbeofgreatsignificanceintheclassificationprogram.

Question1.1.DoesthereexistasemisimpleHopfalgebrawhichisnotgrouptheoretical?

Thecategoricalnatureofthisquestionleadsnaturallytotheprob-lemoffindingandcomputinggaugeinvariantsofgrouptheoreticalquasi-Hopfalgebras,thatis,invariantswhichdependonthegaugeequivalenceclassoftheobjectratherthanontheisomorphismclassitself.

Recently,MasonandNghaveconstructedagaugeinvariant,theFrobenius-Schurindicators,forsemisimplequasi-Hopfalgebras[13].Theyhaveprovedloc.cit.ageneralizationoftheFrobenius-SchurTheoremforfinitegroups,c.f.[20].TheirconstructionextendsresultsofLinchenkoandMontgomeryforsemisimpleHopfalgebras[11];italsoextendsresultsofBantayontheFrobenius-SchurindicatorsfortheDijkgraaf-Pasquier-Rochequasi-HopfalgebraDωG,afterthedefinitionin[1]oftheindicatorsattachedtoconformalfieldtheories.

Essentially,Frobenius-Schurindicatorsweredefinedinacategori-calfashionforanysemisimplerigidtensorcategorywhichispivotal,i.e.,whichadmitsanaturaltensorisomorphismbetweentheidentityandthesecondleftdualityfunctors,intheworkofFuchs,Ganchev,Szlach´anyiandVescerny´es[8].Itisshownin[7]thatrepresentationcategoriesofsemisimplequasi-Hopfalgebrasareinfactpivotal.

Othergaugeinvariantscanbeattachedtoasemisimplequasi-Hopfalgebra.OneofthemoststudiedistheK0-ringofitsrepresentationcategory.Thisinvariantdoesnotdistinguishthegroupalgebrasofthetwononabeliangroupsoforder8:thedihedralgroupD4andthequaternionicgroupQ2.However,byaresultofTambaraandYam-agami[21],thesetwogroupsarenotgaugeequivalent.Intheirpaper,MasonandNghavenotedthattheFrobenius-Schurindicatorsdodis-tinguishthedihedralandquaternionicgroups.Aspointedouttousbythereferee,insomecasesitmayhappenthattheFrobenius-Schur

FROBENIUS-SCHURINDICATORS3

indicatorscontainlessinformationthantheK0-ring,e.g.,inthecaseofdualgroupalgebras.So,insomesense,thesetwoinvariantsareofverydifferentnature.

Inthispaperwegiveanexplicitdescriptionofgroup-theoreticalHopfalgebrasanduseittocomputetheirirreduciblecharactersandFrobenius-Schurindicators.Ourmainoriginalcontributionsinthedescriptionofthequasi-structureareontheonehandtheproofoftheexistenceofacertainnormalizationofthe3-cocycleω(Proposition4.2),andontheotherhand,theconstructionofaquasi-antipode(Theorem4.12)whichis,ofcourse,esentialinthecomputationoftheFrobenius-Schurinvariants.Weobtainthefollowingformulafortheindicatoroftheirreduciblecharacterχ:

󰀂−1

ω(xq,xq,xq)χ(δq(xq)2).(1.2)χ(νAop)=|F|

(q⊳x).q=e

whichinvolvesacertainnormalizationofthe3-cocycleω.SeeCorollary

5.4.Here,qrunsoveranappropriatechoiceofrepresentativesofGmoduloF.

OneinstanceoftheseexamplescomesfromanexactfactorizationG=FQofthegroupGintoitssubgroupsFandQ.Inthiscase,thereisagroupOpext(kF,kQ),whichclassifiestheabelianHopfalgebraextensionsofkQbykF:asaHopfalgebra,theextensioncorrespondingtoanelement[σ,τ]∈Opext(kF,kQ)isabicrossedproductkQ#τσkF,whereσ:F×F→(kQ)×andτ:Q×Q→(kF)×areapairofcompatiblecocycles;see[16,Theorem1.2].Inthiscase,the3-cocycleωinourformulafortheFrobenius-Schurindicatorsistheoneassociatedto[σ,τ]intheKacexactsequence.ThisgivesanalternativecompactexpressionfortheformulafoundbyKashina,MasonandMontgomeryin[10].

Wewouldliketopointoutthatthedescriptionforthequasi-Hopfalgebrastructureforgroup-theoreticalquasi-HopfalgebrasgeneralizestheconstructionofthetwistedquantumdoublesDωGbyDijkgraaf,PasquierandRoche.Thisagreeswiththecharacterizationgiveninourpaper[16]intermsofquantum(orDrinfeld)doubles;soinsomesensethesequasi-HopfalgebrasareallofDPR-type.

Thepaperisorganizedasfollows.InSections2and3werecallthedefinitionoftheindicatorsconstructedbyMasonandNg[13]andthedefinitionandmainpropertiesofgroup-theoreticalcategoriesasgivenbyEtingof,NikshychandOstrik[17],[7].InSection4wegiveade-scription,uptogaugeequivalence,ofthestructureofgroup-theoretical

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quasi-Hopfalgebras,andfinallyinSection5wegiveanexplicitformulafortheFrobenius-Schurindicatorsofgrouptheoreticalcategories.WeconsidersomeexamplesinSection6.Throughoutthispaperweworkoveranalgebraicallyclosedfieldkofcharacteristiczero.

2.Frobenius-Schurindicators

WeshallrecallthedefinitionoftheindicatorsconstructedbyMasonandNg[13].

2.1.Let(H,∆,ǫ,φ,S,α,β)beafinitedimensionalsemisimplequasi-Hopfalgebra[5](lateronindicated(H,Φ)forshort),thatis,Hisanassociativeunitalalgebraoverkwhichissemsimpleandfinite-dimensional;ǫ:H→kand∆:H→H⊗Harealgebramaps;Φ∈H⊗3isaninvertibleelementsuchthat

(2.1)(id⊗id⊗∆)(Φ)(∆⊗id⊗id)(Φ)=(1⊗Φ)(id⊗∆⊗id)(Φ)(Φ⊗1),(2.2)(2.3)(2.4)

(id⊗ǫ⊗id)(Φ)=1⊗1,(ǫ⊗id)∆(h)=h=(id⊗ǫ)∆(h),Φ(∆⊗id)∆(h)Φ−1=(id⊗∆)∆(h),

forallh∈H.ThemapS:H→Hopisanalgebraanti-automorphismofH;α,β∈Haresuchthat(2.5)(2.6)

S(h1)αh2=ǫ(h)α,

h1βS(h2)=ǫ(h)β,

∀h∈H;

Φ(1)βS(Φ(2))αΦ(3)=1=S(Φ(−1))αΦ(−2)βS(Φ(−3)),

whereweareusingtheabbreviatednotationΦ=Φ(1)⊗Φ(2)⊗Φ(3)andΦ−1=Φ(−1)⊗Φ(−2)⊗Φ(−3).

ThecategoryRepH=:Rep(H,Φ)isafusioncategory,intheter-minologyof[7].TheassociativityconstraintisgivenbythenaturalactionofΦ;theleftdualofanobjectVofRepHisthevectorspaceV∗=Hom(V,k)withtheH-action󰀕h.f,v󰀖=󰀕f,S(h)v󰀖;andtheeval-uationandcoevaluationmapsaregiven,respectively,by(2.7)

ev:V∗⊗V→k,

ev(f⊗v)=󰀕f,α.v󰀖,

FROBENIUS-SCHURINDICATORS5

(2.8)coev:k→V⊗V,

∗

1→

󰀂

i

β.vi⊗vi,

forallf∈V∗,v∈V,where(vi)and(vi)aredualbasisofV.Twoquasi-HopfalgebrasH1andH2arecalledgaugeequivalentifthereexistsagaugetransformation,i.e.,aninvertiblenormalizedele-mentF∈H1⊗H1suchthat(H1)FandH2areisomorphicasquasi-bialgebras.

Here,(H1)Fisthequasi-Hopfalgebra(H1,∆F,ǫ,ΦF,SF,αF,βF),where

∆F(h)=F∆(h)F−1,h∈H,

ΦF=(1⊗F)(id⊗∆)(F)Φ(∆⊗id)(F−1)(F−1⊗1),αF=S(F(−1))αF(−2),βF=F(1)βS(F(2));

whereF=F(1)⊗F(2),F−1=F(−1)⊗F(−2).

Thefinitedimensionalquasi-HopfalgebrasH1andH2aregaugeequivalentifandonlyifRepH1isequivalenttoRepH2ask-lineartensorcategories.See[6].

Remark2.1.Itisshownin[7]thatthefusioncategoriesoftheformRep(H,Φ)areexactlythoseforwhichtheFrobenius-Perrondimensionsofsimpleobjectsareintegers.

2.2.Let(H,Φ)beafinitedimensionalquasi-Hopfalgebra.Anor-malizedtwosidedintegralofHisanelementΛ∈Hsuchthat

hΛ=ǫ(h)Λ=Λh,

∀h∈H;

ǫ(Λ)=1.

Supposethat(H,Φ)issemisimple.ThenHcontainsauniquenormal-izedtwosidedintegral[9].

ThefollowingdefinitionisduetoMasonandNg[13].ItgeneralizesapreviousdefinitionforsemisimpleHopfalgebrasgivenbyLinchenkoandMontgomery[11].

Definition2.2.Let(H,Φ)beafinitedimensionalsemisimplequasi-HopfalgebraandletΛ∈Hbeanormalizedtwosidedintegral.Letalsoχ∈H∗beanirreduciblecharacterofH.TheFrobenius-Schurindicatorofχistheelementχ(νH),whereνHisthecanonicalcentralelementofHgivenby(2.9)

νH=m(qL∆(Λ)pL);

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here,m:H⊗H→Histhemultiplicationmap,andqL,pL∈H⊗2aredefinedby

qL:=S(Φ(−1))αΦ(−2)⊗Φ(−3),

pL:=Φ(2)S−1(Φ(1)β)⊗Φ(3).

ThefamilyofFrobenius-Schurindicators{χ(νH)}χisaninvariantofthek-lineartensorcategoryRep(H,Φ).Thismeansthatitisinvariantundergaugetransformationsof(H,Φ).

Also,ifαandβareinvertibleelementsofH,thenthecanonicalcentralelementνHcanbecomputedasfollows[13,Corollary3.5]:(2.10)

νH=(Λ1Λ2)(βα)−1=(βα)−1(Λ1Λ2).

Inanalogywithfinitegroupsituation,theFrobenius-Schurindicatoroftheirreduciblecharacterχ=χVsatisfiesthefollowing:(i)χ(νH)=0,1or−1,andχ(νH)=0ifandonlyifχ=χ∗

(ii)χ(νH)=1(respectively−1)ifandonlyifVadmitsanon-degeneratebilinearform󰀕,󰀖:V⊗V→k,withadjointS,suchthat󰀕x,y󰀖=󰀕y,g−1x󰀖(respectively,󰀕x,y󰀖=−󰀕y,g−1x󰀖),whereg∈HisthesocalledtraceelementofH.

3.Grouptheoreticalfusioncategories

Grouptheoreticalcategorieswereintroducedin[17,Section3]andalsostudiedin[7].Inthissectionwerecalltheirdefinitionandbasicproperties.

3.1.LetGbeafinitegroup,andletF⊆Gbeasubgroup.TheidentityelementofGwillbedenotedbye.Letalsobegiventhefollowingdata:

•anormalized3-cocycleω:G×G×G→k×,thatis,(3.1)(3.2)

ω(ab,c,d)ω(a,b,cd)=ω(a,b,c)ω(a,bc,d)ω(b,c,d),

ω(e,a,b)=ω(a,e,b)=ω(a,b,e)=1,

foralla,b,c,d∈G;

•anormalized2-cochainα:F×F→k×;subjecttothecondition(3.3)

ω|F×F×F=dα.

ConsiderthecategoryVecGωoffinitedimensionalG-gradedvectorspaces,withassociativityconstraintgivenbyω:explicitly,forany

′′′

threeobjectsU,U′andU′′ofVecGω,wehaveaU,U′,U′′:(U⊗U)⊗U→U⊗(U′⊗U′′),givenby(3.4)

aU,U′,U′′((u⊗u′)⊗u′′)=ω(||u||,||u′||,||u′′||)u⊗(u′⊗u′′),

FROBENIUS-SCHURINDICATORS7

onhomogeneouselementsu∈U,u′∈U′,u′′∈U′′,whereweusethesymbol||||todenotethecorrespondingdegreeofhomogeneity.Inotherwords,VecGωisthecategoryofrepresentationsofthequasi-HopfalgebrakG,withassociatorω∈(kG)⊗3.

By(3.3),thetwistedgroupalgebrakαFisan(associativeunital)algebrainVecGω,andonemaynaturallyattachtoitamonoidalcate-gory.Precisely,thecategoryC(G,ω,F,α)isbydefinitionthek-linearmonoidalcategoryofkαF-bimodulesinVecGω:tensorproductis⊗kαFandtheunitobjectiskαF.ThisisafusioncategoryoverkwiththepropertythattheFrobenius-Perrondimensionsofitsobjectsareintegers[7,8.8].

ThecategoriesoftheformC(G,ω,F,α)arecalledgrouptheoretical[7,Definition8.46].Byextension,a(quasi)-HopfalgebraAiscalledgrouptheoreticalifthecategoryRepAofitsfinitedimensionalrepre-sentationsisgrouptheoretical.

3.2.Letη:G×G→k×andχ:F→k×benormalizedcochains,andletω󰀃:G×G×G→k×,α󰀃:F×F→k×begivenby(3.5)

ω󰀃=ω(dη),

α󰀃=α(η|F×F)(dχ).

ThenthecategoriesC(G,ω,F,α)andC(G,ω󰀃,F,α󰀃)areequivalent[7,Remark8.39].

Remark3.1.LetG,F,ωandαbeasabove.LetQbeasetofrepre-sentativesoftheleftcosetsofFinGsuchthate∈Q;sothateveryelementg∈Gwritesuniquelyintheformg=xp,withp∈Q,x∈F.Considerthe2-cochainη:G×G→k×definedintheform(3.6)

η(xp,yq):=α−1(x,y),

p,q∈Q,

x,y∈F.

Then,takingχ=1,weobtainα󰀃=1.ThereforethecategoriesC(G,ω,F,α)andC(G,ω󰀃,F,1)areequivalent,whereω󰀃=ω(dη).Thatis,uptomonoidalequivalence,wemayalwaysassumethatα=1.NotealsothatthecategoriesC(G,ω,F,1)andC(G,ω(dη),F,1)aretensorequivalentforeverynormalized2-cochainη:G×G→k×suchthatη|F×Fisacoboundary.

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3.3.ThefiberfunctorsC(G,ω,F,α)→Vec,inthecasetheyexist,areclassifiedbyconjugacyclassesofsubgroupsΓofG,endowedwitha2-cocycleβ∈Z2(Γ,k×),suchthattheclassofω|Γistrivial;G=FΓandtheclassofthecocycleα|F∩Γβ−1|F∩Γisnon-degenerate[17,Corollary3.1].

Remark3.2.ThecategoryC=C(G,ω,F,α)hasthepropertythattheFrobenius-Perrondimensionsofitsobjectsareintegers.ATannaka-Kreinreconstructionargumentshows,thatCisequivalenttothecat-egoryofrepresentationsofasemisimplequasi-Hopfalgebraoverk[7,Theorem8.33].

Itfollowsfrom[7,8.8]thatduals,opposites,quotientcategories,fullsubcategories,andtensorproductsofgrouptheoreticalcategoriesarealsogrouptheoretical.Also,by[7,Remark8.47],theDrinfeldcenterZ(C)isgrouptheoreticalifandonlyifsoisC.

However,inRemark8.48ofthepaper[7],theauthorsnotethatthereexistsemisimplequasi-Hopfalgebrassuchthattheircategoryofrepresentationsarenotgrouptheroretical:anexplicitexampleisquotedinloc.cit.whichcomesfromtheconstructionofTambaraandYamagami[21].TheanswertothecorrespondingquestionforsemisimpleHopfalgebrasisstillnotknown.

4.Grouptheoreticalquasi-Hopfalgebras

Theaimofthissectionistogiveanexplicitdescription,uptogaugeequivalence,ofthestructureofgroup-theoreticalquasi-Hopfalgebras.ThiswillenableustoexplicitlycomputetheFrobenius-Schurindicatorsofgrouptheoreticalcategoriesinthenextsection.ThedescriptionisbasedonaresultofSchauenburg[18,3.4],whichreconstructsaquasi-bialgebrastructurefromcertainmonoidalcategoriesofbimodulesinamoregeneralcontext.

Ourmainnewresultconcerningthisdescriptionistheexplicitcon-structionofthequasi-antipodeinthegrouptheoreticalcase,whichisrelevantforourpurposes;seeTheorem4.12.

Aninstanceofthisquasi-Hopfalgebraconstruction,forthecasewhereω=1andα=1,wasstudiedbyY.Zhuin[22].Thiscasewasalsostudiedin[2,3],fromthepointofviewofthetensorcategoriesofrepresentations.Throughoutthispaperweshalladoptthenotationin[2].

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InwhatfollowsweshallfixafinitegroupGandasubgroupF⊆G.Following[22],weshallalsofixasetofsimultaneousrepresentativesoftheleftandrightcosetsofFinG,Q⊆G;thisispossiblesinceGisfinite.Thuseveryelementg∈Ghasuniquefactorizationsg=xq=py,wherex,y∈F,q,p∈Q.Weassumethate∈Q.

4.1.TheuniquenessofthefactorizationG=FQimpliesthattherearewelldefinedmaps

⊲:Q×F→F,.:Q×Q→Q,

determinedbytheconditions(4.1)(4.2)

qx=(q⊲x)(q⊳x),pq=θ(p,q)p.q,

q∈Q,x∈F;p,q∈Q.⊳:Q×F→Q,θ:Q×Q→F,

Themainrelationsbetweenthesemapsarestatedinthefollowinglemma.

Lemma4.1.([2,Proposition2.4].)Thefollowingidentitieshold,forallp,q,r∈Q,x,y∈G:(i)p⊳xy=(p⊳x)⊳y,p⊳e=p;(ii)(p.q)⊳x=(p⊳(q⊲x)).(q⊳x);

(iii)p⊲(q⊲x)=θ(p,q)((p.q)⊲x)θ((p⊳(q⊲x),q⊳x)−1,e⊲x=x;(iv)p⊲xy=(p⊲x)((p⊳x)⊲y);

(v)θ(p,q)θ(p.q,r)=(p⊲θ(q,r))θ(p⊳θ(q,r),q.r);(vi)(p⊳θ(q,r)).(q.r)=(p.q).r.(vii)θ(p,e)=θ(e,p)=e.

󰀁

4.2.Letω:G×G×G→k×beanormalized3-cocyclesuchthatω|F×F×Fistrivial.InwhatfollowsweshallfixthegrouptheoreticalcategoryC=C(G,ω,F,1).

Thusthecochainα:F×F→k×asinSubsection3.1willbetrivial.Thisis,uptomonoidalequivalence,nolossofgeneralitythankstoRemark3.1.

Proposition4.2.Thereexistsanormalized2-cochainη:G×G→k×suchthatη|F×F=1andω(dη)|F×G×G=1=ω(dη)|F×F×Q.

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Proof.Recallthatthecoboundarydη:G×G×G→k×isgivenby

(dη)(a,b,c)=η(ab,c)η(a,b)η(b,c)−1η(a,bc)−1,

foralla,b,c∈G.

Theproofwillbedoneinthreesteps.Letfirstη1:G×G→k×bethenormalizedcochaingivenby

η1(xp,yq):=ω(x,y,q),

x,y∈F,

p,q∈Q.

Thenwehaveη1|G×F=1andforallx,y,z∈F,q∈Q,wehave

(dη1)(x,y,zq)=η1(xy,zq)η1(x,y)η1(y,zq)−1η1(x,yzq)−1

=ω(xy,z,q)ω(y,z,q)−1ω(x,yz,q)−1ω(x,y,z)−1=ω(x,y,zq)−1,

thesecondequalitybecauseω|F×F×F=1.Thusω(dη1)|F×F×G=1.Putnowω0=ω(dη1)anddefineη2:G×G→k×intheformη2(xp,yq):=ω0(x,p,yq)ω0(p,y,q)−1,Thenη2|F×G=1andwehave

(dη2)(x,yp,zq)=η2(xyp,zq)η2(x,yp)η2(yp,zq)−1η2(x,ypzq)−1

=ω0(xy,p,zq)ω0(p,z,q)ω0(p,z,q)−1ω0(y,p,zq)−1=ω0(x,yp,zq)ω0(x,y,p)ω0(x,y,pzq)−1=ω0(x,yp,zq),

forallx,y,z∈F,p,q∈Q,whereinthethirdandfourthequalitieswe

−1

haveusedthatω0|F×F×G=1.Henceω0(dη2)|F×G×G=1.

−1Finally,letω1=ω0(dη2).Theconditionω1|F×G×G=1isequivalenttoω1(xt,g,h)=ω1(t,g,h),forallx∈F,t,g,h∈G.Hence

x,y∈F,p,q∈Q.

ω1(zp,x,yq)=ω1(p,x,y)ω1(p,xy,q)ω1(px,y,q)−1,

forallz∈F.

Letη3:G×G→k×bedefinedby

η3(xp,yq):=ω1(xp,y,q),

x,y∈F,

p,q∈Q.

FROBENIUS-SCHURINDICATORS11

Thenη3|F×G=η3|G×F=1,andforallx,y∈F,p,q∈Q,

(dη3)(p,x,yq)=η3(px,yq)η3(p,x)η3(x,yq)−1η3(p,xyq)−1

=η3(px,yq)η3(p,xyq)−1=ω1(px,y,q)ω1(p,xy,q)−1=ω1(p,x,y)ω1(p,x,yq)−1.

Inparticular,(dη3)(p,x,y)=1,andthus(4.3)

ω1(dη3)(p,x,yq)=ω1(dη3)(p,x,y).

Claim4.1.Wehaveω1(dη3)|F×G×G=1.

Proof.Letx,y,z∈F,p,q∈Q.Usingthatω1|F×G×G=1,wecompute

(dη3)(x,yp,zq)=η3(xyp,zq)η3(x,yp)η3(yp,zq)−1η3(x,ypzq)−1

=ω1(xyp,z,q)ω1(yp,z,q)−1=ω1(p,z,q)ω1(p,z,q)−1=1.

Thisprovestheclaim.

󰀁

Inviewoftheclaim,equation(4.3)isequivalenttoω1(dη3)|G×F×Q=1.Thisimpliestheproposition,sincebyconstructionω1(dη3)=ω(dη),forasuitablenormalized2-cochainsuchthatη|F×F=1.󰀁ByRemark3.1,thepropertyη|F×F=1inProposition4.2impliesthatC(G,ω,F,1)istensorequivalenttoC(G,ω(dη),F,1).Thenwemayandshallassumeinwhatfollowsthatthe3-cocycleω:G×G×G→k×satisfiesthenormalizationconditions(4.4)(4.5)

ω|F×G×G=1,ω|G×F×Q=1.

Theseconditionsarenecessaryinordertoapplytheresultsof[18,3.4];seeDefinition3.3.2inloc.cit.

Lemma4.3.Letg,h∈G,x,y∈F,p,q∈Q.Thenwehave(i)ω(xp,g,h)=ω(p,g,h);(ii)ω(g,y,xp)=ω(g,y,x);

(iii)ω(g,x,pq)=ω(g,x,θ(p,q));(iv)ω(pq,g,h)=ω(p.q,g,h).

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Proof.Parts(i)and(ii)followfromthecocyclecondition(3.1)andthenormalizationconditions(4.4)and(4.5).Parts(iii)and(iv)areaconsequenceofparts(i)and(ii),respectively.󰀁4.3.LetA=kQ#σkFbethecrossedproductcorrespondingtotheaction⇀:kF⊗kQ→kQandtheinvertiblemapσ:F×F→(kQ)×defined,respectively,by(4.6)(4.7)

(x⇀f)(p)=f(p⊳x),

x∈F,f∈kQ,p∈Q;

σp(x,y)=ω(p,x,y),x,y∈F,p∈Q;󰀁

whereσ(x,y)=p∈Qσp(x,y)δp,forx,y∈F.Thenormalized3-cocyclecondition(3.1)andthenormalizationassumption(4.4)implythefollowingnormalized2-cocycleconditionforσ:(4.8)(4.9)

σp⊳x(y,z)σp(x,yz)=σp(xy,z)σp(x,y),σe(x,y)=σp(e,y)=σp(x,e)=1,

forallx,󰀁y,z∈F,p∈Q.ThusAisanassociativealgebrawithunitelementp∈Qδp⊗1.Forf∈kQ,x∈F,theelementf⊗x∈Awillbedenotedbyfx.Hence,forallx,y∈F,p,q∈Q,wehave(4.10)

(δpx).(δqy):=δp⊳x,qσp(x,y)δpxy.

Considerthe(nonassociative)crossedproductcoalgebrastructureonAcorrespondingtotheaction⊲andtheinvertiblenormalizedmapτ:Q×Q→(kF)×,givenby(4.11)

τx(p,q)=ω(p,q,x),x∈F,p,q∈Q;

󰀁

whereasbeforeτ(p,q)=x∈Fτx(p,q)δx,forp,q∈Q.Usingagainthenormalized3-cocyclecondition(3.1)andthenormalizationassumption(4.4)onω,wefindthatτsatisfiesthefollowing’twisted’normalized2-cocyclecondition:

τx(p⊳θ(q,t),q.t)τx(q,t)ω(p,q,t)σp(θ(q,t),q.t⊲x)

×σp(q⊲(t⊲x),θ(q⊳(t⊲x),t⊳x))−1

=τx(p.q,t)τt⊲x(p,q)ω(p⊳(q⊲(t⊲x)),q⊳(t⊲x),t⊳x),

τe(p,q)=τx(e,q)=τx(p,e)=1,

forallx∈F,p,q,t∈Q.Explicitly,wehave

󰀂

(4.12)∆(δpx):=τx(s,t)δs(t⊲x)⊗δtx,p∈Q,x∈F.

s.t=p

FROBENIUS-SCHURINDICATORS13

Thecounitforthiscoalgebraisgivenbyǫ⊗ǫ.Bothstructuresarerelatedbythefollowingtheorem.

Theorem4.4.Thesealgebraandcoalgebrastructurescombineintoaquasi-bialgebrastructureonAopwithassociatorΦ∈A⊗3givenby

󰀂

(4.13)Φ=ω(p,q,r)δpθ(q,r)⊗δq⊗δr.

p,q,r∈Q

ThereisamonoidalequivalenceRep(Aop,Φ)∼C(G,F,ω,1).NotethatΦisinvertible,withinverseΦ−1givenbytheformula

󰀂−1Φ=ω(p,q,r)−1σp(θ(q,r),θ(q,r)−1)−1

p,q,r∈Q

δp⊳θ(q,r)θ(q,r)−1⊗δq⊗δr.

Proof.OurdefinitionsaredualtotheonesgiveninDefinitionandLemma3.4.2andTheoremandDefinition3.4.5of[18];notethat,withtheconventionsof[18],ΦisreplacedbyΦ−1incondition(2.4).There-fore,Aopisaquasi-bialgebra.ThemonoidalequivalenceRep(Aop,Φ)∼C(G,F,ω,1)followsfrom[18,Corollary3.4.4].󰀁SinceC(G,ω,F,1)isarigidtensorcategory,itfollowsfrom[19]thatAisaquasi-Hopfalgebra.Weshallgivethequasi-antipodeinthenextsubsection.

NotethatbyRemark3.1everygrouptheoreticalcategoryisequiv-alenttooneoftheformC(G,ω,F,1),forsuitableG,Fandω,whereωsatisfies(4.4),(4.5),inviewofProposition4.2.Thisgivesusthefollowingtheorem.

op

Theorem4.5.Let(H,φ)beafinitedimensionalquasi-Hopfalgebra.Then(H,φ)isgrouptheoreticalifandonlyifitisgaugeequivalenttoaquasi-Hopfalgebraoftheform(Aop,Φ),associatedtosuitabledataG,F,Qandωsatisfying(4.4)and(4.5).󰀁Weshallusethesymbol◦todenotethemultiplicationinAop;sothata◦b=b.a,foralla,b∈Aop.

14SONIANATALE

Remark4.6.UsingthepropertieslistedinLemma4.1andthenormal-izationconditions(4.4)and(4.5),itisnotdifficulttocheckthatAopisaquasi-bialgebra.Forinstance,∆:A⊗A→AisanalgebramapbecauseofLemma4.1-(ii),(iv)andthefollowingrelationshipbetweenσandτ:(4.14)

σt.s(x,y)τxy(t,s)=τx(t,s)τy(t⊳(s⊲x),s⊳x)σt(s⊲x,(s⊳x)⊲y)σs(x,y),foralls,t∈Q,x,y∈F,whichisaconsequenceof(4.5)and(3.1).Comparewith[14,Proposition4.7].

Remark4.7.Identifyσandτ,respectively,withmaps

σ:Q×F×F→k×,

Thenthetuple

(∆G,1,1,.:Q×Q→Q,⊳,⊲,θ,ω|Q×Q×Q,τ,σ),

constitutestheskeletonof(kG,ω)accordingto[18,Definition4.1.1].4.4.Wegiveinthissubsectiontheconstructionofaquasi-antipodeforAop.

Weshallneedtheexistenceofinversesforthe(nonassociative)mul-tiplicationinQ.Thisisguaranteedbythenextlemma.

Lemma4.8.ThesetQhaswell-definedleftandrightinverseswithrespecttothemultiplication.;thatis,foreveryp∈QthereexistuniquepL,pR∈QsuchthatpL.p=e=p.pR.Notethat,bydefinition,wehave(4.15)

ppR=θ(p,pR),

and

pLp=θ(pL,p).τ:Q×Q×F→k×.

Proof.Astoleftinverses,thelemmaiscontainedin[2,Proposition2.3].Toprovethestatementconcerningrightinverses,weshallusetheassumptionthatQisalsoasetofrepresentativesoftheleftcosetsofFinG.

Letp∈Q.ByexactnessofthefactorizationG=QF,thereexistuniques∈Q,x∈F,suchthatp−1=sx.Thenwehave

e=pp−1=psx=θ(p,s)(p.s)x;

FROBENIUS-SCHURINDICATORS15

thus

θ(p,s)−1=(p.s)x.

Becausep.s∈Qandθ(p,s)−1,x∈F,theexactnessofthefactorizationG=QFimpliesthatp.s=e.

Wenowshowtheuniquenessofsuchs,whichgivesthestamentwiths=pR.Supposethats′∈Qissuchthatp.s′=e.Thenps′,ps∈F,andthereforealso(s′)−1s=(ps′)−1ps∈F.Thisimpliesthats′=s,whencetheuniqueness.󰀁Forlateruse,wegiveinthenextlemmasomeoftherelationsbetween()L,()Randtheactions⊲,⊳.Thecontentofthelemmaispartof[2,Section4].

Lemma4.9.Thefollowingrelationshold,forallp∈Q:(i)p−1=θ(pL,p)−1pL=pRθ(p,pR)−1;

(ii)pL⊳θ(p,pR)=pRandpL⊲θ(p,pR)=θ(pL,p);(iii)pLL=p⊳θ(pL,p)−1;(iv)(p⊳x)L=pL⊳(p⊲x).

Proof.TheprooffollowsfromthedefinitionsandLemma4.1.

󰀁

Fornotationalconvenience,weshallconsiderinthesequelthemap≻introducedinthefollowingdefinition.Itsmainpropertiesarelistedinthenextlemma.

Definition4.10.Themap≻:Q×F→Fisdefinedasfollows:

p≻x=θ(p,pR)−1(p⊲x)θ(p⊳x,(p⊳x)R),

forallp∈Q,x∈F.

Lemma4.11.Letp∈Q,x,y∈F.Thenwehave(i)p≻(pR⊲x)=x=pR⊲(p≻x);(ii)(p⊳x)R⊳(p≻x)−1=pR;

(iii)p≻(xy)=(p≻x)((p⊳x)≻y);(iv)p≻θ(pL,p)−1=θ(p,pR)−1.Inparticular,forallp∈Qthemapp≻

:F→F.

16SONIANATALE

Proof.Weshallprovepart(iv),theproofof(i)–(iii)beingstraightfor-ward.ByLemma4.9(iii),wehavepθ(pL,p)−1=(p⊲θ(pL,p)−1)pLL,andontheotherhand,pθ(pL,p)−1=(pL)−1=θ(pLL,pL)−1pLL.

ByexactnessofthefactorizationG=FQ,wegetθ(pLL,pL)−1=p⊲θ(pL,p)−1.Now,bydefinition,

p≻θ(pL,p)−1=θ(p,pR)−1(p⊲θ(pL,p)−1)θ(pLL,pL)=θ(p,pR)−1,

bytheabove.Thisproves(iv).󰀁

Theorem4.12.Thereisaquasi-HopfalgebrastructureonAop,withquasi-antipodeS:Aop→Aopgivenby(4.16)

S(δpx)=τp≻x(p,pR)−1σpR(p≻x,(p≻x)−1)−1δ(p⊳x)R(p≻x)−1.

Wehaveα=1and

β=󰀂

ω(q−1,q,q−1)δqθ(qL,q)

−1

=

󰀂

ω(q,q−1,q)−1δqθ(qL,q)−1.

q∈Q

q∈Q

Comparewiththeformulasgivenin[22],[2]forthecasewhereω=1.

Proof.WeshallfreelyusetherelationsinLemma4.3andthenormal-izationconditions(4.4)and(4.5).Usingrelation(4.14)andLemma4.11,itisstraightforwardtoseethatSisananti-algebramap.TheinjectivityofSfollowsfromtheinjectivityofthemapp≻

FROBENIUS-SCHURINDICATORS17

Wenowcheckcondition(2.5).Letp∈Q,x∈F,andletX=δpx∈Aop.Wehave

S(X1)◦α◦X2=S(X1)◦X2=X2.S(X1)󰀂=τx(s,t)δtx.S(δs(t⊲x))

s.t=p

=

󰀂

τx(s,t)τs≻(t⊲x)(s,sR)−1σsR(s≻(t⊲x),(s≻(t⊲x))−1)−1

s.t=p

×δtx.δ(s⊳(t⊲x))R(s≻(t⊲x))−1

󰀂

τx(s,t)τs≻(t⊲x)(s,sR)−1σsR(s≻(t⊲x),(s≻(t⊲x))−1)−1=

s.t=p

×δt⊳x,(s⊳(t⊲x))R

σt(x,(s≻(t⊲x))−1)δtx(s≻(t⊲x))−1.

ByLemma4.9(iv),wehaveδt⊳x,(s⊳(t⊲x))R=δs,tL.Hence,usingprop-erty(i)inLemma4.11,thelastexpressionequalsδp,e

󰀂

s∈Q

τx(s,sR)τs≻(sR⊲x)(s,sR)−1σsR(s≻(sR⊲x),(s≻(sR⊲x))−1)−1

×σsR(x,(s≻(sR⊲x))−1)δsRx(s≻(sR⊲x))−1󰀂=δp,eδsR=δp,e1=δp,eα.

s∈Q

Thisprovestherighthandsideidentityin(2.5).WenowcomputeX1◦β◦S(X2)=S(X2).β.X1===󰀂󰀂

s.t=p

󰀂

s.t=p

τx(s,t)S(δtx).β.δs(t⊲x)

ω(q,q−1,q)−1τx(s,t)S(δtx).δqθ(qL,q)−1.δs(t⊲x)ω(q,q−1,q)−1τx(s,t)τt≻x(t,tR)−1σtR(t≻x,(t≻x)−1)−1

󰀂󰀂

q

q

s.t=p

×δ(t⊳x)R(t≻x)−1.δqθ(qL,q)−1.δs(t⊲x)󰀂󰀂=ω(q,q−1,q)−1τx(s,t)τt≻x(t,tR)−1σtR(t≻x,(t≻x)−1)−1

s.t=p

q

×δ(t⊳x)R⊳(t≻x)−1,qσ(t⊳x)R((t≻x)−1,θ(qL,q)−1)×δ(t⊳x)R(t≻x)−1θ(qL,q)−1δs(t⊲x).

18SONIANATALE

ByLemma4.11(ii),thisequals

󰀂󰀂

ω(tR,(tR)−1,tR)−1τx(s,t)τt≻x(t,tR)−1σtR(t≻x,(t≻x)−1)−1σσ(t⊳x)R((t≻x)−1,θ(t,tR)−1)δ(t⊳x)R(t≻x)−1θ(t,tR)−1δs(t⊲x)ω(tR,(tR)−1,tR)−1τx(s,t)τt≻x(t,tR)−1σtR(t≻x,(t≻x)−1)−1×σ(t⊳x)R((t≻x)−1,θ(t,tR)−1)σ(t⊳x)R((t≻x)−1θ(t,tR)−1,t⊲x)×δ(t⊳x)R⊳(t≻x)−1θ(t,tR)−1,sδ(t⊳x)R(t≻x)−1θ(t,tR)−1(t⊲x)󰀂=δp,eω(tR,(tR)−1,tR)−1τx(tL,t)τt≻x(t,tR)−1

t

s.t=p

=

s.t=p

×σtR(t≻x,(t≻x)−1)−1σ(t⊳x)R((t≻x)−1,θ(t,tR)−1)×σ(t⊳x)R((t≻x)−1θ(t,tR)−1,t⊲x)δ(t⊳x)Rθ(t⊳x,(t⊳x)R)−1;

thelastequalitybyLemma4.11(ii)andDefinition4.10.Usingthecocyclecondition(4.8)andLemma4.9,wefindσtR(t≻x,(t≻x)−1)−1σ(t⊳x)R((t≻x)−1,θ(t,tR)−1)×σ(t⊳x)R((t≻x)−1θ(t,tR)−1,t⊲x)

=σtR(θ(t,tR)−1,t⊲x)σtR(t≻x,θ(t⊳x,(t⊳x)R)−1)−1=ω(tR,θ(t,tR)−1,t⊲x)

×ω(tR,θ(t,tR)−1(t⊲x)θ(t⊳x,(t⊳x)R),θ(t⊳x,(t⊳x)R)−1)−1

=ω(tR,θ(t,tR)−1,t⊲x)ω((t⊳x)L,θ(t⊳x,(t⊳x)R),θ(t⊳x,(t⊳x)R)−1)−1×ω(tR,θ(t,tR)−1(t⊲x),θ(t⊳x,(t⊳x)R))=ω((t⊳x)L,θ(t⊳x,(t⊳x)R),θ(t⊳x,(t⊳x)R)−1)−1

×ω(tL,t⊲x,θ(t⊳x,(t⊳x)R))ω(tR,θ(t,tR)−1,(t⊲x)θ(t⊳x,(t⊳x)R))=ω((t⊳x)L,θ(t⊳x,(t⊳x)R),θ(t⊳x,(t⊳x)R)−1)−1

×ω((t⊳x)L,t⊳x,(t⊳x)R))−1ω(tL,t,x)−1ω(tL,t,x(t⊳x)R)×ω(tR,θ(t,tR)−1,(t⊲x)θ(t⊳x,(t⊳x)R));

wherewehaveusedthedefinitionofσ(4.7),thecocycleconditiononω,andLemma4.3(iii).

FROBENIUS-SCHURINDICATORS19

Ontheotherhand,

ω((t⊳x)L,θ(t⊳x,(t⊳x)R),θ(t⊳x,(t⊳x)R)−1)−1ω((t⊳x)L,t⊳x,(t⊳x)R))−1=ω((t⊳x)R,θ(t⊳x,(t⊳x)R)−1,(t⊳x)(t⊳x)R)−1ω((t⊳x)L,t⊳x,(t⊳x)R))−1=ω((t⊳x)−1,t⊳x,(t⊳x)R)ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1×ω((t⊳x)L,t⊳x,(t⊳x)R))−1

=ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1.Therefore

ω(tR,(tR)−1,tR)−1τx(tL,t)τt≻x(t,tR)−1σtR(t≻x,(t≻x)−1)−1×σ(t⊳x)R((t≻x)−1,θ(t,tR)−1)σ(t⊳x)R((t≻x)−1θ(t,tR)−1,t⊲x)=ω(tR,(tR)−1,tR)−1ω(t,tR,t≻x)−1ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1×ω(t−1,t,x(t⊳x)R)ω(tR,θ(t,tR)−1,(t⊲x)θ(t⊳x,(t⊳x)R))=ω(tR,(tR)−1,tR)−1ω(t,tRθ(t,tR)−1,(t⊲x)θ(t⊳x,(t⊳x)R))−1

×ω(t,tR,θ(t,tR)−1)−1ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1ω(tL,t,x(t⊳x)R)=ω(tR,(tR)−1,tR)−1ω(t,t−1,tx(t⊳x)R)−1

×ω(t,tR,θ(t,tR)−1)−1ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1ω(t−1,t,x(t⊳x)R)=ω(tR,(tR)−1,tR)−1ω(t,t−1,t)−1ω(t−1,t,x(t⊳x)R)−1

×ω(t,tR,θ(t,tR)−1)−1ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1ω(tL,t,x(t⊳x)R)=ω(tR,(tR)−1,tR)−1ω(t,tRθ(t,tR)−1,t)−1ω(t,tR,θ(t,tR)−1)−1×ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1

=(ω(tR,(tR)−1,tR)ω(t,tR,θ(t,tR)−1t)ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R))−1=ω(tR,(tR)−1,tR)−1ω((tR)−1,tR,(tR)−1)−1ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1=ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1.Henceweget

X1◦β◦S(X2)=δp,e

󰀂

t

ω((t⊳x)R,((t⊳x)R)−1,(t⊳x)R)−1

×δ(t⊳x)Rθ(t⊳x,(t⊳x)R)−1=δp,eβ,

whichgivestherighthandsideidentityin(2.5).

20SONIANATALE

Finally,theproofofconditions(2.6)isstraightforward,usingthepropertieslistedinLemma4.9andthecocycleconditions.Thisfinishestheproofofthetheorem.󰀁

5.Frobenius-SchurindicatorsforC(G,ω,F,1)

LetGbeafinitegroup,F⊆Gasubgroup,andω:G×G×G→k×a3-cocyclesubjecttothenormalizationconditions(4.4)and(4.5).Wekeepthenotationoftheprevioussectionsfortheskeletonmaps⊲,⊳,θ,σandτ.

Wehaveak-linearmonoidalequivalenceRep(Aop,Φ)∼C(G,F,ω,1),where(Aop,Φ)isthequasi-Hopfalgebraattachedtothedata(G,F,ω)inSection4.OuraiminthissectionistogiveanexplicitdescriptionofthecanonicalcentralelementνAop∈AopandthenoftheFrobenius-Schurindicatorsforthequasi-HopfalgebraAop.ItfollowsfromgaugeinvarianceoftheFrobenius-SchurindicatorsthatthesedependonlyonthefusioncategoryC(G,F,ω,1).

5.1.LetΛ0:=|F|x∈Fx∈kFbethenormalizedintegral.ThenormalizedtwosidedintegralΛ∈Aophasthefollowingform:

󰀂−1

(5.1)Λ=δeΛ0=|F|δex.

x∈F

−1

󰀁

Proposition5.1.Theelementβisinvertiblewithinverse

󰀂−1β=ω(pL,p,pR)δpLθ(p,pR).

p∈Q

WehaveinadditionS(β)=β−1.

Proof.Itisnotdifficulttocheckthattheexpression

󰀂

ω(pR,(pR)−1,pR)σpR(θ(p,pR)−1,θ(p,pR))−1δpLθ(p,pR).

p∈Q

definesaninverseofβ.Weclaimthat(5.2)

ω(pR,(pR)−1,pR)σpR(θ(p,pR)−1,θ(p,pR))−1=ω(pL,p,pR),

forallp∈Q.Thiswillimplytheclaimedexpressionforβ−1.Lettingq=pR,equation(4.3)isequivalenttothefollowing:(5.3)

ω(q,q−1,q)σq(θ(qL,q)−1,θ(qL,q))−1=ω(q⊳θ(qL,q)−1,qL,q).

FROBENIUS-SCHURINDICATORS21

Toestablishequation(5.3),wenotethatθ(qL,q)=qLq∈F,forallq∈Q.Then,applyingthecocycleandnormalizationconditionsonω,weget

ω(q⊳θ(qL,q)−1,qL,q)=ω(q,q−1,q)ω(q,θ(qL,q)−1,qLq)−1

=ω(q,q−1,q)ω(q,θ(qL,q)−1,θ(qL,q))−1,

whichistheclaimedidentity.UsingLemma4.11(iv),weget

S(δpθ(pL,p)−1)=τθ(p,pR)−1(p,pR)−1σpR(θ(p,pR)−1,θ(p,pR))−1

×δpLθ(p,pR).

Wenowcompute

τθ(p,pR)−1(p,pR)=ω(p,pR,(ppR)−1)

=ω(ppR,(pR)−1,p−1)−1ω(p,pR,(pR)−1)ω(pR,(pR)−1,p−1)=ω(p,pR,(pR)−1)ω(pR,(pR)−1,p−1),

becauseppR∈F.Similarly,

σpR(θ(p,pR)−1,θ(p,pR))=ω(pR,(pR)−1p−1,ppR)

=ω(pR,(pR)−1,p−1)−1ω((pR)−1,p−1,ppR)−1ω(pR,(pR)−1,pR)=ω(pR,(pR)−1,pR)ω(p,p−1,(ppR))−1.

Hence

τθ(p,pR)−1(p,pR)σpR(θ(p,pR)−1,θ(p,pR))=ω(p,p−1,p)−1ω(p−1,p,pR)−1

=ω(p,p−1,p)−1ω(pL,p,pR)−1.

Thus,

S(β)=

=󰀂󰀂

pp

ω(p−1,p,p−1)S(δpθ(pL,p)−1)ω(pL,p,pR)δpLθ(p,pR)

=β−1.

Thisfinishestheproofoftheproposition.

󰀁

22SONIANATALE

Theorem5.2.ThecanonicalcentralelementνAopisgivenbythefor-mula

󰀂−1

ω(xq,xq,xq)δq(xq)2.(5.4)νAop=|F|

(q⊳x).q=e

WehavealsoνAop=|F|

−1

󰀂

(q⊳x).q=e

τx(q⊳x,q)σq(x,q⊲x)ω((q⊳x)L,q⊳x,q)

δq(xq)2.

×σq(x(q⊲x),(q⊳x)q)

Proof.Sincetheelementβcorrespondingtothequasi-antipodeofAop

isinvertibleandα=1,wehave

νAop=(Λ1◦Λ2)◦β−1=β−1◦(Λ1◦Λ2),

andthus

νAop=β−1◦(Λ1◦Λ2)=(Λ1◦Λ2).β−1.

Usingformula(5.1)fortheintegralΛ,wefind

󰀂󰀂−1

(5.5)∆(Λ)=Λ1⊗Λ2=|F|τx(qL,q)δqL(q⊲x)⊗δqx,

x∈Fq∈Q

thus

(5.6)Λ1◦Λ2=Λ2.Λ1=|F|

−1

󰀂

(q⊳x).q=e

τx(q⊳x,q)σq(x,q⊲x)δqx(q⊲x).

Fromthese,wecomputeβ−1◦(Λ1◦Λ2)andgetthesecondexpressionforνAop.Now,forallx∈Fandq∈Qsuchthat(q⊳x).q=e,wehave

σq(x,q⊲x)σq(x(q⊲x),(q⊳x)q)=σq⊳x(q⊲x,(q⊳x)q)σq(x,qxq)=ω(q⊳x,q⊲x,(q⊳x)q)ω(q,x,qxq)

=ω(q⊳x,q⊲x,(q⊳x)q)ω(q⊳x,q,xq)−1ω(q,xq,xq)

=ω(q⊳x,q⊲x,(q⊳x)q)ω(q⊳x,q,x)−1ω(qx,qx,q)−1ω(q,xq,xq).Thelastequalitybecause

ω(q⊳x,q,xq)=ω(q⊳x,q,x)ω(q⊳x,qx,q)

=ω(q⊳x,q,x)ω(qx,qx,q).

FROBENIUS-SCHURINDICATORS23

Ontheotherhand,

ω((q⊳x)L,q⊳x,q)=ω(qL⊳(q⊲x),q⊳x,q)

=ω(qL,qx,q)ω(qL,q⊲x,(q⊳x)q)−1=ω(q⊳x,qx,q)ω(q⊳x,q⊲x,(q⊳x)q)−1=ω(qx,qx,q)ω(q⊳x,q⊲x,(q⊳x)q)−1.

Therefore

τx(q⊳x,q)σq(x,q⊲x)ω((q⊳x)L,q⊳x,q)σq(x(q⊲x),(q⊳x)q)

=ω(q,xq,xq)=ω(xq,xq,xq).

Thisprovesequation(5.4)andfinishestheproofoftheproposition.󰀁Remark5.3.Computinginstead(Λ1◦Λ2)◦β−1,weget

󰀂−1

τx(q⊳x,q)σq(x,q⊲x)ω((q⊳x)L,q⊳x,q)νAop=|F|

(q⊳x).q=e

ω((q⊳x)L,(q⊳x)q,x(q⊲x))δ(q⊳x)L((q⊳x)(q⊲x))2.

AsaconsequenceofTheorem5.2,wegetthefollowingexpression

fortheFrobenius-Schurindicators.Aftersuitablenormalization,thisexpressionallowstocomputetheFrobenius-Schurindicatorsforeverygrouptheoreticalcategory.

Corollary5.4.SupposeχisanirreduciblecharacterofAop.ThentheFrobenius-Schurindicatorofχisgivenby

󰀂−1

ω(xq,xq,xq)χ(δq(xq)2)χ(νAop)=|F|

(q⊳x).q=e

=|F|

−1

󰀂

τx(q⊳x,q)σq(x,q⊲x)ω((q⊳x)L,q⊳x,q)

󰀁

(q⊳x).q=e

σq(x(q⊲x),(q⊳x)q)χ(δq(xq)2).

5.2.Inthissubsectionweaimtogiveanexplicitdescriptionofthe

irreduciblecharacters(andhenceoftheindicators)ofC(G,ω,F,1)intermsofthegroupsGandF.

AsanalgebraA=kQ#σkFisacrossedproduct.SeeSubsection4.3.HencetheirreducibleleftA-modulescanbedescribedusingCliffordtheory.

Ontheotherhand,toeveryleftA-moduleVonecanassociatetheleftAop-moduleV∗,theactionofa∈Aopbeingthetransposeofthe

24SONIANATALE

actionofa∈AonV.Thisgivesabijectivecorrespondencebetween(ir-reducible)leftA-modulesVand(irreducible)leftAop-modules.More-over,thisbijectionpreservescharacters:χV∗=χV,forallfinitedi-mensionalleftA-moduleV.

LetFp⊆Fdenotetheisotropysubgroupofp∈Q.Thentherestrictionofσpdefinesanormalized2-cocycle

σp:Fp×Fp→k×.

LetkσpFpdenotethecorrespondingtwistedgroupalgebra.

ThespaceofisomorphismclassesofirreducibleA-modulescanbeparametrizedbythemodulesVp,W,where(5.7)

Vp,W=IndkQ#σkFpp⊗W=A⊗kQ#σkFp(p⊗W),

whereprunsoverasetofrepresentativesoftheactionofFonQ,andWrunsoverasystemofrepresentativesofisomorphismclassesofirreducibleleftkσpFp-modules.See[10,Section3].

ThereisanaturalidentificationbetweenQandthespaceF\\G={Fg:g∈G}ofleftcosetsofFinG.Underthisidentification,theactionQ×F→QcorrespondstothenaturalactionofFonF\\Gbyrightmultiplication:Fg.x=F(gx),g∈G,x∈F.

ThisgivesinturnanaturalidentificationbetweenthespaceoforbitsoftheactionQ×F→QandthespaceF\\G/FofdoublecosetsofFinG.Moreover,theisotropysubgroupofanelementp∈QisFp=F∩p−1Fp.Henceweget

Proposition5.5.ThesetofisomorphismclassesofirreducibleAop-modulesisparametrizedbythemodulesUp,W,where(5.8)

∗

=(IndkQ#σkFpp⊗W)∗,Up,W=Vp,W

whereprunsoverasetofrepresentativesofthedoublecosetsofFin

G,Fp=F∩p−1Fp,andWrunsoverasystemofrepresentativesofisomorphismclassesofirreducibleleftkσpFp-modules.

ThecharacteroftheirreducibleAop-moduleUp,Wisgivenbytheformula

󰀂

δp,q⊳yσq(z,y)σq(y,y−1zy)−1χW(y−1zy),(5.9)χp,W(δqz)=

y−1zy∈Fp

wherethesumisoverallyrunningoverasetofrepresentativesofthe

rightcosetsofFpinF,andχWisthecharacterofW.

FROBENIUS-SCHURINDICATORS25

ObservethatdimUp,W=[F:F∩p−1Fp]dimW.SothePropositionimmediatelyimpliesthatthedimensionsoftheirreduciblemodulesofagroup-theoreticalquasi-Hopfalgebradivideitsdimension,i.e.,thatKaplanksy’sconjectureholdsinthiscase.

Proof.Weonlyneedtoprovetheformulaforthecharacter.Thechar-acterofUp,WcoincideswiththecharacterofVp,W.LetYbeasetofrepresentativesoftherightcosetsofFpinF.AbasisofVp,Wisgivenbyy⊗p⊗v,where(v)isabasisofW,andy∈Y.Forallq∈Q,y,z∈F,wehave

(δqz).y=σq(z,y)δqzy

=σq(z,y)δqy(y−1zy)

=σq(z,y)σq(y,y−1zy)−1y.(δq⊳yy−1zy.

Hence,theactionofδqzonthisbasisis

(δqz).y⊗p⊗v=(δqz).y⊗p⊗v

=σq(z,y)σq(y,y−1zy)−1y.(δq⊳yy−1zy)⊗p⊗v.

Thus,inordertocomputethetraceofthisaction,weonlyneedtoconsiderthosebasisvectorsy⊗p⊗v,forwhichy−1zy∈Fp;andforsuchy,wehave

(δqz).y⊗p⊗v=σq(z,y)σq(y,y−1zy)−1y(δq⊳yy−1zy)⊗p⊗v

=δp,q⊳yσq(z,y)σq(y,y−1zy)−1y⊗p⊗(y−1zy).v.

Thisimpliesthedesiredformula.

󰀁

Remark5.6.TheparametrizationinProposition5.5allowstorecoverthestatementintheRemarkafterProposition3.1of[17],forthecate-goryC(G,ω,F,1).Indeed,the2-cocycleψp(x,y)∈Z2(Fp,k×)consid-eredinloc.cit.coincidesinournotationwithσp(y−1,x−1);andthisiscohomologoustoσp(x,y)viad(γ),whereγ(x)=σp(x−1,x),x∈F.

6.Examples

InthissectionwediscusssomespecialcasesoftheresultsinSections4,5.

26SONIANATALE

6.1.Abelianextensions.SupposethatG=FQisanexactfac-torizationofthegroupG;thatis,QisasubgroupofGand(F,Q)isamatchedpairoffinitegroupswiththeactions⊲:Q×F→F,⊳:Q×F→Q.Wereferthereaderto[14,15]forthemainnotionsusedhere,andinparticularforthestudyofthecohomologytheoryassociatedtothematchedpair(F,Q).

Fixarepresentative(τ,σ)ofaclassinOpext(kG,kF);thatis,σ:F×F→(kQ)×andτ:Q×Q→(kF)×arenormalized2-cocyclessubjecttocompatibilityconditions.

Considerthe3-cocycleω:G×G×G→k×givenby(6.1)

ω(τ,σ)(xp,yq,zr)=τz(p⊳y,q)σp(y,q⊲z),

forallx,y,z∈F,p,q,r∈Q.Theclassofthecocycleω=ω(τ,σ)istheimageoftheclassof(τ,σ)intheKacexactsequence[18,15].ItisnotdifficulttoseethatσandτhavethesamemeaningasinSubection4.3.Notethatω|Q×Q×Q=1.

ThereisabicrossedproductHopfalgebraA:=kGτ#σkFcorre-spondingtothisdata.Asiswell-known,thiscorrespondencegivesabijectionbetweentheequivalenceclassesofHopfalgebraextensions

1→kQ→A→kF→1,

andtheabeliangroupOpext(kG,kF).TheHopfalgebraAopcoincideswiththe(quasi-)HopfalgebracorrespondingtoG,Fandω,asinSubsection4.3.

ApplyingCorollary5.4,wefindthefollowingexpressionfortheFrobenius-Schurindicators.

Proposition6.1.LetχbeanirreduciblecharacterofAop.ThentheFrobenius-Schurindicatorofχisgivenby

󰀂−1

τx(q−1,q)σq(x,q⊲x)χ(δqx(q⊲x))χ(νAop)=|F|

q⊳x=q−1

=|F|

−1

󰀂

τx(q−1,q)σq(x,qxq)χ(δq(xq)2).

󰀁

q⊳x=q−1

Thisformulacoincideswiththeexpressionfoundin[10],wherethe

Frobenius-Schurindicatorsofcocentralabelianextensionsarecom-puted,i.e.,extensionsgivingrisetothetrivialaction⊲:F×Q→Q.Corollary5.4givesalsoanalternativeexpressionintermsofthe3-cocycleωattachedtoσandτviatheKacexactsequence.

FROBENIUS-SCHURINDICATORS27

6.2.Twistedquantumdoubles.LetGbeafinitegroupandletωbe3-cocycleonG.ConsidertheDijgraaf-Pasquier-RochequasiHopfalgebraDωG,alsocalledthetwistedquantumdoubleofG[4].Bytheresultsin[16],asemisimplequasi-HopfalgebraHisgrouptheoreti-califandonlyifitsquantumdoubleisgaugeequivalenttoaquasi-HopfalgebraDωG.TheFrobenius-SchurindicatorsforDωGhavebeencomputedin[13],andseentocoincideinthiscasewiththeindicatorsintroducedbyBantay[1].

Itisshownin[17]thatthecategoryRepDωGisequivalenttoC(G×G,ω󰀃,∆(G),1),where∆(G)≃GisthediagonalsubgroupofG×G,

∗−1

andω󰀃isthe3-cocycleonG×Ggivenbyω󰀃=p∗1ω(p2ω);thatis,(6.2)

ω󰀃((a1,a2),(b1,b2),(c1,c2))=ω(a1,b1,c1)ω(a2,b2,c2)−1,

forallai,bi∈G.

ThusourCorollary5.4givesanalternativeformulafortheFrobenius-SchurindicatorsofDωGintermsofanappropriatenormalizationofthe3-cocycleω󰀃.

Acknowledgement

ThisworkwasbeganduringapostdoctoralstayattheDepartment

´ofMathematicsoftheEcoleNormaleSup´erieure,Paris.Theauthoris

gratefultoMarcRossoforhiskindhospitality.ShealsothanksPeterSchauenburgforhelpfulcommentsconcerningTheorem4.12.SpecialthankstoSusanMontgomeryforinterestingdiscussions,andtheDe-partmentofMathematicsoftheUniversityofSouthernCaliforniafortheirsupportandwarmhospitalityduringhervisitinOctober2003.

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FROBENIUS-SCHURINDICATORS29

´tica,Astronom´FacultaddeMatemaıayF´ısica

´rdobaUniversidadNacionaldeCo

CIEM–CONICET

(5000)CiudadUniversitaria´rdoba,ArgentinaCo

E-mailaddress:natale@mate.uncor.eduURL:http://www.mate.uncor.edu/natale