EXTENSIONS OF FINITE QUANTUM GROUPS BY

EXTENSIONS OF FINITE QUANTUM GROUPS

27Vol. 14, No. 1, 2009, pp.1– TransformationGroups, cBirkh¨??auserBoston(2008)

EXTENSIONSOFFINITEQUANTUMGROUPSBY

FINITEGROUPS

N.ANDRUSKIEWITSCH?

FaMAF-CIEM(CONICET)

UniversidadNacionaldeC´ordoba

MedinaAllendes/n,C.Univ.

5000C´ordoba,Argentina

andrus@mate.uncor.eduG.A.GARC´IA*FaMAF-CIEM(CONICET)UniversidadNacionaldeC´ordobaMedinaAllendes/n,C.Univ.5000C´ordoba,Argentinaggarcia@mate.uncor.edu

Abstract.Wegiveanecessaryandsu?cientconditionfortwoHopfalgebraspresentedascentralextensionstobeisomorphic,inasuitablesetting.WethenstudythequestionofisomorphismbetweentheHopfalgebrasconstructedin[AG]asquantumsubgroupsofquantumgroupsatrootsof1.Finally,weapplythe?rstgeneralresulttoshowtheexistenceofin?nitelymanynon-isomorphicHopfalgebrasofthesamedimension,presentedasextensionsof?nitequantumgroupsby?nitegroups.

1.Introduction

Amajordi?cultyintheclassi?cationof?nite-dimensionalHopfalgebrasisthelackofenoughexamples,sothatwearenotevenabletostateconjecturesonthepossiblecandidatestoexhaustdi?erentcasesoftheclassi?cation.Indeed,weareawareatthistimeofthefollowingexamplesof?nite-dimensionalHopfalgebras:groupalgebrasof?nitegroups;pointedHopfalgebraswithabeliangroupclassi?edin[AS2](thesearevariationsofthesmallquantumgroupsintroducedbyLusztig[L1],[L2]);otherpointedHopfalgebraswithabeliangrouparisingfromtheNicholsalgebrasdiscoveredin[G?n1],[He];afewexamplesofpointedHopfalgebraswithnonabeliangroup[MS],[G?n2];combinationsoftheprecedingviastandardoperations:duals,twisting,Hopfsubalgebrasandquotients,extensions.LetGbeaconnected,simplyconnected,simplecomplexalgebraicgroupandlet??beaprimitive??throotof1,??oddand3????ifGisoftypeG2.In[AG]wedeterminedallHopfalgebraquotientsofthequantizedcoordinatealgebraO??(G).(Finite-dimensionalHopfalgebraquotientsofO??(SLN)werepreviouslyobtainedin[M3].)Abyproductofthemaintheoremof[AG]isthediscoveryofmanynewexamplesofHopfalgebraswith?nitedimension,orwith?niteGelfand–Kirillovdimension.Thequotientsin[AG]areparameterizedbydataD(seebelowfortheprecisede?nition)andthecorrespondingquotientAD?tsintothefollowingcommutativediagramwithexactrows:

DOI:10.1007/S00031-008-9039-4

PartiallysupportedbyCONICET,ANPCyT,Secyt(UNC)andMinisteriodeCienciayTecnolog´?adelaProvinciadeC´ordoba.?

Received??August??25,??2007.??Accepted??March??20,??2008.??Published??online??November??18,??2008.

EXTENSIONS OF FINITE QUANTUM GROUPS

2N.ANDRUSKIEWITSCHANDG.A.GARC´IA

1O(G)

resιO??(G)

O??(L)

sResπu??(g)?u??(l)?

HP11O(L)tιLqD?ιπL1(1)1O(Γ)σADπ?1.

Thepurposeofthepresentpaperistostudywhentheseexamplesarereallynew(neithersemisimplenorpointed,nordualtopointed),andwhentheyareisomorphicasHopfalgebras.Inprinciple,itishardtotellwhentwoHopfalgebraspresentedbyextensionsareisomorphic—notasextensionsbutas“abstract”Hopfalgebras(evenforgroupsthereisnogeneralanswer).Webeginbystudying,inSection2,isomorphismsbetweenHopfalgebrasoftheform:

1KιAπH1,(2)

whereKistheHopfcenterofA,andtheHopfcenterofHistrivial.Oneofourmainresults(Theorem2.15)givesanecessaryandsu?cientconditionfortwoHopfalgebrasofthiskindtobeisomorphic,http://wendang.chazidian.comly,weneed:(i)ANoetherianandH-GaloisoverK;and(ii)anyHopfalgebraautomorphismofH“lifts”toA.Thissettingisampleenoughtoincludeexamplesarisingfromquantumgrouptheoryand,inparticular,from[AG].Indeed,thereisanalgebraic??suchthatAD?tsintothefollowingexactsequence:groupΓ

1O(Γ)??ιADπu??(l0)?1,(3)

??istheHopfcenterofAD,andtheHopfcenterofu??(l0)?istrivial.AsawhereO(Γ)??andu??(l0)areinvariantsoftheisomorphismclassofAD,?rstconsequence,bothΓ

seeTheorem3.12.However,theconditionofliftingofautomorphismsremainsanopenquestion,exceptwhenH=u??(l)?,seeCorollary4.2.Nevertheless,theHopfcenterofu??(l)?istrivialifandonlyifl=g.Inthiscase,weclassifythequotientsofO??(G)uptoisomorphisms,seeTheorem4.14.Thenusingsomeresultsoncohomologyofgroups,weprovethattherearein?nitelymanynonisomorphicHopfalgebrasofthesamedimension.Theycorrespondto?http://wendang.chazidian.comingthisfact,weareabletoprovethattheyformafamilyofnonsemisimple,nonpointedHopfalgebraswithnonpointedduals.ForSL2suchanin?nitefamilywasobtainedbyM¨uller[M3].Tryingtounderstandthisresultwasoneofourmainmotivationstostudytheproblemofquantumsubgroups.

1.1.Conventions

OurreferencesforthetheoryofHopfalgebrasare[Mo]and[Sw],forLiealgebras??the[Hu]andforquantumgroups[J]and[BG].IfΓisagroup,wedenotebyΓ

EXTENSIONS OF FINITE QUANTUM GROUPS

EXTENSIONSOFFINITEQUANTUMGROUPS3

charactergroup.Letkbea?eld.TheantipodeofaHopfalgebraHisdenotedbyS.AlltheHopfalgebrasconsideredinthispaperhavebijectiveantipode.SeeRemark3.5.TheSweedlernotationisusedforthecomultiplicationofHbutdroppingthesummationsymbol.Thesetofgroup-likeelementsofacoalgebraCisdenotedbyG(C).WealsodenotebyC+=KerεtheaugmentationidealofC,πwhereε:C→kisthecounitofC.LetA?→HbeaHopfalgebramap,thenAcoH=Acoπ={a∈A|(id?π)?(a)=a?1}denotesthesubalgebraofrightcoinvariantsandcoHA=coπA={a∈A|(π?id)?(a)=1?a}denotesthesubalgebraofleftcoinvariants.

AHopfalgebraHiscalledsemisimple(resp.,cosemisimple)ifitissemisimpleasanalgebra(resp.,ifitiscosemisimpleasacoalgebra).ThesumofallsimplesubcoalgebrasiscalledthecoradicalofHandisdenotedbyH0.Ifallsimplesub-coalgebrasofHareonedimensional,thenHiscalledpointedandH0=k[G(H)].LetHbeaHopfalgebra,AarightH-comodulealgebrawithstructuremapδ:A→A?H,a→a(0)?a(1)andB=AcoH.TheextensionB?AiscalledaHopfGaloisextensionorH-Galoisifthecanonicalmapβ:A?BA→A?H,a?b→ab(0)?b(1)isbijective.See[SS]formoredetailsonH-Galoisextensions.Acknowledgements.ResearchofthispaperwasbegunwhenG.A.G.wasvisitingtheMathematischesInstitutderLudwig-MaximiliansUniversit¨atM¨unchenunderthesupportoftheDAAD.ResultsinthispaperarepartofthePhDthesisofG.A.G.,writtenundertheadviceofN.A.

TheauthorsthankS.NataleforhelpfulindicationsonLemma4.12andJ.Vargasfordiscussionsonreductivesubgroupsofasimplegroup.G.A.G.thanksH.-J.SchneiderforhospitalityandfruitfulconversationsduringhisstayinM¨un-chen.Theyalsothanktherefereesforaverycarefulreadingofthepaper.

2.CentralextensionsofHopfalgebras

2.1.Preliminaries

WerecallsomeresultsonquotientsandextensionsofHopfalgebras.

De?nition2.1([AD]).AsequenceofHopfalgebrasmaps1→B→?A?→H→1,where1denotestheHopfalgebrak,isexactifιisinjective,πissurjective,Kerπ=AB+andB=coπA.

Remark2.2.NotethatAisarightH-GaloisextensionofBby[T],seealso[SS,

3.1.1].

IftheimageofBiscentralinA,thenAiscalledacentralextensionofB.WesaythatAisacleftextensionofBbyHifthereisanH-colinearsectionγofπwhichisinvertiblewithrespecttotheconvolution,see,forexample,[A,3.1.14].By[Sch2,Theorem2.4],a?nite-dimensionalHopfalgebraextensionisalwayscleft.Weshallusethefollowingresult.

Proposition2.3([AG,Prop.2.10]).LetAandKbeHopfalgebras,BacentralHopfsubalgebraofAsuchthatAisleftorrightfaithfully?atoverBandp:B→KaHopfalgebraepimorphism.ThenH=A/AB+isaHopfalgebraandA?tsιπintotheexactsequence1→B→?A?→H→1.IfwesetJ=Kerp?B,thenιπ

EXTENSIONS OF FINITE QUANTUM GROUPS

4N.ANDRUSKIEWITSCHANDG.A.GARC´IA

(J)=AJisaHopfidealofAandAp:=A/(J)isthepushoutgivenbythefollowingdiagram:

B

pιAq

KjAp.

Moreover,Kcanbeidenti?edwithacentralHopfsubalgebraofApandAp?tsintotheexactsequence1→K→Ap→H→1.??

Remark2.4.LetA,BbeasinProposition2.3,thenthefollowingdiagramofcentralexactsequencesiscommutative:

1B

pιAqπHH1(4)1.1KjApπp

Remark2.5.IfdimKanddimHare?nite,thendimApisalso?nite.Indeed,sincetheGaloismapβ:Ap?KAp→Ap?HisbijectivebyRemark2.2andHis?nitedimensional,by[KT,Theorem1.7],Apisa?nitely-generatedprojectiveK-module;inparticulardimApis?nite.

ThefollowinggenerallemmawaskindlycommunicatedtousbyAkiraMasuoka.Lemma2.6.LetHbeabialgebraoveranarbitrarycommutativering,andletA,A??berightH-GaloisextensionsoveracommonalgebraBofH-coinvariants.AssumethatA??isrightB-faithfully?at.ThenanyH-comodulealgebramapθ:A→A??thatisidenticalonBisanisomorphism.

Proof.See[AG,Lemma1.14].??

Remark2.7.Masuoka’sLemma2.6impliesthefollowingfact:letAandA??beHopfalgebraextensionsofBbyHandsupposethatthereisaHopfalgebramapθ:A→A??suchthatthefollowingdiagramcommutes:

1BBAA??θHH11.1

IfA??isrightB-faithfully?at,thenθmustbeanisomorphism;cf.Remark2.2.ThefollowingpropositionisduetoE.M¨uller.

Proposition2.8([M3,3.4(c)]).Let1→B→?A?→H→1beanexactsequenceofHopfalgebras.LetJbeaHopfidealofAof?nitecodimensionandJ=B∩J.Then1→B/J→A/J→H/π(J)→1isexact.??

2.2.Isomorphisms

NowwestudysomepropertiesoftheHopfalgebrasgivenbyProposition2.3.ιπ

EXTENSIONS OF FINITE QUANTUM GROUPS

EXTENSIONSOFFINITEQUANTUMGROUPS5

De?nition2.9([A,2.2.3]).TheHopfcenterofaHopfalgebraAisthemaximalcentralHopfsubalgebraHZ(A)ofA.Italwaysexistsby[A,2.2.2].

Proposition2.10.Fori=1,2,let1→Ki→Ai→Hi→1beexactsequencesofHopfalgebrassuchthatKi=HZ(Ai).Supposethatω:A1→A2isaHopfalgebraisomorphism.Thenthereexistisomorphisms:K1→K2and:H1→H2suchthatthefollowingdiagramcommutes:

1K1

ι1A1ωπ1H1

H211.1K2ι2A2π2

Proof.Straightforward;see[G,2.3.12]fordetails.??

Thefollowinglemmaanditscorollarieswillbeneededlater.

Lemma2.11.Let1→B→A?→H→1beacentralexactsequenceofHopfalgebrassuchthatAisaNoetherian.IfB?C?HZ(A)isaHopfsubalgebrasuchthatπ(C)=HZ(H),thenC=HZ(A).

Proof.LetB?D?HZ(A)beaHopfsubalgebrasuchthatπ(D)=HZ(H).By

[Sch1,Theorem3.3],Aisfaithfully?atoverD.HenceDisadirectsummandofAasaD-module,see,forexample,[SS,3.1.9].SayA=D⊕M.ThenKerπ|D=DB+,sinceKerπ|D=Kerπ∩D=AB+∩D=(D⊕M)B+∩D=DB+.Besides,B?Dcoπ|D?Acoπ=B,whichimpliesthatD?tsintothecentralexactsequence1→B→D→HZ(H)→1.

Moreover,theextensionB?D??HZ(H)isHZ(H)-GaloisbyRemark2.2.NowtakingD=CandD=HZ(A)wegetthefollowingcommutativediagramwithexactrowswhichareHZ(H)-GaloisextensionsofB:

1BCHZ(A)HZ(H)1(5)HZ(H)

??1.π1BHence,byLemma2.6,itfollowsthatC=HZ(A).

Asimmediatecorollariesweget

Corollary2.12.AssumethehypothesisofLemma2.11.IfHZ(H)=k,thenB=HZ(A).

Proof.ThisfollowsbytakingC=BinLemma2.11.

ιπ??Corollary2.13([A,3.3.9]).Let1→K→?A?→H→1beacentralexact

sequenceof?nite-dimensionalHopfalgebras.IfHZ(H)=k,thenHZ(A)=K.??

Wenowgiveasu?cientconditionfortwoHopfalgebrasconstructedviathepushouttobeisomorphic.Let

1BιAπH1