对称线性Gr-范畴中的李代数.pdf 26页

'˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn对称线性Gr-范畴中的李代数黄华林1，杨毓萍21华侨大学数学科学学院，泉州　3620212西南大学数学与统计学院，重庆　400715摘要：本文中，我们主要关注那些具有非平凡结合性质的对称线性Gr-范畴中的李代数。这些李代数乃是具有一般结合线性GR范畴中的color李代数的自然对应。我们给出在对称线性GR范畴中李代数的约化形式，并且对一般李代数、李超代数和color李代数的统一形式的PBW定理。关键词：李代数，对称范畴，辫子，上循环，GR范畴中图分类号：O153,O154LIEALGEBRASINSYMMETRICLINEARGR-CATEGORIESHua-LinHuang1,YupingYang21SchoolofMathematicalSciences,HuaqiaoUniversity,Quanzhou3620212SchoolofMathematicsandStatistics,SouthwestUniversity,Chongqing400715Abstract:Inthispaper,westudyLiealgebrasinsymmetriclinearGr-categorieswithfocusonthosewithnontrivialassociativityconstraints.SuchLiealgebrasarenaturalcounterpartsofthewellknownLiecoloralgebraswhichliveinlinearGr-categorieswithassociativityisomorphismsbeingidentity.WegivethereducedformoftheLiealgebrasinsymmetriclinearGR-categoryandprovethePBWtheoremwhichisaunifiedformofthatforordinaryLiealgebras,LiesuperalgebrasandLiecoloralgebras.Keywords:Liealgebra,symmetriccategory,braiding,cocycle,Gr-categoryFoundations:SRFDP20130131110001AuthorIntroduction:Hua-LinHuang（1975-），male，professor，majorresearchdirection：algebra,Email:hualin.huang@hqu.edu.cn.Correspondenceauthor：YupingYang（1987-），male，assistantprofessor，majorresearchdi-rection：algebra,Email:yupingyang@swu.edu.cn.-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn0IntroductionBraidedtensorcategoriesareinterestedmainlybecauseofthedeepconnectionwithquantumgroupsandphysics.Ontheonehand,alargeclassesofbraidedtensorcategoriesisrepresentationcategoriesofHopfalgebrasandhencethestructureofsuchcategoriescanbeinvestigatedbystudyingthecorrespondingHopfalgebras;ontheanotherhand,thestructureofBraidedtensorcategoriesandthealgebrasinBraidedtensorcategoriesprovidemuchinformationoftherepresentationsandstructuretheoriesofHopfalgebras.ItiswellknownthatthebraidingofBraidedtensorcategoriesprovidesystematicsolutionsofYang-Baxterequation,andthisisoneofthemainreasonswhyBraidedtensorcategoriesandquantumgroupsarewidelyappliedinphysicsandintegrablesystems.Symmetriccategoryisabraidedtensorcategorywithsymmetricbraiding,itisimportantbecauseofavarietyofsymmetriescanbeprovidedfromSymmetriccategories.Inthispaper,westudyLiealgebrasinsymmetriccategories.BraidedlinearGr-categoriesarethemostwidelyusedbraidedtensorcategories.ByalinearGr-categorywemeansatensorcategoryconsistsoffinitedimensionvectorspacesgradedbyagroupGwithusualtensorproductandwithassociativeconstraintgivenbya3-cocycle.Inrecentwork,[1]givesacompletepresentationofthestructuresofbraidedlinearGr-categorieswhenGisabelian,hencethestructureofsymmetriclinearGr-categorycanbecounted.Withthishelp,muchmoreinformationandpropertiesofLiealgebrasinsymmetriclinearGr-categoriescanbeobtainedalthoughwewanttoestablishageneraltheoryofLiealgebrasinabstractsymmetriccategoriesinthispaper.OnecanalsoseethattheordinaryLiealgebras,LiesuperalgebrasandcolorLiealgebrasareallspecialcasesofLiealgebrasinsymmetriclinearGr-categoriesinsection3.1.thereisalonghistoryofthegeneralizationworkofLiealgebras.Liesuperalgebras[2]asfirstgeneralizationofLiealgebras,hasturnedouttobeapowerfultoolinmanycomponentsofmathsandphysicsbecauseofthesupersymmetryiswidelyusedintheseareas.AfterLiesuperalgebras,thegeneralizationisLiecoloralgebras.ThisconceptisassociatedwithanabstractgroupG,andwhenthegroupisZ2wecomebacktotheconceptofLiesuperalgebras,fordetailsthereaderisreferredto[3].TheconceptofLiealgebrasinsymmetriccategoriescanbeseenasthegeneralizationofLiecoloralgebras.TheconceptofLiealgebrasinsymmetriccategoriesorbraidedtensorcategoriesalsoappearedforalongtime.[4]givestheconceptofLiealgebrasinbraidedtensorcategorieswithtrivialassociativeconstraint,hisLiealgebrashaven-arymultiplicationsbetweenvarious-2- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cngradedcomponents.[5]givestheconceptofLiealgebrasinsymmetriccategoriesbutalsowithtrivialassociativeconstraint.Liealgebrasinabstractsymmetriccategoriescanbefoundfirstlyin[6],inthispaper,themainworkistoinvestigatetherelationsbetweensimpleLiealgebrasandquasisimpleLiealgebrasinthecategoriesandproveWestbury’sconjectureforK3-surface.Inthispaper,wemainlyinvestigatethepropertiesofLiealgebrasinsymmetriccategoriessimilartothatofclassicalLiealgebrasandLiesuperalgebras,suchasPBWtheoremandtherelationswithassociativealgebras.Thefollowingisadetaileddescriptionofthepaper.Insection2weintroducesomenecessarydefinitionsandfactsaboutbraidedtensorcategories.Insection3,wegivetheconceptofLiealgebrasandassociativealgebrasinsymmetriccategories,somebasicpropertiesofsuchLiealgebrasandassociativealgebrasareinvestigated.Insection4,weprovethatbraidedtensorfunctorpreservesLiealgebrastructure,andgiveareducedformofLiealgebrasinsymmetriclinearGr-categories.Insection5,representationsaredefined,andaPBWtypetheoremisprovedfortheLiealgebrasinsymmetriclinearGr-categories.1SymmetriccategoriesInthissection,werecallsomebasicdefinitionsonsymmetriccategoriesandinvestigatethesymmetricstructureofsymmetriclinearGr-categories.Thereaderisreferredto[7,8]and[1]fordetailsofsymmetriccategoriesandbraidedlinearGr-categoryrespectively.1.1symmetriccategoriesThroughoutthispaper,tensorcategorywillalwaysmeanamonoidalK-linearcategory,thatistosaytheobjectsofthecategoryareK-vectorspaces.Moreover,weassumethattheunitobjectsofthecategoriesarefieldKandtheleftandrightunitconstraintareidentities.Withtheseassumptions,tensorcategorycanbepresentedbyatriple(C,,a)whereadenotetheassociativityconstraint.ThatistosayforanyobjectsU,V,WofC,thereexistsanaturalisomorphismaU,V,W:(UV)W!U(VW)-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cninthecategorysuchthatthediagramsaU,V,W/(UV)WU(VW)(1.1)(f⊗g)⊗hf⊗(g⊗h)a′′′U′,V′,W′/U(VW)(UV)WandaU,V,W⊗idX/((UV)W)X(U(VW))X(1.2)aU⊗V,W,X(UV)(WX)aU,V⊗W,XaU,V,W⊗XidoU⊗aV,W,XU(V(WX))U((VW)X)commutewheneverf,g,haremorphismsinthecategoryandXisanyobject.Symmetriccategoryisatensorcategorywithasymmetricbraidingt.RecallthattisasymmetricbraidingifthereisanaturalfamilyofisomorphismstU,V:UV!VUindexedbyallcouples(U,V)ofobjectsinCsuchthattV,UtU,V=idU⊗VandthediagramstU,V⊗W/(VW)UU(VW)(1.3)lll6RRRRaU,V,WllllRRaV,W,UllllRRRRllRRR((UV)WV(WU)RRRRlll6RtRU,V⊗idWidV⊗tU,WllllRRRRllllRRR(llaV,U,W/V(UW)(VU)WandtU⊗V,W/W(UV)(UV)W(1.4)a−1lll6RRRRa−1U,V,WllllRRRW,U,VRllllRRRRllR(U(VW)(WU)VRRRRlll6RidRU⊗tV,WtU,W⊗idllVllRRRRllllRRR(a−1llU,W,V/(UW)VU(WV)commuteforallobjectsU,V,Wofthecategory.-4- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn1.2BraidedtensorfunctorNowlet(C,,a,t)and(C′,,a′,t′)aretwosymmetriccategories,(ϕ,φ)isbraidedtensorfunctorfromCtoC′ifϕisafunctorofthetwocategoriesandforallobjectsU,V,WofC,thereisanaturalisomorphismφU,V:ϕ(U)ϕ(V)!ϕ(UV)ofC′suchthatthefollowingtwodiagramscommute:′aϕ(U),ϕ(V),ϕ(W)(ϕ(U)ϕ(V))ϕ(W)/ϕ(U)(ϕ(V)ϕ(W))(1.5)φU,V⊗idϕ(W)idϕ(U)⊗φV,Wϕ(UV)ϕ(W)ϕ(U)ϕ(VW)φU⊗V,WφU,V⊗Wϕ(aU,V,W)/ϕ(U(VW)),ϕ((UV)W)′tϕ(U),ϕ(V)ϕ(U)ϕ(V)/ϕ(V)ϕ(U)(1.6)φU,VφV,Uϕ(tU,V)/ϕ(VU).ϕ(UV)1.3symmetriclinearGr-categoryInthissubsectionweintroducesomeusefulresultsaboutsymmetriclinearGr-categories.Mostofthedetailswillbefoundin[1]and[9].ωLetGbeagroup.ByalinearGr-categoryoverGwemeanatensorcategoryVecGconsistsoffinitedimensionalvectorspacesgradedbyGwiththeusualtensorproductandwithassociativityconstraintgivenbya3-cocycleωonG.Inthispaper,wealwaysletGbeanabeliangroup.ωForallabeliangroupG,thebraidstructureofVecGwasclassifiedin[1].Recallthatafunctionω:GGG7!K∗iscalled3-cocycleifω(ef,g,h)ω(e,f,gh)=ω(e,f,g)ω(e,fg,h)ω(f,g,h)(1.7)foralle,f,g,h2G,anditiscallednormalizedifω(f,1,g)=1foranyf,g2G.TheassociativityconstraintisdeterminedbyaU,V,W((uv)w)=ω(x,y,z)u(vw)for-5- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnu2Ux,v2Vy,w2Wzwherex,y,z2G.SinceGisabeliangroup,wecanalwayssetG=Zm1


Zmn.LetgithegeneratorofZmiandζmbeaprimitivem-throotofunit.DefineAbethesetofallintegersequences(α1,


,αl,


,αn,α12


,aij,


,αn−1,n,α123


,arst,


,αn−2,n−1,n)(1.8)suchthat0αln.Themapisalsodefinedbyγ,andnnobviously(5.20)and(5.21)arealsocorrect.Defineγ:T(L)7!G(L)suchthat∑γjTn(L)=γm.(4.22)0≤m≤nγiswelldefinedby(5.20),(5.21)impliesγisanalgebramorphism.FromthedefinitionofγwecanseethatγissurjectiveandvanishesontheidealI.Soγinducesasurjectivealgebramapβ:S(L)7!G(L).SuchthatthediagramT(L)P/T(L)/I(4.23)uuuγuuuuβzuuG(L)commutewherePisCanonicalprojection.NowwecangivethePBWtypetheoremofiealgebrainsymmetriclinearGr-Category.Theorem4.6.Themapβ:S(L)7!G(L)definedasaboveisanalgebraisomorphismin(Vecw,R).G-24- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnProof.Thediscussionabovehasshowedthatβisansurjectivealgebramap,soweonlyneedtoprovethatβisalsoinjective.Atfirstweshowthatforanyx2Tm(L)J,thehomogenouscomponentxmofxofZ-degreemliesinI.Letxmbethelinearcombinationofxi,whereiistheindexsets.ByProposition5.5,wehaveanLiealgebramapρ:L7!gl(S(L)).Itextendsanalgebramapρ:U(L)7!gl(S(L))satisfyρ(J)=0.Since1
ρ(x)isapolynonimalwhosehighestdegreetermiscombinationofziwhichisthesameasthatofxm,but1
ρ(t)=0impliestheCombinationofziis0inS(L),hencexm2I.Letx2Tm(L),toproveβisinjective,wemustprovethatifβ(P(x))=0impliesx2I.Sinceβ(P(x))=γ(x),soβ(P(x))=0impliesγ(x)=0,henceπ(x)2Um−1(L).Sothereisx′2Tsuchthatπ(x)=π(x′)2U(L),soxx′2Jandhencexx′2TmJ,thism−1m−1forcethehomogenouscomponentofxx′ofdegreemliesinI.Butx2Tm,x′2T,som−1thehomogenouscomponentofxx′ofdegreeMisxandhencex2I.TherearemanycorollariesofPBWtheory.WeonlygivethefollowingtwoandtheproofarethesameasthatofordinaryLiealgebrasandSuperalgebras,soomitted.Corollary4.7.Themapi:L7!U(L)whichiscompositionofnatureinjectionL7!T(L)withtheCannicalprojectionT(L)7!U(L)isinjective.Corollary4.8.Letfx1


xngbetheK-basisifL,thenf((


(xi1xi2)


xim−1)xim)ji1i2


imgalongwith1,isbasisofU(L).Remark4.9.WehaveintroducedordinaryLiealgebra,LieSuperalgebraandLieColoralgebraasspecialcaseofLiealgebrainsymmetricCategoryin3.1.NowifwechoosethecorrespondsymmetriclinearGr-Category,thenthetheorem5.6isthePBWtheoremofsuchLiealgebras.SoTheorem5.6isactuallythegeneralizedversionofPBWtheoremofthatofordinaryLiealgebras,LieSuperalgebrasandLieColoralgebras.参考文献（References）[1]Huang,Hualin;Liu,Gongxiang;Ye,Yu.:OnbraidedlinearGr-categories.Preprint.[2]V.G.Kac,Liesuperalgebras.advanceinMath.26,no.1:8-96,1997.-25- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn[3]Scheunert,M.GeneralizedLiealgebras.J.Math.Physics20:712-720,1979.[4]Pareigis,B.OnLiealgebrasinbraidedcategories.QuantumGroupandQuantumS-pacesBanachCenterpublications,Volum40.IstituteofMathmaticsPolishAcademyofSciencesWarszawa,1997.[5]Kochetov,M.GeneralizedLiealgebrasandandcocycletwists.ComminAlgebra,36:4032-4051,2008[6]Rumynin,D.Liealgebrasinsymmetricmonoidalcatedories.SiberianMath.Jour.2013.arXiv:1205.3705v2[math.QA]25May2012.[7]Etingof,P.;Gelaki,S.;Nikshych,D.;Ostrik,V.:Tensorcategories.LecturenotefortheMITcourse18.769,2009.availableat:www-math.mit.edu/etingof/tenscat.pdf[8]Kassel,C.Quantumgroups.GraduateTextsinMathematics,vol.155,Springer,2000.[9]Majid,S.Foundationsofquantumgrouptheory.CambridgeUniversityPress,Cambridge,2006.-26-'