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一类Weyi型单李超代数 Vot.28(2008) No.4 数学杂志 J.ofMath.(PRC) ACLASSOFS】瓯御P皿mSI瑾·ERAI正理mRA0IFⅥ/脚。TYPE‘ QINYu-fan91,WANGYin91,CHENYu—zhen2 (1.Dept.ofAppliedMath.,DalianUniversityofTechnology,Dalian116024,Chin口) (2.Dept.ofMath.,HenanInstituteofScienceandTechnology,Xinxiang453003.C...

一类Weyi型单李超代数
Vot.28(2008) No.4 数学杂志 J.ofMath.(PRC) ACLASSOFS】瓯御P皿mSI瑾·ERAI正理mRA0IFⅥ/脚。TYPE‘ QINYu-fan91,WANGYin91,CHENYu—zhen2 (1.Dept.ofAppliedMath.,DalianUniversityofTechnology,Dalian116024,Chin口) (2.Dept.ofMath.,HenanInstituteofScienceandTechnology,Xinxiang453003.China) Abs栅I:Inthispaper,westudytheconstructiontheoryofsimpleLiesuperalgebras.By usingthemethodoftensorproduct,aclassofassociativesuperalgebrasandLiesuperalgebrasof Weyltypearedefined.Inaddition,weprovethesufficientandnecessaryconditionsofthe simplicityoftheassociativeandtheLiesuperalgebras. Keywords:Liesuperalgebra;associativesuperalgebra;supercommutative 2000MRSubjectClassification:17850 DocumentCodefA Arl}icleID:0255—7797(2008)04-0373.06 1 Introduction SimpleLiesuperalgehrasandsimpleassociativesuperalgebrasplayanimportantrole intheresearchofLiesuperalgebras.FourseriesoffinitedimensionalLiesuperalgebrasof Caftantypeinprimecharacteristicwereconstructedandtheirsimplicitieswereproredbv ZhangYong—zhenginreferenceE13.MengXian-jiconstructedafinitedimensionalmodular Liealgebra∑andproveditssimplicityinE23.In2005,ZhangYong—zhengconstructeda finitedimensionalmodularLiesuperalgebra—oddHamitonmodularLiesuperalgebraHO andprovedthatitwassimple.InAugust,AlbertoElduqueconstructedanewsimpleLie superalgebraoverafieldofcharacteristic3inE43. RecentlywehaveobtainedmanynewsimpleLiealgebrasthroughgeneralizingsimple LiealgebrasofcartantypeinreferencesE7--一103.In[10],SuYu—caiandZhaoKai—ming studiedLiesuperalgebrasofWeyltypeandassociativesuperalgebrasofWeyltype constructedfromacommutativeassociativealgebrawithanidentityanditscommutative derivationsubspace,andprovedtheirsimplicity.Naturallyweconsiderwheatherwecan applythismethodtoLiesuperalgebras. Inthispaper,weconstrutaclassofLiesuperalgebrasandassociativesuperalgebrasof WeyltypefromasupercommutativeassociativesuperalgebraA(2,2)withanidentityand *Receiveddate:2005-12—23Accepteddate:2006—07-13 Biography:QinYufang(1981一),female-hornatXinxiang.Henan.Master,majorinLiealgebra. 万方数据 374 JournalofMathematics VoL28 theassociativesuperalgebraFED2 generatedbythespecialderivationsubspaceDof A(2,2).Andweprovethattheyarebothsimpleifandonlyif,A(2,2)isD-simpleand F[D]actsfaithfullyonA(2,2). PutFbeafieldofarbitrarycharacteristic.LetA(2,2)=【,(2)OA(2),whereU(2) isadividedalgebrawiththegeneratedset{z:lz≯l口1,a2∈No)onFandA(2)awedge algebrawithindeterminatesz3,z‘onF,then{z}磅z争矸l口l,口2∈No,码,口4∈{0,1}) constitutesaF-basisofassociativealgebraA(2,2).Wewilldenote^(2,2)by^.The multiplicationofA(2,2)isdefinedasfollows: 砰孝砰孝:(口1+届1(口2+届1砷鸲磅鸲,而而一一五而,砰磅鸳才:孝。琦磅. |岛 /t瑶 / LettheZ2一gradationofA(2,2)beinducedbythetrivialZ2一gradationofU(2)andthe canonicalz2一gradationofA(2).ThenA(2,2)isaassociativesuperalgebra.Foreach z:lz≯z争z:‘∈^(2,2),weobtainthatthedegreeofz:1z》z}z:.is(口3+气)1,whichis denotedbyd(z:1z≯z}z:.). LetD1,D2,D3’D4bethespecialderivationsofA(2,2),whereDl’D2∈ Derv(A(2,2)6,D3,D‘∈Derp(A(2,2))T;See[9].LetDbetheF—linearspacespanned byDl,D2,D3,D‘.SinceDfD,=(一1)。‘q’。‘叩D,Di,weknowthatDgeneratesthe associativesuperalgebraF[D]withaF-basis(D“’一研lD;2明3D=·laEJ). Denote J亍{口=(r/1,a2,a3,a4)fm口。∈No涵,口·∈{0’1)).I口I=室¨(:)=血i-I严bi); J(口)={f∈JC‘≤ai,i=1,2,3,4)Ia<6甘lal(z}z孑z;3。:‘)=O. Ifi=j∈{3,4},then(矸z争z:3z:‘)一0.Otherwise,wemayassumethat i=3,.f一4,then(D3,D‘)(z}工;2工}砰)=0.Theclaimisproved. 万方数据 No.4 AclassofsimpleLiesuperalgebraofWeyltype 375 SetA[D]一A(2,2)oF[D].ThenanyelementofA[D]hasadescription ∑扯。OD“’,‰∈^(2,2),D“’∈F[D].WewilldenoteUaOD“’byu。D“’. AndwehaveaZ2一gradationofA[D],i.e. ^[D]6一A(2,2)6oF[D]-④A(2,2)3oF[D]T, A[D]_=A(2,2)3oF[D]-④A(2,2)iOF[D]r. Forallz一∑“。D“’∈A[D],wesaythatzhasleadingdegreedeg(x)一6and 口∈, leadinglevellev(x)一lbI,ifU6≠Oandforalla∈J,ua≠口<6.Definelev(0)=一∞.We definethesupportofzastheset{a∈J}U。4:0}. Nowwedefinealinearmapping: 0:A[D]一Hom(A(2,2),A(2,2)) uD‘。’卜+0(uD‘4’), where日(“D‘4’)(z)=U·D‘4’(z)=口·(D1(Dl⋯Dl(D2⋯(D4(z))⋯),工∈A(2,2). Toobtainanassociativesuperalgebra(A[D],·)SOthat0isahomomorphismof associativesuperalgebra,wedefinetheassociativeproductasfollows: uD“).vD∞=越∑(一1)烈D(”。M㈨-"c4,1d(D(P订’(口)D“’(臼)D‘一。D∞,(1) c∈』(4) 、c, where“,口arethehomogeneouselementsofA(2,2),a,b∈.厂.wecalltheassociative superalgebra(A[D],·)anassociativesuperalgebraofWeyltype. Definethebinaryoperation(<·,·>)astheusualinducedLiesuperoperationon (A[D],·)SOthat(A[D],<·,·>)isLiesuperalgebra. Then0inducesahomomorphismofLiesuperalgebrawhichisstilldenotedas只 ForaU工∈F,uD‘4’∈AED],by(1),wehave(z,uD‘4’>一0,whichshowthatFis containedinthecenterof(A[D3,(·,·>).Let万[D]=^[D]\F. ThenLiesuperalgebraT[D1iscalledLiesuperalgebraofWeyltype. 3 Mainresultsandproof Theorem3.1 Liesuperalgebra万ED] andF[D]actsfaithfullyonA(2,2). issimpleifandonlyifA(2,2)isD—simple, ProofSupposethatJisanonzeroD-stableidealofA(2,2).Byvirtueof(1)andthe equalityJ[D]一JOF[D],wehave(I[D1,AED3>∈虻D]and J[D]=J6OF[D]iOlrOF[D]ToJ6oF[D]T①IrOF[D]6, then虻D]isZ2一gradedsubspaceof^[D]and让D]istheidealofA[D3.Asj≠o,there existso≠uEJ,whichyieldso≠uDl∈J[D]\F.ThusJ[D]can’tbecontainedinF.By thesimplicityofA[D],wehaveJ[D]=AJ-D],whichinturnyields忙A. ThereforeA(2,2)isD—simple. Since0isthehomorphismofLiesuperalgebra,ker0istheidealofA[D].Bythe simplicityof-K[D],wehaveeitherker口=A[D]orker口∈F.Ifker口=A[D],then 万方数据 376 JournalofMathematics V01.28 A[D]actstriviallyonA.Inparticular,DactstriviallyonA,whichcontradictsthefact thatDisthenonzerolinearspace.Itfollowsthatker洋AI-D-I,henceker眶F.Forall uEkerO.then口(u)=o,especiallyO=O(u)(1)=u,soker0=0andAEDlactsfaithfully onA(2,2).Inparticular,F[D]actsfaithfullyonA(2,2). Weprovethesufficientconditionofthetheoremthroughseverallemmas. Lemma3.I A[D]actsfaithfullyonA(2,2). ProofSupposethatkerO≠O.LetPbetheminimalsupportsizeofnonzeroelementin ker良Thenthereexists ∈kerOwithui≠O,i=1,2,⋯p;af∈J.Put , K(2,2)=span{vp∈A(2,2)I了口;∈A(2,2),∑riD‘1’∈Ker0}. Thenwehave‰∈K(2,2).Asker0isthez2一graded ’ ∑riD“;)_ p ∑-iD‘4一,+ i=1 subspaceof^[D],wehave 户 ∑ 云fD‘~’ i=l i=1 i=1 wherethefirsttermofthesummandintheright—handiscontainedinker岛andthesecond containedinker岛.ThenK(2,2)isZ2一graded.HenceK(2,2)isanonzero乙一graded ‘ linearsubspaceofA(2,2).ForallDeD,wehayed=∑k』D』with愚,∈F.Supposethat 』=1 讪beahomogeneouselement.Then i一1 , riD吨’>=∑3(v。)班’∈kerO i=1 thenD(ap)∈K(2,2),whichshowthatK(2,2)isD-stable.ThereforeK(2,2)isa D-stableideal.SinceA(2,2)isD-simple,wehaveK(2,2)一A(2,2),especially 1∈K(2,2),thenj∑口iD‘di’+D‘4,’∈ker乱Fora11D∈D,wehave面 p—l’一1 ker09<西,∑让D‘~’+D‘’’>=∑D(vi)D‘4i), whichcontradictstheminimalsupportsize.ThenD(vi)=0,wherei一1,2,⋯p一1,which p—l impliesthatviEF(i=1,2,⋯p--1).HenceD‘口,’+∑riD‘~’∈F[D]nker0andF[D] actsfaithfullyonA(2,2),acontradiction.Thereforeker0=0,i.e.A[D]actsfaithfully onA(2,2). NowsupposethatLisanidealofA[D]withF≤三LandF≠L. ‘Lemma3.2 If越istheelementwithminimalleadinglevelinL\F,thenu6A(2,2). ProofSupposethat越硭A(2,2).Thenwehavelev(u)>0.Wemayassumethat乱is ahomogeneouselementofL.Asu6L\F,wehave“=让o+M’sothatuo∈A(2,2)andall termsof“’havepositivedegrees.Observethat^[D]actsfaithfullyonA(2,2),then thereexistsahomogeneouselementbofA(2,2)sothatM。(6)≠0.Puty=<“,b)∈L. Thenwehavey=弘+y。,whereY。∈A(2,2)andalltermsofy。havepositivedegrees. Thusy=<“。+uo,6>+<“。,6>.Supposethat口。=∑“。D‘",whereUaisthe “DH ,∑㈦ ,∑,~D< 万方数据 No.4 AclassofsimpleLiesuperalgebraofWeyltype 377 homogeneouselementofA(2,2)anda≠(O,0,0,O).By(1)wehave (H’,6>=∑(M。∑ (一1)。‘。‘一’。(们H4,1d(。‘”。’(Ⅱ1D‘c’(6)D‘一d).(2) arcJ f∈』(d)·f≠(O,0.0,O) 、c/ Observethat口≠(o,0,o,o),wehaveY=芝:UaD‘神(6)=“。(6)≠0,whichshowthatY石 =Y0+Y。≠0.By(2),wehavelev(y)≤lev(u)~1.Bytheminimalchoiceoflev(u).we havey=弘=<甜,b>∈F.Weclaimthat^(2,2)≠F+Fb.Ifbcontainsneitherz3nor丑, thenthereexists623∈^(2,2)\F+F6.If6containseitherz。or2‘,wemayassumethat b=X*11z孑z3,thenthereexistsz}+1z}∈^(2,2),z:1+1工}硭F+F6.Theclaimisproved. WechooseP∈^(2,2)\F+F6,thenz=<乱,P>∈L.Usingthesameargumentasabove. wehavelev(z)≤lev(u)一1.Similarly,bytheminimalchoiceoflev(u),z=<“,P>∈F. Let可一(H,be>∈L.ThenwehaveyoP+(一1)。‘∞。‘。’bz--口=0.Since1,b,PisF-linearly independentandyo≠0,wehaveu圣F,thenu∈L\F.Clearly,口∈A(2,2),thenlev(口)= 0=xDj(u)∈L,thenADi(“)∈L andADl(u)ELnA.LetMCLnAbethemaximalidealof(^,·)containingADi(“). ItisclearthatM+B(M)£L,whichimpliesthatM+Di(M)互LnA.Letb6A(2,2) bethehomogeneouselement.Forallz,y6Mwehaveb(x+Di(y))∈M+Dr(ND,which showthatM+q(M)istheidealofA(2,2).BythemaximalofM,wehaveM+Dj(加 ∈M,thenD(M)∈M,whichinturnyieldsthatMisaD-stableideal.ObservethatM≠ 0,thenM—A(2,2).HenceA(2,2)∈L. Lemma3.4 A[D]=L. ProofItisclearthatA[D]∈L.weclaimthatxD‘曲∈L,foreachzD∽∈^[m3 withaI>0.Fora=(czl,a2,0,0),wetakeb一(口l,口2,1,O).Thenweha【vezD‘。)一 (xD‘∞,工3>∈L.Othercasesaretreatedsimilarly,thenwehavezD‘n1,n2·l,o): ∈L;xD‘4l’42’o’1’=∈L;xD‘4l·a2·1,1)= ∈L.ThenA[D]CLandA[D]一L.Nowthesufficiencyoftheoremfollows. Theorem3.2 TheassociativesuperalgebraA[D]issimpleifandonlyif^(2,2)is D—simple,andF[D]actsfaithfullyonA(2,2). ProofNecessity.SupposethatIisflnonzeroD-stableidealofA(2,2),thenIED3is theidealofA[D].SinceA[D]issimple,thenI[D1=A[D],whichimpliesthatJ=A, henceA isD—simp giveninTheorem3 e.Theproofofthesecondpartiscompletelyanalogoustotheone 1. Sufficiency.SupposethatJ[D]isanonzeroidealoftheassociativesuperalgebraA FD],thenIED]istheidealofLiesuperalgebraA[D].Bytheorem3.1,wehave虻D]∈ ForⅡD]+F=A[D].Intheformercase,itisclearthatⅡD]=AED3.Inthelatter case,sinceDI∈AED]andDl晤F,wehaveD1∈J[D],whichimplies1∈I,then 万方数据 378 JournalofMathematics VoL28 兀D]一A[D].Therefore,AED]issimpleassociativesuperalgebra,asdesired. References: [1]ZhangYongzheng.Finite-dimensionalLiesuperalgebrasofCartantypeoverfieldsofprime characteristic[J].ChineseScienceBulletin:1997,42(7):720-724. [2]MengXianji.Finite-dimensionalLiesuperalgebrasoverfieldsofprimecharacteristic[J].Journalof ShenyangNormalInstitute.2002,20(2):81—85. [3]LiuWende.Finite-dimensionalmodularLiesuperalgebrasofCartantypeHO[J].ActaMathematica Sinica.2005,48(2):319-330. [4]ElduqueA.NewsimpleLiesuperalgebrasincharacteristic3[J].JournalofAlgebra.2006.296(1): 196—233. [5]OsbornJ.M.Newsimpleirdinite-dimentionalLiealgebrasofcharacteristic0[J],JournalofAlgebra. 1996,185(3):820—835. [6]PassmanD.P.SimpleLiealgebrasofWitttype[J],JournalofAlgebra.1998,206(2):682·692. [7]XuXiaoping.NewgeneralizedsimpleLiealgebrasofCartantypeovera fieldwithcharacteristic0 [J],JournalofAlgebra,2000,224(1):23—58. [83SuYucai。ZhaoKai-ming.SimplealgebrasofWeyltype[J].ScienceinChina.2000.30(12):1057— 1062. [9JZhangYongzheng,LiuWende.ModularLieSuperalgebras[M].Beijing:Scientificpublishinghouse, 2004 一类Weyi型单李超代数 秦玉芳1,王 颖1,陈玉珍2 (1.大连理工大学应用数学系,辽宁大连116024) (2.河南科技学院数学系,河南新乡453003) 擅要:本文研究了单李超代数的构造理论.借助于张量积方法,定义了一类Weyl型结合超代数和一 类Weyl型李超代数,并且证明了这类Weyl型结合超代数和Weyl型李超代数是单的充分必要条件. 关键词:单李超代数;单结合超代数;超交换 MR(2000)主题分类号:17850中图分类号:0153.5 万方数据
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