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
本文档为【一类Weyi型单李超代数】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。