the centralizer of an element in a lie algebra of type l.pdf


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854 Science in China Ser.A Mathematics 2004 V01.47 No.6 854_一861 The centralizer of an element in a Lie algebra of type三 LU Caihui&SHA0 Wenwu Department of Mathematics,Capital Normal University,Beijing 1 00037,China Correspondence should be addressed to Shao Wenwu( Received 0ctober 1 5.2003 Abstract In this paper,we investigate the Lie algebra L(A,Q,6)of type L and obtain the respective sufficient conditions for L(A,a,6)to be semisimple,and for z∞)=Fw as well, where 0≠叫C-L(A,Ol,6)and z(“)is the centralizer of u. Keywords:Lie algebra of type L semisimple Lie algebra,centralizer,total ordering. 1 Introduction Recently,a number of new classes of infinite dimensional Lie algebras have been introduced(see refs.[1—12]).Among them,the Lie algebra of type L was defined in ref. [1】as follows: Let A be a torsion free abelian group and F a field of characteristic 0.Set I= {1,2,·一,n]..For each i∈I,letthe nonzero map Ol{:A叫F be additive and瓯∈A, where 51,…,民are distinct.Let a=(O/1,a2,…,OZ。)and J=(J1,如,…,矗).Then under the product 礼 k,e可] ai(x—y)e。+∥一民, (1.1) 三=L(A,OL,5)=0。∈A Fe。becomes a Lie algebra,which is said to be of type L. We define the kernel ofa to be Ker(a)=n垒l Ker(at),while d is said to be non. degenerate if Ker(a)={0).For any nonzero element u=∑m江1 kie。t∈L(A,a,6), where觑∈F+=F一_[o)and Xl,X2,…,z。are distinct,we call the set supp(w)= 1 in the present paper. For convenience,we recall the main results in ref.[1】. Theorem 1.1(Theorem 2.5 of ref.[1】). Suppose that n1,J1:0,and the sub— space((口1(坑),&2(坑),···乜。(坑))fi∈I)c F”is 1-dimensional.Then L(A、a,61 is simple if and only if the following two conditions hold: 1.only one number in(Ofl(62),Of2(62),…,a。(如))is nonzero; 2.Ker(Of)=.[o].. Theorem 1.2(Theorem 3.6 of ref.[1】). Suppose that 6l=0.Then L(A,Q,6)is simple if and only if the following two conditions hold: 1 Z(eo)=Feo; 2 L(A,&,6)is simple. In ref.[11,the authors gave some open problems for the Lie algebras of type L.One of them is:does z(u)=Fu hold for all nonzero elements u in a simple Lie algebras L(A,a,6)?In this paper,we will give an affirmative answer to this question in Section 3. To do this,we need to investigate such Lie algebras and obtain some sufficient conditions for L(A,Q,6)to be semisimple in Section 2. 2 Semisimple Lie algebras of type L In this section,we give a sufficient condition for the Lie algebra L(A,a,6)to be semisimple,where by the semisimple Lie algebras we mean the Lie algebras not contain— ing any nonzero commutative ideals。 The following four lemmas will be used frequently later. Lemma 2.1·Let∑21∑;:1 ai,je酣。,=0.Then 1.V z∈A,we have∑:1∑;:1 nt,je卅%+。=o; 2·if xlk.Hence wehave p,∥]= 薹塾b(酗Y‘Yi-z-zj、)ey+y;+z1 1 1 / 竹文∑∑‰b(∑at(‘ ) 扣,也 {= j= \t= +a%(可+Yi—z—zAeF+蚍+。+勺一以)=0. (3.6) By using the above arguments repeatedly,we get oLt(∥+Y1一z—z1)=0 for t七. rnlen p,∥] =∑∑kihjak(y+玑一z—zj)e可+口。+对勺 i=1 j=l 如)=0. (3.7) By Lemma 2.1,one gets oLk(∥+Y1一z—z1)=0.Noting that ai(y+YI—z—Z1)= ai(可一z)=0 for all i≠k,we conclude Y+Yl—z—Zl∈Ker(a)={0),that is Y+Yl=z+Zl。as is desired. By(1)one can easily check that Z(w)=Fu fbr all nonzero u∈L(A,a,6)in me Lie algebra L(A,oL,6). (2)Show z∞)=Fu,for u=∑讧m 1咒∈L(A,oL,6)with s乱pv(K)c(xi+△). First assume that 4/△is not torsion free.Set/X。={_z∈13n∈Z s.t佗z∈△}. Then it is easy to see that/X C△and△‘is a group.Moreover,one can prove that △’/△is the minimal group containing all torsion elements of A/△and A/△1 is torsion free.By the definition of/X。,we know oLi(/X。)=0 for all i≠k and ak is injective on △。.By replacing/X by△7,we may assume that A/△is a torsion free abelian group. Then the arguments are completely similar to Theorem 3.1(2). The following result follows directly from Theorem 1.1 and Theorem 3.3. CoroHary 3.4. If the conditions of Theorem 3.3 hold,then L(A,ol,6)is simple To end up the paper,we give an example for explaining our Theorem 3.3. Example 2.Let△be a torsion free abelian group generated by 61,如,…,如and let Q1 be an injective map:△一C.Let G be a torsion free abelian group and let Q2,a3,…,an be the nonzero maps:G_C such that n墨2 Kerai=.(o]..Set A=/X×G.Then by regarding the group△and G as the subgroups of A.we define oLl(G)=0 and Qt(△)=o(i=2,3,…,礼).Thus the Lie algebra L(A,a,6)is defined just following the way as shown in the first section.By Theorem 3.3,we have z(u)=Cu for any nonzero element u∈L(A,a,6). Acknowledgements This work was supported by the National Namral Science Foundation of China(Grant No 10271081)and a Fund from Educational Department of Beijing(Grant No.2002KJ-100). 1.Osbom,J.M.,Zhao,K.,Infinite dimensional Lie algebras oftype L,Comm.Alg.,2003,3l(5):2445—2470 Copyright by Science in China Press 2004 万方数据 The centralizer of an element in a Lie algebra of type L 861 2.Osbom,J.M.,Zhao,K.,Generalized caftan type K Lie algebras in characteristic 0,Comm.Alg.,1997,25: 3325—_3360. 3.Osbom,J.M.,Zhao,K.,Infinite dimensional Lie algebras of generalized Block type,Proc.of AMS,1999, 127(6):164l~1650. 4.Djokovic,D.Z.,Zhao,K.,Derivations,isomorphisms,and second cohomology of generalized witt algebras, Trans.Amer.Math.Soc.,1998,350(2):643—664. 5.D10kovic,D.Z.,Zhao,K.,Derivations,isomorphisms,and second cohomology of Block algebras,Algebra Colloquium,1996,3(3):245—272. 6.Djokovic,D.Z.,Zhao,K.,Some infinite dimensional simple Lie algebras related to those of Block,J.Pure and Applied Alg.,1998,127(2):153—165. 7.Djokovic,D.Z.,Zhao,K.,Derivations,genel‘alized canan type S Lie algebras of charateristic 0,J.Alg.,1997, 193:14q一179. 8.Osbom,J.M.,New simple infinite—dimensional Lie algebras of characteristic 0,J.Alg.,1996,185:820一835. 9.Osbom,J.M.,Derivations and isomorphisms of Lie algebras of characteristic of 0,Studies in Advanced Ma血., 1997,49(1):95—108. 10.Osbom.J.M..Automorphisms of the Lie algebras W+in characteristic 0,Canadian J.Math.,1997,49(1): 1 19—132. 11.Osbom,J.M.,Passman,D.S.,Derivations of skew polynomial rings,J.Alg.,1995,176:417—448. 12.Rudakov,A.N.,Groups of automorphisms of infinite dimensional simple Lie algebras,Izv.Nauk SSSR,Ser. 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