Loop代数的有限维不可约模的导子  

The Derivations of the Finite Dimensional Irreducible Modules over the Loop Algebras

作　　者：连海峰[1] 

机构地区：[1]厦门大学数学科学学院,福建厦门361005

出　　处：《厦门大学学报（自然科学版）》2009年第4期464-466,共3页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(10671160)资助

摘　　要：设G是有限维复单李代数,A=C[t±1],GA∶=GCA是loop代数.设a是非零复数,M是有限维不可约G-模,则Ma∶=M是不可约GA-模,其中xf(t)在Ma上的作用为xf(t).v=f(a)xv.首先证明,若李代数L的有限维模都完全可约,那么L的有限维模的导子都是内导子.接着利用有限维复单李代数的有限维模都完全可约这一性质,计算GA-模Ma的导子.证明了当且仅当M是G的伴随模时,Ma存在外导子,这也说明了loop代数的有限维模不是完全可约的.Let G be a simple Lie algebra over the complex field C ,A=C[t^±1]and GA：=G×cA be the loop algebra. For any nonzero complex number a and any finite dimensional irreducible G-module M, Ma 1=M is an irreducible GA-module. Where, the the action of x×f（t） on Ma is defined by sending m to f（a）xm. In this paper, the author firstly proved that if any finite dimensional modules of Lie algebra L is completely reducible, then the derivations of such modules are all inner derivations. Using the fact that any finite dimensional modules of a complex simple Lie algebra is completely reducible, he computed the derivations of GA-module M^a ,and proved that there exists outer derivations of M^a if and only if M is Gs adjoint module.

关 键 词：LOOP代数 不可约模 导子 

分 类 号：O153.5[理学—数学]