Loop代数的有限维不可约模的导子

设G是有限维复单李代数,A=Ct±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.

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

Let G be a simple Lie algebra over the complex field C,A=Ct±1], and GA: =GCA be the loop algebra. For any nonzero complex number a and any finite dimensional irreducible G-module M, Ma: =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 Ma,and proved that there exists outer derivations of Ma if and only if M is G's adjoint module.