中文摘要：

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M^at=M是不可约GA-模，其中x×（t）在M^a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M^a的导子．证明了当且仅当M是G的伴随模时，M^a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．

英文摘要：

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.