系数在复数域C上的Kac-Moody代数的低维上同调

On the Low Dimensional Cohomology of Kac-Moody Algebras With Coefficients in the Complex Field

§1引言和记号 李代数的上同调性质与其本身的结构有着密切的联系.本文我们将给出李代数G(A)的系数在C上的低维上同调.无限维李代数的上同调的计算比有限维的复杂得多.而且即使在有限维的情况,这样的上同调也是难计算的.因此与[1]一样,我们采取直接的方法计算.主要结果是定理3.1.最后指出,此结果推广了[1].

The aim in this paper is to compute the low dimensional cohomology of some infinite dimensional Lie algebras.Let G(A) be the Lie algebra associated to any complex matrix A,G' (A) be its derived Lie algebra. In this paper, the structure of 2-dimensional cohomology H2(G(A)), H2(G'(A)) with coefficients in the complex field,and an evaluation of dimension of 3-dimensional cohomology H3(G(A)) are given. Moreover, if A is a generalized Cartan matrix (then G(A) is called Kac-Moody algebra).The dimensions of H3(G(A)), H3(C'(A)) are accurately given.