典范Kac-Moody代数与可积模的完全可约性

The Standard Kac-Moody Algebra and Completely Reducible of Integrable Module

我们利用g(A)-模引进所谓典范的Kac-Moody代数的定义,证明了Serre关系式是任意一个典范Kac-Moody代数g(A)的生成元定义关系(定理2).证明了g(A)是典范的当且仅当g(A)的任一可积最高权模不可约(定理3).从而直接得出:典范Kac-Moody代数g(A)的属于范畴O的可积模都是完全可约的(定理5).证明了典范Kac-Moody代数g(A)的任一真子代数g(A_1)也是典范的,此处A_1是A的任一主子阵(定理6).

In this paper,we introduce the definition of so called standard Kac-Moody algebra by usingg（A）-module,and prove that Serre relation is the defining relation of any standard Kac-Moody algebra g（A）（see Th.2）.Furthermore,it has been proved that g（A） is standard if and only if any integrable highest weight module of g（A） is irreducible（see Th.3）.Then it follows directly that the integrable module of a standard Kac-Moody algebra,which belongs to category O,is completely reducible（see Th.5）.Also we proved that any proper subalgebra g（A_1） of a standard Kac-Moody algebra g（A） is standard,where A_1 is any principle submatrix of A（see Th.6）.