Vector Of Real Roots And Imaginary Roots Of Kac-moody Algebra Vector And Schubert,-submodule Of Some Nature

There are two parts in this article.Part I is mainly discussing an elementary problem in Kac-Moody Algebra:how to describe the real and imaginary root vectors corresponding to a given real or imaginary root? We reqire the generalized Cartan matrice satisfy the following condition that a- is zero or less than -2 for all 1 i,j2. We specially introduce lemma3.3,an expression of the real set of g(A).Then in chapter 4,we denote n as the sum of some root spaces ga. And we prove that n is a commutative Lie algebra and a Schubert submodule Vw can be expressed by Vw = U(n,}.vw.At last in chapter 5,we give proofs to some properties of real and imaginary roots of g(A) and a guess to express the dimensions of Schubert submodules,whose proof is encountering some difficulty at present.