The Heisenberg Lie Algebra Of The Automorphism Group And A Model Of Kac-moody Algebra With Product Modules Completely Reducible,

This thesis consists of two parts. Part I deals with the automorphism group of Heisenberg Lie algebra and Part II deals with the standard Kac-Moody algebras and the completely reducibility of integrable modules.In Part I , we first present two defining forms of Heisenberg Lie algebra. From these-two forms, we determine the automorphism groupAut(H) of the (2n + 1) -dimensional Heisenberg Lie algebra H (seeTheorem1. 1). Moreover, some subgroups of Aut(H) are obtained, such asthe inner automorphism group, the central automorphism group, the involutional automorphism group, the first and the second extremal automorphism group. More concretely, it has been proved that if n = 0,then every element of Aut(H) is an inner automorphism and Aut(H) C' (see Theorem 1.12) ; if n = 1, then every element of Aut(H) is the multiplicationof finitely many inner automorphisms, central automorphisms, involutionary automorphisms , and the first extremal automorphisms (seeTheorem 1.13); and if n = 2 , then every element of Aut(H) is themultiplication of finitely many inner automorphisms, central automorphisms, involutionary automorphisms, and the first and the second extremal automorphisms (see Theorem 1.14).In Part II , we introduce the definition of so called standard Kac-Moody algebra by using g(A)-module, and prove that Serre relationis the defining relation of any standard Kac-Moody algebrag(A) . Furthermore, it has been proved that g(A)is standard if and only if any integrable highest weight module of g(A) is irreducible. Then itfollows directly that the integrable module of a standard Kac-Moody algebra, which belongs to category 6, is completely reducible (see Theorem 2.5). Also we proved that any proper subalgebra of a standardKac-Moody algebra g(A) is standard, where A1 is any principle submatrix of A (see Theorem 2.6).