Some Properties Of L-type Lie Algebras And Lie Algebra G (a) Derivation Algebras

In recent years, the theory of infinite dimensional simple Lie algebras has become an important branch in Lie theory. The structure theory,such as derivations ,isomorphic classes, the second cohomology group, the automorphisms group, and the classification of Harish-Chardra modules. The infinite dimensional Lie algebras developed fastly in 1990's, especially in the area of Kac - Moody algebras generalized Yirasoro algebras , generalized Block algebras . generalized Cartan type algebras and Weyl type algebras.This paper consists of two parts. In part I, we investigate the Lie algebra of type L. These algebras arose as one subclass in the recent calssfication of generalized Block algebra. They are defined as follows. Let A be a torsion free abelian group and F be a field of characteristic 0. Set I = {1. 2, .... n}. For each i S /, let the nonzero map ai : A - F be additive. Set a = (a1,a2,...., an) and , where are distinct.. Then under the product Fex becomes a Lie algebra, we call the Lie algebra L(A.a. ) a Lie algebra of type L. In this part ,we obtain that Z(w) = F(w) holds for all simple Lie algebra of type L . where and Z(w) is the centralizer of w. Otherwise,we shall prove that the center of the Lie algebras of type L is 0. Finally,we prove an important result that all the Lie algebra of type L are scmisimple algebras. In part II,we discuss the derivation algebra Der(g(A)) of the contragre-dient Lie algebra g(A),associated to any complex n x n matrix A. When A is a generalized Cartan matrix, g(A) is the Kac - Moodey algebra assoiated to A. So the result obtained in this part also holds for any Kae- Moody algebra g(A).