可换环上一些上三角矩阵李代数的导子.pdf

273 Vol.27, No.3 20078 JOURNAL OF MATHEMATICAL RESEARCH AND EXPOSITION Aug., 2007 Article ID: 1000-341X(2007)03-0474-05 Document code: A Derivations of Certain Lie Algebras of Upper Triangular Matrices over Commutative Rings WANG Deng-yin, YU Qiu, OU Shi-kun (Department of Mathematics, China University of Mining and Technology, Jiangsu 221008, China ) (E-mail: wdengyin@126.com) Abstract: Let R be an arbitrary commutative ring with identity. Denote by t the Lie algebra over R consisting of all upper triangular n by n matrices and let b be the Lie subalgebra of t consisting of all matrices of trace 0. The aim of this paper is to give an explicit description of the derivation algebras of the Lie algebras t and b, respectively. Key words: derivations of Lie algebras, commutative rings. MSC(2000): 17B CLC number: O152.2 1. Introduction Let R be a commutative ring with identity and R∗ the group of invertible elements of R. Let M (R) be the R-algebra of n by n matrices over R that has a structure of a Lie algebra over n R with the bracket [x, y] = xy − yx. We denote by t (resp., n) the subset of M (R) consisting of n all upper triangular (resp., strictly upper triangular) matrices. When n 1, let b be the subset of t consisting of all matrices of trace 0. Cao[1−3] described the automorphism groups of t, n and b respectively, when they are viewed as Lie algebras. Jϕndrup[4] gave a complete description of the derivations of t, when t is viewed as a ring. 2. Preliminaries Following the notations in [1] mainly, we denote