A Problem About The Intersection Of Orthogonal And Symplectic Algebras

Let V be an n-dimensional vector space over an algebraically closed field F of characteristic 0, where n = 2m. We call C the intersection of an orthogonal and a sym-plectic subalgebra of gl(n,F). In this paper, we mainly use complex semisimple algebra's knowledge. First, we give the structure formula of C. Second, we show that any n x n symmetric matrix A is congruent under the symplectic group to the direct sum of a identity matrix and a tridiagonal matrix. At last, we express the condition under which the dimension of C can get the maximal and minimal value.It is organized as follows:Chapter 1: we introduce the the development of Lie algebra and the background of this paper.Chapter 2: we first express a proposition about the intersection of orthogonal group and a symplectic group. And then, we talk about the maximal and minimal dimension of the simple Lie algebras which are the intersection of orthogonal algebras and symplectic algebras over an algebraically closed field F of characteristic 0.Chapter 3: Take a further look at the development of the complete Lie algebra which conclude the simple Lie algebra.