The Structure Of Upper Triangular Matrix Lie Algebra T(n, F)

Let F be a field of numbers. The set t(n,F) consisting of all upper triangular matrixes of order n over F, is a sub-algebra of general linear Lie Algebra gl(n,F). This article focuses on the study of t(n,F). In the first section, we introduce the main concept. In the second section,we discuss the ideal of t(n,F), nilpotent radical and radical. In the third section, we discuss the Cartan sub-algebra, killing form, structural formula and some properties of t(n,F).When we do the research of the ideal, we determine two special forms of the ideal at first; and then, through the two forms of the ideal, we intersect and sum to get more ideals. We use the block matrix to simplify the calculation on the study of nilpotent radical, which provide a lot of skill to prove the maximality, We start from the definition and calculate to get the Cartan sub-algebra and killing form.