Contragredient Lie algebras and Lie algebras associated with a standard pentad

Abstract

From a given standard pentad, we can construct a finite or infinite-dimensional graded Lie algebra. In this paper, we will define standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. Using pentads of Cartan type, we can obtain arbitrary contragredient Lie algebras with an invertible symmetrizable Cartan matrix. Moreover, we can use pentads of Cartan type in order to find the structure of a Lie algebra. When a given standard pentad consists of a finite-dimensional reductive Lie algebra, its finite-dimensional completely reducible representation and a symmetric bilinear form, we can find the structure of its corresponding Lie algebra under some assumptions.