Reduced contragredient Lie algebras and PC Lie algebras

Abstract

Using the theory of standard pentads, we can embed an arbitrary finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation into some larger graded Lie algebra. However, it is not easy to find the structure of the ``larger graded Lie algebra'' from the definition in general cases. Under these, the first aim of this paper is to show that the ``larger graded Lie algebra'' is isomorphic to some PC Lie algebra, which are Lie algebras corresponding to special standard pentads called pentads of Cartan type. The second aim is to find the structure of a PC Lie algebra.