Cartan matrices for restricted Lie algebras

Abstract

Consider a finite dimensional restricted Lie algebra over a field of prime characteristic. Each linear form on this Lie algebra defines a finite dimensional quotient of its universal enveloping algebra, called a reduced enveloping algebra. This leads to a Cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra. In this paper we show how to compare such Cartan matrices belonging to distinct linear forms. As an application we rederive and generalise the reciprocity formula first discovered by Humphreys for Lie algebras of reductive groups. For simple Lie algebras of Cartan type we see, for example, that the Cartan matrices for linear forms of non-positive height are submatrices of the Cartan matrix for the zero linear form.