Representations of Finite Lie Algebras

The notion of Frobenius morphisms for finite dimensional associative algebras is introduced in [2] and applied there to the study of representations of finite dimensional algebras over finite fields. In this paper, we develop a parallel theory for Lie algebras. Thus, a connection between representations over finite fields Fq and over their algebraic closures k = F̄q is established analogously. This enables us to understand irreducible representations of a Lie algebra over Fq through those over k. We further show that Frobenius morphisms can be used to easily determine the Fq-forms of classical simple Lie algebras, and hence reobtain a classical result given in [15]. Finally, we illustrate the theory with the example of classifying simple modules of sl(2,Fq).