Representations of finite Lie algebras

We develop an approach to investigate representations of finite Lie algebras gF over a finite field Fq through representations of Lie algebras g with Frobenius morphisms F over the algebraic closure . As an application, we first show that Frobenius morphisms on classical simple Lie algebras can be used to determine easily their Fq-forms, and hence, reobtain a classical result given in [G.B. Seligman, Modular Lie Algebras, Springer-Verlag, Berlin, 1967]. We then investigate representations of finite restricted Lie algebras gF, regarded as the fixed-point algebra of a restricted Lie algebra g with restricted Frobenius morphism F. By introducing the F-orbital reduced enveloping algebras associated with a reduced enveloping algebra Uχ(g), we partition simple gF-modules via F-orbits of their p-characters χ. We further investigate certain relations between the categories of g-modules with p-character χ, gF-modules with p-character , and gF-modules with p-character , for an automorphism σ of g. Finally, we illustrate the theory with the example of sl(2,Fq).