Representations of finite Lie algebras

We develop an approach to investigate representations of finite Lie algebras g^F over a finite field F_q through representations of Lie algebras g with Frobenius morphisms F over the algebraic closure k = over(F, ̄)_q. As an application, we first show that Frobenius morphisms on classical simple Lie algebras can be used to determine easily their F_q-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 g^F, 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 U_{under(χ, {combining low line})} (g) associated with a reduced enveloping algebra U_χ (g), we partition simple g^F-modules via F-orbits under(χ, {combining low line}) of their p-characters χ. We further investigate certain relations between the categories of g-modules with p-character χ, g^F-modules with p-character under(χ, {combining low line}), and g^F-modules with p-character under(σ {white diamond suit} χ, {combining low line}), for an automorphism σ of g. Finally, we illustrate the theory with the example of sl (2, F_q).