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 = (F) over bar (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-(chi) under bar(g) associated with a reduced enveloping algebra U-chi(g), we partition simple g(F)-modules via F-orbits (chi) under bar of their p-characters chi. We further investigate certain relations between the categories of g-modules with p-character chi, g(F)-modules with p-character (chi) under bar, and g(F)-modules with p-character (sigma lozenge chi) under bar, for an automorphism sigma of g. Finally, we illustrate the theory with the example of s1(2, F-q). (C) 2008 Elsevier Inc. All rights reserved.