Representations of finite Lie algebras

Abstract

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 ¯ 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 χ ̲ ( g ) associated with a reduced enveloping algebra U χ ( g ) , we partition simple g F -modules via F-orbits χ ̲ 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 χ ̲ , and g F -modules with p-character σ ⋄ χ ̲ , for an automorphism σ of g . Finally, we illustrate the theory with the example of sl ( 2 , F q ) .