A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Abstract

Let G be a simple algebraic group over an algebraically closed field of characteristic p>0 and suppose that p is a very good prime for G. In this paper we prove that any maximal Lie subalgebra M of g=Lie(G) with rad(M)≠0 has the form M=Lie(P) for some maximal parabolic subgroup P of G. This means that Morozov's theorem on maximal subalgebras is valid under mild assumptions on G. We show that such assumptions are necessary by providing a counterexample to Morozov's theorem for groups of type E8 over fields of characteristic 5. Our proof relies on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields of prime characteristic.