Double automorphisms of graded Lie algebras

We introduce the concept of a double automorphism of an A-graded Lie algebra L. Roughly, this is an automorphism of L which also induces an automorphism of the group A. It is clear that the set of all double automorphisms of L forms a subgroup in Aut(L). In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. One of the obtained results is as follows. Let A be a torsion-free abelian group and L an A-graded Lie algebra in which [L, L₀, …, L₀] = 0, with L₀ occurring k times. Assume that L admits a finite group of double automorphisms H such that C_A(h) = 0 for all nontrivial h ∈ H and C_L(H) is nilpotent of class c. Then L is nilpotent and the class of L is bounded in terms of |H|, k and c only. We also give an application of our results to groups admitting a Frobenius group of automorphisms.