Double automorphisms of graded Lie algebras

Abstract

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 AutL. 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︸k]=0. Assume that L admits a finite group of double automorphisms H such that CA(h)=0 for all non-trivial h∈H and CL(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.