Double automorphisms of graded Lie algebras

TL;DR

This paper introduces double automorphisms in graded Lie algebras, explores their properties, and establishes nilpotency criteria for such algebras with finite automorphism groups, with applications to Frobenius groups.

Contribution

It defines double automorphisms of graded Lie algebras and proves nilpotency criteria for algebras with finite groups of these automorphisms, extending understanding of symmetry in Lie algebras.

Findings

Nilpotency criteria for graded Lie algebras with finite double automorphism groups

Characterization of automorphisms inducing automorphisms of the grading group

Application to groups with Frobenius automorphism groups

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 Aut(L). In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. We also give an application of our results to groups admitting a Frobenius group of automorphisms.