Nilpotency of Lie type algebras with metacyclic Frobenius groups of automorphisms

TL;DR

This paper proves that Lie type algebras with certain Frobenius automorphism groups are nilpotent, extending previous results to broader classes by establishing bounds on their nilpotency class.

Contribution

It extends known theorems by showing that Lie type algebras with metacyclic Frobenius automorphisms are nilpotent with bounded class, under specific fixed-point conditions.

Findings

L is nilpotent under given automorphism conditions

Nilpotency class is bounded in terms of |H| and c

Generalizes previous results on Lie algebras with Frobenius automorphisms

Abstract

Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is nilpotent of class c. If the ground field K contains a primitive n-th root of unity, then L is nilpotent and the nilpotency class of L is bounded in terms of |H| and c. The result extends the known theorem of Khukhro, Makarenko and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms.