Arithmetically-free group-gradings of Lie algebras

TL;DR

This paper generalizes classical results on Lie algebra gradings by cyclic groups to arbitrary groups with arithmetically-free support, providing bounds on nilpotency class and applications to automorphisms of groups satisfying identities.

Contribution

It introduces a broader framework for Lie algebra gradings by arbitrary groups with arithmetically-free support, extending known results and deriving new bounds.

Findings

Lie algebras with such gradings are nilpotent with bounded class

Classical results are extended to more general group gradings

Applications to automorphisms of groups satisfying identities

Abstract

A Lie algebra L is known to be nilpotent if it admits a grading by (Zp, +) with support X not containing 0. It is also known that the class of L can be bounded by some explicit function of |X|. We generalise this and other classical results to gradings of Lie algebras by arbitrary groups with arithmetically-free support. We then apply these results to automorphisms of groups satisfying an identity.