The structure of Lie algebras with a derivation satisfying a polynomial identity

TL;DR

This paper investigates the structure of Lie algebras with derivations satisfying polynomial identities, establishing nilpotency conditions, bounds, and classifications, including extensions to Lie rings over integers.

Contribution

It provides new nilpotency results for Lie algebras with derivations satisfying polynomial identities, including bounds and classifications, and extends findings to Lie rings over integers.

Findings

Nilpotency results for Lie algebras with derivations satisfying polynomial identities.

Optimal bounds on nilpotency class in characteristic p.

Characterization of orders of periodic derivations in finite-dimensional Lie algebras.

Abstract

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a periodic derivation of order $n$. We find an optimal bound on the nilpotency class in characteristic $p$ if $p$ does not divide a certain invariant $\rho_n$. We give a new description of the set $\mathcal{N}_p$ of positive integers $n$, introduced by Shalev, which arise as the order of a periodic derivation of a finite-dimensional non-nilpotent Lie algebra in characteristic $p>0$. Finally we generalize the results to Lie rings over $\Bbb Z$.