On regular Lie algebras

TL;DR

This paper investigates regular Lie algebras, where every nonzero element is regular, and explores their connection to an open problem about expressing trace-zero elements as commutators in simple associative algebras.

Contribution

It establishes a link between regular Lie algebras and the open problem of representing trace-zero elements as commutators in simple associative algebras.

Findings

Characterization of regular Lie algebras

Connection to the commutator problem in associative algebras

Insights into the structure of regular elements

Abstract

We study so called regular Lie algebras, i.e. Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.