Finite generation of Lie algebras associated to associative algebras

TL;DR

This paper investigates conditions under which the Lie algebras derived from associative algebras, specifically the commutator algebra and skew-symmetric elements, are finitely generated, expanding understanding of their algebraic structure.

Contribution

It provides new sufficient conditions for the finite generation of Lie algebras associated with associative algebras and their involutions.

Findings

Identifies conditions for finite generation of [R,R]

Establishes criteria for [K,K] to be finitely generated

Advances understanding of Lie algebra structures from associative algebras

Abstract

Let $F$ be a field of characteristic not $2$ . An associative $F$-algebra $R$ gives rise to the commutator Lie algebra $R^{(-)}=(R,[a,b]=ab-ba).$ If the algebra $R$ is equipped with an involution $*:R\rightarrow R$ then the space of the skew-symmetric elements $K=\{a \in R \mid a^{*}=-a \}$ is a Lie subalgebra of $R^{(-)}.$ In this paper we find sufficient conditions for the Lie algebras $[R,R]$ and $[K,K]$ to be finitely generated.