On the simplicity of Lie algebras associated to Leavitt algebras

TL;DR

This paper characterizes when the Lie algebra derived from matrix rings over Leavitt algebras is simple, revealing a precise condition involving the characteristic of the field and the parameters n and d.

Contribution

It provides a complete characterization of the simplicity of Lie algebras associated with matrix rings over Leavitt algebras, linking algebraic properties to field characteristics.

Findings

$[S^-,S^-]$ is simple iff char$( ext{K})$ divides $n-1$ and does not divide $d$

When $d=1$, $[L_ ext{K}(n)^-,L_ ext{K}(n)^-]$ is simple iff char$( ext{K})$ divides $n-1$

The simplicity depends critically on the characteristic of the base field and the parameters $n$ and $d$.

Abstract

For any field $\K$ and integer $n\geq 2$ we consider the Leavitt algebra $L_\K(n)$; for any integer $d\geq 1$ we form the matrix ring $S = M_d(L_\K(n))$. $S$ is an associative algebra, but we view $S$ as a Lie algebra using the bracket $[a,b]=ab-ba$ for $a,b \in S$. We denote this Lie algebra as $S^-$, and consider its Lie subalgebra $[S^-,S^-]$. In our main result, we show that $[S^-,S^-]$ is a simple Lie algebra if and only if char$(\K)$ divides $n-1$ and char$(\K)$ does not divide $d$. In particular, when $d=1$ we get that $[L_\K(n)^-,L_\K(n)^-]$ is a simple Lie algebra if and only if char$(\K)$ divides $n-1$.