# On the simplicity of Lie algebras associated to Leavitt algebras - Mathematics > Rings and Algebras

On the simplicity of Lie algebras associated to Leavitt algebras - Mathematics > Rings and Algebras - Download this document for free, or read online. Document in PDF available to download.

Abstract: For any field $\K$ and integer $n\geq 2$ we consider the Leavitt algebra$L \Kn$; for any integer $d\geq 1$ we form the matrix ring $S =M dL \Kn$. $S$ is an associative algebra, but we view $S$ as a Lie algebrausing 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 showthat $S^-,S^-$ 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$L \Kn^-,L \Kn^-$ is a simple Lie algebra if and only if char$\K$divides $n-1$.

Author: ** Gene Abrams, Darren Funk-Neubauer**

Source: https://arxiv.org/