On the simplicity of Lie algebras associated to Leavitt algebras

Abstract

For any field $\K$ and integer $n\ge 2$ we consider the Leavitt algebra $L_\K(n)$; for any integer $d\ge 1$ 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 $n-1$.

arXiv comment16 pages. Replaces earlier version. New version contains complete classification of simple Lie algebras of form $[S^{-},S^{-}]$ where $S$ is a matrix ring over a Leavitt algebra.