On the simplicity of Lie algebras associated to Leavitt algebras

Details On the simplicity of Lie algebras associated to Leavitt algebras On the simplicity of Lie algebras associated to Leavitt algebras

2009-08-15

arxiv.org/abs/0908.2179v2

Mathematics

Rings and Algebras

16 pages. Replaces earlier version. New version contains complete classification of simple Lie algebras of form $[S^-,S^-]$ wh

Scientific paper

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 $n-1$.

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