On the simplicity of Lie algebras associated to Leavitt algebras

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For any field and integer n≥ 2 we consider the Leavitt algebra L(n); for any integer d≥ 1 we form the matrix ring S = Md(L(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a,b]=ab-ba for a,b ∈ 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() divides n-1 and char() does not divide d. In particular, when d=1 we get that [L(n)-,L(n)-] is a simple Lie algebra if and only if char() divides n-1.