On the Simplicity of Lie Algebras Associated to Leavitt Algebras

Abstract

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L 𝕂(n); for any integer d ≥ 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 ∈ 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.