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.

Key Words:

• Leavitt algebra
• Leavitt path algebra
• Simple Lie algebra

2000 Mathematics Subject Classification:

• Primary: 16D30
• 16S99
• 17B65

ACKNOWLEDGMENT

The authors are grateful to the referee for a very thorough and quite helpful report.

The first author is partially supported by the U.S. National Security Agency under grant number H89230-09-1-0066.

Notes

Communicated by R. Wiegand.