On the Simplicity of Lie Algebras Associated to Leavitt Algebras |
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| Authors: | Gene Abrams Darren Funk-Neubauer |
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| Institution: | 1. Department of Mathematics , University of Colorado , Colorado Springs, Colorado, USA abrams@math.uccs.edu;3. Department of Mathematics , Colorado State University , Pueblo, Colorado, USA |
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| 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. |
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| Keywords: | Leavitt algebra Leavitt path algebra Simple Lie algebra |
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