Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras

Abstract

Let $K$ be a field and $E$ be a graph. Let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$ with the standard involution ${}^{\star }$. We investigate the set of skew-symmetric elements, ${\mathbf{K}}_{L_K(E)}=\{x\in L_K(E):x^{\star }=-x\}$, and show that for any simple $L_K(E)$ containing a cycle, $[{\mathbf{K}}_{L_K(E)},{\mathbf{K}}_{L_K(E)}]\ne {\mathbf{K}}_{L_K(E)}$. This provides a negative answer to a question posed by Herstein raised in 1961.