Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras
Nguyen Huynh Thao Nhi, Huynh Viet Khanh
TL;DR
The paper addresses whether the skew-symmetric elements $oldsymbol{K}_{L_K(E)}$ of a Leavitt path algebra $L_K(E)$, under the standard involution, are generated by commutators as in Herstein's question for simple rings of the first kind. By providing a detailed description of $oldsymbol{K}_{L_K(E)}$ and distinguishing cases by ${ m char}(K)$, it shows that when $L_K(E)$ is simple and contains a cycle, $[\boldsymbol{K}_{L_K(E)}, \boldsymbol{K}_{L_K(E)}] \neq \boldsymbol{K}_{L_K(E)}$, giving a negative answer to Herstein's question in this setting. The authors give explicit constructions, including the rose graphs $R_n$ (with $L_K(R_n)$ simple) and a simple, locally finite Leavitt path algebra from the graph $A_{\mathbb{Z}}$, where the skew-symmetric space is not generated by commutators. These results advance the understanding of Lie-theoretic structures in Leavitt path algebras and provide concrete counterexamples to a long-standing conjecture. The findings highlight the limitations of transferring Herstein's conjecture to the realm of Leavitt path algebras with cycles.
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.