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arXiv · 2503.19291

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

Nguyen Huynh Thao Nhi·Huynh Viet Khanh
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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.

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Nguyen Huynh Thao Nhi, Huynh Viet Khanh. 2025-03-25. Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras. https://arxiv.org/abs/2503.19291

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