Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909551400321024 |
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| author | Nhi, Nguyen Huynh Thao Khanh, Huynh Viet |
| author_facet | Nhi, Nguyen Huynh Thao Khanh, Huynh Viet |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19291 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras Nhi, Nguyen Huynh Thao Khanh, Huynh Viet Rings and Algebras 16S88, 17B65, 17B20, 17B30 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. |
| title | Lie Algebras of Skew-Symmetric Elements in Simple Leavitt path algebras |
| topic | Rings and Algebras 16S88, 17B65, 17B20, 17B30 |
| url | https://arxiv.org/abs/2503.19291 |