Non-commuting graphs of projective spaces over central quotients of Lie algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913815986176000 |
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| author | Sriwongsa, Songpon |
| author_facet | Sriwongsa, Songpon |
| contents | Let $L$ be a finite-dimensional non-abelian Lie algebra with the center $Z(L)$. In this paper, we define a non-commuting graph associated with $L$ as the graph whose vertex set is the projective space of the quotient algebra $L/Z(L)$, and two vertices $span \{ x + Z(L) \}$ and $span \{ y + Z(L) \}$ are adjacent if $x$ and $y$ do not commute under the Lie bracket of $L$. We present several theoretical properties of this graph. For certain classes of Lie algebras, we show that if the non-commuting graphs from two Lie algebras are isomorphic, then these Lie algebras themselves must be isomorphic. Furthermore, we discuss a relation between graph isomorphisms between non-commuting graphs of Lie algebras over finite fields and the size of the algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-commuting graphs of projective spaces over central quotients of Lie algebras Sriwongsa, Songpon Rings and Algebras 05C25, 17B99 Let $L$ be a finite-dimensional non-abelian Lie algebra with the center $Z(L)$. In this paper, we define a non-commuting graph associated with $L$ as the graph whose vertex set is the projective space of the quotient algebra $L/Z(L)$, and two vertices $span \{ x + Z(L) \}$ and $span \{ y + Z(L) \}$ are adjacent if $x$ and $y$ do not commute under the Lie bracket of $L$. We present several theoretical properties of this graph. For certain classes of Lie algebras, we show that if the non-commuting graphs from two Lie algebras are isomorphic, then these Lie algebras themselves must be isomorphic. Furthermore, we discuss a relation between graph isomorphisms between non-commuting graphs of Lie algebras over finite fields and the size of the algebras. |
| title | Non-commuting graphs of projective spaces over central quotients of Lie algebras |
| topic | Rings and Algebras 05C25, 17B99 |
| url | https://arxiv.org/abs/2505.00726 |