The nilpotent graph of a finite0-dimensional Lie algebra
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913910427222016 |
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| author | Towers, David Gutierrez, Ismael Fernandez, Luis |
| author_facet | Towers, David Gutierrez, Ismael Fernandez, Luis |
| contents | Let $L$ be a finite-dimensional Lie algebra over a field $F$. In This paper we introduce the \emph{nilpotent graph} $Γ_\mathfrak{N}(L)$ as the graph whose vertices are the elements of $L \setminus \nil(L)$, where \[\nil(L) = \{x \in L \mid \langle x, y \rangle \text{ is nilpotent for all } y \in L\},\] and where two vertices $x, y$ are adjacent if the Lie subalgebra they generate is nilpotent. We give some characterizations of $\nil(L)$ and its connection with the hypercenter $Z^*(L)$, for example, they are equal when $F$ has characteristic zero. We prove that the nilpotentizer behaves well under direct sums, allowing a decomposition of $Γ_\mathfrak{N}(L)$ between components. The paper also investigates the structural and combinatorial properties of $Γ_\mathfrak{N}(L)$, including the conditions under which the graph is connected. We characterize the existence of strongly self-centralizing subalgebras in relation to connectivity and vertex isolation. Explicit computations are carried out for the algebra $\mathfrak{t}(2,\mathbb{F}_q)$, where $Γ_\mathfrak{N}(L)$ decomposes into $q+1$ components, each of size $q(q-1)$, forming a $(q^2-q-1)$-regular graph. We conclude with algorithms for constructing $Γ_\mathfrak{N}(L)$ in SageMath, and pose open problems concerning bipartiteness, regularity, and structural implications in higher dimensions over finite fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19758 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The nilpotent graph of a finite0-dimensional Lie algebra Towers, David Gutierrez, Ismael Fernandez, Luis Rings and Algebras Combinatorics Group Theory 17B30, 17B45, 05C40, 05C25 Let $L$ be a finite-dimensional Lie algebra over a field $F$. In This paper we introduce the \emph{nilpotent graph} $Γ_\mathfrak{N}(L)$ as the graph whose vertices are the elements of $L \setminus \nil(L)$, where \[\nil(L) = \{x \in L \mid \langle x, y \rangle \text{ is nilpotent for all } y \in L\},\] and where two vertices $x, y$ are adjacent if the Lie subalgebra they generate is nilpotent. We give some characterizations of $\nil(L)$ and its connection with the hypercenter $Z^*(L)$, for example, they are equal when $F$ has characteristic zero. We prove that the nilpotentizer behaves well under direct sums, allowing a decomposition of $Γ_\mathfrak{N}(L)$ between components. The paper also investigates the structural and combinatorial properties of $Γ_\mathfrak{N}(L)$, including the conditions under which the graph is connected. We characterize the existence of strongly self-centralizing subalgebras in relation to connectivity and vertex isolation. Explicit computations are carried out for the algebra $\mathfrak{t}(2,\mathbb{F}_q)$, where $Γ_\mathfrak{N}(L)$ decomposes into $q+1$ components, each of size $q(q-1)$, forming a $(q^2-q-1)$-regular graph. We conclude with algorithms for constructing $Γ_\mathfrak{N}(L)$ in SageMath, and pose open problems concerning bipartiteness, regularity, and structural implications in higher dimensions over finite fields. |
| title | The nilpotent graph of a finite0-dimensional Lie algebra |
| topic | Rings and Algebras Combinatorics Group Theory 17B30, 17B45, 05C40, 05C25 |
| url | https://arxiv.org/abs/2506.19758 |