2025/06/24 by David A. Towers, Towers, David, Gutierrez, Ismael +2 · 1 citation
Mathematics · #Finite Group Theory Research #Graph theory and applications #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2506.19758
Let L be a finite-dimensional Lie algebra over a field F. In This paper we introduce the nilpotent graph Γ_\mathfrakN(L) as the graph whose vertices are the elements of L ∖ \nil(L), where \nil(L) = \x ∈ L | ⟨ x, y ⟩ is nilpotent for all y ∈ 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_\mathfrakN(L) between components. The paper also investigates the structural and combinatorial properties of Γ_\mathfrakN(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 \mathfrakt(2,\mathbbFq), where Γ_\mathfrakN(L) decomposes into q+1 components, each of size q(q-1), forming a (q2-q-1)-regular graph. We conclude with algorithms for constructing Γ_\mathfrakN(L) in SageMath, and pose open problems concerning bipartiteness, regularity, and structural implications in higher dimensions over finite fields.