The nilpotent graph of a finite0-dimensional Lie algebra

TL;DR

This paper introduces the nilpotent graph of a finite-dimensional Lie algebra, exploring its properties, structure, and computational aspects, with specific examples and open problems in algebraic graph theory.

Contribution

It defines the nilpotent graph for Lie algebras, characterizes its properties, and provides explicit computations and algorithms, advancing the understanding of algebraic structures through graph theory.

Findings

The nilpotent graph decomposes into components under direct sums.

The graph's connectivity relates to strongly self-centralizing subalgebras.

Explicit structure of the nilpotent graph for (2, \u1EF3_q) is provided.

Abstract

Let $L$ be a finite-dimensional Lie algebra over a field $F$. In This paper we introduce the \emph{nilpotent graph} $\Gamma_\mathfrak{N}(L)$ 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 $\Gamma_\mathfrak{N}(L)$ between components. The paper also investigates the structural and combinatorial properties of $\Gamma_\mathfrak{N}(L)$, 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 $\Gamma_\mathfrak{N}(L)$ 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 $\Gamma_\mathfrak{N}(L)$ in SageMath, and pose open problems concerning bipartiteness, regularity, and structural implications in higher dimensions over finite fields.