Non-commuting graphs of projective spaces over central quotients of Lie algebras

TL;DR

The paper introduces a projective-space based non-commuting graph $\Gamma_L$ for finite-dimensional non-abelian Lie algebras with center $Z(L)$, where vertices are the projective points of $L/Z(L)$ and two vertices are adjacent when their representatives do not commute. It derives foundational graph-theoretic properties and shows how the base field and algebra structure influence finiteness, regularity, completeness, diameter, planarity, and domination in $\Gamma_L$. It then investigates isomorphism rigidity, presenting a counterexample that same graphs need not imply isomorphic Lie algebras, yet obtaining positive rigidity results for CT Lie algebras and related AC and finite-field cases, including size-detection relations via center sizes. Collectively, the results establish that graph properties of $\Gamma_L$ encode meaningful information about the underlying Lie algebra and highlight directions for extending the approach to broader classes, including nilpotent and simple algebras.

Abstract

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.

Theorems & Definitions (66)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof