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

TL;DR

This paper introduces a new non-commuting graph based on projective spaces of Lie algebra quotients, exploring its properties and establishing conditions under which graph isomorphisms imply algebra isomorphisms, especially over finite fields.

Contribution

It defines a novel non-commuting graph for Lie algebras and proves that graph isomorphisms can determine algebra isomorphisms for certain classes.

Findings

Non-commuting graphs encode algebraic structure.

Graph isomorphism implies algebra isomorphism in specific cases.

Relation between graph isomorphisms and algebra size over finite fields.

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.