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

2025/04/27 by Songpon Sriwongsa, Sriwongsa, Songpon
Mathematics · #05C25 #17B99 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2505.00726

openalex publication_date 2025/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• On the non-commuting graph associated to a finite-dimensional Lie algebra
• A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs
• The non-commuting, non-generating graph of a finite simple group
• Some Theorems on Abstract Graphs

Related

• On the non-commuting graph associated to a finite-dimensional Lie algebra
• Complete and cocomplete Lie algebras with injective- and projective-type properties
• On non-commuting graph of a finite ring
• Non-commuting graphs of nilpotent groups
• Proof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups