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arXiv · 2505.00726

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

Songpon Sriwongsa
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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.

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Songpon Sriwongsa. 2025-04-27. Non-commuting graphs of projective spaces over central quotients of Lie algebras. https://arxiv.org/abs/2505.00726

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