On reducible Killing forms for groups of Lie type

TL;DR

This paper investigates the properties of Killing forms associated with finite groups of Lie type, focusing on their non-degeneracy and irreducibility, and explores connections with character theory and commuting graphs.

Contribution

It provides new insights into the reducibility and non-degeneracy of Killing forms for specific conjugacy classes in finite simple groups of Lie type and Lie rank one.

Findings

Killing forms are non-degenerate for certain conjugacy classes.

Irreducibility of Killing forms depends on the class type and group structure.

Connections established with character theory and commuting graphs.

Abstract

Killing forms on finite groups arise as examples of braided Killing forms on braided Lie algebras. For a finite group $G$ and a $G$-stable subset $\mathcal{C}$, the Killing form associated with $\mathbb{C}[\mathcal{C}]$ is given by $K_{\mathcal{C}}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b\in \mathcal{C}$. Motivated by Cartan's criterion for semisimplicity of Lie algebras, and previous work of L\'opez Pe\~na, Majid, and Rietsch, we study the non-degeneracy and irreducibility of $K_{\mathcal{C}}$ when $\mathcal{C}$ is a conjugacy class of involutions or unipotent elements in a finite simple group of Lie type and Lie rank one. Our approach suggests interesting connections with character theory, related counting formulas, and the study of commuting graphs.