On reducible Killing forms for groups of Lie type

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 López Peña, 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.

36 pages, comments are welcome