Killing forms on finite simple groups

We study Killing forms on finite groups arising from the extension of the theory of Killing forms on Lie algebras to braided-Lie algebras. For certain braided structures associated with a conjugation-stable subset of a finite group, these forms admit an expression in terms of the character of the conjugation action on the group. Motivated by Cartan’s criterion and earlier work by López Peña, Majid, and Rietsch, we investigate the non-degeneracy and irreducibility of such Killing forms defined on conjugation-stable subsets of finite groups. We will particularly focus on conjugacy classes of involutions and unipotent elements in simple groups of Lie type of rank one. Our methods reveal interesting connections with counting formulas in character theory, generation properties, and commuting graphs. This talk is based on a recent preprint in collaboration with Kevin Piterman.