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 braided structures associated with finite groups, these forms admit an expression in terms of the character of the conjugation action on a certain conjugation-stable subset of 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, commuting graphs and generation of permutation modules.
This talk is based on a preprint in collaboration with Kevin Piterman and recent work with Carsten Dietzel.