Are we counting or measuring something?

Abstract

Let $H$ be a semisimple Hopf algebras over an algebraically closed field $k$ of characteristic $0.$ We define Hopf algebraic analogues of commutators and their generalizations and show how they are related to $H',$ the Hopf algebraic analogue of the commutator subgroup. We introduce a family of central elements of $H',$ which on one hand generate $H'$ and on the other hand give rise to a family of functionals on $H.$ When $H=kG,\,G$ a finite group, these functionals are counting functions on $G.$ It is not clear yet to what extent they measure any specific invariant of the Hopf algebra. However, when $H$ is quasitriangular they are at least characters on $H.$