Are we counting or measuring something?

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 = k G, 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.