Are we counting or measuring something?

TL;DR

This paper introduces Hopf algebraic analogues of commutators, explores their properties, and investigates how associated functionals relate to invariants and characters, especially in the quasitriangular case.

Contribution

It defines new commutator analogues in Hopf algebras and studies their relation to the Hopf algebra's structure and invariants, extending classical group concepts.

Findings

Defined Hopf algebraic commutators and their generalizations.

Introduced central elements generating the commutator analogue.

Established that these elements give rise to characters in the quasitriangular case.

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 $H.$