Probabilistically nilpotent Hopf algebras (Q2790623)

The notion of nilpotency of a fusion category was introduced by \textit{S. Gelaki} and \textit{D. Nikshych} [Adv. Math. 217, No. 3, 1053--1071 (2008; Zbl 1168.18004)]. A semisimple Hopf algebra is called nilpotent if its category of finite dimensional representations is nilpotent. In this paper, the authors give a formulation of this notion of nilpotency of a semisimple Hopf algebra \(H\) in terms of a descending chain of commutators, which are defined as certain normal left coideal subalgebras of \(H\). They also introduce a family of central iterated commutators, which are elements of \(H\), and give criteria for \(H\) to be nilpotent in terms of this family. Based on these criteria, they introduce the notion of probabilistically nilpotent semisimple Hopf algebra. One of the main results of the paper states that every semisimple quasitriangular Hopf algebra over an algebraically closed field of characteristic zero is probabilistically nilpotent, extending a result for finite groups in the paper of \textit{A. Amit} and \textit{U. Vishne} [J. Algebra Appl. 10, No. 4, 675--686 (2011; Zbl 1246.20030)].