A trace-like invariant for representations of Hopf algebras

TL;DR

This paper introduces a new trace-like invariant for irreducible representations of finite dimensional complex Hopf algebras, computed explicitly for certain non-semisimple examples, aiding in understanding their representation theory.

Contribution

It proposes a novel trace-like invariant based on the antipode's action, extending tools for analyzing representations of Hopf algebras.

Findings

Defined the trace-like invariant for irreducible representations.

Computed the invariant for u_q(sl_2) and D(H_n(q)).

Provided insights into the structure of non-semisimple Hopf algebra representations.

Abstract

In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra.