On integrals and cointegrals for quasi-Hopf algebras

TL;DR

This paper explores the structure of integrals and cointegrals in quasi-Hopf algebras using Frobenius algebra techniques, providing explicit formulas for quantum doubles and addressing a longstanding conjecture.

Contribution

It introduces formulas linking antipodes to cointegrals in quasi-Hopf algebras and derives explicit forms for integrals in quantum doubles, solving a conjecture from the 1990s.

Findings

Explicit formulas for cointegrals using Frobenius algebra methods

Derived integrals for quantum doubles of quasi-Hopf algebras

Confirmed a conjecture by Hausser and Nill from the 1990s

Abstract

Using the machinery provided by a Frobenius algebra we show how the antipode of a quasi-Hopf algebra $H$ carries out left or right cointegrals for $H$. These formulas will allow us to find out the explicit form of an integral and a cointegral for the quantum double $D(H)$ of $H$ in terms of those of $H$, and so to answer to a conjecture of Hausser and Nill raised at the end of the nineties.