Zestings of Hopf Algebras

TL;DR

This paper generalizes zesting techniques from fusion categories to tensor categories of comodules over Hopf algebras, providing explicit constructions and formulas for associators and structures, especially in the pointed case.

Contribution

It extends zesting methods to general tensor categories of comodules over Hopf algebras, with explicit formulas and systematic approaches for cyclic gradings.

Findings

Zesting yields coquasi-Hopf algebras with modified structures.

Explicit formulas for zesting in pointed Hopf algebras.

Systematic approach for cyclic group gradings.

Abstract

We extend the previously established zesting techniques from fusion categories to general tensor categories. In particular we consider the category of comodules over a Hopf algebra, providing a detailed translation of the categorical zesting construction into explicit Hopf algebraic terms: we show that the associative zesting of the category of comodules yields a coquasi-Hopf algebra whose comodule category is precisely the zested category. We explicitly write the modified multiplication and the associator, as well as the structures involved in the braided case. For pointed Hopf algebras, we derive concrete formulas for constructing zestings and establish a systematic approach for cyclic group gradings, providing explicit parameterizations of the zesting data.