Zestings of Hopf Algebras
Iván Angiono, César Galindo, Giovanny Mora
TL;DR
This work develops a comprehensive framework translating categorical zesting from fusion categories to general tensor categories, focusing on comodules $^{H}{\mathcal{M}}$ over Hopf algebras. It shows that associative zesting produces a coquasi-Hopf algebra $H^{\lambda}$ whose comodules reproduce the zested category, giving explicit formulas for the modified multiplication $m^{\lambda}$ and associator $\Omega$, and extends to braided zesting when $H$ is coquasitriangular with a twisted $r$-form $r^{\lambda}$. The authors apply the theory to pointed Hopf algebras, deriving a universal grading framework $U(H)$ and detailing cyclic-group zestings with concrete data $(\gamma,\lambda,\omega)$, yielding families of new coquasi-Hopf algebras including examples from diagonal and non-diagonal Nichols algebras such as super type $A(1|2)$ and Fomin-Kirillov algebras. This provides a robust algebraic counterpart to categorical zesting, enabling systematic construction of non-semisimple, braided tensor categories with controlled associators and braidings, and broadens the toolkit for exploring modular-like structures in Hopf-algebraic settings.
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.