Graded twisting of categories and quantum groups by group actions

TL;DR

This paper introduces a method called graded twisting for Hopf algebras and monoidal categories, exploring its properties, examples, and implications for quantum groups and their subgroups.

Contribution

It defines graded twisting of Hopf algebras and categories, linking it to pseudo-2-cocycles, and analyzes its effects on quantum groups and representation categories.

Findings

Graded twisting can produce new quantum groups with distinct properties.

Certain semisimple categories cannot be realized as quantum group representations after twisting.

Full classification of quantum subgroups for cyclic prime order twistings.

Abstract

Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and $q$-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the Kazhdan-Wenzl categories in the general semisimple case cannot be always realized as representation categories of compact quantum groups, and for genuine compact groups, we analyze quantum subgroups of the new twisted compact quantum groups, providing a full description when the twisting group is cyclic of prime order.