Graded twisting of categories and quantum groups by group actions

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.

v3: minor corrections, to appear in Ann. Inst. Fourier; v2: compilation error fix and minor phrasing changes; v1: 25 pages

References in corpus (5)

• A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries
• Twisting the q-deformations of compact semisimple Lie groups
• Corepresentation theory of universal cosovereign Hopf algebras
• Quantum subgroups of the compact quantum group SU_{-1}(3)
• Classification of non-Kac compact quantum groups of SU(n) type

Cited by in corpus (4)

• L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups
• Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups
• Quantum coordinate ring in WZW model and affine vertex algebra extensions
• Towards a classification of compact quantum groups of Lie type