Graded twisting of categories and quantum groups by group actions

Bichon, Julien; Neshveyev, Sergey; Yamashita, Makoto

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2299-2338.

À une algèbre de Hopf $A$ graduée par un groupe et munie d’une action de ce même groupe préservant cette graduation, nous associons une nouvelle algèbre de Hopf, que nous appelons le twist gradué de $A$. Quand l’action est de type adjoint, cette nouvelle algèbre de Hopf est un twist de $A$ par un pseudo-$2$-cocycle. Une construction similaire est effectuée au niveau des catégories monoïdales. Nous étudions les exemples des algèbres de Hopf des formes bilinéaires non dégénérées, leurs produits libres, les groupes quantiques hyperoctaédraux, et les $q$-déformations des groupes de Lie compacts semi-simples. En application, nous montrons que les analogues des catégories de Kazhdan–Wenzl dans le cas semi-simple général ne peuvent pas toujours être réalisées comme catégories de représentations de groupes quantiques compacts, et pour les groupes compacts usuels, nous décrivons complètement les sous-groupes quantiques du nouveau groupe quantique twisté, dans le cas où le groupe twisteur est d’ordre premier.

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 by adjoint maps, 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.

Classification : 16T05, 46L65, 18D10, 20G42
Keywords: quantum group, monoidal category, grading, pseudo-2-cocycle
Mot clés : groupe quantique, catégorie monoïdale, graduation, pseudo-2-cocycle

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