Given a Hopf algebra 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 . If the action is by adjoint maps, this new Hopf algebra is a twist of by a pseudo--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 -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.
À une algèbre de Hopf 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 . Quand l’action est de type adjoint, cette nouvelle algèbre de Hopf est un twist de par un pseudo--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 -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.
Revised:
Accepted:
Published online:
Keywords: quantum group, monoidal category, grading, pseudo-2-cocycle
Mot clés : groupe quantique, catégorie monoïdale, graduation, pseudo-2-cocycle
Bichon, Julien 1; Neshveyev, Sergey 2; Yamashita, Makoto 3
@article{AIF_2016__66_6_2299_0, author = {Bichon, Julien and Neshveyev, Sergey and Yamashita, Makoto}, title = {Graded twisting of categories and quantum groups by group actions}, journal = {Annales de l'Institut Fourier}, pages = {2299--2338}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {6}, year = {2016}, doi = {10.5802/aif.3064}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3064/} }
TY - JOUR AU - Bichon, Julien AU - Neshveyev, Sergey AU - Yamashita, Makoto TI - Graded twisting of categories and quantum groups by group actions JO - Annales de l'Institut Fourier PY - 2016 SP - 2299 EP - 2338 VL - 66 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3064/ DO - 10.5802/aif.3064 LA - en ID - AIF_2016__66_6_2299_0 ER -
%0 Journal Article %A Bichon, Julien %A Neshveyev, Sergey %A Yamashita, Makoto %T Graded twisting of categories and quantum groups by group actions %J Annales de l'Institut Fourier %D 2016 %P 2299-2338 %V 66 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3064/ %R 10.5802/aif.3064 %G en %F AIF_2016__66_6_2299_0
Bichon, Julien; Neshveyev, Sergey; Yamashita, Makoto. Graded twisting of categories and quantum groups by group actions. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2299-2338. doi : 10.5802/aif.3064. https://aif.centre-mersenne.org/articles/10.5802/aif.3064/
[1] Quantum deformations of , Comm. Pure Appl. Math., Volume 44 (1991) no. 8-9, pp. 879-895 | DOI
[2] Théorie des représentations du groupe quantique compact libre , C. R. Acad. Sci. Paris Sér. I Math., Volume 322 (1996) no. 3, pp. 241-244
[3] Le groupe quantique compact libre , Comm. Math. Phys., Volume 190 (1997) no. 1, pp. 143-172 | DOI
[4] Representations of compact quantum groups and subfactors, J. Reine Angew. Math., Volume 509 (1999), pp. 167-198 | DOI
[5] Half-liberated manifolds, and their quantum isometries (2015) (to appear in Glasg. Math. J., http://arxiv.org/abs/1505.00646)
[6] The hyperoctahedral quantum group, J. Ramanujan Math. Soc., Volume 22 (2007) no. 4, pp. 345-384
[7] Classification results for easy quantum groups, Pacific J. Math., Volume 247 (2010) no. 1, pp. 1-26 | DOI
[8] Liberation of orthogonal Lie groups, Adv. Math., Volume 222 (2009) no. 4, pp. 1461-1501 | DOI
[9] Quantum isometries of the finite noncommutative geometry of the standard model, Comm. Math. Phys., Volume 307 (2011) no. 1, pp. 101-131 | DOI
[10] Free wreath product by the quantum permutation group, Algebr. Represent. Theory, Volume 7 (2004) no. 4, pp. 343-362 | DOI
[11] Co-representation theory of universal co-sovereign Hopf algebras, J. Lond. Math. Soc., Volume 75 (2007) no. 1, pp. 83-98 | DOI
[12] Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys., Volume 262 (2006) no. 3, pp. 703-728 | DOI
[13] Hopf algebras having a dense big cell, Trans. Amer. Math. Soc., Volume 368 (2016) no. 1, pp. 515-538 | DOI
[14] Quantum subgroups of the compact quantum group , Bull. Lond. Math. Soc., Volume 46 (2014) no. 2, pp. 315-328 | DOI
[15] Grothendieck rings of universal quantum groups, J. Algebra, Volume 349 (2012), pp. 80-97 | DOI
[16] Braided bialgebras and quadratic bialgebras, Comm. Algebra, Volume 21 (1993) no. 5, pp. 1731-1749 | DOI
[17] Quasi-Hopf algebras, Algebra i Analiz, Volume 1 (1989) no. 6, pp. 114-148 Translation in Leningrad Math. J. 1 (1990), no. 6, 1419–1457
[18] The quantum group of a nondegenerate bilinear form, Phys. Lett. B, Volume 245 (1990) no. 2, pp. 175-177 | DOI
[19] Deformation of a Kac algebra by an abelian subgroup, Comm. Math. Phys., Volume 178 (1996) no. 3, pp. 571-596 http://projecteuclid.org/getRecord?id=euclid.cmp/1104286767 | DOI
[20] Module categories over representations of and graphs, Math. Res. Lett., Volume 11 (2004) no. 1, pp. 103-114 | DOI
[21] Quantum subgroups of , J. Algebra, Volume 324 (2010) no. 6, pp. 1392-1428 | DOI
[22] Nilpotent fusion categories, Adv. Math., Volume 217 (2008) no. 3, pp. 1053-1071 | DOI
[23] Amenability and strong amenability for fusion algebras with applications to subfactor theory, Internat. J. Math., Volume 9 (1998) no. 6, pp. 669-722 | DOI
[24] Reconstructing monoidal categories, I. M. Gelʼfand Seminar (Adv. Soviet Math.), Volume 16, Amer. Math. Soc., Providence, RI, 1993, pp. 111-136
[25] Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997, xx+552 pages
[26] On the center of a compact group, Int. Math. Res. Not. (2004) no. 51, pp. 2751-2756 | DOI
[27] Compact quantum groups and their representation categories, Cours Spécialisés [Specialized Courses], 20, Société Mathématique de France, Paris, 2013, vi+169 pages
[28] Poisson boundaries of monoidal categories (2014) (http://arxiv.org/abs/1405.6572)
[29] Classification of Non-Kac Compact Quantum Groups of Type, Int. Math. Res. Not. (2015) (e-print, http://dx.doi.org/10.1093/imrn/rnv241)
[30] Twisting the -deformations of compact semisimple Lie groups, J. Math. Soc. Japan, Volume 67 (2015) no. 2, pp. 637-662 | DOI
[31] Symmetries of quantum spaces. Subgroups and quotient spaces of quantum and groups, Comm. Math. Phys., Volume 170 (1995) no. 1, pp. 1-20 http://projecteuclid.org/getRecord?id=euclid.cmp/1104272946 | DOI
[32] Hopf bi-Galois extensions, Comm. Algebra, Volume 24 (1996) no. 12, pp. 3797-3825 | DOI
[33] A two-parameter quantization of , Proc. Japan Acad. Ser. A Math. Sci., Volume 66 (1990) no. 5, pp. 112-114 http://projecteuclid.org/euclid.pja/1195512514 | DOI
[34] Cocycle deformations of coordinate rings of quantum matrices, J. Algebra, Volume 189 (1997) no. 1, pp. 23-33 | DOI
[35] Invariants and semi-direct products for finite group actions on tensor categories, J. Math. Soc. Japan, Volume 53 (2001) no. 2, pp. 429-456 | DOI
[36] A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys., Volume 275 (2007) no. 1, pp. 271-296 | DOI
[37] Homotopy quantum field theory, EMS Tracts in Mathematics, 10, European Mathematical Society (EMS), Zürich, 2010, xiv+276 pages | DOI
[38] Universal quantum groups, Internat. J. Math., Volume 7 (1996) no. 2, pp. 255-263 | DOI
[39] Free products of compact quantum groups, Comm. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 http://projecteuclid.org/getRecord?id=euclid.cmp/1104272163 | DOI
[40] Ergodic actions of compact groups on operator algebras. I. General theory, Ann. of Math. (2), Volume 130 (1989) no. 2, pp. 273-319 | DOI
[41] Crossed products of -algebras, Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007, xvi+528 pages | DOI
Cited by Sources: