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Some isomorphism results for graded twistings of function algebras on finite groups
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 501-544.

We provide isomorphism results for Hopf algebras that are obtained as graded twistings of function algebras on finite groups by cocentral actions of cyclic groups. More generally, we also consider the isomorphism problem for finite-dimensional Hopf algebras fitting into abelian cocentral extensions. We apply our classification results to a number of concrete examples involving special linear groups over finite fields, alternating and symmetric groups, and dihedral groups.

Nous obtenons des résultats de classification à isomorphisme près pour les algèbres de Hopf obtenues comme twists gradués d’algèbres de fonctions sur des groupes finis par des actions cocentrales de groupes cycliques Plus généralement, nous considérons le problème d’isomor-phisme pour les algèbres de Hopf de dimension finie s’insérant dans des extensions cocentrales abéliennes. Nous appliquons ensuite nos résultats de classification à divers exemples concrets impliquant les groupes spéciaux linéaires sur des corps finis, les groupes symétriques et alternés, et les groupes diédraux.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1701
Classification: 16T05
Keywords: Finite-dimensional Hopf algebras, graded twist, abelian cocentral extension
Julien Bichon 1; Maeva Paradis 1

1 Université Clermont Auvergne, CNRS, LMBP, 63000 Clermont-Ferrand, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Julien Bichon; Maeva Paradis. Some isomorphism results for graded twistings of function algebras on finite groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 501-544. doi : 10.5802/afst.1701. https://afst.centre-mersenne.org/articles/10.5802/afst.1701/

[2] Nicolás Andruskiewitsch; Monique Müller Examples of extensions of Hopf algebras, Rev. Colomb. Mat., Volume 49 (2015) no. 1, pp. 193-211 | DOI | MR | Zbl

[3] Julien Bichon; Sonia Natale Hopf algebra deformations of binary polyhedral groups, Transform. Groups, Volume 16 (2011) no. 2, pp. 339-374 | DOI | MR | Zbl

[4] Julien Bichon; Sergey Neshveyev; Makoto Yamashita Graded twisting of categories and quantum groups by group actions, Ann. Inst. Fourier, Volume 66 (2016) no. 6, pp. 2299-2338 | DOI | Numdam | MR | Zbl

[5] Julien Bichon; Sergey Neshveyev; Makoto Yamashita Graded twisting of comodule algebras and module categories, J. Noncommut. Geom., Volume 12 (2018) no. 1, pp. 331-368 | DOI | MR | Zbl

[6] Alexandru Chirvasitu Centers, cocenters and simple quantum groups, J. Pure Appl. Algebra, Volume 218 (2014) no. 8, pp. 1418-1430 | DOI | MR | Zbl

[7] Alexandru Chirvasitu; Paweł Kasprzak On the Hopf (co)center of a Hopf algebra, J. Algebra, Volume 464 (2016), pp. 141-174 | DOI | MR | Zbl

[8] Jean A. Dieudonné La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5, Springer, 1971, viii+129 pages | MR

[9] Yukio Doi Braided bialgebras and quadratic bialgebras, Commun. Algebra, Volume 21 (1993) no. 5, pp. 1731-1749 | MR | Zbl

[10] Nobuyuki Fukuda Semisimple Hopf algebras of dimension 12, Tsukuba J. Math., Volume 21 (1997) no. 1, pp. 43-54 | DOI | MR | Zbl

[11] César Galindo; Yiby Morales A five-term exact sequence for Kac cohomology, Algebra Number Theory, Volume 13 (2019) no. 5, pp. 1121-1144 | DOI | MR | Zbl

[12] Peter J. Hilton; Urs Stammbach A course in homological algebra, Graduate Texts in Mathematics, 4, Springer, 1997, xii+364 pages | DOI | MR

[13] Georgiĭ I. Kac Extensions of groups to ring groups, Math. USSR, Sb., Volume 5 (1969), pp. 451-474

[14] Gregory Karpilovsky The Schur multiplier, London Mathematical Society Monographs. New Series, 2, Clarendon Press, 1987, x+302 pages | MR

[15] Yevgenia Kashina Classification of semisimple Hopf algebras of dimension 16, J. Algebra, Volume 232 (2000) no. 2, pp. 617-663 | DOI | MR | Zbl

[16] Yevgenia Kashina On two families of Hopf algebras of dimension 2 m , Commun. Algebra, Volume 31 (2003) no. 4, pp. 1643-1668 | DOI | MR | Zbl

[17] Yevgenia Kashina On semisimple Hopf algebras of dimension 2 m , II, Algebr. Represent. Theory, Volume 19 (2016) no. 6, pp. 1387-1422 | DOI | MR | Zbl

[18] David Kazhdan; Hans Wenzl Reconstructing monoidal categories, I. M. Gelfand Seminar (Advances in Soviet Mathematics), Volume 16, American Mathematical Society, 1993, pp. 111-136 | DOI | MR | Zbl

[19] Leonid Krop On the classification of finite-dimensional semisimple Hopf algebras, Groups, rings, group rings, and Hopf algebras (Contemporary Mathematics), Volume 688, American Mathematical Society, 2017, pp. 181-218 | DOI | MR | Zbl

[20] Akira Masuoka Self-dual Hopf algebras of dimension p 3 obtained by extension, J. Algebra, Volume 178 (1995) no. 3, pp. 791-806 | DOI | MR | Zbl

[21] Akira Masuoka Calculations of some groups of Hopf algebra extensions, J. Algebra, Volume 191 (1997) no. 2, pp. 568-588 | DOI | MR | Zbl

[22] Akira Masuoka Extensions of Hopf algebras, Trabajos de Matemática. Serie B., 41, Universidad Nacional de Córdoba, 1999 | Zbl

[23] Akira Masuoka; Yukio Doi Generalization of cleft comodule algebras, Commun. Algebra, Volume 20 (1992) no. 12, pp. 3703-3721 | DOI | MR | Zbl

[24] Susan Montgomery Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, American Mathematical Society, 1993, xiv+238 pages | DOI | MR

[25] Sonia Natale On semisimple Hopf algebras of dimension pq 2 , J. Algebra, Volume 221 (1999) no. 1, pp. 242-278 | DOI | MR | Zbl

[26] Sergey Neshveyev; Makoto Yamashita Twisting the q-deformations of compact semisimple Lie groups, J. Math. Soc. Japan, Volume 67 (2015) no. 2, pp. 637-662 | DOI | MR | Zbl

[27] Piotr Podleś Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU (2) and SO (3) groups, Commun. Math. Phys., Volume 170 (1995) no. 1, pp. 1-20 | DOI | MR | Zbl

[28] Hans-Jürgen Schneider Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra, Volume 152 (1992) no. 2, pp. 289-312 | DOI | MR | Zbl

[29] James J. Zhang Twisted graded algebras and equivalences of graded categories, Proc. Lond. Math. Soc., Volume 72 (1996) no. 2, pp. 281-311 | DOI | MR | Zbl

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