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Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension \(p^{3}\) and \(pq^{2}\). (English) Zbl 1409.16027
Summary: Let \(p\) and \(q\) be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension \(p^3\) and of dimension \(p q^2\). We obtain that the \(p + 1\) non-isomorphic self-dual semisimple Hopf algebras of dimension \(p^3\) classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension \(p^3\), established by the third-named author in an appendix.

MSC:
16T05 Hopf algebras and their applications
16S35 Twisted and skew group rings, crossed products
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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[1] Andruskiewitsch, N.; Natale, S., Examples of self-dual Hopf algebras, J. Math. Sci. Univ. Tokyo, 6, 181-215, (1999) · Zbl 0934.16033
[2] Cartan, H.; Eilenberg, S., Homological algebra, Princeton Math. Series, (1956), London · Zbl 0075.24305
[3] Dijkgraaf, R.; Pasquier, V.; Roche, P., Quasi-quantum groups related to orbifold models, (Proc. Modern Quantum Field Theory, (1990), Tata Institute Bombay), 375-383 · Zbl 0854.17012
[4] Doi, Y., Braided bialgebras and quadratic bialgebras, Commun. Algebra, 21, 1731-1749, (1993) · Zbl 0779.16015
[5] Doi, Y.; Takeuchi, M., Cleft comodule algebras for a bialgebra, Commun. Algebra, 14, 5, 801-817, (1986) · Zbl 0589.16011
[6] Etingof, P., On Vafa’s theorem for tensor categories, Math. Res. Lett., 9, 651-657, (2002) · Zbl 1035.18004
[7] Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V., Tensor categories, AMS Mathematical Surveys and Monographs, vol. 205, (2015) · Zbl 1365.18001
[8] Etingof, P.; Nikshych, D.; Ostrik, V., On fusion categories, Ann. Math., 162, 581-642, (2005) · Zbl 1125.16025
[9] Etingof, P.; Nikshych, D.; Ostrik, V., Weakly group-theoretical and solvable fusion categories, Adv. Math., 226, 176-205, (2011) · Zbl 1210.18009
[10] Gelaki, S.; Naidu, D., Some properties of group-theoretical categories, J. Algebra, 322, 2631-2641, (2009) · Zbl 1209.18007
[11] Günther, R., Crossed products for pointed Hopf algebras, Commun. Algebra, 27, 9, 4389-4410, (1999) · Zbl 0947.16024
[12] Kac, G. I., Finite ring groups, Dokl. Akad. Nauk SSSR, 147, 21-24, (1962)
[13] Kreimer, H.; Takeuchi, M., Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J., 30, 675-692, (1981) · Zbl 0451.16005
[14] Masuoka, A., Semisimple Hopf algebras of dimension 6, 8, Isr. J. Math., 92, 361-373, (1995) · Zbl 0839.16036
[15] Masuoka, A., Self dual Hopf algebras of dimension \(p^3\) obtained by extension, J. Algebra, 178, 791-806, (1995) · Zbl 0840.16031
[16] Masuoka, A., Extensions of Hopf algebras, Trabajos de Matemática, Fa.M.A.F., 41, 99, (1999) · Zbl 0942.16043
[17] Masuoka, A., Cocycle deformations and Galois objects for some cosemisimple Hopf algebras of finite dimension, Contemp. Math., 267, 195-214, (2000) · Zbl 0985.16025
[18] Masuoka, A., Hopf algebra extensions and cohomology, Math. Sci. Res. Inst. Publ., 43, 167-209, (2002) · Zbl 1011.16024
[19] Montgomery, S., Hopf algebras and their actions on rings, Reg. Conf. Ser. Math., vol. 82, (1993), Amer. Math. Soc. Providence, Rhode Island · Zbl 0804.16041
[20] Naidu, D., Categorical Morita equivalence for group-theoretical categories, Commun. Algebra, 35, 3544-3565, (2007) · Zbl 1143.18009
[21] Natale, S., On semisimple Hopf algebras of dimension \(p q^2\), J. Algebra, 221, 242-278, (1999) · Zbl 0942.16045
[22] Natale, S., On group theoretical Hopf algebras and exact factorizations of finite groups, J. Algebra, 270, 199-211, (2003) · Zbl 1040.16027
[23] Natale, S., Semisolvability of semisimple Hopf algebras of low dimension, Mem. Am. Math. Soc., vol. 186, (2007), 123 pp · Zbl 1185.16033
[24] Natale, S., On the equivalence of module categories over a group-theoretical fusion category, SIGMA, 13, (2017), 9 pp · Zbl 1437.18011
[25] Ostrik, V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups, 8, 177-206, (2003) · Zbl 1044.18004
[26] Ostrik, V., Module categories over the Drinfeld double of a finite group, Int. Math. Res. Not., 2003, 27, 1507-1520, (2003) · Zbl 1044.18005
[27] Schauenburg, P., Hopf bigalois extensions, Commun. Algebra, 24, 3797-3825, (1996) · Zbl 0878.16020
[28] Schauenburg, P., Hopf-Galois and bi-Galois extensions, Galois theory, Hopf algebras, and Semiabelian categories, Fields Inst. Commun., vol. 43, 469-515, (2004), Amer. Math. Soc. Providence, RI · Zbl 1091.16023
[29] Schauenburg, P., Hopf bimodules, coquasibialgebras, and an exact sequence of Kac, Adv. Math., 165, 194-263, (2002) · Zbl 1006.16054
[30] Tahara, K., On the second cohomology groups of semidirect products, Math. Z., 129, 365-379, (1972) · Zbl 0238.20068
[31] Ulbrich, K.-H., Galois extensions as functors of comodules, Manuscr. Math., 59, 391-397, (1987) · Zbl 0634.16027
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