Semi-simple Hopf algebras with restrictions on irreducible modules of dimension exceeding $1$
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V. A. Artamonov
Translated by: N. A. Vavilov - St. Petersburg Math. J. 26 (2015), 207-223
- DOI: https://doi.org/10.1090/S1061-0022-2015-01337-X
- Published electronically: February 3, 2015
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Abstract:
The semisimple finite-dimensional Hopf algebras are considered all of whose irreducible representations of each dimension exceeding $1$ are isomorphic. The Hopf algebras with a unique irreducible representation of dimension exceeding $1$ are described, provided this dimension is equal to the order of the group of group-like elements of the dual Hopf algebra. Under some additional restrictions, it is shown that a Hopf algebra cannot have two irreducible representations of dimension exceeding $1$.References
- V. A. Artamonov, On semisimple finite-dimensional Hopf algebras, Mat. Sb. 198 (2007), no. 9, 3–28 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 9-10, 1221–1245. MR 2360805, DOI 10.1070/SM2007v198n09ABEH003880
- È. M. Žmud′, Symplectic geometries on finite abelian groups, Mat. Sb. (N.S.) 86(128) (1971), 9–33 (Russian). MR 0292962
- È. M. Žmud′, Symplectic geometries and projective representations of finite abelian groups, Mat. Sb. (N.S.) 87(129) (1972), 3–17 (Russian). MR 0292963
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- V. A. Artamonov and I. A. Chubarov, Dual algebras of some semisimple finite-dimensional Hopf algebras, Modules and comodules, Trends Math., Birkhäuser Verlag, Basel, 2008, pp. 65–85. MR 2742621, DOI 10.1007/978-3-7643-8742-6_{4}
- V. A. Artamonov and I. A. Chubarov, Properties of some semisimple Hopf algebras, Algebras, representations and applications, Contemp. Math., vol. 483, Amer. Math. Soc., Providence, RI, 2009, pp. 23–36. MR 2497948, DOI 10.1090/conm/483/09432
- V. A. Artamonov, R. B. Mukhatov, and R. Wisbauer, On the category of modules over some semisimple bialgebras, Arab. J. Math. (Springer) 1 (2012), no. 1, 29–38 (English, with English and Arabic summaries). MR 3040910, DOI 10.1007/s40065-012-0010-9
- V. A. Artamonov, On semisimple Hopf algebras with few representations of dimension greater than one, Rev. Un. Mat. Argentina 51 (2010), no. 2, 91–105. MR 2840164
- R. Frucht, Über die Darstekkung endlicher abeischer Gruppen durch Kollineationen, J. Reine Angew. Math. 166 (1932), 16–29.
- Bertram Huppert, Character theory of finite groups, De Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. MR 1645304, DOI 10.1515/9783110809237
- Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
- Sonia Natale, Semisolvability of semisimple Hopf algebras of low dimension, Mem. Amer. Math. Soc. 186 (2007), no. 874, viii+123. MR 2294999, DOI 10.1090/memo/0874
- Sonia Natale and Julia Yael Plavnik, On fusion categories with few irreducible degrees, Algebra Number Theory 6 (2012), no. 6, 1171–1197. MR 2968637, DOI 10.2140/ant.2012.6.1171
- S. Yu. Spiridonova, Generalized commutativity of some Hopf algebras and their relation to finite fields, Algebra i Analiz 25 (2013), no. 5, 202–220 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 25 (2014), no. 5, 855–868. MR 3184611, DOI 10.1090/S1061-0022-2014-01319-2
Bibliographic Information
- V. A. Artamonov
- Affiliation: Department of Mechanics and Mathematics, Moscow State University, Vorobievy Gory 1, 119991 Moscow, Russia
- Email: artamon@mech.math.msu.su
- Received by editor(s): August 14, 2013
- Published electronically: February 3, 2015
- Additional Notes: Supported by RFBR (grant no. 12-01-00070)
- © Copyright 2015 American Mathematical Society
- Journal: St. Petersburg Math. J. 26 (2015), 207-223
- MSC (2010): Primary 16T05
- DOI: https://doi.org/10.1090/S1061-0022-2015-01337-X
- MathSciNet review: 3242035