Abstract
In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2n +1 with the group of grouplikes isomorphic to ℤ2 n−1×ℤ2. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.
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References
Andruskiewitsch, N.: Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996), 3–42.
Andruskiewitsch, N. and Devoto, J.: Extensions of Hopf algebras, Algebra i Analiz 7(1) (1995), 22–61.
Benson, D. J.: Representations and Cohomology. I, Basic Representation Theory of Finite Groups and Associative Algebras, 2nd edn, Cambridge Stud. Adv. Math. 30, Cambridge Univ. Press, Cambridge, 1998.
Etingof, P. and Gelaki, S.: On the exponent of finite-dimensional Hopf algebras, Math. Res. Lett. 6(2) (1999), 131–140.
Hofstetter, I.: Extensions of Hopf algebras and their cohomological description, J. Algebra 164 (1994), 264–298.
James, G. and Liebeck, M.: Representations and Characters of Groups, Cambridge Math. Textbooks, Cambridge Univ. Press, Cambridge, 1993.
Kashina, Y.: On the order of the antipode of Hopf algebras in HH Y D, Comm. Algebra 27(3) (1999), 1261–1273.
Kashina, Y.: A generalized power map for Hopf algebras, In: Hopf Algebras and Quantum Groups (Brussels, 1998), Lecture Notes Pure Appl. Math. 209, Dekker, New York, 2000, pp. 165–182.
Kashina, Y.: Studies of semisimple finite-dimensional Hopf algebras, Dissertation, USC, 1999.
Kashina, Y.; Classification of semisimple Hopf algebras of dimension 16, J. Algebra 232 (2000), 617–663.
Kashina, Y.: Examples of Hopf algebras of dimension 2m, MSRI Preprint #2000- 003.
Kashina, Y., Mason, G. and Montgomery, S.: The Schur indicator for some finite-dimensional Hopf algebras, J. Algebra 251 (2002), 888–913.
Kobayashi, T. and Masuoka, A.: A result extended from groups to Hopf algebras, Tsukuba J. Math. 21(1) (1997), 55–58.
Linchenko, V. and Montgomery, S.: A Frobenius- Schur theorem for Hopf algebras, Algebras and Representation Theory 3(4) (2000), 347–355.
Masuoka, A.: Coideal subalgebras of finite Hopf algebras, J. Algebra 163 (1994), 819–831.
Masuoka, A.: Selfdual Hopf algebras of dimension p 3 obtained by extension, J. Algebra 178(3) (1995), 791–806.
Masuoka, A.: Calculations of some groups of Hopf algebra extensions, J. Algebra 191(2) (1997), 568–588; Corrigendum, J. Algebra 191(2) (1997), 656.
Masuoka, A.: Cocycle deformations and Galois objects for some cosemisimple Hopf algebras of finite dimension, Contemporary Math.267, 195–214.
Montgomery, S.: Hopf Algebras and their Actions on Rings, CBMS Lectures 82, Amer. Math. Soc., Providence, RI, 1993.
Montgomery, S. and Witherspoon, S. J.: Irreducible representations of crossed products, J. Pure Appl. Algebra 129(3) (1998), 315–326.
Natale, S.: On semisimple Hopf algebras of dimension pq 2, J. Algebra 221 (1999), 242–278.
Nichols, W. and Richmond, M. B.: The Grothendieck group of a Hopf algebra, J. Pure Appl. Algebra 106 (1996), 297–306.
Nichols, W. and Zoeller, M. B.: A Hopf algebra freeness theorem, Amer.J.Math. 111 (1989), 381–385.
Schneider, H.-J.: Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), 289–312.
Serre, J.-P.: Groups de Grothendieck des schemas en groups réductifs déployés, Publ. Math. I.H.E.S. 34 (1968), 37–52.
Zhu, Y.: Hopf algebras of prime dimension, Internat. Math. Res. Notices 1 (1994), 53–59.
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Kashina, Y. On Semisimple Hopf Algebras of Dimension 2m . Algebras and Representation Theory 6, 393–425 (2003). https://doi.org/10.1023/B:ALGE.0000003540.74953.97
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DOI: https://doi.org/10.1023/B:ALGE.0000003540.74953.97