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On pointed Hopf algebras of dimension $p^n$

Authors: M. Beattie, S. Dascalescu and L. Grünenfelder
Journal: Proc. Amer. Math. Soc. 128 (2000), 361-367
MSC (1991): Primary 16W30
Published electronically: July 6, 1999
MathSciNet review: 1622781
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we describe nonsemisimple Hopf algebras of dimension $p^n$ with coradical isomorphic to $kC$, $C$ abelian of order $p^{n-1}$, over an algebraically closed field $k$ of characteristic zero. If $C$ is cyclic or $C=(C_p)^{n-1}$, then we also determine the number of isomorphism classes of such Hopf algebras.

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  • 1. N. Andruskiewitsch and H.-J. Schneider, Hopf algebras of order $p^2$ and braided Hopf algebras of order $p$, J. Algebra 199 (1998), 430-454. CMP 98:06
  • 2. M. Beattie, S. D\u{a}sc\u{a}lescu, L. Grünenfelder, C. N\u{a}st\u{a}sescu, Finiteness conditions, co-Frobenius Hopf algebras and quantum groups, J. Algebra 200 (1998), 312-333. CMP 98:08
  • 3. M. Beattie, S. D\u{a}sc\u{a}lescu, L. Grünenfelder, Constructing pointed Hopf algebras by Ore extensions, preprint.
  • 4. S. Gelaki, Quantum groups of dimension $pq^2$, Israel J. Math. 102 (1997), 227-267. CMP 98:06
  • 5. S. Gelaki, On Pointed Ribbon Hopf Algebras, J. Algebra 181 (1996), 760-786. MR 97d:16044
  • 6. C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155 (1995), Springer Verlag. MR 96e:17041
  • 7. R. Larson and D. Radford, Semisimple Hopf algebras, J. Algebra 171 (1995), 5-35. MR 96a:16040
  • 8. A. Masuoka, Semisimple Hopf algebras of dimension 6,8, Israel J. Math. 92 (1995), 361-373. MR 96j:16045
  • 9. A. Masuoka, Self-dual Hopf algebras of dimension $p^3$ obtained by extension, J. Algebra 178 (1995), 791-806. MR 96j:16046
  • 10. A. Masuoka, The $p^n$ theorem for semisimple Hopf algebras, Proc. Amer. Math. Soc. 124 (1996), 735-737. MR 96f:16046
  • 11. A. Masuoka, Semisimple Hopf algebras of dimension $2p$, Comm. Algebra 23 (1995), 1931-1940. MR 96e:16050
  • 12. S. Montgomery, Hopf algebras and their actions on rings, CBMS no. 82, Amer. Math. Soc., 1993. MR 94i:16019
  • 13. D. E. Radford, Operators on Hopf algebras, Amer. J. Math. 99 (1977), 139-158. MR 55:10505
  • 14. D. E. Radford, Irreducible representations of ${\mathcal U}_q(g)$ arising from Mod$_{C^{1/2}}^\cdot$, Israel Math. Conference Proceedings 7 (1993), 143-170. MR 95b:17020
  • 15. D. E. Radford, On Kauffman's knot invariants arising from finite-dimensional Hopf algebras, in ``Advances in Hopf Algebras", Lecture Notes in Pure and Appl. Math., vol. 158, 205-266, Marcel Dekker, New York, 1994. MR 96g:57013
  • 16. D. Stefan, Hopf subalgebras of pointed Hopf algebras and applications, Proc. Amer. Math. Soc. 125 (1997), 3191-3193. MR 97m:16076
  • 17. Y. Zhu, Hopf algebras of prime dimension, Int. Math. Research Notices 1 (1994), 53-59. MR 94j:16072

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Additional Information

M. Beattie
Affiliation: Department of Mathematics and Computer Science, Mount Allison University, Sack- ville, New Brunswick, Canada E4L 1E6

S. Dascalescu
Affiliation: Faculty of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest 1, Romania

L. Grünenfelder
Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5

Received by editor(s): October 7, 1997
Received by editor(s) in revised form: April 3, 1998
Published electronically: July 6, 1999
Additional Notes: The first and third authors research was partially supported by NSERC
Communicated by: Ken Goodearl
Article copyright: © Copyright 1999 American Mathematical Society

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