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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Crossed products of semisimple cocommutative Hopf algebras

Author: William Chin
Journal: Proc. Amer. Math. Soc. 116 (1992), 321-327
MSC: Primary 16W30; Secondary 16S30, 16S35
MathSciNet review: 1100646
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Abstract: We provide a short proof of an analog of Nagata's theorem for finite-dimensional Hopf algebras. The result, proved Hopf-algebraically by Sweedler and using group schemes by Demazure and Gabriel, says that a finite-dimensional cocommutative semisimple irreducible Hopf algebra is commutative. With mild base field assumptions such a Hopf algebra is just the dual of a $ p$-group algebra. We give en route an easy proof of a version of Hochschild's theorem on semisimple restricted enveloping algebras.

Let $ R{\char93 _t}H$ denote a crossed product with an invertible cocycle $ t$, where $ H$ is a semisimple cocommutative Hopf algebra $ H$ over a perfect field. The result above is applied to show that $ R{\char93 _t}H$ is semiprime if and only if $ R$ is $ H$-semiprime. The approach relies on results on ideals of the crossed product that are stable under the action of the dual of $ H$ and the Fisher-Montgomery theorem for crossed products of finite groups.

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  • [A] E. Abe, Hopf algebras, Cambridge Univ. Press, Cambridge, 1980. MR 594432 (83a:16010)
  • [BCM] R. J. Blattner, M. Cohen, and S. Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986), 671-711. MR 860387 (87k:16012)
  • [BM1] R. J. Blattner and S. Montgomery, A duality theorem for Hopf module algebras, J. Algebra 95 (1985), 153-172. MR 797661 (87h:16016)
  • [BM2] -, Crossed products and Galois extensions of Hopf algebras, Pacific J. Math. 137 (1989), 37-54. MR 983327 (90a:16007)
  • [BeC] J. Bergen and M. Cohen, Actions of commutative Hopf algebras, Bull. London Math. Soc. 18 (1986), 159-164. MR 818820 (87e:16052)
  • [BeM] J. Bergen and S. Montgomery, Smash products and outer derivations, Israel J. Math. 53 (1986), 321-345. MR 852484 (87i:16065)
  • [Ch1] W. Chin, Crossed products and generalized inner actions of Hopf algebras, Pacific J. Math. 150 (1991), 241-259. MR 1123442 (92h:16030)
  • [Ch2] -, Spectra of smash products, Israel J. Math. 72 (1990). MR 1098981 (92g:16045)
  • [CM] M. Cohen and S. Montgomery, Group-graded rings, smash products and group actions, Trans. Amer. Math. Soc. 586 (1986), 237-258. MR 728711 (85i:16002)
  • [CF] M. Cohen and D. Fischman, Hopf algebra actions, J. Algebra 100 (1986), 363-379. MR 840582 (87i:16012)
  • [DG] M. Demazure and P. Gabriel, Groupes algébriques, Tome I, North-Holland, Amsterdam, 1970. MR 0302656 (46:1800)
  • [DT] Y. Doi and M. Takeuchi, Hopf Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras, J. Algebra 121 (1989), 488-516. MR 992778 (90b:16015)
  • [FS] R. Farnsteiner and H. Strade, Modular Lie algebras and representation theory, Pure Appl. Math., vol. 116, Dekker, New York, 1986.
  • [H] G. Hochschild, Representations of restricted Lie algebras of characteristic $ p$, Proc. Amer. Math. Soc. 5 (1954), 603-605. MR 0066361 (16:562d)
  • [L] R. Larson, Cocommutative Hopf algebras, Canad. J. Math. 19 (1967), 350-360. MR 0209206 (35:108)
  • [LR] R. Larson and D. Radford, Semisimple Hopf algebras J. Algebra (to appear). MR 1314091 (96a:16040)
  • [M] S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Math., vol. 818, Springer-Verlag, Berlin, 1980. MR 590245 (81j:16041)
  • [NZ] W. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 11 (1989), 381-385. MR 987762 (90c:16008)
  • [P] D. S. Passman, Infinite crossed products, Pure Appl. Math., vol. 135, Academic Press, San Diego, 1989. MR 979094 (90g:16002)
  • [Q] D. Quinn, Integral extensions of noncommutative rings, Israel J. Math. 73 (1991), 113-121. MR 1119933 (92h:16025)
  • [R] D. Radford, Freeness for pointed irreducible Hopf algebras, J. Algebra 45 (1977), 266-273. MR 0437582 (55:10506)
  • [S1] M. Sweedler, Hopf algebras with one grouplike element, Trans. Amer. Math. Soc. 127 (1967), 515-526. MR 0210748 (35:1634)
  • [S2] -, Connected fully reducible affine group schemes in positive characteristic are abelian, J. Math. Kyoto Univ. 11 (1971), 51-70. MR 0280499 (43:6219)
  • [S3] -, Hopf algebras, Benjamin, New York, 1969. MR 0252485 (40:5705)

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