Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

Author(s): Akira Masuoka
Journal: Proc. Amer. Math. Soc. 137 (2009), 1925-1932.
MSC (2000): Primary 16W30
Posted: February 9, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We prove that a finite-dimensional irreducible Hopf algebra $ H$ in positive characteristic is semisimple if and only if it is commutative and semisimple if and only if the restricted Lie algebra $ P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes in the special but essential situation with a finiteness assumption added.

References:

[DG]
M. Demazure and P. Gabriel,
``Groupes Algébriques'',
North-Holland, Amsterdam, 1970. MR 0302656 (46:1800)

[EG]
P. Etingof and S. Gelaki,
On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic,
Internat. Math. Res. Notices, 1998, No. 16, 851-864. MR 1643702 (99i:16068)

[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. G. Larson,
Characters of Hopf algebras,
J. Algebra 17(1971), 352-368. MR 0283054 (44:287)

[LR]
R. G. Larson and D. E. Radford,
Semisimple cosemisimple Hopf algebras,
Amer. J. Math. 110(1988), 187-195. MR 926744 (89a:16011)

[M]
A. Masuoka,
Classification of semisimple Hopf algebras,
in: M. Hazewinkel (ed.), ``Handbook of Algebra'', Vol. 5, Elsevier, 2008, pp. 429-455.

[Mo]
S. Montgomery,
``Hopf Algebras and Their Actions on Rings'',
CBMS Regional Conference Series in Math., Vol. 82, Amer. Math. Soc., Providence, RI, 1993. MR 1243637 (94i:16019)

[SF]
H. Strade and R. Farnsteiner,
``Modular Lie Algebras and Their Representations'',
Monogr. Textbooks Pure Appl. Math., Vol. 116, Marcel Dekker, New York and Basel, 1988. MR 929682 (89h:17021)

[Sw1]
M. Sweedler,
``Hopf Algebras'',
Benjamin, New York, 1969. MR 0252485 (40:5705)

[Sw2]
M. Sweedler,
Connected fully reducible affine group schemes in positive characteristic are abelian.
J. Math. Kyoto Univ., 11(1970), 51-70. MR 0280499 (43:6219)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16W30

Retrieve articles in all Journals with MSC (2000): 16W30

Additional Information:

Akira Masuoka
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan
Email: akira@math.tsukuba.ac.jp

DOI: 10.1090/S0002-9939-09-09863-3
PII: S 0002-9939(09)09863-3
Keywords: Hopf algebra in positive characteristic, restricted Lie algebra
Received by editor(s): July 14, 2008
Posted: February 9, 2009
Communicated by: Gail R. Letzter
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google