|
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
MathSciNet review:
2480272
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We prove that a finite-dimensional irreducible Hopf algebra  in positive characteristic is semisimple if and only if it is commutative and semisimple if and only if the restricted Lie algebra  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  ,
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.
|
