A quasi-Hopf algebra freeness theorem

Schauenburg, Peter

doi:10.1090/S0002-9939-03-07181-8

A quasi-Hopf algebra freeness theorem

Author: Peter Schauenburg
Journal: Proc. Amer. Math. Soc. 132 (2004), 965-972
MSC (2000): Primary 16W30
DOI: https://doi.org/10.1090/S0002-9939-03-07181-8
Published electronically: July 7, 2003
MathSciNet review: 2045410
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

1. V. G. Drinfel'd, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457. MR 91b:17016
2. Frank Hausser and Florian Nill, Diagonal crossed products by duals of quasi-quantum groups, Rev. Math. Phys. 11 (1999), 553-629. MR 2000d:81069
3. -, Doubles of quasi-quantum groups, Comm. Math. Phys. 199 (1999), 547-589. MR 2000a:16075
4. -, Integral theory for quasi-Hopf algebras, preprint (math.QA/9904164).
5. Irving Kaplansky, Bialgebras, University of Chicago Lecture Notes, 1975. MR 55:8087
6. Christian Kassel, Quantum groups, Graduate Texts in Math., vol. 155, Springer-Verlag, 1995. MR 96e:17041
7. Akira Masuoka, Hopf algebra extensions and cohomology, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 167-209. MR 2003d:16050
8. Akira Masuoka, Cohomology and coquasi-bialgebra extensions associated to a matched pair of bialgebras, Adv. Math. 173 (2003), 262-315.
9. Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Amer. Math. Soc., Providence, Rhode Island, 1993. MR 94i:16019
10. Warren D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111 (1989), 381-385. MR 90c:16008
11. U. Oberst and H.-J. Schneider, Untergruppen formeller Gruppen von endlichem Index, J. Algebra 31 (1974), 10-44. MR 50:13057
12. Florin Panaite, A Maschke-type theorem for quasi-Hopf algebras, Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), Marcel Dekker, New York, 1998, pp. 201-207. MR 99k:16085
13. Bodo Pareigis, Non-additive ring and module theory I. General theory of monoids, Publ. Math. Debrecen 24 (1977), 189-204. MR 56:8656
14. -, Non-additive ring and module theory II. $\mathcal C$ -categories, $\mathcal C$ -functors and $\mathcal C$ -morphisms, Publ. Math. Debrecen 24 (1977), 351-361. MR 58:16834a
15. Peter Schauenburg, Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), 3349-3378. MR 2003c:16050