Hopf modules and the double of a quasi-Hopf algebra

Schauenburg, Peter

doi:10.1090/S0002-9947-02-02980-X

Hopf modules and the double of a quasi-Hopf algebra

Author: Peter Schauenburg
Journal: Trans. Amer. Math. Soc. 354 (2002), 3349-3378
MSC (2000): Primary 16W30
DOI: https://doi.org/10.1090/S0002-9947-02-02980-X
Published electronically: April 1, 2002
MathSciNet review: 1897403
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras. We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case. The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.

1. Alistair K. Macpherson, Atomic mechanics of solids, Mechanics and Physics of Discrete Systems, vol. 2, North-Holland Publishing Co., Amsterdam, 1990. MR 1140707
2. V. G. Drinfel′d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989), no. 6, 114–148 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 6, 1419–1457. MR 1047964
3. Frank Hausser and Florian Nill, Diagonal crossed products by duals of quasi-quantum groups, Rev. Math. Phys. 11 (1999), no. 5, 553–629. MR 1696105, https://doi.org/10.1142/S0129055X99000210
4. Frank Hausser and Florian Nill, Doubles of quasi-quantum groups, Comm. Math. Phys. 199 (1999), no. 3, 547–589. MR 1669685, https://doi.org/10.1007/s002200050512
5. HAUSSER, F., AND NILL, F. Integral theory for quasi-Hopf algebras. preprint (math. QA/9904164).
6. JOYAL, A., AND STREET, R.
Braided tensor categories.
Adv. in Math. 102 (1993), 20-78.
7. André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, https://doi.org/10.1006/aima.1993.1055
8. V. V. Lyubashenko, Hopf algebras and vector-symmetries, Uspekhi Mat. Nauk 41 (1986), no. 5(251), 185–186 (Russian). MR 878344
9. Shahn Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995. MR 1381692
10. S. Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), no. 1, 1–9. MR 1631648, https://doi.org/10.1023/A:1007450123281
11. Bodo Pareigis, Non-additive ring and module theory. I. General theory of monoids, Publ. Math. Debrecen 24 (1977), no. 1-2, 189–204. MR 450361
12. B. Pareigis, Non-additive ring and module theory. II. \cal𝐶-categories, \cal𝐶-functors and \cal𝐶-morphisms, Publ. Math. Debrecen 24 (1977), no. 3-4, 351–361. MR 498792
13. Peter Schauenburg, Hopf modules and Yetter-Drinfel′d modules, J. Algebra 169 (1994), no. 3, 874–890. MR 1302122, https://doi.org/10.1006/jabr.1994.1314
14. SCHAUENBURG, P.
Hopf algebra extensions and monoidal categories.
preprint (2001).
15. Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
16. D. Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 2, 425–456. MR 1071429
17. S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), no. 1, 125–170. MR 994499