Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.

Key Words:

• Quantum groupoids
• Weak Doi–Hopf bimodules
• Cibils–Rosso's theorem

Mathematical Subject Classification:

• 16W30
• 16W50

Acknowledgments

The author would like to thank S. Caenepeel and E. De Groot for their valuable help while the author was visiting the Free University of Brussels (VUB), supported by the bilateral project BIL99/43 “New computational, geometric and algebraic methods applied to quantum groups and differential operators” of the Flemish and Chinese governments. In particular, the author is very grateful to S. Caenepeel for helpful discussions and suggestions. He would like to thank the Free University of Brussels for its warm hospitality. This work also was supported in part by Science Foundation of Henan Province for Distinguished Young Scholars and Natural Science Foundation of China. Finally, he is very grateful to thank the referee for his/her helpful comments on this paper.

Notes

^#Communicated by C. Cibils.