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Abstract
The main aim of this article is to generalize results from Pflaum and Schauenburg (Citation1997) to quantum groupoids (weak Hopf algebras). We first introduce the concept of a weak bicovariant differential calculus on quantum groupoids following some ideas of Woronowicz (Citation1989). Then we establish such a noncommutative differential calculus on weak smash product over quantum groupoids in the sense of Nikshych (Citation2000) and study the notion of a connection with respect to such differential calculus. Finally, we study the notion of a higher order differential calculus in an analogous concept for quantum groupoids in the sense of Schauenburg (Citation1996).
ACKNOWLEDGMENT
The authors would like to thank the referee for his/her valuable comments on this article. This work was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20060286006), the FNS of CHINA (10571026), and the Southeast University Fund (XJ0707273).
Notes
Communicated by E. R. Puczylowski.