The Generalized C.M.Z.-Theorem and a Drinfel'd Double Construction for WT-Coalgebras and Graded Quantum Groupoids

Abstract

Let π be a group. In this article, we introduce the notions of a weak Doi–Hopf π-module and a weak π-twisted smash product. We show that the Yetter–Drinfel'd π-modules over a weak crossed Hopf π-coalgebra (WT-coalgebra) are special cases as these new weak Doi–Hopf π-modules, generalizing the main result by Caenepeel et al. (1997) and that the Drinfel'd double for WT-coalgebras (Van Daele and Wang, 2008) appears as, a type of such a weak π -twisted smash product, respectively. Finally, starting from a weak Hopf algebra endowed with an action of a group π by weak Hopf automorphisms, we construct a quasitriangular weak Hopf π -coalgebra by a twisted double method, generalizing the main result in Virelizier (2005). This method allows us to obtain nontrivial examples of quasitriangular weak Hopf π-coalgebras.

Key Words:

• Drinfel'd double
• Graded quantum groupoids
• Weak Doi–Hopf π-module
• Weak Hopf π -coalgebra
• Weak π -twisted smash product

2000 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENT

The authors have many thanks for the referee's helpful comments. This work was partially supported by the FNS of CHINA (10571026), the Southeast University Fund (XJ0707273), and the SRFDPHE (20060286006).

Notes

Communicated by L. Ein.