New Braided Crossed Categories and Drinfel'd Quantum Double for Weak Hopf Group Coalgebras

Abstract

In order to construct a class of new braided crossed categories in the sense of Turaev (cf. Turaev, 1994, 2000), we first study some properties of a weak Hopf group-coalgebra introduced in Van Daele and Wang (2004). Then, we develop the fundamental theorem of weak Hopf group-comodules, generalizing the ones both in Böhm et al. (1999) and in Virelizier (2002), and the concept of Yetter–Drinfel'd module over weak crossed structures, generalizing the ones both in Böhm (2000) and Zunino (2004a) by using an approach of Turaev categorical theory introduced in their article by Caenepeel and De Lombaerde (2006). Finally, over a weak crossed Hopf group-coalgebra we introduce an analog of a Drinfel'd quantum double construction and show that the category of modules over the such Drinfel'd quantum doubles is isomorphic to the category of Yetter–Drinfel'd modules as a class of new braided crossed categories.

Key Words:

• Braided crossed category
• Drinfel'd quantum double
• Turaev category
• Unifying Hopf group-coalgebra
• Weak Hopf algebra
• Weak Hopf group-coalgebra
• Weak Hopf group-comodule
• Yetter–Drinfel'd module

2000 Mathematics Subject Classification:

• 16W30
• 57R56
• 18D10

ACKNOWLEDGMENTS

The authors would like to thank the referee for his/her valuable comments on this article and to thank Prof. S. Caenepeel for his private communications about this topic. The second author was supported by the Research Council of the KU Leuven. He is very grateful to Prof. J. Quaegebeur and J. Kustermans for their help. in many matters, and to the research group of K.U. Leuven for providing a good atmosphere to work. He would like to thank the K.U. Leuven for its warm hospitality. This work was also partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20060286006), the FNS of CHINA (10571026) and Jiangsu Province (BK2005207).

Notes

Communicated by R. Wisbauer.