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.