Hopf Group-Coalgebras

We establish the one-to-one correspondence between group-algebras and group-graded algebras. We generalize the notion of weak Hopf algebra,quasi-braided Hopf algebra and quasi-braided tensor category to the setting of group-coalgebras,and show that their main properties still hold.We give the necessary and sufficient conditions of the category of representations of H to be a tensor category, where H in Turaev category is both an algebra and a coalgebra with its comultiplication an algebra morphism. We finally improve the notion of a Yetter-Drinfeld module over a crossed Hopf group-coalgebra,prove that the general center of the category of representations of a crossed Hopf group-coalgebra and the category of Yetter-Drinfeld modules over H are tensor equivalent, and give a sufficient condition of the general center to be the center.