CONSTRUCTION OF A BRAIDED MONOIDAL CATEGORY FOR BRZEZINSKI CROSSED COPRODUCTS OF HOPF π-ALGEBRAS

Let pi be a group, C, H Hopf pi-algebras, and g(alpha) : H. and T-alpha : C-alpha circle times H-alpha -> H-alpha circle times C-alpha, families of linear maps. We give necessary and sufficient conditions for the family of Brzezinski crossed coproduct coalgebras {C-alpha #(g alpha)(T alpha) H-alpha}(alpha is an element of pi), to be a Hopf pi-algebra. Moreover, necessary and sufficient conditions for the Brzezinski crossed coproduct Hopf pi-algebra C(sic)(T)(g) (pi) H to be quasitriangular are derived, and in this case, the left pi-module category c(sic)(T)(6) M-pi(Pi) is a braided monoidal category.