Radford’s Biproducts for Hopf Group-Coalgebras and Its Quasitriangular Structures

摘要

Let π be a group and H={H α } α∈π be a semi-Hopf π-coalgebra in the sense of Virelizier (J. Pure Appl. Algebra 171:75–122, 2002). Let H coact weakly on a coalgebra B and λ={λ α,β :B⟶H α ⊗H β } be a family of k-linear maps. Then in this paper we first introduce the notion of a π-crossed coproduct (Btimes_{lambda }^{pi}H={Btimes_{lambda }H_{alpha }}_{alpha in pi }) and find some sufficient and necessary conditions making it into a π-coalgebra, generalizing the main construction in Lin (Commun. Algebra 10:1–17, 1982). Secondly, we find a sufficient and necessary condition for (B_{#^{pi}}^{times_{lambda }^{pi}} H), with the π-crossed coproduct (Btimes_{lambda }^{pi}H) and π-smash product B# π H to form a semi-Hopf π-coalgebra, if λ is convolution invertible dual 2-cocycle, which generalizes the well-known Radford’s biproduct in Radford (J. Algebra 92:322–347, 1985). Furthermore, we derive some sufficient conditions for (B_{#^{pi}}^{times_{lambda }^{pi}} H) to be a Hopf π-coalgebra. Finally, we construct a quasitriangular structure on the Hopf π-coalgebra (Btimes_{lambda }^{pi}H) (with the usual tensor product).