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

Abstract

Let π be a group and H={Hα}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={λα,β:B⟶Hα⊗Hβ} be a family of k-linear maps. Then in this paper we first introduce the notion of a π-crossed coproduct \(B\times_{\lambda }^{\pi}H=\{B\times_{\lambda }H_{\alpha }\}_{\alpha \in \pi }\) 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 \(B\times_{\lambda }^{\pi}H\) and π-smash product B#πH 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 \(B\times_{\lambda }^{\pi}H\) (with the usual tensor product).