FS-coalgebras and crossed coproducts

In this paper, we introduce FS-coalgebras, which provide solutions of FS-equations and also solution of braid equations considered by Caenepeel, Militaru and Zhu. FS-coalgebras are constructed by using FS-equations and Harrison cocycles. As applications, we prove that every bialgebra H is an FS-bialgebra if and only if there is a two-sided integral α in {H^{\ast}} such that {\varepsilon(\alpha)=1} , and we show that the crossed coproduct {H^{R}} introduced by the Harrison cocycle R is an FS-coalgebra when {(H,R)} is a finite-dimensional quasitriangular Hopf algebra or a Long copaired bialgebra.

An extended form of Majid’s double biproduct

Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

QUASITRIANGULARITY OF BRZEZIŃSKI'S CROSSED COPRODUCTS

In continuation of our recent work about the quasitriangular structures for the twisted tensor biproduct, we give the necessary and sufficient conditions for Brzeziński crossed coproduct coalgebra, including the twisted tensor coproduct introduced by Caenepeel, Ion, Militaru and Zhu and crossed coproduct as constructed by Lin, equipped with the usual tensor product algebra structure to be a Hopf algebra. Furthermore, the necessary and sufficient conditions for Brzeziński crossed coproduct to be a quasitriangular Hopf algebra are obtained.

Twistings, crossed coproducts, and Hopf-Galois coextensions

LetHbe a Hopf algebra. Ju and Cai (2000) introduced the notion of twisting of anH-module coalgebra. In this paper, we study the relationship between twistings, crossed coproducts, and Hopf-Galois coextensions. In particular, we show that a twisting of anH-Galois coextension remainsH-Galois if the twisting is invertible.