Hopf¦Ð-coalgebra And ¦Ð-smash Product

Hopf algebra is an interesting subject for algebraic people, and it has been widely and deeply studied. Recently, for a groupπ, the notions ofπ-coalgebra and Hopfπ-coalgebra were introduced by Turaev in [15, Section11.2]. Hopfπ- coalgebra also is an algebraic structure and the notion of a Hopfπ-coalgebra generalizes that of a Hopf algebra. Let K be a fixed field. Aπ-coalgebra over K is a family C = {Cα}α∈πof K -spaces endowed with a comultiplicationΔand a counitε, whereΔ= {Δα,β:Cαβ→Cα-Cβ}α,β∈π,ε:C1→k. AndΔis coassociative,εis counitary. A Hopfπ-coalgebra H = ({ Hα}α∈π,Δ,ε) is a family of K - algebras, and H = { Hα}α∈πis aπ- coalgebra endowed with an antipode 1S = {Sα: Hα→Hα- }α∈πwhich satisfies some compatibility conditions. Alexis Virelizier also studied some algebraic properties of Hopfπ- coalgebra and gave the notion of a Hopfπ-comodule. Meanwhile, he also generalized the main properties of quasitriangular Hopfπ-coalgebras, see [1].Before Alexis Virelizier's study for Hopfπ-coalgebra, Susan Montgomery in [4] had studied module algebras, comodule algebras, and smash product algebras for usual Hopf algebras, and D.E. Radford had studied smash coproduct coalgebra, smash biproduct in [9]. Upon the background above, in this paper, we mainly studyπ-smash product,π-smash coproduct, andπ- smash biproduct with respect to Hopfπ- coalgebra. First, we give definitions ofπ- H-module algebra, andπ-smash product which is between the Hopfπ-coalgebra and theπ-H-module algebra. We describe a family of algebra structure of theπ-smash product. Second, we introduce definitions ofπ-H-comodule coalgebra, andπ-smash coproduct which is between the Hopfπ-coalgebra and theπ-H-comodule coalgebra. Then, we give some conclusions of theπ-smash coproduct. Third, we defineπ- smash biproduct which is both aπ- smash product and aπ- smash coproduct. Some necessary and sufficient conditions are given in this paper.The paper is organized as follows. In Section 1, we review some basic notions about π- coalgebras, Hopfπ- coalgebras, Coopposite Hopfπ- coalgebras,π- module andπ-comodule.In Section 2, First of all, we define a leftπ-H -module,π-H-module algebra andπ-smash product A #H which is between a Hopfπ-coalgebra H and aπ- H -module algebra. We prove one of main results in this paper that theπ- smash product A # H = { A # Hα}α∈πis a family of K-algebras, see definition 2.3 and theorem 2.4. Also, we obtain the similar conclusion to algebraic structure on A # Hcop = {A#Hαc op}α∈π, in which H cop is a coopposite Hopfπ- coalgebra.In Section 3, we first give the definitions of rightπ- H - comodule, rightπ- H-comodule coalgebra andπ-smash coproduct A H cop= {AHαc op}α∈πwhich is between the Hopfπ-coalgebra and theπ-H-comodule coalgebra. Theπ-smash coproduct is a generalization of Molnar's smash coproduct. We get another main result in this paper, which show that theπ- smash coproduct A H cop= {AHαc op}α∈πis aπ- coalgebra, see definition 3.3 and theorem 3.4.In Section 4, we defineπ- smash biproduct A * H cop which is both aπ- smash product and aπ-smash coproduct, and then we investigate some properties ofπ-smash biproduct, see theorem 4.5. This is the last main result in which we give some necessary and sufficient conditions for theπ-smash biproduct A * H cop to be a Hopfπ-coalgebra.