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 groupÏ€, the notions ofÏ€-coalgebra and HopfÏ€-coalgebra were introduced by Turaev in [15, Section11.2]. HopfÏ€- coalgebra also is an algebraic structure and the notion of a HopfÏ€-coalgebra generalizes that of a Hopf algebra. Let K be a fixed field. AÏ€-coalgebra over K is a family C = {Cα}α∈πof K -spaces endowed with a comultiplicationΔand a counitε, whereΔ= {Δα,β:Cαβâ†'Cα-Cβ}α,β∈π,ε:C1â†'k. AndΔis coassociative,εis counitary. A HopfÏ€-coalgebra H = ({ Hα}α∈π,Δ,ε) is a family of K - algebras, and H = { Hα}α∈πis aÏ€- coalgebra endowed with an antipode 1S = {Sα: Hαâ†'Hα- }α∈πwhich satisfies some compatibility conditions. Alexis Virelizier also studied some algebraic properties of HopfÏ€- coalgebra and gave the notion of a HopfÏ€-comodule. Meanwhile, he also generalized the main properties of quasitriangular HopfÏ€-coalgebras, see [1].Before Alexis Virelizier's study for HopfÏ€-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 studyÏ€-smash product,Ï€-smash coproduct, andÏ€- smash biproduct with respect to HopfÏ€- coalgebra. First, we give definitions ofÏ€- H-module algebra, andÏ€-smash product which is between the HopfÏ€-coalgebra and theÏ€-H-module algebra. We describe a family of algebra structure of theÏ€-smash product. Second, we introduce definitions ofÏ€-H-comodule coalgebra, andÏ€-smash coproduct which is between the HopfÏ€-coalgebra and theÏ€-H-comodule coalgebra. Then, we give some conclusions of theÏ€-smash coproduct. Third, we defineÏ€- smash biproduct which is both aÏ€- smash product and aÏ€- 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 Ï€- coalgebras, HopfÏ€- coalgebras, Coopposite HopfÏ€- coalgebras,Ï€- module andÏ€-comodule.In Section 2, First of all, we define a leftÏ€-H -module,Ï€-H-module algebra andÏ€-smash product A #H which is between a HopfÏ€-coalgebra H and aÏ€- H -module algebra. We prove one of main results in this paper that theÏ€- 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 HopfÏ€- coalgebra.In Section 3, we first give the definitions of rightÏ€- H - comodule, rightÏ€- H-comodule coalgebra andÏ€-smash coproduct A H cop= {AHαc op}α∈πwhich is between the HopfÏ€-coalgebra and theÏ€-H-comodule coalgebra. TheÏ€-smash coproduct is a generalization of Molnar's smash coproduct. We get another main result in this paper, which show that theÏ€- smash coproduct A H cop= {AHαc op}α∈πis aÏ€- coalgebra, see definition 3.3 and theorem 3.4.In Section 4, we defineÏ€- smash biproduct A * H cop which is both aÏ€- smash product and aÏ€-smash coproduct, and then we investigate some properties ofÏ€-smash biproduct, see theorem 4.5. This is the last main result in which we give some necessary and sufficient conditions for theÏ€-smash biproduct A * H cop to be a HopfÏ€-coalgebra.