On Hopf Group Coalgebra's G (?) π-Crossed Products

As a promotion of the group crossed products,this paper mainly conducts two aspects of research:On the one hand,we give the structure of G(?)Ï€- crossed products,and get the G(?)Ï€- coalgebra; on the other hand,we promote the general crossed products' duality theory to G(?)Ï€- crossed products'.This article organized as follows:Firstly,let A={Ag}g∈G and H={Hα}α∈πbe two hopf group coalgebra,in which G andÏ€are two discrete groups,in the way of crossed product's definition, H={Hα}α∈πacts weakly on A={Ag}g∈G:Hα(?)Agâ†'Ag,eachα∈π,g∈G,give the definition ofσ={σg)g∈G as a family of linear mapσg:H1Ï€(?)H1Ï€â†'Hα,as the definitin of general crossed products,then we define G(?)Ï€- crossed product's multiplication as follows: M(g,α):(a(?)h)(b(?)g)=a(h(1,1Ï€)·b)σg(h(2,1Ï€),g(1,1Ï€))(?)h(3,α)g(2,α) eachα∈π,g∈G,11G(?)11Ï€as the unit,and we build G(?)Ï€-crossed products,further we get GA(?)σπH={Ag(?)σgHα)(g∈G,α∈π)as a G(?)Ï€-crossed products' necessary and sufficient condition.Next,we give the comultiplicationâ–³={â–³(g,α)(g',β)}(g,g'∈G,α,β∈π)and counitε=ε(1G,1Ï€)=ε1G(?)ε1Ï€ofG(?)Ï€- crossed products GA#σπH={Ag#σg Ha)(g∈G,α∈π)as follows:â–³(g,α)(g',β):Agg'#σgg'Hαβâ†'(Ag#σgHα)(?)(4g'#σg'Hβ) a#σgg'h(?)(a(1,g)#σgh(1α))(?)(n(2,g')#σg'(2,β))ε(1G,1Ï€):A1G#σ1G H1Ï€â†'k further step,we get GA#Hσπ={Ag#σgHα)(g∈G,α∈π)as a semi-Hoopf G(?)Ï€-coalgebra's necessary and sufficient condition.Further more,we reasch semi-Hopf G(?)Ï€- coalgebra in the further way,on which we define antipode S=S(g,α)(g∈G,α∈π) as follows:S(g,α):Ag#σgHαâ†'Ag-1#σg-1Hα-1, a#σgh(?)(Sg-1(σg-2(S1Ï€(h(2,1Ï€)),h(3,1Ï€)))#σg-1Sα(h(1,α)))(Sg(a)#σg-11α-1) and we have made a series of stipulations,in which we makes it as a Hopf G(?)Ï€- coalgebra,as the suppliments,we give the relations between Hopf G(?)Ï€- coalgebra and A={Ag)g∈G,H={Hα}α∈π.Finally,reference the knowledge of mold algebra,define two mapsαg andβg:α9:(Ag#σgHα)#H1*â†'End(Ag#σgHα)gαg((x#σgh)#f)(y#σg9))=(x#σgh)(y#σgfâ†')=(x#σgh)(y#σg<f,g(2,1Ï€))g(1,α))βg:End(Ag#σgHα)gâ†'(Ag#σgHα)#H1Ï€*.βg:T(?)∑[T(σg-1,(fi(3,1),S1Ï€-1(fi(3,1))#σgfi(4,α))1gσgSα-1(fi(1,α-1))]#ψi then we promute general crossed products' duality theory to G(?)Ï€- crossed products.