Coquasitriangular HOPF Group-Coalgebra

In this article,we generalize the concept of quasitriangular Hopf algebra to Hopf group-coalgebra,and give the definition of coquasitriangular Hopf group-coalgebra.H= ({Hα)α∈π,△,ε,S,σ)is a coquasitriangular Hopf group-coalgebra,then(Hασ,·is a new algebra in the following multiplication:(?) h,k∈Hασ,α∈π,h·k=∑(σ1,1(h(1,1),k(1,1))h(2,α)k(2,α)σ1,1-1(h(3,1),k(3,1)); Hσ=({Hασ}α∈π,△σ,εσ,Sσ)is a Hopf group-coalgebra when Sασ(h)=∑σ1,1(h(1,1),S1(h(2,1))Sα(h(3,α))σ1,1-1(S1(h(4,1)),h(5,1)). In particular,H1σis a Hopf algebra.Asasume that H1is commutative,then(H1σ,σ1,1*)is a symmetrical braid double algebra,whereσ1,1*(h,k)=∑σ(1,1)(k1,1,h(1,1))σ1,1-1(h(2,1),k(2,1)). And further discussed the sufficient and necessary conditions for crossed product Hopf group-coalgebra A#ρπH existing coquasitriangular structure.That is:Hopf group-coalgebra A#ρπH is coquasitriangular if and only if there exist elements T={Tα,β)α,β∈π∈(H(?) H)*,σ-={σα,β1}∈(A(?)H)*,σ2=(σα,β2)∈(H(?)A)*,σ3={σα,β3}∈(A(?)A)* satisfying the following conditions:a)(A,σ3)be a Hopf group-coalgebra on algebra A associated to Hopf group-coalgebra H;b)(H,T)be a weak group-coquasitriangular Hopf group-coalgebra on H associated to(A,σ1,σ2);c)(A,H)be a group-compatible pair associated toσ1;d)(A,H)be a group-skew compatible pair associated t0σ2;e)T,σ1,σ2,σ3satisfied the conditions C1)-C5)in proposition3.2.8.Moreover,the coquasitriangular structureσο={σα,β)α,γ∈πhas a unique decomposition: (?) a,c∈A,k∈Hα,r∈Hγσα,γο(a#ρk,c#ρr)=∑σα,11(a1,r(1,1))Tα,γ(k(1,a),r(2,γ))σ1,13(a2,c1)σ1,γ2(k(2,1),c2).