A Quasitriangular Stucture On HOPF π-ALGEBRA

The article is divided into two sections.In the first section,Letπbe a group and let H={Hα)α∈πbe a Hopfπalgebra. First assuption C is coalgera. The paper introdues the notion of aπ-crossed coproduct C(?)απ:H={C(?)απHa)α∈πand finds some sufficient and necessary condiyions under C(?)απH formsπ-crossed coalgera.Futher assuption C is Hopf algebra,and find some sufficient and necessary condiyions under C(?)απH forms Hopfπ-algebra.In the second section,we discuss quasitriangular Hopfπ-algebra,giving some prop-erties of quasitriangular Hopfπ-algebra and find some sufficient and necessary conditions under Hopfπ-algebra C(?)απH forms a quasitriangular Hopfπ- algebra.That is:Hopfπ- algebra C(?)απH is a quasitriangular if and only if there exist elements T={Tα,β)α,β∈π=∑{Tα(1)(?)Tβ(2)}α,β∈π∈{Hα(?)Hβ)α,β∈π,V{Vα,β}α,β∈π=∑{Vα(1)(?) Vβ(2)}α,β∈π∈{C(?)Hβ}β∈π,U-{uα,β}α,β∈π-{∑Uα(1)(?)Uβ(2)}α,β∈π∈{Hα(?)C}α∈π且Q={Qα,β)α,β∈π∈=∑{Qα(1)(?)Qβ(2)}α,β∈π∈=∑{C(?)C}.sudh that(H,T)is a quasitriangular Hopfπ- algebra,(C,Q)is a quasitriangular algebra,(C,H)is a compatible pair a,ssociated to (T,U),(G,H)is a shew compatible pair associated to(T,U)and T,Q,V,U satisfied the conditions C1-C10.Moreover,the quasitriangular structure R has decomposition:...