Distortion Of The Hopf Algebra

This article starts off with a twisted partiial action of H on A.The notion of twisted partial Hopf coactiion is introducced through the dual method. Then the condiitions on partial cocycles are established in order to construct partial crossed coproducts.Finally, the classification of partial crossed coproducts is discussed,and we prove the following conclusion.Theorem 4.1.Let A be a coalgebra and H a Hopf algebra with two twisted partial coactions on A,ρ:a�'a(-1](?)a[0] and ρ’:a�'a[-1]’(?)a[0]’ with cocycles ω and σ, respectively.Then there exists a coalgebra isomorphism (?):A(?)(ρ,ω)H�'A(?)(ρ’,σ)H which is also a left A-comodule and right.H-module map if and ony if there exist linear maps Φ,ψ∈Hom(A,H)such that εH.oΦ=EH,εH oψ=εEH and (a)ψ*Φ(a)=a[-1]εA(a[0]),Φ*ψ(a)=a[-1]’εA(a[0]’), (b)ψ(a)=a(1)[-1]εA(a(1)[0])Φ(a(2))=ψ(a(1))a(2)[-1]εA(a(2)[0]), (c)Φ(a)=a(1)[-1]’εA(a(1)[0]’)Φ(a(2))=Φ(a(1))a(2)[-1]’εA(a(2)[0],), (d)a[-1]’(?)[0],=Φ(a(1))a(2)[-1]ψ(a(3))(?)a(2)[0], (e)σ(a)1(?)σ(a)2=a(1)[-1]’Φ(a(2))ω(a(3))1ψ(a(4))(1)(?)Φ(a(1)[0]’)ω(a(3))2ψ(a(4))(2), (f)(?)(a(?)h)=a(1)4(?)(a(2))h, for all a∈A and h∈H.