Weak Cross-plot Of The Weak Hopf Algebra And Projection

Crossed products of an algebra A with a Hopf algebra H were introduced by R. J. Blattner, M. Cohen, S. Montgomery [1][3] in 1986. Let H be a Hopf algebra and A an algebra. Assume that H mesures A and that σ is an invertible map in Homk(H (?)H,A). The crossed products A#_σH of A with H is the set A (?)H as a vector space with multiplication: (a#x)(b#y) = a(x_x(?) b)σ (x₂, y₁)#x₃y₂, for all x,y ∈ H,a,b ∈ A. Here we have written a#x for a (?) x. A#_σH is an associative algebra with identity element 1_A#1_H if and only if the following two conditions are satisfied: 1. A is a twisted H-module. 2. σ is a normal cocycle.In this paper, we mainly study weak crossed products, weak twisted products, weak twisted coproducts and projective representation of weak Hopf algebras.We introduce crossed products to a weak Hopf algebra. We name it weak crossed products. We obtain a sufficient and necessary condition for a weak crossed products A#_σH to be an associative algebra with identity element 1_a#1_H:2. σ is a normal cocycle:For the special case of weak crossed products A#_σH, we show a weak twisted products A_σ[H] is a weak bialgebra if and only if the following conditions are satisfied:If H be weak Hopf algebra and hold then weak bialgebra A_σ[H] is a weak Hopf algebra. It generalizes I.Boca [8] and zhangliangyun's[6] results. For weak Hopf algebras A_σ[H] and invariant subalgebraAH, we show that AH and End (a4h]A) are anti-algebra morphism. We obtain the semisimplicity of weak Hopf algebra Ar[#] by the semisimplicity of A and H.Weak crossed coproducts C xQH and weak twisted coproducts Ca(H) are dual of weak crossed products A#aH and weak twisted products Aa[H] respectively. We give a sufficient and necessary condition for a weak twisted coproducts Ca(H) to be a weak bialgebra.If H be a weak Hopf algebra and a e Hom(C, H?H) satisfies that ai(c2) \1L (/i)5?(af1(ci)a2(c2))a21(ci) = ec{c) nL (h) for all c G C, h e H then weak twisted coproducts CQ(H) is a weak Hopf algebra.We introduce projective representation to weak Hopf algebra H. We obtain some properties about projective representation T. Finally we show that we can obtain a projective representation X" from a projective representation T of H and give a condition for an ordinary representation of Aa[H) to be a projective representation of weak Hopf algebra H. It generalizes I.Boca's ' ' results.