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Weak crossed product and projection indicates weak Hopf algebra

Author: ZhangJianYong
Tutor: ZhangZiLong
School: Hebei Normal
Course: Basic mathematics
Keywords: Weak Hopf algebra weak crossed products A#_σH weak twisted products A_σ[H] weak twisted coproducts C_α(H) projective representation
CLC: O153
Type: Master's thesis
Year: 2006
Downloads: 18
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Abstract


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（xx（?） 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 1A#1H 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 1a#1H:2. σ is a normal cocycle:For the special case of weak crossed products A#σH, we show a weak twisted products Aσ[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 Aσ[H] is a weak Hopf algebra. It generalizes I.Boca [8] and zhangliangyun’s[6] results. For weak Hopf algebras Aσ[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.

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