Cocycle deformations for weak Hopf algebras

Abstract

In this paper we introduce a theory of multiplication alteration by two-cocycles for weak Hopf algebras. We show that, just like it happens for Hopf algebras, if H a weak Hopf algebra and H? its weak Hopf algebra deformation by a 2-cocycle ?, there is a braided monoidal category equivalence between the categories of left-right Yetter-Drinfel?d modules HYDH and H?YDH?. As a consequence we get in this context that the category Rep(D(H)) of left modules over the Drinfel?d double D(H) can be identified with the category Rep(D(H?)) of left modules over the Drinfel?d double D(H?).