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 $H^{\sigma}$ its weak Hopf algebra deformation by a $2$-cocycle $\sigma$, there is a braided monoidal category equivalence between the categories of left-right Yetter-Drinfel'd modules $_{H}{\mathcal {YD}}^{H}$ and $_{H^{\sigma}}{\mathcal {YD}}^{H^{\sigma}}$. 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 $Rep(D(H^{\sigma}))$ of left modules over the Drinfel'd double $D(H^{\sigma})$.