Cocycle deformations for liftings of quantum linear spaces

Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose there is a coalgebra projection $\pi$ from $A$ to a sub-Hopf algebra $H$ that splits the inclusion. If the projection is $H$-bilinear, then $A$ is isomorphic to a biproduct R #_{\xi}H where $(R,\xi)$ is called a pre-bialgebra with cocycle in the category $_{H}^{H}\mathcal{YD}$. The cocycle $\xi$ maps $R \otimes R$ to $H$. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points $\Gamma$ as classified by Andruskiewitsch and Schneider [AS1]. One asks when such an $A$ can be twisted by a cocycle $\gamma:A\otimes A\rightarrow K$ to obtain a Radford biproduct. By results of Masuoka [Ma1, Ma2], and Gr\"{u}nenfelder and Mastnak [GM], this can always be done for the pointed liftings mentioned above. In a previous paper [ABM1], we showed that a natural candidate for a twisting cocycle is {$\lambda \circ \xi$} where $\lambda\in H^{\ast}$ is a total integral for $H$ and $\xi$ is as above. We also computed the twisting cocycle explicitly for liftings of a quantum linear plane and found some examples where the twisting cocycle we computed was different from {$\lambda \circ \xi$}. In this note we show that in many cases this cocycle is exactly $\lambda\circ\xi$ and give some further examples where this is not the case. As well we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension

Quasi-bialgebra Structures and Torsion-free Abelian Groups

We describe all the quasi-bialgebra structures of a group algebra over a torsion-free abelian group. They all come out to be triangular in a unique way. Moreover, up to an isomorphism, these quasi-bialgebra structures produce only one (braided) monoidal structure on the category of their representations. Applying these results to the algebra of Laurent polynomials, we recover two braided monoidal categories introduced in \cite{CG} by S. Caenepeel and I. Goyvaerts in connection with Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras)