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 \cite{ASannals}. 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 \cite{Mas2, Mas1}, and Gr\"{u}nenfelder and Mastnak \cite{Grunenfelder-Mastnak}, this can always be done for the pointed liftings mentioned above. In a previous paper \cite{ABM}, 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