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 π 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# ξ H where (R, ξ) is called a pre-bialgebra with cocycle in the category . The cocycle ξ maps R\otimes R to H. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points Γ as classified by Andruskiewitsch and Schneider. One asks when such an A can be twisted by a cocycle γ: A\otimes A → K to obtain a Radford biproduct. By results of Masuoka, and Grünenfelder and Mastnak, this can always be done for the pointed liftings mentioned above. In a previous article, we showed that a natural candidate for a twisting cocycle is λ ○ ξ where λ belongs to H* and it is a total integral for H and ξ 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 λ ○ ξ. In this note, we show that in many cases this cocycle is exactly λ ○ ξ and give some further examples where this is not the case. Also, we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension.