Cocycle Deformations for Hopf Algebras with a Coalgebra Projection

Let H be a Hopf algebra over a field K of characteristic 0 and let A be a bialgebra or Hopf algebra such that H is isomorphic to a sub-Hopf algebra of A and there is an H-bilinear coalgebra projection π from A to H which splits the inclusion. Then A is isomorphic to R#ξH where R is the pre-bialgebra of coinvariants. In this paper we study the deformations of A by an H-bilinear cocycle. If γ is a cocycle for A, then γ can be restricted to a cocycle γR for R, and A^γ is isomorphic to R^γR#ξγ H. As examples, we consider liftings of B(V)#K[Γ] where Γ is a finite abelian group, V is a quantum plane and B(V) is its Nichols algebra, and explicitly construct the cocycle which twists the Radford biproduct into the lifting.