Lifting in a non-semisimple world

This is a contribution to the problem of classifying all deformations – a. k. a. liftings – of the bosonization of a Nichols algebra B ( V ) \mathfrak {B}(V) over a cosemisimple and non-semisimple Hopf algebra H H . Such a situation arises when the underlying field has positive characteristic or when H H is infinite-dimensional. Given an H H -module M M that is an extension of V V by H H , we first introduce an algebra T ( V ) # M H T(V)\#_MH which generalizes the usual bosonization T ( V ) # H T(V)\#H . Indeed, these two objects coincide when M M is a trivial extension. We provide necessary conditions for T ( V ) # M H T(V)\#_MH to be a Hopf algebra and a cocycle deformation of T ( V ) # H T(V)\#H . These conditions appear particularly natural when H H is a group algebra. We then prove that every lifting is a quotient of