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 $\mathfrak{B}(V)$ over a cosemisimple and non-semisimple Hopf algebra $H$. Such a situation arises when the underlying field has positive characteristic or when $H$ is infinite-dimensional. Given an $H$-module $M$ that is an extension of $V$ by $H$, we first introduce an algebra $T(V)\#_MH$ which generalizes the usual bosonization $T(V)\#H$. Indeed, these two objects coincide when $M$ is a trivial extension. We provide necessary conditions for $T(V)\#_MH$ to be a Hopf algebra and a cocycle deformation of $T(V)\#H$. These conditions appear particularly natural when $H$ is a group algebra. We then prove that every lifting is a quotient of $T(V)\#_MH$ for some extension $M$. Echoing Archimedes, $T(V)\#_MH$ stands as a fulcrum over which we can pivot to lift the relations of the Nichols algebra. From this point, one can replicate the strategy proposed by Andruskiewitsch, Angiono, Garcia I., Masuoka and the second author to show that every lifting is a cocycle deformation of $\mathfrak{B}(V)\#H$. We illustrate this idea with two examples of different nature. We classify all pointed liftings of the Fomin-Kirillov algebra $\mathcal{FK}_3$ in characteristic $2$, and prove they are all cocycle deformations one another. We also prove that the Jordanian enveloping algebra of $\mathfrak{sl}(2)$ defined by Andruskiewitsch, Angiono and Heckenberger is a cocycle deformation of the bosonization of the Jordan plane over the infinite cyclic group.