Liftings of Nichols algebras of diagonal type II: all liftings are cocycle deformations

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type V with a principal realization in the category of Yetter–Drinfeld modules of a cosemisimple Hopf algebra H and such that the Nichols algebra $$\mathfrak {B}(V)$$ B(V) is finite-dimensional, thus presented by a finite set $${{\mathcal {G}}}$$ G of relations, we define a family of Hopf algebras $$\mathfrak {u}(\varvec{\lambda })$$ u(λ) , $$\varvec{\lambda }\in \Bbbk ^{{{\mathcal {G}}}}$$ λ∈kG , which are cocycle deformations of $$\mathfrak {B}(V)\# H$$ B(V)#H and such that $${\text {gr}}\mathfrak {u}(\varvec{\lambda })\simeq \mathfrak {B}(V)\# H$$ gru(λ)≃B(V)#H .