A geometric realization of liftings of Cartan type

Abstract

We introduce a novel approach to compute liftings of bosonizations of Nichols algebras of diagonal braided vector spaces of Cartan type which replaces heavy computations with structural maps related to quantum groups. This provides an answer to a question posed by Andruskiewitsch and Schneider, who classified finite-dimensional complex pointed Hopf algebras over finite abelian groups whose order is coprime with 210. As application and in order to give not-too-technical examples, we recover with our method the liftings of type $A_{n}$ computed by Andruskiewitsch and Schneider, of type $B_2$ computed by Beattie, Dascalescu and Raianu, and of type $B_3$ computed by the authors, for Drinfeld-Jimbo type braidings. Moreover, we present all liftings of type $B_{\theta}$ and $D_{\theta}$, for $\theta \geq 2$, giving in this way new explicit infinite families of liftings for Drinfeld-Jimbo type braidings.