On the Lifting of Nichols Algebras

Abstract

Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. Based on recent work of Heckenberger about Nichols algebras of diagonal type we compute explicitly the liftings of all Nichols algebras with generalized Cartan matrix of type A 2, some Nichols algebras with generalized Cartan matrix of type B 2, and some Nichols algebras of two Weyl equivalence classes of non-standard type giving new classes of finite-dimensional pointed Hopf algebras.