On the lifting of Nichols algebras

TL;DR

This paper explicitly computes all possible deformations (liftings) of certain Nichols algebras, advancing the classification of finite-dimensional pointed Hopf algebras.

Contribution

It provides explicit liftings for Nichols algebras of specific types, including A_2, B_2, and some non-standard types, expanding the known classes.

Findings

Explicit liftings of Nichols algebras of type A_2

Liftings of some B_2 type Nichols algebras

New classes of finite-dimensional pointed Hopf algebras discovered

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 Cartan matrix of type A_2, some Nichols algebras with 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.