Lifting of Nichols Algebras
- Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classi cation program of nite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider [6] is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classi cation was carried out in this way in [11] when the group of group-like elements is abelian and the prime divisors of the order of the group are > 7. In this case the appearing Nichols algebras are of Cartan type.
Based on recent work of Heckenberger about diagonal Nichols algebras [29, 28, 27] we compute explicitly the liftings of some Nichols algebras not treated in [11]; namely we lift
- all Nichols algebras with Cartan matrix of type A2 (Theorem 6.3.3),
- some Nichols algebras with Cartan matrix of type B2 (Theorem 6.4.3), and
- some Nichols algebras of two Weyl equivalence classes of non-standard type (Theorem 6.5.3),
giving new classes of nite-dimensional pointed Hopf algebras.
Crucial is the knowledge of a good presentation of the Nichols algebra and its liftings: We want to have an explicit description in terms of generators and (non-redundant) relations, and a basis; this requires new ideas and methods that generalize those in [11].
In this spirit, we describe Hopf algebras generated by skew-primitive elements and an abelian group with action given via characters (including Nichols algebras and their liftings) in Theorem 5.4.1. The relations form a Grobner basis and are given by a combinatorial property involving the theory of Lyndon words.
Furthermore, in Theorem 7.3.1 we give a necessary and su cient criterion to check whether a given set of iterated q-commutators establishes a restricted PBW basis for a given realization of the relations. Also with the help of this criterion we determine the redundant relations in the examined Nichols algebras and their liftings.
