Abstract
This article serves a two-fold purpose. On the one hand, it is asurvey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of Cartan type B2.
Highlights
A pointed Hopf algebra A is characterized by the fact that its coradical A0 coincides with the subalgebra kG generated by its group-like elements G = G(A)
The lifting method developed by Andruskiewitsch and Schneider to classify finite-dimensional pointed Hopf algebras A with abelian coradical A0 kΓ consists of the several steps
The main contribution of [5, 4, 16], which is the main focus of this survey, was to provide a strategy to complete (4) and to prove that it provided a full classification. This strategy is built on cocycle deformations of graded Hopf algebras, as suggested by a result of Masuoka [28], who showed that the class of liftings given in [10] were cocycle deformations of the associated graded algebras
Summary
When considering the classification problem, this group G is a first invariant. Associated to it there is a braided structure at the heart of A: the so-called infinitesimal braiding; this is an object V in the category of YetterDrinfeld modules kkGGYD. We shall review the development of this classification and give detailed instructions about how to carry on this program on each example. This step-by-step guide is the selfcontained Section 4 and can be extracted by a potential user
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