The class of finite-dimensional pointed Hopf algebras is a field of
current active research. The classication of these algebras has seen
substantial progress since the development of the so-called "Lifting
method" by Andruskiewitsch and Schneider [AS]. With this tool, the
case in which the group of group-like elements is abelian is almost
completed and recent results by these and other authors
such as Angiono [Ang] and Heckenberger [H] seem to lead to a very
On the other hand, the non-abelian case, though also approachable
via this method, is far from being completed. One of the main
obstacles is the little amount of known examples of
finite-dimensional Nichols algebras (a key structural component of
these Hopf algebras) and the lack of intuition on how these objects
should behave. However examples do exist and it is expectable that
sufficient information about them could throw some light on the
In this talk, we will review the classication of pointed Hopf
algebras over the symmetric groups S_3 and S_4, as found in a joint
work with G. Garcia [GG]. We will also construct Hopf-Galois
extensions of these algebras and use them to show that all of them
are actually cocycle deformations of their graded versions, which
are Radford bi-products of the corresponding Nichols algebra and
group algebra. This will allow us to give a classication of the
module categories over the categories of representations of these
algebras. These results were proved in a joint work with M.
[AndS]Annals of Mathematics 171 No. 1 (2010).
[Ang] Submitted (2011).
[H] Advances in Mathematics 220 (2009).
[GG] Israel Journal of Mathematics 183 (2011).
[GM] Pacic Journal of Mathematics, to appear.