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 general case.

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. Mombelli[GM].

[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.