Liftings of Nichols algebras over racks via Doi twist

Question

As a more nontrivial example for my Dissertation thesis, I'd require some example of the following type (of course I'll "cite" ;-) ), so thanx in advance:

Andruskiewitsch/Grana have by a new construction given very interesting new liftings of finite dimensional Nichols algebras e.g. over $S_4$ ("Examples of liftings of Nichols algebras over racks", 2004).

On the other hand Masuoka has shown, that all the "well-known" liftings of Nichols algebras over abelian groups (finite dimensional, all prime divisors>7, by Andruskiewitsch/Schneider) are Doi/Cocycle-twists of the unlifted/graded one's ("Abelian and non-abelian second cohomologies of quantum enveloping algebras", 2008)

....is there an easy way to see that this is also true for the "new" liftings over nonabelian groups?...and how could I easily read off the precise 2-cocycle?

Answer 1

At the moment it is not easy to see that this will also hold for the non-abelian case, or even for the abelian cases yet to be computed, although there are no counter examples. It holds in most known examples as shown in

A. G. I. and Mombelli, M. Representations of the category of modules over pointed Hopf algebras over S_3 and S_4 (joint work with Martín Mombelli). Pacific Journal of Mathematics, 252 (2) (2011), pp. 343–378. Available at arXiv:1006.1857v5.

for liftings over S4 and S3, and in

G. A. García and M. Mastnak. Deformation by cocycles of pointed Hopf algebras over non-abelian groups. Preprint: arXiv:1203.0957v1

for liftings over D4. In this last paper, moreover, the cocycles are explicitly computed.