Cocycle deformations for Hopf algebras with a coalgebra projection. (Q994278)

Let \(K\) be a field of characteristic zero. Let \(A\) be Hopf algebra (or a bialgebra) over \(K\) and let \(H\) be subbialgebra of \(A\), such that \(H\) is a Hopf algebra, and there is an \(H\)-bilinear coalgebra projection \(\pi\colon A\to H\) which splits the inclusion. In this case, the space \(R\) of coinvariants of \(A\) under \(\pi\) is a coalgebra in the category of Yetter-Drinfeld modules over \(H\) and \(A\), called a prebialgebra with cocycle \(\xi\), and \(A=R\#_\xi H\) a structure studied in the paper [Trans. Am. Math. Soc. 359, No. 3, 991-1044 (2007; Zbl 1125.16022)], by \textit{A. Ardizzoni, C. Menini} and \textit{D. Ştefan}, which generalizes Radford-Majid bosonization. In this paper the authors study the behavior of the above construction under cocycle deformations. An example is discussed related to the classification of pointed Hopf algebras with Abelian group of group-like elements, as developed in the work of \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [Ann. Math. (2) 171, No. 1, 375-417 (2010; Zbl 1208.16028)]. The authors consider the Radford biproduct \(A=\mathcal B(V)\#k[\Gamma]\) of the group algebra of a finite Abelian group \(\Gamma\) and the Nichols algebra \(\mathcal B(V)\) of a quantum plane. It has been shown by \textit{A. Masuoka} [J. Algebra 320, No. 1, 1-47 (2008; Zbl 1157.17005)] and \textit{L. Grunenfelder} and \textit{M. Mastnak} [Pointed and copointed Hopf algebras as cocycle deformations, preprint \url{arXiv:0709.0120v2}], that the families of so called liftings of \(A\), appearing in the above mentioned classification results, are cocycle deformations of \(A\). In the paper under review, an explicit description of the cocycle which twists this Hopf algebra into the liftings is given.