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We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensions. The infinitesimal part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered.
We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does a
We study actions of pointed Hopf algebras in the $ZZ$-graded setting. Our main result classifies inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the $ZZ$-grading. We also show that generically th
The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal en
In this work we study the deformations of a Hopf algebra $H$ by partial actions of $H$ on its base field $Bbbk$, via partial smash product algebras. We introduce the concept of a $lambda$-Hopf algebra as a Hopf algebra obtained as a partial smash p
We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.