Related papers: Pointed Hopf algebras as cocycle deformations

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…

We show that a large class of finite dimensional pointed Hopf algebras is quasi-isomorphic to their associated graded version coming from the coradical filtration, i.e. they are 2-cocycle deformations of the latter. This supports a slightly…

We present an explicit description of the Hopf 2-cocycles involved in the classification of finite dimensional pointed and copointed Hopf algebras over the symmetric group on three letters. We determine which cocycles are exponentials of…

A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a "non-abelian" cohomology theory, which gives a…

For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the…

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all…

Hopf algebras appear in connection with various problems in Pure Mathematics and Theoretical Physics, mainly through their categoriesof representations, which are examples of tensor categories. In recent years, there have been major…

We compute the Hopf 2-cocycles involved in the classification of pointed Hopf algebras of diagonal type $A_2$. When the quantum Serre relations are deformed, we characterize those cocycles that can be recovered from Hochschild cohomology,…

We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification…

We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work…

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions over a dihedral group D_m, with m=4a> 11. We obtain this…

We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close…

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type…

We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order…

We describe quantum groups given by multiparametric deformations of enveloping algebras of Kac-Moody algebras as a family of pointed Hopf algebras introduced by Andruskiewitsch and Schneider associated to a generalized Cartan matrix. We…

For each $\ell\geq 1$ and $\lambda,\mu\in\Bbbk$, we study the representations of a family of pointed Hopf algebras $\mathcal{A}_{\lambda,\mu}$. These arise as Hopf cocycle deformations of the graded algebra $\mathcal{FK}_3\#\Bbbk…

This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of…

We provide isomorphism results for Hopf algebras that are obtained as graded twistings of function algebras on finite groups by cocentral actions of cyclic groups. More generally , we also consider the isomorphism problem for…

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf…

We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by…