Lifting via cocycle deformation

Abstract

Abstract. We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.

Pointed Hopf algebras: a guided tour to the liftings

“…One of the main purposes of this article is to present the recipe to construct liftings in full detail; this is presented in §4 and an example is developed in §5. Although the strategy developed in [5] applies in a more general level, we restrict ourselves for pedagogical reasons to the following setting:…”

“…Starting with G and V , the classification of finitedimensional Hopf algebras with abelian coradical has been achieved throughout the collaborative work of many authors, specially those of Andruskiewitsch and Schneider [8,10], Heckenberger [26] and Angiono [14,13]. The final step was recently completed by the authors in [16], based on a strategy to construct Hopf algebras -the liftings-developed in [5] and [4]. In this article, we shall review the development of this classification and give detailed instructions about how to carry on this program on each example.…”

Representations of copointed Hopf algebras arising from the tetrahedron rack

“…Then the multiplication in A induces an isomorphism R⊗k G −→ A of k G -bimodules [AAGMV,Lemma 4.1]. Hence we can think of R as a left k G -submodule of A and therefore…”

“…Any spherical Hopf algebra H has an associated tensor category Rep(H) which is a quotient of Rep(H), see [AAGMV,BaW1,BaW2] for the background of this subject. Moreover, Rep(H) is semisimple but rarely is a fusion category in the sense of [ENO], i. e. Rep(H) rarely has a finite number of irreducibles.…”

Hopf cocycle deformations and invariant theory

“…This classification was completed by Angiono and Garcia Iglesias in [AG19], where it is also shown that all Hopf algebras whose coradical is an abelian group algebra are cocycle deformations of their associated graded Hopf algebras. For more classification results of non-semisimple Hopf algebras in which cocycle deformations appear, see [AAIMV14], [FGM19], [GarM15], [GruM] and [AV11].…”