Non-Abelian Hopf Cohomology of Radford Products

Abstract

Abstract. We study the non-abelian Hopf cohomology theory of Radford products with coefficients in a comodule algebra. We show that these sets can be expressed in terms of the non-abelian Hopf cohomology theory of each factor of the Radford product. We write down an exact sequence relating these objects. This allows to compute explicitly the non-abelian Hopf cohomology sets in large classes of examples. Key-words: non-abelian cohomology, Radford products, Hopf comodule algebra, cosimplicial nonabelian groups, Taft algebras. INTRODUCTION.The present article is devoted to the study of the non-abelian Hopf cohomology theory of Radford products. The cohomology theory for Hopf algebras with coefficients in comodule algebras was defined by the authors in [4]. It generalizes the non-abelian cohomology theory for groups ([6], [7]) and is closely related to descent cohomology [3]. Recall that the Radford product of two Hopf algebras (also called in the literature Radford biproducts or bosonization) was introduced in [5] as a generalisation of the semi-direct product of groups (see also Majid's interpretation in terms of braided categories [2]).The Radford product H⋆E, which is a Hopf algebra, is defined for a Hopf algebra H and any algebra E in the braided Yetter-Drinfeld category over H [11]. Our main result allows to describe the cohomology sets of H⋆E in terms of those of H and of a braided version of the cohomology sets of E. This decomposition has a counterpart in the non-abelian cohomology theory of a semi-direct product of groups. We now detail the organisation of the article.

Crossed homomorphisms and Cartier-Kostant-Milnor-Moore theorem for difference Hopf algebras

“…Crossed homomorphisms on Hopf algebras [36] can be interpreted as 1-cocycles of the Sweedler cohomology of cocommutative Hopf algebras with coefficients in commutative module algebras. In the study of non-abelian Hopf cohomology [25,26,27], 1-descent cocycles of a Hopf algebra H with coefficients in a relative Hopf module M serve the purpose of crossed homomorphisms on Hopf algebras. Among the recent studies, bijective Hopf algebra crossed homomorphisms were applied to construct solutions of the quantum Yang-Baxter equation [1], originated from the works of Etingof-Schedler-Soloviev and Lu-Yan-Zhu on the set-theoretical solutions of the quantum Yang-Baxter equation [7,17].…”