Weak Hopf algebras with projection and weak smash bialgebra structures

AbstractIn this paper we study weak Hopf algebras with projection. If f:H→B, g:B→H are morphisms of weak Hopf algebras such that g∘f=idH, we prove that it is possible to find an object BH, in the new category of weak Yetter–Drinfeld modules, that verifies similar conditions to the ones include in the definition of weak Hopf algebra. Finally, we define weak smash bialgebra structures and prove that, under central and cocentral conditions, BH and H determine an example of them

Cocycle deformations for Hom-Hopf algebras

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this paper we introduce the Hom-analogue of the definition of 2-cocycle for Hopf algebras, called Hom-2-cocycle, and study its properties in order to give a theory of multiplication alteration by Hom-2-cocycles for Hom-Hopf algebras. We show that, just like in the classical setting, if H is a Hom-Hopf algebra with associated endomorphism α and σ is a convolution invertible Hom-2-cocycle, it is possible to define a new product in H to get a new Hom-Hopf algebra if . Moreover we introduce the notions of matched pair and skew pairing in the Hom case and, by the close connection between Hom-2-cocycles and Hom-skew pairings, we show that a special case of Hom-matched pair can be obtained as a deformation of a Hom-Hopf algebra by a Hom-2-cocycle built by a Hom-skew pairing.Xunta de Galicia | Ref. ED431C 2019/10Agencia Estatal de Investigación | Ref. PID2020-115155GB-I0

Yetter–Drinfeld modules and projections of weak Hopf algebras

AbstractIn this paper we prove that if g:B→H is a morphism of weak Hopf algebras which is split as an algebra–coalgebra morphism, then the subalgebra of coinvariants BH of B is a Hopf algebra in the category of Yetter–Drinfeld modules associated to H

Some remarks about monoidal Hom-algebras

In this paper, for C a strict braided monoidal category with tensor product  , we improve the denition of Hom-associative algebra by removing the multiplicativity condition of the automorphism . After that we state the close connection between the classical notions of (co)algebra, (co)module and Hopf algebra and the corresponding ones in the Hom-world. As a consequence we can study the questions about the Hom-world by passing to the classical one, and the main contribution of this paper is to illustrate the suitable procedure to move from one side to the other. Finally, we apply our techniques to get in the Hom setting some classical results about cleft and Galois extensions.Key words: Monoidal category, Hom-(co)algebra, Hom-Hopf algebra