Yetter–Drinfel’d modules over weak Hopf quasigroups

In this paper, we first give more interested properties of a weak Hopf quasigroup [Formula: see text]. Next, we introduce the notion of Yetter–Drinfel’d weak quasimodule over [Formula: see text] and prove that the category [Formula: see text] of right-right Yetter–Drinfel’d weak quasimodules is braided with left and right dualities under suitable conditions. Finally, we describe the notion of coquasitriangular weak Hopf quasigroup [Formula: see text], and study there exist a relation between Yetter–Drinfel’d weak quasimodule and the coinvariant space of right [Formula: see text]-comodule over [Formula: see text].

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

Maschke Type Theorems for Weak Hopf Quasigroups

In this paper we give necessary and sufficient conditions for a comodule magma over a weak Hopf quasigroup to have a total integral, thus extending the theories developed in the Hopf algebra, weak Hopf algebra and non-associative Hopf algebra contexts. From this result we also deduce a version of Maschke’s theorems for right (H, B)-Hopf triples associated to a weak Hopf quasigroup H and a right H-comodule magma B.

Generalized Yetter–Drinfeld (quasi)modules and Yetter–Drinfeld–Long bi(quasi)modules for Hopf quasigroups

As generalizations of Yetter–Drinfeld module over a Hopf quasigroup, we introduce the notions of Yetter–Drinfeld–Long bimodule and generalize the Yetter–Drinfeld module over a Hopf quasigroup in this paper, and show that the category of Yetter–Drinfeld–Long bimodules [Formula: see text] over Hopf quasigroups is braided, which generalizes the results in Alonso Álvarez et al. [Projections and Yetter–Drinfeld modules over Hopf (co)quasigroups, J. Algebra 443 (2015) 153–199]. We also prove that the category of [Formula: see text] having all the categories of generalized Yetter–Drinfeld modules [Formula: see text], [Formula: see text] as components is a crossed [Formula: see text]-category.

Cleft comodules over Hopf quasigroups

In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, ϱA) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product AH♯σAHH, where AH is the coinvariants subalgebra of A and σAH is a morphism between H ⊗ H and AH. As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.