Yetter–Drinfel’d modules over weak Hopf quasigroups

Abstract

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].