Search

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 ℒ ℛ (H) over Hopf quasigroups is braided, which generalizes the results in Alonso Álvarez et al. [Projections and Yetter–Drinfeld modules over Hopf (co)quasigroups, J. Algebra443 (2015) 153–199]. We also prove that the category of gen − 𝒴 𝒟 having all the categories of generalized Yetter–Drinfeld modules l − 𝒴 𝒟 , l ∈ ℤ as components is a crossed ℤ -category. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the category of -Yetter-Drinfeld modules () over a Hom-Hopf algebra. We first prove that every category of -Yetter-Drinfeld modules over a Hom-Hopf algebra with a bijective antipode is a braided tensor category and that every -Yetter-Drinfeld module can provide the solution of the Hom-Yang-Baxter equation. Secondly, we find sufficient and necessary conditions for to be symmetric and pseudosymmetric, respectively. Finally, we construct examples of -Yetter-Drinfeld modules by a quasitriangular Hom-Hopf algebra and study their relationship. [ABSTRACT FROM AUTHOR]

We consider two new algebras from anH-biquasimodule algebraAand a Hopf quasigroupH: twisted smash productA ⊛ Hand L-R smash productA⋇H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, ifHis a cocommutative Hopf quasigroup, we prove thatA ⊛ His isomorphic toA⋇Has Hopf quasigroups. [ABSTRACT FROM AUTHOR]

The notion of a group C-ring is introduced. Furthermore, the relationships of weak group entwining structures, invertible weak group entwining structures and group C-rings are investigated, which generalize the recent work of Brzezinski [5] and Brzezinski and Turner [11]. As applications, we finally construct two kinds of examples of weak entwining structures by weak group algebra Galois coextensions and weak Hopf group algebras, respectively. [ABSTRACT FROM AUTHOR]

We show that the antipode of a braided dual quasi-Hopf algebra is inner, and, a fortiori, bijective. This improves a result of Li [10]. [ABSTRACT FROM AUTHOR]