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To unify different definitions of smash Hom-products in a Hom-bialgebra H , we firstly introduce the notion of 𝜃 -twisted smash Hom-product B # 𝜃 H. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product B # 𝜃 H and the twisted smash Hom-coproduct B × ω H to afford a Hom-bialgebra, which generalize the well-known Radford's biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math.137 (2014) 43–65]. Furthermore, we introduce the notion of the category of ϖ -Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math.137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the (𝜃 , ω) -twisted Radford's Hom-biproduct B × ω # 𝜃 H is a Hom-bialgebra if and only if (B , α B) is a Hom-bialgebra in the category of 𝜃 − 1 ω − 1 α H 2 -Yetter–Drinfeld modules H H ℋ 𝒴 𝒟 𝜃 − 1 ω − 1 α H − 2 , generalizing the well-known Majid's conclusion. [ABSTRACT FROM AUTHOR]

In order to construct a class of new Turaev-braided group category with nontrivial associativity, the concept of a quasitriangular quasi-Turaev group coalgebras was recently introduced. Inside the definition, the conditions of invertibility of the R-matrix R and bijectivity of the antipode S are required. In this article, we prove that the antipode of a quasitriangular quasi-Turaev group coalgebra without the assumptions about invertibility of the antipode and R-matrix is inner, and a fortiori, bijective. As an application, we prove that for a quasitriangular quasi-Turaev group coalgebra, two conditions mentioned above are unnecessary. [ABSTRACT FROM AUTHOR]

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 an equivalence relation on quasitriangular quasi-Turaev group coalgebras such that the categories of representations of two quasitriangular quasi-Turaev group coalgebras are tensor -equivalent. As an application, we construct a new Turaev braided group category by a gauge transformation and give a new solution of Yang-Baxter type equation. [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]

Abstract: Zn-catalyzed [4 + 2] benzannulation of 2-ethynylbenzaldehydes with alkynes is described for the selective synthesis of naphthalene derivatives. In the presence of ZnCl2, a variety of 2-ethynylbenzaldehydes underwent the [4 + 2] benzannulation reactions with alkynes to selectively afford the corresponding naphthalene derivatives in moderate to good yields. [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]

The notions of a “weak group entwining structure” and an “invertible weak group entwining structure” are introduced. Furthermore, the relationship between them and group corings is also investigated. It turns out that (weak) group entwining structures and (weak) group entwined modules can be viewed as group corings and comodules over group corings, respectively, which generalizes the recent work of Brzezinski (2002) and Brzezinski et al. (2006). [ABSTRACT FROM AUTHOR]