Let C n(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if C n+1(A,B) is isomorphic to a direct summand of a multiple of C n(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B....

We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_q\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on ${\left(ℂ³\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group $U_q\left(2\right)$. The purpose of this note is to present the construction.

Two new results concerning complements in a semisimple Hopf algebra are proved. They extend some well-known results from group theory. The uniqueness of a Krull-Schmidt-Remak type decomposition is proved for semisimple completely reducible Hopf algebras.

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product, and consider some examples. Furthermore, we introduce the notion of a generalized partial twisted smash product and discuss a necessary condition under which such partial smash product forms a Hopf algebra. Based on these notions and properties, we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra.

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.

We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are...

The main purpose of the present paper is to study representations of BiHom-Hopf algebras. We first introduce the notion of BiHom-Hopf algebras, and then discuss BiHom-type modules, Yetter-Dinfeld modules and Drinfeld doubles with parameters. We get some new $n$-monoidal categories via the category of BiHom-(co)modules and the category of BiHom-Yetter-Drinfeld modules. Finally, we obtain a center construction type theorem on BiHom-Hopf algebras.

Quantum quasigroups and loops are self-dual objects that provide a general framework for the nonassociative extension of quantum group techniques. They also have one-sided analogues, which are not self-dual. In this paper, natural quantum versions of idempotence and distributivity are specified for these and related structures. Quantum distributive structures furnish solutions to the quantum Yang-Baxter equation.

A twisted generalization of quasitriangular Hopf algebras called quasitriangular Hom-Hopf algebras is introduced. We characterize these algebras in terms of certain morphisms. We also give their equivalent description via a braided monoidal category $̃{(}_Hℳ)$. Finally, we study the twisting structure of quasitriangular Hom-Hopf algebras by conjugation with Hom-2-cocycles.

Let $\pi$ be a group, and $H$ be a semi-Hopf $\pi$-algebra. We first show that the category ${}_Hℳ$ of left $\pi$-modules over $H$ is a monoidal category with a suitably defined tensor product and each element $\alpha$ in $\pi$ induces a strict monoidal functor $F_{\alpha }$ from ${}_Hℳ$ to itself. Then we introduce the concept of quasitriangular semi-Hopf $\pi$-algebra, and show that a semi-Hopf $\pi$-algebra $H$ is quasitriangular if and only if the category ${}_Hℳ$ is a braided monoidal category and $F_{\alpha }$ is a strict braided monoidal functor for any $\alpha \in \pi$. Finally,...

We study bialgebra structures on quiver coalgebras and monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed...

We first describe the Sekine quantum groups ${𝒜}_k$ (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of ${𝒜}_k$ and describe their representation rings $r\left({𝒜}_k\right)$. Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of $r\left({𝒜}_k\right)$.

The classical Serre-Swan's theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially...