We propose a definition of a quantised ${\mathrm{𝔰𝔩}}_2$-differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${\mathrm{𝔰𝔩}}_2$ are natural examples of such algebras.

The goal of this expository paper is to give a quick introduction to $q$-deformations of semisimple Lie groups. We discuss principally the rank one examples of ${𝒰}_q\left({\mathrm{𝔰𝔩}}_2\right)$, $𝒪\left({\mathrm{SU}}_q\left(2\right)\right)$, $𝒟\left({\mathrm{SL}}_q(2,ℂ)\right)$ and related algebras. We treat quantized enveloping algebras, representations of ${𝒰}_q\left({\mathrm{𝔰𝔩}}_2\right)$, generalities on Hopf algebras and quantum groups, $*$-structures, quantized algebras of functions on $q$-deformed compact semisimple groups, the Peter-Weyl theorem, $*$-Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations...

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...