Related papers: Two-Parameter Quantum Groups and Ringel-Hall algeb…

From the theory of finite-dimensional weight modules, we get the basic braided $R$-matrix $\widehat R$ of $U_{r, s}(\mathfrak{so}_{2n+1})$. For its FRT presentation $U(\widehat R)$, we achieve two word-formation methods of quantum Lyndon…

Let $r,s$ be two parameters chosen from $\C^{\ast}$ such that $r^{m}s^{n}=1$ implies $m=n=0$. We compute the derivations of the two-parameter quantized enveloping algebra $\U$ and calculate its first degree Hochschild cohomology group. We…

In this note, we construct dual PBW bases of the positive and negative subalgebras of the two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ in classical types, as used in our earlier work arXiv:2407.01450. Following the ideas of Leclerc…

Let A be a finite dimensional Hopf algebra and (H, R) a quasitriangular bialgebra. Denote by H^*_R a certain deformation of the multiplication of H^* via R. We prove that H^*_R is a quantum commutative left H\otimes H^{op cop}-module…

Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an analogue…

Originally motivated by connections to integrable systems, two natural subalgebras of the rational Cherednik algebra have been considered in the literature. The first is the subalgebra of all degree zero elements and the second is the Dunkl…

We introduce $*$-structures on braided groups and braided matrices. Using this, we show that the quantum double $D(U_q(su_2))$ can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski…

We describe the quasitriangular structure (universal $R$-matrix) on the non-standard quantum group $U_q(H_1,H_2,X^\pm)$ associated to the Alexander-Conway matrix solution of the Yang-Baxter equation. We show that this Hopf algebra is…

This paper is devoted to investigating the centre of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ via establishing the Harish-Chandra homomorphism. Based on the Rosso form and the representation theory of weight modules, we prove…

The Heisenberg double $D_q(E_2)$ of the quantum Euclidean group $\mathcal{O}_q(E_2)$ is the smash product of $\mathcal{O}_q(E_2)$ with its Hopf dual $U_q(\mathfrak{e}_2)$. For the algebra $D_q(E_2)$, explicit descriptions of its prime,…

We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of $\mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their…

Let R be an integral domain, h non-zero in R such that R/hR is a field, and HA the category of torsionless (or flat) Hopf algebras over R. We call any H in HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal…

We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras,…

We introduce uniparametric and multiparametric quantisations of the general linear supergroup, in the form of "quantised function algebras", both in a formal setting - yielding "quantum formal series Hopf superalgebras", a` la Drinfeld -…

In this article we use a parametrized version of the FRT construction to construct two new coquasitriangular Hopf algebras. The first one, $\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine $SL(2)$. We show that there…

We define for real $q$ a unital $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ quantizing the universal enveloping $*$-algebra of $\mathfrak{sl}(2,\mathbb{R})$. The $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ is realized as a…

Let $U$ be a connected, simply connected compact Lie group with complexification $G$. Let $\mathfrak{u}$ and $\mathfrak{g}$ be the associated Lie algebras. Let $\Gamma$ be the Dynkin diagram of $\mathfrak{g}$ with underlying set $I$, and…

We discuss two-parameter deformations of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). These deformations are Hopf algebras.…

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{\pi i/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families…

We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the…