Related papers: Duality for the Jordanian Matrix Quantum Group $GL…

Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a complex semisimple Lie algebra $\mathfrak{g}$. We apply a modification of the $R$-matrix construction for quantum groups to the evaluation of the universal…

Some duality properties for induced representations of enveloping algebras involve the character $Trad_{\goth g}$. We extend them to deformation Hopf algebras $A_{h}$ of a noetherian Hopf $k$-algebra $A_{0}$ satistying $Ext^{i}_{A_{0}}(k,…

We consider the algebra isomorphism found by Frenkel and Ding between the RLL and the Drinfeld realizations of $U_q(\widehat{gl(2)})$. After we note that this is not a Hopf algebra isomorphism, we prove that there is a unique Hopf algebra…

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.…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebras ${\bf B}^{\vee}$ of $sl(N)$ the explicit expressions are obtained for the twist element ${\cal F}$, universal ${\cal…

We find a self-dual noncommutative and noncocommutative Hopf algebra acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to…

In this paper, we give an geometric description of the Schur-Weyl duality for two-parameter quantum algebras $U_{v, t}(gl_n)$, where $U_{v, t}(gl_n)$ is the deformation of $U_v(I, \cdot)$, the classic Shur-Weyl duality $(U_{r, s}(gl_n),…

We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a…

The study of the pentagon (fusion) equation leds to the Structure and the Classification theorem for finite dimenasional Hopf algebras: there exists a one to one correspondence between the set of types of n-dimensional Hopf algebtras and…

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,…

A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like finite-dimensional Hopf algebras and their duals, or quantisations of function algebras and of universal enveloping algebras of Poisson-Lie…

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H_1, the second term in the…

Starting with a given generalized boson algebra U_<q>(h(1)) known as the bosonized version of the quantum super-Hopf U_q[osp(1/2)] algebra, we employ the Hopf duality arguments to provide the dually conjugate function algebra Fun_<q>(H(1)).…

The generators of the Jordanian quantum algebra ${\cal U}_h(sl(2))$ are expressed as nonlinear invertible functions of the classical $sl(2)$ generators. This permits immediate explicit construction of the finite dimensional irreducible…

Representation theory for the Jordanian quantum algebra $U=U_h(sl(2))$ is developed. Closed form expressions are given for the action of the generators of U on the basis vectors of finite dimensional irreducible representations. It is shown…

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of $U_q \hat{gl}_N$ in the limit $N \to \infty$. The resulting Hopf algebra $Rep U_q \hat{gl}_\infty$ is a tensor product of its…

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras…

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras which have some geometrical content. When the ground field has characteristic zero, the first pair is made by a function algebra over a…

We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}(2)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf…