Related papers: Schur-Weyl duality for $U_{v,t}(sl_{n})$

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…

The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\Bbb Z / r\Bbb Z)\wr \frak S_n$ introduced by Ariki and Koike. Let $K=\Bbb Q(q,u_1,\dots ,u_r)$ be the field of rational…

We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$ and the Hopf algebra…

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…

Let $U_q(\mathfrak{g})$ denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra $\mathfrak{g}$. Let $\lambda$ be a nonzero dominant integral weight of $\mathfrak{g}$, and let $V$ be the…

We establish a new Howe duality between a pair of quantum queer superalgebras $(\mathrm{U}_{q^{-1}}(\mathfrak{q}_n), \mathrm{U}_q(\mathfrak{q}_m))$. The key ingredient is the construction of a non-commutative analogue…

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…

In this paper, we completely determine the group of algebra automorphisms for the two-parameter Hopf algebra ${\check U}_{r,s}^{\geq 0}({\mathfrak sl_{3}})$. As a result, the group of Hopf algebra automorphisms is determined for $\V$ as…

We define a natural concept of duality for the h-Hopf algebroids introduced by Etingof and Varchenko. We prove that the special case of the trigonometric SL(2) dynamical quantum group is self-dual, and may therefore be viewed as a…

We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [VV96] to the super case.

We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of quantum differential operators on the quantum symplectic space $\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a…

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…

Let g be a complex, simple Lie algebra and t a Cartan subalgebra of g. A new unitary, flat connection on t with values in any finite-dimensional g-module V and simple poles along the root hyperplanes was recently introduced by J. Millson…

The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig's small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and…

This paper gives a Schur-Weyl duality approach to the representation theory of the affine Hecke algebras of type C with unequal parameters. The first step is to realize the affine braid group of type $C_k$ as the group of braids on $k$…

This article introduces the duplex Hecke algebra, which is an infinite dimensional algebra generated by two Hecke algebras. This concept originates from the degenerate duplex Hecke algebra in the theory of Schur-Weyl duality related to…

By means of the notions of cross product algebras of the theory of quantum groups, in the context of classical Hopf algebra structures, we deduce some known structures of Weyl algebras type (as the Drinfeld quantum double, the restricted…

We show that some factors of the universal R-matrix generate a family of twistings for the standard Hopf structure of any quantized contragredient Lie (super)algebra of finite growth. As an application we prove that any two isomorphic…

Let $(H, \sigma)$ be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras,…

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…