Related papers: The algebra $U_q(\hat{sl}_\infty)$ and application…
We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge…
Recently, there has been considerable progress in classifying the irreducible representations of Iwahori--Hecke algebras at roots of unity. Here, we present an application of these results to $\ell$-modular Harish--Chandra series for a…
In this paper, we extend the Reshetikhin-Semenov-Tian-Shansky formulation of quantum affine algebras to the two-parameter quantum affine superalgebra $U_{p, q}(\widehat{\mathfrak{gl}}(m|n))$ and obtain its Drinfeld realization. We also…
Let $G$ be the complex symplectic or special orthogonal group and $\g$ its Lie algebra. With every point $x$ of the maximal torus $T\subset G$ we associate a highest weight module $M_x$ over the Drinfeld-Jimbo quantum group $U_q(\g)$ and a…
The Jacobi-Trudi formula implies some interesting quadratic identities for characters of representations of $gl_n$. Earlier work of Kirillov and Reshetikhin proposed a generalization of these identities to the other classical Lie algebras,…
We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We…
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…
We prove a version of the Poincare-Birkhoff-Witt theorem for the twisted quantized enveloping algebra U'_q(sp_2n). This is a subalgebra of U_q(gl_2n) and a deformation of the universal enveloping algebra U(sp_2n) of the symplectic Lie…
The properties of the quantum Minkowski space algebra are discussed. Its irreducible representations with highest weight vectors are constructed and relations to other quantum algebras: $su_{q}(2)$, $q$-oscillator, $q$-sphere are pointed…
We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple…
Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the…
We study the nonstandard $q$-deformation $U'_q({\rm so}_4)$ of the universal enveloping algebra $U({\rm so}_4)$ obtained by deforming the defining relations for skew-symmetric generators of $U({\rm so}_4)$. This algebra is used in quantum…
The positive-energy unitary irreducible representations of the $q$-deformed conformal algebra ${\cal C}_q = {\cal U}_q(su(2,2))$ are obtained by appropriate deformation of the classical ones. When the deformation parameter $q$ is $N$-th…
In this paper, we enlarge the language of MTL-algebras by a unary operation $\forall$ equationally described so as to abstract algebraic properties of the universal quantifier "for any" in its original meaning. The resulting class of…
We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized…
We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that…
Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in $h$. They are derived from the quantized enveloping algebras $\uqg$. The quantum Lie bracket satisfies a generalization of antisymmetry.…
In this study, an integrable vertex model based on the quantum affine superalgebra $U_q\bigl(\hat{gl}(2|2)\bigr)$ is constructed. The model is characterized by a particular assignment of spectral parameters and lowest as well as highest…
We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In…
We consider the cyclic representations $\Omega_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $\Omega_{rs}$ is related to the tensor product…