Related papers: $RLL$-realization of two-parameter quantum affine …

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…

The purpose of this paper is to compute the Drinfel'd polynomials for two types of evaluation representations of quantum affine algebras at roots of unity and construct those representations as the submodules of evaluation Schnizer modules.…

The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the…

The quantum GL(n) of Faddeev-Reshetikhin-Takhtajan and Dipper-Donkin are realized geometrically by using double partial flag varieties. As a consequence, the difference of these two Hopf algebras is caused by a twist of a cocycle in the…

Finite dimensional irreducible modules of the two-parameter quantum enveloping algebra $U_{r,s}(\mathfrak{sl}_n)$ are explicitly constructed using the fusion procedure when $rs^{-1}$ is generic. This provides an alternative and…

This is a follow-up to the paper in which we categorified the affine quantum Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification of the affine quantum sl_n. In this paper we categorify S(n,n) for n…

In this paper we study the algebra of quantum observables of the Chern-Simons matrix model which was originally proposed by Susskind and Polychronakos to describe electrons in fractional quantum Hall effects. We establish the commutation…

We give explicit constructions of quantum symplectic affine algebras at level 1 using vertex operators.

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of…

Quantum affine reflection algebras are coideal subalgebras of quantum affine algebras that lead to trigonometric reflection matrices (solutions of the boundary Yang-Baxter equation). In this paper we use the quantum affine reflection…

We give a $K$-theoretic realization of all affine Hecke algebras with two unequal parameters including exceptional types. This extends the celebrated work of Kazhdan and Lusztig, who gave a $K$-theoretic realization of affine Hecke algebras…

We consider a quantum affine algebra realized in two-dimensional non-linear sigma models with target space three-dimensional squashed sphere. Its affine generators are explicitly constructed and the Poisson brackets are computed. The…

We study the crystal of quantum nilpotent subalgebra of $U_q(D_n)$ associated to a maximal Levi subalgebra of type $A_{n-1}$. We show that it has an affine crystal structure of type $D_n^{(1)}$ isomorphic to a limit of perfect…

The factorization of the universal R-matrix corresponding to so called Drinfeld Hopf structure is described on the example of quantum affine algebra $U_q(\hat{sl}_2)$. As a result of factorization procedure we deduce certain differential…

We construct an irreducible representation for the extended affine algebra of type $sl_2$ with coordinates in a quantum torus. We explicitly give formulas using vertex operators similar to those found in the theory of the infinite rank…

We compute $t$--analogs of $q$--characters of all $l$--fundamental representations of the quantum affine algebras of type $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ by a supercomputer. In particular, we prove the fermionic formula for…

Infinite dimensional representations of the real form U_q(u_{n,1}) of the Drinfeld--Jimbo algebra U_q(gl_{n+1}) are defined. The principal series of representations of U_q(u_{n,1}) is studied. Intertwining operators for pairs of the…

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define the affine analogue of the Elias-Khovanov and the…

The generalized quantum group $\mathcal{U}(\epsilon)$ of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra $\mathfrak{gl}_{M|N}$. We prove that there exists a unique $R$ matrix on tensor product…

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…