Related papers: Wedge modules for two-parameter quantum groups
Let $U_q(\hat{\cal G})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, braid…
We construct $2^n$-families of solutions of the Yang-Baxter equation from $n$-products of three-dimensional $R$ and $L$ operators satisfying the tetrahedron equation. They are identified with the quantum $R$ matrices for the Hopf algebras…
Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases,…
In this paper we discuss representations of the Birman-Wenzl-Murakami algebra as well as of its dilute extension containing several free parameters. These representations are based on superalgebras and their baxterizations permit us to…
We construct a wide class of finite W-algebras as truncations of Yangians. These truncations correspond to algebra homomorphisms and allow to construct the W-algebras as exchange algebras, the R-matrix being the Yangian's one. As an…
We introduce a generalization of degenerate affine Hecke algebra, called wreath Hecke algebra, associated to an arbitrary finite group G. The simple modules of the wreath Hecke algebra and of its associated cyclotomic algebras are…
The Baxterization process for the dynamical Yang-Baxter equation is studied. We introduce the local dynamical Hecke ,Temperley-Lieb and Birman-Murakami-Wenzl operators, then by inserting spectral parameters, from each representation of…
We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affine algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental…
Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the…
We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the…
We derive a structure of $\mathbb{Z}[t,t^{-1}]$-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove…
On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension. The Gauss decomposition of T-matrices gives us the so-called Drinfel'd generators.…
Studying the algebraic structure of the double ${\cal D}Y(g)$ of the yangian $Y(g)$ we present the triangular decomposition of ${\cal D}Y(g)$ and a factorization for the canonical pairing of the yangian with its dual inside ${\cal D}Y(g)$.…
Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the…
Based on the formulation of Drinfel'd, Chari, and Pressley, a technique to analyze the structure of tensor products of the Yangian algebra representations is presented. We then apply the results to the $S$-matrix theory of the $G\otimes…
We consider the quantum affine vertex algebra $\mathcal{V}_{c}(\mathfrak{gl}_N)$ associated with the rational $R$-matrix, as defined by Etingof and Kazhdan. We introduce certain subalgebras $\textrm{A}_c (\mathfrak{gl}_N)$ of the completed…
In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding…
Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang-Baxter equation, we determine the corresponding quantum group deformations to all orders in $\hbar$ by deducing their RTT…
We introduce the notion of deformed quantum vertex algebra module associated with a braiding map. We construct two families of braiding maps over the Etingof-Kazhdan quantum vertex algebras associated with the rational $R$-matrices of…
A theory of canonical basis for a two-parameter quantum algebra is developed in parallel with the one in one-parameter case. A geometric construction of the negative part of a two-parameter quantum algebra is given by using mixed perverse…