Related papers: Two-Parameter Quantum Groups and $R$-Matrices: Cla…

Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the existence of two meromorphic R-matrices associated to any pair of representations of Y_hg in the category O. They are related by a unitary constraint and constructed…

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The…

The exotic quantum double and its universal R-matrix for quantum Yang-Baxter equation are constructed in terms of Drinfeld's quantum double theory.As a new quasi-triangular Hopf algebra, it is much different from those standard quantum…

Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ and its universal quantum $R$-matrix are explicitely constructed as functionals of the associated classical $r$-matrix. In this…

A general scheme for construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to WDVV associativity equations is formulated. The advantage is taken from the Rota-Baxter identity and some…

In this paper we construct a new quantum double by endowing the l-state bosonalgebra with a non-trivial Hopf algebra structure,which is not a q-deformation of the Lie algebra or superalgebra.The universal R-matrix for the Yang-Baxter…

We further define two-parameter quantum affine algebra $U_{r,s}(\widehat{\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of importance for the quantum…

We suggest a formula for quantum universal $R$-matrices corresponding to quasitriangular classical $r$-matrices classified by Belavin and Drinfeld for all simple Lie algebras. The $R$-matrices are obtained by twisting the standard universal…

In this paper a class of new quantum groups is presented: deformed Yangians. They arise from rational solutions of the classical Yang-Baxter equation of the form $c_2 /u + const$ . The universal quantum $R$-matrix for a deformed Yangian is…

We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r, s}(\mathrm{C}_n^{(1)})$ and the $R$-matrix realization \'a la Faddeev, Reshetikhin and Takhtajan.

We construct a special principal series representation for the modular double $U_{q\tilde{q}}(g_R)$ of type $A_r$ representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also…

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…

Every unitary involutive solution of the quantum Yang-Baxter equation ("R-matrix") defines an extremal character and a representation of the infinite symmetric group $S_\infty$. We give a complete classification of all such Yang-Baxter…

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 consider Yang-Baxter equations arising from its associative analog and study corresponding exchange relations. They generate finite-dimensional quantum algebras which have form of coupled ${\rm GL}(N)$ Sklyanin elliptic algebras. Then we…

Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from…

Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation.…

Yang-Baxterising a braid group representation associated with multideformed version of $GL_{q}(N)$ quantum group and taking the corresponding $q\rightarrow 1$ limit, we obtain a rational $R$-matrix which depends on $\left ( 1+ {N(N-1) \over…

R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov…

Let $\mathcal{V}^c(\mathfrak{gl}_N)$ be Etingof--Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $\phi$-coordinated…