We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang–Baxter equation. The method is based on an affine realization of certain seaweed algebras and their quantum analogues. We also propose a method of ω-affinization, which enables us to quantize rational r-matrices of sl(3).

The degenerate affine and affine BMW algebras arise naturally in the context of SchurWeyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we...

In this paper, we classify the irreducible integrable modules for nullity 2 toroidal extended affine Lie algebras with finite dimensional weight spaces.

Abstract We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a $2$-cocycle for connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and if unipotent supported on $G$, then ${}_J\mathcal{O}(G)_{J}\co...

Recent interests in quantum groups are stimulated by their marvelous relations with quantum Yang-Baxter equations, conformal field theory, invariants of links and knots, and q-hypergeometric series. Besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q-deformations or quantum analogues of known st...

Vertex operators discovered by physicists in string theory have turned out to be important objects in mathematics. One can use vertex operators to construct various realizations of the irreducible highest weight representations for affine Kac-Moody algebras. Two of these, the principal and homogeneous realizations, are of particular interest. The principal vertex operator construction for the a...

Any Lie algebra equipped with a symplectic form can be equipped with an affine structure. On the other hand there exist (2p + 1)-dimensional Lie algebras with contact form and no affine structure. But each nilpotent contact Lie algebra is a one-dimensional central extension of a symplectic algebra. The aim of this work is to study how we can extend, under certain conditions, the symplectic stuc...

We construct spectral parameter dependent R-matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral parameter dependent quantum Yang-Baxter equation. q-alg/9508012 KCL-TH-95-8 YITP/K-1119

In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by...

We propose integral representations for wave functions of Bn, Cn, and Dn open Toda chains at zero eigenvalues of the Hamiltonian operators thus generalizing Givental representation for An. We also construct Baxter Q-operators for closed Toda chains corresponding to Lie algebras B∞, C∞, D∞, affine Lie algebras B (1) n , C (1) n , D (1) n and twisted affine Lie algebras A (2) 2n−1 and A (2) 2n . ...