Elliptic Quantum Groups $E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )$ and Quasi-Hopf Algebras

Abstract

Abstract. We construct an algebra morphism from the elliptic quantum group E τ,η (sl 2 ) to a certain elliptic version of the "quantum loop groups in higher genus" studied by V. Rubtsov and the first author. This provides an embedding of E τ,η (sl 2 ) in an algebra "with central extension". In particular we construct L ± -operators obeying a dynamical version of the Reshetikhin-Semenov-TianShansky relations. To do that, we construct the factorization of a certain twist of the quantum loop algebra, that automatically satisfies the "twisted cocycle eqaution" of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang-Baxter equation.Introduction. The aim of this paper is to compare the sl 2 -version of the elliptic quantum groups introduced by the second author ([14]) with quasi-Hopf algebras introduced by V. Rubtsov and the first author ([11, 12]). Elliptic quantum groups are presented by exchange (or "RLL") relations, whereas the algebras of [11] are "quantum loop algebras". Our result can be viewed as an elliptic version of the results of J. Ding and I. Frenkel ([4]) and of S. Khoroshkin ([21]), where Drinfeld's quantum current algebra ([6]) was shown to be isomorphic with the Reshetikhin-Semenov L-operator algebra of [22,13], in the trigonometric and rational case respectively.Elliptic quantum groups are based on a matrix solution R(z, λ) of the dynamical Yang-Baxter equation (YBE). Here "dynamical" means that in addition to the spectral parameter z (belonging to an elliptic curve E), R depends on a parameter λ, which undergo certain shifts in the Yang-Baxter equation. The RLL relations defining the elliptic quantum groups E τ,η (sl 2 ) are then an algebraic variant of the dynamical YBE.In [3], O. Babelon, D. Bernard and E. Billey studied the relation beween the dynamical and quasi-Hopf Yang-Baxter equations. They showed that given a family of twists of a quasi-triangular Hopf algebra, satisfying a certain "twisted cocycle equation", the quasi-Hopf

A commutative algebra on degenerate CP1 and Macdonald polynomials

“…See [Fe,EF,Fr1,Fr2,ABRR,JKOS1,JKOS2] for example and references therein. We aim at making the representation theory of the Ding-Iohara algebra U(q, t) compatible with the elliptic algebra A(p), and apply the quasi-Hopf twisting prescribed by Babelon, Bernard and Billey [BBB].…”

Classical elliptic current algebras. I

“…To make these differences more transparent we shall consider only the simplest case of Lie algebra a = sl 2 defined as a three-dimensional complex Lie algebra with commutation relations [h, e] = 2e, [h, f ] = −2f and [e, f ] = h. We denote the constructed current algebraĝ for the case K = K 0 as e τ ( sl 2 ) and for K = K = K(Cyl) as u τ ( sl 2 ). These current algebras may be identified with classical limits of the quantized currents algebra E τ,η (sl 2 ) of [7] and U p,q ( sl 2 ) of [18] respectively. The Green distributions appear in the algebras e τ ( sl 2 ) and u τ ( sl 2 ) as a regularization of the same meromorphic quasi-doubly periodic functions but in different spaces: (K 0 ⊗ K 0 ) and (K ⊗ K) respectively.…”

“…developed a theory of quantum current algebras related to arbitrary genus complex curves (in particular to an elliptic curve) as a quantization of certain (twisted) Manin pairs [9] using Drinfeld's new realization of quantized current algebras. Further, it was shown in [7] that the Felder algebra can be obtained by twisting of the Enriquez-Rubtsov elliptic algebra. This twisted algebra will be denoted by E τ,η and it is a quasi-Hopf algebra.…”

The Elliptic Algebra and the Drinfeld Realization of the Elliptic Quantum Group

“…In recent papers [1,2,3,4,5], the notion of elliptic quantum groups has been proposed. There are two types of elliptic quantum groups, the vertex type A q,p ( sl N ) and the face type B q,λ (g), where…”