Quasi-Hopf twistors for elliptic quantum groups

Abstract

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al. [1], Felder [2]). Frønsdal [3,4] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U q (g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U q (g). We also prove the shifted cocycle condition for the twistors, thereby completing Frønsdal's findings.This construction entails that, for generic values of the deformation parameters, representation theory for U q (g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebra A q,p ( sl 2 ).

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

“…Hopf algebra equipped with the standard coproduct ∆, counit ε, antipode S and universal R matrix R. Our conventions on the coalgebra structure follows [5]. Let h andh be the Cartan subalgebras of sl N and sl N , respectively.…”

“…[5]). Hence just one dynamical RLL-relation (2.13) characterizes the algebra B q,λ ( sl N ) completely in the sense of Reshetikhin and Semenov-TianShansky [19].…”

“…construction of the twistor F (λ) is given in [5]. A quasi-Hopf deformation means that as an associative algebra, B q,λ ( sl N ) is isomorphic to U q ( sl N ), but the coalgebra structure is deformed.…”

“…In this section, we give a review on the face type elliptic quantum group B q,λ ( sl N ) based on the results in [5].…”

“…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…”