Quasi-Hopf twistors for elliptic quantum groups

TL;DR

This paper provides explicit formulas for twistors in elliptic quantum groups derived from quantum affine algebras, demonstrating their quasi-Hopf structure and implications for representation theory.

Contribution

It explicitly constructs the twistors as infinite products of the universal R matrix and proves the shifted cocycle condition, completing previous theoretical insights.

Findings

Explicit formula for the twistors as infinite products of the universal R matrix.

Proof of the shifted cocycle condition for the twistors.

Representation theory of U_q(g) extends to elliptic algebras, confirming conjectures.

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., Felder). Fronsdal 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 Fronsdal'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).