Notes on Highest Weight Modules of the Elliptic Algebra $\mathfrak{A}_{q, p}(\hat{sl_2})$

Abstract

We discuss a construction of highest weight modules for the recently defined elliptic algebra A q,p ( sl 2 ), and make several conjectures concerning them. The modules are generated by the action of the components of the operator L on the highest weight vectors. We introduce the vertex operators Φ and Ψ * through their commutation relations with the L-operator. We present ordering rules for the Land Φ-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of sl 2 -modules. IntroductionIn our previous paper [1] we defined the elliptic quantum affine algebra A q,p (ĝ) with g = gl 2 or sl 2 . The present paper is an attempt toward understanding the correct elliptic analogues of the highest weight modules and vertex operators. Our aim here is to present conjectures concerning their existence and expected properties, along with some experimental computations to support them. *

A Family of Integral Transformations and Basic Hypergeometric Series

“…(As for the definition of the intertwiner Φ(ζ), see [5,6] and [1].) More precisely, in [1] the matrix elements Φ(ζ 1 )Φ(ζ 2 ) · · · Φ(ζ n ) were represented by applying the n-fold integral transformation to a basic hypergeometric series.…”

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

“…For this purpose, we remind the reader of the fact that in the trigonometric case, i.e. U q ( sl N ), the L-operator can be constructed as a composition of type I and II vertex operators [17,18]. The following theorem is an elliptic analogue of such a construction.…”

“…We thus give a representation theoretical meaning to the vertex operators of the A N −1 type face model [16]. Conversely, as a composition of the type I and type II intertwiners, one can construct a L-operator which satisfies the RLL-relations [17,18]. As a check of our free field realization, we investigate a connection between the two L-operators, the one constructed by a composition of the vertex operators and the other by the half currents, in the level-one representation.…”

Generalized Quantum Current Algebras

“…Further investigations along this line showed that there are several kinds of elliptic quantum groups [7,8,9,10,11,12,14,15,18,21], some are recognized as quasi-triangular quasi-Hopf algebras and some are still not. Due to the lack of co-algebraic structures for some of the elliptic…”

“…From a pure mathematical point of view, the most important significance for the discovery of elliptic quantum groups might be that it reveals the possibility for the existence of non-co-associative and non-commutative non-co-commutative associative algebras. Such algebras also happen to describe the dynamical symmetry of some statistical models and hence greatly attracted the attention of theoretical physicists.Further investigations along this line showed that there are several kinds of elliptic quantum groups [7,8,9,10,11,12,14,15,18,21], some are recognized as quasi-triangular quasi-Hopf algebras and some are still not. Due to the lack of co-algebraic structures for some of the elliptic…”