On highest weight modules over elliptic quantum groups

The purpose of this note is to define and construct highest weight modules for Felder’s elliptic quantum groups. This is done by using exchange matrices for intertwining operators between modules over quantum affine algebras. A similar problem for the elliptic quantum group corresponding to Belavin’s R-matrix was posed in [7]. This problem, as well as its analogue for Felder’s R-matrix was solved in a recent paper [9]. Thus our goal is to suggest another solution of this problem. Our approach is similar to that of [9] but somewhat different. Namely, we construct the quasi-Hopf twist (which is necessary to pass from the quantum affine algebra to elliptic algebra) not as an infinite product of R-matrices, but axiomatically as a fusion matrix for intertwining operators (this forces us to consider intetwining operators taking values in arbitrary, not necessarily finite dimensional modules). For finite dimensional Lie algebras and quantum groups, a similar construction was introduced in [5]. The equivalence of the two approaches (ours and that of [9]) follows from the fact that the fusion matrix satisfies a version of the quantum KZ equations, which implies that it is a one-sided infinite product of (modified) R-matrices.