Weight-Finite Modules Over the Quantum Affine and Double Quantum Affine Algebras of Type $\mathfrak a_{1}$

Abstract

We define the categories of weight-finite modules over the type \(\mathfrak a_{1}\) quantum affine algebra \(\dot {\mathrm {U}}_{q}(\mathfrak a_{1})\) and over the type \(\mathfrak a_{1}\) double quantum affine algebra \(\ddot {\mathrm {U}}_{q}(\mathfrak a_{1})\) that we introduced in Mounzer and Zegers (2019). In both cases, we classify the simple objects in those categories. In the quantum affine case, we prove that they coincide with the simple finite-dimensional \(\dot {\mathrm {U}}_{q}(\mathfrak a_{1})\)-modules which were classified by Chari and Pressley in terms of their highest (rational and ℓ-dominant) ℓ-weights or, equivalently, by their Drinfel’d polynomials. In the double quantum affine case, we show that simple weight-finite modules are classified by their (t-dominant) highest t-weight spaces, a family of simple modules over the subalgebra \(\ddot {\mathrm {U}}_{q}^{0}(\mathfrak a_{1})\) of \(\ddot {\mathrm {U}}_{q}(\mathfrak a_{1})\) which is conjecturally isomorphic to a split extension of the elliptic Hall algebra \(\mathcal E_{q^{-4}}{q^{2}},{q^{2}}\). The proof of the classification, in the double quantum affine case, relies on the construction of a double quantum affine analogue of the evaluation modules that appear in the quantum affine setting.