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 a previous paper. 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 $\ell$-dominant) $\ell$-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. 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.

46 pages

References in corpus (1)

• On double quantum affinization: 1. Type ${\mathfrak{a}}_1$