We define the categories of weight-finite modules over the type alpha(1) quantum affine algebra U-q (alpha(1)) and over the type alpha(1) double quantum affine algebra U-q. (alpha(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 finitedimensional U-q (alpha(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 U-q(0) (alpha(1)) of U-q (alpha(1)) which is conjecturally isomorphic to a split extension of the elliptic Hall algebra epsilon(-4)(q)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.
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