2020/07/04 by Elie Mounzer, Mounzer, Elie, Robin Zegers +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2007.02030
openalex publication_date 2020/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the categories of weight-finite modules over the type \mathfrak a1 quantum affine algebra Uq(\mathfrak a1) and over the type \mathfrak a1 double quantum affine algebra Uq(\mathfrak a1) 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 Uq(\mathfrak a1)-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 Uq0(\mathfrak a1) of Uq(\mathfrak a1) 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.