Representations of shifted quantum affine algebras

David Hernandez

Published 2020-10-14Version 1

We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our direct approach is based on relations that we establish with the category $\mathcal{O}$ of representations of the quantum affine Borel algebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ and on associated quantum integrable models we have previously studied. We introduce the category $\mathcal{O}^\mu$ of representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we classify its simple objects. For $\mathfrak{g} = sl_2$ we prove the existence of evaluation morphisms to $q$-oscillator algebras. We establish the existence of a fusion product and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}^\mu)$. We introduce induction and restriction functors to the category $\mathcal{O}$ of $\mathcal{U}_q(\mathfrak{b})$. As a by product we classify simple finite-dimensional representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We establish a necessary condition for a simple representation to descend to a truncation, which is also sufficient for $\mathfrak{g} = sl_2$. We introduce a related partial ordering on simple modules and we prove a truncation has only a finite number of simple representations. We state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.