Representations of shifted quantum affine algebras

TL;DR

This paper develops the representation theory of shifted quantum affine algebras, introducing new techniques, classifying simple modules, and establishing algebraic structures relevant to quantum integrable models and Coulomb branches.

Contribution

It introduces a novel framework for shifted quantum affine algebras, including classification of simple modules and connections to cluster algebras and Langlands duality.

Findings

Classification of simple objects in category _

Existence of fusion products and Grothendieck ring structure

Finite simple representations in truncations and related conjectures

Abstract

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 approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well : realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal{U}_q(\hat{\mathfrak{g}})$, induction and restriction functors to the category $\mathcal{O}$ of representations of the Borel subalgebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ of $\mathcal{U}_q(\hat{\mathfrak{g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal{O}_\mu$ of representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}_\mu)$. 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 prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, 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.