Representations of Shifted Quantum Affine Algebras

Abstract

Abstract We develop the representation theory of shifted quantum affine algebras $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ 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, and parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal {O}_\mu $ of representations of $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ 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 $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ 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.