Representations of shifted quantum affine algebras - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Int.Math.Res.Not. Année : 2023

Representations of shifted quantum affine algebras

Résumé

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.
Fichier principal
Vignette du fichier
2010.06996.pdf (696.67 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte

Dates et versions

hal-02987492 , version 1 (27-02-2024)

Identifiants

Citer

David Hernandez. Representations of shifted quantum affine algebras. Int.Math.Res.Not., 2023, 2023 (13), pp.11035-11126. ⟨10.1093/imrn/rnac149⟩. ⟨hal-02987492⟩
68 Consultations
4 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More