On double quantum affinization: 1. Type $ \mathfrak{a}_1$

Zegers, Robin; Mounzer, Elie

doi:10.5802/aif.3691

On double quantum affinization: 1. Type

𝔞_{1}

[De la Double Affinisation Quantique : 1. En type

𝔞_{1}

]

Zegers, Robin ¹ ; Mounzer, Elie ¹

¹ Université Paris-Saclay, CNRS, IJCLab 91405, Orsay (France)

Annales de l'Institut Fourier, Online first, 64 p.

Résumé
Abstract

Nous définissons la double affinisation quantique ${\ddot{U}}_{q} (𝔞_{1})$ de type $𝔞_{1}$ comme une algèbre de Hopf topologique. Nous démontrons qu’elle admet une sous-algèbre ${\ddot{U}}_{q}^{'} (𝔞_{1})$ dont la complétion est (bicontinûment) isomorphe à la complétion de l’algèbre quantique toroïdale ${\dot{U}}_{q} ({\dot{𝔞}}_{1})$ , elle-même définie comme l’affinisaton quantique (simple) de l’algèbre de Kač–Moody affine non-torsionnée ${\dot{𝔰𝔩}}_{2}$ de type ${\dot{𝔞}}_{1}$ , munie d’une certaine topologie héritée de sa $ℤ$ -graduation naturelle. L’isomorphisme est construit au moyen d’une action bicontinue par automorphismes d’une version affinisée $\ddot{𝔅}$ – techniquement une extension scindée $\ddot{𝔅} ≅ \dot{𝔅} ⋉ P^{\lor}$ par le réseau des co-poids $P^{\lor}$ – du groupe des tresses affine $\dot{𝔅}$ de type ${\dot{𝔞}}_{1}$ sur cette complétion de ${\dot{U}}_{q} ({\dot{𝔞}}_{1})$ . Il peut être vu comme une version affinisée de l’isomorphisme de Damiani–Beck, bien connu dans le cadre des algèbres quantiques affines. Nous prouvons finalement la décomposition triangulaire correspondante de ${\ddot{U}}_{q} (𝔞_{1})$ et discutons brièvement les conséquences sur la théorie des représentations des algèbres quantiques toroïdales.

We define the double quantum affinization ${\ddot{U}}_{q} (𝔞_{1})$ of type $𝔞_{1}$ as a topological Hopf algebra. We prove that it admits a subalgebra ${\ddot{U}}_{q}^{'} (𝔞_{1})$ whose completion is (bicontinuously) isomorphic to the completion of the quantum toroidal algebra ${\dot{U}}_{q} ({\dot{𝔞}}_{1})$ , defined as the (simple) quantum affinization of the untwisted affine Kač–Moody Lie algebra ${\dot{𝔰𝔩}}_{2}$ of type ${\dot{𝔞}}_{1}$ , equipped with a certain topology inherited from its natural $ℤ$ -grading. The isomorphism is constructed by means of a bicontinuous action by automorphisms of an affinized version $\ddot{𝔅}$ – technically a split extension $\ddot{𝔅} ≅ \dot{𝔅} ⋉ P^{\lor}$ by the coweight lattice $P^{\lor}$ – of the affine braid group $\dot{𝔅}$ of type ${\dot{𝔞}}_{1}$ on that completion of ${\dot{U}}_{q} ({\dot{𝔞}}_{1})$ . It can be regarded as an affinized version of the Damiani–Beck isomorphism, familiar from the quantum affine setting. We eventually prove the corresponding triangular decomposition of ${\ddot{U}}_{q} (𝔞_{1})$ and briefly discuss the consequences regarding the representation theory of quantum toroidal algebras.

Reçu le : 2020-10-27
Accepté le : 2023-10-02
Première publication : 2025-02-10

DOI : 10.5802/aif.3691

Classification : 17B37, 17B67
Keywords: Quantum Affine Algebras, Quantum Toroidal Algebras, Representation Theory
Mots-clés : Algèbres auantiques affines, algèbres quantiques toroïdales, théorie des représentations

Affiliations des auteurs :

Zegers, Robin ¹ ; Mounzer, Elie ¹

¹ Université Paris-Saclay, CNRS, IJCLab 91405, Orsay (France)

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     title = {On double quantum affinization: 1. {Type} $ \mathfrak{a}_1$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
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     language = {en},
     note = {Online first},
}

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Zegers, Robin; Mounzer, Elie. On double quantum affinization: 1. Type $ \mathfrak{a}_1$. Annales de l'Institut Fourier, Online first, 64 p.

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