no. 6
On double quantum affinization: 1. Type $ \mathfrak{a}_1$
[De la Double Affinisation Quantique : 1. En type $\mathfrak{a}_1$]
Zegers, Robin1 ; Mounzer, Elie1
1 Université Paris-Saclay, CNRS, IJCLab 91405, Orsay (France)
Annales de l'Institut Fourier, Tome 75 (2025) no. 6, pp. 2359-2422.

We define the double quantum affinization $\ddot{\mathrm{U}}_q(\mathfrak{a}_1)$ of type $\mathfrak{a}_1$ as a topological Hopf algebra. We prove that it admits a subalgebra $\ddot{\mathrm{U}}_q^{\prime }(\mathfrak{a}_1)$ whose completion is (bicontinuously) isomorphic to the completion of the quantum toroidal algebra $\dot{\mathrm{U}}_q ( \dot{\mathfrak{a}}_1 ) $, defined as the (simple) quantum affinization of the untwisted affine Kač–Moody Lie algebra $\dot{\mathfrak{sl}}_2$ of type $\dot{\mathfrak{a}}_1$, equipped with a certain topology inherited from its natural $\mathbb{Z}$-grading. The isomorphism is constructed by means of a bicontinuous action by automorphisms of an affinized version $\ddot{\mathfrak{B}}$ – technically a split extension $\ddot{\mathfrak{B}}\cong \dot{\mathfrak{B}} \ltimes P^\vee $ by the coweight lattice $P^\vee $ – of the affine braid group $\dot{\mathfrak{B}}$ of type $\dot{\mathfrak{a}}_1$ on that completion of $\dot{\mathrm{U}}_q ( \dot{\mathfrak{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 $\ddot{\mathrm{U}}_q(\mathfrak{a}_1)$ and briefly discuss the consequences regarding the representation theory of quantum toroidal algebras.

Nous définissons la double affinisation quantique $\ddot{\mathrm{U}}_q(\mathfrak{a}_1)$ de type $\mathfrak{a}_1$ comme une algèbre de Hopf topologique. Nous démontrons qu’elle admet une sous-algèbre $\ddot{\mathrm{U}}_q^{\prime }(\mathfrak{a}_1)$ dont la complétion est (bicontinûment) isomorphe à la complétion de l’algèbre quantique toroïdale $\dot{\mathrm{U}}_q ( \dot{\mathfrak{a}}_1 ) $, elle-même définie comme l’affinisaton quantique (simple) de l’algèbre de Kač–Moody affine non-torsionnée $\dot{\mathfrak{sl}}_2$ de type $\dot{\mathfrak{a}}_1$, munie d’une certaine topologie héritée de sa $\mathbb{Z}$-graduation naturelle. L’isomorphisme est construit au moyen d’une action bicontinue par automorphismes d’une version affinisée $\ddot{\mathfrak{B}}$ – techniquement une extension scindée $\ddot{\mathfrak{B}}\cong \dot{\mathfrak{B}} \ltimes P^\vee $ par le réseau des co-poids $P^\vee $ – du groupe des tresses affine $\dot{\mathfrak{B}}$ de type $\dot{\mathfrak{a}}_1$ sur cette complétion de $\dot{\mathrm{U}}_q ( \dot{\mathfrak{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 $\ddot{\mathrm{U}}_q(\mathfrak{a}_1)$ et discutons brièvement les conséquences sur la théorie des représentations des algèbres quantiques toroïdales.

Reçu le :
Accepté le :
Première publication :
Publié le :
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

Zegers, Robin 1 ; Mounzer, Elie 1

1 Université Paris-Saclay, CNRS, IJCLab 91405, Orsay (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Zegers, Robin; Mounzer, Elie. On double quantum affinization: 1. Type $ \mathfrak{a}_1$. Annales de l'Institut Fourier, Tome 75 (2025) no. 6, pp. 2359-2422. doi : 10.5802/aif.3691. https://aif.centre-mersenne.org/articles/10.5802/aif.3691/

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