Simplicity of vacuum modules and associated varieties

Tomoyuki Arakawa; Cuipo Jiang; Anne Moreau

doi:10.5802/jep.144

Simplicity of vacuum modules and associated varieties
[Simplicité des algèbres vertex affines et variétés associées]

Tomoyuki Arakawa ¹ ; Cuipo Jiang ² ; Anne Moreau ³

¹ Research Institute for Mathematical Sciences, Kyoto University Kyoto, 606-8502, Japan
² School of Mathematical Sciences, Shanghai Jiao Tong University Shanghai, 200240, China
³ Faculté des Sciences d’Orsay, Université Paris-Saclay 91405 Orsay, France

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 169-191.

Résumé
Abstract

Dans cet article, nous démontrons que l’algèbre vertex affine universelle associée à une algèbre de Lie simple $𝔤$ est simple si et seulement si la variété associée à son unique quotient simple est égale à $𝔤^{*}$ . Nous en déduisons un résultat analogue pour la réduction quantique de Drinfeld-Sokolov appliquée à l’algèbre vertex affine universelle.

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $𝔤$ is simple if and only if the associated variety of its unique simple quotient is equal to $𝔤^{*}$ . We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.

Reçu le : 2020-04-03
Accepté le : 2021-01-08
Publié le : 2021-01-19

MR

DOI : 10.5802/jep.144

Classification : 17B69
Keywords: Associated variety, affine Kac-Moody algebra, affine vertex algebra, singular vector, affine $W$-algebra
Mot clés : Variété associée, algèbre de Kac-Moody, algèbre vertex affine, vecteur singulier, $W$-algèbre affine

Affiliations des auteurs :

Tomoyuki Arakawa ¹ ; Cuipo Jiang ² ; Anne Moreau ³

¹ Research Institute for Mathematical Sciences, Kyoto University Kyoto, 606-8502, Japan
² School of Mathematical Sciences, Shanghai Jiao Tong University Shanghai, 200240, China
³ Faculté des Sciences d’Orsay, Université Paris-Saclay 91405 Orsay, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Tomoyuki Arakawa and Cuipo Jiang and Anne Moreau},
     title = {Simplicity of vacuum modules and associated~varieties},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {169--191},
     publisher = {\'Ecole polytechnique},
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     year = {2021},
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Tomoyuki Arakawa; Cuipo Jiang; Anne Moreau. Simplicity of vacuum modules and associated varieties. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 169-191. doi : 10.5802/jep.144. https://jep.centre-mersenne.org/articles/10.5802/jep.144/

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