Simplicity of vacuum modules and associated varieties
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 169-191.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.144
Classification: 17B69
Keywords: Associated variety, affine Kac-Moody algebra, affine vertex algebra, singular vector, affine W-algebra
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Tomoyuki Arakawa; Cuipo Jiang; Anne Moreau. Simplicity of vacuum modules and associated varieties. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 169-191. doi : 10.5802/jep.144. https://jep.centre-mersenne.org/articles/10.5802/jep.144/

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