Comptes Rendus
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Azumaya locus over certain quantum symplectic spaces II
[Lieu d’Azumaya sur certains espaces symplectiques quantiques II]
Snehashis Mukherjee1
1 Stat-Math Unit, Indian Statistical Institute, Bangalore Center, 8th Mile, Mysore Rd, RVCE Post, Bengaluru, Karnataka 560059, India
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 383-405.

This article undertakes an exploration of a particular variant of multiparameter quantum symplectic algebras, focusing specifically on the quantum Heisenberg algebra at the roots of unity. In this context, the algebra undergoes a transformation into Polynomial Identity algebra, where the dimensions of the simple modules are restricted by their PI degree. We conduct an extensive examination of all possible maximal dimensional simple modules associated with this algebra. Additionally, we present a condition that is both necessary and sufficient for the attainment of maximal dimensional simple modules, thereby facilitating the classification of its Azumaya locus.

Cet article explore une variante particulière des algèbres symplectiques quantiques multiparamétriques, en se concentrant spécifiquement sur l’algèbre de Heisenberg quantique aux racines de l’unité. Dans ce contexte, l’algèbre subit une transformation en algèbre à identité polynomiale, où les dimensions des modules simples sont restreintes par leur degré PI. Nous effectuons un examen approfondi de tous les modules simples de dimension maximale possibles associés à cette algèbre. En outre, nous présentons une condition qui est à la fois nécessaire et suffisante pour l’obtention de modules simples de dimension maximale, facilitant ainsi la classification de son lieu d’Azumaya.

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DOI : 10.5802/crmath.723
Classification : 16D60, 16D70, 16S85
Keywords: Quantum Heisenberg algebra, polynomial identity algebra, Smith normal form, eigenvalue analysis, simple modules, Azumaya locus
Mots-clés : Algèbre de Heisenberg quantique, algèbre à identité polynomiale, forme normale de Smith, analyse des valeurs propres, modules simples, lieu d’Azumaya

Snehashis Mukherjee 1

1 Stat-Math Unit, Indian Statistical Institute, Bangalore Center, 8th Mile, Mysore Rd, RVCE Post, Bengaluru, Karnataka 560059, India
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Azumaya locus over certain quantum symplectic spaces {II}},
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Snehashis Mukherjee. Azumaya locus over certain quantum symplectic spaces II. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 383-405. doi : 10.5802/crmath.723. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.723/

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