The Rank-One property for free Frobenius extensions

Gwyn Bellamy; Ulrich Thiel

doi:10.5802/crmath.502

Théorie des représentations

The Rank-One property for free Frobenius extensions

Gwyn Bellamy ¹ ; Ulrich Thiel ²

¹ School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ.
² Department of Mathematics, University of Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1341-1348.

Résumé
Abstract

Une conjecture du deuxième auteur, qui a été prouvée par Bonnafé–Rouquier, dit que la matrice de multiplicité des bébé modules de Verma de l’algèbre rationnelle restreinte de Cherednik est de rang un sur $ℚ$ lorsqu’elle est restreinte à chaque bloc de l’algèbre.

Dans cet article nous montrons que si $H$ est une algèbre première qui est une extension libre de Frobenius sur une sous-algèbre centrale régulière $R$ , et si le centre de $H$ est Gorenstein normal, alors chaquequotient central $A$ de $H$ par un idéal maximal $𝔪$ de $R$ satisfait la propriété de rang un par rapport à la matrice de Cartan de $A$ . Les exemples où le résultat est applicable incluent les algèbres de Hecke graduées, les algèbres de Hecke affines étendues, les algèbres enveloppantes quantifiées aux racines de l’unité, les résolutionscrépantes non commutatives des domaines de Gorenstein et les algèbres PI Sklyanin à dimension $3$ et $4$ .

En particulier, puisque la matrice de multiplicité pour les algèbres de Cherednik rationnelles restreintes a la propriété de rang un si et seulement si sa matrice de Cartan l’a aussi, notre résultat fournit une preuve différente de la conjecture originale.

A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $ℚ$ when restricted to each block of the algebra.

In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$ , and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $𝔪$ of $R$ satisfies the rank-one property with respect to the Cartan matrix of $A$ . Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras.

In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.

Reçu le : 2023-01-19
Révisé le : 2023-03-27
Accepté le : 2023-03-29
Publié le : 2023-10-31

DOI : 10.5802/crmath.502

Affiliations des auteurs :

Gwyn Bellamy ¹ ; Ulrich Thiel ²

¹ School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ.
² Department of Mathematics, University of Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The {Rank-One} property for free {Frobenius} extensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1341--1348},
     publisher = {Acad\'emie des sciences, Paris},
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Gwyn Bellamy; Ulrich Thiel. The Rank-One property for free Frobenius extensions. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1341-1348. doi : 10.5802/crmath.502. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.502/

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