Koszulity of dual braid monoid algebras via cluster complexes

Matthieu Josuat-Vergès; Philippe Nadeau

doi:10.5802/ambp.420

Koszulity of dual braid monoid algebras via cluster complexes
[Koszulité de l’algèbre du monoïde dual de tresses via les complexes d’amas]

Matthieu Josuat-Vergès ¹ ; Philippe Nadeau ²

¹ Université de Paris, CNRS, IRIF (Institut de Recherche en Informatique Fondamentale, UMR8243), France
² Université Claude Bernard Lyon 1, CNRS, École Centrale de Lyon, INSA Lyon, Universié Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France

Annales mathématiques Blaise Pascal, Tome 30 (2023) no. 2, pp. 141-188.

Résumé
Abstract

Le monoïde dual des tresses a été introduit par Bessis dans le contexte des arrangements d’hyperplans complexes. Le but de ce travail est de montrer que la dualité de Koszul fournit une interaction remarquable avec le complexe d’amas introduit par Fomin et Zelevinsky. Premièrement, nous démontrons la koszulité de l’algèbre du monoïde dual des tresses, en donnant explicitement la résolution libre minimale du corps de base. Cette construction utilise des complexes de chaînes définis grâce à la partie positive du complexe d’amas. Deuxièmement, nous examinons diverses propriétés de l’algèbre quadratique duale. Nous démontrons qu’elle est naturellement graduée par le treillis des partitions non-croisées. Nous obtenons une base explicite, indicée par les faces positives du complexe d’amas. Les constantes de structure peuvent être décrites explicitement en termes de l’éventail des amas. Enfin, nous réalisons cette algèbre duale comme un quotient d’une algèbre de Nichols. Ce dernier point se relie aux travaux de Zhang, qui a utilisé cette algèbre pour un calcul d’homologie des fibres de Milnor d’un arrangement de Coxeter.

The dual braid monoid was introduced by Bessis in his work on complex reflection arrangements. The goal of this work is to show that Koszul duality provides a nice interplay between the dual braid monoid and the cluster complex introduced by Fomin and Zelevinsky. Firstly, we prove koszulity of the dual braid monoid algebra, by building explicitly the minimal free resolution of the ground field. This is done by using some chains complexes defined in terms of the positive part of the cluster complex. Secondly, we derive various properties of the quadratic dual algebra. We show that it is naturally graded by the noncrossing partition lattice. We get an explicit basis, naturally indexed by positive faces of the cluster complex. Moreover, we find the structure constants via a geometric rule in terms of the cluster fan. Eventually, we realize this dual algebra as a quotient of a Nichols algebra. This latter fact makes a connection with results of Zhang, who used the same algebra to compute the homology of Milnor fibers of reflection arrangements.

Publié le : 2024-04-30

DOI : 10.5802/ambp.420

Affiliations des auteurs :

Matthieu Josuat-Vergès ¹ ; Philippe Nadeau ²

¹ Université de Paris, CNRS, IRIF (Institut de Recherche en Informatique Fondamentale, UMR8243), France
² Université Claude Bernard Lyon 1, CNRS, École Centrale de Lyon, INSA Lyon, Universié Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Koszulity of dual braid monoid algebras via cluster complexes},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {141--188},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Matthieu Josuat-Vergès; Philippe Nadeau. Koszulity of dual braid monoid algebras via cluster complexes. Annales mathématiques Blaise Pascal, Tome 30 (2023) no. 2, pp. 141-188. doi : 10.5802/ambp.420. https://ambp.centre-mersenne.org/articles/10.5802/ambp.420/

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