The fundamental group of a triangular algebra without double bypasses

Patrick Le Meur

doi:10.1016/j.crma.2005.07.004

Algebra

The fundamental group of a triangular algebra without double bypasses

Presented by: Michèle Vergne

Patrick Le Meur ¹

¹ Institut de mathématiques et de modélisation de Montpellier, UMR CNRS 5149, université Montpellier 2, case courier 051, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 211-216.

Abstract
Résumé

Let A be a basic connected finite dimensional algebra over a field of characteristic zero. A fundamental group depending on the presentation of A has been defined by several authors [see R. Martínez-Villa, J.A. de La Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983) 277–292]. Assuming the quiver of A has no oriented cycles and no double bypasses, we show there exists a suitable presentation of A with quiver and admissible relations, with fundamental group denoted by $π_{1} (A)$ , such that the fundamental group of any other presentation of A with quiver and admissible relations is a quotient of $π_{1} (A)$ .

Soit A une algèbre basique connexe et de dimension finie sur un corps de caractéristique nulle. Plusieurs auteurs [voir R. Martínez-Villa, J.A. de La Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983) 277–292] ont défini pour A un groupe fondamental dépendant du choix d'une présentation de A. En supposant que le carquois de A n'a pas de cycle orienté et n'a pas de double raccourci, nous démontrons qu'il existe une présentation privilégiée de A par carquois et relations admissibles, de groupe fondamental noté $π_{1} (A)$ , telle que le groupe fondamental de toute autre présentation de A par carquois et relations admissibles est un quotient de $π_{1} (A)$ .

Received: 2005-03-16
Accepted: 2005-06-28
Published online: 2005-08-15

DOI: 10.1016/j.crma.2005.07.004

Author's affiliations:

Patrick Le Meur ¹

¹ Institut de mathématiques et de modélisation de Montpellier, UMR CNRS 5149, université Montpellier 2, case courier 051, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

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Patrick Le Meur. The fundamental group of a triangular algebra without double bypasses. Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 211-216. doi : 10.1016/j.crma.2005.07.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.004/

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