Galois coverings and fundamental groups of finite dimensional algebras ; Revêtements galoisiens et groupe fondamental d'algèbres de dimension finie

Thèse effectuée de septembre 2003 à février 2006 ; This thesis is devoted to the study of the Galois coverings of finite dimensional algebras. In particular, it investigates the existence of a universal cover and a fundamental group for finite dimensional connected and basic algebras over an algebraically closed field. For this purpose, we start from an already existing notion: the fundamental group associated with any presentation of such an algebra A by its ordinary quiver Q and admissible relations. We first compare the different presentations of A. The automorphisms of the path algebra kQ allow one to link two arbitrary presentations of A. Among these automorphisms, we distinguish the dilatations and the transvections: they generate the group of automorphisms of kQ, moreover, the fundemantal groups of two presentations of A linked by a transvection or a dilatation are related by a quotient relation. These properties allow us to exhibit a fundamental group for A when the ground field has characteristic zero and when Q has no double bypass. These considerations can be translated in terms of Galois coverings since any presentation gives rise to a Galois covering with group the fundamental group of the presentation. Hence, under the above hypotheses granting a fundamental group, A admits a universal cover. This last result is extended to the case where A is monomial, Q has no oriented cycles and no multiple arrows (but may have double bypasses) and where the ground field may have any characteristic. ; Cette thèse est consacrée à l'étude des revêtements galoisiens et à la recherche du revêtement universel et du groupe fondamental pour les algèbres de dimension finie, connexes et basiques sur un corps algébriquement clos. Pour ce faire, nous partons d'une construction déjà existante: le groupe fondamental associé à toute présentation d'une telle algèbre A par son carquois ordinaire Q et des relations admissibles. Nous commençons par comparer les différentes présentations de A. Les automorphismes de l'algèbre kQ ...