Friday, 6 April, 2018 - 15:00
Speaker: Pierre-Guy Plamondon (Université de Paris Sud XI)
Room: Aula 1E
Title: Surface coverings and skew-group algebras of Jacobian algebras
Abstract: The theory of cluster algebras has sparked various interactions between combinatorial geometry and representation theory. One instance of this is as follows: to any oriented Riemann surface with boundary and marked points, one can associate an associative algebra called a Jacobian algebra. The algebra and the surface are very much related; for instance, it has been shown that if all the marked points lie on the boundary of the surface, then indecomposable representations of the Jacobian algebra are parametrized by curves and arcs on the surface. In this talk, we will be interested in the case of a surface with marked points in its interior. We will construct a covering of any such surface by a surface without punctures, together with an action of Z_2. We will then show that the group Z_2 acts on the Jacobian algebra of the covering surface, and that taking the skew-group algebra of the Jacobian algebra of the covering yields the Jacobian algebra of the surface with punctures. This strong algebraic link between the two algebras will allow us to describe all indecomposable representations in terms of curves on the surface with punctures.
This is a joint work with Claire Amiot.