Algebraický seminář

Lucas Toury (IRMA Strasbourg): Hall algebra of directed graphs

Abstract: Interest in Hall algebras has grown significantly following Ringel’s foundational work on Hall algebras of quivers. While module categories provide the classical framework for constructing Hall algebras, recent work of Dyckerhoff and Kapranov has considerably broadened this perspective, leading to new classes of examples, particularly in combinatorial contexts. In this talk, we introduce and study a Hall algebra associated with a directed graph. We define its product and coproduct structures and investigate its primitive elements. These primitive elements encode combinatorial features of the graph, namely strongly connected orientations. The associated Poincaré polynomial defines a graph invariant that shares relations with the Tutte polynomial. Finally, we establish a bialgebra structure theorem for this Hall algebra, analogous to Green’s theorem in the classical setting that allows us to give a presentation of the algebra.