Abstract
This lecture series is an introduction to the link between cluster algebras and representations of quivers or, more generally, quivers with potential. This link goes back to work by Marsh-Reineke-Zelevinsky (2003) and has evolved into what is now known as "additive categorification" of cluster algebras (in contrast to "monoidal categorification", invented by Hernandez-Leclerc in 2010). We will begin by explaining Caldero-Chapoton's formula (2006), which allows to compute the cluster variables of a (cluster-finite, acyclic) quiver from the geometry of its indecomposable representations. We will then introduce the cluster category of an acyclic quiver (Buan-Marsh-Reineke-Reiten-Todorov 2006) and explain how it allows to lift the combinatorics of the associated cluster algebra to the level of a triangulated 2-Calabi-Yau category. To extend these constructions to quivers with oriented cycles, we will endow our quivers with potentials, following the work of Derksen-Weyman-Zelevinsky (2008, 2010). As an application, we will construct "generic bases" for large classes of upper cluster algebras (Geiss-Leclerc-Schroeer, 2012, and Plamondon, 2013). Finally, if time permits, we will present the generalization of the theory to ice quivers with potential (Yilin Wu, 2023), with motivations from higher Teichmuller theory as approached by Fock-Goncharov (2006) and Goncharov-Shen (preprint 2019).