On cluster theory and quantum dilogarithm identities

Abstract

These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin–Zelevinsky’s theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock–Goncharov, Kontsevich–Soibelman and Nagao.