On cluster theory and quantum dilogarithm identities
Bernhard Keller
TL;DR
This paper explores identities involving quantum dilogarithms linked to Dynkin quivers and extends these to quivers with potential, connecting them to cluster algebra theory and highlighting recent advances in the field.
Contribution
It provides an overview of quantum dilogarithm identities for Dynkin quivers and their generalization to quivers with potential within the framework of cluster algebras.
Findings
Identifies identities between quantum dilogarithm products for Dynkin quivers.
Extends these identities to quivers with potential.
Links quantum dilogarithm identities to cluster algebra structures.
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.
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