Cluster algebras, quiver representations and triangulated categories

TL;DR

This paper introduces the connections between cluster algebras, quiver representations, and triangulated categories, highlighting recent advances such as the periodicity conjecture and mutations as derived equivalences.

Contribution

It provides an overview of classical and recent results linking cluster algebras with representation theory and triangulated categories, including new insights into the periodicity conjecture and mutation interpretations.

Findings

Outline of proof for the periodicity conjecture for Dynkin diagram pairs

Recent results on mutations as derived equivalences

Connections between cluster algebras and Calabi-Yau triangulated categories

Abstract

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.