Cluster algebras: an introduction

TL;DR

This paper introduces cluster algebras, a combinatorial algebraic structure with diverse applications in mathematics and physics, highlighting their origins, connections, and significance.

Contribution

It provides an accessible introduction to cluster algebras and illustrates their natural emergence in Teichmüller theory and their role in proving the Zamolodchikov periodicity conjecture.

Findings

Cluster algebras connect to various mathematical fields.

They were used to prove the Zamolodchikov periodicity conjecture.

The paper offers an accessible overview of the theory.

Abstract

Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichm\"uller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichm\"uller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture in mathematical physics.