Introduction to Cluster Algebras

TL;DR

This paper introduces the concept of cluster algebras, illustrating their definitions through examples like Markov triples and Grassmannians, and explores their structures, quantization, and applications in integrable systems.

Contribution

It provides a comprehensive introduction to cluster algebras, including their definitions, fundamental results, Poisson structures, quantization, and applications in integrable systems.

Findings

Cluster algebras are defined through examples such as Markov triples and Grassmannians.

Fundamental results are proved in the rank 2 case.

Applications include Zamolodchikov periodicity and the pentagram map.

Abstract

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.