An introduction to representation-theoretic canonical bases of cluster algebras

TL;DR

This paper introduces the representation-theoretic canonical bases of cluster algebras, reviewing their construction, properties, and examples, with a focus on their relation to Lie theory and quantum affine algebras.

Contribution

It provides a comprehensive introduction to the canonical bases of cluster algebras from a representation-theoretic perspective, including recent developments and detailed examples.

Findings

Cluster algebras have Kazhdan-Lusztig-type canonical bases.

Tools like seed mutations and base change facilitate their study.

Examples from Lie groups and quantum affine algebras illustrate the theory.

Abstract

This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds associated with signed words, cluster operations (freezing and base change), the extension technique from finite to infinite rank, which are convenient tools to study these algebras and their bases. We present recent results on the topic and provide examples. An appendix collects detailed examples of some cluster algebras from Lie theory, including those from Lie groups and representations of quantum affine algebras.