An introduction to representation-theoretic canonical bases of cluster algebras

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.