Cluster algebras of infinite rank

Abstract

Holm and Jorgensen have shown the existence of a cluster structure on a certain category $D$ that shares many properties with finite type $A$ cluster categories and that can be fruitfully considered as an infinite analogue of these. In this work we determine fully the combinatorics of this cluster structure and show that these are the cluster combinatorics of cluster algebras of infinite rank. That is, the clusters of these algebras contain infinitely many variables, although one is only permitted to make finite sequences of mutations. The cluster combinatorics of the category $D$ are described by triangulations of an $\infty$-gon and we see that these have a natural correspondence with the behaviour of Plucker coordinates in the coordinate ring of a doubly-infinite Grassmannian, and hence the latter is where a concrete realization of these cluster algebra structures may be found. We also give the quantum analogue of these results, generalising work of the first author and Launois. An appendix by Michael Groechenig provides a construction of the coordinate ring of interest here, generalizing the well-known scheme-theoretic constructions for Grassmannians of finite-dimensional vector spaces.