Ind-cluster algebras and infinite Grassmannians

Abstract

A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Pl\"ucker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Pl\"ucker coordinates via mutations. Fioresi and Hacon studied a specific colimit of the coordinate rings of finite Grassmannians and its link with the infinite Grassmannian introduced by Sato and independently by Segal and Wilson in connection with the Kadomtsev-Petiashvili (KP) hierarchy, an infinite set of nonlinear partial differential equations which possess soliton solutions. In this article we prove that this ring is a cluster algebra of infinite rank with the structure induced by the colimit construction. More generally, we prove that cluster algebras of infinite rank are precisely the ind-objects of a natural category of cluster algebras.