Quasi-cluster algebras from non-orientable surfaces

TL;DR

This paper introduces quasi-cluster algebras from non-orientable surfaces, classifies those with finitely many variables, and links their solutions to type A cluster variables, expanding the theory of cluster algebras.

Contribution

It defines quasi-cluster algebras for non-orientable surfaces, classifies finite cases, and connects their solutions to integrable systems and type A cluster variables.

Findings

Quasi-cluster algebras generalize cluster algebras to non-orientable surfaces.

Finite quasi-cluster algebras have a basis formed by quasi-cluster monomials.

Solutions to associated integrable systems relate to type A cluster variables.

Abstract

With any non necessarily orientable unpunctured marked surface (S,M) we associate a commutative algebra, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the set of arcs and one-sided simple closed curves in (S,M). Quasi-cluster variables are naturally gathered into possibly overlapping sets of fixed cardinality, called quasi-clusters, corresponding to maximal non-intersecting families of arcs and one-sided simple closed curves in (S,M). If the surface S is orientable, then the quasi-cluster algebra is the cluster algebra associated with the marked surface (S,M) in the sense of Fomin, Shapiro and Thurston. We classify quasi-cluster algebras with finitely many quasi-cluster variables and prove that for these quasi-cluster algebras, quasi-cluster monomials form a linear basis. Finally, we attach to (S,M) a family of discrete integrable systems satisfied by quasi-cluster variables associated to arcs in the quasi-cluster algebra and we prove that solutions of these systems can be expressed in terms of cluster variables of type A.