Quasi-cluster algebras from non-orientable surfaces

Dupont, Grégoire ; Palesi, Frédéric

Journal of Algebraic Combinatorics, Tome 42 (2015) no. 2, pp. 429-472.

Résumé

With any non-necessarily orientable unpunctured marked surface $(\mathbf{S,M})$, we associate a commutative algebra $\mathcal A_{(\mathbf{S,M})}$, 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 $(\mathbf{S,M})$. 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 $(\mathbf{S,M})$. If the surface $\mathbf S$ is orientable, then $\mathcal A_{(\mathbf{S,M})}$ is the cluster algebra associated with the marked surface $(\mathbf{S,M})$ in the sense of S. Fomin et al. [Acta Math. 201, No. 1, 83--146 (2008; Zbl 1263.13023)]. 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 $(\mathbf{S,M})$ a family of discrete integrable systems satisfied by quasi-cluster variables associated to arcs in $\mathcal A_{(\mathbf{S,M})}$ and we prove that solutions of these systems can be expressed in terms of cluster variables of type $A$.

Détail

Classification : 13F60, 16G70, 53A35
Keywords: cluster algebra, triangulations, hyperbolic geometry, non-orientable surfaces