Quasi-cluster algebras from non-orientable surfaces

Grégoire Dupont (1) , Frédéric Palesi (2)

1 ESPE Guadeloupe - École supérieure du professorat et de l'éducation - Académie de Guadeloupe
2 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: cluster algebras; triangulation of surfaces; non-orientable surfaces; hyperbolic geometry

Domaines: Anneaux et algèbres [math.RA]; Topologie géométrique [math.GT]

Grégoire Dupont, Frédéric Palesi. Quasi-cluster algebras from non-orientable surfaces. Journal of Algebraic Combinatorics, 2015, pp.10.1007. ⟨10.1007/s10801-015-0586-1⟩. ⟨hal-00821878⟩