Quasi-cluster algebras from non-orientable surfaces

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.Comment: 38 pages, 14 figure

Modules over cluster-tilted algebras determined by their dimension vectors

We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.Comment: 9 page

Compactifying Exchange Graphs I: Annuli and Tubes

We introduce the notion of an \emph{asymptotic triangulation} of the annulus. We show that asymptotic triangulations can be mutated as the usual triangulations and describe their exchange graph. Viewing asymptotic triangulations as limits of triangulations under the action of the mapping class group, we compactify the exchange graph of the triangulations of the annulus. The cases of tubes are also considered.Comment: 14 page

Quantum frieze patterns in quantum cluster algebras of type A

We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the explicit polynomials arising from the lower bound phenomenon in these quantum cluster algebras.Comment: 10 page

On a category of cluster algebras

We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we describe interactions between this category and the geometry of the surfaces.Comment: 37 page

Mutations de carquois

En 2001, Sergey Fomin et Andrei Zelevinsky ont introduit un procédé combinatoire appelé mutation modifiant localement un carquois, c'est à dire un graphe orienté fi ni. L'application récursive de ce procédé à un carquois donné génère une liste de carquois qui peut être finie ou infinie. Le problème de la classification des carquois donnant une liste fi nie, bien que de nature simple, a demandé plusieurs années de travail avant d'être résolu par Anna Felikson, Michael Shapiro et Pavel Tumarkin en novembre 2008. Dans cet article, nous introduisons de manière élémentaire la notion de mutation et présentons la classification de Felikson, Shapiro et Tumarkin d'un point de vue à la fois mathématique et épistémologique. Cet article fait suite à un exposé donné au Club Mathématique de l'université de Sherbrooke en septembre 2009

A homological interpretation of the transverse quiver Grassmannians

In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second-named author to introduce a variety called transverse quiver Grassmannian and the first-named and third-named authors to consider the smooth loci of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext^1(N,M/N)=0. As a corollary we prove that the transverse quiver Grassmannian coincides with the smooth locus of the irreducible components of minimal dimension in the quiver Grassmannian.Comment: final version, 7 pages, corollary 1.2 has been modifie

Generic cluster characters

Let \CC be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set \mathcal G^T(\CC) of generic values of the cluster character associated to $T$. If \CC has a cluster structure in the sense of Buan-Iyama-Reiten-Scott, \mathcal G^T(\CC) contains the set of cluster monomials of the corresponding cluster algebra. Moreover, these sets coincide if $\mathcal C$ has finitely many indecomposable objects. When \CC is the cluster category of an acyclic quiver and $T$ is the canonical cluster-tilting object, this set coincides with the set of generic variables previously introduced by the author in the context of acyclic cluster algebras. In particular, it allows to construct $\Z$-linear bases in acyclic cluster algebras.Comment: 24 pages. Final Version. In particular, a new section studying an explicit example was adde

Passive leg raising can predict fluid responsiveness in patients placed on venovenous extracorporeal membrane oxygenation

International audienceABSTRACT: INTRODUCTION: In ICUs, fluid administration is frequently used to treat hypovolaemia. Because volume expansion (VE) can worsen acute respiratory distress syndrome (ARDS) and volume overload must be avoided, predictive indicators of fluid responsiveness are needed. The purpose of this study was to determine whether passive leg raising (PLR) can be used to predict fluid responsiveness in patients with ARDS treated with venovenous extracorporeal membrane oxygenation (ECMO). METHODS: We carried out a prospective study in a university hospital surgical ICU. All patients with ARDS treated with venovenous ECMO and exhibiting clinical and laboratory signs of hypovolaemia were enrolled. We measured PLR-induced changes in stroke volume (ΔPLRSV) and cardiac output (ΔPLRCO) using transthoracic echocardiography. We also assessed PLR-induced changes in ECMO pump flow (ΔPLRPO) and PLR-induced changes in ECMO pulse pressure (ΔPLRPP) as predictors of fluid responsiveness. Responders were defined by an increase in stroke volume (SV) > 15% after VE. RESULTS: Twenty-five measurements were obtained from seventeen patients. In 52% of the measurements (n = 13), SV increased by > 15% after VE (responders). The patients' clinical characteristics appeared to be similar between responders and nonresponders. In the responder group, PLR significantly increased SV, cardiac output and pump flow (P 10% ΔPLRSV may predict fluid responsiveness. ΔPLRPP and ΔPLRPO cannot predict fluid responsiveness