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 Boris Beaude EspacesTemps.net , 2003, Abstract: Où et quand est le carnaval, populaire et subversif ? Plus qu'un moment, le carnaval est devenu un lieu. Rio, Venise, ou Nice sont le carnaval. Moment privilégié lors duquel l'anonymat fut prétexte à toutes les extravagances, le carnaval est aujourd'hui un moment de divertissement parmi d'autres. Le temps est passé, la mairie a remplacé l'église et le carnaval n'a gardé que ses apparats. Si la modernité en a terminé avec le carnaval et les bals masqués, la ...
 Mathematics , 2012, Abstract: Motivated by the representation theory of quivers with potentials introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of Dynkin quivers, we associate a Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra. By definition, the Caldero-Chapoton algebra is (as a vector space) generated by the Caldero-Chapoton functions of the decorated representations of the basic algebra. The Caldero-Chapoton algebra associated to the Jacobian algebra of a quiver with potential is closely related to the cluster algebra and the upper cluster algebra of the quiver. The set of generic Caldero-Chapoton functions, which conjecturally forms a basis of the Caldero-Chapoton algebra) is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra and was introduced by Geiss, Leclerc and Schr\"oer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. Thanks to the decomposition theorem, all generic Caldero-Chapoton functions can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton algebras lead to several general conjectures on cluster algebras.
 Dylan Rupel Mathematics , 2010, DOI: 10.1093/imrn/rnq192 Abstract: Let $Q$ be any invertible valued quiver without oriented cycles. We study connections between the category of valued representations of $Q$ and expansions of cluster variables in terms of the initial cluster in quantum cluster algebras. We show that an analogue of the Caldero-Chapoton formula holds for all quantum cluster algebras of finite type and for any cluster variable in an almost acyclic cluster.
 Mathematics , 2010, Abstract: We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all objects of the triangulated category, but we show that it is a (weak) cluster map in the sense of Buan-Iyama-Reiten-Scott. As a corollary, it induces a surjection from the set of exceptional objects which can be reached from the cluster tilting subcategory to the set of cluster variables of an associated cluster algebra. Along the way, we study the interaction between Calabi-Yau reduction, cluster structures, and the Caldero-Chapoton map. We apply our results to the cluster category D of Dynkin type A infinity which has a rich supply of cluster tilting subcategories with infinitely many indecomposable objects. We show an example of a cluster map which cannot be extended to all of D. The case of D also permits us to illuminate results by Assem-Reutenauer-Smith on SL_2-tilings of the plane.
 G. Dupont Mathematics , 2007, Abstract: Buan, Marsh and Reiten proved that if a cluster-tilting object $T$ in a cluster category $\mathcal C$ associated to an acyclic quiver $Q$ satisfies certain conditions with respect to the exchange pairs in $\mathcal C$, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to $Q$ has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of $T$. In this paper, we give an alternative proof of this result using the Caldero-Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.
 Mathematics , 2012, DOI: 10.1016/j.jalgebra.2013.10018 Abstract: The Caldero-Chapoton formula relates for hereditary algebras of Dynkin type the cluster characters of the end terms of an Auslander-Reiten sequence with the cluster character of the middle term. We extend this result to generalized cluster categories with cluster tilting object by considering Auslander-Reiten triangles.
 Dorothée Chouitem Amerika : Mémoires, Identités, Territoires , 2010, DOI: 10.4000/amerika.729 Abstract: En période de répression, la société est amenée à réinventer ses codes de communication. L'Uruguay (1973 - 1985) a vu se transformer le carnaval en véritable voix bruyante de la conscience collective Davis (1979). En effet, ce lieu d'expression populaire par excellence n'est pas à considérer comme une simple soupape de s reté qui détournerait l'attention de la réalité sociale mais plut t comme un moyen par lequel une communauté perpétue certaines de ses valeurs, par lequel aussi elle peut contester un ordre politique Davis (1979). Ainsi, le carnaval s'est transcendé pour donner naissance à un espace tiers qui abritera en son sein l'un des vecteurs de la liberté de parole. Et, la murga va muter, devenir transgressive et subversive puis finira par être revendiquée comme l'un des éléments identificatoires d'une société en rupture. Nous nous proposons de présenter et commenter ce teatro de los tablados qui, sous étroite surveillance, va devenir le porte-parole d'une vérité autre en s'affranchissant de la censure. Nous analyserons comment la situation politique va amener certains paroliers à réinvestir et à travestir le genre pour parler au spectateur et lui donner la possibilité de manifester une forme d’engagement. During times of repression, society is made to develop new codes of communication. From 1973 to 1985, Uruguay has seen carnivals become true “representatives for collective conscience” Davies (1979). In fact, this highly regarded form of popular expression ought not to be considered as a simple “safety-valve” that would divert attention from the realities of society. It should be regarded instead as a “means through which a community perpetuates certain traditions, by which political orders may be challenged.” Davies (1979) Thus, the transcending of carnivals gives birth to a vector for freedom of speech. Moreover, the murga will evolve, becoming transgressive and subversive, before being claimed an identifying element in a collapsing society. We intend to introduce and comment on this teatro de los tablados which, under strict surveillance, will represent a new truth once censorship is abolished. We will analyse how the political situation will cause librettists to review and alter their style to convey a clear message to their audience and to give them the possibility to become involved.
 Thomas A. Fisher Mathematics , 2015, Abstract: Frieze patterns of integers were studied by Conway and Coxeter. Let $\mathscr{C}$ be the cluster category of Dynkin type $A_n$. Indecomposables in $\mathscr{C}$ correspond to diagonals in an $(n+3)$-gon. Work done by Caldero and Chapoton showed that the Caldero-Chapoton map (which is a map dependent on a fixed object $R$ of a category, and which goes from the set of objects of that category to $\mathbb{Z}$), when applied to the objects of $\mathscr{C}$ can recover these friezes. This happens precisely when $R$ corresponds to a triangulation of the $(n+3)$-gon. Later work by authors such as Bessenrodt, Holm, Jorgensen and Rubey generalised this connection with friezes further, now to $d$-angulations of the $(n+3)$-gon with $R$ basic and rigid. In this paper, we extend these generalisations further still, to the case where the object $R$ corresponds to a general Ptolemy diagram, i.e. $R$ is basic and $\textrm{add}(R)$ is the most general possible torsion class.
 Mathematics , 2013, Abstract: The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps "reachable" indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a "categorification" of the cluster algebra. The definition of the Caldero-Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster tilting object in the category. We study a modified version of the Caldero-Chapoton map which only requires the category to have a Serre functor, and only depends on a rigid object in the category. It is well-known that the usual Caldero-Chapoton map gives rise to so-called friezes, for instance Conway-Coxeter friezes. We show that the modified Caldero-Chapoton map gives rise to what we call generalised friezes, and that for cluster categories of Dynkin type A, it recovers the generalised friezes introduced by combinatorial means by Bessenrodt and us.
 Reyes Morris Victor Revista Colombiana de Sociología , 2011, Abstract: En este artículo, al retomar el viejo concepto de Anomia, el autor pretende hacer un aporte a su desarrollo teórico a través de la introducción de los conceptos sucedáneos de “espacio anómico” y “tiempo anómico”. Los ilustra, a la manera de émile Durkheim en La División del Trabajo Social, con tres ejemplos para cada uno de estos conceptos. Precisamente en este artículo se hacen algunas nuevas reflexiones acerca de esa “ilustración” del tiempo anómico con respecto al Carnaval de Barranquilla, fiesta emblemática de la Costa Caribe colombiana.
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