Cluster ensembles, quantization and the dilogarithm

Abstract

A. -A cluster ensemble is a pair (X , A) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ. The space A is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p : A −→ X . The space A is equipped with a closed 2-form, possibly degenerate, and the space X has a Poisson structure. The map p is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. We define a non-commutative q-deformation of the X -space. When q is a root of unity the algebra of functions on the q-deformed X -space has a large center, which includes the algebra of functions on the original X -space.The main example is provided by the pair of moduli spaces assigned in [6] to a topological surface S with a finite set of points at the boundary and a split semisimple algebraic group G. It is an algebraicgeometric avatar of higher Teichmüller theory on S related to G.We suggest that there exists a duality between the A and X spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.R. -Un ensemble amassé est une paire (X , A) d'espaces positifs (i.e. de variétés munies d'un atlas positif) munis de l'action d'un groupe discret. L'espace A est relié au spectre d'une algèbre amassée [12]. Les deux espaces sont liés par un morphisme p : A −→ X . L'espace A est muni d'une 2-forme fermée, éventuellement dégénérée, et l'espace X est muni d'une structure de Poisson. L'application p est compatible avec ces structures. Le dilogarithme avec ses avatars motiviques et quantiques joue un rôle fondamental dans la structure d'un ensemble amassé. Nous définissons une déformation non-commutative de l'espace X . Nous montrons que, dans le cas où le paramètre de la déformation q est une racine de l'unité, l'algèbre déformée a un centre qui contient l'algèbre des fonctions sur l'espace X originel.Notre exemple principal est celui de l'espace des modules associé dans [6] à une surface topologique S munie d'un nombre fini de points distingués sur le bord et à un groupe algébrique semi-simple G. C'est un avatar algébro-géométrique de la théorie de Teichmüller d'ordre supérieur sur la surface S à valeurs dans G.

Moduli spaces of convex projective structures on surfaces

“…Based on the FominZelevinsky classification theorem [9], we showed in [6] that cluster X -spaces of finite type are also parametrised by Dynkin diagrams of the Cartan-Killing type. In this paper, we defined cluster X -spaces under a simplifying assumption that the matrix ε ij is skew-symmetric.…”

“…To introduce the action of the Thompson group on the non-commutative field Frac(T q 3 ), let us recall some facts about the quantum cluster ensembles from [6]. …”

“…We introduce and study the universal higher Teichmüller space T 3 which contains the moduli spaces T + 3 (S) for all surfaces S. The Thompson group T acts by its automorphisms. We show that the universal higher Teichmüller space T 3 is the set of R >0 -points of certain infinite dimensional cluster X -variety, as defined in [6]. Moreover, the Thompson group is a subgroup of the mapping class group T 3 of this cluster variety.…”

Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

“…An important class of cluster algebras is given by cluster algebras of surface type [9][10][11][12]16]. From a classification point of view, this class is very important, since it has been shown in [6] that almost all (skew-symmetric) mutation finite cluster algebras are of surface type.…”

“…10 In each case, the vertex x of the tile G s−1 in G 3 is matched in ϕ 3 (P 56 ) by an edge in G s−1 (namely e(x) in case 1 and the edge a in the other cases). This implies that (1) the completed part ϕ 3 (P 56 ) ∩ G 3 (s, s − 1) is the same in each case, (2) ϕ 4 (P 56 ) does not depend on P 56 , (3) the matching ϕ 4 (P 56 ) does not contain the edge a.…”