Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil-Petersson form for one of these spaces. It is related to the motivic dilogarithm.

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.

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex.) Our aim is however to study the Poisson structure on the moduli space of locally flat vector bundles on a Riemann surface with holes (i.e. with boundary). It is shown that this moduli space can be obtained as a quotient of the space of graph connections by the Poisson action of a lattice gauge group endowed with a Poisson-Lie structure. The present paper contains as a part an updated version of a 1992 preprint [11] which we decided still deserves publishing. We have removed some obsolete inessential remarks and added some newer ones.

We describe two spaces related to Riemann surfaces -the Teichmüller space of decorated surfaces and the Teichmüller space of surfaces with holes. We introduce simple explicit coordinates on them. Using these coordinates we demonstrate the relation of these spaces to the spaces of measured laminations, compute Weil-Petersson forms, mapping class group action and study properties of lamination length function. Finally we use the developed technique to construct a noncommutative deformation of the space of functions on the Teichmüller spaces and define a class of unitary projective mapping class group representations (conjecturally a modular functor). One can interpret the latter construction as quantisation of 3D or 2D Liouville gravity. Some theorems concerning Markov numbers as well as Virasoro orbits are given as a by-product.

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