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For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors construct a geometric realization in terms of suitable decorated Teichmüller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. The authors' model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from the authors' previous paper, removing unnecessary assumptions on the surface.

Cluster algebras.

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Cluster algebras. --- Poisson algebras.

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This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on mathcal{G} corresponds to a cluster structure in mathcal{O}(mathcal{G}). The authors have shown before that this conjecture holds for any mathcal{G} in the case of the standard Poisson-Lie structure and for all Belavin-Drinfeld classes in SL_n, n.

Cluster algebras. --- Quantum groups. --- Poisson algebras. --- Representations of Lie algebras. --- Lie algebras.

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Cluster algebras --- Quantum groups. --- Poisson algebras --- Representations of Lie algebras. --- Lie algebras. --- Algèbres amassées --- Groupes quantiques --- Algèbres de Poisson --- Représentations des algèbres de Lie --- Algèbres de Lie --- Quantum groups --- Representations of Lie algebras --- Lie algebras --- Poisson, Algèbres de --- Représentations d'algèbres de Lie --- Lie, algèbres de --- Algèbres amassées --- Algèbres de Poisson --- Représentations des algèbres de Lie --- Algèbres de Lie --- Groupes quantiques. --- Poisson, Algèbres de. --- Représentations d'algèbres de Lie. --- Lie, Algèbres de.