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Cluster ensembles, quantization and the dilogarithm. (English) Zbl 1180.53081
Cluster ensemble is a pair of positive spaces $$(X, A)$$ related by a map $$p: A \to X$$. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the $$A$$-space. The authors develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. The authors define a $$q$$-deformation of the $$X$$-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. The authors support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmüller theory, that is by the pair of moduli spaces corresponding to a split reductive group $$G$$ and a surface $$S$$ defined in [V. V. Fock and A. B. Goncharov, Handbook of Teichmüller theory. Volume I. Zürich: European Mathematical Society (EMS). IRMA Lectures in Mathematics and Theoretical Physics 11, 647–684 (2007; Zbl 1162.32009)]. The authors suggest that cluster ensembles provide a natural framework for higher quantum Teichmüller theory.

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 14T05 Tropical geometry (MSC2010) 53D05 Symplectic manifolds, general 53D55 Deformation quantization, star products 53D30 Symplectic structures of moduli spaces
##### Keywords:
Cluster ensemble; dilogarithm; quantization; Poisson structure
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