Cluster ensembles, quantization and the dilogarithm

Details Cluster ensembles, quantization and the dilogarithm Cluster ensembles, quantization and the dilogarithm

2003-11-14

arxiv.org/abs/math/0311245v7

Mathematics

Algebraic Geometry

Version 7: Final version. To appear in Ann. Sci. Ecole Normale. Sup. New material in Section 5. 58 pages, 11 pictures

Scientific paper

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.

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