Cluster ensembles, quantization and the dilogarithm. (English) Zbl 1180.53081

Fock, Vladimir V.; Goncharov, Alexander B.

Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865-930 (2009).

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.

Reviewer: Benjamin Cahen (Metz)

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
14T05	Tropical geometry (MSC2010)
53D05	Symplectic manifolds (general theory)
53D55	Deformation quantization, star products
53D30	Symplectic structures of moduli spaces

Keywords:

Cluster ensemble; dilogarithm; quantization; Poisson structure

Citations:

Zbl 1162.32009