Cluster Ensembles, Quantization and the Dilogarithm

Cluster algebras are a remarkable discovery of S. Fomin and A. Zelevinsky [FZI]. Cluster algebras are certain commutative algebras defined by a very simple and general data. Below we consider only cluster algebras of geometric origin, which are quite general and probably the most important examples of cluster algebras. We show that a cluster algebra of geometric origin is part of a richer structure, which we call a cluster ensemble. Roughly speaking a cluster ensemble is a pair (X ,A) of positive spaces equipped with an action of a certain discrete symmetry group Γ. These two spaces are related by a morphism p : A −→ X , which in general, as well as in many interesting examples, is neither injective nor surjective. The space A has a degenerate symplectic structure, and the space X has a Poisson structure. The map p relates the Poisson and degenerate symplectic structures in a natural way. Amazingly the dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. The space A is closely related to the spectrum of the cluster algebra. However it is the space X which in many situations is the most interesting part of the structure. One of our results is quantization of cluster ensembles. We suggest that there exists a remarkable duality between the A and X spaces. One of its manifestations is our package of duality conjectures in Chapter 5. These conjectures assert that the tropical points of the A/X -space parametrise a basis in a certain class of functions on the X/A-space of the Langlands dual cluster ensemble. Below we describe the key ingredients of the cluster ensemble structure in an important special case when there are no so–called tropical variables. Our main example is provided by the (X ,A)