The quantum dilogarithm and representations of quantized cluster varieties

To David Kazhdan for his 60th birthday " Loxadь sostoit iz trh neravnyh polovin ". 4 The quantum dilogarithm and its properties 26 4.1 The quantum logarithm function and its properties. . 1 " A horse consists of three unequal halves ". cf. A. de Barr, Horse doctor. Moscow 1868. Cluster varieties [FG2] are relatives of cluster algebras [FZI]. Cluster modular groups act by automor-phisms of cluster varieties. They admit natural extensions, saturated cluster modular groups, which include classical modular groups of punctured surfaces, also known as the mapping class/Teichmüller groups. Our main result is a construction of series of *-representations of quantized cluster X-varieties. By this we mean the following. A cluster X-variety is a Poisson variety. It admits a canonical non-commutative q-deformation [FG2]. The latter gives rise to a non-commutative *-algebra L(X q), the algebra of regular functions on the q-deformation. We consider its Langlands modular double, a *-algebra L X := L(X q) ⊗ L(X ∨ q ∨). Here q = e iπ and the second factor is the algebra of regular functions on the non-commutative Langlands dual cluster X-variety for q ∨ := e iπ/. We define a Freschet linear space S X , the Schwartz space of L X , and a *-representation of L X in S X. The (saturated) cluster mapping class group Γ of X acts by automorphisms of L X. Using the quantum dilogarithm function we construct explicitly a projective representation of the group Γ in the Schwartz space S X , intertwining the action of Γ on L X. The space S X is a dense subspace of a Hilbert space V X. Assuming |q| = 1, the group Γ acts by unitary operators in V X. Summarizing: A *-representation of quantized cluster X-variety is a triple (This representation is decomposed according to the unitary characters of the center of algebra L X. Cluster modular groups are automorphism groups of objects of cluster modular groupoids. We run the construction of (1) for groupoids rather then groups. The constructed unitary projective representation of the group Γ in the Hilbert space V X can be viewed as a rather sophisticated analog of the Weil representation. In both cases representations are given by integral operators, intertwining representations of certain Heisenberg-type algebras. The kernels of these intertwiners in our case are given by the quantum dilogarithm function. The program of quantization of cluster X-varieties, …