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The quantum dilogarithm and representations of quantum cluster varieties. (English) Zbl 1183.14037
Cluster varieties are relatives of cluster algebras on which cluster modular groups act by automorphisms. Certain extensions of these groups, called saturated modular groups, are used. The program of quantization of cluster $$\mathcal{X}$$-varieties, including a construction of intertwiners, was initiated in [V. V. Fock and A. B. Goncharov, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865–930 (2009; Zbl 1180.53081)]. A cluster $$\mathcal{X}$$-variety is equipped with a natural Poisson structure. One of the main results is a construction of series of $$*$$-representations of quantum cluster $$\mathcal{X}$$-varieties. In the paper under review, the authors underline that in the previously quoted paper some ingredients are missing, including a proof of a crucial relation for intertwiners. The new features of the present paper are a new construction of intertwiners. A Schwartz space $$\mathcal S_{\mathcal X}$$ is introduced. Since the Langlands modular double $$*$$-algebra L$$_{\mathcal X}$$ actually acts in $$\mathcal S_{\mathcal X}$$, the claim that the intertwiners indeed intertwine this action makes sense. It is shown that this implies the relations for the intertwiners. In the quasiclassical limit they give functional equations for the classical dilogarithm. A. B. Goncharev [in: Geometry and dynamics of groups and spaces. In memory of Alexander Reznikov. Partly based on the international conference on geometry and dynamics of groups and spaces in memory of Alexander Reznikov, Bonn, Germany, September 22–29, 2006. Basel: Birkhäuser. Progress in Mathematics 265, 415–428 (2008; Zbl 1139.81055)] developed the simplest example of this program, quantization of the moduli space $$\mathcal M^{\text{cyc}}_{0,5}$$. One of the applications of the construction is quantum higher Teichmüller theory. Let $$\widehat{S}$$ be a surface $$S$$ with holes and a finite collection of marked points at the boundary, considered modulo isotopy. Let $$G$$ be a split reductive group. The pair $$(G, \widehat{S})$$ gives rise to a moduli space $$\mathcal X_{G,\widehat{S}}$$ related to the moduli space of $$G$$-local systems on $$S$$ The modular group $$\Gamma_S$$ of $$S$$ acts on $$\mathcal X_{G,\widehat{S}}$$. The moduli space $$\mathcal X_{G,\widehat{S}},$$ in the case when $$G$$ has connected center, has a natural cluster $$\mathcal X$$-variety structure. The authors’ construction provides a family of infinite-dimensional unitary projective representations of the saturated cluster modular group $$\widehat{\Gamma}_{G,\widehat{S}}$$ related to the pair $$(G,\widehat{S})$$. The group $$\widehat{\Gamma}_{G,\widehat{S}}$$ includes, as a subquotient, the classical modular group $$\Gamma_S$$ of $$S,$$ but can be bigger. To prove relations for the intertwiners, the authors introduce and study a geometric object encapsulating their properties: the symplectic double of a cluster $$\mathcal X$$-variety and its noncommutative $$q$$-deformation, the quantum double.

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B37 Quantum groups (quantized enveloping algebras) and related deformations 53D55 Deformation quantization, star products 33B30 Higher logarithm functions
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