2025/01/04 by Ishibashi, Tsukasa
#FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2501.02316
Based on the pioneering ideas of Kashaev [Kas98,Kas00], we present a fully explicit construction of a finite-dimensional projective representation of the dotted Ptolemy groupoid when the quantum parameter q is a root of unity, which reproduces the central charge of the SU(2) Wess--Zumino--Witten model. A basic ingredient is the cyclic quantum dilogarithm [FK94]. A notable contribution of this work is a reinterpretation of the relations among the parameters in the cyclic quantum dilogarithm to ensure its pentagon identity in terms of the mutations of coefficients. In particular, we elucidate the dual roles of these parameters: as coefficients in quantum cluster algebras and as the central characters of quantum cluster variables. We introduce the quantum intertwiner associated with a mapping class as a composite of cyclic quantum dilogarithm operators, whose trace defines a quantum invariant. We prove that it gives an intertwiner of local representations of quantum Teichmüller space in the sense of Bai--Bonahon--Liu [BBL07], and also coincides with the transpose of the reduced quantum hyperbolic operator of Baseilhac--Benedetti [BB18]. We provide a geometric method to decompose the space of quantum states into irreducible modules of the Chekhov--Fock algebra. The reduced version of quantum intertwiner conjecturally coincides with the Bonahon--Liu intertwiner [BL07].