Quantization of the Teichmüller space of a non-compact Riemann surface has emerged in 1980s as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston’s shear coordinate functions on the edges form a coordinate system for the Teichmüller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic–analytic theory of the Shale–Weil intertwiners for the Schrödinger representations, as well as Faddeev–Kashaev’s quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster $\mathcal{X}$-varieties.

非紧致黎曼表面的Teichmüller空间的量化是1980年代出现的一种用于三维量子引力的方法。对于表面的理想三角剖分的任何选择，边缘上的Thurston剪切坐标函数形成Teichmüller空间的坐标系，应在希尔伯特空间上用合适的自伴算子代替它们。改变三角剖分后，必须在希尔伯特空间之间构造一个operator算子，该ing算子将量子坐标算子交织在一起，并满足高达乘法相常数的组成恒等式。在契kh夫（Chekhov），福克（Fock）和贡恰洛夫（Goncharov）的著名结构中，量子坐标算符形成了一系列可约化的表示形式，并且相位常数都是不重要的。在本文中，我们采用Shale-Weil孪生子的谐解析理论进行Schrödinger表示，以及Faddeev-Kashaev的量子对数函数，以构造一类不可约的量子剪切坐标函数表示，以及相应的交织器来改变三角剖分。相位常数是由辛矢量空间的Lagrangian子空间的Maslov索引以及三角剖分的五边形关系明确计算和描述的。目前的工作可以推广到集群 构造一族不可逆的量子剪切坐标函数表示法，以及相应的交织对三角剖分的影响。相位常数是由辛矢量空间的Lagrangian子空间的Maslov索引以及三角剖分的五边形关系明确计算和描述的。目前的工作可以推广到集群 构造一族不可逆的量子剪切坐标函数表示法，以及相应的交织对三角剖分的影响。相位常数是由辛矢量空间的Lagrangian子空间的Maslov索引以及三角剖分的五边形关系明确计算和描述的。目前的工作可以推广到集群$\mathcal{X}$-品种。