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Quantum Teichmüller spaces and Kashaev’s \(6j\)-symbols. (English) Zbl 1131.17005

This paper establishes an intriguing connection between two seemingly different objects, Kashaev’s \(6j\) symbols and the quantum Teichmüller space. In general, \(6j\) symbols are entries of a transition matrix between two bases in a \(\operatorname{Hom}\) space of a representation category. They satisfy just the algebraic properties (tetrahedron, pentagon, unitarity) that allow one to define topological invariants of 3-manifolds by state summing them over triangulations. In Kashaev’s case one considers representations of the Weyl algebra, a Borel subalgebra of \(U_q(\mathfrak{sl}_2\mathbb{C})\) at \(q\) a root of unity.
On the other hand, the quantum Teichmüller space of a surface is a \(q\)-deformation of the algebra of rational functions on the classical Teichmüller space. Bonahon-Liu used it to construct quantum invariants of surface diffeomorphisms. It turned out that expressions for Kashaev’s and Bonahon-Liu invariants share many similarities and the main goal of this paper is to explain them.
The author notices that the representation theory of the Weyl algebra is very similar to that of the triangle algebra appearing in the construction of the quantum Teichmüller space. The latter is defined using ideal triangulations of a surface and naturally embeds into the tensor product of triangle algebras of their elements. Geometric invariance of the definition is realized via intertwiners between algebras defined from different triangulations. The author proves that these intertwiners for the closed disk with four punctures are essentially scalar multiples of Kashaev’s \(6j\) symbols generalizing a previous observation of Kashaev and Baseilhac.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
32G81 Applications of deformations of analytic structures to the sciences
57R56 Topological quantum field theories (aspects of differential topology)
53D50 Geometric quantization
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References:

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