• pseudo-Anosov diffeomorphisms
• intertwining operators
• local representations
• quantum Teichmuller space

The quantum Teichmüller space is an algebraic object associated with a punctured surface admitting an ideal triangulation. Two somewhat different versions of it have been introduced, as a quantization by deformation of the Teichmüller space of a surface, independently by Chekhov and Fock and by Kashaev. As in the article [1], we follow the exponential version of the Chekhov-Fock approach, whose setting has been established in [2]. In this way, the study is focused on non-commutative algebras and their finite-dimensional representations, instead of Lie algebras and self-adjoint operators on Hilbert spaces, as in [3] and [4].

Given $S$ a surface admitting an ideal triangulation $\lambda$, we can produce a non-commutative $\C$-algebra $\mathcal{T}^q_\lambda$, generated by variables $X_i^{\pm 1}$ corresponding to the edges of $\lambda$ and endowed with relations $X_i X_j = q^{2 \sigma_{i j}} X_j X_i$, where $\sigma_{i j}$ is an integer number, depending on the mutual position of the edges $\lambda_i$ and $\lambda_j$ in $\lambda$, and $q \in \C^*$ is a complex number. The algebra $\mathcal{T}^q_\lambda$ is called the Chekhov-Fock algebra associated with the surface $S$ and the ideal triangulation $\lambda$. Varying $\lambda$ in the set $\Lambda(S)$ of all the ideal triangulations of $S$, we obtain a collection of algebras, whose fraction rings $\widehat{\mathcal{T}}^q_\lambda$ are related by isomorphisms $\mappa{\Phi^q_{\lambda \lambda'}}{\widehat{\mathcal{T}}^q_{\lambda'}}{\widehat{\mathcal{T}}^q_\lambda}$. This structure allows us to consider an object realized by "gluing" all the $\widehat{\mathcal{T}}^q_\lambda$ through the maps $\Phi^q_{\lambda \lambda'}$. The result of this procedure is an intrinsic algebraic object, called the quantum Teichmüller space of $S$ and denoted by $\mathcal{T}^q_S$, which does not depend on the chosen ideal triangulation any more. The explicit expressions of the $\Phi_{\lambda \lambda'}^q$ reveal the geometric essence of this algebraic object. These isomorphisms are designed in order to be a non-commutative generalization of the coordinate changes on the ring of rational functions on the classical Teichmüller space $\mathcal{T}(S)$ of a surface $S$ (here $\mathcal{T}(S)$ denotes the space of isotopy classes of complete hyperbolic metrics on $S$).

The main purpose of this thesis is the study of the quantum Teichmüller space and the investigation of its finite-dimensional representations. A necessary condition for the existence of a finite-dimensional representation of any Chekhov-Fock algebra $\mathcal{T}^q_\lambda$ is that $q^2$ is a root of unity, hence we always assume that $q^2$ is a primitive $N$-th root of unity, for a certain $N \in \N$. Firstly, we focus on the Chekhov-Fock algebras $\mathcal{T}^q_\lambda$ and the classification of their local and irreducible representations. Then, following [1], we give a notion of finite-dimensional representation of the quantum Teichmüller space, defined as a collection
\[
\rho = \set{\rho_\lambda \vcentcolon \mathcal{T}^q_\lambda \rightarrow \End(V_\lambda) }_{\lambda \in \Lambda(S)}
\]
of representations of all the Chekhov-Fock algebras associated with the surface $S$, such that $\rho_\lambda \circ \Phi_{\lambda \lambda'}^q$ is isomorphic to $\rho_{\lambda'}'$ for every $\lambda, \lambda' \in \Lambda(S)$. Local (and irreducible) representations of the quantum Teichmüller space turn out to be classified by conjugacy classes of homomorphisms from the fundamental group of the surface $S$ to the group of orientation preserving isometries of the $3$-dimensional hyperbolic space, together with some additional data.

In the last part, we study certain linear isomorphisms through which local representations are connected. Given $\rho, \rho'$ two isomorphic local representations of the quantum Teichmüller space of $S$, for every $\lambda, \lambda' \in \Lambda(S)$ the representations $\rho_\lambda \circ \Phi_{\lambda \lambda'}^q$ and $\rho_{\lambda'}'$ are isomorphic through a linear isomorphism $L^{\rho \rho'}_{\lambda \lambda'}$. Such a $L^{\rho \rho'}_{\lambda \lambda'}$ is called an intertwining operator. One of the main purposes of [4] was to select a unique intertwining operator $L^{\rho \rho'}_{\lambda \lambda'}$ for every choice of $\rho, \rho'$ local representations and $\lambda, \lambda'$ ideal triangulations, requiring that the whole system of operators (for varying $\lambda, \lambda' \in \Lambda(S)$, the representations $\rho, \rho'$ and the surface $S$) verifies some natural Fusion and Composition properties, concerning their behaviour with respect to the fusion of representations and changing of triangulations. However, in our investigation of the ideas exposed in [5], we have found a difficulty that compromises the original statement [5,Theorem 20], in particular the possibility to select a unique intertwining operator for every choice of $\rho, \rho', \lambda, \lambda'$. We prove that the best that can be done is the selection of sets of intertwining operators $\mathscr{L}^{\rho \rho'}_{\lambda \lambda'}$, instead of a single linear isomorphism. Each set $\mathscr{L}^{\rho \rho'}_{\lambda \lambda'}$ is endowed with a natural free and transitive action of $H_1(S;\Z_N)$, so its cardinality is always finite, but it goes to $\infty$ by increasing the complexity of the surface $S$ and the number $N \in \N$. In conclusion, we reformulate the theory of invariants for pseudo-Anosov diffeomorphisms developed in [5] in light of these facts.

We also expose an explicit calculation of an intertwining operator when $\lambda$ and $\lambda'$ differ by diagonal exchange and $S$ is an ideal square, which is basically the elementary block needed to express a generic intertwining operator.

[1] Bonahon, Francis, and Xiaobo Liu. "Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms." Geometry & Topology 11.2 (2007): 889-937.
[2] Liu, Xiaobo. "The quantum Teichmüller space as a noncommutative algebraic object." Journal of Knot Theory and its Ramifications 18.05 (2009): 705-726.
[3] Fock, Vladimir V., and Leonid O. Chekhov. "A quantum Teichmüller space." Theoretical and Mathematical Physics 120.3 (1999): 1245-1259.
[4] Kashaev, Rinat M. "A link invariant from quantum dilogarithm." Modern Physics Letters A 10.19 (1995): 1409-1418.
[5] Bai, Hua, Francis Bonahon, and Xiaobo Liu. "Local representations of the quantum Teichmuller space." arXiv preprint arXiv:0707.2151 (2007).