Intertwining operators of the quantum Teichm\"uller space

In arXiv:0707.2151 the authors introduced the theory of local representations of the quantum Teichm\"uller space $\mathcal{T}^q_S$ ($q$ being a fixed primitive $N$-th root of $(-1)^{N + 1}$) and they studied the behaviour of the intertwining operators in this theory. One of the main results [Theorem 20, arXiv:0707.2151] was the possibility to select one distinguished operator (up to scalar multiplication) for every choice of a surface $S$, ideal triangulations $\lambda, \lambda'$ and isomorphic local representations $\rho, \rho'$, requiring that the whole family of operators verifies certain Fusion and Composition properties. By analyzing the constructions of arXiv:0707.2151, we found a difficulty that we eventually fix by a slightly weaker (but actually optimal) selection procedure. In fact, for every choice of a surface $S$, ideal triangulations $\lambda, \lambda'$ and isomorphic local representations $\rho, \rho'$, we select a finite set of intertwining operators, naturally endowed with a structure of affine space over $H_1(S;\mathbb{Z}_N)$ ($\mathbb{Z}_N$ is the cyclic group of order $N$), in such a way that the whole family of operators verifies augmented Fusion and Composition properties, which incorporate the explicit behavior of the $\mathbb{Z}_N$-actions with respect to such properties. Moreover, this family is minimal among the collections of operators verifying the "weak" Fusion and Composition rules (in practice the ones considered in arXiv:0707.2151). In addition, we adapt the derivation of the invariants for pseudo-Anosov diffeomorphisms and their hyperbolic mapping tori made in arXiv:0707.2151 and arXiv:math/0407086 by using our distinguished family of intertwining operators.