Intertwining operators of the quantum Teichmüller space

• Published in: arXiv.org 2016-10
• Main Author: Mazzoli, Filippo
• Format: Article
• Language: English
• Subjects: Composition; Isomorphism; Mapping; Multiplication; Operators; Properties (attributes); Representations; Toruses
• ISSN: 2331-8422

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.