Irreducible decomposition for local representations of quantum Teichmüller space

Abstract

We give an irreducible decomposition of the so-called local representations [BBL07] of the quantum Teichmüller space Tq(Σ) where Σ is a punctured surface of genus g > 0 and q is a N -th root of unity with N odd. As an application, we construct a family of representations of the Kauffman bracket skein algebra of the closed surface Σ.

The quantum trace as a quantum non-abelianization map

“…They are classified, up to isomorphism, by a classical shadow [ρ ab ] ∈ X σ C * ( Σ) and a complex number h C ∈ C * named its central charge. The following corollary was proven by Toulisse in [Tou16] when Σ is a closed surface.…”

On the quantum Teichmüller invariants of fibred cusped 3-manifolds

“…obtained by restriction to the isotypical components ρ λ (µ) of ρ λ , associated to irreducible representations µ of T q λ . Any irreducible representation of T q λ can be embedded as a summand of some local representation ( [13]), and so has a corresponding isotypical component.…”

“…In perspective, another application of Theorem 1.1 would be to study the representations of the Kauffman bracket skein algebras defined by means of the QHI intertwiners, by using the results of [13].…”

“…Now we have: w). Note that we used the relation (13) in the fourth equality. Hence, using (27) we see that (25) holds true for X = X 4 whenever y 0 0 = w −1 1 y 3 2 .…”

“…Once again, H red N (M φ , r, κ) is determined by φ and the isomorphism class of the local representation ρ λ , because it is the trace of H red N (T C φ , b, w). Each isotypical component of ρ λ is the intersection of one eigenspace of each of the so called puncture elements of T q λ (see [7] and [13]). Hence it is determined by the parameters ({x 1 , .…”