Intertwining operators of the quantum Teichmüller space

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Intertwining operators of the quantum Teichmüller space

Filippo Mazzoli October 11, 2016

Abstract

In [BBL07] the authors introduced the theory oflocal representations of the quantum Teichmüller spaceT_S^q(qbeing a fixed primitiveN-th root of(−1)^N+1) and they studied the behaviour of the intertwining operators in this theory. One of the main results [BBL07, Theorem 20] was the possibility to select one distinguished operator (up to scalar multiplication) for every choice of a surfaceS, ideal triangulationsλ, λ⁰ and isomorphic local representations ρ, ρ⁰, requiring that the whole family of operators verifies certain Fusion and Composition properties. This selection was also used to produce invariants for pseudo-Anosov diffeomorphisms and their hyperbolic mapping tori (extending to local representations what had been done in [BL07] for irreducible ones). However, by analyzing the constructions of [BBL07], we found a difficulty that we eventually fix by a slightly weaker (but actually optimal) selection procedure. In fact, for every choice of a surfaceS, ideal triangulationsλ, λ⁰ and isomorphic local representationsρ, ρ⁰, we select afinite set of intertwining operators, naturally endowed with a structure of affine space overH1(S;ZN) (ZN

is the cyclic group of orderN), in such a way that the whole family of operators verifiesaugmented Fusion and Composition properties, which incorporate the explicit behavior of theZ^N-actions with respect to such properties. Moreover, this family isminimal among the collections of operators verifying the "weak" Fusion and Composition rules (in practice the ones considered in [BBL07]). In addition, we adapt the derivation of the invariants for pseudo-Anosov diffeomorphisms and their hyperbolic mapping tori made in [BBL07] and [BL07] by using our distinguished family of intertwining operators.

Introduction

The quantum Teichmüller space is an algebraic object associated with a punc- tured 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 [CF99] and by Kashaev [Kas95]. As in the articles [BL07] and [BBL07], we follow the exponential version of the Chekhov-Fock approach, whose setting has been established in [Liu09].

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 [CF99] and [Kas95].

Given S a surface admitting an ideal triangulation λ, we can produce a non-commutative C-algebra T_λ^q generated by variables X_i^±1 corresponding to the edges of λand endowed with relations XiXj =q^2σ^ijXjXi, whereσij is an integer number, depending on the mutual position of the edges λi andλj in λ, and q ∈C^∗ is a complex number. The algebraT_λ^q is called theChekhov-Fock algebra associated with the surface S and the ideal triangulation λ. Varying λin the set Λ(S)of all the ideal triangulations of S, we obtain a collection of algebras, whose fraction ringsTb_λ^q are related by isomorphismsΦ^q_λλ0:Tb_λ^q0 →Tb_λ^q. This structure allows us to consider an object realized by "gluing" all the Tb_λ^q

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through the maps Φ^q_λλ0. The result of this procedure is an intrinsic algebraic object, called the quantum Teichmüller space of S and denoted by T_S^q, which does not depend on the chosen ideal triangulation, but just on S and on the N-th rootq.

In [BL07] the authors have given a suitable notion of finite-dimensional representations of the quantum Teichmüller space and have studied its irreducible representations. A representation of the quantum Teichmüller space is a collection

ρ={ρλ:T_λ^q −→End(Vλ)}_λ∈Λ(S)

of representations of all the Chekhov-Fock algebras associated with the surface S, verifying a compatibility condition in terms of the isomorphismsΦ^q_λλ0. More precisely, two representations ρ_λ: T_λ^q → End(V_λ) and ρ_λ0: T_λ^q0 → End(V_λ0) are compatible if ρ_λ◦Φ^q_λλ0 makes sense and it is isomorphic to ρ_λ⁰. A necessary condition for the existence of a finite-dimensional representation of any Chekhov-Fock algebra is thatq²is a root of unity, hence we always assume that qis a primitive N-th root of (−1)^N⁺¹.

Later, Bai, Bonahon, and Liu [BBL07] introduced a new type of representations of the quantum Teichmüller space ofS, calledlocal representations, which are constructed as "fusion" of irreducible representations of the Chekhov-Fock algebras associated with the triangles composing an ideal triangulationλ. These representations have a simpler set of invariants than the irreducible ones and they can be glued in order to construct local representations of surfaces obtained by identifying couples of edges in (S, λ). Even in this case, the authors of [BBL07] have found a good notion of local representations of the quantum Teichmüller space, defined as collections of compatible local representations of the Chekhov-Fock algebras. In addition, they have developed a theory of intertwining operators between couples of isomorphic local representations of the quantum Teichmüller space.

More precisely, let

ρ={ρ_λ:T_λ^q →End(V_λ)}_λ∈Λ(S) ρ⁰={ρ⁰_λ:T_λ^q→End(V_λ⁰)}_λ∈Λ(S) be two isomorphic local representations of the quantum Teichmüller spaceT_S^q. By definition, for every λ, λ⁰ ∈Λ(S), the representations ρ_λ◦Φ^q_λλ0 andρ_λ⁰ are isomorphic and ρ_λ⁰ itself is isomorphic to ρ⁰_λ0. Therefore, there exists a linear isomorphism L^ρρ_λλ⁰0:V_λ⁰0 →Vλ such that

(ρλ◦Φ^q_λλ0)(X⁰) =L^ρρ_λλ⁰0◦ρ_λ⁰0(X⁰)◦(L_λλ^ρρ⁰0)⁻¹ ∀X⁰ ∈ T_λ^q0

Such aL^ρρ_λλ⁰0 is called anintertwining operator. In general, fixed ρ, ρ⁰ andλ, λ⁰, there is not a unique intertwining operator, not even up to scalar multiplication (in the following, we denote by .

=the equality between two linear isomorphisms up to non-zero scalar multiplication). Indeed, in light of the irreducible decom- position of local representations, described in [Tou14] for surfaces with genus g≥1andp+1punctures, they admit a lot of non-trivial automorphisms (see for example the caseρ=ρ⁰ andλ=λ⁰). One of the main purposes of [BBL07] was to select a unique intertwining operatorLb^ρρ_λλ⁰0 for every ρ, ρ⁰, λ, λ, by requesting some additional properties on them. More precisely, one of the results stated in [BBL07] was the following theorem:

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Theorem ([BBL07, Theorem 20]). For every surfaceS there exists a unique family of intertwining operators Lb^ρρ_λλ⁰0, indexed by couples of isomorphic local representations ofT_S^q and by couples of ideal triangulationsλ, λ⁰ ∈Λ(S), indi- vidually defined up to scalar multiplication, such that:

Composition relation: for everyλ, λ⁰, λ⁰⁰∈Λ(S)and for every triple of isomorphic local representationsρ, ρ⁰, ρ⁰⁰, we haveLb^ρρ_λλ⁰⁰00

=. Lb^ρρ_λλ⁰0◦Lb^ρ_λ⁰0^ρλ⁰⁰⁰⁰; Fusion relation: letS be a surface obtained from another surfaceR by fu-

sion, and let λ, λ⁰ be two triangulations of S obtained by fusion of two triangulationsµ, µ⁰ ofR. Ifη, η⁰ are two isomorphic local representations ofT_R^q and ρ, ρ⁰ are local representations ofT_S^q obtained by fusion respectively fromη andη⁰, then we haveLb^µµ_ηη0⁰

=. Lb^ρρ_λλ⁰0.

However, in our study of the ideas exposed in [BBL07], we have found a difficulty that compromises this statement, in particular the possibility to select a unique intertwining operator for every choice ofρ, ρ⁰, λ, λ⁰.

Let us try to describe this obstruction. Let λ be an ideal triangulation of S. Denote by S₀ the surface obtained by splitting S along all the edges of the ideal triangulation λand by λ0 its unique ideal triangulation. In [BBL07]

the procedure to select the isomorphism L^ρρ_λλ⁰ was the following: we fix two representatives(ρj)^m_j=1and(ρ⁰_j)^m_j=1respectively ofρλandρ⁰_λthat are isomorphic to each other, as local representations of the Chekhov-Fock algebra T_λ^q

0. As observed in the proof of [BBL07, Lemma 21], a local representation of T_λ^q

0 is irreducible, so there exists a unique isomorphism M_λλ^ρρ⁰ between (ρ_j)^m_j=1 and (ρ⁰_j)^m_j=1, up to scalar multiplication. Then the isomorphism Lb^ρρ_λλ⁰ was defined in [BBL07, Lemma 22] as Lb^ρρ_λλ⁰ :=M_λλ^ρρ⁰. The problem we will observe is that this choice does depend on the selected representatives (ρj)j and (ρ⁰_j)j. In other words, the representation ρλ has representatives that are non-trivially isomorphic to each other, so different choices of (ρj)^m_j=1 and (ρ⁰_j)^m_j=1 lead us to a (finite) collection of intertwining operators, in general not to a unique element. We will focus on this problem in the first Section and in particular in Subsubsection 1.4.2.

The main purpose of this paper is to understand this phenomenon and to try to recover a result on intertwining operators similar to the one in [BBL07, Theorem 20]. We will be able to select just a setLλλ^ρρ⁰⁰ of intertwining operators, instead of a unique linear isomorphism. Moreover, the selected sets L_λλ^ρρ0⁰ will be naturally endowed with a transitive and free action of H₁(S;ZN). This fact implies that, fixed λ, λ⁰, the setL_λλ^ρρ⁰ is finite, but its cardinality goes to infinity by increasing the number N ∈N and the complexity of the surface S. In order to provide a statement analogous to [BBL07, Theorem 20], a few steps will be necessary: we will produce the fundamental objects in Section 2, we will investigate on their properties in Section 3 and finally we will complete the procedure in Section 4, where we will prove the following Theorem:

Theorem. For every surfaceSthere exists a collection{(L_λλ^ρρ⁰⁰, ψ^ρρ_λλ⁰0)}, indexed by couples of isomorphic local representationsρ, ρ⁰ of the quantum Teichmüller spaceT_S^q and by couples of ideal triangulationsλ, λ⁰ ∈Λ(S)such that

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Intertwining: for every couple of isomorphic local representations ρ={ρλ:T_λ^q →End(Vλ)}_λ∈Λ(S) ρ⁰ ={ρ⁰_λ:T_λ^q →End(V_λ⁰)}_λ∈Λ(S) and for everyλ, λ⁰ ∈Λ(S),Lλλ^ρρ⁰⁰ is a set of linear isomorphismsL^ρρ_λλ⁰0 from V_λ⁰0 toV_λ such that

(ρλ◦Φ^q_λλ0)(X⁰) =L^ρρ_λλ⁰0 ◦ρ_λ⁰0(X⁰)◦(L_λλ^ρρ⁰0)⁻¹ ∀X⁰ ∈ T_λ^q0

for everyX⁰ ∈ T_λ^q0;

Action: everyLλλ^ρρ⁰⁰is endowed with a transitive and free actionψ_λλ^ρρ⁰0ofH1(S;ZN); Fusion property: letR be a surface andS obtained by fusion fromR. Fix

η={ηµ:T_µ^q →End(Wµ)}_µ∈Λ(R) η⁰ ={η_µ⁰ :T_µ^q →End(W_µ⁰)}_µ∈Λ(R) two isomorphic local representations ofT_R^q and

ρ={ρλ:T_λ^q →End(Vλ)}_λ∈Λ(S) ρ⁰ ={ρ⁰_λ:T_λ^q →End(V_λ⁰)}_λ∈Λ(S) two isomorphic local representations of T_S^q, with ρ and ρ⁰ respectively obtained by fusion fromη andη⁰. Then for every µ, µ⁰ ∈Λ(R), ifλ, λ⁰ ∈ Λ(S)are the corresponding ideal triangulations onS, there exists a natural inclusion j: Lµµ^ηη⁰⁰ → L_λλ^ρρ⁰⁰ such that for every L ∈ Lµµ^ηη⁰⁰ the following holds

(j◦ψ_µµ^ηη⁰0)(c, L) =ψ_λλ^ρρ⁰0(π_∗(c), j(L))

for everyc∈H1(R;ZN), whereπ:R→S is the projection map;

Composition property: for everyρ, ρ⁰, ρ⁰⁰isomorphic local representations of T_S^q and for everyλ, λ⁰, λ⁰⁰∈Λ(S), the composition map

L_λλ^ρρ⁰⁰×L_λ^ρ⁰⁰_λ^ρ⁰⁰⁰⁰ −→ L_λλ^ρρ⁰⁰⁰⁰ (L, M) 7−→ L◦M is well defined and it verifies

(c·L)◦(d·M) = (c+d)·(L◦M)

Observe that in the reviewed assertion the Fision and Composition properties have been modified incorporating the affine structure onH₁(S;ZN).

Moreover, we will prove that the collection{Lλλ^ρρ⁰⁰}of intertwining operators we will exhibit is minimal in the family of collections of intertwining operators verifying "weak" Fusion and Composition properties (in practice the ones of the original statement in [BBL07]).

It turns out that the basic building block of our family{Lλλ^ρρ⁰⁰}is the unique intertwining operator of local representations carried by a square equipped with two triangulations related by a diagonal exchange. In Subsection 2.3 we provide explicit formulas for such basic operators.

An important application of [BBL07, Theorem 20] was the construction of invariants of pseudo-Anosov surface diffeomorphisms and their hyperbolic mapping tori, extending to local representations what had been done in [BL07] for

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irreducible ones (for which the uniqueness of the intertwining operator up to scalar multiples is automatic). There are good reasons to conjecture that such

"local" invariants eventually are instances of quantum hyperbolic invariants of cusped hyperbolic 3-manifolds defined in [BB05], [BB07] and [BB15] by devel- oping seminal Kashaev’s work [Kas94]. This is indeed a reason of the interest of local representations. We will show that this construction can be adapted rather straightforwardly by replacing [BBL07, Theorem 20] with our main theorem, although the resulting invariant is a bit more complicated. This could also indicate that establishing an actual relation between "quantum Teichmüller"

and quantum hyperbolic invariants for fibred cusped hyperbolic 3-manifolds can be a bit subtler than one expects.

Aknowledgement This article is extracted from my Master’s thesis at the University of Pisa. I would like to thank my advisor Prof. R. Benedetti for his help and support all along this work.

1 Preliminaries

In this paper we will always assume that a surfaceS is oriented and obtained, from a compact oriented surfaceSwith genusgandbboundary components, by removingp≥1puncturesv₁, . . . , v_p, with at least a puncture in each boundary components. Assume further that 2χ(S)≤p_∂−1, where p_∂ is the number of punctures lying in ∂S. These are the hypotheses in which S admits an ideal triangulation (see below for the Definition).

Moreover,q∈C^∗ will always be a certain primitiveN-th root of(−1)^N+1.

1.1 The Chekhov-Fock algebra

Definition 1.1. Let S be a surface. We define an ideal triangulation λ of S as a triangulation of S having as set of vertices exactly the set of punctures {v₁, . . . , v_p}. We identify two ideal triangulations ofSif they are isotopic. Also, we denote byΛ(S)the set of all ideal triangulations ofS.

Given λ ∈ Λ(S), let n be the number of 1-cells in the ideal triangulation λ and m the number of faces ofλ. Easy calculations show that the following relations hold:

n=−3χ(S) + 3p−p∂

=−3χ(S) + 2p_∂ m=−2χ(S) +p∂

In particularmandndo not depend on the ideal triangulationλ∈Λ(S). Since S is oriented, on each triangle ofλwe have a natural induced orientation. With respect to this orientation, it is reasonable to speak about a left and a right side of each spike of a triangle. We select an orderλ₁, . . . , λ_n on the set of1-cells of the triangulation λ. Given λ_i andλ_j 1-cells ofλ, we denote bya_ij the number of spikes of triangles in λ in which we find λ_i on the left side and λ_j on the right. Now we set

σij :=aij−aji

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Definition 1.2. Given q∈C^∗, we define theChekhov-Fock algebra associated with the ideal triangulation λ and the parameter q as the non-commutative C-algebra T_λ^q, generated byX₁^±1, . . . , X_n^±1 and endowed with the following relations

XiXj=q^2σ^ijXjXi

for every i, j= 1, . . . , n.

Givenλ∈Λ(S), we designate the freeZ-module generated by the 1-cells of the triangulationλasH(λ;Z). A choice of an ordering on the1-cells ofλgives us a natural isomorphism of H(λ;Z) with Zⁿ and through it we can define a bilinear skew form onH(λ;Z)given by

σ



 X

i

a_iλ_i,X

j

b_iλ_j



:=X

i,j

a_ib_jσ_ij (1)

Observe that σ_ij is determined by the mutual positions of λ_i and λ_j, hence σ is indeed independent from the choice of an ordering on λand it is a bilinear skew form intrinsically defined onH(λ;Z).

Now we fix an ordering in λ and we choose α = (α₁, . . . , α_n) and β = (β1, . . . , βn)inZⁿ∼=H(λ;Z). We can associate withαa monomial inT_λ^q, that we briefly denote byX^α, defined by

X^α:=X₁^α¹· · ·X_n^αⁿ Introduce also the following notation

X^α:=q⁻^Pî<j^αⁱ^α^j^σîjX^α=q⁻^Pî<j^αⁱ^α^j^σîjX₁^α¹· · ·X_n^αⁿ

Lemma 1.3. For every α, β ∈ Zⁿ ∼= H(λ;Z), the following relations hold in T_λ^q:

X^αX^β=q^2σ(α,β)X^βX^α=q²⁽^P^i>j^αⁱ^β^j^σ^ij⁾X^α+β (2) X^αX^β =q^2σ(α,β)X^βX^α=q^σ(α,β)X^α+β (3) Furthermore, for every permutationτ∈Sl

q⁻^P^h<k^σ^ihikXi₁· · ·Xi_l=q⁻

P

h<kσ_iτ(h)iτ(k)

Xi_τ(1)· · ·Xi_τ(l) (4) Proof. The first relations easily follow by direct calculations, it is sufficient to control how the coefficients change by the appropriate permutation. We will see how to deduce the relations in 3 using 2. The first equality is obvious, it is enough to multiply both the members of 2 by q⁻^Pî<j^αⁱ^α^j^σîjq⁻^Pî<j^βⁱ^β^j^σîj. We would like to show that X^αX^β =q^σ(α,β)X^α+β holds. By virtue of 2, it is sufficient to prove that the exponents ofq, multiplying the elementsX^αX^β and q²⁽^Pî>j^αⁱ^β^j^σîj⁾X^α+β respectively, coincide. It is simple to show we can reduce to prove the following equality

−X

i<j

αiαjσij−X

i<j

βiβjσij =−X

i<j

(α+β)i(α+β)jσij+σ(α, β)−2X

i>j

αiβjσij

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Using the fact thatσis skew-symmetric, we deduce

−X

i<j

(α+β)_i(α+β)_jσ_ij+σ(α, β)−2X

i>j

α_iβ_jσ_ij=

=−X

i<j

(α+β)_i(α+β)_jσ_ij+X

i<j

α_iβ_jσ_ij+X

i>j

α_iβ_jσ_ij−X

i>j

α_iβ_jσ_ij+

−X

i>j

αiβjσij

=−X

i<j

(α+β)i(α+β)jσij+X

i<j

αiβjσij+X

i<j

βiαjσij

=−X

i<j

αiαjσij−X

i<j

βiβjσij

that concludes the proof.

For what concerns the relation 4, it is simple to prove it whenτ is a trans- position of consecutive elements and the general case easily follows.

1.2 Representations of the Chekhov-Fock algebra

Following Definition 1.2, we see that, ordering the edges with respect to the orientation as in Figure 1, the Chekhov-Fock algebra associated with an ideal triangle is isomorphic toC[X1, X2, X3] endowed with the relations

XiXi+1=q²Xi+1Xi

for everyi= 1,2,3, where the indices are mod (3). Unless specified differently, we will always assume to be in this situation, with the edges ordered clockwise with respect to the orientation.

Proposition 1.4. Letρ: T_T^q→End(V)be an irreducible representation, with V a C-vector space of dimension d and with T_T^q denoting the Chekhov-Fock algebra associated with the ideal triangleT, which admits a unique ideal triangulation. Thend=N and there existx1, x2, x3, h∈C^∗ such that

ρ(X_i^N) =xiidV fori= 1,2,3 ρ(H) =h id_V

where the X_i denote the generators associated with the edges of T and H = q⁻¹X₁X₂X₃. In addition, the following holds

h^N =x₁x₂x₃

λ₁ λ₃

λ2

Figure 1: Standard indexing on a triangle

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Moreover, two irreducible representations ρ: T_T^q → End(V) and ρ⁰: T_T^q → End(V⁰) are isomorphic if and only if x_i = x_i⁰ and h = h⁰, where the x_i, h are the above quantities related to ρand the x⁰_i, h⁰ are related to ρ⁰. Further- more, for every choice of valuesx1, x2, x3, h∈C^∗ verifyingh^N =x1x2x3, there exists a representation, unique up to isomorphism, realizing them as invariants.

Proof. See [BBL07, Lemma 2].

Remark 1.5. Following the proof of the previous Proposition, we see that, for every irreducible representation ρ: T_T^q → End(V), there exists a basis of V in which the elements ρ(X1), ρ(X2), ρ(X3) are respectively represented by the following matrices:

y1B1=y1





 1

q² ...

q^2(N⁻¹⁾







y2B2=y2







0 · · · 0 1 0 IN−1 ...

0







y3B3=y3







0 q^1−2(2−1)

... ...

0 q^1−2(N⁻¹⁾

q 0 · · · 0







where y_i is an N-th root of x_i for every i = 1,2,3 and h = y₁y₂y₃. When a representationρ:T_T^q→End(V)has this form, that isρ(X_i) =y_iB_i∈End(C^N), we will say that ρis in standard form.

The standard form is not unique. More precisely, for every choice ofN-th roots yi of the xi such that h =y1y2y3, there exists a basis B, unique up to common scalar multiplication, such that ρ is represented as described above with respect to B.

Definition 1.6. When two linear isomorphismsA,Bdiffer by a non-zero scalar, we will briefly writeA .

=B. 1.2.1 Fusion of surfaces

LetS be a surface and letλbe an ideal triangulation of it. We can construct, starting fromSand the choice of certain internal1-cellsλi₁, . . . , λi_h, a surfaceR obtained by splittingSalong these selected edgesλi₁, . . . , λi_h, i. e. by removing the identifications alongλi_j between the edges of the ideal triangles havingλi_j

as edge. On R we can clearly find an ideal triangulationµ and an orientation induced byλand by the orientation on S. In these circumstances, we will say

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thatSis obtained fromRby fusion, and analogously that the ideal triangulation λ∈Λ(S)is obtained fromµ∈Λ(R)by fusion.

Fix an indexing on the 1-cells of λ and one on the 1-cells of µ. We can construct a natural inclusionjµλ:H(λ;Z)→ H(µ;Z), which associates to each ei∈Zⁿ ∼=H(λ;Z), corresponding to the edgeλi, the vectorvi= (vi1, . . . , vis)∈ Z^s∼=H(µ;Z), defined by

v_ik:=

(1 ifµk goes by fusion inλi

0 otherwise

It is easy to verify that map jµλ is indeed an inclusion. In fact, every vectorvi

has at most two non-zero components and, ifi6=j, the supports of the vectors vi and vj are disjoint. Denote by σ and η the skew-symmetric bilinear forms respectively on H(λ;Z) and H(µ;Z), defined as in Subsection 1.1. Then the following holds

σ(v, w) =η(j_µλ(v), j_µλ(w))

In order to prove it, it is sufficient to verify the equality on a set of generators, so it is enough to prove that, for everyi, j, we have σ_ij =η(v_i, v_j). Recalling the definition, we can expressσij as follows

X

T triangle ofλ

X

cspike ofT

sλ(c, λi, λj)

wheres_λ(c, λ_i, λ_j)is equal to+1ifchasλ_i on the left andλ_j on the right,−1 ifchasλ_j on the left andλ_ion the right, and is equal to 0otherwise. Suppose that the edgeλ_i is the result of the identification of the edgesµ_i₁, µ_i₂ inµ, and that analogously λ_j is the result of the identification of the edgesµ_j₁, µ_j₂. The faces of the ideal triangulation λ are in natural bijection with the faces of µ, and consequently the respective spikes too. Hence it is sufficient to prove that

sλ(c, λi, λj) =sµ(c, µi₁, µj₁) +sµ(c, µi₁, µj₂) +sµ(c, µi₂, µj₁) +sµ(c, µi₂, µj₂) for every spikecof the ideal triangulation. In the right member there exists at most one non-zero term and it is immediate to see that, thanks to the coherent choice of orientations on the faces of both triangulations, the equality holds.

Denote now byX1, . . . , Xn the generators of T_λ^q associated with the edges ofλand byY1, . . . , Ysthe ones ofT_µ^q associated with the edges ofµ. Define the map

ι_µλ : T_λ^q −→ T_µ^q X^α 7−→ Y^j^µλ^(α)

Using the relation 3 and what just shown, we can verify that ιµλ(X^αX^β) =ιµλ(q^σ(α,β)X^α+β) =q^σ(α,β)Y^j^µλ^(α+β)

=q^η(j^µλ^(α),j^µλ^(β))Y^j^µλ^(α)+j^µλ^(β)=Y^j^µλ^(α)Y^j^µλ^(β)

=ι_µλ(X^α)ι_µλ(X^β)

The injectivity of j_µλ: H(λ;Z)→ H(µ;Z) immediately implies the injectivity of ι_µλ. Hence we have proved that, if (S, λ)is obtained from (R, µ) by fusion,

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then there exists an inclusion of algebrasι_µλ: T_λ^q → T_µ^q, carrying monomials of T_λ^q in monomials of T_µ^q. Moreover, these maps, for varying S, R, λ and µas above, verify a sort of composition property. More precisely, assume that(S, λ), (S⁰, λ⁰)and(S⁰⁰, λ⁰⁰)are three surfaces endowed with ideal triangulations, with (S, λ)obtained from(S⁰, λ⁰)by fusion and with(S⁰, λ⁰)obtained from(S⁰⁰, λ⁰⁰) by fusion. Then, by definition, the homomorphismsjλ⁰⁰λ:H(λ;Z)→ H(λ⁰⁰;Z) andjλ⁰⁰λ⁰◦jλ⁰λ:H(λ;Z)→ H(λ⁰⁰;Z)coincide, so on the Chekhov-Fock algebras the following relation holds

ι_λ⁰⁰_λ=ι_λ⁰⁰_λ⁰◦ι_λ⁰_λ (5) Now let us focus on a more specific situation. GivenSa surface as above and λ∈Λ(S)an ideal triangulation of it,Scan be obtained by fusion from a surface S0 realized by splittingSalong all its internal edges or, in other words, realized as the disjoint union of the trianglesTicomposingλ, fori= 1, . . . , m. S0admits a unique ideal triangulation λ0 and its Chekhov-Fock algebra associated with λ0 is naturally isomorphic to

T_T^q

1⊗ · · · ⊗ T_T^q

m

In this case we will denote simply by ιλ the inclusion mapιλ₀λ: T_λ^q →N

iT_T^q

i, defined as above.

Each triangle Ti is endowed with an orientation, determined by the one on S. Order the edges λ⁽ⁱ⁾₁ , λ⁽ⁱ⁾₂ , λ⁽ⁱ⁾₃ of Ti ⊆ S0 clockwise, as in Figure 1, and denote respectively byX₁⁽ⁱ⁾, X₂⁽ⁱ⁾, X₃⁽ⁱ⁾the generators ofT_λ^q associated with these edges. For the sake of simplicity, given F⁽¹⁾⊗ · · · ⊗F^(m) a monomial in N

iT_T^q

i, we will omit in it the tensor product by those terms verifyingF⁽ⁱ⁾= 1. Recalling the definition of ι_λ, we want to give an explicit description of the image, under this inclusion, of the generators X_i associated with the edges of the ideal triangulationλofS:

• ifλi is a boundary edge, then it is side of a single ideal triangle Tk_i, so we haveιλ(Xi) =Xa^(kiⁱ⁾∈N

iT_T^q

i, whereai is the index of the edge ofTk_i

identified inS withλi;

• ifλ_iis an internal edge and it is side of two distinct trianglesT_l_i andT_r_i, then we have ι_λ(X_i) = Xa^(l_iⁱ⁾⊗X_b^(rⁱ⁾

i ∈ N

iT_T^q

i, where a_i and b_i are the indices of the edges ofT_l_i andT_r_i, respectively, identified inS withλ_i;

• ifλi is an internal edge and it is side of a single triangleTk_i, then we have ιλ(Xi) =q⁻¹Xa^(kiⁱ⁾X_b^(kⁱ⁾

i =q X_b^(kⁱ⁾

i Xa^(kiⁱ⁾∈N

iT_T^q

i, whereai andbiare the indices of the edges ofTk_i identified inS with λi and λa^(kiⁱ⁾, λ^(k_b ⁱ⁾

i lie on the left and on the right, respectively, of their common spike.

1.2.2 Local representations

Definition 1.7. GivenSa surface andλan ideal triangulation ofS, we denote byΓ_S,λthe dual graph ofλ. More precisely,Γ_S,λis a CW-complex of dimension 1, whose verticesT_b^∗correspond to the trianglesTbofλand, for everyλainternal edge, there is a1-cellλ^∗_athat connects the vertices corresponding to the triangles on the sides of λa, even if the triangles coincide.

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It follows from the definition that all the vertices have valency ≤ 3. In particular, the valency of a vertexT_b^∗ inΓ_S,λ is equal to the number of sides of Tb that are internal edges inλ.

GivenΓ a graph, we will denote with Γ^(k) the set of the k-cells composing Γ.

Fixλ∈Λ(S)an ideal triangulation ofSand, for every edgeλiofλ, choose an arbitrary orientation on it. Now orient the edges of Γ = ΓS,λ as follows: the1- cellλ^∗_i, dual of the internal edgeλi, is oriented in such a way that the intersection numberi(λi, λ^∗_i)inSis equal to+1(remember that we are considering oriented surfaces). Moreover, we assume that all the verticesT_l^∗ have positive sign.

Given a= 1, . . . , n with λa an internal edge having two different triangles on its sides, we define

ε(a, b) :=







+1 ifTb is on the left of λa

−1 ifT_b is on the right of λ_a 0 otherwise

for everyb= 1, . . . , m. Ifλ_a is internal and it has the same triangle on its sides, we define ε(a, b) = 0 for every b = 1, . . . , m. Observe that a triangle Tb is on the left ofλa if and only if the previously fixed orientation onλa coincides with the orientation determined as boundary ofTb. Moreover, it is immediate to see that the definition ofε(a, b)can be reformulated as follows

ε(a, b) :=







+1 if λ^∗_a goes towardsT_b^∗

−1 if λ^∗_a comes from T_b^∗ 0 otherwise

ifλ^∗_a has different ends, otherwiseε(a, b) = 0for everyb= 1, . . . , m.

Fixed an orientation on an ideal triangulationλ, the definition of local representation ofT_λ^q given in [BBL07] can be reformulated as follows:

Definition 1.8. Let λ ∈ Λ(S) be an ideal triangulation, with triangles la- belled asT1, . . . , Tm, and let(ρ1, . . . , ρm),(ρ₁, . . . , ρ_m)be twom-tuples in which, for every j = 1, . . . , m, ρj:T_T^q

j → End(Vj) and ρ_j: T_T^q

j → End(Wj) are irreducible representations of the triangle algebraT_T^q

j. The elements(ρ₁, . . . , ρ_m), (ρ₁, . . . , ρ_m)arelocally equivalent if the following hold

• for everyj= 1, . . . , m the vector spacesV_j andW_j are equal;

• for everyi= 1, . . . , n, we have:

– ifλ_i is a boundary edge, then there is a unique triangleT_s_i that has λi on its side. In this case we ask that

ρs_i(X_a^(s_iⁱ⁾) =ρ_s_i(X_a^(s_iⁱ⁾)

where aiis the index of the edge in Ts_i that is identified toλi; – if λi is an internal edge and Tl_i, Tr_i are distinct triangles on the

left and on the right, respectively, of λi (recall that we have fixed

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orientations on the edges λ_i), then there exists α_i ∈C^∗ such that ρl_i(X_a^(l_iⁱ⁾) =α_i⁺¹ρ_l_i(X_a^(l_iⁱ⁾) =α^ε(i,l_i ⁱ⁾ρ_l_i(X_a^(l_iⁱ⁾)

ρr_i(X_b^(rⁱ⁾

i ) =α_i⁻¹ρ_r_i(X_b^(rⁱ⁾

i ) =α^ε(i,r_i ⁱ⁾ρ_r_i(X_b^(rⁱ⁾

i )

whereaiandbiare the indices of the edges inTl_iandTr_i, respectively, that are identified toλi;

– ifλiis an internal edge and it has the triangleTk_ion both sides, then there exists αi ∈C^∗such that

ρki(X_a^(k_iⁱ⁾) =α⁺¹_i ρ_k_i(X_a^(k_iⁱ⁾) ρki(X_b^(kⁱ⁾

i ) =α⁻¹_i ρ_k_i(X_b^(kⁱ⁾

i )

wherea_i andb_i are the indices of the edges inT_k_i that are identified to λ_i and thea_i-th side, unlike theb_i-th one, has the orientation as boundary of T_k_i coherent with the orientation ofλ_i.

A local representation ρ of T_λ^q is a local equivalence class of m-tuples of representations(ρ1, . . . , ρm)as above.

Remark 1.9. Given [ρ1, . . . , ρm] a local representation of T_λ^q, we can define a representation ofT_λ^q as follows

ρ:= (ρ1⊗ · · · ⊗ρm)◦ιλ:T_λ^q −→End(V1⊗ · · · ⊗Vm)

By definition of the locally equivalence relation between m-tuples of representations, thisρdoes not depend on the choice of the representative of the equivalence class[ρ1, . . . , ρm].

We will confuse a representative(ρ₁, . . . , ρ_m)of a local representationρwith the obvious corresponding representation ρ₁⊗ · · · ⊗ρ_m on the Chekhov-Fock algebra T_λ^q

0 of the surface S₀, where S₀ is the surface obtained by splitting S along all its edges and λ₀ is its ideal triangulation.

Definition 1.10. Letλ∈Λ(S)be an ideal triangulation ofS and letρ:T_λ^q → End(V)be a local representation of T_λ^q. We denote byFS₀(ρ)the set of representatives of ρ as local representation, which are local (and irreducible, see Proposition 1.14) representations of the Chekhov-Fock algebra T_λ^q

0 of the surface S0, obtained by splitting S alongλ. Moreover, fixed an orientation on λ and givenρ1⊗ · · · ⊗ρmandρ₁⊗ · · · ⊗ρ_mtwo elements ofFS₀(ρ), we will write

ρ₁⊗ · · · ⊗ρ_m−→^αⁱ ρ1⊗ · · · ⊗ρm

ifρ₁⊗ · · · ⊗ρmandρ₁⊗ · · · ⊗ρ_mare related by the numbers(α_i)_ias described in Definition 1.8. The(α_i)_i are called thetransition constants fromρ₁⊗ · · · ⊗ρ_m toρ₁⊗ · · · ⊗ρ_m.

It is very simple to see that, with these notations introduced, the following holds:

Lemma 1.11. Letζ=N

iρi,ζ⁰ =N

iρ⁰_i andζ⁰⁰=N

iρ⁰⁰_i be three representatives of a local representation ρ. Then the following properties hold:

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1. there exists a unique collection of transition constants, depending on the chosen orientation onλ, such thatζ−→^αⁱ ζ⁰;

2. ifζ→^αⁱ ζ⁰ andζ⁰→^βⁱ ζ⁰⁰, then ζ^α−→ⁱ^βⁱζ⁰⁰ 3. ifζ→^αⁱ ζ⁰, thenζ⁰^α

−1

→i ζ.

Definition 1.12. Given S a surface as above,λ∈Λ(S)an ideal triangulation and two local representations[ρ1, . . . , ρm],[ρ₁⁰, . . . , ρ⁰_m]ofT_λ^q, with

ρ_j :T_T^q_j −→End(V_j) ρ⁰_j :T_T^q

j −→End(V_j⁰)

we will say that [ρ₁, . . . , ρ_m] and [ρ⁰₁, . . . , ρ⁰_m] are isomorphic if there exist respectively representatives(ρ₁, . . . , ρ_m)and(ρ⁰₁, . . . , ρ⁰_m)of them and there exist linear isomorphismsL_j:V_j→V_j⁰ such that, for everyj= 1, . . . , mwe have

Lj◦ρj(X)◦L⁻¹_j =ρ⁰_j(X) ∀X ∈ T_T^q

j

Assume that[ρ1, . . . , ρm] e[ρ⁰₁, . . . , ρ⁰_m]are isomorphic and let(ρ1, . . . , ρm) and (ρ⁰₁, . . . , ρ⁰_m) be two representatives of them such that there exist linear isomorphismsLj:Vj →V_j⁰ withLj◦ρj◦L⁻¹_j =ρ⁰_j. Then, for any other choice of a representative ( ¯ρ₁, . . . ,ρ¯_m)of [ρ₁, . . . , ρ_m], the m-tuple of representations

¯

ρ⁰_j := L_j ◦ρ¯_j ◦L_j⁻¹ is ∼S-locally equivalent to (ρ⁰₁, . . . , ρ⁰_m). From this fact immediately follows that the isomorphism relation defined above is indeed an equivalence relation.

Now we recall some results of [BBL07] that will be useful in the following analysis.

Lemma 1.13. Let [ρ1, . . . , ρm] be a local representation of T_λ^q. Then, for every generator Xi ∈ T_λ^q associated with the edgeλi, the representation ρλ:=

(ρ1⊗ · · · ⊗ρm)◦ιλ verifies

ρ(X_i^N) =xiidV₁⊗···⊗Vm

for a certainxi∈C^∗. In addition, there existsh∈C^∗ such that ρ(H) =h idV1⊗···⊗Vm

Proof. See [BBL07, Lemma 5].

The following result was firstly stated in [BBL07], for the sake of complete- ness we provide a proof in Appendix A.

Proposition 1.14. Let S be an ideal polygon with p ≥ 3 vertices. Then, for every λtriangulation ofS and for every local representation[ρ1, . . . , ρm] of the Chekhov-Fock algebra T_λ^q, the representation ρ:= (ρ1⊗ · · · ⊗ρm)◦ιλ is irreducible.

Denote byRloc(T_λ^q)the set of isomorphism classes, as local representations, of local representations ofT_λ^q.

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Theorem 1.15. Let S be a surface and fix q ∈ C^∗ a primitive N-th root of (−1)^N⁺¹. Then the setRloc(T_λ^q)is in bijection with the set of elements((x_i)_i;h) in (C^∗)ⁿ×C^∗ verifying

h^N =x1· · ·xn

and the correspondence associates with the isomorphism class as local representations of[ρ1, . . . , ρm], the element((xi)i;h)defined by the relations

ρ(X_i^N) =xiidV₁⊗···⊗Vm

ρ(H) =h id_V₁_{⊗···⊗V}_m fori= 1, . . . , n, whereρ= (ρ1⊗ · · · ⊗ρm)◦ιλ. Proof. See [BBL07, Proposition 6].

Corollary 1.16. Let[ρ1, . . . , ρm]and[ρ⁰₁, . . . , ρ⁰_m]be two local representations of T_λ^q. Then they are isomorphic to each other as local representations if and only ifρ:= (ρ1⊗ · · · ⊗ρm)◦ιλ andρ⁰ := (ρ⁰₁⊗ · · · ⊗ρ⁰_m)◦ιλ are isomorphic as representations.

1.2.3 An action of H₁(S;ZN)

Denote by (C_•(Γ;Z^N), ∂_•) the cellular chain complex of Γ = ΓS,λ. Then C0(Γ;ZN) is the ZN-module freely generated by the vertices T_b^∗ of Γ and C1(Γ;ZN) is the ZN-module freely generated by the oriented 1-cells λ^∗_a of Γ. Because Γ has dimension1, all the other Ci(Γ;ZN) are equal to zero. Thanks to what observed, we can describe the boundary∂1in terms of the triangulation λ. Givenλ^∗_a a1-cell ofΓ, the boundary∂1(λ^∗_a)verifies

∂1(λ^∗_a) =

m

X

b=1

ε(a, b)T_b^∗∈C0(Γ;ZN)

where theε(a, b)are the numbers defined in 1.2.2. Hence the first group of cellular homology H1(Γ;ZN)is equal to the subgroupKer∂1ofC1(Γ;ZN), whose elements are theZN-combinationsPn

a=1caλ^∗_a such that, for everyb= 1, . . . , m the following holds:

n

X

a=1

ε(a, b)ca = 0∈ZN

The dual graphΓis a deformation retract ofS, so the groupH₁(Γ;ZN)can be identified to H1(S;Z^N) via a certain inclusion ofΓ in S, which is well defined up to homotopy.

Given λ ∈ Λ(S) an oriented ideal triangulation and ρ:T_λ^q → End(V) a local representation, we can define an action ofH1(S;ZN)on the setFS₀(ρ)as follows:

Definition 1.17. Given c = P

iciλ^∗_i an element of H1(Γ;ZN) ∼= H1(S;ZN) and fixed a representative ρ1⊗ · · · ⊗ρm ofρ, we can produce anotherm-tuple of representationsρ₁⊗ · · · ⊗ρ_mofρdefined as follows: for everyl= 1, . . . , m

Intertwining operators of the quantum Teichmüller space