Fusion and monodromy in the Temperley-Lieb category

Graham and Lehrer (1998) introduced a Temperley-Lieb category $\mathsf{\widetilde{TL}}$ whose objects are the non-negative integers and the morphisms in $\mathsf{Hom}(n,m)$ are the link diagrams from $n$ to $m$ nodes. The Temperley-Lieb algebra $\mathsf{TL}_{n}$ is identified with $\mathsf{Hom}(n,n)$. The category $\mathsf{\widetilde{TL}}$ is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on $\mathsf{\widetilde{TL}}$. We introduce a module category ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}$ whose objects are functors from $\mathsf{\widetilde{TL}}$ to $\mathsf{Vect}_{\mathbb C}$ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for $\mathsf{\widetilde{TL}}$ to induce the structure of a ribbon category on ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}(\beta=-q-q^{-1})$, when $q$ is not a root of unity. We discuss how the braiding on $\mathsf{\widetilde{TL}}$ and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.

The (original) family of the Temperley-Lieb algebras was cast into a categorical framework by Graham and Lehrer [1] in 1998. A decade later Read and Saleur [2] introduced a product − 1 × f − 2 between two modules over two (maybe distinct) Temperley-Lieb algebras. Still later this product was computed between several families of modules by Gainutdinov and Vasseur [3], and Belletête [4]. Their recent results (obtained in 2012 and 2015 respectively) lead to natural questions: how can one define the module category over Graham and Lehrer's category? Does the natural braiding that exists on Graham and Lehrer's category (described for example by Turaev [5]) extend to this module category? And how many of the defining properties of tensor categories does the module category satisfy? The present paper answers these questions.
Date: October 4, 2018. Statistical models in two dimensions are defined by an evolution operator or a transfer matrix acting on finitedimensional vector spaces. The sizes of both the lattice and the vector spaces are parameters of the formulation. The limit when these parameters go to infinity is known in some cases (numerically or rigorously) to be a conformal field theory. For several XXZ and loop models [6,7], the Hamiltonian is first defined as an element of a Temperley-Lieb algebra TL n , or one of its generalizations, and the actual linear operator is obtained as the representative of this element in some representations over the algebra. The fusion product is an algebraic construction, actually a bifunctor, that associates to two modules M and N over TL m and TL n respectively a module over TL m+n . As said above, for the Temperley-Lieb algebra, such a fusion product − 1 × f − 2 was introduced by Read and Saleur [2] and computed in many cases by Gainutdinov and Vasseur [3] and Belletête [4]. It is associative and commutative, and the braiding gives the isomorphism between M × f N and N × f M.
There are reasons to believe that algebraic information obtained from the finite algebras, either the Temperley-Lieb family, its dilute counterpart or any other one, is intimately related to analogous structures of the CFTs and should help understand them. First there is compelling evidence that, in the limit when the size of the lattice goes to infinity, the spectrum of the Hamiltonian, properly scaled, reproduces characters of the Virasoro algebra. Second, in some representations of the TL family, the Hamiltonian has Jordan blocks (of size 2 × 2) [7,8], indicating a possible link to logarithmic CFTs. Third, when restricted to TL-modules that are known to give rise to the Virasoro modules appearing in minimal CFTs, the highly non-trivial fusion product defined between TL modules does reproduce the simple fusion rules of these minimal CFTs. Since the operator product expansion of CFTs leads to a fusion product between modules over the Virasoro algebra whose many properties are captured into a tensor category, it is natural to ask how many of these properties are shared by Read's and Saleur's fusion.
The (original) family of the Temperley-Lieb algebras was cast into a categorical framework by Graham and Lehrer [1] while they were actually studying another family, the periodic (or affine) Temperley-Lieb algebras. The construction brings together all algebras TL n (β ), n ≥ 0, in the same category TL(β ). Their formulation will be the starting point of Section 2 where the requirements for a category to be monoidal and then braided will be fulfilled for the TL family.
Even though the braiding on TL(β ) is already known, the twist presented in this section is new to our knowledge. Section 3 defines a module category Mod TL whose objects are functors from TL to Vect C . The associator, commutor and twist defined on TL are shown to induce similar natural transformations on Mod TL . Section 4 then shows how the integrability of two-dimensional statistical models and the components of the commutor η r,s defining the braiding are related. Section 5 extends the results to the family of dilute Temperley-Lieb algebras dTL n . A short conclusion follows.

THE TEMPERLEY-LIEB CATEGORY
Graham and Lehrer [1] showed that the algebras TL n (β ), n ≥ 0, can be studied as a whole and given the structure of a category TL. The goal of this section is to recall the definitions of monoidal and braided categories and show that the Temperley-Lieb category TL is braided. Some of the data necessary to define a braided category are known for the TL family (see for example [5]). However giving the details here fulfills several goals. It provides a pedagogical introduction to these structures with detailed proofs. It also establishes many of their properties that will play a crucial role in Section 3.
2.1. TL(β ) as a monoidal category. The first step is to cast the family of algebras TL n , n ≥ 0, into a category and show that the additional requirements of a monoidal category are easily fulfilled.
We take the convention that morphisms and functors acts on the left, so that (FG)(X) ≡ F(G(X)).
The sets of morphisms Hom(n, m) from n to m is empty if n and m do not have the same parity and, if they do, are defined as the sets of formal C-linear combinations of (m, n)-diagrams. A (m, n)-diagram α ∈ Hom(n, m) is composed of two vertical columns of m nodes on the left, and n nodes on the right, linked pairwise by non-intersecting strings. For instance, here are a pair of (4, 2)-diagrams and one (2, 4)-diagram: , , .
Note that the third diagram can be obtained from the first by reflection through a vertical line midway between the two columns of nodes. We will call the result of this reflection the transpose of the diagram. The identity morphism 1 n ∈ Hom(n, n) is the (n, n)-diagram where every point on the left is linked to the one at the same height on the right.
The associativity of the composition of diagrams is easily verified. The depiction of 1 0 by a simple space is consistent with the depiction of the composition by concatenation of diagrams. For example the following product of the (2, 0)- could equally be understood as d • 1 0 • e. Note finally that End(n) ≡ TL n (β ) is the usual Temperley-Lieb algebra TL n (β ).
The category TL can be easily enriched to become a monoidal one. A category C is said to be monoidal if it is equipped with the following structures [9,10,11]: (1) A bifunctor − ⊗ − : C × C −→ C, called the tensor product; (2) An object I ∈ Ob(C) called the identity; (3) Three natural isomorphisms 1 : , called the associator.
Moreover these structures have to satisfy the triangle and the pentagon axioms. These axioms require that the diagrams in Figures 1 and 2 commute for all A, B,C, D ∈ Ob C. Finally, if the associator, the left and right unitors are all identity isomorphisms, the category is said to be strict. 1 We recall that a natural isomorphism µ : F → G between two functors F,G : C 1 → C 2 associates to each A ∈ Ob(C 1 ) an invertible morphism n, m ∈ Ob TL, simply set 2 n ⊗ m ≡ n + m where "+" stands for the addition in N 0 and, thus, the identity object is This is then expanded bilinearly to all morphisms. The associator α m,n,k is the isomorphism (m + n) + k → m + (n + k) and the unitors are 0 + m → m and m + 0 → m respectively.
Since (N 0 , +) is a monoid, the axioms are trivially verified for the objects. It is easy to verify that the axioms also hold for the morphisms and, thus, TL is a strict monoidal category.
2.2. TL(β ) as a braided category. Let C be a monoidal category and let the opposite tensor product between two objects A, B ∈ Ob C be defined as 2 The category TL is not additive. Indeed one of the requirements for additivity is the existence of direct sum objects for any finite set of objects. The direct sum of two objects m,n ∈ Ob TL would be given by an object (m ⊕ n) ∈ Ob TL together with maps m qm −→ (m ⊕ n) and n qn −→ (m ⊕ n) satisfying some universal property. However if m and n are of different parity, one of the two sets Hom(m,(m ⊕ n)) and Hom(n,(m ⊕ n)) is empty and the basic requirements for the existence of the direct sum cannot be met. To endow TL with a braiding requires more work than to define its monoidal structure. We start by outlining the strategy. Since TL is strict, the hexagon diagrams are equivalent to It follows that, if we can find η 1,1 , the other η n,m , n, m ≥ 1, will be uniquely defined by these two conditions, provided that they are consistent, that is, if η n,m satisfy the above two conditions, then so does η n+1,m ≡ (η n,m ⊗ 1 1 ) • (1 n ⊗ η 1,m ), for instance. Proposition 2.1 will establish this consistency. We shall then build the η n,m recursively. It will then remain to prove that these η m,n define natural isomorphisms. This will require several steps: Lemma 2.2 will express the morphisms η r,s in terms of η 1,1 only and a short computation will express η 1,1 in terms of the generators e i of Temperley-Lieb algebras. Lemmas 2.3 to 2.5 show how the η r,s braid with the e i and some diagrams in Hom(n, 0) and Hom(0, n). Then Proposition 2.6 proves that the η r,s form together a commutor for the category TL.
Proof. We only give the proof for η r+1,s and equation (4), as the other checks are similar. Suppose that n + m = r + 1 with 0 ≤ n ≤ r and 1 ≤ m ≤ s. The steps are the following: Steps 1 and 4 are obtained by using the fact that η m,s and η r,s satisfy the hexagon identity (4). Steps 2 and 3 use the where t i (n) ≡ 1 i−1 ⊗ η 1,1 ⊗ 1 n−i−1 ∈ Hom(n, n) and the factors in a product are listed starting from the right, that is, Proof. The proof of the first part is obtained by induction on r and s. Taking the induction first on r, then on s gives the first expression, while doing the inductions in the reverse order yields the second. The proof of the former is given as example. When r = s = 1, the first expression is simply η 1,1 and the statement is trivially true. Assume therefore that the result stands for η r,1 . If η r+1,1 satisfies equation (4), then in particular , which is η r+1,1 as given by the first expression in (6). Assume then that the result stands for some r, s ≥ 1. Then (3) gives which is the expression for η r,s+1 given in (6). The converse can be obtained as follows. The first expression gives and, thus, satisfies (3). The second expression is shown similarly to satify (4). The proof that the first expression satisfies (4) is harder and it is then easier, though tedious, to prove that the two expressions are equal. It is done using the identity t i t j = t j t i for |i − j| > 1, that follows from the definition of the t i (n). Here is an example. The two expressions for η 2,3 are (t 3 t 4 )(t 2 t 3 )(t 1 t 2 ) and (t 3 t 2 t 1 )(t 4 t 3 t 2 ) and those for η 2,2 are (t 2 t 3 )(t 1 t 2 ) and (t 2 t 1 )(t 3 t 2 ). Assuming that the latter are equal, the former are shown to be equal by where the two expressions for η 2,2 gives the first equatlity while the commutativity of t 4 with t 2 and t 1 gives the second. The argument can be extended into a proof by induction on the sum r + s of the indices of η r,s . The next step is to find an expression for η 1,1 . Since η 1,1 : 1 ⊗ 1 → 1 ⊗ 1 is an element of End(2) ≃ TL 2 (β ), which is two-dimensional, there exists α, γ ∈ C such that η 1,1 = α1 2 + γe 1 (2), where the notation is used. It can be checked directly from this definition that the e i satisfy the Temperley-Lieb defining relations: In fact, it can be proved that the set {e i (n)} 1≤i<n generates End(n) = TL n (β ). Using these relations, it can be seen that η 1,1 is invertible provided that α = 0. Now, if the family of η r,s is to define a commutor then, in particular, it must verify where z = ∈ Hom TL (0, 2) and η 1,2 = (1 1 ⊗ η 1,1 ) • (η 1,1 ⊗ 1 1 ) = α 2 1 3 + αγ(e 1 (3) + e 2 (3)) + γ 2 e 2 (3)e 1 (3). The first equation of (9) will be satisfied if and only if α 2 + β αγ + γ 2 = 0, while the second will be if and only if αγ = 1. Solving these equations yields where q ∈ C × is such that β = −q − q −1 and the two ± signs are independent. There are thus four solutions. Note that one of the ± is responsible for an overal sign on η 1,1 while the remaining one mirrors the invariance of β under q → q −1 . Without loss of generalities, we shall concentrate on the following choice: and η 1,1 : 1 ⊗ 1 → 1 ⊗ 1 is simply η 1,1 = t 1 (2). These building blocks t i (n) of the η r,s have appeared numerous times in the literature. The identity (16) below was recognized by Chow [12] as crucial to identify the center of braid groups. Much later Martin [13] used the t i (up to a factor) to construct central elements of the Temperley-Lieb algebra.
It can also be useful to introduce diagrams representing t i (n) and t i (n) −1 ; we choose the following The other t i (n) can be built from these by using the tensor product of morphisms. These diagrams are concatenated using the same rules as for the other diagrams representing morphisms in the category, so diagrams with isotopic strings are equivalent. Note however that diagrams related through a Reidemeister move of type I are not necessarily equivalent; for instance, The following lemmas will give the behaviour of these crossings under the Reidermeister moves of the two other types.
It now remains to show that this choice does defines a braiding on TL, but doing so requires a few lemmas. From now on, we shall omit the arguments specifying the Hom-space, unless they are needed to avoid confusion, and assume that these arguments are large enough for the expressions to make sense. For example the next lemma proves that t i t i+1 e i = e i+1 e i . The statement stands for t i (n)t i+1 (n)e i (n) = e i+1 (n)e i (n) for all i + 2 ≤ n as t i+1 (n) and e i+1 (n) act non-trivially on nodes i + 1 and i + 2 of the elements of Hom(n, n). The next three lemmas prepare the proof that the η r,s 's are natural isomorphisms and thus define a braiding on TL. The first is obtained by direct computation.
Lemma 2.3. The morphisms t i and e i satisfy In terms of diagrams, this lemma can be written Combining these identities with the definition of the braiding η n,m gives its diagrammatic picture, for instance The next one is almost as easy.
Thus, for all f ∈ End(n) and g ∈ End(m), Proof. If 1 ≤ k ≤ i and thus k ≤ i < k + n − 1, the preceding lemma and equation (8) give

It follows that
The second identity in (20) is proved similarly using the second expression of (6). Finally, (21) follows from the fact that End(n) ≃ TL n is generated by the e i .
where η 0,n = η n,0 = 1 n , z is defined in equation (10), (z) t is its transpose, and Proof. We prove the first identity only as both proofs are nearly identical. We proceed first by induction on n and then on p. If p = n = 1, the equation is the second of the two equations in (9) that were solved to construct the t i and obtain (11). Suppose therefore that the result stands for p = 1 and some n ≥ 1. The hexagon identity (4) gives Assume then that the result stands for some p ≥ 1 and all n ≥ 1. The hexagon identity (3) gives In terms of diagrams, this lemma simply states that the two points linked together on the right side of the diagrams in equation (17) can be moved over the underlying links.
With these three lemmas, we are now ready to prove the main result of this section.
Proposition 2.6. The category TL is braided with a commutor having components and t i (n) = q 1/2 (1 n + q −1 e i (n)).
Proof. The category TL will be braided if the components η r,s are natural isomorphisms satisfying the hexagon axioms. Lemma 2.2 has already showed that the proposed expressions for the components η r,s satisfy the hexagon axioms. Moreover, since t i (n) is invertible, so are the morphisms η r,s . There remains only the naturality condition to prove. It states the following: For all pairs (n, m) and (r, s) in Ob TL × TL and all pairs of morphisms (c, d) ∈ Hom(n, r) × Hom(m, s), the following diagram commutes Since the Hom-spaces are spanned by diagrams and that the η r,s are bilinear, it is sufficient to prove that for any (r, n)-diagram c and (s, m)-diagram d.
Consider then c ∈ Hom TL (n, r) a diagram having k through lines, that is, precisely k nodes on the left side of c are connected to k nodes on its right side. Any such diagram can be expressed as where a ∈ End r and b ∈ End n. The hexagon identities and lemma 2.5 give The same arguments also give Using lemma 2.4, it follows that The same steps are used to prove that any diagram d ∈ Hom(m, s) with ℓ through lines satisfies η n,s (1 n ⊗ d) = (d ⊗ 1 n )η n,m . Then, for any (r, n)-diagram c and (s, m)-diagram d, these identities give which closes the proof.

2.3.
The twist θ . The previous section established that the category TL is braided. It has even more structure: It has a twist.
A twist θ on a braided category C is a natural isomorphism of the identity functor whose components{θ This section constructs such a natural isomorphism for the Temperley-Lieb category TL. The first step toward this goal is to solve a subset of these equations, namely those that have either r or s equal to 1. The next lemma is a corollary of Lemma 2.7. The morphisms t i satisfy Proof. Lemma 2.3 provides the cases t i t i+1 t i = t i+1 t i t i+1 for all i ≥ 1. Then, for a fixed i, induction on n gives The solution of (26) when either r or s is 1 is given by a family of central elements c n whose main properties are proved in Appendix A. (To our knowledge, as an element of TL n , the element c n appeared first in Martin's book (see section 6.1 of [13]), but was actually defined much earlier by Chow [12] to study braid groups.) and Proof. Note first that the two equations are trivial for n = 0 and, for n = 1, they both give as claimed. Suppose now that the c k , k ≤ n, all satisfy both equations. Then where the identity (27) was used repeatedly to transform the sequences of generators between square brackets. Since, by Proposition A.2 (d), the central element c n can be written both as q 3n/2 (t n−1 . . .t 2 t 1 ) n and q 3n/2 (t 1 t 2 . . .t n−1 ) n , a similar argument using the latter form may be used to show that the family {c n } also solves the second identity in (28).
The diagram representing c n is quite convoluted, but we nevertheless illustrate the first identity in (28) for n = 3 and with the symbol ∼ meaning "equal up to a power of q": The final identification of the twist {θ n , n ∈ N 0 } requires yet another technical lemma.
Lemma 2.9. The commutor {η r,s } satisfies for all r and s ∈ N 0 such that all indices in these identities are non-negative.
Proof. The proof of the first identity proceeds as follows and uses the explicit form (23) of the commutor: and then moving the leftmost t r ,t r+1 , . . . , and t r+s−1 to their rightmost possible positions within the square brackets The second identity is obtained from the first by the substitutions r − 1 → s, s + 1 → r.
In terms of diagrams, the identity (30) (with s = r = 3) is This lemma simplifies considerably the proof of the existence of the twist.
Proposition 2.10. The morphisms in End(n) given by the multiplication by θ n = c n , n ∈ N 0 , define a natural isomorphism θ between the identity functor and itself, and is a twist for the commutor η , .
Proof. 3 The first step is to prove that the family {θ n = c n , n ∈ N 0 } satisfy (26), which amounts to establish Indeed Lemma 2.8 has proved η 1,n η n,1 (θ n ⊗ θ 1 ) = θ n+1 and, starting from the latter, the above identity proves recursively the identities (26) for all r and s such that n + 1 = r + s. Note that equation (28) holds also for the components θ n = c n . We first rewrite (θ r−1 ⊗ θ s+1 ) in terms of (θ r ⊗ θ s ): again by (28), It is then sufficient to prove With the technical Lemma 2.9, this is now straightforward: by the naturality (24) of η, (30) and (31), which ends the proof of (26).
It remains to prove that θ defines a natural isomorphism of the identity functor, i.e. that for all f ∈ Hom (m, n) Let f ∈ Hom (m, n) be a diagram with k through lines. As before (see equation (25) of the proof of proposition 2.6), such a diagram can be written as for some a ∈ End (n), b ∈ End (m). Since θ n is central in End (n) by proposition A.2, Steps 2 and 5 were obtained by using equation (26). Steps 3 and 4 rest on two observations. (It is here that the factor q 3n/2 plays an essential role!) First η is a natural transformation and thus Hom (n, 0) ≃ S n,0 as left End (n)-modules.

THE CATEGORY OF MODULES OVER TEMPERLEY-LIEB ALGEBRAS
3.1. Braiding modules. The representation theory of the family of TL n , n ∈ N 0 , was cast into a categorical framework by Graham and Lehrer [1] as follows. Let F ∈ Funct( TL, Vect C ) be a functor from the category TL to that of finitedimensional vector spaces over C. Then each F(n), n ∈ N 0 , is a vector space and each F(α), for α ∈ Hom(n, n) ≃ TL n is a linear map F(n) → F(n). Since F preserves composition, F(n) is naturally a TL n -module, but the functor F is somewhat richer than a choice of a TL n -module for each n ≥ 0. Indeed the functor F also gives linear maps F(γ) : for all γ ∈ Hom(n, m) between modules over distinct algebras of the Temperley-Lieb family. These linear maps must also preserve the composition of diagrams. We now give examples of such functors taken from [1].
Let k ∈ N 0 . A functor S k ∈ Funct( TL, Vect C ) is defined as follows. If the parities of n and k are distinct, then the vector space S k (n) is set to 0. If their parities coincide, then S k (n) is the formal span of (n, k)-diagrams with exactly k through lines. If α ∈ Hom(n, m) with m, n and k having the same parity, then S k (α) : S k (n) → S k (m) is the linear map defined by its action on (n, k)-diagrams with k through lines. If γ ∈ S k (n) is such diagram, then S k (α)γ ∈ Hom(k, m) is α • γ if α • γ has k through lines and 0 otherwise. For all other α ∈ Hom(m ′ , n ′ ), that is with m ′ or n ′ not sharing the parity of k, the linear map S k (α) is zero. It is straightforward to check that S k is a functor and that S k (n) is the usual with the action on morphisms f : n → k given by for α ∈ Hom(m, n) and x ∈ M.
Note that, given a pair (m, M), the functor F m,M is isomorphic to a functor F m,M ′ for a certain TL m -module M ′ . Indeed Note that, since clearly Mod TL ⊂ Funct( TL, Vect C ), the direct sums are defined in the same way on both. In particular, Here is a simple example of a natural transformation between for any c ∈ TL m . The naturality of φ f is easily checked. For b ∈ Hom(k, l) Hom(k, l). Other examples are given below.
Let M and N be a TL m -and a TL n -module, respectively. A fusion product M × f N was first defined by Read and Saleur [2] and later on computed systematically by Gainutdinov and Vasseur [3] and Belletête [4]. To endow Mod TL with a braided structure, 4 we extend their definition to pairs (m, M) and (n, N) of partitions and choices of modules: The fusion M × f N is thus a left TL m+n -module. This (slightly more general) definition makes the introduction of a bifunctor − 1 × f − 2 on Mod TL straightforward: which can be rewritten as The bifunctor's action on morphism is defined in the obvious way, that is, for g : F m,M → F u,U a morphism, then the morphism g × f F n,N is defined through its components (removing the overhead bars for simplicity) for all α ∈ Hom(m + n, k), x ∈ M, y ∈ N, and {γ z,  Proof. Let ϕ : F 2,S 2,0 → F 0,TL 0 be defined by where f ∈ Hom TL (2, k). Then, if a ∈ Hom(k, l): that is, ϕ is a morphism or, in other words ϕ ∈ Nat(F 2,S 2,0 , F 0,TL 0 ) Clearly ϕ has an inverse defined by for c ∈ C = TL 0 and f ∈ Hom(0, k), and ϕ −1 is also a natural transformation. Thus ϕ is a natural isomorphism.
The set of natural transformations from the functor F n,TL n , where TL n is seen here as the left regular module, to the functor F m,M can be made explicit. Indeed, by definition of F n,TL n (−) = Hom(n, −) ⊗ TL n TL n ≃ Hom(n, −). Thus Nat(F n,TL n , F m,M ) ≃ Nat (Hom(n, −) as vector spaces, where the last isomorphism follows from Yoneda's lemma. In particular again as vector spaces. The definition (36) of objects in Mod TL allows for an easy definition of an associator, also noted α, on Mod TL . Let F l,L , F m,M and F n,N be three objects in Mod TL . Then Let k ∈ N 0 , f ∈ Hom(l + m + n, k) and x ∈ L, y ∈ M, z ∈ N. Then the associator α (l,L),(m,M),(n,N) must act as The verification of the triangle and pentagon axioms for these unitors λ , ρ and associator α then mimics that for the usual tensor product of vector spaces.
A lighter notation will be used from now on. The tensor signs ⊗ C and ⊗ TL m ⊗ TL TL n will be replaced by ⊗ and ⊗ m,n respectively and the functor F m,M by F M . Furthermore, when appearing in indices, we shall write only M instead of F M .
The braiding η of TL(β ) induces one on Mod TL as follows: for all x ∈ M, y ∈ N, a ∈ Hom(n + m, k), k ∈ TL, and extended linearly in the natural way. The components of η are well-defined natural morphisms in Mod TL . Suppose indeed that c ∈ TL m , d ∈ TL n , b ∈ Hom(k, s). Then (η M,N ) s (ba ⊗ m,n (cx ⊗ dy)) = baη n,m ⊗ n,m (dy ⊗ cx) It is straightforward (but tedious) to show that η , is natural in both entries. (An example of such verification is done below for the twist θ.) The check of the hexagonal axiom is the last step. Again any element of (( is a linear combination of terms of the form b ⊗ l,m,n ((x ⊗ y) ⊗ z) where b ∈ Hom(l + m + n, k) for some k, and x ∈ L, y ∈ M, z ∈ N.
Then the upper part of the hexagon gives (we dropped the k index to lighten the notation) = bη m+n,l ⊗ m,n,l (y ⊗ (z ⊗ x)) and the lower one = bη m+n,l ⊗ m,n,l (y ⊗ (z ⊗ x)), by (1).
Since the two expressions coincide, the family of commutors η M,N , for (m, M), (n, N) ∈ Ob Mod TL , satisfies the first hexagon axiom. The proof of the second is similar and the above discussion establishes the braided structure of Mod TL .

Proposition 3.3. The category
Mod TL together with the bifunctor × f , the identity I, the associator α, the unitors λ , ρ and the braiding η is a braided category.

3.2.
Rigidity and Mod TL . The representation theory of rational conformal field theories offers examples of modular categories. These categories satisfy even more structural constraints than the braided tensor ones. This section identifies conditions under which the braided tensor categories Mod TL can be endowed with some of these additional structures.
A braided category is rigid if, for every object F in the category, there correpond two others, its right and left duals F * and * F, and morphisms such that the compositions are the identity morphism, on F and F * , respectively. If such morphisms exists, they are unique up to isomorphism.
Similar axioms hold for e ′ F and ι ′ F . The rigidity axiom insures, in CFT, that any primary field φ has a right partner φ * and a left one * φ such that the correlation functions (φ * )φ and φ ( * φ ) are non-zero. Often the left and right partners coincide. Actually in all rigid braided category, the two duals are always isomorphic, since once can show that (ι F ) ′ ≡ η −1 F,F * • ι F : 1 → F * ⊗ F, and (e F ) ′ ≡ e F • η F,F * : 1 → F ⊗ F * , also satisfy the axioms for the right dual. Note that the category TL is rigid, with n * ≡ * n ≡ n for all objects n ∈ TL, and duality morphisms ι m =   The rigidity axiom is fulfilled for Mod TL when the algebras TL n (β ) are semisimple for all n, that is when q is not a root of unity (recall that β = −q − q −1 ). For these values of β , the cellular modules S n,k , 0 ≤ k ≤ n with k = n mod 2, form a complete set of non-isomorphic irreducible modules of TL n . All other (finite) modules of TL n are direct sums of these. Their fusion product is known [3,4].

Proposition 3.4. If q is not a root of unity, then
and where ⊕ ′ stands for a direct sum with a step equal to 2.
This simple fusion rule leads to the following result.

as vector spaces.
In other words F m × f F n has a direct summand isomorphic to I = is non-zero. Thus define the components ι m (k) : where f ∈ Hom(0, k) and α ∈ C a constant to be fixed. A quick check shows that this is a morphism. (This morphism ι m exists even for q a root of unity.) Note that the definition of ι m usesē m ∈ Hom(2m, 0) and, as will be seen below, that of e m usesῑ m ∈ Hom(0, 2m), a fact that could be confusing. The definition of e m : F m × f F m → I requires the primitive idempotent wj m ∈ TL m , known as the Wenzl-Jones projector.
It is the unique non-zero element of TL m such that wj m · wj m = wj m and wj m · e i = e i · wj m = 0, 1 ≤ i < m [16]. Thus for g ∈ Hom(2m, k) some α ′ ∈ C. The two constants α and α ′ are tied by the rigidity axioms (47) and (48). For example (47) gives for f ∈ Hom(m, k) Proposition 3.6. If q is not root of unity, the category Mod TL together with the bifunctor × f , the identity I, the associator α, the unitors λ , ρ, the braiding η and the morphisms e, ι, e ′ and ι ′ , is rigid.
A ribbon category is a rigid category endowed with a functorial isomorphism θ and a dual isomorphism θ * satisfying (θ F ) * = θ F * and * (θ F ) = θ * F . The isomorphism θ is called a twist. The duals are defined by the composition (omitting associators and unitors) and similarly for the left duals. The natural isomorphism θ for Mod TL will first be constructed and then the compatibility between duals checked.
The twist on TL induces a twist on Mod TL as follows. For c ∈ Hom(m, k) and Since the elements θ m ∈ TL m are central, θ M (k) is well-defined. Since its (unique) eigenvalue on an indecomposable module is never zero, θ M (k) is also invertible. The next step is to prove that it is a natural transformation of the identity functor of Mod TL ; to see this, one must prove that for all functors F N , F M ∈ Mod TL , and all natural transformation µ : F N → F M ∈ Hom Mod TL , the components of (µ • θ N ) and (θ M • µ) must be equal on all objects k ∈ TL. Let a ⊗ TL x be some element of F N (k), so a ∈ Hom(n, k), x ∈ N, and write µ k (a ⊗ TL x) ≡ ∑ i b i ⊗ TL y i , where the sum is over some finite set, the b i are in TL m and the y i in M. One then verifies that where we used the fact that θ is a natural transformation on TL and µ is natural on Mod TL . Since (µ • θ N ) k and (θ M • µ) are both linear, it follows that they must be equal on F N (k). Finally, let a ∈ Hom(m + n, k) and x ∈ M and y ∈ N such that a ⊗ m,n (x ⊗ y) ∈ F M× f N . Then ,n (x ⊗ y)), and the twist θ on Mod TL verifies the axiom (26).
It simply remains to verify that the twist is compatible with the duals; we show that it is compatible with right duals, as the proof for the left ones is very similar. First note that for all 1 ≤ i ≤ m, whereῑ m ,ē m are the duality morphisms from TL, introduced in equation (49). It therefore follows that where e m and ι m are now the components of e F and ι F . Using this observation with the definition of the twist and duals in Mod TL , one quickly sees that for all F ∈ Mod TL The right dual of θ F is thus where we have used the fact that F * ≡ F and, to get the last line, that Mod TL is rigid. The twist θ in Mod TL is thus compatible with its duals.
These checks on θ holds for any q. However, for a category to be a ribbon category, it needs to be rigid and thus, for Mod TL , q may not be a root of unity.
Note that we also proved that the twist in TL is also compatible with its duals, so TL (with the appropriate data) is also a ribbon category.
The categories appearing naturally in minimal conformal field theories are the modular tensor categories. Beside being ribbon categories, the modular ones require among other things that all objects can be written as a finite direct sum of simple objects and that the number of (isomorphic classes of) simple objects be finite. (An object A in an abelian category C is simple if any injective morphism B → A is either 0 or an isomorphism.) When q is not a root of unity, the simple objects are the standard modules S n,k , n ∈ N 0 , and 0 ≤ k ≤ n with n = k mod 2. Corollary 3.2 shows that F n,S n,n , n ∈ N 0 , are all simple objects and non-isomorphic. However their number is infinite and the ribbon category Mod TL cannot be a modular one.
The category Mod TL cannot be rigid when q is a root of unity. Indeed rigidity would imply the exactness of the bifunctor × f which it is not when q is a root of unity [4]. But it turns out that × f is closed when restricted to projective modules, even when q is a root of unity. It is thus natural to consider the full subcategory Mod proj TL with objects restricted to projective modules (or, more precisely, functors of the form F n,P where P is projective TL n -module). The fusion coefficients are more intricate than those of Corollary 3.5. (See Proposition 4.1.1 of [3] or Section 4.1 and the quick computational tool 5.4 of [4].) However the Corollary's key feature, the one that allows for the introduction of morphisms e F , e ′ F , ι F and ι ′ F , still holds: as vector spaces, when q is a root of unity such that the smallest positive integer ℓ such that q 2ℓ = 1 is larger than 2.
Here F n stands now for F n,P n,n . The morphisms (45) and (46) thus exist. We have checked on a few cases that the rigidity conditions (47) hold for proper choices of the e's and ι's. The full subcategory Mod proj TL has an unexpected feature however: it is not any more an abelian category, a characteristic that is usually assumed in the study of fusion.
Recall that, in an abelian category, every morphism has a kernel and a cokernel in the category. But, when a projective module P n,k has three or four composition factors, there are morphisms P n,k → P n,k whose kernels and cokernels are not projective and thus absent from the subcategory Mod The nature of the monodromy is easy to describe in the semisimple case, i.e. when q is not a root of unity. However the morphism may have a nilpotent part when q is a root of unity, as will be seen below. In the following we use fairly standard notations, writing I n,k , S n,k and P n,k for the irreducible, standard and projective modules over TL n . The standard S n,k was described at the beginning of Section 3.1, the irreducible I n,k is its irreducible quotient and P n,k the projective cover of I n,k . (See [14,4], and also [1] where S n,k is denoted by W k (n).) The twist θ defines isomorphisms of modules over the Temperley-Lieb algebras. Indeed they are defined through the invertible central elements θ n of TL n and thus define isomorphisms of modules by left multiplication. The defining property (26) of the twist θ gives a rather explicit expression for the monodromy. We choose to state this (obvious) fact in a proposition to underline its crucial character and ease further references.
When q is generic (not a root of unity), then the algebras TL n are semisimple for all n and the standard modules S n,k , with 0 ≤ k ≤ n and k ≡ n mod 2, provide a complete list of non-isomorphic irreducible modules. Their fusion was given in terms of the associated functor in Proposition 3.4. Here is a simpler statement in terms of the modules themselves. Proposition 3.9 ( [3,4]). Let n 1 , n 2 ≥ 1 and k 1 , k 2 such that 0 ≤ k i ≤ n i and k i ≡ n i mod 2. Then when q is not a root of unity. Again ⊕ ′ indicates a direct sum whose index step is two.
Proposition 3.8 then gives a complete characterization of the monodromy of standard modules in this generic case.
Proposition 3.10. The monodromy η S n 2 ,k 2 ,S n 1 ,k 1 • η S n 1 ,k 1 ,S n 2 ,k 2 acts on the direct summand S n 1 +n 2 ,k of S n 1 ,k 1 × f S n 2 ,k 2 as the identity times when q is not a root of unity.
Proof. The central elements c n = θ n act as multiples of the identity on the irreducible S n,k by Schur's lemma. The eigenvalues γ n,k of c n on the standard modules S n,k are obtained in Proposition A.2. Thus The restriction of the monodromy to the direct summand S n 1 +n 2 ,k of the fusion is then γ n 1 +n 2 ,k /(γ n 1 ,k 1 γ n 2 ,k 2 ). A direct computation leads to the desired expression.
It is worthwhile to note that the multiple of the identity is independent of n 1 and n 2 whose only role here is to fix the parities of k 1 and k 2 .
When q is a root of unity, the monodromy η N,M • η M,N is still given by Proposition 3.8, but it might not be a multiple of the identity. Indeed the action of the central elements c n on TL n -modules is in general not such a multiple. The Corollary A.3 and the paragraph leading to it show that such a non-trivial action, i.e. a non-diagonalisable action, might occur only on projective modules with three or four composition factors. Let P n.k be such a projective module. Then Hom(P n,k , P n,k ) is two-dimensional. Beside the identity id, there is a map sending the head of P n,k to its socle, both being isomorphic to I n,k . Let f be this map, that is, a map such that im f is the socle of P n,k . This map is nilpotent: We now give two examples of such non-trivial action of the monodromy.
One of the simplest cases occurs when q 2ℓ = 1 for ℓ = 3. Then, even though the standard modules S 2,2 and S 1,1 are irreducible, their fusion S 2,2 × f S 1,1 is not. Using the expressions computed in [3,4], one finds S 2,2 × f S 1,1 ≃ P 3,3 where P 3,3 is an indecomposable projective module. This projective is three-dimensional, has three one-dimensional composition factors: I 3,1 once and I 3,3 twice. The latter composition factors are isomorphic to the socle and head of the module. Hence, even though θ 2 and θ 1 act as multiples of the identity on S 2,2 and S 1,1 , the morphism defined by c 3 is not diagonalisable on P 3,3 . In fact a direct computation shows that the monodromy in this case is η S 1,1 ,S 2,2 • η S 2,2 ,S 1,1 = e 4πi/3 · id +ν f if q is chosen to be e 2πi/3 . Here ν is a non-zero constant (that depends on the basis) and f is the map described above. Note that the (unique) eigenvalue of the monodromy is still correctly predicted by (52), as it should be: µ 2,1,3 = q 2 = e 4πi/3 .
Our second example is more intricate: we shall study the monodromy on the product TL 2 × f TL 2 for q generic and q = √ −1. If q is generic, then TL 2 ≃ S 2,0 ⊕ S 2,2 . The linearity of the fusion together with Proposition 3.9 gives Since q is generic, the monodromy η TL 2 ,TL 2 • η TL 2 ,TL 2 is diagonalisable with eigenvalues given by Proposition 3.10. The eigenvalues on the two copies isomorphic to S 4,0 are q −8 and 1, on the three S 4,2 they are q −4 , 1 and 1, and on S 4,4 it is q 4 . (Note that η TL 2 ,TL 2 • η TL 2 ,TL 2 does not take the same eigenvalues on isomorphic copies of the standard modules in TL 4 , since the multiple µ k 1 ,k 2 ,k depends also on the modules begin fused.) If q = √ −1 and thus β = 0, then TL 2 ≃ P 2,2 and the fusion is then TL 2 × f TL 2 ≃ P 2,2 × f P 2,2 ≃ P 4,2 ⊕ P 4,2 ⊕ P 4,4 .
At this value of q, the isomorphism η TL 2 ,TL 2 • η TL 2 ,TL 2 is even more complicated because none of the three isomorphisms in θ TL 4 (θ TL 2 ⊗ θ TL 2 ) −1 is a multiple of the identity on the modules they act upon. Still it has a unique eigenvalue as µ 2,2,0 = µ 2,2,2 = µ 2,2,4 = 1. While η TL 2 ,TL 2 • η TL 2 ,TL 2 is non-diagonalisable, it is possible to find two subspaces A and B, both isomorphic to P 4,2 that allows for an easy description of the morphism. Let C be the other summand P 4,4 . Then δ = η TL 2 ,TL 2 • η TL 2 ,TL 2 can be broken down into its action on each summand as  They are where σ , ν, ρ are non-zero constants and id A,B stands for the isomorphism between A and B. From these maps, it is straighforward to compute the Jordan form of η. Its non-trivial Jordan blocks are 2 blocks 3 × 3 and 2 blocks 2 × 2.
The root q = √ −1 is somewhat special in the representation theory of the algebra TL n : It is the only value for which the semisimplicity of TL n varies with the parity of n. (For all other roots q 2ℓ = 1 with ℓ ≥ 3, the algebra TL n (β = −q − q −1 ), n ≥ ℓ, is never semisimple.) Although the example above was given at this particular value q = √ −1, it seems to be representative of what happens at other values of q.

BRAIDING AND INTEGRABILITY
One of the most profound uses of Temperley-Lieb algebras in physics is in the study of solvable models, like the XXZ Hamiltonians or loop models on two-dimensional lattices. The goal of the present section is to tie braiding and integrability in some of these statistical models. The former will appear through the elementary brainding η 1,1 (or t i (n)) that was used in Proposition 2.6 to write all other components η r,s of the braiding natural isomorphism. The latter will also be cast in terms of a fundamental "face operator" that must satisfy three identities. The physical object, that is, the Hamiltonian or the transfer matrix, is then defined in terms of several copies of this face operator.
In the literature on statistical models, the face operator X i (q, u) is also an element of one of the algebras TL n (β ). It depends on several parameters: The spectral parameter λ , tied to β by β = −q − q −1 and q = e iλ , and the anisotropy parameter u that measures the ratio of the interaction constants along two linearly independent vectors spanning the lattice. As for the t i , the X i is usually a linear combination of TL n generators and it is represented graphically by Since all faces will be evaluated at the same value of the parameter q (or λ ), this parameter is often omitted. In terms of the face X i (q, u), the transfer matrix D n (λ , u) ∈ Hom(n, n) on n sites is constructed out of 2n tiles organized in diagonal lines. For example the case n = 3 is depicted as follows: In the notation of the previous section, it is thus Its physical properties are revealed through its spectrum in some representations. It was recognized by Behrend, Pearce and O'Brien [17] that some algebraic conditions on the face operator X i (q, u) ensure that the transfer matrix, constructed from it, will have the properties that D n (λ , u) • D n (λ , v) − D n (λ , v) • D n (λ , u) = 0 in TL n . This means that, in any representation φ : TL n → gl(V) with V some vector space, the matrices φ (D n (λ , u)) and φ (D n (λ , v)) will commute for all values u and v. The modes φ (D n (λ , u)) in any expansion with respect to u (Taylor's expansion, Fourier's, ...) will commute, that is, they will be integrals of motions. Thus the integrability of the models based on such a transfer matrix D n follows from these algebraic conditions. Here they are.
These conditions are found in the literature drawn as follows: It is not too difficult to construct such a face operator X i out of the elementary braiding element η 1,1 = t i . As an intermediary step, consider i . Both products y i (u)y i+1 (v)y i (w) and y i+1 (w)y i (v)y i+1 (u) contain eight terms, each cubic in the generators t i , t i+1 and their inverses. The identity (16) gives rise to the six following ones: . Thanks to these identities, all sixteen terms of the difference of triple products of the y i cancel pairwise, but four: Moreover it is easily verified that So the difference (57) will be zero if quv −1 w = u −1 vw −1 . This is easily achieved with the following definition of the x i .
Proposition 4.2. Let n ≥ 2. The x i (q, u) defined by satisfy the three conditions (54)-(56) with ρ(q, u) = ((q 2 + q −2 ) − (u 2 + u −2 )). The solution x i of the three conditions in Proposition 4.1 is well-known. For example, Section 3 of [7] is devoted to this solution and its relationship with the Temperley-Lieb algebra. (Note that their λ and our q is related by q = e iλ and their u and ours is also related by an exponential. Their β is q + q −1 while ours is −q − q −1 . Finally they consider a larger class of boundary conditions that those above.) However the above discussion shows how the braiding of the Temperley-Lieb category TL and integrability of statistical models are intimately related.

THE DILUTE CATEGORY dTL
The dilute Temperley-Lieb algebras dTL n (β ) are a family of algebras defined through diagrams similar to those appearing in the original algebras TL n (β ). This family can be cast into a category dTL similar to the category TL introduced in Section 2.2. This new category can also be given a braided structure. This section introduces this structure and discusses the relationship between the braiding on dTL and the integrability of dilute statistical models.
We start by giving the definition of the category dTL, while recalling the definitions of the algebras dTL n themselves. (See [18] for further details on the dilute family.) The objects of the dilute Temperley-Lieb category dTL are the nonnegative integers. The morphisms between two integers n and m are defined as linear combinations of dilute (n, m)diagrams. These dilute diagrams are defined in the same way as the (n, m)-diagrams appearing in TL except that nodes on either sides of the diagrams are now allowed to be free of strings; a node without a string is called a vacancy. For example, the following are all acceptable dilute diagrams: , and .
The first two are elements of Hom (2,4) and the last of Hom (4,5). Composition of morphisms is defined by extending bilinearly the following composition rule. For b and c dilute (m, n)-and (k, m)-diagrams, the composition b • c is a dilute (k, n)-diagram defined by first drawing b on the left of c, identifying the points on the m neighbouring sites, joining the strings that meets there, and then removing the points on this side. If a string is closed in this process, it is removed and the diagram obtained is multiplied by β = −q − q −1 ∈ C. If a string is attached to only one of its extremities (because it was joined to a vacancy during the composition), the result b • c is the zero morphism 5 . Here are few examples of these compositions. In the first b ∈ Hom(4, 5), c ∈ Hom(2, 4) and b • c ∈ Hom(2, 5).
For a stricly positive integer n, the algebra dTL n (β ) is identified to the set Hom(n, n) with the product being the composition just defined.
Endowing this category with a monoidal structure is a straigthforward generalisation of the one on TL. Again define n ⊗ m ≡ n + m. For morphisms, if a and b are dilute (n, m)-and (r, s)-diagrams, define their tensor product a ⊗ b as the (n + r, m + s)-diagram obtained by simply putting a on top of b, as in TL, and extend this bilinearly to all morphisms.
With the associator and the unitors as identities, this tensor product makes dTL into a strict monoidal category. 5 Note that the case of a string ending at a vacancy can also be resolved by simply removing it and replacing its two ends by vacancies. This then yields a different structure, the planar rook algebras (see, for example, [19]).
The commutor for dTL is obtained similarly to that of TL. We only outline the computation of η 1,1 . The space End 2 is spanned by the following 9 diagrams: and the elementary commutor η 1,1 is a linear combination of these nine diagrams. A line in any diagram will mean the sum of two diagrams, the first with a straight line between the two nodes, the second with nothing between these nodes that are then vacancies. The identity 1 1 ∈ Hom(1, 1) is thus such a dashed line and the identity in End 2 is the sum of the first four of the nine diagrams above. Four of the coefficients of η 1,1 are easily set to zero by the following requirements: The commutor η 1,1 is thus found to be a sum of the remaining five diagrams: As in Section 2.2, we define t i (n) ≡ 1 i−1 ⊗ η 1,1 ⊗ 1 n−i−1 ∈ Hom(n, n) and write η 1,2 = t 2 t 1 and η 2,1 = t 1 t 2 . The conditions are then η 1,2 (a ⊗ b) = (b ⊗ a)η 1,2 and η 2, for all a ∈ dTL 1 and b ∈ dTL 2 . Choosing a as 1 1 and b as one of the following ones gives the algebraic equations a 2 1 + a 1 a 5 β + a 2 5 = 0 and a 2 2 = a 2 3 = a 2 4 = a 1 a 5 .
Proposition 5.1. The category dTL is braided for a commutor with components η r,s given by (23), t i (n) ≡ 1 i−1 ⊗ η 1,1 ⊗ 1 n−i−1 and η 1,1 and η −1 1,1 now given by It follows that a disjoint module category Mod dTL can be defined along the lines introduced in Section 3.1 and that it is also braided.
In the case of the original Temperley-Lieb algebras, the construction of x i (q, u) satisfying the three conditions (54)-(56) rested on the identities (16) and (58). These can be shown to be satisfied by the t i defined with η 1,1 in (60). The elementary braiding (60) thus leads again to the following non-trivial solution of the Yang-Baxter equation (54): where η 1,1 is now given by (60) and the x i are understood as elements of Hom(n, n). Does this solution also satisfy the two other conditions (55) and (56)? For the latter, one has first to decide what is to replace the "boundary terms" (z ⊗ z).
A direct calculation shows that (56) is satisfied by the dilute x i for only three boundary conditions, namely , and .
Finally the dilute x i does not satisfy the inversion relation (55): While this non-trivial solution only partially satisfies (54)-(56), there is another one that solves the three conditions. It uses Boltzmann weights discovered by Izergin and Korepin [20], and Nienhuis [21]. It iŝ where Like x i , thisx i solves the Yang-Baxter equation. But it also satisfies the inversion relation (with a new functionρ) and the boundary Yang-Baxter equation with particular boundary conditions [22,23]. Notice that, up to a global factor, the y ± are the commutors η ±1 1,1 that would have been obtained if a 1 would have been chosen as −q 1 2 . The other three terms w + , w − and z are however completely different. It is not clear whether the integrable model it defines is related to a braiding for a different bifunctor − ⊗ ′ −.

CONCLUSION
The main results of this article lie in Sections 2 and 3. In Section 2, the category TL was given the structure of a braided category. Even though several of the functors and natural morphisms were already known, casting them in TL pursues Graham and Lehrer's goal of understanding the family of Temperley-Lieb algebras as a whole. This goal highlights properties of the family that are shared by all TL n , independently of n. It is also very natural physically speaking as the continuum limit of the lattice models defined using the TL family is often their raison d'être. Section 3 introduced the category Mod TL of modules over TL and used the functors and natural morphisms of TL to induce the structure of a ribbon category on Mod TL when q is generic. The tools developed showed how non-trivial can the monodromy be, even for the finite associative TL algebras. Section 3 also explained that rigidity cannot be implemented straightforwardly on Mod TL when q is a root of unity.
The subcategory Mod proj TL of projective modules might satisfy the axioms (47) and (48), but it fails to be abelian. Another possibility would be to consider the subcategory of irreducible modules I n,k with k on the left of the first critical line. Theorem 6.11 of [4] shows that × f is closed on this subset of modules. But a more central question is what are "approriate" weaker forms of rigidity for Mod TL at q a root of unity.
The question also arises of the existence of a commutor for other family of algebras and its eventual link to integrable models defined using them. Section 5 showed that the "elementary braiding η 1,1 " does not reproduce the transfer matrix defining dilute loop models. Is it possible to understand better the link between this η 1,1 and the integrability of the dilute models? There are other algebras physically relevant in statistical physics, for example the one-boundary TL family (also known as the blob algebra [24]) and the affine (periodic) TL family. It is not known whether one can define a fusion product between their modules or even make a braided category out of their link diagrams.

APPENDIX A. THE CENTRAL ELEMENT c n
We gather in this appendix the properties of the central element c n ∈ TL n together with their proofs. Some of these results are to be found in [13].
The two elements of TL n ρ n = t 1 t 2 . . .
The definition and properties of the elements c n ∈ TL n that are used in the study of the twist θ are contained in the next Proposition. The statement refers to the standard modules S n,k , 0 ≤ k ≤ n with k ≡ n mod 2, over TL n . These were defined in Section 3.1. Again a basis for S n,k can be chosen to be the (n, k)-diagrams in Hom(k, n) with k through lines.
(More information on these modules can be found in [13,25,14,15].) Proposition A.2. The elements c n def = q 3n/2 ρ n n and d n def = q 3n/2 λ n n ∈ TL n satisfy the following properties: (a) c n and d n are invertible central elements of TL n ; (b) c n and d n act on the standard modules S n,k as γ n,k = q 1 2 k(k+2) times the identiy; (c) if q is not a root of unity, then the powers {1 n , c n , c n 2 , . . . , c n ⌊n/2⌋ } of c n form a basis of the center of TL n ; (d) c n = d n .
The factor q 3n/2 is the definition of c n and d n will be useful in the identification of the component θ n of the natural isomorphism θ with c n .
Proof. Statement (a) follows from the invertibility of the t i 's and the cyclic property proved in the previous Lemma. For (b), note that c n and d n being central, they define endomorphisms of the standard modules S n,k by left multiplication.
To compute the determinant of t 1 , choose a basis where (n, k)-diagrams with an arc between position 1 and 2 appear first. There are dim S n−2,k such diagrams and e 1 acts on them as β times the identity. Moreover, on any other (n, k)diagrams, e 1 acts either as zero or gives a diagram with an arc between 1 and 2. Therefore e 1 takes the form The dimension of the standard module S n,k is n (n−k)/2 − n (n−k)/2−1 and a direct computation shows that n(n − 1) dim S n−2,k = 1 4 (n − k)(n + k + 2) dimS n,k which gives γ n,k = ω × q 1 2 k(k+2) where ω is a root of unity such that ω dim S n,k = 1. The central element c n is a Laurent polynomial in q. (There are n(n − 1) factors t i = q 1 2 (1 n + q −1 e i ) in c n = (t 1 t 2 . . .t n−1 ) n and their factors q 1 2 are thus in even numbers.) The eigenvalue γ n,k will thus be continuous, except maybe at q = 0 or ∞. The root ω must thus be constant. At q = 1, the Temperley-Lieb algebra TL n (β = 2) is known to be a quotient of the group algebra of the symmetric group S n and the elements t i are then the transposition (i, i + 1). Thus ρ n is the permutation (1, 2, . . . , n) and the central element c n is the identity permutation. Hence γ n,k = 1 at q = 1 and the only possible choice for ω is 1.
If q is not a root a unity, then TL n (β = −q − q −1 ) is semisimple and is known to have ⌊n/2⌋ + 1 inequivalent irreducible modules. By Wedderburn's theorem the algebra then decomposes into two-sided ideals, one for each inequivalent irreducible, and each ideal is isomorphic to the algebra of d × d-matrices, if the corresponding irreducible is of dimension d. The dimension of the center of TL n at such a q is thus ⌊n/2⌋ + 1. Moreover the eigenvalues γ n,k , 0 ≤ k ≤ n with k ≡ n mod 2, computed in (b) are distinct when q is not a root of unity. Therefore the minimal polynomial of c n in the (faithful) regular representation has degree ⌊n/2⌋ + 1 and the powers {1 n , c n , c n 2 , . . . , c n ⌊n/2⌋ } are linearly independent.
They must form a basis of the center of TL n .
The computation of the determinant of d n follows the same line than that of c n . Since c n and d n contain the same number of generators t i , their eigenvalues on the standard modules thus coincide. When q is not a root of unity, these eigenvalues completely determine the linear decomposition in the basis obtained in (c) and d n and c n must be equal. By continuity, they must also be equal at roots of unity.
When q is a root of unity, the algebra TL n (β = −q − q −1 ) is in general not semisimple. A list of its indecomposable modules is known ( [15], see also [25]) and the only indecomposable modules M whose endomorphism groups Hom(M, M) are larger than C are the projective modules P n,k that have three or four composition factors. Statement (b), above, can be extended to all others. Corollary A.3. Let M be a module over TL n such that Hom(M, M) ≃ C and let I n,k be one of its composition factors. 6 Then c n acts on M as γ n,k · id.
If M is an indecomposable projective such that Hom(M, M) ≃ C 2 , then c n will still have a single eigenvalue on M, but it might not be a multiple of the identity. Such possibility will occur in the examples of the monodromy η N,M • η M,N given in Section 3.3.