Patrick Bas

A QUANTIZATION WATERMARKING TECHNIQUE ROBUST TO LINEAR AND NON-LINEAR VALUMETRIC DISTORTIONS USING A FRACTAL SET OF FLOATING QUANTIZERS

This paper presents an extension of the classical Quantization Index Modulation (QIM) data-hiding scheme in the context of valumetric distortions. This scheme uses a fractal quantization structure during the detection but also a content dependent quantization grid to achieve both global constant robustness and the ability to recover the watermark after non-linear valumetric distortions. Previous works are first presented. Then the construction of a floating quantizer that addresses the problem of non-linear transformations is introduced. The embedding and detection schemes for digital image watermarking are afterward introduced, the main characteristic of this scheme is the fact that the detection scheme can use a hierarchical set of quantizers to deal with non-linear valumetric transforms while preserving an average constant quantization step. Finally the performance of this scheme and the comparison with other robust quantization schemes considering valumetric transforms and noise addition are presented. Histogram analysis: Eggers et. al have presented a scheme to address the case of an affine valumetric distortion f (p) = αp + β [2]. This method is based

Introduction

Quantization watermarking techniques 1 , first introduced by Chen and Worwell [START_REF] Chen | Quantization index modulation: A class of provably good methods for digital watermarking and information embedding[END_REF] are widely used in watermarking applications because they provide both robustness to the AWGN channel and high capacity capabilities while preserving the fidelity of the host document. Basically, the Quantization Index Modulation (QIM) scheme uses one different quantizer for each code word that is transmitted and the set of quantizer is span on the range of possible values that can be taken by each sample. For example, using a two stages codeword (e.g. a bit of information) requires two disjoints quantizers. The embedding rule of a bit b in a coefficient C is the following:

If b[k] = 1 : C w = 2∆E C+∆/2 2∆ -∆/2 If b[k] = 0 : C w = 2∆E C-∆/2 2∆ + ∆/2 (1)
Where ∆ represents the distance between one quantized value and the next one, and E(x) is the integer part of x. For security issues, it is also possible to add a key-dependant random signal on the future quantized sample before performing the quantization step and to remove the random signal (call the dither signal) on the quantized value. Chen et. al have first introduced such a method called dither modulation (DM) and have combined it with a distortion compensation module.

However the usage of quantization techniques such as QIM or DM are not straightforward in real application scenarios such as video watermarking where valumetric distortions are often present during the broadcasting process.

QIM/DM and valumetric distortions

A valumetric distortions can be defined by any function f (p) that modifies the value of the original pixel p. In practical cases, f () can be either linear (e.g. f (p) = α(p)) and represents a gain, or non-linear to represents more widely used distortions. For example, the gamma correction function Γ γ (p), which has to be especially considered in video watermarking applications when the watermarked signal has to suffer DA/AD conversions, is given by:

Γ γ (p) = p M AX p p M AX γ
where p M AX is the maximum value of a pixel. A valumetric transform will consequently alter the quantized value in such a way that, if p w -f (p w ) > ∆/2, the decoding step will probably lead to an error. Such an effect is depicted on Fig. 1 which outlines the displacement of the different quantized values after linear and non-linear valumetric transforms.

Previous works

The goal of this subsection is to present a brief overview of the other works that address the problem of QIM/DM detection in presence of valumetric distortions.

Fig. 1. Effects of valumetric transforms on quantized values: the initial quantization grid (first axis) cannot be used to decode the watermark neither after a linear transform (second axis) nor after a non linear transform such as the Gamma transform (third axis). In many cases, the decoding will lead to erroneous results.

on the analysis of a two-dimentional probability density function (pdf) that depends on the received sample r and the range of the possible deviations for one secret key k. The resulting pdf presents periodical components for each deviation and the period is proportional to α∆. Consequently the authors propose to apply a DFT and a phase shift function for each deviation of the two-dimensional PDF before summing the spectra of the pdfs on the k axis. The maximum of the overall spectrum reveals then the term α and the term β is derived from the phase at the location of the maxima. In [START_REF] Eggers | Estimation of amplitude modifications before scs watermark detection[END_REF] the authors have also addressed the problem of the non-linear gamma transform by trying to invert the gamma transform before the estimation of α and β but this method need a full search of the γ parameter and consequently increase the complexity of the algorithm.

ML estimator: Lagendijk et. al have proposed a similar approach than the previous one that also uses the Fourier transform of the histogram to estimate α [START_REF] Shterev | Statistical amplitude scale estimation for quantization-based watermarking[END_REF]. They also have presented in [START_REF] Shterev | Statistical amplitude scale estimation for quantization-based watermarking[END_REF] and [START_REF] Lagendijk | Estimation of attacker's scale and noise variance for qim-dc watermark embedding[END_REF] an estimation of parameter α that is based on the Maximum Likelihood (M.L.) estimator. The pdf of the watermarked image after scaling and noise addition is first expressed as f (p, α, σ 2 n ) where σ 2 n denotes the noise variance. The estimates α and σ2 n parameters are calculated performing a search on the Likelihood function:

(α, σ2 ) = arg max α,σ 2 n L(α, σ 2 n ) = arg max α,σ 2 n log f p (p, α, σ 2 n )
It is important to notice that the likelihood function is expressed for an host signal and a noise that have both gaussian distributions. The model of the host signal does not represent typical distributions of natural images. This can be seen as a limitation of this approach. Moreover this approach is based on a fixed valumetric distortion model (in this case linear), and the use of ML estimator for nonlinear functions is not straightforward.

Host proportional embedding: Another way to deal with the problem of valumetric transforms is to use a quantification step ∆ p that is constructed in such a way that it is proportional to a valumetric feature of the signal. Pérez-Gonzaález et. al. [START_REF] Perez-Gonzalez | Rational dither modulation: a novel data-hiding method robust to value-metric scaling attacks[END_REF] have proposed to use a function g(p) which has the property that g(αp i ) = αg(p i ) as a weighting function of the initial quantization step2 . This function can be for example the l n vector norm given by:

g(p i ) =   1 L i-1 j=i-L |p j | n   1/n
For example if n = 1 and p j ≥ 0 for all j, the quantization step is proportional to the mean of the vector p i . If n = 2, it is then proportional to the standard deviation of p i . The authors give the proof that when L → +∞, the presented scheme, called Rational Dither Modulation, is equivalent to the classical dither modulation scheme presented by Eggers. Another similar scheme has been also proposed at the same time by Oostveen et. al [START_REF] Oostveen | Adaptive quantization watermarking[END_REF], in this scheme the proportional function is chosen as the averaging3 function (n = 1). As we will see in section 3, host proportional embedding is adapted for linear valumetric functions when L is large enough. Nevertheless the performances of proportional embedding quantization schemes are poor for non-linear transforms beacause the function g(p) has not special properties to cope with a nonlinearity). Additionally for affine transforms, such schemes cannot handle the addition of an offset value. Another important drawback of proportional embedding schemes is the fact that the quantization step ∆ is not constant. If L +∞ and n = 2 for example, the quantization step is less important in homogeneous areas than in textured areas. This last point can be a serious handicap considering addition of noise because homogeneous areas will be more sensitive to provide detection errors than textured areas.

A floating quantization scheme with a fractal structure

As we have mentioned previously, the presented schemes have several drawbacks: histogram analysis enables to identify a linear distortion but its extension to non linear distortions is not straightforward, ML estimator depends both on a host model and a distortion model, host proportional embedding is adapted for linear distortion but not to affine or non linear distortions and moreover the quantization step ∆ p is constant only for stationary iid images. The objectives of the presented scheme are twice, the first is to provide a quantization scheme that is robust both to linear and non-linear distortions and the second is to provide a constant quantization step for each watermarked coefficient, or at least to bound the range of values that can be taken by the quantization step ∆. In this paper, we propose first to use a floating quantization grid that is based on the local features of the host coefficient to provide robustness to both linear and non linear transforms. A nearly constant quantization step is secondly guaranteed using a fractal set of quantizers. These ideas are presented in the following sections.

Dealing with non-linear distortions

A classical solution to cope with a non-linear distortion f (p) is to locally approximate the function f () by a linear function whose the slope is an approximation of the derivate of f (p). To estimate this derivate, we have can use the fact that images are not idd signals but are more often highly correlated. For example adjacent pixels of an image have highly similar values. We choose in this paper the two neighbours of a given pixel because these three values are often highly similar for natural images. The quantization step ∆ f is afterward constructed by taking the minimum p M in and the maximum pixel p M ax of these three values respectively as the lower and upper bounds of the quantization grid. Because such a grid relies on the host signal and is specific for each pixel, we have named it a floating quantization grid. The embedding of one bit of information is after done quantizing the middle value p M id . Fig. 2 depicts the principle of the floating quantizer. For clarity purposes only two quantized values p 1 and p 0 are used in this figure. We can notice that if the distorted values p 1 and p 0 are reliable approximations of their equivalent linear transformations, the respective approximation errors 1 and 2 are small under the condition that the nonlinearity is not too important.

A floating quantizer has however to deal with several limitations that have to be addressed. The first one in the fact that the embedding rate is limited to one third of the total number of sample in the host signal. This is due to the fact that three samples are needed to embed one bit. The second point to adress is the case where p M ax = p M in . It is however easy to solve this issue by increasing p M ax or decreasing p M in in such a way that p M ax -p M in = ∆ min . The last limitation of this proposal is the fact that the resilience against additive noise is not constant because we choose a quantization step ∆ f that is proportional to the range distance r = p M ax -p M in . This fact is by nature contradictory with the essence of a quantization based-scheme which aims to give an equal quantization step for each watermarked coefficient. We have consequently decided to adopt a specific quantization rule R that is chosen according to the value of the quantization step r: ∆ f = R(r). This last point is addressed in the next subsection.

2.2

Using a fractal quantization structure:

The main idea is to build a set of N hierarchiezed quantizers Q = {Q 1 , ..., Q N } such that it is possible to have a bijection between one quantizer and one specific value r. In order to have a nearly constant quantization step ∆ f for each quantizer one might fix the minimum and maximum possible values of ∆ f by setting

α 1 ∆ ≤ ∆ f ≤ α 2 ∆ where α 1 ≤ 1, α 2 ≥ 1 and
∆ is an arbitrary fixed quantization step. To guaranty that there are no overlapping or impossible affectation between on quantizer and its adjacent quantizers these parameters are naturally linked by the relation α 2 = 2α 1 . The choice of α 1 depends of different kinds of applications. For example if we want to guarantee a minimum quantization step ∆, this leads to α 1 = 1. On the other hand, if we consider an uniform host distribution and want an average quantization step equal to ∆, we then have to choose α 1 = 0.66. The value of the floating quantization step ∆ f is given by the relation:

∆ f = r 2 N (r) if N (r) > 0 (2)
∆ f = ∆ if N (r) = 0 (3)
where the power function N () is given using the positive integer part function E + ():

N (r) = E + log 2 r α 1 ∆
The calculus of ∆ f naturally leads to the construction of the set of quantizers Q because each different value of N (r) is associated to a number of quantization values. Fig. 3 depicts the set of quantizers which mainly presents a fractal structure: the basic quantization pattern Q 1 is repeated on Q 2 , Q 3 and Q 4 using a contraction factor respectively equal to 1/2; (1/2) 2 and (1/2) 3 . Only the quantizer Q 0 is not a member of the fractal set and this quantizer can be additionally used when initially p M ax = p M in . This fractal structure has the main advantage to avoid to have different quantization cells on the same location on the different axes. Fig. 4 represents the repartition of the set of quantizer Q in function of the range value r. We have decided to note each upper border of the quantizer Q i by q i . Note also that the quantizer Q 0 corresponds to the case where N (r) = 0 and in this case, because the value of r is considered as too small, the quantizer step is forced to be equal to ∆.

Embedding scheme

Based on the building of the specific floating quantizer, we can afterward apply a classical quantization scheme. The presented scheme is basic but it can be easily improved by using a dither signal or a distortion compensation module as proposed by Chen. The embedding of a message b(k) of size N is done by embedding one bit b for each triplet of the original image. The main steps of the embedding procedure are listed below:

-Select a triplet of pixels (p i , p j , p k ). Note that better performance for nonlinear distortion will be obtained if this triplet corresponds to adjacent pixels.

-Order the triplet in (p M in , p M id , p M ax ) and compute r = p M ax -p M in , ∆ f from ( 2) and (3) that will provide Q i .

-

If Q i = Q 0 quantize p M id according to Q i and b.
-Else quantize both p M id and p M in for b = 0 or p M id and p M ax for b = 1.

-Select another triplet that was not selected before and embed another bit b.

Detection scheme

The detection procedure has to consider that the received image has undergone a valumetric transform f (). This implies that a given pixel that has been quantized using quantizer Q i may be decoded using another quantizer. To deal with this issue we have first to constrain the possible variations of f () assuming that

β 1 ≤ f (p) ≤ β 2 . Assuming that (β 1 , β 2 ) ∈ [1/2; 1] × [1; 2]
, the pixels that are watermarked using a given quantizer Q i may be decoded using Q i and Q i-1 or Q i and Q i+1 . Note that in these particular cases, because we use a set a 2 quantizers, the robustness may be divided by 2 or 4. Therefore we can also consider another alternate decoding scheme that consists in using only the decoder Q i . This is due to the fact that after a valumetric distortion, an initial pixel watermarked using Q i may still be decoded using Q i if the value of r has not changed of quantization cell. The differences between these two decoders will be assessed in the next section. Finally, the different steps of the decoding algorithm are listed below:

-Select (p i , p j , p k ).

-Compute r and ∆ f .

-Using the fractal detection: if r > β 1 q i-1 perform the detection using both Q i and Q i-1 and choose the nereast quantization cell considering the two quantizers.

-Using the fractal detection: if r < β 2 q i+1 perform the detection using both Q i and Q i+1 and choose the nereast quantization cell considering the two quantizers.

-Else or for the classical floating quantizer: perform the detection using Q i .

In this section we outline the capability of the presented scheme to achieve robustness for linear and non-linear distortions and noise addition. We also compare these performances with classical and proportional QIM/DM embedding and detection schemes. In each cases, 1/3 of the pixels are watermarked for both the lena image and baboon image. This is done to achieve the same capacity of embedding. For proportional quantization, we have chosen to use a proportional function that skips the future quantized coefficients to improve the estimation of quantization step. Note that in [START_REF] Perez-Gonzalez | Rational dither modulation: a novel data-hiding method robust to value-metric scaling attacks[END_REF], the authors compute the function g(p i ) using watermarked samples which is an equivalent way to cope with this problem. We have tested two neighbourhood size: N = 10 and N = 100 pixels and we also used n = 2 which means that the quantization step is proportional to the standard deviation of the selected samples. The floating quantization scheme is tested using both a fractal decoder with β 1 = 0.7 and β 2 = 1.42 and a nonfractal decoder. Fig. 5 In Fig. 5, we can notice that the possibility to use multiple quantizers (named in the figure by Fractal quantizer) improve the decoding performances comparing to only one quantizer. Moreover this figure illustrates the fact that the proportional embedding is not adapted to non linear distortions even for little neighbourhood size. In this last case, this is mainly due to the fact that the quantization step of proportional schemes can be too small. For example, if the neighbourhood represents a constant portion of the image, the distance between watermarked samples representing 0 and 1 will be null and will consequenlty lead to decoding errors. The effect of the used valumetric distortion is illustrated in Fig. 8. Table 1 presents the robustness of the floating quantization scheme after histogram equalizations that are equivalent to an image dependent nonlinear transform. Fig. 9 represents these functions for the two images and Fig. 8 shows the result on the Lena image. We can notice that the performance of the floating quantization is once again superior to the use of proportional embedding but also depends of the nature of the image. For example, the baboon image, which is more textured than the lena image, offer weaker performance: as mentioned before, this is due to the fact that the linear approximation is not reliable when r is too important, e.g. when the triplet belong to a textured area. Fig. 6 presents the performance of the different schemes when the valumetric transform is linear for different DW Rs. A scaling factor equal to 0.5 has been applied on the pixels of the watermarked image and no noise is added. Here the proportional quantization scheme can outperform the floating quantization scheme if the size of the neighbourhood is important enough (N = 100). Note however that contrary to floating quantization, proportional quantization is robust to an affine transform f (x) = ax + b only when b = 0. outperforms all the other schemes, and that the floating quantization scheme is weaker than the proportional quantization scheme for a large neighbourhood size (N = 100). This is due to the fact that ∆ f is proportional to r that is also subject to noise. When the neighbourhood size is not enough important however, the proportional quantizer scheme is less efficient that the floating quantizer because a minimum quantization step is not guaranteed with the first solution.

Fig. 2 .

 2 Fig. 2. Effect of a floating quantizer with the function Γ () considering γ = 2.

Fig. 3 .

 3 Fig. 3. Building of the set of quantizers. Quantizers that are in the gray area represent a fractal structure. The choice of the quantizer is determined by the value of r.

Fig. 4 .

 4 Fig. 4. Repartition of Q in function of r for α1 = 1.

Fig. 5 .Fig. 6 .

 56 Fig. 5. BER results after gamma correction γ = 0.5 for the lena image.

Fig. 7 .

 7 Fig. 7. BER results after gaussian noise addition for the lena image.

Fig. 7

 7 Fig. 7 compare the robustness after Gaussian noise addition for a DW R equal to 25.35dB. In this case, we can see that the classical QIM/DM scheme

  and Table.1 presents the performances of the floating quantization scheme to non-linear distortions (gamma correction and histogram equalization) for different Document to Watermark Ratios.

Table 1 .

 1 BER results after histogram equalization for two different images (WDR=27db, neighbourhood size=200).

	BER (%) fractal non fractal proportional
	lena	3.0	3.0	36.2
	baboon 12.3	15.5	58.8

The work described in this paper has been supported (in part) by the European Commission through the IST Programme under Contract IST-2002-507932 ECRYPT and the National French project Fabriano.

pi represents a vector composed by i neighbour pixel values that have been centered.

In this work, the authors do not use the absolute value term because they only consider pixel (positive) values otherwise the two schemes can be considered as similar.

Concluding remarks and perspectives

We have presented in this paper a practical implementation of a quantization watermarking scheme that is more robust to non-linear distortions than previous proposed schemes. This performance gain is mainly due to the use of a floating quantizer and the adoption of a fractal set of quantizers. Future works will address the capacity limitation of this scheme by performing an iterative embedding on each sample to achieve a capacity equal to one bit per sample.


K Iohara
email: iohara@math.univ-lyon1.fr

G I Lehrer

R B Zhang

TEMPERLEY-LIEB ALGEBRAS AT ROOTS OF UNITY, A FUSION CATEGORY AND THE JONES QUOTIENT

Introduction

The Temperley-Lieb algebras TL n (q) (see §4 below) are algebras over a ring R which depend on a parameter q ∈ R. They occur in many areas of mathematics and physics, and may be characterised as the endomorphism algebras of the objects in the Temperley-Lieb category (see [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF]). In this work we shall generally take R = C. These algebras are well known to have a cellular structure [START_REF] Graham | Cellular algebras[END_REF] and their representation theory may be analysed using this structure.

For generic vaues of q, the algebra TL n (q) is semisimple, and its simple modules are the cell modules W t (n), for t ∈ Z, 0 ≤ t ≤ n and t ≡ n(mod 2). However when q is a root of unity, the cell modules are often no longer simple, but have a simple head L t (n). The modules L t (n), where t runs over the same values as above, form a complete set of simple modules for TL n (q) in this case.

In this work, our first purpose is to give explicit formulae for the dimensions of the modules L t (n). This will be done by deriving, for each t ∈ Z ≥0 , an explicit formula for the generating function The algebra TL n (q) has a trace tr n : TL n (q) → C, identified by Jones, whose associated bilinear form is generically non-degenerate (see (2.2) below). If q is a root of unity, and the order |q 2 | = , then tr n has a radical of dimension 1 if n = -1, the generating element being the Jones-Wenzl idempotent E -1 ∈ TL -1 (q). An explicit formula for E -1 is given in [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF]. Jones has shown [START_REF] Jones | Hilbert space representations of the annular Temperley-Lieb algebra[END_REF]Thm. 2.1] that in this case, for any n ≥ -1, the radical R n (q) of tr n is generated by E -1 ∈ TL -1 (q) ⊆ TL n (q). Moreover, for n ≥ , the algebra TL n (q) has the canonical semisimple quotient Q n ( ) := TL n (q)/R n (q), which we refer to as the Jones algebra.

As a consequence of our analysis, we deduce a complete description of the simple representations of the Jones algebras Q n ( ), as well as a generating function for its dimension, which recovers a result of [START_REF] Goodman | Coxeter graphs and towers of algebras[END_REF]. Note that Q n (q) is far from being the maximal semisimple quotient of TL n (q), as our work shows.

In §8 we relate Q n ( ) to the fusion category C red introduced by Reshetikhin, Turaev and Andersen [START_REF] Reshetikhin | Invariants of 3-manifolds via link polynomials and quantum groups[END_REF][START_REF] Henning Haahr Andersen | Tensor products of quantized tilting modules[END_REF] whose objects are sums of the indecomposable tilting modules of non-zero quantum dimension for the quantum group U q (sl 2 ), when q 2 is a primitive th root of unity. The category C red has a fusion product ⊗, and if ∆ q (1) is the indecomposable (in fact simple) tilting module with highest weight 1, we show that Q n ( ) ∼ = End Uq(sl 2 ) (∆ q (1) ⊗n ). Together with our earlier results, this recovers a formula for the dimension of Q n ( ) due to Jones [START_REF] Goodman | Coxeter graphs and towers of algebras[END_REF].

We note also that our results are related to those of [START_REF] Erdmann | Tensor products and dimensions of simple modules for symmetric groups[END_REF], which could be thought of as treating the more complicated positive characteristic analogue of some of our material. Acknowledgement The authors would like to thank the referee for a very thorough reading of the manuscript, and in particular for detecting an error in the original version. The paper has benefited significantly from the referee's input.

2. The Temperley-Lieb algebras.

2.1. Definitions. In this work, all algebras will be over C. Much of the theory we develop applies over more general domains, but since we will be concerned here with connections to the theory of operator algebras and mathematical physics, we limit our discussion to C-algebras. For n ∈ N, the Temperley-Lieb algebra TL n (q) is defined as follows.

Definition 2.1. Let q ∈ C. TL n = TL n (q) is the associative C-algebra with generators f 1 , f 2 , . . . , f n-1 and relations (2.1)

f 2 i = -(q + q -1 )f i for all i f i f i±1 f i =f i for all i f i f j =f j f i if |i -j| ≥ 2.
2.2. The Jones form. In his seminal work [START_REF] Jones | Index for subfactors[END_REF] on subfactors of a factor, Jones showed that certain projectors {e 1 , . . . , e n-1 } (n = 1, 2, , 3, . . . ) in a von Neumann algebra satisfy the Temperley-Lieb-like relations, a fact that led to the definition of the "Jones polynomial" of an oriented link. In the notation of [17, p. 104, (I)-(VI)], Jones showed that if f i = (q + q -1 )e i , then the f i satisfy the relations (2.1), where Jones' parameter t is replaced by q 2 . If q 2 = -1, Jones' form on TL n (q) is defined as the unique (invariant) trace tr n = tr on TL n (q) which satisfies (2.2) tr(1) = 1 and tr

(xf i ) = -(q + q -1 ) -1 tr(x) for x ∈ TL i-1 ⊂ TL i ⊆ TL n for 1 ≤ i ≤ n -1.
This trace on TL n (q) is non-degenerate if and only if q 2 is not a root of unity, or, if [13, (3.8)]. Thus the discrete set of values of q 2 for which Jones' sequence (A n ) of algebras is infinite coincides precisely with the set of values of q 2 for which the trace form above on TL n (q) is degenerate.

|q 2 | = , n ≤ -2 (

2.3.

Cell modules and forms. Let us fix n and consider the representation theory of TL n .

Recall [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF] that the Temperley-Lieb category T over C has object set N and for t, n ∈ N, Hom T (t, n) is the vector space with basis the set of Temperley-Lieb diagrams from t to n, i.e., which have t lower vertices, and n upper vertices. Composition is by concatenation of diagrams, with free circles replaced by -(q + q -1 ). In speaking of Temperley-Lieb diagrams, we shall freely use the well known terminology which refers to upper arcs (which join two upper vertices), lower arcs (which join two lower vertices) and through strings (which join an upper vertex to a lower vertex).

A diagram from t to n is monic if it has a left inverse. This means (assuming q + q -1 = 0) that there are no lower arcs. The algebra Hom T (n, n) is the Temperley-Lieb algebra TL n = TL n (q). By [START_REF] Graham | Cellular algebras[END_REF] TL n has cell modules W t := W t (n) whose basis is the set of monic Temperley-Lieb morphisms from t to n, where t ∈ T (n), and

T (n) = {t ∈ Z | 0 ≤ t ≤ n and t + n ∈ 2Z}.
Now W t has an invariant form ( , ) t which may be described as follows. For monic diagrams The form ( , ) t is evidently equivariant for the TL n (q)-action; That is, we have for any element a ∈ TL n (q) and elements v, w ∈ W t (n), (av, w) t = (v, a * w) t . Hence the radical Rad t of the form ( , ) t is a submodule of W t . Let L t := W t /Rad t . The general theory asserts that the L t are simple, and represent all the distinct isomorphism classes of simple TL n (q)-modules.

D 1 , D 2 : t → n, we form the diagram D * 1 D 2 : t → t. If D * 1 D 2 is monic (i.
Remark 2.2 (Notation). Note that W t , Rad t and L t may be regarded as functors : T → vect, where vect is the category of finite dimensional C-vector spaces. Thus W t (n) is the evaluation of the functor at n. However, when the context makes it clear, we shall abuse notation by writing W t for W t (n), etc.

Semisimplicity and non-degeneracy.

Clearly, if the trace (2.2) is non-degenerate, the algebra TL n is semisimple. The converse is true except for one single case (see [ Moreover we have very precise information concerning the radical of the invariant trace form.

3.1. Radical of the trace form. The radical of the trace form above is given by the following result (see [13, §3], [START_REF] Jones | Hecke algebra representations of braid groups and link polynomials[END_REF]). Proposition 3.1. ([18, Thm. 2.1]) If q is not a root of unity then tr n is nondegenerate and TL n is semisimple for all n.

Suppose the order of q 2 is . Then there is a unique idempotent

E -1 ∈ TL -1 (the Jones-Wenzl idempotent) such that f i E -1 = E -1 f i = 0 for 1 ≤ i ≤ -2.
Moreover for n ≥ -1 the radical of tr n is generated as ideal of TL n by E -1 . Remark 3.2. cf. [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF]Remark (3.8)] It follows from Proposition 3.1 that the trace tr n is non-degenerate if and only if n ≤ -2, where = |q 2 |. It follows that the case n = -1 is uniquely characterised as the one where the form tr n is degenerate, but TL -1 is semisimple.

Note also that E -1 is regarded as an element of TL n for n ≥ -1 in the usual way, by thinking of it as E -1 ⊗ I ⊗n-+1 .

The following formula for the idempotent E -1 was proved in [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF]Cor. 3.7]. To prepare for its statement, recall that if F is a finite forest (i.e. a partially ordered set in which x ≤ a, x ≤ b =⇒ a ≤ b or b ≤ a), then we define a Laurent polynomial

(3.1) h F (x) = [|F |] x ! a∈F [|F ≤a |] x , where, for m ∈ N, [m] x = x m -x -m x-x -1 and [m] x ! = [m] x [m -1] x .
. . [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF] x [START_REF] Henning Haahr Andersen | Tensor products of quantized tilting modules[END_REF] x . Theorem 3.3. For any Temperley-Lieb diagram a : 0 → 2n we have an associated forest F a , which is simply the poset of arcs, ordered by their nesting. For any Temperley-Lieb diagram D : t → n, one obtains a unique diagram D : 0 → t + n by rotating the bottom line clockwise by π. With this notation, if |q 2 | = , we have

(3.2) E -1 = D h F D (q)D,
where the sum is over the diagrams from -1 to -1, i.e. over the diagram basis of TL -1 .

Example 3.4. If = 4, we may take q = -exp πi 4 , so that q 2 = i and the element E 3 ∈ TL 3 is easily shown to be equal to

E 3 = 1 + f 1 f 2 + f 2 f 1 - √ 2(f 1 + f 2 ).
Note that our defining parameter for TL n in this case is -(q + q -1 ) = √ 2, and the above element is the familiar one which occurs in the study of the two-dimensional Ising lattice model.

3.2.

Properties of the Jones-Wenzl idempotent E -1 . It is well-known that the Jones-Wenzl idempotent is harmonic. This means that in the Temperley-Lieb category T, if a cap is placed above or below E -1 , one obtains zero. Diagramatically, this is depicted in Figure 1.

The following result will be needed for the proof of Theorem 7.5 below. (1) Let |q 2 | > n, and let E i be the Jones-Wenzl idempotent in TL i for each i ≤ n. Let A (resp. U ) be the unique diagram from 2 to 0 (resp. 0 to 2) in T, and let I be the unique diagram 1 to 1. Then

(I ⊗(n-1) ⊗ A)(E n ⊗ I)(I ⊗(n-1) ⊗ U ) = [n -1] q [n] q -[2] q E n-1 .
Diagramatically, this equation may be depicted as in Figure 2.

... E n ... = [n-1]q [n]q -[2] q E n-1 Figure 2 (2) When |q 2 | = , (I ⊗( -2) ⊗ A)(E -1 ⊗ I)(I ⊗( -2) ⊗ U ) = 0. Diagramatically this is depicted in Figure 3. ... E -1 ... = 0 Figure 3
Proof. It is a result of Wenzl (cf. [START_REF] Frenkel | Canonical bases in tensor products and graphical calculus for U q (sl 2 )[END_REF]Thm. 3.3,p.461]) that the Jones-Wenzl idempotents satisfy the recursion

(3.3) E n = E n-1 ⊗ I + [n -1] q [n] q (E n-1 ⊗ I)f n-1 (E n-1 ⊗ I),
where f i is the i th generator of TL n . Now denote the operator

D → (I ⊗(n-1) ⊗ A)(D ⊗ I)(I ⊗(n-1) ⊗ U ) by γ : TL n → TL n-1 . It is easily verified that γ(E n-1 ⊗I) = -[2] q E n-1 and that γ(f n-1 ) = I ⊗(n-1) . Hence γ((E n-1 ⊗ I)f n-1 (E n-1 ⊗ I)) =(I ⊗(n-1) ⊗ A)(E n-1 ⊗ I ⊗ I)(f n-1 ⊗ I)(E n-1 ⊗ I ⊗ I)(I ⊗(n-1) ⊗ U ) =E n-1 (I ⊗(n-1) ⊗ A)(f n-1 ⊗ I)(I ⊗(n-1) ⊗ U )E n-1 =E n-1 γ(f n-1 )E n-1 =E n-1 .
Part (1) now follows immediately by applying the operator γ to the relation (3.3).

To obtain the relation in part [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF], observe that if |q 2 | = , then by (1), the left side is equal to (

[ -2]q [ -1]q -[2] q )E -2
. But in our case, [ -1] q = -q and [ -2] q = -q [2] q , which proves (2).

3.3. The Jones quotient. We now wish to consider the quotient of TL n by the ideal generated by E -1 . Definition 3.6. Assume that |q 2 | = for a fixed integer ≥ 3. Let R n = R n (q) = E -1 be the ideal of TL n (q) generated by the idempotent E -1 ∈ TL -1 (q), where TL -1 (q) is thought of as a subalgebra of TL n (q) for n ≥ -1 in the obvious way. If n < -1, we set R n = 0.

The algebra

Q n = Q n ( ) (n = -1, , + 1, . . . ) is defined by Q n ( ) = TL n R n (q) .
This algebra will be referred to as the "Jones (projection) algebra".

Since we are taking the quotient by the radical of the trace form tr n , it follows that Q n has a non-degenerate invariant trace, and hence that (3.4) Q n ( ) is semisimple.

Remark 3.7.

• The algebras Q n ( ) are not the maximal semisimple quotients of TL n (q), as the results of the next section will show.

• We remark that E 2 = 1 -f 1 , from which it follows that Q n (3) ∼ = C for all n.
4. Representation theory of TL n (q). We shall apply the basic results of [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF] to obtain precise information about the simple modules for TL n (q) from the general results in §2.3 about cell modules. The following description of the composition factors of W t = W t (n) was given in [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF]Thm. 5.3], and in the formulation here in [2, Thm. 6.9].

t = a + b ∈ N , 0 ≤ b ≤ -2, define g(t) = (a + 1) + -2 -b. Notice that g(t) -t = 2( -b -1)
, so that g(t) ≥ t + 2 and g(t) ≡ t(mod 2).

(1) For t ∈ T (n) ∩ N such that g(t) ∈ T (n), there is a non-zero homomorphism θ t : W g(t) (n) → W t (n). These are explicitly described in [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF]Thm 5.3], and are the only non-trivial homomorphisms between the cell modules of TL n . (2) If t ∈ T (n) is such that t ∈ N and g(t) ∈ T (n), then W t (n) has composition factors L t and L g(t) , each with multiplicity one. All other cell modules for TL n (q) are simple.

(3) If ≥ 3, all the modules L t (n), t ∈ T (n), are non-zero, and form a complete set of simple TL n (q)-modules.

Definition 4.2. For t ∈ Z ≥0 define functions w t and l t :

N → N by w t (n) = dim(W t (n)) and l t (n) = dim(L t (n)). Note that if t > n, w t (n) = l t (n) = 0. Further w t (n) = l t (n) = 0 if n ≡ t(mod 2). Proposition 4.3. Let |q 2 | = .
We have, for t ∈ N (defined as in Theorem 4.1):

(4.1) l t (n) = ∞ i=0 (-1) i w g i (t) (n).
Proof. Note that since g is a strictly increasing function on N , for any particular n, the sum on the right side of (4.1) is finite. It is evident from Theorem 4.1 [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF], that for any t ∈ N , (4.2)

l t (n) = w t (n) -l g(t) (n).
Applying (4.2) with t replaced by g(t) gives l t (n) = w t (n) -w g(t) (n) + l g 2 (t) (n). Applying this repeatedly, and noting that there is an integer t 0 ∈ N such that t 0 ≤ n and g(t 0 ) > n, we obtain the relation (4.1).

This may be made a little more explicit by the following observation. Fix = |q 2 | and t ≥ 0, write b(t) = b, where t = a + b, with 0 ≤ b ≤ -1. Then for t ∈ N we have

(4.3) g(t) =t + 2b(t) and g 2 (t) =t + 2 , where b(t) := -1 -b(t).
The equation (4.1) may therefore be written as follows.

Corollary 4.4. We have the following equality of functions on N:

(4.4) l t = ∞ i=0 w t+2i - ∞ i=0 w t+2b(t)+2i = ∞ i=0
(w t+2i -w t+2b(t)+2i ).

5.

Generating functions for the cell modules.

In this section we recall explicit generating functions for the dimensions of the cell modules of TL n (q) (cf. [START_REF] Martin | Potts models and related problems in statistical mechanics[END_REF]Ch. 6]).

a 1 a 2 a t b 1 b 2 b t+2(k+1) b 3 ... ... Figure 4. Monic diagram t → t + 2(k + 1) as a diagram 0 → 2t + 2(k + 1).
5.1. Cell modules for TL n . Recall that the cell module W t (n) has a basis consisting of the monic TL-diagrams D : t → n. Since such diagrams exist only when t ≡ n(mod 2), we may write n = t + 2k, k ≥ 0.

Definition 5.1. For t, k ≥ 0, we write w(t, k) := dim W t (t + 2k). By convention, W 0 (0) = 0, so that w(0, 0) = 0. Note that by Definition 4.2, w(t, k) = w t (t + 2k).

Proposition 5.2. We have the following recursion for w(t, k). For integers t, k ≥ 0:

(5.1)

w(t, k + 1) = w(t -1, k + 1) + w(t + 1, k).
Proof. The proof is based on the interpretation of w(t, k) as the number of monic TL-diagrams from t to t + 2k. Consider first the case t = 0. The assertion is then that w(0, k +1) = w(1, k). But all TL-diagrams D : 1 → 1 + 2k are monic, as are all diagrams 0 → 2 (any ). It follows that w(1, k) = dim(Hom T (1, 1 + 2k)) = dim(Hom T (0, 2 + 2k)) = w(0, k + 1). Thus the assertion is true for t = 0 and all k ≥ 0. Similarly, if k = 0, the assertion amounts to w(t, 1) = w(t -1, 1) + w(t + 1, 0). If t > 0, the left side is easily seen to be equal to t + 1, while w(t -1, 1) = t and w(t + 1, 0) = 1. If t = 0, the left side is equal to dim(Hom T (0, 2k + 2)) = dim(Hom T (1, 2k + 1)) = w(1, k). So the recurrence is valid for k = 0 and all t. Now consider the general case. Our argument will use the fact that w(t, k + 1) may be thought of as the number of TL-diagrams 0 → 2t + 2(k + 1) of the form depicted in Fig. 4, which illustrates the case k = 0.

The condition that the diagram be monic is simply that each a i is joined to some b j , i.e. that each arc crosses the dotted line; of course distinct arcs are nonintersecting.

Evidently such diagrams fall into two types: those in which [a t , b 1 ] is an arc, and the others. Now the number of diagrams in which [a t , b 1 ] is an arc is clearly equal to w(t -1, k + 1), while those in which [a t , b 1 ] is not an arc are in bijection with the monic diagrams from t + 1 to t + 1 + 2k, as is seen by shifting the dotted line one unit to the right. Hence the number of the latter is w(t + 1, k), and the recurrence (5.1) is proved.

A binomial expression for w(t, k).

Definition 5.3. For integers t, k ≥ 0, define

(5.2) F (t, k) = t + 2k k - t + 2k k -1 .
This definition is extended to the domain Z × Z by stipulating that

F (t, k) = 0 if t < 0 or k < 0.
It is easily seen that

(5.3) F (t, k) = (t + 1)(t + 2k)(t + 2k -1) . . . (t + k + 2) k! = t + 1 t + k + 1 t + 2k k ,
and that Lemma 5.4. We have the following recursion for F (t, k). For t, k ≥ 0:

(5.4)

F (t, k + 1) = F (t -1, k + 1) + F (t + 1, k).

Catalan calculus-generating functions.

For n ≥ 0, write c(n

) := w(0, 2n), c(0) = 1.
It is easily seen by inspecting diagrams that for n > 0,

(5.5) c(n) = n k=1 c(k -1)c(n -k). Writing c(x) := ∞ n=0 c(n)x n , the recursion (5.5) translates into (5.6) xc(x) 2 -c(x) + 1 = 0, from which it is immediate that (5.7) c(x) = 1 -(1 -4x) 1 2
2x , and applying the binomial expansion, that

(5.8) c(n) = 1 n + 1 2n n . Now define W t (x) = ∞ k=0 w(t, k)x k = ∞ k=0 w t (t + 2k)x k .
Inspection of diagrams shows that the w(t, k) satisfy the following recursion.

(5.9)

w(t, k) = k =0 w(t -1, )c(k -),
which translates into the recursion W t (x) = W t-1 (x)c(x) for the generating function W t (x). Using the fact that W 0 (x) = c(x), we have proved Proposition 5.5 below.

Proposition 5.5. For t = 0, 1, 2, . . . , we have

W t (x) = ∞ k=0 w(t, k)x k = c(x) t+1 . Corollary 5.6. We have the following equation in Z[[x, y]]. (5.10) W (x, y) := ∞ t,k=0 w(t, k)y t x k = c(x) 1 -yc(x)
.

A closed expression for w(t, k).

The following result is an easy consequence of the recurrences above.

Theorem 5.7. For integers t, k ≥ 0, we have

(5.11) w(t, k) = F (t, k).
That is,

(5.12) dim(W t (t + 2k)) = t + 1 t + k + 1 t + 2k k .
6. Generating functions for the simple TL n (q)-modules.

We assume throughout this section that the order

|q 2 | = ∈ N. Recall (Definition 4.2) that l t (n) = dim(L t (n)
) is non-zero only if n = t + 2k for some integer k ≥ 0. For any integer t ≥ 0 define (6.1)

L t (x) = L ( ) t (x) = ∞ k=0 l t (t + 2k)x k .
In this section we shall give explicit formulae for the power series L ( ) t (x).

6.1.

A recurrence for the functions l t . We maintain the following notation, which was introduced in §4. Proposition 6.1. Let t ∈ N and assume below that n ≡ t -1(mod 2).

Notation. Recall that N = {t ∈ N | t ≡ -1(mod )}, and for t ∈ N , b(t) = b, where t = a + b with 0 ≤ b ≤ -2. Write R = {0, 1, 2, . . . , -2}, R = {1,
(

) If b(t) ∈ R (i.e. t ∈ N ) and b(t) = -2, then 1
l t (n + 1) = l t-1 (n) + l t+1 (n). (6.2)
(2) For t ∈ N with b(t) = -2, we have, for n ≡ t -1(mod 2), (6.3)

l t (n + 1) = l t-1 (n).
Proof. The relation (5.1) may be written as follows. For all t, n ∈ N, we have (6.4)

w t (n + 1) = w t-1 (n) + w t+1 (n).
Now observe that if b(t) ∈ N , then applying (6.4) twice, we obtain (6.5)

w t (n + 1) -w t+2b(t) (n + 1) = w t-1 (n) + w t+1 (n) -w t+2b(t)-1 (n) -w t+2b(t)+1 (n).
We shall combine the terms of the right side of (6.5) in different ways, depending on the value of b(t). First take t such that 0

< t < -2. Note that b(t) ± 1 = b(t) ∓ 1, and for t such that 0 < b(t) < -2 we have b(t ± 1) = b(t) ± 1. Hence (6.6) w t (n + 1) -w t+2b(t) (n + 1) =(w t-1 (n) -w t+2b(t)+1 (n)) + (w t+1 (n) -w t+2b(t)-1 (n)) =(w t-1 (n) -w t-1+2b(t)+2 (n)) + (w t+1 (n) -w t+1+2t-2 (n)) =(w t-1 (n) -w t-1+2b(t-1) (n)) + (w t+1 (n) -w t+1+2b(t+1) (n)).
The same relation holds when t in (6.6) is replaced by t + 2i (i ≥ 0). That is, for i ≥ 0 we have (6.7)

w t+2i (n + 1) -w t+2i +2b(t) (n + 1) = (w t+2i -1 (n) -w t+2i -1+2b(t-1) (n)) + (w t+2i +1 (n) -w t+1+2i +2b(t+1) (n)).
Now given the second line of (4.4), summing both sides of (6.7) over i ≥ 0 yields the relation (6.2).

Next take t ≡ 0(mod ), i.e. b(t) = 0. Then (6.5) may be written as follows. For t ≡ 0(mod ), note that b(t) = -1, and we have (6.8)

w t (n + 1)-w t+2( -1) (n + 1) = (w t-1 (n) -w t-1+2 (n)) + (w t+1 (n) -w t+1+2( -2) (n)) = (w t-1 (n) -w t-1+2 (n)) + (w t+1 (n) -w t+1+2(b(t+1)) (n)).
The same relation (6.8) holds when t is replaced by t + 2i (i ≥ 0).

Summing both sides of (6.8) over i ≥ 0, we see that the first summand on the right is w t-1 (n) since all other summands cancel, while the second summand is l t+1 (n) by (4.4). Now observe that when b(t) = 0, then t -1 ≡ -1(mod ), whence by Theorem 4.1(2), we have w t-1 (n) = t-1 (n). This completes the proof of (1).

Finally, take t ≡ -2(mod ), i.e. b(t) = -2, so that b(t) = 1 and g(t) = t + 2. In this case (6.5) reads as follows. (6.9)

w t (n + 1) -w t+2 (n + 1) =w t-1 (n) + w t+1 (n) -(w t+1 (n) + w t+3 (n)) =w t-1 (n) -w t+3 (n)) =w t-1 (n) -w t-1+2b(t-1) (n)).
The relation (6.9) remains true when t is replaced by t + 2i for any i ≥ 0, so that (6.10)

w t+2i (n + 1) -w g(t+2i ) (n + 1) = w t+2i -1 (n) -w t+2i -1+2b(t+2i -1) (n)),
and summing both sides of (6.10) over i ≡ -2(mod ) yields the relation (6.3) and completes the proof of the proposition.

6.2. Generating functions. We continue to assume that q 2 has finite order . In this subsection, we give explicit generating functions for the dimensions l t (n) of the simple modules L t (n) of the algebras TL n (q). Specifically, we give explicit formulae for the power series L ( ) t (x) defined in (6.1). Recall (5.7) that c(x) = 1 + ∞ n=1 1 n+1 2n n x n , and define (6.11)

d(x) = c(x) -1 = xc(x) 2 .
Notice that the relation (5.6) may be written (6.12)

x(d(x) + 1) 2 = d(x).
Recall also that for t ∈ N, b(t) is defined by t = a + b(t), where 0 ≤ b(t) ≤ -1, and that R = {0, 1, . . . , -2}.

We shall prove Theorem 6.2. Maintain the above notation and let

t ∈ N. If b(t) = -1, then L ( ) t (x) = W t (x) = c(x) t+1 . If b(t) ∈ R then (6.13) L ( ) t (x) = (d(x) + 1) t+1 (1 -d(x) -1-b(t) ) 1 -d(x) .
Proof. If b(t) = -1 then by Theorem 4.1(2), W t (n) is simple for all n. Hence in this case L ( )

t (x) = ∞ k=0 w t (t + 2k)x k = W t (x) = c(x) t+1 by Proposition 5.5. Now assume that b(t) ∈ R.
It follows from (4.4) that for k ≥ 0, (6.14)

l t (t + 2k) = ∞ i=0 w g 2i (t) (t + 2k) - ∞ i=0 w g 2i+1 (t) (t + 2k) = ∞ i=0 w t+2i (t + 2k) - ∞ i=0 w t+2i +2b(t) (t + 2k).
where g is the function defined in Theorem 4.1.

Now multiply each side of (6.14) by x k and sum over k. We evaluate the two summands separately. We first have

∞ k=0 ∞ i=0 w t+2i (t + 2k)x k = ∞ k=0 ∞ i=0 w t+2i (t + 2i + 2(k -i ))x k = ∞ k=0 ∞ i=0 w t+2i (t + 2i + 2(k -i ))x k-i x i = ∞ i=0 x i ∞ k=0 w t+2i (t + 2i + 2(k -i ))x k-i = ∞ i=0 x i W t+2i (x) since w t (n) = 0 for n < t = ∞ i=0 x i c(x) t+2i +1 by Proposition 5.5 =c(x) t+1 ∞ i=0 (xc(x) 2 ) i = c(x) t+1 1 -(xc(x) 2 ) = (d(x) + 1) t+1 1 -d(x) A similar calculation yields that ∞ k=0 ∞ i=0 w t+2i +2b(t) (t + 2k)x k = (d(x) + 1) t+1 d(x) -1-b(t) 1 -d(x) ,
and using (6.14), the proof is complete.

6.

3. An alternative formula for L ( ) t (x). We give in this section a formula for L ( ) t (x) in terms of the polynomials p i (x) defined below. Definition 6.3. Define a sequence of polynomials p i (x) ∈ Z[x], i = 1, 2, 3, . . . by (6.15) p 1 (x) = p 2 (x) = 1 and

p i+1 (x) = p i (x) -xp i-1 (x) for i ≥ 2. Thus p 3 (x) = 1 -x, p 4 (x) = 1 -2x, p 5 (x) = 1 -3x + x 2 and p 6 (x) = 1 -4x + 3x 2 , etc.
Lemma 6.4. Let y be an indeterminate over Z and j a positive integer.

(1) For each j ≥ 1 there are unique integers c j i such that

(6.16) 1 + y + y 2 + • • • + y j-1 = [ j-1 2 ] i=0 c j i y i (y + 1) j-1-2i .
(2) The integers c j i satisfy the recurrence

c j+1 i = c j i -c j-1 i-1 . (3) We have [ j-1 2 ] i=0 c j i x i = p j (x). Proof. A polynomial f (y) ∈ C[y] is said to be n-palindromic (n ≥ 0) if y n f (y -1 ) = f (y).
The n-palindromic polynomials form a vector space of dimension 1 + n 2 . For fixed j ≥ 1, the polynomials y i (y+1) j-1-2i , 0 ≤ i ≤ j-1 2 are all (j -1)-palindromic, and since they have leading terms of different degrees, they form a basis of the space of (j -1)-palindromic polynomials.

The statement (1) follows easily.

If we write σ j = 1 + y + y 2 + • • • + y j-1 , note that (1 + y)σ j = σ j+1 + yσ j-1 . Applying (6.16), we obtain

[ j-1 2 ] i=0 c j i y i (y + 1) j-2i = [ j 2 ] i=0 c j+1 i y i (y + 1) j-2i + [ j-2 2 ] i=0 c j-1 i y i+1 (y + 1) j-2-2i .
Comparing the coefficients of y i (y + 1) j-2i yields the relation [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF].

Write C j (x) = [ j-1 2 ] i=0 c j i x i .
The recurrence [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF] shows that C j+1 (x) = C j (x) -xC j-1 (x). Further, it is easily checked that C 1 (x) = C 2 (x) = 1, which, by comparison with (6.15), completes the proof that C j (x) = p j (x).

t (x) = c(x) a p (x)
,

where p (x) is the polynomial defined in (6.15).

Proof. First observe that by (6.3),

l t (t + 2k) = l t-1 (t -1 + 2k) if b(t) = -2, so that L ( )
t (x) will be the same for the two nominated values of t. Now take t = a + -2. Applying the formula (6.13), one sees easily that L ( )

t (x) = (d(x) + 1) a + -1 1 + d(x) + d(x) 2 + • • • + d(x) -1 .
Now using the relation x(d(x) + 1) 2 = d(x) repeatedly, together with Lemma 6.4, one sees that p -1 (x)L ( )

t (x) = (d(x) + 1) a .
The next result is a generalisation of [18, Thm. 2.3], which deals essentially with the case 0 ≤ t ≤ -2 of the Theorem. Theorem 6.6. With the above notation, we have, for t = a + b with 0 ≤ b ≤ -2, (6.17) L ( )

t (x) = p -1-b (x) p (x) c(x) a ,
where c(x) is the Catalan series (5.7) and the p i (x) are defined in (6.15).

Proof. The recurrence (6.2) may be written as follows: for t such that b(t) = 0, -2, we have l t (t + 2k) = l t-1 (t -1 + 2k) + l t+1 (t + 1 + 2(k -1)). Multiplying this relation by x k and summing over k ≥ 0, we obtain, after rearrangement, (6.18) L ( )

t-1 (x) = L ( ) t (x) -xL ( ) t+1 (x).
Now fix a ∈ N and consider the power series L ( ) t (x) for t = a + b, 0 ≤ b ≤ -2. We have seen in Lemma 6.5 that when b = -2 or -3, then L ( )

a + -2 (x) = L ( ) a + -3 (x) = c(x) a p (x)
. Now fix b such that 0 ≤ b ≤ -3 and assume that for all b with -2 ≥ b ≥ b, there are polynomials r -1-b (x) such that L ( )

a +b (x) = r -1-b (x)L ( )
a + -2 (x). Then r 1 (x) = r 2 (x) = 1 and from the recurrence (6.18) we have

L ( ) a +b-1 (x) = L ( ) a +b (x) -L ( ) a +b+1 (x) = r -1-b (x)L ( ) a + -2 (x) -xr -2-b (x)L ( ) a + -2 (x) = r -1-(b-1) (x)L ( ) a + -2 (x), where r -1-b+1 (x) = r -1-b (x) -xr -2-b (x).
It follows that for b = 0, 1, 2, . . . , -2, r -1-b (x) = p -1-b (x) where p i (x) is as in (6.15), and that L ( )

a +b (x) = p -1-b (x)L ( ) a + -2 (x).
The Theorem now follows by using the expression for L ( ) a + -2 (x) in Lemma 6.5. Remark 6.7. It is clear that the Jones quotient Q n ( ) is not generally the largest semisimple quotient of TL n (q). In fact this is true precisely when R n annihilates all the simple TL n -modules, which happens if and only if n < .

For example if = 3, TL 3 (q) has two simple modules L 1 (3) and L 3 (3) of dimension 1, so its largest semisimple quotient has dimension 2, while

Q 3 (3) has dimension 1.
Other examples include TL 7 where = 5. We have dim(Q 7 ) = F 13 = 233 while the maximal semisimple quotient of TL 8 has dimension 270. 6.3.1. Some examples. We give several examples of the application of Theorem 6.6.

(1

) When = 3, L t (x) = c(x) 3a 1-x for t = 3a + b, 0 ≤ b ≤ 1. (2) When = 4, L 1 (x) = L 2 (x) = 1 1-2x , while L 0 (x) = p 3 (x) p 4 (x) = 1-x 1-2x . So dim L 0 (2n) = 2 n-1 , dim L 1 (2n + 1) = 2 n and dim L 2 (2n) = 2 n-1 .
(3) Take = 5. We shall determine L i (x) for i = 0, 1, 2, 3. We have

L 2 (x) = L 3 (x) = 1 1-3x+x 2 , L 1 (x) = 1-x 1-3x+x 2 and L 0 (x) = 1-2x 1-3x+x 2 . Note that L 0 (x)-1 x = L 1 (x). Let us write 1 -x 1 -3x + x 2 = ∞ n=0 a n x n and x 1 -3x + x 2 = ∞ n=0 b n x n . Then a 0 = 1, a 1 = 2, a 2 = 5, a 3 = 13 and b 0 = 0, b 1 = 1, b 2 = 3, b 3 = 8. Let F 1 , F 2 , F 3 , • • • = 1, 1, 2,
a 0 , b 1 , a 1 , b 2 , a 2 , b 3 , a 3 , • • • = F 1 , F 2 , F 3 , . . . ,
i.e. that for i = 0, 1, 2, . . . , we have a i = F 2i+1 and b i = F 2i .

To prove (6.19), given the initial values of the a i and b i , it suffices to show that (i) b n + a n = b n+1 for n ≥ 0, and (ii)

a n + b n+1 = a n+1 for n ≥ 0. For (i), observe that ∞ n=0 (a n +b n )x n = 1-x+x 1-3x+x 2 = 1 1-3x+x 2 = ∞ n=0 b n+1 x n . Similarly, for (ii), we have ∞ n=0 (a n + b n+1 )x n = 2-x 1-3x+x 2 , which is readily shown to be equal to ∞ n=0 a n+1 x n . It follows that L 3 (x) = L 2 (x) = ∞ n=0 b n+1 x n We have therefore shown that dim(L 2 (2+2n)) = dim(L 3 (3+2n)) = b n+1 = F 2n+2 , dim(L 1 (1+2n)) = a n = F 2n+1 and for n > 0, dim(L 0 (2n)) = a n -b n = a n-1 = F 2n-1 .
(4) Again take = 5, and consider L 8 (x). This gives the dimension of the simple TL n -modules L 8 (n), which are not modules for Q n . We have

L 8 (x) = c(x) 5 p 5 (x) = 1+5x+20x 2 +... 1-3x+2x 2 . So, for example, 8 (12)
= 40, while dim(W 8 (12)) = 54. This shows that the latter module has radical of dimension 14.

(5) Take = 6. We shall compute L t (x) for t = 0, 1, 2, 3, 4. Note first that L 3 (x) = L 4 (x) = 1 1-4x+3x 2 , and since 1 -4x + 3x 2 = (1 -x)(1 -3x), we have

L 3 (x) = L 4 (x) = ∞ n=0 3 n+1 -1 2 x n It follows that L 2 (x) = (1-x)L 3 (x) = ∞ n=0 3 n x n = 1 1-3x . Similarly, L 1 (x) = 1-2x 1-4x+3x 2 = ∞ n=0 3 n +1 2 x n and L 0 (x) = 1 + xL 1 (x) = 1 + ∞ n=1 3 n-1 +1 2
x n . (6) Take = 7. An easy but tedious calculation shows that in this case

L 4 (x) = L 5 (x) =1 + 5x + 19x 2 + 66x 3 + 221x 4 + 728x 5 + . . . L 3 (x) =1 + 4x + 14x 2 + 47x 3 + 155x 4 + 507x 5 + . . . L 2 (x) =1 + 3x + 9x 2 + 28x 3 + 89x 4 + 286x 5 + . . . L 1 (x) =1 + 2x + 5x 2 + 14x 3 + 42x 4 + 131x 5 + . . . L 0 (x) = 1 + xL 1 (x) =1 + x + 2x 2 + 5x 3 + 14x 4 + 42x 5 + . . .

The algebras Q n ( ).

We assume throughout this section that is fixed and |q 2 | = . Recall (Definition 3.6) that Q n ( ) TL n (q)/R n (q), where R n is generated by the Jones-Wenzl idempotent E -1 ∈ TL -1 (q). We have seen (3.4) that, although they are not the maximal semisimple quotients of the TL n (q), the algebras Q n are semisimple and we therefore focus on a description of their simple modules.

7.1. Classification of the simple Q n ( )-modules. The next statement is elementary.

Proposition 7.1. Let n ≥ -1. With R n as above, the simple Q n -modules are precisely those simple TL n -modules L t , t ∈ T (n), such that R n L t = 0. Remark 7.2. If N is a TL n module, then since R n = TL n E -1 TL n , it follows that R n N = 0 if and only if E -1 N = 0. Thus the condition in the Proposition is relatively straightforward to check.

Remark 7.3 (Remark concerning notation). Although a priori E -1 ∈ TL -1 , we have regarded it as an element of TL n for any n ≥ -1. The strictly correct notation for E -1 ∈ TL n , where n ≥ , is E -1 ⊗ I ⊗(n-+1) , where the tensor product is in the Temperley-Lieb category T, as described in [START_REF] Graham | The representation theory of affine Temperley-Lieb algebras[END_REF] or [START_REF] Lehrer | The Brauer Category and Invariant Theory[END_REF]; that is, it is described diagrammatically as juxtaposition of diagrams, and I is the identity diagram from 1 to 1. We shall use this notation freely below. Then the idempotent E -1 ⊗ I ⊗(n-+1) acts non-trivially on L t . Thus Q n has at most 2 isomorphism classes of simple modules. Proof. We begin by showing that if t ∈ T (n) and t ≥ -1 then (E -1 ⊗I ⊗(n-+1) )W t contains all diagrams of the form I ⊗t ⊗ D , where D is any monic diagram from 0 to n -t.

To see this, note that W t is spanned by monic diagrams from t to n in T. Take D = I ⊗t ⊗D ∈ W t , where D is any (monic) diagram from 0 to n-t. By the formula in Theorem 3.3, the coefficient of I ⊗( -1) in E -1 is 1. Since all the other summands act trivially on D (because they reduce the number of 'through strings'), it follows that (E -1 ⊗ I ⊗(n-+1) )D = D in W t , and hence that D ∈ (E -1 ⊗ I ⊗(n-+1) )W t . Now if D = I ⊗t ⊗ D as above and ( , ) t is the canonical bilinear form on W t (see [12, §2]), then (D, D) t is a power of -(q + q -1 ), and hence is non-zero. It follows that D ∈ Rad t , and hence that E -1 L t = 0.

It follows from the above result that the only possible simple Q n -modules are the L t with t < -1.

Theorem 7.5. The simple Q n modules are the L t with t ≤ -2.

Proof. In view of Proposition 7.1 and Theorem 7.4, it suffices to show that (E -1 ⊗

I ⊗(n-+1) )L t = 0 for t ≤ -2.
For this, it suffices to show that (E -1 ⊗ I ⊗(n-+1) )W t ⊆ Rad t for t in the relevant range, and this latter statement will follow if we prove that for any two monic diagrams D 1 , D 2 ∈ W t , we have

(7.1) (E -1 ⊗ I ⊗(n-+1) )D 1 , D 2 t = 0.
To see (7.1), observe that since (E -1 ⊗ I ⊗(n-+1) ) is an idempotent, it follows from the invariance of the form ( , ) t , that

(E -1 ⊗ I ⊗(n-+1) )D 1 , D 2 t = (E -1 ⊗ I ⊗(n-+1) )D 1 , (E -1 ⊗ I ⊗(n-+1) )D 2 t .
Now for any monic diagram D ∈ W t , (E -1 ⊗ I ⊗(n-+1) )D = 0 unless any upper arc of D whose left end i ≤ -1 has right end j ≥ , for otherwise D is annihilated by

E -1 ⊗ I ⊗(n-+1) because E -1 is harmonic. It follows, again from the harmonic nature of E -1 , that if (E -1 ⊗ I ⊗(n-+1) )D = 0, we must have D = I ⊗t 1 ⊗ D ⊗ I ⊗t 2 ,
where t 1 + t 2 = t and D : 0 → n -t 1 -t 2 is a diagram such that each of its arcs (it has only upper arcs) has right end in { , + 1, . . . , n}. Now let D 1 and D 2 be two such diagrams. Then to show that

((E -1 ⊗ I ⊗(n-+1) )D 1 , (E -1 ⊗ I ⊗(n-+1) )D 2 ) t = 0, it suffices to show that D * 1 (E -1 ⊗ I ⊗(n-+1
) )D 2 = 0. This latter fact will follow from the harmonic nature of E -1 , as well as the property of E -1 proved in Proposition 3.5 [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF]. To see it, note that if D * 1 (E -1 ⊗ I ⊗(n-+1) )D 2 is depicted diagramatically in a way similar to Fig. 1,2 or 3, the string emanating from the ( -1) st top vertex (i.e. the rightmost upper vertex of E -1 ) either ultimately returns to an upper vertex of E -1 , in which case we have the zero element by harmonicity, or else it joins the corresponding string emanating from the ( -1) st bottom vertex. In this case we have the zero element by Proposition 3.5 [START_REF] Henning | Cellularity of certain quantum endomorphism algebras[END_REF].

This completes the proof of the theorem.

7.2. Dimensions of the simple Q n ( )-modules. Since the simple Q n ( )-modules are just the L ( )

t (n) 0 ≤ t ≤ -2, t ≡ n(mod 2)
, their dimensions are given by the formula (6.17). That is,

(7.2) ∞ k=0 dim(L ( ) t (t + 2k))x k = p -1-t (x) p (x) .
7.3. The case = 4. Clifford algebras. We have seen in (6.3.1)(2) that Q 2n+1 (4) has just one simple module, whose dimension is 2 n and that Q 2n (4) has two simple modules, both of dimension 2 n-1 . It follows (see also the general formula (8.13)) that dim Q n (4) = 2 n-1 for n ≥ 1. We shall see in this section that in this case, Q n is actually the even subalgebra of a Clifford algebra. Because of its connection to the Ising model in statistical mechanics [START_REF] Martin | Potts models and related problems in statistical mechanics[END_REF], we shall refer to the Q n (4) as the Ising algebras.

Let U be a complex vector space of finite dimension n, with a non-degenerate symmetric bilinear form -, -. Then U has an orthonormal basis u 1 , . . . , u n . If γ i = 1 √ 2 u i for i = 1, . . . , n, then for any i, j,

(7.3) γ i , γ j = 1 2 δ i,j .
The Clifford algebra C n = C(U, -, -) (for generalities about Clifford algebras we refer the reader to [START_REF] Garling | Clifford algebras: an introduction[END_REF]) is defined as

(7.4) C n = T (U ) I ,
where T (U ) = ⊕ ∞ i=0 U ⊗i is the free associative C-algebra (or tensor algebra) on U , and I is the ideal of T (U ) generated by all elements of the form u ⊗ u -u, u 1 (u ∈ U ). This last relation may equivalently be written (omitting the ⊗ in the multiplication) (7.5) uv

+ vu = 2 u, v 1.
The algebra C n is evidently generated by any basis of U , and hence by (7.3) and (7.5) has the presentation

(7.6) C n = γ 1 , . . . , γ n | γ i γ j + γ j γ i = δ ij for 1 ≤ i, j ≤ n .
For any subset J = {j 1 < j 2 < • • • < j p } ⊆ {1, . . . , n}, write γ J = γ j 1 γ j 2 . . . γ jp . It is evident that {γ J | J ⊆ {1, 2, . . . , n}} is a basis of C n which is therefore Z 2 -graded (since the relations are in the even subalgebra of the tensor algebra), the even (resp. odd) subspace being spanned by those γ J with |J| even (resp. odd).

The following statement is now clear.

Proposition 7.6. The Clifford algebra C(U, -, -) has dimension 2 n , where n = dim(U ). Its even subalgebra C 0 n has dimension 2 n-1 . The next theorem is the main result of this section; it asserts that the Ising algebra is isomorphic to the even subalgebra of the Clifford algebra.

Theorem 7.7. We continue to assume = 4 and that q = -exp( πi 4 ). Other notation is as above. For n = 3, 4, . . . there are surjective homomorphisms φ n : TL n (q) → C 0 n which induce isomorphisms

φ n : Q n -→ C 0 n . Proof. Define φ n (f j ) = 1 √ 2 (1+2iγ j γ j+1
). It was remarked by Koo and Saleur [19, §3.1 eq. (3.2)] (see also [START_REF] Connes | Embedding of U (1)-current algebras in noncommutative algebras of classical statistical mechanics[END_REF]) that the φ n (f j ) satisfy the relations (2.1) in C n , and therefore that φ n defines a homomorphism from TL n to C n , and further that E 3 ∈ ker(φ n ).

It is evident that the image of φ n is C 0 n , and therefore that φ n : Q n → C 0 n is surjective. But by Prop. 7.6 these two algebras have the same dimension, whence φ n is an isomorphism. 7.3.1. Canonical trace. Let TL n (q) be the n-string Temperley-Lieb algebra as above, and assume δ := -(q + q -1 ) = 0 is invertible. The canonical Jones trace tr n on TL n (q) was defined in (2.2). As pointed out in (3.4), this trace descends to a nondegenerate trace on Q n , satisfying similar properties. In the case = 4 this amounts to the following statement. Proposition 7.8. There is a canonical trace tr n on C 0 n , given by taking the constant term (coefficient of 1) of any of its elements. This trace corresponds to the Jones trace above in the sense that for x ∈ Q n , tr n (x) = tr n (φ(x)). It is therefore nondegenerate.

The proof is easy, and consists in showing that tr n satisfies the analogue of (2.2) in C 0 n .

The spinor representations of so(n).

We give yet another interpretation of the algebra in terms of the spin representations of so(n). Let SO(n) be the special orthogonal group of the space (U, -, -) above. Its Lie algebra has basis the set of matrices (with respect to the orthogonal basis (γ i ))

J ij := E ij -E ji , 1 ≤ i < j ≤ n,
where the E ij are the usual matrix units. This basis of so(n) satisfies the commutation relations

(7.7) [J ij , J kl ] = δ jk J il -δ jl J ik -δ ik J jl + δ il J jk .
Proposition 7.9. For n ≥ 2, there are surjective homomorphisms

ψ n : U(so(n)) → C 0 n ∼ = Q n , such that ψ n (J ij ) = ω ij := 1 2 (γ i γ j -γ j γ i ).
The irreducible spin representations of so(n) are realised on the simple Q n -modules L 0 and L 2 when n is even and on L 1 when n is odd.

Proof. As this is well known, we give merely a sketch of the argument. To show that ψ n defines a homomorphism, it suffices to observe that the ω ij satisfy the same commutation relations (7.7) as the J ij , and this is straightforward. The surjectivity of ψ n is evident from the observation that ω ij = γ i γ j , which shows that the image of ψ n contains the whole of C 0 n Q n .

8. The algebras Q n ( ) and the Reshetikhin-Turaev-Andersen fusion category.

We show in this section that Q n ( ) is the endomorphism algebra of a certain truncated tensor product of modules for U q = U q (sl 2 ), where q is such that q 2 is a primitive th root of unity. An observation about the relevant fusion category permits the determination of the dimension of Q n ( ). We assume throughout that ≥ 3. 8.1. Tilting modules for U q (sl 2 ). For n ∈ N, let ∆ q (n) be the Weyl module (cf. [2, §1]) of the quantum group U q = U q (sl 2 ) and let T q (n) be the unique indecomposable tilting module for U q with highest weight n [2, §5].

It follows from [2, Thm. 5.9] that for n ∈ N,

(8.1) ∆ q (1) ⊗n t∈N l t (n)T q (t),
where l t (n) = dim(L t (n)) is the dimension of the simple TL n (q)-module L t (n). Note that l t (n) is non-zero only if t ≡ n(mod 2). Further, the structure of the tilting modules T q (m) is described in [2, Prop. 6.1] as follows. (8.2) (1) If m < or m ≡ -1(mod ), then T q (m) ∆ q (m) is a simple U q -module.

(2) If m = a + b with a ≥ 1 and 0 ≤ b ≤ -2, then T q (m) has a submodule isomorphic to ∆ q (m) such that

T q (m) ∆ q (m) ∆ q (g -1 (m)),
where g is the function defined in (4.3).

8.2.

Andersen's fusion category. Andersen proved in [START_REF] Henning Haahr Andersen | Tensor products of quantized tilting modules[END_REF]Thm. 3.4] a general result for quantised enveloping algebras at a root of unity, which implies in the case of U q (sl 2 ) that the tilting modules T q (m) with 0 ≤ m ≤ -2 are precisely those indecomposable tilting modules with non-zero quantum dimension. This may easily be verified directly in our case using the description (8.2) of the tilting modules. Andersen's result [START_REF] Henning Haahr Andersen | Tensor products of quantized tilting modules[END_REF]Cor. 4.2] (see also [START_REF] Reshetikhin | Invariants of 3-manifolds via link polynomials and quantum groups[END_REF]) implies that in our case, we have the following result.

Proposition 8.1. Let M, N be tilting modules for U q (sl 2 ). Write

M ⊗ N = n∈N m n T q (n),
and define the fusion product ⊗ by

(8.3) M ⊗N = -2 n=0 m n T q (n).
Then the fusion product ⊗ is associative.

This implies that we have a semisimple tensor category C reg with objects the tilting modules

⊕ -2 n=0 m n T q (n)(= ⊕ -2 n=0 m n ∆ q (n)) (m n ∈ N)
, and tensor product ⊗. Definition 8.2. For modules

M = ⊕ -2 n=0 m n T q (n) and M = ⊕ -2 n=0 m n T q (n), define (8.4) (M, M ) Uq = dim(Hom Uq (M, M )) = -2 n=0 m n m n . Lemma 8.3. Let M, N ∈ C red . Then (8.5) (M ⊗∆ q (1), N ) Uq = (M, N ⊗∆ q (1)) Uq
Proof. Since both sides of (8.5) are additive in M and N , it suffices to take M = T q (s) = ∆ q (s) and N = ∆ q (t) for s, t ∈ {0, 1, 2, . . . , -2}. The "reduced Clebsch-Gordan formula" asserts that for m ∈ {0, 1, 2, . . . , -2}, (

∆ q (m)⊗∆ q (1)

     ∆ q (m -1) ⊕ ∆ q (m + 1) if m = 0 or -2 ∆ q (1) if m = 0 ∆ q ( -3) if m = -2.
It is now easily verified that Hom Uq (M ⊗∆ q (1), N ) ∼ = Hom Uq (M, ∆ q (1)⊗N ), and the result follows.

The following result provides an explicit description of the operation ⊗ in the category C red . Proposition 8.4. Suppose s, t ∈ Z are such that 0 ≤ s, t ≤ -2. Then (8.7) ∆ q (s)⊗∆ q (t)

∼ = ∆ q (|s -t|) ⊕ ∆ q (|s -t| + 2) ⊕ • • • ⊕ ∆ q (m),
where m = m(s, t) = min{s + t, 2( -2) -(s + t)}.

Proof. Note first that by the commutativity of ⊗, it suffices to prove (8.7) for s, t such that 0 ≤ t ≤ s ≤ -2. Further, observe that (8.7) holds for t = 0, 1. The case t = 0 is trivial, while if t = 1(≤ s), we have

∆ q (s)⊗∆ q (1) ∼ = ∆ q (s -1) ⊕ ∆ q (s + 1) if s < -2 ∆ q (s -1) if s = -2,
which is precisely the assertion (8.7) in this case. We next show that (8.7) holds when s = -2. This assertion amounts to (8.8) ∆ q ( -2)⊗∆ q (t) ∼ = ∆ q ( -2 -t) for all t.

We prove (8.8) by induction on t; the statement holds for t = 0, 1, as already observed. For 1 < t ≤ -2, we have ∆ q (t -1)⊗∆ q (1) ∼ = ∆ q (t) ⊕ ∆ q (t -2), whence by induction, ∆ q ( -2)⊗∆ q (t-1)⊗∆ q (1) ∼ = ∆ q ( -2)⊗∆ q (t)⊕∆ q ( -2)⊗∆ q ( -2-(t -2)). But again by induction, the left side is equal to ∆ q ( -2

-(t -1))⊗∆ q (1) ∼ = ∆ q ( -2 -(t -2)) ⊕ ∆ q ( -2 -t)
, which proves (8.8).

We may therefore now assume that -3 ≥ s ≥ t ≥ 2, and proceed by induction on t. Using (8.6), and (8.5), we see easily that for any r with 0 ≤ r ≤ -2, we have (8.9) (∆ q (s)⊗∆ q (t), ∆ q (r)) Uq = (∆ q (s)⊗∆ q (t -1), ∆ q (1)⊗∆ q (r)) Uq -(∆ q (s)⊗∆ q (t -2), ∆ q (r)) Uq .

We shall show, using (8.9), that for 0 ≤ r ≤ -2, Assertion 8.5. The multiplicity of ∆ q (r) in both sides of (8.7) is the same.

If r = 0, the right side of (8.9) is zero unless s -t + 1 = 1, i.e. s = t, in which case it is 1. This proves the assertion for r = 0. If r = -2, the first summand on the right side of (8.9) is (∆ q (s)⊗∆ q (t -1), ∆ q ( -3)) Uq , which is 1 if s + t -1 = -1 or -3 and zero otherwise. If s + t -1 = -1, then the second summand on the right side of (8.9) is 1, whence the right side is zero unless s + t = -2, in which case it is 1. This proves Assertion 8.5 when r = -2.

We may therefore assume that 0 < r < -2, so that (8.9) may be written as follows.

(8.10) (∆ q (s)⊗∆ q (t), ∆ q (r)) Uq = (∆ q (s)⊗∆ q (t -1), ∆ q (r -1)) Uq + (∆ q (s)⊗∆ q (t -1), ∆ q (r + 1)) Uq -(∆ q (s)⊗∆ q (t -2), ∆ q (r)) Uq .

Now by induction, we have (8.11) ∆ q (s)⊗∆ q (t -1) ∼ =∆ q (s -t + 1) ⊕ ∆ q (s -t + 3) ⊕ • • • ⊕ ∆ q (m(s, t -1))

and ∆ q (s)⊗∆ q (t -2) ∼ =∆ q (s -t + 2) ⊕ ∆ q (s -t + 4) ⊕ • • • ⊕ ∆ q (m(s, t -2)).

We consider three cases. Case 1: s + t -1 > -2. In this case it is clear that m(s, t -1) = m(s, t) + 1 and m(s, t + 2) = m(s, t) + 2. Hence in equation (8.10), the last two summands cancel, and we are left with (∆ q (s)⊗∆ q (t), ∆ q (r)) Uq = (∆ q (s)⊗∆ q (t -1), ∆ q (r -1)) Uq (for 0 < r < -2). Bearing in mind that m(s, t -1) = m(s, t) + 1, this completes the proof of Assertion 8.5 in this case. Case 2: s+t-1 ≤ -2 and s+t = -1. When s+t-1 ≤ -2, a short calculation shows that m(s, t -1) = m(s, t) -1 and m(s, t -2) = m(s, t) -2, except in the single case when s + t = -1, with which we shall deal separately. We therefore assume for the moment that s + t = -1, and using (9.7), evaluate each of the three terms in the right side of (8.10). The first term is 1 for r (of the correct parity) such that s -t + 2 ≤ r ≤ m(s, t), and zero otherwise. The second term is 1 for r (of the correct parity) such that s -t ≤ r ≤ m(s, t) -2, and zero otherwise, while the third term is -1 for r (of the correct parity) such that s -t + 2 ≤ r ≤ m(s, t) -2, and zero otherwise. This proves Assertion 8.5 in this case. Case 3: We consider finally the remaining case s + t = -1. In this case we have m(s, t) = -3 = m(s, t -2), and m(s, t -1) = -2. Using this we again evaluate the three terms on the right side of (8.10), recalling that r ≤ -3. The first term is 1 for r (of the correct parity) such that s -t + 2 ≤ r ≤ -3, and zero otherwise. The second term is 1 for r (of the correct parity) such that s -t ≤ r ≤ -3, and zero otherwise while the third term is -1 for r (of the correct parity) such that s -t + 2 ≤ r ≤ -3, and zero otherwise.

This completes the proof of Proposition 8.4.

8.3.

Connection with the algebra Q n ( ). We start with the following observation.

Proposition 8.6. We have

End Uq (∆ q (1) ⊗ n ) ∼ = Q n ( ).
Proof. It follows from the definition and from (8.1) that (8.12) ∆ q (1) ⊗n -2 t=0 l t (n)∆ q (t).

Since the ∆ q (t) are simple for 0 ≤ t ≤ -2, it follows that End Uq (∆ q (1) ⊗ n ) is the direct sum of matrix algebras of degree l t (n), for t such that 0 ≤ t ≤ -2 and t ≡ n(mod 2). But this latter set of integers is precisely the set of degrees of the simple modules for the semisimple algebra Q n ( ). The result follows.

Remark 8.7. The results in this paper actually suffice to give a decomposition of T q (m) ⊗ T q (n) as a sum of tilting modules. This would give an alternative proof of Proposition 8.4. Proposition 8.4 may be used to deduce the dimension of Q n ( ).

( ) (x) := ∞ n=0 dim(Q n+1 ( ))x n . Then (8.13) Q ( ) (x) = p -2 (x) p (x) ,
where the polynomials p i (x) are defined in (6.15).

Proof. It follows from Proposition 8.6 that in the notation of Definition 8.2, dim(Q n+1 ( )) = (∆ q (1) ⊗(n+1) , ∆ q (1) ⊗(n+1) ) Uq .

But by n applications of Lemma 8.3, we see that (∆ q (1) ⊗(n+1) , ∆ q (1) ⊗(n+1) ) Uq = (∆ q (1) ⊗(2n+1) , ∆ q (1)) Uq = l 1 (2n + 1).

Finally, by (7.2), we have ∞ n=0 l 1 (2n + 1)x n = p -2 (x) p (x) , and the proof is complete.

Fusion algebras and fusion categories.

In this section we investigate some structures which are related to the constructions above. We start with a fusion structure on the representation rings of the algebras Q n ( ). Throughout this section we take = |q 2 | ≥ 3 as fixed, unless otherwise specified. 9.1. Fusion structure on the Jones algebras. Let Q n = Q n ( ) be as above. This is a semisimple algebra, and if we write R(n) := {t ∈ Z | t ≡ n(mod 2) and 0 ≤ t ≤ min{n, -2}}, then writing K 0 for the Grothendieck ring, (9.1)

K 0 (Q n ) ∼ = ⊕ t∈R(n) Z[L t (n)]. Define the algebra K(Q) := n≥1 K 0 (Q n ),
where multiplication is given by

(9.2) [L s (m)] • [L t (n)] := [Ind Q m+n Qm⊗Qn (L s (m) L t (n))
]. Remark 9.1. Here Q m ⊗Q n is the subalgebra of Q m+n which is generated by the image of TL m (q) ⊗ TL n (q) ⊆ TL m+n (q) under the canonical map TL m+n (q) → Q m+n ( ). The induced representation Ind TL m+n TLm⊗TLn (L s (m) L t (n)) may have summands which are not acted upon trivially by R m+n (q). To obtain a representation of Q m+n , we consider the submodule of this induced representation of TL m+n consisting of elements annihilated by R m+n (q). The multiplication defined above on K(Q) is bilinear, associative and commutative.

Theorem 9.2. We have

(9.3) [L s (m)] • [L t (n)] =
|s-t|≤r≤m(s,t)

[L r (m + n)].
where m(s, t) = min{s + t, 2( -2) -(s + t)}, as in Proposition 8.4.

Proof. It follows from Proposition 8.6 that for m ≥ 1, as U q ⊗ Q m -module, ∆ q (1) ⊗m ∼ = ⊕ s∈R(m) ∆ q (s) L s (m).

It follows that as U q ⊗ (Q m ⊗ Q n )-module, we have (9.4) ∆ q (1) ⊗(m+n) ∼ = s∈R(m), t∈R(n)

(∆ q (s)⊗∆ q (t)) (L s (m) L t (n)) .

But as a module for U q ⊗ Q m+n , (9.5) ∆ q (1) ⊗(m+n) ∼ = r∈R(m+n)

∆ q (r) L r (m + n).

Moreover by Proposition 8.4, we may expand (9.4) as follows.

(∆ q (s)⊗∆ q (t)) (L s (m) L t (n)) ∼ = r∈R(m+n),|s-t|≤r≤m(s,t)

∆ q (r) (L s (m) L t (n)) .

Comparing this last equation with (9.5), we see that given r ∈ R(m + n), we have As h r,s = h m-r,m+1-s , it is suffices to take, 1 ≤ s ≤ r < m in the latter case. The "if" part of this statement was proved by Goddard, Kent and Olive [START_REF] Goddard | Unitary representations of the Virasoro and uper-Virasoro Algebras[END_REF] and the "only if " part was proved by Langlands [START_REF] Langlands | On unitary representations of the Virasoro algebra[END_REF]. This result bears a superficial resemblance to Jones' result on the range of values of the index of a subfactor as was mentioned in the preamble. Thus it might be expected that case (2) is somehow connected with our algebras Q n ( ) for = 3, 4, 5, . . . . Further, there are several instances in the literature (see, e.g. [START_REF] Gainutdinov | Fusion and braiding in finite and affine Temperley-Lie categories[END_REF][START_REF] Koo | Representations of the Virasoro algebra from lattice models[END_REF][START_REF] Nichols | The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models[END_REF]) which hint at a connection between Q n ( ) and the minimal unitary series of L with central charge c . Our work may provide some further evidence along those lines. For = 3, c = 0, and there is just one irreducible representation, viz. the trivial one. This is 'consistent' with Q n (3) = C. For = 4, c = By the usual properties of inverse limits, we have maps τ n : K(Q ∞ ) -→ K 0 (Q n ( )), given by

τ n ([L t ]) = [L t (n)] if n -t ∈ 2Z ≥0 0 otherwise.
Theorem 9.2 implies that the ring K(Q ∞ ) is a 'stable limit' or 'completion' of the Grothendieck ring K(Q)(= ⊕ ∞ n=1 K 0 (Q n ( ))) in the sense that for all m, n, s and t, (9.

2010

  Mathematics Subject Classification. 81R15, 16W22, 46L37 . The present work was initiated during an Australian Research Council funded visit of K. I. to the University of Sydney in October-November 2016. He gratefully acknowledges the support and hospitality extended to him.

  t (t + 2k))x k .

  e. a multiple of id t ), then we write D * 1 D 2 = (D 1 , D 2 )id t ; otherwise we say (D 1 , D 2 ) = 0. Here D * denotes the diagram obtained from D by reflection in a horizontal, extended to W t by linearity.

Figure 1 Proposition 3

 13 Figure 1

4. 1 .

 1 Review of the representation theory of TL n at a root of unity. Let |q 2 | = , where ≥ 3.

Theorem 4 . 1 .

 41 Let |q 2 | = , fix n ≥ and let T (n) be as above. Let N = {i ∈ N | i ≡ -1(mod )}. Define g : N → N as follows: for

  2, 3, . . . , -1} and b → b for the bijection R → R given by b = -1 -b.

Lemma 6 . 5 .

 65 Suppose t = a + b with b = -3 or b = -2. Then L ( )

Theorem 7 . 4 .

 74 With notation as in Theorem 4.1, let t ∈ T (n) satisfy t ≥ -1.

Corollary 8 . 8 .

 88 (see [11, Thm. 2.9.8]) Define Q

  r (m + n)) ∼ = |s-t|≤r≤m(s,t) (L s (m) L t (n)) .But by Frobenius reciprocity, the multiplicity of Ls (m) L t (n) in Res Q m+n Qm⊗Qn (L r (m+ n)) is equal to that of L r (m + n) in Ind Q m+n Qm⊗Qn (L s (m) L t (n)). It follows that (9.6) is equivalent to: (9.7) Ind Q m+n Qm⊗Qn (L s (m) L t (n)) ∼ = r:|s-t|≤r≤m(s,t) L r (m + n),which is the required statement.

9. 2 .

 2 Connections with the Virasoro algebra. We conclude with some observations and speculations about possible connections of our results with Virasoro algebras. Recall that the Virasoro algebra L = ⊕ i∈Z CL i ⊕ CC has irreducible highest weight modules L(c, h) with highest weight (c, h), where c, h(∈ C) are respectively the central charge and the eigenvalue of L 0 . It was conjectured by Friedan, Qiu and Schenker[START_REF] Friedan | Conformal Invariance, Unitarity and Critical Exponents in Two Dimensions[END_REF] that L(c, h) is unitarisable if and only if either (1) c ≥ 1 and h ≥ 0, or (2) there exist integers m ≥ 2, r and s with 0 < r < m and 0 < s < m + 1 such thatc = c m := 1 -6 m(m + 1)and h = h r,s := ((m + 1)r -ms) 2 -1 4m(m + 1) .

1 2 .

 2 This case is the Ising model, or equivalently, the 2-state Potts model, as we have already observed. Now the abelian groups K 0 (Q 2n ) n = 1, 2, 3, . . . form an inverse system, as do theK 0 (Q 2n+1 ), via the maps [L t (n + 2)] → [L t (n)] if n -t ∈ 2Z ≥0 0 otherwise .Define the abelian groups K(Q even ( )) := lim← (K(Q 2n ( ))) and K(Q odd ( )) := lim ← (K(Q 2n+1 ( ))). Then K(Q ∞ ) := K(Q even ( )) ⊕ K(Q odd ( )) has a Z-basis which may be written {[L t ] | t = 0, 1, 2, . . . , -2}. Define a multiplication on K(Q ∞ ) by [L s ] • [L t ] = r≡s+t(2) |s-t|≤r≤m(s,t) [L r ].

  8) τ m ([L s ]) • τ n ([L t ]) = τ m+n ([L s ] • [L t ]). Moreover the ring K(Q ∞ ) is isomorphic[27, (4.6), p.369] to the fusion ring of sl 2 at level -2, which in turn is isomorphic to the subring of the fusion algebra of L with central charge c -1 generated by [L(c l-1 , h 1,s )] (1 ≤ s ≤ l -1) (cf. [14, §9.3]).