John Susainathan

Florent Eyma

Christophe Bouvet

Emmanuel De Luycker

Arthur Cantarel

Bruno Castanie
email: bruno.castanie@insa-toulouse.fr

Experimental investigation of compression and compression after impact of wood-based sandwich structures

Keywords:

Experimental investigation of compression and compression after impact of wood-based sandwich structures

Introduction

Low impact resistance is the main issue of laminate or sandwich structures in the structural design of components in aeronautics or, more generally, in the transportation sector.

A tremendous decrease in the residual strength of laminated or sandwich structure under compression loading may occur when it is subjected to low-velocity / low-energy impacts. This kind of impact may occur when a tool is dropped on the assembly line or during maintenance, causing minor apparent damage. It leads to the concept of damage tolerance, which was mainly developed in the early 1980s by Airbus engineers for the certification of the ATR 72 outer wing box [START_REF] Tropis | Certification of the Composite Outer Wing of the ATR72[END_REF], the first carbon primary structure ever certified in the field of civil aviation. In order to check the residual strength, a CAI (Compression After Impact) test was designed to ensure the most conservative design and find allowables in the form of maximum strains [START_REF] Bouvet | Mechanics of Aeronautical Composite Material[END_REF]. Numerous studies concerning impact and compression after impact on sandwich structures are available in the literature [START_REF] Abrate | Dynamic Failure of Composite and Sandwich Structures[END_REF]. Studies of CAI focus on modeling the residual strength with analytical [START_REF] Xiong | A prediction method for the compressive strength of impact damaged composite laminates[END_REF][START_REF] Naik | Estimation of compressive strength of delaminated composites[END_REF] or semi-empirical approaches [START_REF] Qi | An engineering approach for predicting residual strength of carbon/epoxy laminates after impact and hygrothermal cycling[END_REF] or, more usually, with a finite element strategy [START_REF] Suemasu | A numerical study on compressive behaviour of composite plates with multiple circular delaminations considering delamination propagation[END_REF][START_REF] Moura | Modeling compression failure after low velocity impact on laminated composites using interface elements[END_REF][START_REF] Hawyes | Effect of impact damage on the compressive response of composite laminates[END_REF][START_REF] Yan | Compression-after-impact response of woven fibre-reinforced composites[END_REF][START_REF] González | Simulation of drop-weight impact and compression after impact tests on composite laminates[END_REF][START_REF] Tan | Predicting low velocity impact damage and Compression-After-Impact (CAI) behaviour of composite laminates[END_REF]. The physical mechanisms behind the failure modes that lead to the final rupture have been clearly explained in [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF] and [START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF]. The response is a combination of three nonlinearities: a nonlinear geometric behavior due to the shift of the neutral plane because of the residual dent, material nonlinear behavior of the core, and plastic or damage behavior for metallic or composite skins, respectively. Once this behavior was understood, an original core crush criterion was proposed to compute the residual strength [START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF]. Other studies focus on the influence of material properties, such as fiber and resin type (thermoset or thermoplastic) [START_REF] Cartie | Effect of resin and fibre properties on impact and compression after impact performance of CFRP[END_REF][START_REF] Ishikawa | Some experimental findings in compression-after-impact (CAI) tests of CF/PEEK (APC-2) and conventional CF/epoxy flat plates[END_REF][START_REF] Vieille | Influence of matrix toughness and ductility on the compression-after-impact behavior of woven-ply thermoplastic-and thermosettingcomposites: a comparative study[END_REF], transverse reinforcement like stitching [START_REF] Aymerich | Characterization of fracture modes in stitched and unstitched cross-ply laminates subjected to low velocity impact and compression after impact loading[END_REF][START_REF] Tan | Effect of stitch density and stitch thread thickness on compression after impact strength and response of stitched composites[END_REF] or Zpinning [START_REF] Zhang | Improvement of low-velocity impact and compression-after-impact performance by z-fibre pinning[END_REF], fabric as an alternative to unidirectional tape [START_REF] Dale | Low velocity impact and compression after impact characterization of woven carbon/vinylester at dry and water saturated conditions[END_REF][START_REF] Kinsey | Post-impact compressive behaviour of low temperature curing woven CFRP laminates[END_REF] and aging [START_REF] Berketis | The influence of long term water immersion ageing on impact damage behaviour and residual compression strength of glass fibre reinforced polymer (GFRP)[END_REF][START_REF] Qi | An engineering approach for predicting residual strength of carbon/epoxy laminates after impact and hygrothermal cycling[END_REF], on the evolution of residual strength. The influence of a protective layer [START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with cork thermal shield[END_REF] or preloads [START_REF] Zhang | Impact damage with compressive preload and post-impact compression of carbon composite plates[END_REF] has also been studied.

All the papers cited above are restricted to laminates or classical sandwich structures with foam or honeycomb cores and composite skins. Only sandwiches with balsa core have been widely studied, because of their practical application in marine structures (for example [START_REF] Ozdemir | Single and repeated impact behaviors of biosandwich structures consisting of thermoplastic face sheets and different balsa core thicknesses[END_REF][START_REF] Atas | On the impact response of sandwich composites with cores of balsa wood and PVC foam Compos[END_REF]). In a very interesting study, the beneficial effect of wood used as a shield against hypervelocity impacts was also demonstrated by Wen et al. [START_REF] Wen | Preliminary study on shielding performance of wood stuffed shield[END_REF]. Sandwiches with classical foam cores but with intermediate layers made of ash wood or rubber cork have been analyzed under low-velocity impact and the benefit of the additional layers was demonstrated by Demircioğlu et al. [START_REF] Demircioğlu | Experimental investigation on low-velocity impact response of wood skinned sandwich composites with different core configurations[END_REF]. The same authors also studied the compression after impact response of this kind of sandwich [START_REF] Balıkoğlu | Compression after low velocity impact tests of marine sandwich composites: Effect of intermediate wooden layers[END_REF] and this is the only reference found on this subject to date.

The authors of this paper have recently studied the manufacturing and the static behavior of sandwiches made with plywood core and different skins made of aluminum, glass, carbon and flax fiber laminates [START_REF] Susainathan | Manufacturing and quasistatic bending behavior of wood-based sandwich structures[END_REF], and their low velocity, low energy impact response [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF]. A first attempt to model the impact damage has been proposed [START_REF] Susainathan | Numerical Modeling of impact on wood-based sandwich structures[END_REF] and the present paper focuses on compression and compression after impact (CAI) tests performed on the nine types of wood-based sandwich structures studied in [START_REF] Susainathan | Manufacturing and quasistatic bending behavior of wood-based sandwich structures[END_REF][START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF][START_REF] Susainathan | Numerical Modeling of impact on wood-based sandwich structures[END_REF]. Materials used in this study will be recalled briefly in the form of some impact results.

Materials and methods

Sandwich manufacturing

In comparison with simple wood, plywood structures have much better physical properties, due to the material's better in-plane behavior. In order to develop a new structure and to compare configurations, we used two different plywood cores with four different skins [START_REF] Susainathan | Manufacturing and quasistatic bending behavior of wood-based sandwich structures[END_REF][START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF][START_REF] Susainathan | Numerical Modeling of impact on wood-based sandwich structures[END_REF][START_REF] Prakash | Development and characterization of Wood Based Ecostructure[END_REF]: two different configurations of plywood, named "A" and "B" and composed of different plies obtained from poplar and okoume wood that were bonded together using Melamine Urea Formaldehyde (MUF) resin. The stacking sequences of okoume and poplar plies are given in Table . 1.

Plywoods A and B had the same thickness and were used as the core with different skins or face sheet materials made of aluminum, or fiber-reinforced polymer composite such as carbon, glass, or flax. A constant core thickness was chosen to prevent any effect of geometry on the bending stiffness. Samples were manufactured by thermo-press or vacuum molding using prepreg (Table 2) in order to evaluate the influence of the manufacturing process [START_REF] Susainathan | Manufacturing and quasistatic bending behavior of wood-based sandwich structures[END_REF].

The skin materials were chosen to be representative of the different types of face sheets used in sandwich construction. For all the plywood with aluminum or composite skin containing flax, glass and carbon, the skins were bonded on each side of the core material using an epoxy resin. For plywood A, with glass or carbon skin, they were fabricated by vacuum bag molding using carbon or glass composite prepreg. In the case of plywood B, with various skins such as flax, glass and carbon skin, the sandwiches were manufactured by thermo-compression using composite prepreg. The thermo-press process led to better adhesion between skin and core due to its high working pressure [START_REF] Prakash | Development and characterization of Wood Based Ecostructure[END_REF]. The size of the specimen was 100 x 150 mm² after trimming according to Airbus (AITM 1-010) or Boeing standards for low energy impact tests and compression after impact tests.

Experimental investigation

Impact

Impact tests were performed using a drop weight apparatus (Fig. 1) followed by tomographic analysis [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF][START_REF] Prakash | Development and characterization of Wood Based Ecostructure[END_REF]. Low velocity impact tests were carried out at three energy levels (5 J, 10 J and 15 J), corresponding to energy levels generally encountered in industry, to simulate tools falling onto a floor. The principle of the falling weight is to drop an instrumented mass and guide it in a tube onto a sample plate held by a support window.

Compression after impact

The purpose of these tests was to determine the loss of compression strength of an impacted specimen compared with a pristine specimen. The CAI tests were carried out at room temperature using an MTS machine as shown in Fig. 2 with a loading cell of 150 kN. This assembly was developed in order to meet the AIRBUS standard AITM 1-0010 and was the same as in references [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF][START_REF] Vieille | Influence of matrix toughness and ductility on the compression-after-impact behavior of woven-ply thermoplastic-and thermosettingcomposites: a comparative study[END_REF][START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with cork thermal shield[END_REF][START_REF] Prakash | Development and characterization of Wood Based Ecostructure[END_REF]. The use of clamping knives on each side of the test piece enabled "all-supported" boundary conditions to be applied and thus prevented an overall buckling process before the impact damage propagated. The pristine and impacted specimens of wood-based sandwich structures were compressed along the length direction at a constant displacement rate of 1.2 mm/min. The impacted face was painted white and sprayed with black dots so that the evolution of the depth and shape of the dent could be observed by 3D digital image stereo correlation (D.I.C.). Two CCD cameras were placed in front of the impacted face of the specimen (see Fig. 2a) to capture changes in the in-plane and out-ofplane displacement of the impacted face.

The rear side of specimens (corresponding to the bottom face of the impacted sample) was instrumented with an LVDT sensor to measure the evolution of deformations during the experiment (see Figure 2b). The CAI test rig is recalled Figure 2c. The compression force was measured directly by the load cell. The test matrix for the 36 tests performed is shown Table 3. The reference specimen called HC aramid was a floor actually used in aircraft. It had carbon skins and a 10 mm thick Nomex honeycomb core [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF].

Results and discussion

Impact behavior.

In this subsection, some key points of the impact behavior of these sandwich structures are recalled. Details are presented in [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF]. The force-displacement curves for all the specimens tested for impacts at 15 J are recalled in Fig. 3. The initial stiffenesses differed for the 8 configurations tested, due to the different materials of the skins. Concerning plywood structures, plywood A was found to yield slightly better results than plywood B in terms of absorbed energy and indentation, as it had a longer plateau. In terms of specific energy absorption, the sandwich floor gave the highest results, thanks to its low density, but the low density of the plywood also enabled it to give good results regarding specific energy absorption [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF].

For plywood structures with aluminum skins, the response was comparable to those of classical sandwiches with foam or honeycomb cores: there was no plateau and peak force oscillation occurred due to the high strength and stiffness of the skins. This structure showed comparable energy absorption and better resistance to indentation than the reference material but was not as good in terms of specific energy absorption, because of the high density of the aluminum skin. Moreover, this structure resulted in a deeper indentation than any of the other structures with composite skin and this can be undesirable in some applications.

The sandwiches with CFRP skins showed weak results. There was a small dent due to the elastic behavior of the skins, with a large delamination area in the skin and severe crushing of the plywood core. With glass fiber reinforced composite skins, the perfect adhesion and the spring back effect of the skin limited delamination, resulting in an absorbed energy comparable to that observed with flax fibers. Whatever the impact energy, this material proved to be the best compromise between absorbed energy and indentation. When flax fiber reinforcement was used in the skins, the composite behaved similarly to plywood with aluminum skins in terms of absorbed energy and specific energy absorption but showed smaller indentation as the plastic deformation was lower than for aluminum. There was minimum delamination and debonding between skin and core because of the moderate elastic spring back effect. All these results are presented in more detail in [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF].

Compression After Impact

To analyze the experimental results, the structural strength (or failure) was calculated from the load at which a change of slope (typically more than 5%) occurred in the loaddisplacement plot as illustrated in Fig. 4. In practice, this value is important for design and was introduced in [START_REF] Adam | Discrete ply model of circular pull-through test of fasteners in laminates[END_REF]. Residual strength corresponded to the maximum compressive load reached.

Plywoods A and B.

The load/displacement curves obtained for plywoods A and B are shown Fig. 5 (a) and (b). The evolution of residual strength, structural failure and residual stiffness are given Fig. 6. In general, plywood (A) had better mechanical properties than plywood (B). The degradation of the mechanical characteristics was more progressive and the residual strength reductions were 28% and 32.6% respectively. It can also be noted that, in the pristine state, plywood A had a very long plateau (Fig. 5 (a)) of practically 3 mm. This mechanical characteristic could be of interest in a crash. For plywood B in new condition, the length of the plateau was only 0.5 mm because the test was stopped prematurely. The compression failure scenarios of the two new plywoods were quite different. Plywood A gradually deformed out of plane following a mode II (in the sense of Euler beam theory), see Fig 7 (a) and the upper fold of okoume at 0° ended up breaking in traction at the tip of wrinkle (Fig. 7 (b)).

For pristine plywood B, it would appear that the same mode of out-of-plane displacement was activated at the beginning of the loading but, towards 17.13 MPa, a weakness (local buckling?) appeared at the end of the specimen near the zone of loading (see Fig 8 (a)) and changed the deformed shape. The post-mortem analysis showed a rupture of the outer, okoume ply at 0°, see Fig 8 (b). Since the plywood had not been locally reinforced, it is possible that the rupture was premature. Unwanted local buckling in this zone is common in conventional sandwiches [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF] and the outer okoume ply was superimposed on 2 poplar plies oriented at 90°, and therefore not rigid (Table 1), which may have caused local buckling.

The post-impact behavior of the two plywoods is shown in Figure 9. Typical post-impact fracture patterns are visible in Figure 9 (a) and (b), as well as the evolution of out-of-plane displacement measured by DIC in an off-impact zone for plywood B at 5 J impact. For the 6 different configurations, the failure was generally brutal in compression, as shown in Fig. 9 (b). In this case, we have a rupture scenario close to that for laminates [START_REF] Moura | Modeling compression failure after low velocity impact on laminated composites using interface elements[END_REF][START_REF] Hawyes | Effect of impact damage on the compressive response of composite laminates[END_REF][START_REF] Yan | Compression-after-impact response of woven fibre-reinforced composites[END_REF][START_REF] González | Simulation of drop-weight impact and compression after impact tests on composite laminates[END_REF][START_REF] Tan | Predicting low velocity impact damage and Compression-After-Impact (CAI) behaviour of composite laminates[END_REF]. Sometimes it can be preceded by an increase in depth of the impact dent, as shown in Fig. 9b, and even a local buckling of the upper ply. In this case, the scenario is closer to sandwich behavior [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF][START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF].

Reference sandwich

The overall load/displacement results and the strength reduction vs impact energy for this sandwich are shown in Figure 10. The strength reduction reaches 70 %, which is very great but also almost classical [START_REF] Castanié | Combined shear/compression structural testing of asymmetric sandwich structures[END_REF]. In this sandwich, the core is made of Nomex honeycomb, which is weak with respect to out-of-plane loads, and the skins are very thin (about 0.5 mm), two features that can explain these results. The failure scenario for this sandwich is classical [START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF][START_REF] Castanié | Combined shear/compression structural testing of asymmetric sandwich structures[END_REF] and will not be recalled here.

Plywood structures with aluminum skin

The overall load/displacement results and the strength reduction vs impact energy are shown in Figure 11. The sandwich with aluminum skins exhibits interesting results. For the pristine specimen, a very long plateau area is found. This behavior should be useful in crash situations. Moreover, even with impact, the loss of strength is less than 1% for 5 and 10 J impacts and reaches 15 % for the 15 J impact. This behavior may be due partially to the very ductile behavior of 1000 series aluminum. The pristine specimen fails by local skin buckling, located at the center of the specimen (see Fig. 12 (a)). For the impacted specimens, two phenomena seem to compete: on the one hand, the increase in depth and extension perpendicular to the compression direction of the residual indentation in the center of the plate, a phenomenon already described in [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF][START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF][START_REF] Castanié | Combined shear/compression structural testing of asymmetric sandwich structures[END_REF] and, on the other hand, an appearance giving the impression of local buckling at several places on the aluminum skin (see Fig 12 (b)), on 3 of the 4 sides of the dent. It has been observed that, during impact, a delamination occurs at the edge of the dent [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF], which may explain these occurrences. Lateral buckling, perpendicular to the compression direction, ultimately produces skin failure. This scenario is corroborated by the DIC measurements (Fig. 13) for an impact of 15 J. In Figure 13 (a), an increase in depth and an extension of the dent can be observed in a qualitative way as can the 3 points of local buckling around it. Figure 13 (b) shows the evolution of the displacements in the median plane of the specimen.

Plywood structures with composite skins. a. Carbon composite skins

The overall load/displacement results and the strength reduction vs impact energy are shown in Figures 14 and15, respectively, for the two types of sandwich, i.e. plywood A + carbon skins Vacuum Molded (VM) and plywood A + carbon skins obtained by Thermo-Compression (TC). The behavior is almost the same for pristine specimens and for impacted specimens, with nearly linear behavior followed by a sudden drop. The plywood A specimen impacted at15 J has different behavior with a large plateau area -probably because of the large extent of the damage. The residual strength falls by 30 % for plywood B/ carbon skins TC and by 35 % for plywood A/ carbon skins VM. It is noticeable that this drop remains limited in comparison to the reference sandwich (up to 70%) and may be interesting in terms of sizing, despite the heavier weight. Also, the sensitivity to the manufacturing process highlighted in [START_REF] Susainathan | Manufacturing and quasistatic bending behavior of wood-based sandwich structures[END_REF] is not found here.

After the impact, the carbon skin is delaminated due to its high stiffness and has strong spring-back, so the permanent indentation is slight. Nevertheless, the wood is still crushed below the skin and the skin becomes unsupported [START_REF] Susainathan | Experimental investigation of impact behavior of wood-based sandwich structures[END_REF]. This failure pattern is very similar to that in impacts on sandwich structures with Nomex honeycomb cores and carbon skins, where the honeycomb is crushed under the residual dent [START_REF] Aminanda | Experimental analysis and modeling of the crushing of honeycomb cores[END_REF][START_REF] Castanié | Modelling of low energy/low velocity impact on Nomex honeycomb sandwich structures with metallic skins[END_REF]. So the behavior is almost the same as described in [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF][START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF] and explains the shape of the curves in Figure 14 and the local skin buckling visible in Figure 16 . The only significant difference is the shape of the buckling found for the 15 J impact of plywood A with carbon skin (see Fig. 17) where the load/displacement curve has a plateau. The buckling is so large that a second mode appears (in an Euler beam sense), which is probably due to a large crushing area combined with debonding (due to a poor manufacturing process) between carbon skin and plywood after the impact.

b. Glass composite skins

The overall load/displacement results and the strength reduction vs impact energy are shown in Figures 18 and19 respectively. Unlike the other configurations, this one experiences a rapid drop in strength, even for the 5 J impact, which also reaches 35 %. The loss in stiffness and in structural failure is nearly inexistent. The failure scenario is a local bucking of the unsupported skin near the impact (in a direction away from the plywood) followed by a brittle failure at the tip of the wrinkle (see Fig. 20 for the 15 J impact).

c. Flax composite skins

The overall load/displacement results and the strength reduction vs impact energy are shown in Figures 21 and22, respectively. These sandwiches exhibit an exceptional and unexpected behavior with a drop in strength and in stiffness limited to less than 10 %, even for the 15 J impact. Moreover, as for the plywood with aluminum skins, a plateau area is observed. The failure scenario is also similar to that found in [START_REF] Aminanda | Experimental and numerical study of compression after impact response of sandwich structures with metallic skins[END_REF][START_REF] Castanié | Core crush criterion to determine the strength of sandwich composite structures subjected to compression after impact[END_REF] with a final brittle failure of the flax skin.

Summary

Compression After Impact tests results are presented in Figure 23 a,b,c & d in terms of force-displacement response for the nine different structures when pristine and after 5 J, 10 J and 15 J impact. Their main interest is that they enable comparisons to be made between the values at failure and the various behaviors. Strength reduction and normalized specific strength are presented for the nine different structures in Table 4 and Figure 24 a andb.

Whatever the method of manufacture, in pristine state or impacted, it is the sandwich with fiberglass skins that resists the best, by a factor of 2. It is also the one made with the cheapest materials. Carbon skin sandwiches come after fiberglass sandwiches. This result is a little unexpected but the manufacturing method is not stabilized and, given the observed failure modes (buckling of the skins over a large surface area), it is likely that the bonding between skins and plywood is not yet optimum. For these two sandwiches, the method of manufacture is important and thermo-compression gives worse results than vacuum molding. For these four configurations, the rupture is fragile and the potential for use in crash energy absorption is limited.

(b)

Conversely, the reference sandwich, except in pristine condition, almost always has the lowest compressive strength, even compared to plywood alone. However, this can be explained by a density that is half that of the plywood specimens. Another peculiarity is that, at 15 J, the strength drop reaches 70% for this sandwich (Fig 24 a) and, from this point of view, this material is very poor.

Sandwiches with flax and aluminum skins have very similar and particularly interesting mechanical behaviors: they have a very large plateau area and a quasi-insensitivity to impact (see Fig. 24 a) with less than 10% reduction. For example, at 15 J, they are comparable to carbon skin sandwiches. For a sandwich to be effective in compression, the skins thus need to have a certain flexibility (like glass fibers) or a high plastic area, like flax or aluminum skins.

In any case, it would be necessary to "tune" the skins to the plywood.

In Fig. 24 (b), the specific characteristics have been normalized with respect to the reference sandwich in the pristine state (see Table 4 calculations) and the results in this form are very interesting since the fiberglass skin sandwiches have a specific resistance 20% higher than the reference sandwich in compression in the pristine state. In this configuration, the other sandwiches are, on the other hand, about half as efficient. However, by 10 J, all sandwiches with a core in plywood become more efficient than the reference sandwich. Here, we find an intrinsic performance of the wood under this stress [START_REF] Wouts | Strain rate effects on the compressive response of wood and energy absorption capabilities -Part A: Experimental investigations[END_REF][START_REF] Wouts | Strain rate effects on the compressive response of wood and energy absorption capabilities -Part B: Experimental investigation under rigid lateral confinement[END_REF] provided that it is confined, a role that is probably performed by the skins of the sandwich.

Conclusions

Compression and compression after impact of nine combinations of sandwiches with plywood cores, a reference sandwich commonly used for aircraft flooring, and plywood alone were investigated. The failure scenarios have been identified and are caused by breakages in the skin, often after local buckling. One of the most interesting points highlighted by this study is a long plateau for plywood, flax composite skin and aluminum skin structures, which characterizes progressive damage in both pristine and impacted samples and shows great accommodation to the stress. These results suggest that ductile materials should be used in skins to dissipate a maximum of energy, especially for crash-type applications. Plywood structure with glass skin is found to provide the best compromise between residual strength and stiffness reduction, which may be explained by factors such as higher strength and thickness of the glass skin and better adhesion attained through the thermo-compression process, in which lower pressure is required for glass than carbon skin. Plywood with flax skin, which could be produced from bio-resources, is identified as another good compromise solution based on factors such as residual stiffness, strength reduction, specific properties (stiffness and strength) and eco-aspects.

In general, these structures have a very good specific compressive strength and a very good specific compressive strength after impact, both of which are higher than in the reference sandwich. These results show that these structures with plywood cores have great potential for use in the transport field -and for a cost 20 times lower than the reference sandwich.

Optimization is needed with regard to both stacking and the choice of plywood species, as well as with regard to the manufacturing methods and the bonding of skins to plywood in particular. A very strong potential also exists for weight reduction, by simply piercing the core of the plywood.

FiguresFig. 1

 1 Figures

Figure 2 :Figure 3 :

 23 Fig. 1: Drop-weight impact test set-up,

Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :

 456789 Figure 4: Structural strength (or failure[START_REF] Adam | Discrete ply model of circular pull-through test of fasteners in laminates[END_REF]) and residual strength.

Figure 10 :Figure 11 :Figure 12 :Figure 13 :

 10111213 Figure 10: Test results (a) and evolution of mechanical characteristics (b) for reference sandwich.

Figure 14 :Figure 15 :

 1415 Figure 14: Tests results for sandwiches of plywoods A (a) and B (b), with carbon skins.

Figure 16 :

 16 Figure 16: Typical local skin buckling for plywood with carbon skins.

Figure 17 :

 17 Figure 17: Out-of-plane displacement of plywood A with carbon skins during CAI after a 15 J impact.

Figure 18 :

 18 Figure 18: Test results for sandwiches with plywood B and glass skins.

Figure 19 :Figure 20 :

 1920 Figure 19: Evolution of mechanical characteristics for sandwiches with plywood A and glass

Figure 21 :

 21 Figure 21: Tests results for sandwiches with plywood B and flax skins.

Figure 22 :

 22 Figure 22: Evolution of mechanical characteristics for sandwiches with plywood A and flax

Figure 23 :

 23 Figure 23: Overall compression force vs displacement curves for the nine configurations of

Table 3 :

 3 Test Matrix for CAI tests.

Table 4 :

 4 Summary of strengths for the nine sandwich configurations.

	Materials	New	5 Compressive strength (MPa) 10 15	Density (kg/m3)	0	5 Specific strength (MPa.m3/kg) 10	15	0	5 Normalized Specific strength 10	15
	Plywood A	25.9	24.3	22.4	18.5	461.0	0.0562	0.0527	0.0485	0.0402	0.55	0.52	0.48	0.39
	Plywood B	17.9	17.7	17.0	12.1	433.0	0.0413	0.0410	0.0391	0.0279	0.40	0.40	0.38	0.27
	Plywood A/Al	31.6	31.5	31.0	28.8	678.0	0.0466	0.0465	0.0457	0.0425	0.46	0.45	0.45	0.42
	Plywood A/Glass	66.0	64.1	50.3	45.3	579.0	0.1140	0.1108	0.0868	0.0782	1.12	1.08	0.85	0.77
	Plywood A/ carbon	40.0	37.9	37.5	25.9	638.0	0.0627	0.0594	0.0588	0.0406	0.61	0.58	0.58	0.40
	Plywood B/ Flax	25.5	24.2	24.2	23.5	614.0	0.0415	0.0394	0.0394	0.0382	0.41	0.39	0.39	0.37
	Plywood B/ Carbon	33.9	26.9	26.2	23.7	609.0	0.0556	0.0442	0.0431	0.0390	0.54	0.43	0.42	0.38
	Plywood B/ Glass	60.7	44.2	40.1	39.7	488.0	0.1244	0.0907	0.0821	0.0814	1.22	0.89	0.80	0.80
	Reference Sandwich	23.8	14.4	7.87	6.48	233.0	0.1021	0.0618	0.0338	0.0278	1.00	0.61	0.33	0.27

Acknowledgements

The authors would like to acknowledge the French Ministry of Education and Research for funding this study and AIRBUS for providing the reference materials.

Tables


Michel To

for his spetsial intuitions. Broué

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An asymptotic cell category for cyclic groups

Cédric Bonnafé, Raphaël

Broué, Malle and Michel [BMM] have constructed a combinatorial version of Lusztig's theory of unipotent characters of finite groups of Lie type for certain complex reflection groups ("spetsial groups"). The case of spetsial imprimitive complex reflection groups has been considered by Malle in [Ma]: Malle defines a combinatorial set which generalizes the one defined by Lusztig to parametrize unipotent characters of the associated finite reductive group when W is a Weyl group. Malle generalizes also the partition of this set into Lusztig families. To each family, he associates a -fusion datum: a -fusion datum is a structure similar to a usual fusion datum (which we will call a + -fusion datum) except that the structure constants of the associated fusion ring might be negative.

It is a classical problem to find a tensor category with suitable extra-structu--res (pivot, twist) corresponding to a given + -fusion datum. The aim of this paper is to provide an ad hoc categorification of the -fusion datum associated with the non-trivial family of the cyclic complex reflection group of order d : it is provided by a quotient category of the representation category of the Drinfeld double of the Taft algebra of dimension d 2 (the Taft algebra is the positive part of a Borel subalgebra of quantum sl 2 at a d -th root of unity).

The constructions of this paper have recently been extended by Lacabanne [START_REF] Lacabanne | Slightly degenerate categories and -modular data[END_REF], [START_REF] Lacabanne | Drinfeld double of quantum groups, tilting modules and -modular data associated to complex reflection groups[END_REF] to some families of the complex reflection groups G (d , 1, n). His construction involves super-categories instead of triangulated categories, as suggested by Etingof. In the particular case studied in our paper, Lacabanne's construction gives a reinterpretation as well as some clarifications of our results.

In the first section, we recall some basic properties of the Taft algebra and its Drinfeld double. The second section is devoted to recalling some of the structure of its category of representations: simple modules, blocks and structure of projective modules. We summarize some elementary facts on the tensor structure in the third section: invertible objects and tensor product of simple objects by the defining two-dimensional representation. The fourth section provides generators and characters of the Grothendieck group of D (B ), a commutative ring.

Our original work starts in the fifth section, with the study of the stable module category of D (B ) and a further quotient and the determination of their Grothendieck rings. In the sixth section, we define and study a pivotal structure on D (B )-mod and determine characters of its Grothendieck group associated to left or right traces. This gives rise to positive and negative S and T -matrices. We proceed similarly for the small quotient triangulated category. These are our fusion data.

We recall in the final section the construction of Malle's fusion datum associated with cyclic groups and we check that it coincides with the fusion data we defined in the previous section.

In Appendix A, we provide a description of S -matrices in the setting of pivotal tensor categories.

The Drinfeld double of the Taft algebra

From now on, ⊗ will denote the tensor product ⊗ . We fix a natural number d 2 as well as a primitive d -th root of unity ζ ∈ × . We denote by µ d = 〈ζ〉 the group of d -th roots of unity.

Given n 1 a natural number and ξ ∈ , we set

(n) ξ = 1 + ξ + • • • + ξ n -1 and (n)! ξ = n i =1 (i ) ξ .
We also set (0)! ξ = 1.

1.

A. The Taft algebra. -We denote by B the -algebra defined by the following presentation:

• Generators: K , E .

• Relations:

   K d = 1, E d = 0, K E = ζE K .
It follows from [START_REF] Kassel | Quantum groups[END_REF]Proposition IX.6.1] that:

(∆) There exists a unique morphism of algebras ∆ : B → B ⊗ B such that

∆(K ) = K ⊗ K and ∆(E ) = (1 ⊗ E ) + (E ⊗ K ).
(ǫ) There exists a unique morphism of algebras ǫ : B → such that

ǫ(K ) = 1 and ǫ(E ) = 0.
(S) There exists a unique anti-automorphism S of B such that

S (K ) = K -1
and

S (E ) = -E K -1 .
With ∆ as a coproduct, ǫ as a counit and S as an (invertible) antipode, B becomes a Hopf algebra, called the Taft algebra [START_REF] Etingof | Tensor categories[END_REF]Example 5.5.6]. It is easily checked that (1.1)

B = d -1 i , j =0 K i E j = d -1 i , j =0 E i K j .
1.B. Dual algebra. -Let K * and E * denote the elements of B * such that

K * (E i K j ) = δ i ,0 ζ j and E * (E i K j ) = δ i ,1 .
Recall that B * is naturally a Hopf algebra [START_REF] Kassel | Quantum groups[END_REF]Proposition III.3.3] and it follows from [START_REF] Kassel | Quantum groups[END_REF]Lemma IX.6.3] that (1.2)

(E * i K * j )(E i ′ K j ′ ) = δ i ,i ′ (i )! ζ ζ j (i + j ′ ) .
We deduce easily that

(E * i K * j ) 0 i , j d -1 is a -basis of B * .
We will give explicit formulas for the coproduct, the counit and the antipode in the next subsection. We will in fact use the Hopf algebra (B * ) cop , which is the Hopf algebra whose underlying space is B * , whose product is the same as in B * and whose coproduct is opposite to the one in B * . 1.C. Drinfeld double. -We denote by D (B ) the Drinfeld quantum double of B , as defined for instance in [START_REF] Kassel | Quantum groups[END_REF]Definition IX.4.1] or [START_REF] Etingof | Tensor categories[END_REF]Definition 7.14.1]. Recall that D (B ) contains B and (B * ) cop as Hopf subalgebras and that the multiplication induces an isomorphism of vector spaces [START_REF] Kassel | Quantum groups[END_REF]Proposition IX.6.4]. We shall slightly modify it by setting

(B * ) cop ⊗ B ∼ -→ D (B ). A presentation of D (B ), with generators E , E * , K , K * is given for instance in
z = K * -1 K and F = ζE * K * -1 .
Then [START_REF] Kassel | Quantum groups[END_REF]Proposition IX.6.4] can we rewritten as follows:

Proposition 1.3. -The -algebra D (B ) admits the following presentation:

• Generators: E , F , K , z ;

• Relations:

                 K d = z d = 1, E d = F d = 0, [z , E ] = [z , F ] = [z , K ] = 0, K E = ζE K , K F = ζ -1 F K , [E , F ] = K -z K -1 .
The next corollary follows from an easy induction argument:

Corollary 1.4. -If i 1, then [E , F i ] = (i ) ζ F i -1 (ζ 1-i K -z K -1 ) and [F, E i ] = (i ) ζ E i -1 (ζ 1-i z K -1 -K ).
The algebra D (B ) is endowed with a structure of Hopf algebra, where the comultiplication, the counit and the antipode are still denoted by ∆, ǫ and S respectively (as they extend the corresponding objects for B ). We have [START_REF] Kassel | Quantum groups[END_REF]Proposition IX.6.2]:

(1.5)

         ∆(K ) = K ⊗ K , ∆(z ) = z ⊗ z , ∆(E ) = (1 ⊗ E ) + (E ⊗ K ), ∆(F ) = (F ⊗ 1) + (z K -1 ⊗ F ),          S (K ) = K -1 , S (z ) = z -1 , S (E ) = -E K -1 , S (F ) = -ζ -1 F K z -1 , (1.6) ǫ(K ) = ǫ(z ) = 1 and ǫ(E ) = ǫ(F ) = 0.
ǫ ξ (K ) = ξ, ǫ ξ (z ) = ξ 2 and ǫ ξ (E ) = ǫ ξ (F ) = 0.
It is easily checked that the ǫ ξ 's are the only morphisms of algebras D (B ) → . Note that ǫ 1 = ǫ is the counit.

1.E. Group-like elements. -It follows from (1.5) that K and z are grouplike, so that K i z j is group-like for all i , j ∈ . The converse also holds (and is certainly already well-known).

Lemma 1.7. -If g ∈ D (B ) is group-like, then there exist i , j ∈ such that g = K i z j .

Proof. -Let g ∈ D (B ) be a group-like element. Let us write

g = d -1 i , j ,k ,l =0 α i , j ,k ,l K i z j E k F l .
We denote by (k 0 , l 0 ) the biggest pair (for the lexicographic order) such that there exist i , j ∈ {0, 1, . . ., d -1} such that α i , j ,k 0 ,l 0 = 0. The coefficient of

K i z j E k 0 F l 0 ⊗ K i z j E k 0 F l 0 in g ⊗ g is equal to α 2 i , j ,k 0 ,l 0 , so it is different from 0.
But, if we compute the coefficient of

K i z j E k 0 F l 0 ⊗ K i z j E k 0 F l 0 in g ⊗ g = ∆(g ) = d -1 i , j ,k ,l =0 α i , j ,k ,l ∆(K ) i ∆(z ) j ∆(E ) k ∆(F ) l
using the formulas (1.5), we see that it is equal to 0 if (k 0 , l 0 ) = (0, 0). Therefore (k 0 , l 0 ) = (0, 0), and so g belongs to the linear span of the family (K i z j ) i , j ∈ . Now the result follows from the linear independence of group-like elements.

1.F. Braiding. -For 0 i , j d -1, we set

β i , j = E * i d • (i )! ζ d -1 k =0 ζ -k (i + j ) K * k . It follows from (1.2) that (β i , j ) 0 i , j d -1 is a dual basis to (E i K j ) 0 i , j d -1 . We set now R = d -1 i , j =0 E i K j ⊗ β i , j ∈ D (B ) ⊗ D (B ).
Note that R is a universal R -matrix for D (B ) and it endows D (B ) with a structure of braided Hopf algebra [START_REF] Kassel | Quantum groups[END_REF]Theorem IX.4.4]). Using our generators E , F , K , z , we have:

(1.8) R = 1 d d -1 i , j ,k =0 ζ (i -k )(i + j )-i (i +1)/2 (i )! ζ E i K j ⊗ z -k F i K k . 1.G. Twist. -Let us define τ : D (B ) ⊗ D (B ) -→ D (B ) ⊗ D (B ) a ⊗ b -→ b ⊗ a .
Following [START_REF] Kassel | Quantum groups[END_REF]§VIII.4], we set

u = d -1 i , j =0 S (β i j )E i K j ∈ D (B ).
Recall that u is called the Drinfeld element of D (B ). It satisfies several properties (see for instance [START_REF] Kassel | Quantum groups[END_REF]Proposition VIII.4.5]). For instance, u is invertible and we will recall only three equalities:

(1.9)

ǫ(u) = 1, ∆(u) = (τ(R )R ) -1 (u ⊗ u) and S 2 (b ) = u b u -1 for all b ∈ D (B ). A straightforward computation shows that (1.10) S 2 (b ) = K b K -1
for all b ∈ D (B ). We now set

θ = K -1 u.
The following proposition is a consequence of (1.9) and (1.10).

Proposition 1.11. -The element θ is central and invertible in D (B ) and satisfies

ǫ(θ ) = 1 and ∆(θ ) = (τ(R )R ) -1 (θ ⊗ θ ).
Let us give a formula for θ :

(1.12) [START_REF] Kassel | Quantum groups[END_REF]Theorems III.3.4 and VIII.2.4], it follows from Proposition 1.11 that g is central and group-like. Hence, by Lemma 1.7, there exists l ∈ such that S (θ ) = θ z l . So, by (1.12), we have

θ = 1 d d -1 i , j ,k =0 (-1) i ζ (i -k )(i + j )-i (i )! ζ z k -i F i E i K i + j -k -1 . Corollary 1.13. -We have S (θ ) = z θ . Proof. -Let g = S (θ )θ -1 . Since ∆ • S = τ • (S ⊗ S ) • ∆ and (S ⊗ S )(R ) = R (see for instance
(♯) S (θ )E d -1 = θ z l E d -1 = 1 d d -1 j ,k ∈ /d ζ -j k z k +l K j -k -1 E d -1 .
Let us now compute S (θ )E d -1 by using directly (1.12). We get

S (θ )E d -1 = E d -1 S (θ ) = 1 d j ,k ∈ /d ζ -j k E d -1 z -k K 1+k -j = 1 d j ,k ∈ /d ζ -j k ζ 1+k -j z -k K 1+k -j E d -1 = 1 d j ,k ∈ /d ζ (1-j )(1+k ) z -k K 1+k -j E d -1 . So, if we set j ′ = 1 -j and k ′ = -1 -k , we get S (θ )E d -1 = 1 d j ′ ,k ′ ∈ /d ζ -j ′ k ′ z k ′ +1 K j ′ -k ′ -1 E d -1 .
Comparing with (♯), we get that z l = z .

D (B )-modules

Most of the result of this section are due to Chen [START_REF] Chen | Irreducible representations of a class of quantum doubles[END_REF] or Erdmann, Green, Snashall and Taillefer [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF], [START_REF] Erdmann | Representation theory of the drinfeld doubles of a family of hopf algebras II: corrections and new results[END_REF]. By a D (B )-module, we mean a finite dimensional left D (B )-module. We denote by D (B )-mod the category of (finite dimensional left) D (B )-modules. Given α 1 ,. . . , α l -1 ∈ , we set

J + l (α 1 , . . ., α l -1 ) =        0 α 1 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 α l -1 0 • • • • • • • • • 0        and J - l (α 1 , . . ., α l -1 ) = t J + l (α 1 , . . ., α l -1 ).
Given ) 1 i l where the action of z , K , E and F in the basis (l ,p ) its given by the following matrices:

z | M l ,p = ζ 2p +l -1 Id M l ,p , K | M l ,p = ζ p diag(ζ l -1 , ζ l -2 , . . ., ζ, 1), E | M l ,p = ζ p J + l ((1) ζ (ζ l -1 -1), (2) ζ (ζ l -2 -1), . . ., (l -1) ζ (ζ -1)), F | M l ,p = J - l (1, . . ., 1).
It is readily checked from the relations given in Proposition 1.3 that this defines a

D (B )-module of dimension l . The next result is proved in [Ch1, Theo- rem 2.5]. Theorem 2.1 (Chen). -The map {1, 2, . . ., d } × /d -→ Irr(D (B )) (l , p ) -→ M l ,p
is bijective. We also set

( /d ) # = ( /d ) \ {0} and Λ # (d ) = ( /d ) # × /d . Finally, let Λ 0 (d ) = {0} × /d be the complement of Λ # (d ) in Λ(d ). Define ι : Λ(d ) -→ Λ(d ) (l , p ) -→ (-l , p + l ).
We have

ι 2 = Id Λ(d ) and Λ 0 (d ) is the set of fixed points of ι. Given a ι-stable subset of Λ(d ), we denote by [ /ι] a set of representatives of ι-orbits in . The next result is proved in [EGST1, Theorem 2.26]. Theorem 2.2 (Erdmann-Green-Snashall-Taillefer). -Let λ, λ ′ ∈ Λ(d ).
Then M λ and M λ ′ belong to the same block of D (B ) if and only if λ and λ ′ are in the same ι-orbit.

We have constructed in §1.G a central element, namely θ . Note that (2.3)

The element θ acts on M l ,p by multiplication by ζ (p -1)(l +p -1) .

Proof. -It is sufficient to compute the action of θ on the vector e (l ,p ) 1

. Note that E i e (l ,p ) 1 = 0 as soon as i 1. Therefore, for computing θ e (l ,p ) 1 using the formula (1.12), only the terms corresponding to i = 0 remain. Consequently,

ω l ,p (θ ) = 1 d d -1 j ,k =0 ζ -j k ζ (2p +l -1)k ζ (p +l -1)( j -k -1) = ζ 1-l -p d d -1 k =0 ζ p k d -1 j =0 ζ (p +l -1-k ) j
The term inside the big parenthesis is equal to d if p +l -1-k ≡ 0 mod d , and is equal to 0 otherwise. The result follows. (a) If λ ∈ Λ # (d ), then dim (P λ ) = 2d , Rad 3 (P λ ) = 0 and the Loewy structure of P λ is given by:

P λ / Rad(P λ ) ≃ M λ Rad(P λ )/ Rad 2 (P λ ) ≃ M ι(λ) ⊕ M ι(λ) Rad 2 (P λ ) ≃ M λ (b) P d ,p = M d ,p has dimension d .

Tensor structure

We mainly refer here to the work of Erdmann, Green, Snashall and Taillefer [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF], [START_REF] Erdmann | Representation theory of the drinfeld doubles of a family of hopf algebras II: corrections and new results[END_REF]. Since D (B ) is a finite dimensional Hopf algebra, the category D (B )-mod inherits a structure of a tensor category. We will compute here some tensor products between simple modules. We will denote by M l the simple module M l ,0 .

3.A. Invertible modules. -We denote by

V ξ = v ξ the one-dimensional D (B )-module associated with the morphism ǫ ξ : D (B ) → defined in §1.D: b v ξ = ǫ ξ (b )v ξ for all b ∈ D (B ). We have (3.1) V ζ p ≃ M 1,p .
An immediate computation using the comultiplication ∆ shows that

(3.2) M l ,p ⊗ V ζ q ≃ V ζ q ⊗ M l ,p ≃ M l ,p +q
as D (B )-modules. The V ξ 's are (up to isomorphism) the only invertible objects in the tensor category D (B )-mod.

3.B. Tensor product with M

2 . -We set e i = e
(2,0) i for i ∈ {1, 2}, so that (e 1 , e 2 ) is the standard basis of M 2 . The next result is a particular case of [START_REF] Erdmann | Representation theory of the Drinfeld doubles of a family of Hopf algebras[END_REF]Theorem 4.1].

Theorem 3.3 (Erdmann-Green-Snashall-Taillefer). -Let (l , p ) be an element of {1, 2, . . ., d } × /d . (a) If l d -1, then M 2 ⊗ M l ,p ≃ M l +1,p ⊕ M l -1,p +1 . (b) M 2 ⊗ M d ,p ≃ P d -1,p .

Grothendieck rings

We denote by Gr(D (B )) the Grothendieck ring of the category of (left) D (B )modules. We set

m λ = [ M λ ], m l = [ M l ,0 ]
and

v ξ = [ V ξ ] ∈ Gr(D (B )), Recall that v ζ p = m 1,p .It follows from (3.2) and Theorem 3.3 that (4.2) v ζ q m l ,p = m l ,p +q and m 2 m l ,p = m l +1,p + m l -1,p +1 if l d -1, 2(m d -1,p + m 1,p -1 ) if l = d .
Gr(D (B )) -→ [ M ] -→ Tr(b | M )
is a morphism of rings. Here, Tr denotes the usual trace (not the quantum trace) of an endomorphism of a finite dimensional vector space. Recall from Lemma 1.7 that the only group-like elements of D (B ) are the K i z j , where (i , j ) ∈ Λ(d ). We set

χ i , j : Gr(D (B )) -→ [ M ] -→ Tr(K i z j | M ).
An easy computation yields (4.4)

χ i , j (m l ,p ) = ζ p i +(2p +l -1) j • (l ) ζ i .
Note that the χ i , j 's are not necessarily distinct:

Lemma 4.5. -Let λ and λ ′ be two elements of Λ(d ). Then χ λ = χ λ ′ if and only if λ and λ ′ are in the same ι-orbit.

Proof. -Let us write λ = (i , j ) and λ ′ = (i ′ , j ′ ). The "if" part follows directly from (4.4). Conversely, assume that χ i , j = χ i ′ , j ′ . By applying these two characters to v ζ and m 2 , we get:

ζ i +2 j = ζ i ′ +2 j ′ , ζ j (1 + ζ i ) = ζ j ′ (1 + ζ i ′ ).
It means that the pairs (ζ j , ζ i + j ) and (ζ j ′ , ζ i ′ + j ′ ) have the same sum and the same product, so

(ζ j , ζ i + j ) = (ζ j ′ , ζ i ′ + j ′ ) or (ζ j , ζ i + j ) = (ζ i ′ + j ′ , ζ j ′ ).
In other words, (i ′ , j ′ ) = (i , j ) or (i ′ , j ′ ) = ι(i , j ), as expected. In particular, its Grothendieck group (as a triangulated category), which will be denoted by Gr st (D (B )), is a ring and the natural map

Triangulated categories

Gr(D (B )) -→ Gr st (D (B )) m -→ m st
→ I (M ) ⊕ N , m → (i M (m), f (m)). Then the cone of f is isomorphic in D (B )-stab to (I (M ) ⊕ N )/ Im(∆ f ). But ∆ f is injective, M is an injective B -module and so I (M ) ≃ M ⊕ (I (M ) ⊕ N )/ Im(∆ f ) as a B -module, which shows that (I (M ) ⊕ N )/ Im(∆ f ) is a projective B -module.
We denote by Gr If λ ∈ Λ # (d ), then it follows from [START_REF] Erdmann | Representation theory of the drinfeld doubles of a family of hopf algebras II: corrections and new results[END_REF]Property 1.4] that there exists a D (B )-module P B λ which is projective as a B -module and such that there is an exact sequence

0 -→ M ι(λ) -→ P B λ -→ M λ -→ 0.
It then follows that

(5.3)

m st B λ + m st B ι(λ) = 0.
Also, we still have Recall that Lemma 4.5 shows that, through the χ i , j 's, only d (d + 1)/2 different characters of the ring Gr(D (B )) have been defined. It is not clear if Gr(D (B )) is semisimple in general but, for d = 2, it can be checked that it is semisimple (of dimension 4), so that there is a fourth character Gr(D (B )) → which is not obtained through the χ i , j 's. Now, a character χ : Gr(D (B )) → factors through Gr st B (D (B )) if and only if its kernel contains the m λ 0 's (where λ 0 runs over Λ 0 (d )) and the m λ + m ι(λ) 's (where λ runs over Λ # (d )). This implies the following result.

5.C. Complements. -Given

a monoidal category, we denote by Z( ) its Drinfeld center (see [START_REF] Kassel | Quantum groups[END_REF]§XIII.4]) and we denote by For : Z( ) → the forgetful functor.

There is an equivalence between Z(B -mod) and D (B )-mod such that the forgetful functor becomes the restriction functor Res Since any D (B )-module that is projective as a B -module is sent to the zero object of Z(B -stab) through , the functor factors through D (B )-proj B and we obtain a commutative diagram of functors

D (B )-mod Res D (B ) B / / can D (B ) st B w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ B -mod can B st D (B )-stab B ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ Z(B -stab) For B -stab / / B -stab .
M ⊗ N ∼ -→ N ⊗ M is given by c M ,N (m ⊗ n) = τ(R )(n ⊗ m).
Recall that τ :

D (B ) ⊗ D (B ) ∼ -→ D (B ) ⊗ D (B ) is given by τ(a ⊗ b ) = b ⊗ a . In particular, (6.1) c N ,M c M ,N : M ⊗ N ∼ -→ M ⊗ N is given by the action of τ(R )R . Given i ∈ , we have S 2 (b ) = (z -i K )b (z -i K ) -1
for all b ∈ D (B ) and z -i K is group-like, so the algebra D (B ) is pivotal with pivot z -i K . This endows the tensor category D (B )-mod with a structure of pivotal category (see Appendix A) whose associated traces Tr (i ) + and Tr (i ) -are given as follows: given M a D (B )-module and f ∈ End D (B ) (M ), we have

Tr (i ) + ( f ) = Tr(z -i K f )
and

Tr (i ) -( f ) = Tr( f K -1 z i ).
Recall that Tr denotes the "classical" trace for endomorphisms of a finite dimensional vector space. So the pivotal structure depends on the choice of i (modulo d ). The corresponding twist is θ i = z i θ , which endows D (B )-mod with a structure of balanced braided category (depending on i ).

Hypothesis and notation.

From now on, and until the end of this paper, we assume that the Hopf algebra D (B ) is endowed with the pivotal structure whose pivot is z -1 K . The structure of balanced braided category is given by θ 1 = z θ and the associated quantum traces Tr

(1)

± are denoted by Tr ± . Given M a D (B )-module, we set dim ± (M ) = Tr ± (Id M ).

We define

dim(D (B )) = M ∈Irr D (B ) dim -(M ) dim + (M ).
We have (6.2)

dim(D (B )) = 2d 2 (1 -ζ)(1 -ζ -1 )
.

This follows easily from the fact that

(6.3) dim + M l ,p = ζ 1-l -p (l ) ζ and dim -M l ,p = ζ p +l -1 (l ) ζ -1 = ζ p (l ) ζ .
6.B. Characters of Gr(D (B )) via the pivotal structure. -As in Appendix A, these structures (braiding, pivot) allow to define characters of Gr(D (B )) associated with simple modules (or bricks). Given λ ∈ Λ(d ), we set

s + M λ : Gr(D (B )) -→ [ M ] -→ (Id M λ ⊗ Tr M + )(c M ,M λ c M λ ,M )
and

s - M λ : Gr(D (B )) -→ [ M ] -→ (Tr M -⊗ Id M λ )(c M ,M λ c M λ ,M )
These are morphism of rings (see Proposition A.4). The main result of this section is the following.

Theorem 6.4. -Given λ ∈ Λ(d ), we have

s + M λ = χ -λ and s - M (l ,p ) = χ (0,1)-λ .
Proof.

-Write λ = (l , p ) and

γ i , j ,k = ζ (i -k )(i + j )-i (i +1)/2 (i )! ζ .
We have

τ(R )R = 1 d 2 d -1 i ,i ′ , j , j ′ ,k ,k ′ =0 γ i , j ,k γ i ′ , j ′ ,k ′ ζ i (k ′ -j ′ ) (z -k ′ F i ′ E i K k ′ + j ) ⊗ (z -k E i ′ F i K j ′ +k ).
We need to compute the endomorphism of M l ,p given by

(Id M l ,p ⊗ Tr M + )(τ(R )R | M l ,p ⊗M ).
Since M l ,p is simple, this endomorphism is the multiplication by a scalar ̟, and so it is sufficient to compute the action on e (l ,p ) 1 ∈ M l ,p . Therefore, all the terms (in the big sum giving τ(R )R ) corresponding to i = 0 disappear (because E e (l ,p ) 1 = 0). Also, since we are only interested in the coefficient on e (l ,p ) 1 of the result (because the coefficients on other vectors will be zero), all the terms corresponding to i ′ = 0 also disappear. Therefore,

̟ = 1 d 2 j , j ′ ,k ,k ′ ∈ /d ζ -k j -k ′ j ′ (ζ 2p +l -1 ) -k ′ (ζ p +l -1 ) k ′ + j Tr(z -1 K z -k K j ′ +k | M ).
So it remains to compute the element

b = 1 d 2 j , j ′ ,k ,k ′ ∈ /d ζ -k j -k ′ j ′ (ζ 2p +l -1 ) -k ′ (ζ p +l -1 ) k ′ + j z -k -1 K j ′ +k +1 of D (B ). Since b = 1 d 2 j ′ ,k ∈ /d j ,k ′ ∈ /d ζ l (p +l -1-k ) ζ k ′ (-p -j ′ ) z -k -1 K j ′ +k +1 , only the terms corresponding to k = l + p -1 and j ′ = -p remain, hence b = z -l -p K l . So s + M λ = χ -ι(λ) = χ -λ
, as expected. The other formula is obtained via a similar computation.

We denote by ± = ( ± λ,λ ′ ) λ,λ ′ ∈Λ(d ) the square matrix defined by

± λ,λ ′ = Tr ± (c M λ ′ ,M λ • c M λ ,M λ ′ ).
Similary, we define ± to be the diagonal matrix (whose rows and columns are indexed by λ ∈ Λ(d )) and whose λ-entry is

± λ = ω λ (θ ∓1 1 ).
Let us first give a formula for ± λ,λ ′ and ± λ .

Corollary 6.5. -Let (l , p ), (l ′ , p ′ ) ∈ Λ(d ). We have

+ (l ,p ),(l ′ ,p ′ ) = ζ 1 -ζ ζ -l l ′ -l p ′ -p l ′ -2p p ′ (1 -ζ l l ′ ), + (l ,p ) = ζ -p (p +l ) , - (l ,p ),(l ′ ,p ′ ) = ζ 2p +l +2p ′ +l ′ -1 1 -ζ ζ -l l ′ -l p ′ -p l ′ -2p p ′ (1 -ζ l l ′ )
and

- (l ,p ) = ζ p (p +l ) .
Proof. -This follows immediately from formulas (2.3), (4.4), (6.3) and Theorem 6.4.

dim st B (D (B )) = (l ,p )∈ dim -(M l ,p ) dim + (M l ,p ).
This can be understood in terms of super-categories, as explained recently by Lacabanne [START_REF] Lacabanne | Slightly degenerate categories and -modular data[END_REF]. We have

dim st B (D (B )) = 1 2 dim(D (B )) = d 2 (1 -ζ)(1 -ζ -1 )
.

So dim st B D (B ) is a positive real number and we denote by dim st B D (B ) its positive square root. Since

1 -ζ -1 = -ζ -1 (1 -ζ), there exists a unique square root -ζ of -ζ such that dim st B (D (B )) = d -ζ 1 -ζ .
We denote by st B = ( λ,λ ′ ) λ,λ ′ ∈ the square matrix defined by

st B λ,λ ′ = + λ,λ ′ dim st B (D (B ))
.

We denote by st B the diagonal matrix whose λ-entry is + λ (for λ ∈ ). It follows from Corollary 6.5 that (6.6)

st B (l ,p ),(l ′ ,p ′ ) = -ζ d ζ -l l ′ -l p ′ -p l ′ -2p p ′ (ζ l l ′ -1) and st B (l ,p ) = ζ -p (l +p ) .
The root of unity -ζ appearing in this formula has been interpreted in terms of super-categories by Lacabanne [La1]: it is due to the fact that our category is not spherical. Finally, note that (6.7)

st B (l ,p ),(l ′ ,p ′ ) = - st B ι(l ,p ),(l ′ ,p ′ ) = - st B (l ,p ),ι(l ′ ,p ′ ) = st B ι(l ,p ),ι(l ′ ,p ′ ) .

Comparison with Malle -fusion datum

We refer to [Ma] and [Cu] for most of the material of this section. We denote by (d ) the set of pairs (i , j ) of integers with 0 i < j d -1.

π(i ) = 1 if i ∈ {0, 1}, 0 if i 2.
We denote by Ψ(Y , π) the set of maps f : Y → {0, 1, . . ., d -1} such that f is strictly increasing on π -1 (0) = {2, 3, . . ., d } and strictly increasing on π -1 (1) = {0, 1}. Since f is injective on {2, 3, . . ., d }, there exists a unique element k( f ) ∈ {0, 1, . . ., d -1} which does not belong to f ({2, 3, . . ., d }). Note that, since f is strictly increasing on {2, 3, . . ., d }, the element k( f ) determines the restriction of f to {2, 3, . . ., d }. So the map (7.1)

Ψ(Y , π) -→ (d ) × {0, 1, . . ., d -1} f -→ ( f (0), f (1), k( f )) is bijective. For f ∈ Ψ(Y , π), we set ǫ( f ) = (-1) |{(y ,y ′ )∈Y ×Y | y <y ′ and f (y )< f (y ′ )}| .
We put by V = d -1 i =0

v i and we denote by the square matrix (ζ i j ) 0 i , j d -1 , which will be viewed as an automorphism of V . Note that is the character stable of the cyclic group µ d . We set δ(d

) = det( ) = 0 i < j d -1 (ζ j -ζ i ). Recall that δ(d ) 2 = (-1) (d -1)(d -2)/2 d d . Given f ∈ Ψ(Y , π), let v f = (v f (0) ∧ v f (1) ) ⊗ (v f (2) ∧ v f (3) ∧ • • • ∧ v f (d ) ) ∈ 2 V ⊗ d -1 V . Note that (v f ) f ∈Ψ(Y ,π) is a -basis of 2 V ⊗ d -1 V . Given f ′ ∈ Ψ(Y , π), we put 2 ⊗ d -1 (v f ′ ) = f ∈Ψ(Y ,π) S f , f ′ v f . In other words, (S f , f ′ ) f , f ′ ∈Ψ(Y ,π) is the matrix of the automorphism 2 ⊗ d -1 of 2 V ⊗ d -1 V in the basis (v f ) f ∈Ψ(Y ,π) . Lemma 7.2. -Let f , f ′ ∈ Ψ(Y , π). We define i = f (0), j = f (1), k = k( f ), i ′ = f ′ (0), j ′ = f ′ (1), k ′ = k( f ′ ).
We have

S f , f ′ = (-1) k +k ′ δ(d ) d ζ -k k ′ (ζ i i ′ + j j ′ -ζ i j ′ + j i ′ ).
Proof. -The computation of the action of 2 is easy, and gives the term ζ i i ′ + j j ′ -ζ i j ′ + j i ′ . It remains to show that the determinant of the matrix (k , k ′ ) obtained from by removing the k -th row and the k ′ -th column is equal to (-1) k +k ′ ζ -k k ′ δ(d )/d . For this, let ′ (k ) denote the matrix whose k -th row is equal to (1, t , t 2 , . . ., t d -1 ) (where t is an indeterminate) and whose other rows coincide with those of . Then (-1)

k +k ′ det( (k , k ′ )) is equal to the coefficient of t k ′ in the polynomial det( ′ (k ))
. This is a Vandermonde determinant and

det( ′ (k )) = 0 i < j d -1 i =k , j =k (ζ j -ζ i ) • k i =0 (t -ζ i ) • d -1 i =k +1 (ζ i -t ) = δ(d ) d -1 i =0 i =k (t -ζ i ) (ζ k -ζ i ) . Since d -1 i =0 i =k (t -ζ i ) = t d -1 t -ζ k = d -1 i =0 t i ζ (d -1-i )k , we have det( (k , k ′ )) = (-1) k +k ′ δ(d ) ζ (d -1-k ′ )k d ζ (d -1)k = (-1) k +k ′ δ(d ) d ζ -k k ′ ,
as desired.

7.B. Malle -fusion datum

. -Let Ψ # (Y , π) = { f ∈ Ψ(Y , π) | y ∈Y f (y ) ≡ d (d -1) 2 mod d }. Given f ∈ Ψ # (Y , π), we define Fr( f ) = ζ d (1-d 2 ) * y ∈Y ζ -6(f (y ) 2 +d f (y )) * , where ζ * is a primitive (12d )-th root of unity such that ζ 12 * = ζ.
We denote by diagonal matrix (whose rows and column are indexed by

Ψ # (Y , π)) equal to diag(Fr( f )) f ∈Ψ # (Y ,π) . We denote by = ( f ,g ) f ,g ∈Ψ # (Y ,π) the square matrix defined by f ,g = (-1) d -1 δ(d ) ǫ( f )ǫ(g ) S f ,g . Note that f , f 0,1 = 0 for all f ∈ Ψ # (Y , π) (see Lemma 7.2).
Proposition 7.3 (Malle [Ma], Cuntz [Cu]). -With the previous notation, we have:

(a) 4 = ( ) 3 = [ 2 , ] = 1. (b) t = and t = 1. (c) For all f , g , h ∈ Ψ # (Y , π), the number N h f ,g = i ∈Ψ # (Y ,π) i , f i ,g i ,h i , f 0,1 belongs to .
The pair ( , ) is called the Malle -fusion datum.

7.C.

Comparison. -We wish to compare the -fusion datum ( , ) with the ones obtained from the tensor categories D (B )-mod and D (B )-stab. For this, we will use the bijection (7.1) to characterize elements of Ψ # (Y , π). Given k ∈ , we denote by k res the unique element in {0, 1, . .

., d -1} such that k ≡ k res mod d . Lemma 7.4. -Let f ∈ Ψ(Y , π). Then f ∈ Ψ # (Y , π) if and only if k( f ) = ( f (0) + f (1)) res . Consequently, the map Ψ # (Y , π) -→ (d ) f -→ ( f (0), f (1)) is bijective Proof. -We have y ∈Y f (y ) = f (0) + f (1) + d (d -1) 2 -k( f )
and the result follows.

Given (i , j ) ∈ (d ), we denote by f i , j the unique element of Ψ # (Y , π) such that f i , j (0) = i and f i , j (1) = j . We have (7.5)

Fr( f i , j ) = ζ i j and, if (i , j ), (i ′ , j ′ ) ∈ Λ(d ), then (7.6) f i , j , f i ′ , j ′ = (-1) (i + j ) res +(i ′ + j ′ ) res d ǫ( f i , j )ǫ( f i ′ , j ′ )(ζ i j ′ + j i ′ -ζ i i ′ + j j ′ ).
Proof. -The second equality follows immediately from Lemmas 7.2 and 7.4. Let us prove the first one. By definition, Fr( f i , j ) = ζ α * , where

α = d (1 -d 2 ) -6 y ∈Y ( f i , j (y ) 2 + d f i , j (y )).
The construction of f i , j shows that

α = d (1 -d 2 ) -6(i 2 + d i ) -6( j 2 + d j ) -6 d -1 k =0 (k 2 + d k ) + 6(((i + j ) res ) 2 + d (i + j ) res ).
Write i + j = (i + j ) res + ηd , with η ∈ {0, 1}. Then η 2 = η and so (i + j ) 2 + d (i + j ) = ((i + j ) res ) 2 + d (i + j ) res + 2ηd (i + j ) + 2ηd 2 ≡ ((i + j ) res ) 2 + d (i + j ) res mod 2d .

Therefore,

α ≡ 12i j + d (1 -d 2 ) -6 d -1 k =0 (k 2 + d k ) mod 12d ≡ 12i j mod 12d . So Fr( f i , j ) = ζ 12i j * = ζ i j .
We define ϕ : (d ) -→ Λ # (d ) (i , j ) -→ ( ji , i ).

Note that ϕ( (d )) is a set of representatives of ι-orbits in Λ # (d ). We set φ(i , j ) = ϕ(i , j )

if (-1) (i + j ) res ǫ( f i , j ) = 1, ι(ϕ(i , j )) if (-1) (i + j ) res ǫ( f i , j ) = -1.

Then φ( (d )) is also a set of representatives of ι-orbits in Λ # (d ) and the pairs of matrices ( st B , st B ) and ( , ) are related by the following equality (which follows immediately from Corollary 6.5 and formulas (6.7), (7.5) and (7.6)):

(7.7)

f i , j , f i ′ , j ′ = -ζ st B φ(i , j ), φ(i ′ , j ′ )
and

f i , j = st B φ(i , j ) .
Therefore, up to the change of ζ into ζ -1 , we obtain our main result.

Theorem 7.8. -Malle -fusion datum ( , ) can be categorified by the monoidal category D (B )-stab B , endowed with the pivotal structure induced by the pivot z -1 K and the balanced structure induced by z θ .

A. Appendix. Reminders on S -matrices

We follow closely [START_REF] Etingof | Tensor categories[END_REF]Chapters 4 and 8]. Let be a tensor category over , as defined in [START_REF] Etingof | Tensor categories[END_REF]Definition 4.1.1]: is a locally finite -linear rigid monoidal category (whose unit object is denoted by 1) such that the bifunctor ⊗ : × → is -bilinear on morphisms and End (1) = . If X is an object in , its left (respectively right) dual is denoted by X * (respectively * X ) and we denote by coev X : 1 -→ X ⊗ X * and ev X : X * ⊗ X -→ 1 the coevealuation and evaluation morphisms respectively.

We assume that is braided, namely that it is endowed with a bifunctorial family of isomorphisms c X ,Y :

X ⊗ Y ∼ -→ Y ⊗ X such that (A.1) c X ,Y ⊗Y ′ = (Id Y ⊗c X ,Y ′ ) • (c X ,Y ⊗ Id Y ′ ) and (A.2) c X ⊗X ′ ,Y = (c X ,Y ⊗ Id X ′ ) • (Id X ⊗c X ′ ,Y ).
for all objects X , X ′ , Y and Y ′ in (we have omitted the associativity constraints).

Finally, we also assume that is pivotal [START_REF] Etingof | Tensor categories[END_REF]Definition 4.7.8], i.e. that it is equipped with a family of functorial isomorphisms a X : X → X * * (for X running over the objects of ) such that a X ⊗Y = a X ⊗ a Y . Given f ∈ End (X ), the pivotal structure allows to define two traces:

Tr + ( f ) = ev X * •(a X f ⊗ Id X * ) • coev X ∈ End (1) = and Tr -( f ) = ev X •(Id X * ⊗ f a -1 X ) • coev X * ∈ End (1) = .
We will sometimes write Tr X + ( f ) or Tr X -( f ) for Tr + ( f ) and Tr -( f ). We define two dimensions dim + (X ) = Tr + (Id X ) and dim -(X ) = Tr -(Id X ).

To summarize, we will work under the following hypothesis: Hypothesis and notation. We fix in this section a braided pivotal tensor category as above. We denote by Gr( ) its Grothendieck ring. Given X is an object in , we denote by [ X ] its class in Gr( ). The set of isomorphism classes of simple objects in will be denoted by Irr( ). If X ∈ Irr( ) and Y is any object in , we denote by [Y : X ] the multiplicity of X in a Jordan-Hölder series of Y .

  1.D. Morphisms to . -Given ξ ∈ µ d , we denote by ǫ ξ : D (B ) → the unique morphism of algebras such that

  M a D (B )-module and b ∈ D (B ), we denote by b | M the endomorphism of M induced by b . For instance, E | M and F | M are nilpotent and K | M and z | M are semisimple. 2.A. Simple modules. -Given 1 l d and p ∈ /d , we denote by M l ,p the D (B )-module with -basis (l ,p ) = (e (l ,p ) i

  We put Λ(d ) = /d × /d , a set in canonical bijection with {1, 2, . . ., d } × /d , which parametrizes the simple D (B )-modules. Given λ ∈ Λ(d ), we denote by M λ the corresponding simple D (B )-module.

  2.C. Projective modules. -Given λ ∈ Λ(d ), we denote by P λ a projective cover of M λ . The next result is proved in [EGST1, Corollary 2.25]. Theorem 2.4 (Erdmann-Green-Snashall-Taillefer). -Let λ ∈ Λ(d ).

  4.A. Structure. -Since D (B ) is a braided Hopf algebra (with universal Rmatrix R ), (4.1) the ring Gr(D (B )) is commutative. Given M a D (B )-module, we denote by [ M ] the class of M in Gr(D (B )).

  Proposition 4.3. -The Grothendieck ring Gr(D (B )) is generated by v ζ and m 2 . Proof. -We will prove by induction on l that m l ,p ∈ [v ζ , m 2 ]. Since m 1,p = (v ζ ) p , this is true for l = 1. Since m 2,p = (v ζ ) p m 2 , this is also true for l = 2. Now the induction proceeds easily by using (4.2). 4.B. Some characters. -If b ∈ D (B ) is group-like, then the map

5

  .A. Stable category. -As B is a Hopf algebra, it is selfinjective, i.e., B is an injective B -module. Recall that the stable category B -stab of B is the additive category quotient of B -mod by the full subcategory B -proj of projective modules. Since B is selfinjective, the category B -stab has a natural triangulated structure. Similarly, the category D (B )-stab is triangulated. Note that a Bmodule (resp. a D (B )-module) is projective if and only if it is injective. Since the tensor product of a projective D (B )-module by any D (B )-module is still projective[START_REF] Etingof | Tensor categories[END_REF] Proposition 4.2.12], it inherits a structure of monoidal category (such that the canonical functor D (B )-mod → D (B )-stab is monoidal).

  is a morphism of rings. Given M a D (B )-module, we denote by [ M ] st its class in Gr st (D (B )). It follows from Theorem 2l ,p + m st d -l ,p +l ) = 0 if l d -1. 5.B. A further quotient. -We denote by D (B )-proj B the full subcategory of D (B )-mod whose objects are the D (B )-modules M such that Res D (B ) B M is a projective B -module. Since D (B ) is a free B -module (of rank d 2 ), D (B )-proj is a full subcategory of D (B )-proj B . We denote by D (B )-stab B the additive quotient of the category D (B )-mod by the full subcategory D (B )-proj B : it is also the quotient of D (B )-stab by the image of D (B )-proj B in D (B )-stab. Lemma 5.2. -The image of D (B )-proj B in D (B )-stab is a thick triangulated subcategory. In particular, D (B )-stab B is triangulated. Proof. -Given M a D (B )-module, we denote by π M : P (M ) ։ M (resp. i M : M → I (M )) a projective cover (resp. injective hull) of M . We need to prove the following facts: (a) If M belongs to D (B )-proj B , then Ker(π M ) and I (M )/ Im(i M ) also belong to D (B )-proj B . (b) If M ⊕N belongs to D (B )-proj B , then M and N also belong to D (B )-proj B . (c) If M and N belong to D (B )-proj B and f : M → N is a morphism of D (B )modules, then the cone of f also belong to D (B )-proj B . (a) Assume that M belongs to D (B )-proj B . Since it is a projective B -module, there exists a morphism of B -modules f : M → P (M ) such that π M • f = Id M . In particular, P (M ) ≃ Ker(π M ) ⊕ M , as a B -module. So Ker(π M ) is a projective B -module. On the other hand, I (M ) is a projective D (B )-module since D (B ) so it is a projective B -module and so it is an injective B -module. So, again, I (M ) ≃ M ⊕ I (M )/ Im(i M ), so I (M )/ Im(i M ) is a projective B -module. This proves (a). (b) is obvious. (c) Let M and N belong to D (B )-proj B and f : M → N be a morphism of D (B )-modules. Let ∆ f : M

  st B (D (B )) the Grothendieck group of D (B )-stab B , viewed as a triangulated category. If M belongs to D (B )-proj B and N is any D (B )module, then M ⊗ N and N ⊗ M are projective B -modules [EGNO, Proposition 4.2.12], so D (B )-stab B inherits a structure of monoidal category, compatible with the triangulated structure. This endows Gr st B (D (B )) with a ring structure. The natural map Gr(D (B )) → Gr st B (D (B )) will be denoted by m → m st B : it is a surjective morphism of rings that factors through Gr st (D (B )).

and

  The ring Gr st B (D (B )) is generated by v Gr st B (D (B )) is a free -module of rank d (d -1)/2.

  Theorem 5.6. -The character χ λ : Gr(D (B )) → factors through Gr st B (D (B )) if and only if λ ∈ Λ # (d ). So the (χ λ ) λ∈[Λ # (d )/ι] are all the characters of Gr st B (D (B )) and the -algebra Gr st B (D (B )) is semisimple.

.

  The canonical functors between these categories will be denoted by canD (B ) st : D (B )-mod → D (B )-stab, can B st : B -mod → B -stab and can st B : D (B )-mod → D (B )-stab B . The functor can B st • Res D (B ) B : D (B )-mod -→ B -stabfactors through Z(B -stab) (this triangulated category needs to be defined in a homotopic setting, for example that of stable ∞-categories).

Question.

  Is : D (B )-stab B → Z(B -stab) an equivalence of categories? traces. -The element R ∈ D (B ) ⊗ D (B ) defined in §1.F is a universal R -matrix which endows D (B ) with a structure of braided Hopf algebra. The category D (B )-mod is braided as follows: given M and N two D (B )-modules, the braiding c M ,N :

  datum associated with D (B )-stab B . -Let denote a set of representatives of ι-orbits in {1, 2, . . ., d -1} × /d . We define

  7.A. Set-up. -Let Y = {0, 1, . . ., d } and let π : Y → {0, 1} be the map defined by

ACKNOWLEDGEMENTS -We wish to thank warmly Abel Lacabanne for the fruitful discussions we had with him and his very careful reading of a preliminary version of this work.

The first author is partly supported by the ANR (Project No ANR-16-CE40-0010-01 GeRep-Mod). The second author was partly supported by the NSF (grant DMS-1161999 and DMS-1702305) and by a grant from the Simons Foundation (#376202).

Given X , Y two objects in , we set

and

These induce two morphisms of abelian groups

For instance, a simple object is a brick (and 1 is also a brick, but 1 is simple in a tensor category [START_REF] Etingof | Tensor categories[END_REF]Theorem 4.3.1]). Note also that a brick is indecomposable. So if is moreover semisimple, then an object is a brick if and only if it is simple.

If X is a brick, then we will view s + X ,Y and s - X ,Y as elements of = End (X ).

Proposition A.4. -If X is a brick, then s + X : Gr( ) → and s - X : Gr( ) → are morphisms of rings.

Proof. -Assume that X is a brick. We will only prove the result for s + X , which amounts to show that

First, note that the following equality

holds by (A.1) and (A.2). Taking

this proves ( * ).

Proposition A.5. -Let X be a brick and let X ′ be a subquotient of X which is also a brick. Then

Proof. -Indeed, the endomorphism (

) of X is multiplication by a scalar, and this scalar can be computed on any non-trivial subquotient of X .

Corollary A.6. -Let X and X ′ be two bricks in belonging to the same block. Then

Proof.

-By Proposition A.5, we may asume that X and X ′ are simple. We may also assume that X is not isomorphic to X ′ and that Ext 1 (X , X ′ ) = 0. Let X ∈ such that there exists a non-split exact sequence 0 -→ X ′ -→ X -→ X -→ 0.

Since X ≃ X ′ , we have End (X) = , hence X is a brick. It follows from Proposition A.5 that

as desired.