Bertrand Rémy

Annette Werner

AN INTRINSIC CHARACTERIZATION OF BRUHAT-TITS BUILDINGS INSIDE ANALYTIC GROUPS

Keywords: algebraic group, valued field, Berkovich analytic geometry, Bruhat-Tits building, 14L15, 14G22

Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

INTRODUCTION

Bruhat-Tits theory ( [START_REF] Bruhat | Groupes réductifs sur un corps local, I. Données radicielles valuées[END_REF], [START_REF]Groupes réductifs sur un corps local, II. Schémas en groupes. Existence d'une donnée radicielle valuée[END_REF]) deals with reductive groups over valued fields. Given such a group G over a complete non-Archimedean field k, the theory provides a certain cell complex B(G, k), called a Euclidean or affine building, on which the group of rational points G(k) acts in a very balanced way (see [START_REF]Groupes réductifs sur un corps local, II. Schémas en groupes. Existence d'une donnée radicielle valuée[END_REF]Introduction] for more precise conditions on the ground field). The space B(G, k) is large enough with respect to the group G(k) in the sense that the group action is proper (i.e. cell stabilizers are compact subgroups), and the group G(k) is large enough with respect to the space B(G, k) in the sense that the group action is strongly transitive (i.e. the action is transitive on the inclusions of a maximal cell -called an alcove -in a maximal flat subspace -called an apartment).

The latter property implicitly uses the fact that Euclidean buildings are complete metric spaces with non-positive curvature properties, which makes them relevant to geometric group theory [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF]: this is the geometric side of Bruhat-Tits theory. The second aspect of the theory is of arithmetic nature. Namely, in order to perform the construction of the buildings, F. Bruhat and J. Tits define and study integral models of the group G: a posteriori, each cell in the building -called a facet -gives rise to such a model whose group of integral points is, possibly up to finite index, the stabilizer of the facet. These integral models turned out to be very useful in various mathematical fields, such as representation theory ( [START_REF] Moy | Unrefined minimal K-types for p-adic groups[END_REF], [START_REF]Jacquet functors and unrefined minimal K-types[END_REF]), so that the viewpoint of integral models can be chosen as main viewpoint for Bruhat-Tits theory, as shown by the upcoming book [START_REF] Kaletha | Bruhat-Tits theory: a new approach[END_REF].

Bruhat-Tits buildings are singular spaces (from the viewpoint of differential geometry for instance) since they are products of simplicial complexes. Therefore it was a big surprise when V. Berkovich noticed that they could be related to non-Archimedean analytic geometry from his perspective [START_REF] Vladimir | Spectral theory and analytic geometry over non-Archimedean fields[END_REF]Chapter 5]. We developed this approach and this led us to the construction of families of compactifications by analogy with those associated to Riemannian symmetric spaces ( [START_REF] Rémy | Bruhat-Tits theory from Berkovich's point of view. I: realizations and compactifications of buildings[END_REF], [START_REF]Bruhat-Tits theory from Berkovich's point of view. II: Satake compactifications of buildings[END_REF], [START_REF]Wonderful compactifications of Bruhat-Tits buildings, Épijournal[END_REF], [START_REF] Chanfi | Wonderful compactifications of Bruhat-Tits buildings in the non-split case[END_REF]). For a semisimple group G over a complete non-Archimedean local field k, these compactifications are defined to be the closures of embedding maps from B(G, k) to analytifications of proper G-schemes (flag varieties G/P or the wonderful compactification G associated to G). More precisely, the required maps are obtained by constructing first a canonical G(k)-equivariant embedding ϑ : B(G, k) → G an , where G(k) acts on the analytification G an by conjugation, and then by composing it with the analytication of the maps G → G/P (where P is a suitable parabolic subgroup) or of the de Concini-Procesi map G ֒→ G.

In this paper, we go back to the canonical G(k)-equivariant embedding ϑ : B(G, k) → G an itself and address the following basic problem about the relationship between Bruhat-Tits and Berkovich theories: give an intrinsic description of the building B(G, k) as a subspace of the analytic group G an . Our answer is the content of Theorem 3.4; it is also rephrased in Theorem 4.2. It is intrinsic in the sense that the information we use from a point z ∈ G an to decide whether it belongs to the image of ϑ or not, is its holomorphic envelope in G an defined as G

(z) = {y ∈ G an ; ∀ f ∈ O(G), | f (y)| | f (z)|}.
THEOREM A. -The image of the canonical embedding ϑ : B(G, k) ֒→ G an is the subset of points z ∈ G an whose holomorphic envelope G(z) is a k-affinoid subgroup potentially of Chevalley type for which there exists a maximal torus T of G containing a maximal split torus and such that G(z) ∩ T an is the maximal affinoid subgroup T 1 of T an .

Being potentially of Chevalley type is our first (natural) necessary condition; it is explained in Section 1. In Section 2, we show why this condition cannot be sufficient: this is related to the fact, well known when performing descent of the ground field in Bruhat-Tits theory, that Galois fixed points sets are in general bigger than the expected rational buildings. The second condition deals with tori, or equivalently with apartments in the building: the existence of a torus intersecting nicely with G(z) corresponds to the fact that the point z does not belong to the undesirable part of Galois fixed point sets. In Section 3 we explain that the combination of these two conditions is a sufficient one in order to belong to the image of ϑ . In Section 4, we provide a reformulation of the above theorem according to which the building of G(k) has a natural realization as a space of multiplicative k-norms on the coordinate ring O(G) of G.

In what follows, we denote by k a field which is complete with respect to a given non-Archimedean absolute value and we denote by G a semisimple group defined over k. We use the functoriality of the building B(G, k) with respect to non-Archimedean extensions of k; it is satisfied at least when G is split or when k is discretely valued with perfect residue field (see [START_REF] Rémy | Bruhat-Tits theory from Berkovich's point of view. I: realizations and compactifications of buildings[END_REF]1.3.3]). Each of the latter two conditions also allow us to use the following fact: for any maximal split torus S in G there are a maximal torus T containing S and a finite Galois extension k ′ /k splitting T such that, for the map given by the previous functoriality, the apartment of T ⊗ k k ′ contains the apartment of S (this is clear when G is split and it follows from [BrT84, Corollaire 5.1.12] in the second case). At last, we denote by G an the analytic space attached to G: it is a k-analytic space in the sense of Berkovich whose underlying topological space consists of the multiplicative seminorms on the coordinate ring O(G) inducing the initial absolute value on k.

Acknowledgement: The third author was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124.

AFFINOID SUBGROUPS POTENTIALLY OF CHEVALLEY TYPE

As already mentioned in the introduction, recall [RTW10, Theorem 2.1] that to any point x of B(G, k) can be attached a k-affinoid subgroup G x of G an satisfying the following condition: for any non-Archimedean extension K/k, the subgroup G x (K) of G(K) is the stabilizer of x seen in the building B(G, K). By definition, the point ϑ (x) is the unique element of the Shilov boundary of G x , i.e. the only point of G x such that | f (y)| | f (ϑ (x))| for any y ∈ G x and any f ∈ O(G). Conversely, one can recover G x from ϑ (x) as its holomorphic envelope [RTW10, Proposition 2.4 (ii)], which is to say:

G x = {y ∈ G an ; ∀ f ∈ O(G), | f (y)| | f (ϑ (x))|}.
This can be phrased equivalently in terms of multiplicative norms on O(G) by saying that one recovers the affinoid algebra of G x as the completion of the normed k-algebra (O(G), |.|(ϑ (x)).

Let us say that a k-affinoid subgroup H of G an is of Chevalley type (or a Chevalley k-affinoid subgroup) if there exists a k • -Chevalley semi-simple group H [SGA 3, Exposé XXIII §5] with H ⊗ k • k ≃ G and such that H is the generic fibre of the formal completion of H along its special fibre. More generally, we will say that an affinoid subgroup H of G an is potentially of Chevalley type if there exists an affinoid extension K/k such that H ⊗ k K is a Chevalley affinoid subgroup of G an ⊗ k K ; by an affinoid extension, we simply mean that K is a non-Archimedean field which is a k-affinoid algebra (see [RTW10, Appendix A]; this restriction allows to recover k-affinoid algebras from K-affinoid algebras equipped with a descent datum). By construction, the k-affinoid subgroup G x attached to a point x of B(G, k) is always potentially of Chevalley type.

For a point z of G an , let us define its holomorphic envelope by

G(z) = {y ∈ G an ; ∀ f ∈ O(G), | f (y)| | f (z)|}.
The above discussion brings out a first condition fulfilled by any point of G an belonging to the image of ϑ .

FIRST CONDITION.-The holomorphic envelope of z is a k-affinoid subgroup potentially of Chevalley type.

It is easily checked that a point satisfying this condition does appear in the image of ϑ over some non-Archimedean extension of k, as the next Lemma shows. LEMMA 1.1. -Let z be a point of G an whose holomorphic envelope is a k-affinoid subgroup potentially of Chevalley type.

(i) For every non-Archimedean extension K/k, the Shilov boundary of the K-affinoid domain G(z) ⊗ k K is a singleton {z K }. (ii) There exists a non-Archimedean extension K/k such that the point z K belongs to the image of ϑ K .

Proof.-(i) Let K/k be a non-Archimedean extension. By assumption, there exists an affinoid extension

K 0 /k such that G(z) ⊗ k K 0 is a K 0 -affinoid subgroup of Chevalley type in G an ⊗ k K 0 .
We consider a non-Archimedean extension L/k containing both K 0 and K. The set

G(z) ⊗ k L = (G(z) ⊗ k K 0 ) ⊗ K 0 L
is a L-affinoid subgroup of Chevalley type in G an ⊗ k L, hence is the generic fibre of a formal scheme with geometrically integral special fibre; it implies that the Shilov boundary of G(z) ⊗ k L is a singleton {z L }. Since the canonical projection map sends the Shilov boundary of G(z) ⊗ k L onto the Shilov boundary of G(z) ⊗ k K, the latter is also a singleton.

(ii) We consider again an affinoid extension

K/k such that G(z) ⊗ k K is a K-affinoid subgroup of Chevalley type in G an ⊗ k K. The K-affinoid Chevalley subgroup G(z) ⊗ k K is the stabilizer of a unique point x of B(G, K), hence (G K ) x = G(z) ⊗ k K and therefore ϑ K (x) = z K .
We used the fact that any K-affinoid Chevalley subgroup of G an ⊗ k K occurs as the stabilizer of some point in the building. To see this, just pick a special vertex; its stabilizer is a K-affinoid Chevalley subgroup of G an ⊗ k K, and any two of them are G(K)-conjugate.

GALOIS-FIXED POINTS IN BUILDINGS

It is clear that the above condition does not suffice to characterize the image of ϑ . Indeed, consider a finite Galois extension k ′ /k and pick a point x ′ in B(G, k ′ ) which is fixed under the natural action of Gal(k ′ |k) on the building. Let z denote the image of ϑ k ′ (x ′ ) under the canonical projection of G an

k ′ onto G an . The k ′ -affinoid subgroup (G k ′ ) x ′ is equipped with a Galois descent datum, from which one deduces: (G k ′ ) x ′ = G(z) ⊗ k k ′ . It follows that G(z) is a k-affinoid subgroup potentially of Chevalley type. Now, if the field extension k ′ /k is wildly ramified, then the inclusion of B(G, k) into the set of Galois-fixed points in B(G, k ′ )
is strict in general (see example below); therefore, if we pick a Galois-fixed point x ′ outside B(G, k), then z does not belong to the image of ϑ . Therefore we see, up to illustrating the strict inclusions mentioned above, that this argument shows that we need additional conditions to describe the image of ϑ in G an .

We want to illustrate this discussion by looking at an elementary example. Let us consider the group G = SL 2 over some discretely valued field k and pick a finite Galois extension k ′ of k. Via its canonical embedding in P 1,an k ′ , the building B(G, k ′ ) can be identified with the convex hull of P 1 (k ′ ) inside P 1,an k ′ with P 1 (k ′ ) omitted, i.e. with the subset

a∈k η a (R >0 ) ,
where η a denotes the map from R >0 to A 1,an k sending r to the maximal point of the ball of radius r centered in a. The Galois action on B(G, k ′ ) is induced by the Galois action on P 1,an k ′ , and the subbuilding B(G, k) is the image of paths η a with a ∈ k. Since the field k is discretely valued -hence spherically complete -there exists a well-defined Galois-equivariant retraction τ : P 1,an k ′ -P 1,an (k) -→ B(G, k) defined by sending a point x to the maximal point of the smallest ball with center in k containing x.

Using this picture, one easily sees how a Galois-fixed point can appear in B(G, k ′ ) -B(G, k). It suffices to find an element α of k ′ such that all the paths η α g (R >0 ) issued from conjugates α g of α intersect at some point distinct form τ(α); since the Galois action permutes these paths, their meeting point x ′ will be fixed. Note that we have x ′ = η α (r) and τ(α) = η α (r ′ ), where r = max{|α g -α| ; g ∈ Gal(k ′ |k)} is the diameter of the Galois orbit of α and r ′ = min{|α -a| ; a ∈ k} is the distance from α to k.

Let k ′ be any totally ramified finite Galois extension of k. It is well-known that k ′ can be realized as the splitting field of some Eisenstein polynomial P(T) = T e + a e-1 T e-1 + . . . + a 1 T + a 0 , where

|a i | |a 0 | < 1 for all i |k × | = |a 0 | Z (
Q(U) = 1 a 0 P(αU) = α e a 0 U e + a e-1 α e-1 a 0 U e-1 + . . . + a 1 a 0 αU + 1. Since |a i |.|α| i < |a i | |a 0 | for any i ∈ {1, . . . , e -1}, the polynomial Q reduces to α e a 0 U e + 1 = 1 -U e in k ′ [U].
α g ′ α g η 0 (|α|) Case e = 0 in k α α g α g ′ α η 0 (|α|) Case e = 0 in k 0 0 ∞ ∞
In particular, for any totally ramified (finite) Galois extension k ′ /k, the building B(G, k) is strictly smaller than the set of Galois-fixed points in B(G, k ′ ) if and only if [k ′ : k] is divisible by the residue characteristic. We refer to [START_REF] Prasad | Finite group actions on reductive groups and buildings and tamely-ramified descent in Bruhat-Tits theory[END_REF] for a recent contribution to the question of describing fixed-point sets of finite group actions in Bruhat-Tits theory. EXAMPLE 2.1. -Let k = Q 2 and k ′ = Q 2 (α), where α 2 = 2. The two paths η α (R >0 ) and η -α (R >0 ) intersect B(G, k) along the image of [2 -1/2 , ∞), whereas they meet along the image of [2 -3/2 , ∞). The whole interval η α [2 -3/2 , 2 -1/2 ) consists of Galois-fixed points lying outside B(G, k). In general, Rousseau gave an upper bound for the distance of a Galois-fixed point in B(G, k ′ ) to B(G, k) in terms of the ramification of k ′ /k [Rou77, Proposition 5.2.7].

APARTMENTS

As we have just seen, the characterization of the image of the Bruhat-Tits building of G inside G an requires an additional condition. The goal of this section is to formulate such a condition which involves maximal tori of G. We will have to make use of the following fact. LEMMA 3.1. -Let T be a torus over k.

(i) Its analytification T an contains a largest bounded subgroup T 1 . This is an affinoid subgroup, which coincides with the affinoid domain cut out by the equations |χ| = 1 for all χ ∈ X * (T) if T is split. (ii) The Shilov boundary of T 1 is reduced to a single point.

We will write o T for the unique Shilov boundary point of T 1 .

Proof.-We consider first the case of a split torus. If χ 1 , . . . , χ n is a basis of characters of T, then the equations |χ 1 | = 1, . . . , |χ n | = 1 cut out a k-affinoid subgroup T 1 of T an over which |χ| = 1 for any character χ ∈ X * (T). Let K/k be a non-Archimedean extension and Γ a bounded subgroup of T(K). For any character χ of T, both χ(Γ) and (-χ)(Γ) = χ(Γ) -1 are bounded subgroups of K × , hence |χ(Γ)| = 1 and Γ ⊂ T 1 (K). The Shilov boundary of T 1 is reduced to a point since the reduction of T 1 , a torus over k, is irreducible.

In general, pick a finite Galois extension k ′ /k splitting T and set

T k ′ = T ⊗ k k ′ . The affinoid sub- group T 1 k ′ of T an k ′ is stable under the natural Galois action on T an k ′ , hence its descends to a k-affinoid subgroup T 1 of T such that T 1 k ′ = (T 1 ) ⊗ k k ′ .
Finally, since the Shilov boundary of T 1 ⊗ k k ′ is the preimage of the Shilov boundary of T 1 under the canonical projection, we see that T 1 contains a unique Shilov boundary point. ✷ Let us now go back to our discussion toward a characterization of the building inside G an . We assume temporarily that the group G is split. Given a point z in G an whose holomorphic envelope, again denoted by G(z), is a k-affinoid subgroup potentially of Chevalley type, let us consider a non-Archimedean extension K/k such that the canonical lifting z K of z belongs to the image of ϑ K and denote by x its preimage: z K = ϑ K (x). Since the group G is split, the embedding B(G, k) ֒→ B(G, K) identifies the left-hand side with the union of apartments of all maximal split tori in G K which are defined over k. Therefore, in order to guarantee that the point z itself belongs to the image of ϑ , we should require that x belongs to the apartment of a maximal split torus defined over k. The next proposition translates this additional condition into appropriate terms. PROPOSITION 3.2. -Let S be a maximal split torus and let x be a point of B(G, k). The following conditions are equivalent:

(i) the point x belongs to the apartment A(S, k);

(ii) for every non-Archimedean extension K/k, the point x is fixed by the S 1 (K)-action on B(G, K);

(iii) the affinoid subgroup G x of G an contains S 1 . Note that the proof will provide another equivalent statement, namely: (ii) ′ for every finite unramified extension k ′ /k, the point x is fixed by the S 1 (k ′ )-action on B(G, k ′ ).

Proof.-Equivalence of assertions (ii) and (iii) follows immediately from the definition of the affinoid group G x .

For any non-Archimedean extension K/k, the action of S(K) on B(G, K) preserves the apartment A(S, K) and the induced action of the maximal bounded subgroup S 1 (K) is trivial [RTW15, Proposition 4.18 (iii)]. Hence (i) implies (ii).

The only non trivial point to prove is that (ii) implies (i). We argue by contraposition: given a point x of the building which does not belong to the apartment A = A(S, k), we will exhibit a finite unramified extension k ′ /k and an element s of S 1 (k ′ ) such that s

• x = x in B(G, k ′ ).
Recall that the building B(G, k) is also a complete metric space satisfying a non-positive curvature property [BrT72, 2.5 and 3.2], called by now the CAT(0)-property (see [BH99, Part II]). Since the apartment A is a closed and convex subset, it contains a unique point x ′ with d(x, A) = d(x, x ′ ).

Moreover, there exists a unique geodesic segment [x, x ′ ] between those two points and

[x, x ′ ] ∩ A = {x ′ }.
Consider any apartment A ′ containing both x and x ′ ; it contains [x, x ′ ] and intersects A along a closed convex subset C. The point x ′ coincides with the projection in A ′ of x to C, hence it lies in the boundary of C since x ∈ A. Finally, since C is the intersection of a finite number of half-apartments [Rou09, Proposition 9.1], we conclude that x ′ belongs to the wall H in A defined by the vanishing of some affine root a. Let H + denote the half-apartment on which a is non-negative and write a = α + λ , where the linear part α belongs to the root system R(T, G) and λ is an element of log|k × |.

Recall now that the unipotent root group U α is endowed with a separated, exhaustive and decreasing filtration {U α,s } s∈R by affinoid subgroups [RTW15, 4.2.1]. The subgroup U a = U α,λ corresponding to the affine root a = α + λ has the following geometric interpretation: for every non-Archimedean extension K/k, the action of U a (K) on B(G, K) fixes pointwise the half-apartment H + and is transitive on the set of apartments which intersect A along H + . Moreover, if we set

U + α,λ = r>λ U α,r , then U α,λ (K)/U + α,λ (K)
is in bijection with equivalence classes of apartments in B(G, K) containing H + , where two such apartment are said to be equivalent they intersect along a neighborhood of H + .

The torus S acts on U α by conjugation. The bounded subgroup S 1 preserves each step of the filtration and there is a non-canonical bijection U α,λ (K)/U + α,λ (K) ≃ K such that the action of an element s ∈ S 1 (K) on the left-hand side corresponds to multiplication by α(s) ∈ K × on the right-hand side. Note that this condition implies that the unit element on the lefthand side corresponds to 0 on the right-hand side.

To conclude the proof, observe that there exists a unique element ξ ∈ k satisfying the following property: for every u ∈ U α,λ (k) whose class modulo U + α,λ (k) corresponds to ξ , the apartment u • A contains a germ of [x, x ′ ] at x ′ , i.e. we may write u • A ∩ [x, x ′ ] = [x, y] for some y ∈ [x, x ′ ). We have ξ = 0 since x does not belong to A. Now, if x is fixed by S 1 (k), then α(s) • ξ = ξ and therefore α(s) = 1 for any s ∈ S 1 (k). This means that the character α of the k-torus S 1 is trivial at the level of k-points. Since the character α is non-trivial, there exists a finite separable extension ℓ of k such that α = 1 on S(ℓ). Denoting by k ′ the corresponding unramified extension of k, we conclude that x is not fixed by the whole group S 1 (k ′ ). REMARK 3.3. -The arguments above are close to the ones proving that the apartment A(S, k) coincides with the fixed-point set of S 1 (k) if k contains at least 4 elements [BrT84, 5.1.37].

In the split case, the discussion above shows precisely which additional condition is required in order to characterize the image of ϑ in G an : there exists a maximal split torus S in G such that G(z) ∩ S = S 1 . In general, we impose the following: SECOND CONDITION.-There exists a maximal torus T in G which contains a maximal split torus and such that G(z) ∩ T = T 1 .

Once this second condition has been introduced naturally, we are in good position to characterize the image of ϑ in the analytic space G an of G. THEOREM 3.4. -The image of the canonical embedding ϑ : B(G, k) ֒→ G an is the subset of points z satisfying the following two conditions:

1. the holomorphic envelope G(z) of z is a k-affinoid subgroup potentially of Chevalley type; 2. there exists a maximal torus T of G containing a maximal split torus and such that G(z) ∩ T an is the maximal affinoid subgroup T 1 of T an .

Proof.-We have already seen that the first condition is necessary. The same holds for the second one. Given a point x ∈ B(G, k), there exists a maximal split torus S and a maximal torus T containing S such that x ∈ A(S, k) ⊂ A(T, k ′ ), where k ′ is a finite extension of k which splits T. It follows that

G x ⊗ k k ′ contains the bounded torus T 1 ⊗ k k ′ , hence T 1 ⊂ G x .
Now, let us show that the two conditions are sufficient. We consider a point z ∈ G an satisfying the two conditions and pick a finite Galois extension k ′ /k spliting the maximal torus T given by the second condition, as well as a non-Archimedean extension K/k ′ and a point w ∈ B(G, K) such that z K = ϑ K (w), where z K denotes the (unique) Shilov boundary point of G(w)

⊗ k K. Since T an K ∩ (G K ) x = T an K ∩ G(z) K = (T an ∩ G(z)) K = T 1 K = (T K ) 1
, it follows from Proposition 3.2 that w belongs to the apartment of T K . The canonical embedding of B(G, k ′ ) into B(G, K) induces a bijection between A(T, k ′ ) and A(T, K), hence w comes from a point

y of B(G, k ′ ) contained in A(T, k ′ ). Moreover (G k ′ ) y = H k ′ ,
for both sides coincide with (G K ) w after base change to K. In particular, this shows that y is fixed by Gal

(k ′ |k). Since A(T, k ′ ) Gal(k ′ |k) is the image of A(S, k) in B(G, k ′ ), we get a point x in A(S, k) such that (G z k ′ ) ⊗ k k ′ = (G k ′ ) y = H k ′ , hence H = G x and z = ϑ (x). ✷

A REFORMULATION IN TERMS OF NORMS

The above characterization of points of G an lying inside the building B(G, k) (identified with its image by the canonical map ϑ ) can be conveniently rephrased in terms of (multiplicative) k-norms on the coordinate algebra O(G). As we are going to explain, we need here to make the additional assumption that the group G splits over a tamely ramified extension of k.

Let us start by recalling the notion of a universal point (1) . Let z be a point in G an , seen as a multiplicative k-seminorm on O(G). For a given non-Archimedean field extension K/k, there is a

natural K-seminorm ||.|| = x ⊗ 1 on O(G) ⊗ k K, defined by ||a|| = inf max i |a i (z)| • |λ i |
where the infimum is taken over the set of all expressions ∑ i a i ⊗ λ i representing a, with a i ∈ O(G) and λ i ∈ K. The point z is said to be universal if, for any non-Archimedean field extension K/k, the above K-seminorm on O(G) ⊗ k K is multiplicative. One then writes z K for the corresponding point in G an ⊗ k K. REMARKS 4.1. -1. Obviously, points of G an coming from k-rational points of G are universal. 2. Let x ∈ G an be universal. For any finite Galois extension k ′ /k, the canonical extension x k ′ of x to G an ⊗ k k ′ is invariant under the action of Gal(k ′ /k): indeed, the k

′ -norm x ⊗ 1 on O(G) ⊗ k k ′ is Galois invariant. 3. If G is split, then ϑ (x)
(G) → O(G) ⊗ k O(G)
. This binary operation is associative, with unit the element 1 ∈ G(k); moreover, it is obvious that we recover the group law if g and h belong to G(k).

More generally, given an (analytic) action of G an on some k-analytic space X, one can define in a similar way an action of the monoid G an u on the topological space underlying X, which extends the action of G(k). In particular, x * y is well-defined for any two points x, y in G an such that one of them is universal.

Finally, we also recall that one can define a partial order on the set underlying G an as follows:

x y if and only if ∀a ∈ O(G), |a(x)| |a(y)|.

We can now give the following description of the building inside G an . THEOREM 4.2. -Let x ∈ G an . Then the point x belongs to the image of the canonical embedding ϑ : B(G, k) ֒→ G an if, and only if, the following conditions are satisfied:

(i) x is universal;

(ii) x * x x and inv(x) x;

(iii) there exists a maximal torus T containing a maximal split torus such that o T x;

(iv) x is maximal with respect to the three conditions above. REMARKS 4.3. -1. Conditions (i)-(iv) imply that x is a multiplicative k-norm on O(G). 2. One way to understand condition (ii) is to say that x defines a k-multiplicative norm on the commutative Hopf algebra O(G) with respect to which commultiplication and the antipode are bounded. One should observe that (ii) obviously implies 1 G x, so that the counit is also bounded. 3. By Lemma 4.4 below, the condition 1 G x (resp. inv(x) x) implies x = 1 G * x x * x (resp.

x inv(x)), hence condition (ii) could be replaced by x * x = x and inv(x) = x. LEMMA 4.4. -Let x, y, x ′ , y ′ be four points in G an such that both sets {x, y} and {x ′ , y ′ } contain at least one universal point (so that x * y and x ′ * y ′ are well-defined). If x x ′ and y y ′ , then x * y x ′ * y ′ and inv(x) inv(x ′ ).

Proof.-This follows directly from the formulas We assume now that x is a universal point of G an satisfying the conditions of (ii). Obviously, G(x) contains the k-rational point 1 G . Given a non-Archimedean extension K/k and elements g, h ∈ G(x)(K), we have for all a ∈ O(G):

∀a ∈ O(G), |a(x * y)| = inf max i |b i (x)| • |c i (y)| and |a(x ′ * y ′ )| = inf max i |b i (x ′ )| • |c i (y ′ )|,
|a(g -1 )| = |inv(a)(g)| |inv(a)(x)| = |a(inv(x))| and |a(gh)| = |∆(a)(g, h)| inf max i |b i (g)| • |c i (h)| inf max i |b i (x)| • |c i (x)| = |a(x * x)|,
where the infimum is taken over the set of all expressions ∑ i b i ⊗c i representing ∆(a). x by Lemma 4.5. Moreover, there exists a maximal torus T containing a maximal split torus and such that G(x) ∩ T an = T 1 , which amounts to saying that x dominates the distinguished point o T of T. Finally, consider a universal point z ∈ G an satisfying condition (ii) and dominating x (which implies that z dominates 1 G ). For any non-Archimedean extension K/k, Lemma 4.5 implies that G(z)(K) is a bounded subgroup of G(K) containing G(x)(K); by maximality of the latter, we deduce G(z)(K) = G(x)(K), hence G(x) = G(z) and z = x. We have thus checked that x satisfies conditions (i)-(iv).

Conversely, let x be a point in G an satisfying conditions (i)-(iv). We observe that condition (iv) implies that x dominates 1 G , hence it follows from Lemma 4.5 that G(x)(K) is a bounded subgroup of G(K) for any non-Archimedean extension K/k. We are going to see that all these subgroups fix a common point in B(G, k). By condition (iii), there exists a maximal torus T of G containing a maximal split torus S and such that σ T x, hence T 1 (K) ⊂ G(x)(K) for every non-Archimeden extension K/k. We first assume that T is split and that k contains at least 4 elements. For any non-Archimedean field extension K/k, we identify A(T, k) with A(T, K) in B(G, K), and we let F (K) denote the fixed-point set of G(x)(K) in B(G, K). This is a non-empty closed subset, which is contained in the apartment A(T, k) since G(x)(K) contains the group of units T 1 (K) and there are at least 4 elements in K. Considering in particular the extension H (x)/k, we get a point z ∈ A(T, k) fixed under the canonical element x ∈ G(H (x)) lifting x. This means that x belongs to G(z)(H (x)), or equivalently that x is contained in the k-affinoid subgroup G(z). This amounts to saying that z dominates x, hence x = z by maximality.

In general, we consider a finite Galois extension k ′ /k which splits T and such that k ′ contains at least four elements. It follows from the argument above that the canonical extension x k ′ of x to G an ⊗ k k ′ belongs to the apartment of T ⊗ k k ′ in B(G, k ′ ). Since this point is invariant under the action of Gal(k ′ /k) by Remark 4.1 2, x k ′ belongs to the image of A(S, k) in A(T, k ′ ), and therefore x belongs to A(S, k). ✷

  see for example Chapter 7, Theorem 7.1, in[START_REF] John | Local fields, London mathematical society students text 3[END_REF]). The group |k ′ × | is generated by |α| = |a 0 | 1/e for any root α of P. We have d(α, k) = |α| and all conjugates of α are contained in the closed ball E(α, |α|) = E(0, |α|). The endomorphism of A 1,an k ′ defined by P(T) maps this ball onto the closed ball E(0, |a 0 |). In order to study the induced map E(0, |α|) → E(0, |a 0 |), set U = T/α and write

  It follows that the following four conditions are equivalent: -all paths η α g (R >0 ), for g ∈ Gal(k ′ |k), intersect outside B(G, k); -all roots of P are contained in the open ball D(α, |α|); -all roots of Q are contained in the open ball D(1, 1); e vanishes in k.

  where the infimum is taken over the set of all expressions ∑ i b i ⊗ c i representing ∆(a), and∀a ∈ O(G), |a(inv(x))| = |inv(a)(x)| |inv(a)(x ′ )| = |a(inv(x ′ ))|. ✷ LEMMA 4.5.-Let x be a universal point of G an . We again denote by G(x) = {z ∈ G an ; z x} its holomorphic envelope. Then the following conditions are equivalent:(i) The envelope G(x) is a subgroup object of G an , i.e. G(x)(K) is a subgroup of G(K)for any non-Archimedean extension K/k; (ii) The point x satisfies: 1 G x, inv(x) x and x * x x. Moreover, G(x) is bounded in G an . Proof.-Assume that G(x) is a subgroup object of G an . Since G(x)(k) is a subgroup of G(k), it contains the unit element 1 G and therefore 1 G x. Now we consider the canonical point x ∈ G(H (x)) lying over x; we have |a(x)| = |a(x)| for any a ∈ O(G), as well as |a(x * x)| = |a(x • x)| and |a(inv(x))| = |a(x -1 )|. Since x • x and x -1 belong to G(x)(H (x)), it follows that |a(x * x)| |a(x)| and |a(inv(x))| |a(x)| for all a ∈ O(G), which is to say x * x x and inv(x) x.

  Since inv(x) x and x * x x, we deduce |a(g -1 )| |a(x)| and |a(gh)| |a(x)|, hence g -1 , gh ∈ G(x)(K). This proves that G(x) is a subgroup object of G an . Finally, boundedness is obvious: if f 1 , . . . , f n is a finite set generating O(G) as a k-algebra, then we have | f i (y)| max i | f i (x)| for any point y ∈ G(x). ✷ Proof of Theorem 4.2.-In what follows, we identify the building B(G, k) with its image in G an by the embedding ϑ . If a point x of G an belongs to B(G, k), then x is universal by Remark 4.1 and G(x) is a k-affinoid subgroup of G an , hence x * x and inv(x)

  is universal for every point x of B(G, k), as follows readily from the explicit description of ϑ (x) as a norm on the coordinate algebra of a big cell [RTW10, Proposition 2.6]. More generaly, the same is true if G splits over a tamely ramified extension k ′ /k by [RTW10, Lemma A.10] and [RTW15, Erratum]. 4. Given a torus T over k, we denote by σ T the (unique) Shilov boundary point of the maximal affinoid subtorus T 1 of T an . This point is universal if T splits over a tamely ramified extension of k, but not in general as checked by Mayeux [May20, Proposition 10.2].SETTING -As already mentioned, the semisimple group G considered in this last section is assumed to split over a tamely ramified extension of k, so that all points in the image of ϑ are universal.Let G an u denote the subset of universal points in G an . Following Berkovich [Ber90, 5.2], there is a natural monoid structure on G an u extending the group structure on G(k). Given any two points g, h ∈ G an u , the seminorm g ⊗ h on O(G) ⊗ k O(G) is multiplicative and one defines g * h as the induced multiplicative seminorm on O(G) via the comultiplication map ∆ : O

(1) This notion was introduced by Berkovich, who used the adjective peaked [Ber90, 5.2]; its study was carried on by Poineau, who preferred the adjective universal[START_REF] Poineau | Les espaces de Berkovich sont angéliques[END_REF].


Ikram Chraibi Kaadoud
email: ikram.chraibi-kaadoud@imt-atlantique.fr

Lina Fahed
email: lina.fahed@imt-atlantique.fr

Philippe Lenca
email: philippe.lenca@imt-atlantique.fr

Explainable AI: a narrative review at the crossroad of Knowledge Discovery, Knowledge Representation and Representation Learning

Keywords: XAI, Knowledge Discovery, Knowledge representation, Representation learning, State representation learning, Manifold representation learning, Multi-view representation learning, Network representation learning

EXplainable Artificial Intelligence (XAI) has recently become a very active domain, mainly due to the extensive development of black-box models such as neural networks. Recent XAI objectives have been defined in the state-of-the-art, for which specific approaches have been proposed. Implicit links can be found between XAI and other domains, especially related to knowledge and neural networks. We here aim to highlight these implicit links. We present a narrative review of research works in two domains: (i) Knowledge domain with focus on Knowledge Discovery and Representation, and (ii) Representation Learning. We discuss the similarity and joining points between these domains and XAI. We conclude that, in order to make black-boxes more transparent, XAI approaches should be more inspired and take advantage of past and recent works in Knowledge and Representation Learning domains. Through this paper, we offer an entry point to the domain of XAI for both multidisciplinary researchers and specialists in AI, as well for AI knowledgeable users.

Introduction: XAI

During the last few years, eXplainable Artificial Intelligence (XAI), has become a very active domain 1 facing the high development of black-box models, such as neural networks [START_REF] Guidotti | [END_REF]. A new generation of XAI approaches have been proposed, for which several new concepts and terms are specific to application domains, data types or modeling. Application domains of XAI are multiple: machine learning, robotics, multi-agent systems, computer vision, Knowledge Representation and Reasoning, etc.

[Barredo Arrieta et al., 2020] defined "Given an audience, an explainable Artificial Intelligence is the one that produces details or reasons to make its functioning clear or easy to understand". Indeed, XAI aims to make Artificial Intelligence (AI) models more intelligible and accessible or to directly design explainable models and results [Buchanan and Shortliffe, 1984;[START_REF] Guidotti | [END_REF]Barredo Arrieta et al., 2020]. When the first case arises, XAI provides an explanation of the internal mechanisms and/or the reasons behind the AI model behavior i.e. its functioning and performance: an explanation is thus an interface between the AI model to explain and the target audience [Gunning, 2017]. We define an explanation as an information in a semantically complete format, which is self-sufficient and chosen according to the target audience regarding its knowledge, its expectations and the context. Hence, the purpose of an explanation is to clarify the cause, context and consequences of described facts through a set of statements or information [Van Fraassen, 1988].

It is important to underline that an explanation by its very nature is contextual: it is specific to a given target audience and also to a given context [START_REF]Douglas Walton. A new dialectical theory of explanation[END_REF]. This makes XAI more challenging as automatic context understanding is still a very challenging task [START_REF] Brézillon | Patrick Brézillon. Context in Artificial Intelligence: I. A survey of the literature[END_REF]Lim et al., 2009;Augusto et al., 2017;Hollister et al., 2017] and no unified way for modelling context in intelligent environments has yet been proposed in the literature [START_REF] Brenon | [END_REF]. We emphasize that the context (i.e users context, goal context, etc.) is important to take into account in XAI. However, this point is not the focus of the paper.

In the state-of-the-art, an explanation can take different formats (e.g. visual, natural language, features relevance explanations, etc.) and combine several representations of the same information [Barredo Arrieta et al., 2020]. Two main XAI techniques are proposed: (i) Ante-hoc techniques which consist in optimizing an already transparent AI model (e.g. linear regression, decision trees, etc.) by adding constraints or features in order to increase transparency through metrics, data visualisation, etc. (ii) Post-hoc techniques that aim to explain already built black-box AI models (mainly deep neural networks). Among famous Post-hoc techniques: LIME [START_REF] Ribeiro | [END_REF], SHAP [START_REF] Lundberg | [END_REF], visual explanations, saliency mapping, etc. XAI has recently been covered by several reviews that reveal its complex and intrinsically multidisciplinary aspects from a technical, user or Human-Interaction viewpoint [START_REF] Guidotti | [END_REF][START_REF] Gilpin | [END_REF]Barredo Arrieta et al., 2020;[START_REF] Vilone | Explainable artificial intelligence: a systematic review[END_REF]. As examples, we can note technicalbased reviews, as those related to reinforcement learning [START_REF] Puiutta | [END_REF]Heuillet et al., 2021], data-based reviews as those related to time series [START_REF] Schlegel | [END_REF][START_REF] Rojat | [END_REF] and application-based reviews related to healthcare [START_REF] Adadi | Amina Adadi and Mohammed Berrada. Explainable ai for healthcare: From black box to interpretable models[END_REF] and banking [START_REF] Burgt | Joost van der Burgt. Explainable AI in banking[END_REF]. Other reviews are inspired by social science, human psychology, sociology or cognitive sciences [Miller, 2019;[START_REF] Capone | [END_REF] in order to build ethical and fair models [Barredo Arrieta et al., 2020].

One key issue that have not been discussed in the above cited reviews and that we would like to highlight, is the importance of knowledge in XAI. As an interface between an AI and a target audience, an explanation can be considered as an interpreter between the AI knowledge and the human target audience knowledge. Since knowledge domain is historical in AI, this raises in turn important questions about the impact of domains such as Knowledge Discovery and Representation on XAI. Furthermore, regarding black-box models and especially neural networks, it is important to mention that in recent papers, concepts like representation learning, knowledge/latent/hidden/abstract representation, latent space, etc. have been studied in order to tackle issues such as dimensionality, running time, algorithmic complexity, etc. However, to the best of our knowledge, no explicit relation has been defined between these concepts and XAI. We consider that as these concepts are increasingly recurrent in the literature, with no consensual definitions across fields, it becomes, in turn, more difficult to apprehend the XAI domain.

To address this issue, we propose a narrative review that, contrary to the above cited literature reviews, does not review XAI techniques. Our paper is a narrative review across several domains: a literature-based review that synthesizes technical research works related to domains that implicitly inspire XAI works. Our goal is to bring original insights, formulate new research questions and highlight promising future directions of XAI. More precisely, in this narrative review, we aim to address three questions. First, to centralize and clarify concepts recurrently used in AI domains but not always clear for XAI specialists. Second, to bring a new light to XAI by making explicit the links between XAI and two other domains: (i) Knowledge domain including Knowledge Discovery Process (KDP) and Knowledge Representation (KR), and (ii) Representation Learning (RL) more associated to deep learning domain. Third, to offer an entry point to the XAI domain for multidisciplinary or specialists in these domains.

Actually, these domains are often perceived as disconnected as most of the research is currently concentrated on only one of them [Sallinger et al., 2020]. Despite this, we believe that it is important to enhance the links and the implicit relations that can be found between them. We thus consider that XAI has been indirectly inspired by these domains.

Figure 1 shows our vision as a schematic representation of XAI domain and both KDP, KR and RL domains. Table 1 lists the acronyms used. The paper is organized as follows: definitions are presented in section 2, KDP and KR in section 3, and RL in section 4. At the end of both last sections, we discuss the relation between the highlighted points, related to KDP, KR, RL and XAI. Finally, in section 5, we discuss future directions and perspectives related to XAI.

Definitions

This section is dedicated to the definition of several concepts related to Knowledge and Representation learning domains. Several definitions are inspired from state-of-the-art works.

Definition 2.1 The raw material that represents the input of an algorithm is called data. Data can be noisy, partial/complete, un/structured and of different types [START_REF] Grazzini | Collaborative research-grade software for crowd-sourced data exploration: from context to practice -part I: Guidelines for scientific evidence provision for policy support based on big data and open technologies[END_REF][START_REF] Malhotra | [END_REF].

Definition 2.2 A data set is a collection of data that describes real-word objects (such as cars, documents, animal, etc.) through multiple properties called features [START_REF] Bishop | [END_REF].

Definition 2.3 Once data is analyzed and correlated, it represents information. Information can be reproduced from data and its importance depends on the context it is generated from/for [START_REF] Grazzini | Collaborative research-grade software for crowd-sourced data exploration: from context to practice -part I: Guidelines for scientific evidence provision for policy support based on big data and open technologies[END_REF][START_REF] Malhotra | [END_REF].

Definition 2.4 Knowledge is a set of information that is assessed by a human, i.e. human adds a value and semantics according to his/her own background and context [START_REF] Grazzini | Collaborative research-grade software for crowd-sourced data exploration: from context to practice -part I: Guidelines for scientific evidence provision for policy support based on big data and open technologies[END_REF][START_REF] Malhotra | [END_REF].

Definition 2.5 In the data mining domain, a "pattern is an expression in some language describing a subset of the data or a model applicable to the subset" [START_REF] Fayyad | [END_REF].

Hence, Pattern extraction designates the process of finding structures in data, fitting a model to data, or finding a highlevel description of a data set.

Many data modeling approaches have been proposed in the state-of-the-art. We can cite reinforcement learning, graphbased approaches, neural networks, etc. We now define some important concepts related to these approaches. Definition 2.6 Reinforcement learning is an approach in which an intelligent agent interacts with its environment through trial-and-errors actions in order to reach a goal. Each action leads to a modification of the state of the agent and the environment and the increase or decrease of a cumulative reward value. Actions are chosen according to a strategy that is called a policy [Barto and Sutton, 1995].

Definition 2.7 A Manifold is a topological structure of ndimensions. For example, a one-dimensional manifold is a curve, a two-dimensional manifold is a surface, a threedimensional manifold is a sphere. Definition 2.8 A network is a collection of discrete objects called nodes, which are connected through links: it can be viewed as a graph with vertices and edges, both with attributes/weights or not [Fletcher et al., 1991].

Definition 2.9 Neural networks are machine learning models with several architectures, that are usually structured by one or several layers (input, hidden and output). Each layer is composed of one or several computational units called artificial neurons -conceptually derived from biological neurons [McCulloch and Pitts, 1943;[START_REF] Abraham | Ajith Abraham. Artificial neural networks[END_REF]. Computational units can also be a Long Short Term Memory (well known also as LSTM) [Hochreiter and Schmidhuber, 1997] or Gated recurrent units [Cho et al., 2014]. A deep neural network have many hidden layers, units, and edges with weights. Units of layer n can be all or partially connected to units of layer n + 1. Due to this inner complexity, deep neural networks are a typical example of black-boxes. Definition 2.10 In neural networks, an activation pattern refers to units activation values of one of the layers. An activation pattern is a numerical vector of the size of the layer it is associated with. A hidden pattern refers to the activation pattern of a hidden layer.

In the literature of neural networks, concepts like latent space and latent representation have been developed and widely used. However, to the best of our knowledge, no complete definitions have been clearly proposed for such concepts. Due to the importance of both concepts in the rest of this paper, we choose to formulate their definition next.

Definition 2.11 Latent space refers to the abstract multidimensional space associated to each layer of a neural network where the representation of the learned data is implicitly built. Latent space contains the meaningful internal features (definition 2.2) representations of learned data, which makes it not directly interpretable. In a deep neural network (definition 2.9), each hidden layer, whether it has the same number of units or not, has its own latent space. It is thus possible to extract several implicit representations from this network. The latent space can be used to achieve a data dimensionality reduction, when the hidden layer is smaller than the input layer. This is the case for example with autoencoders and variational autoencoders [Kingma and Welling, 2014], models that can reduce high-dimensional inputs into efficient and representative low-dimensional representations [Roberts et al., 2018b].

Definition 2.12 Latent or hidden representation refers to the data representation implicitly encoded by a neural network during the learning task and thus is hidden-layerdependant [START_REF] Bengio | [END_REF]. It is a machine-readable data representation that contains features of the original data that have been learned by associated hidden layer. One key property of latent space (definition 2.11) is that real-world objects (definition 2.2) that are semantically close (e.g. cars of different brands), will end up grouped together in one latent space: their respective hidden representation in the corresponding layer, will be close to each other compared to other objects that are not semantically close (e.g. cats) [Roberts et al., 2018a]. Thus, a latent representation is useful for pattern analysis (definition 2.5) and for similarity detection between objects (definition 2.2) using clustering methods.

Knowledge: discovery and representation

We now present two active research domains: KDP (section 3.1) and KR (section 3.2). Then, we discuss the relation between them and XAI in section 3.3.

Knowledge Discovery Process (KDP)

KDP is a human-centered domain that seeks useful knowledge (definition 2.4) through an iterative and interactive process that involves humans [START_REF] Lenca | Philippe Lenca. Human centered processes[END_REF][START_REF][END_REF]. As the domains KDP, data mining, and Knowledge Discovery in Databases (referred to as KDD) are often used in a confused way, we consider that it is important to present a clarification about them, as follows:

• According to [START_REF][END_REF], KDP and KDD designate the same process. However, KDP can be generalized to non-databases sources of data, while KDD emphasizes databases as a primary source of data. • KDP and data mining are related to each other as well as to other domains like machine learning and statistics, but are clearly distinct. Indeed, according to [START_REF] Fayyad | [END_REF] and [START_REF][END_REF], KDP is the global process of discovering useful knowledge from data, whereas data mining is a particular step within the KDP process that consists in applying algorithms to extract patterns (definition 2.5) or to build a model that fits the data.

There is no consensus about the steps of a KDP: nine steps in [START_REF] Fayyad | [END_REF], eight steps in [START_REF] Anand | [END_REF], six steps in [Wirth, 2000;[START_REF][END_REF] and five steps in [Cabena et al., 1998]. However, we emphasize that globally KDP consists of three common main steps:

1. A pre-processing step for data collection or generation, data preparation, cleaning, curing, etc. 2. A data processing step where several techniques from statistics/machine learning/data mining, etc. communities can be used. 3. A post-processing step for visualisation, evaluation and validation.

At each step, the extracted information (definition 2.3) is usually evaluated by the human, given the context, to form knowledge 2 (definition 2.4). Thus, the target audience of the KDP is the human: application domain experts and decision makers. In addition, it is important to underline that two mains goals of KDP are usually defined [START_REF] Fayyad | [END_REF]: (i) verification of a user hypothesis, and (ii) discovery of valid and useful new knowledge that is understandable with respect to the data (definition 2.1) from which it is derived. These goals are thoroughly discussed in section 3.3.

Knowledge Representation (KR)

KR is a crucial question in AI [START_REF] Malhotra | [END_REF]. Also known as "Knowledge Representation and Reasoning", KR aims at finding ways to efficiently structure specific domain knowledge for automated reasoning. In this way, intelligent machines can learn, draw inferences, make decision and answer questions related to this knowledge [START_REF] Davis | [END_REF][START_REF] Shapiro | [END_REF]Davis, 2015]. Thus, seen in such a way, KR can be considered as a machine-oriented domain. The purpose of KR is neither about storing data, nor making actions but it is about allowing "thinking by reasoning" [START_REF] Davis | [END_REF]. Consequently, KR has been a key component for the conception of intelligent knowledge-based systems.

KR is also, according to [START_REF] Malhotra | [END_REF], closely related to the Knowledge retrieval in the shape of ontologies (concepts for representing, storing and accessing knowledge [START_REF] Guarino | [END_REF]). KR techniques have also been widely developed and applied to semantic web [Hagedorn et al., 2020], semantic networks [START_REF] Malhotra | [END_REF]], text interpretation and cognitive robotics [Davis, 2015]. In addition, from a user viewpoint, KR is important during the development of software systems in order to perform particular tasks, as well as for broader community of cognitive science whose goal is to constitute and organize knowledge from humans and machine perspectives [Das, 2003].

Knowledge Representation Learning (KRL) KRL is the process of making AI algorithms model and learn a structured representation of domain-specific knowledge. As a consequence, concepts, relations between them and their representations can be encoded in a lowdimensional semantic space [START_REF][END_REF]. For example, when knowledge is represented as a graph, the KRL process allows graph embedding and preserves semantic similarities [START_REF] Xie | [END_REF]. Notice that the development of deep learning algorithms and their performance on distributed representations (i.e. representations that describe features of the same data across layers) that reduce the computational complexity has contributed to the emergence of several KRL applications such as recommendation system [Zhang et al., 2016], language modeling [START_REF] Ahn | [END_REF] and question answering [START_REF] Yin | [END_REF]. We consider that KR has recently become a more central domain in AI, and by extension in XAI. This is mainly due to the development of Representation Learning in neural networks (introduced in section 4).

2 Notice that recent approaches like AutoML tend to perform all these steps automatically without user intervention [He et al., 2021] 3.3 Discussion: relation between KDP, KR and XAI

We now discuss and highlight several links and common points between KDP, KR and XAI. As mentioned in sections 3.1 and 3.2, KDP is a human-centered domain, whereas KR is a machine-oriented domain. However, both domains are complementary: in KDP, the main question is "How to efficiently discover new or retrieve existing knowledge?", whereas in KR the tackled question is "How to represent the knowledge efficiently to be able to reason on it?".

It is important to highlight that both KDP and KR questions are also addressed and are crucial in XAI. Recall that the objective of XAI is to make the reasons behind AI behavior simple and accessible to a target audience regarding a given task and context. We consider that this XAI objective can be viewed and divided into two sub-objectives: (i) to discover the reasons behind AI behavior -which is the same as in a KDP problem -, (ii) to represent these reasons in a way that is intelligible for the human target audience, but also sometimes for an artificial one -which is the same as for a KR problem.

Let us first detail the links between KDP and XAI. In XAI, for black-boxes like deep neural networks [START_REF] Guidotti | [END_REF][START_REF] Gilpin | [END_REF], technical approaches are used to search the behavior of AI and make it explainable by providing an explanation that can take several forms and be multimodal [Barredo Arrieta et al., 2020]. Explaining an AI model is therefore very inspired by KDP. The particular point is that in XAI, the input data (definition 2.1) is related to the blackbox AI model. This input data can be of several types, e.g. activation patterns of hidden layers (definition 2.10), features or representations, and require the same techniques as in KDP.

Figure 2 represents a schematic representation of the transformation of data into knowledge, in KDP and XAI domains. It clarifies the similarities between both domains regarding the human intervention, and the role of the technical part, i.e. data mining and explainable methods.

Let us now go into deep details about knowledge representation in XAI. Two cases can be highlighted according to target audience: (i) human who uses the knowledge representation to reason and understand the situation, e.g. the decision maker and the application domain expert, depending on their expertise, role and goals, (ii) another AI system for which the input data is provided from a complex AI architecture. Let us take an example in the domain of computer vision and especially classification using deep neural networks. Researchers have proposed approaches that exploit different AI algorithms and their latent representation (definition 2.12) as an input to the neural networks. The objective of such approaches is to perform both classification and explainability tasks through saliency masks applied to images and text generation [START_REF] Lecun | [END_REF]. This is one strategy among multiple others for the representation of knowledge in order to favor the explainability of the behavior of the initial AI model.

In addition, notice that KRL has been basically associated with deep learning algorithms, especially with techniques like graph representation learning [Hamilton, 2020] and concept learning [START_REF] Dolgikh | Serge Dolgikh. Spontaneous concept learning with deep autoencoder[END_REF], which are both studied in the XAI domain [START_REF] Xu | [END_REF]Fazi, 2020].

Finally, it is important to highlight the importance of the target audience in both KDP/KR and XAI domains. Actually, the role of the target audience is decisive: knowledge is usually retrieved and shaped in order to answer a question of a target audience related to a given task and context such as verifying an hypothesis, inference and decision making. Knowledge representation and content in both KDP/KR and XAI domains are thus target and context dependant.

As a conclusion, XAI is closely related to both KDP and KR, and future works in XAI should take advantage of recent works in both domains, as well as older works.

Representation Learning (RL)

We now present the RL domain, its importance in deep neural networks, and RL sub-domains that are recurrent and popular.

RL introduction and definition

RL has been discussed as a key challenge related to different machine learning domains [START_REF] Dietterich | [END_REF] especially to neural networks. As first demonstrated by [START_REF] Rumelhart | [END_REF], in neural networks, back-propagation algorithms can generate useful internal representations of data in hidden layers. Since then, different approaches have been proposed in order to learn, analyze and visualize latent data representations [START_REF] Gilpin | [END_REF][START_REF] Guidotti | [END_REF]. Thus, RL has become an active research domain for which the objective is to study of latent representations in order to improve deep neural network efficiency [START_REF] Bengio | [END_REF].

RL -and synonyms like Data RL or Feature Learning [Zhong et al., 2016] -focuses on "learning representations of the data that make it easier to extract useful information when building classifiers or other predictors" [START_REF] Bengio | [END_REF]. In other words, RL is designed to learn abstract features that characterize data [START_REF] Lesort | [END_REF].

RL algorithms can be classified into two categories: global and local RL algorithms. While the first ones tend to pre-serve the data global information in the learned feature space, the second ones focus more on preserving local similarity between data during learning the new representations [Zhong et al., 2016]. Representations are not task-specific but are useful to machine learning algorithms to solve tasks, as well as to humans to comprehend the behavior of these last algorithms [START_REF] Bengio | [END_REF]. One of the reasons that makes RL popular is that representations express priors about the data. The expressed priors can vary within a single learning algorithm. Consequently, the characteristic of the priors variations leads to different RL approaches, that we classify into two categories: problems-oriented RL and data-oriented RL.

In the following section, we first present the concept of hierarchical representation in deep neural networks, a key property of RL. Then we present examples of particular cases of RL that are problems-oriented and data-oriented.

Hierarchical representations in deep neural networks

One key property of the RL domain in deep neural networks is the ability to provide both high level features and low level features for the same learned data. Recall that a deep neural network will encode a latent representation at each hidden layer (definitions 2.9, 2.12). Since the layer n units can be all or partially connected to the layer n + 1 units, each layer uses the previous layer as input. If the previous layer is a hidden layer, then the input is already a latent representation, i.e. an abstract feature representation that characterizes the data. Thus, each layer extracts an abstract feature representation of the previous layer. As a result, a deep neural network learns multiple levels of abstraction and implicitly encodes a hierarchy of latent and abstract representations that are built progressively, layer by layer. The layers that are close to the input layer will encode a low-level feature representation, whereas those deeper inside the architecture will encode a high level feature representation. In other words, the closer the considered layer is to the output layer, the more the representation is abstract [START_REF] Bengio | [END_REF]Zhong et al., 2016;[START_REF] Lesort | [END_REF], as represented in Figure 3. It has also been shown that, in deep learning algorithms, hidden representations tend to keep dominant information and propagate them across hidden layers, regardless the width or depth increase of the deep neural networks [Nguyen et al., 2021]. This characteristic of RL is also a key one for XAI: by extracting and comparing the low-level and the high-level representations of a deep architecture, we consider that it is possible to explicit the inner mechanism of the architecture by observing the differences between the representations. This will be discussed further in section 5.

Problems-oriented RL approaches

Recall that the objective of RL algorithms is to learn abstract features that characterize data. This objective can be challenging according the issues that one could face such as high dimensionnality of data or RL application to another AI paradigm like reinforcement learning (definition 2.6). In the following sub-sections, we describe two RL sub-domains: Manifold RL and State RL, that have recently shown great performances in deep learning and that deal with our core questions. The links with XAI are also briefly discussed.

Manifold RL

Manifold RL is particularly suited for dealing with highdimensional data sets that are very difficult to visualize and less intuitive. However, within such data sets, data can locally belong to a subset that can be represented by a manifold. As stated in definition 2.7, a manifold is a topological structure of n-dimensions. Thus, Manifold RL corresponds to the learning of complex data representation in several dimensions while preserving the topological properties of the considered manifold. We consider Manifold RL as a non-linear dimensionality reduction approach, that can help to discover similarities in data for which dimensions have been reduced [Cayton, 2005;Bengio, 2009;[START_REF] Zhang | [END_REF]. The Manifold RL domain aims at discovering manifold structure hidden in high dimensional data. It seeks to discover the intrinsic structure of a given manifold. Notice that when many manifolds are considered, we refer to this as multi-manifold RL [Lee et al., 2016;Torki et al., 2010]. It allows to both preserve the local geometric structure within distinct manifolds while ensuring the discriminability between them [START_REF] Wu | [END_REF].

When more neural networks transparency is required, the visualisation of latent representations is essential: it allows to develop an intuition about the distance between subsets of data represented by their associated latent manifold representations. Consequently, we consider that this dimensionreduction characteristic is therefore of great practical interest for XAI. Indeed, reducing the complexity due to the high dimensions can strongly contribute in understanding the inner mechanisms of models exploiting the data, but also the role of the data subsets on the models behaviors.

State RL (SRL)

In addition, RL can also concern domains where data are in a low dimensional space. SRL is "is a particular type of representation learning that aims at building a low-dimensional and meaningful representation of a state space, by processing high-dimensional raw observation data (e.g., learn a position (x, y) from raw image pixels)." [Heuillet et al., 2021]. This domain is thus particularly suited for learning features in reinforcement learning, robotics and control scenarios. Thus, learning in SRL for an artificial agent is rather related to building a latent model of the environment and the task to perform through interactions [START_REF] Lesort | [END_REF]. In addition, it has been shown that SRL provides three main advantages for several research domains [Heuillet et al., 2021]:

• The learned features are of low dimensions which improves speed and generalization of deep learning models [Lesort et al., 2017].

• SRL helps improving performance in some reinforcement learning steps such as policy learning [Heuillet et al., 2021].

• Learning representations of states (definition 2.6), actions or policies provide meaning to explain a reinforcement learning algorithms. Indeed, SRL allows to learn representations that capture the variation in the environment generated by the action of the agent [Lesort et al., 2017;Heuillet et al., 2021].

It has been shown that SRL is particularly suitable to make the behavior of an artificial agent and the reasons of this behavior accessible for humans [Lesort et al., 2017;[START_REF] Lesort | [END_REF]Heuillet et al., 2021]. Consequently, we can consider SRL as an example of domains used for explanation goals in reinforcement learning.

Data-oriented RL approaches

In RL, several approaches tackle the problem of increasing data volumes, their heterogeneity and the multiplicity of their sources. We can consider them as data-oriented approaches and present two of them: the Multi-view RL and the Network RL. We also highlight the link between RL applied to realworld data-oriented problems and XAI domain.

Multi-view RL

In real-world applications, each object can be described by multiple features (definition 2.2) [START_REF] Xu | [END_REF]. It is thus referred as Multi-view data. These features, also referred to as views, constitute complementary and diverse information of the same data [START_REF] Xu | [END_REF]. For example, one information can be obtained through multiple sources, which is the case in the application where different people are talking about the same thing. Another example can be an image that is described via a set of visual features such as color, shape and textures. Multi-view RL is thus concerned with the problem of the integration of information from multiple views and uncovers the latent structure shared by multiple views, while preserving the original information and the global meaning [Zhu et al., 2014;[START_REF] Xu | [END_REF]. It has been shown that Multi-view RL can facilitate extracting useful information when developing prediction models [Li et al., 2018] and also helps encoding concepts and semantics in deep neural network [START_REF] Xu | [END_REF]. Recently, Multi-view RL has been used to design an explainable recommendation system [START_REF] Gao | [END_REF], where authors claim that "it is difficult to model the relationships between high-level and low-level features since they have overlapping meaning". To overcome this issue, a Multi-view learning approach has been proposed by considering different levels of features as different views. The learned representation can then be a representation of different levels of features of the input data. Accordingly, we consider that Multi-view RL can be employed for explainability tasks.

Network RL

Network RL is a learning paradigm proposed to analyze networks such as graphs, and thus allows users to deeply understand the hidden features of graphs [Sun et al., 2020]. This domain aims at learning in a low-dimensional space of network vertices (definition 2.8), while preserving the structure of the network topology, the content of the vertices and other information as vertices attributes and links attributes. Network RL can be considered as a dimensionality reduction technique and an intermediate step to solve a target task [START_REF] Zhang | [END_REF]. Since the information of the original network is preserved in a new vector-based representation, conventional vector-based machine learning algorithms can be applied. Thus, Network analysis and mining tasks become easier as there is no more need to use complex algorithms directly designed for graphs.

Consequently, Network RL has multiple applications such as: vertex classification, link prediction, clustering, visualization and recommendations [Dong et al., 2020;[START_REF] Zhang | [END_REF]. Network RL approaches have been widely applied to information networks [Sun et al., 2020;[START_REF] Zhang | [END_REF] and are becoming increasingly popular for capturing complex relationships in various real-world applications [START_REF][END_REF]Sun et al., 2020;[START_REF] Zhang | [END_REF], such as social networks, citation networks, telecommunication networks, biological networks, recommender systems, etc.

In addition, Network RL is essential in the study of heterogeneous information networks (i.e. where vertices are of different types), in order to capture semantic proximity between vertices representations [Dong et al., 2020]. Given the high scale of some networks that can range from hundred to billions of vertices and the heterogeneity of information, we believe that Network RL and XAI should be considered together in order to perform efficient and explainable analytical tasks. Also, in related applications, an in depth analysis using XAI techniques and Network RL can help interpreting empirical results and providing a deep understanding of the applied black-box model. To conclude, Network RL should be considered as a dimensionality reduction technique whenever graph-data structure is involved in the design of XAI.

Discussion: relation between RL and XAI

We have presented several research works in RL (Manifold RL, State RL, Multi-view RL and Network RL) and we next highlight common points between RL and XAI.

First, let us discuss the contribution of the hierarchical RL on XAI modeling. Recall that while RL focuses on learning a data representation in order to get a better performance of the AI model [START_REF] Bengio | [END_REF], XAI is interested in exploring this representation to explain the performance and behavior of the model. This representation varies according to the techniques used in the involved AI models (e.g. an artificial agent or a neural network). In the case of deep neural networks models, the hierarchical level of representations is important for XAI, as it allows to extract different types of information that can be used in several ways:

• The study of low-level representations can help to detect important features used by the deep network to make a prediction. This contributes to the explanation and understanding of the deep network by determining features involved in a particular output (i.e. a prediction). • The study of high-level representations can help to detect groups of features involved in a prediction, and how and where a deep neural architecture deals with these groups. This is interesting to explain relevant hidden information and their location within the architecture. For example, a hierarchical multi-scale deep recurrent network approach has been proposed for data sequences [START_REF][END_REF]: in order to discover temporal dependencies in data, the latent hierarchical structure in the sequences has been exploited without using explicit boundary information. Accordingly, we consider that the hierarchical structure of the latent representations is an important characteristic of deep networks in order to propose a model-specific XAI modeling.

Second, we focus now on the contribution of problemsoriented and data-oriented RL approaches discussed above on the explainability of AI models.

• Recall that for high-dimensional data sets, Manifold RL allows to perform dimension reduction in the latent space while preserving the distance or similarities between data. Consequently, one of the main advantages is that visualisation of the data representation inside the latent space allows to get a better intuition and understanding of the inner mechanisms of models. • Recall that in reinforcement learning, SRL allows to explicit the agent state changes while performing a task in a given environment. This is similar to the XAI objective as it makes the behavior of an artificial agent explicit and more intelligible for a given target audience. Also, recent works have mentioned that State RL can be viewed as a mean for XAI in reinforcement learning [Heuillet et al., 2021]. Other works describe State RL as an approach for robotics and control scenarios that provides easier interpretation of the variation in the environment [Lesort et al., 2017]. Consequently, we can consider that the goals of SRL are in line with those of XAI. • Through the presentation of Multi-view RL and Network RL in section 4.4, we have shown that real-world applications of RL techniques that can be more specific to a particular data type or data organisation, are also linked to XAI. Indeed, an AI model can learn from multiple data sets of complex data representation such as networks (e.g. social network modeling, biological networks). The complexity of the learned data can also impact the behavior of the AI model. Consequently, this allows us to conclude that adopting RL approaches that take into account the type of learned data, is a way to make AI models more explicit and explainable. Figure 4 summarizes the above conclusions and questions tackled throughout the section 4. Multi-view RL Concept learning [START_REF] Xu | [END_REF] Image processing [START_REF] Su | [END_REF] Recommender systems explainability [START_REF] Gao | [END_REF] Network RL Networks of concepts [START_REF][END_REF]Qi et al., 2020] Identification of genes in biology [START_REF] Ietswaart | [END_REF] Community detection in social networks [START_REF] Tu | [END_REF] Table 2: A Summary of RL approaches, examples of application domains (NLP stands for Natural language processing).

Discussion and conclusion

We now summarize the highlighted points presented in previous sections. We also present promising directions related to the XAI domain. Since our paper is a multidisciplinary one at the crossroad of several domains, we have first (in section 2) centralized and clarified definitions of several concepts, that could indeed seem basic and well-known to involved AI experts, but are important to bridge the discussed domains. A special focus has been made on latent space, latent representation and hierarchical representation which are essential for knowledge extraction in deep neural networks and thus in XAI. To the best of our knowledge, no previous work has established a clear definition of these concepts for XAI community. This is necessary to allow the collaboration between the different domains necessary to build XAI. Second, we analysed and highlighted the existence of relations between Knowledge domains (KDP, KR), RL and XAI.

As we have shown in section 1, the goal of XAI is to convey the most semantically complete explanation to a target audience in order to answer a particular question within a given context. This explanation should take into account two important points: (i) the prior knowledge of the target audience regarding the application context, and (ii) the technical aspects of the AI used model that provided solutions to a specific task, and that thus contributed, due to its complexity/opacity, to the emergence of the question behind the need of XAI, i.e in short, "What are the reasons behind the results and/or how the AI model reaches these results?".

We consider that XAI is technically at the crossroad of at least two domains: (i) KDP and KR when viewed from a human perspective, and (ii) RL that tackles implicitly the same objectives as XAI, from a technical and algorithmic perspectives. KDP, KR, RL domains, while distinct, are overlapped. They do and should have an explicit impact on XAI approaches:

• First, as we have previously mentioned, several XAI approaches are indirectly inspired by the domain of Knowledge (KDP, KR and data mining) as both tend to express information from data. However, it is important to recall that, in XAI the input data reflects the internal mechanisms of the AI model, its predictions, and/or its behavior. The evolution of the Knowledge domain is therefore an inspiration area for XAI.

• Second, the development of AI approaches and in particular of deep learning, has blurred the boundaries between KR and RL, since several KR approaches involve RL and deep learning. In addition, recall that while RL is interested in features modeling for algorithmic issues (performance, dimensionality, etc.), XAI is interested in features since it contributes to explicit the inner mechanisms behind the results. This implies that KR, RL and XAI are indeed interested in the data representation in order to answer different but related questions. We thus consider that, in order to make a significant progress, XAI future works should not forget KR and RL past and recent works as inspirations.

KDP, KR and RL have been extensively confronted with, first, issues related to providing a data-driven explanation to different stakeholders according to their expectations and context, and second, issues related to biases and fairness in AI [Nelson, 2019]. This highlights the human significant role on data processing and bias detection in AI towards XAI. We believe that this review is all the more topical and important as works about the alliance between symbolic AI and connectionist AI should be more and more important in the next years3 , e.g. injecting a priori knowledge into neural networks to limit unethical AI [START_REF] Goebel | [END_REF] and biases [START_REF] Gordon | [END_REF]Leavy, 2018;Lepri et al., 2018;Nelson, 2019]. We are convinced that very promising directions can be taken in XAI future works by taking advantage of KDP, KR and RL development to help design ethical, unbiased and human-centered XAI. To conclude, we point out that other domains, not discussed in this paper, also impact XAI directions such as cognitive psychology [START_REF] Saux | [END_REF], cognitive sciences for biases studies [START_REF] Soleimani | [END_REF], social sciences [Miller, 2019] and Human Machine Interaction field [START_REF] Saux | [END_REF]Mueller et al., 2021;[START_REF] Ehsan | [END_REF].

Acknowledgments

Thanks to the Conseil régional de Bretagne and the European Union for funding this work via the FEDER program.

Figure 1 :

 1 Figure 1: A schematic representation of XAI and its positioning at the crossroads of other domains.

Figure 2 :

 2 Figure 2: Two schematic ways for data transformation into knowledge: On the left, within a Knowledge Discovery process, and on the right within a XAI process.

Figure 3 :

 3 Figure 3: Illustrative and schematic representation of the position of a low level representation and a high level representation in a deep neural network. hx refers to the x th hidden layer in the network.

Figure 4 :

 4 Figure4: Questions addressed throughout the paper in section 4. Associated reference to each example: (i)[START_REF] Madumal | [END_REF], (ii)[Torki et al., 2010], (iii)[START_REF] Gao | [END_REF], (iv)[Qi et al., 2020]

Table 1 :

 1 Acronyms of research domains discussed in this paper.

	Acronym Research domain
	XAI	eXplainable Artificial Intelligence
	KDP	Knowledge Discovery process
	KR	Knowledge Representation
	RL	Representation Learning
	SRL	State Representation Learning

  Table 2 summarizes RL domains and some examples of application domains.

The alliance of symbolic AI and connectionist approaches have been proposed a long time ago, e.g.[Honavar, 1995].


Modeste Assemien

Alain Le Bot

Vibration level induced by the friction of two rough surfaces weakly loaded

This paper presents an experiment to measure the vibration level generated when a light solid slides over a dry and rough surface. The experimental set-up is based on linear constant motion with speed range from 1 to 1000 mm/sec. The aim is to study the evolution of the vibration level as a function of the sliding speed and the friction area. Measurements show that the vibration level is a logarithmic increasing function of the sliding speed with a transient velocity. It is also observed that two regimes exist for the evolution of vibration level versus apparent contact area. On the one hand the vibration level is proportional to the friction area, but on the other hand this level is constant.

INTRODUCTION

The noise produced by the friction of two rough and dry surfaces weakly loaded, also called roughness noise, is a wideband noise with generally a small sound level, it is characterized by light contact pressure and percussion at the top of antagonist asperities. The micro-impact behaves like small hammer strokes, excite the structure which produces vibration responsible of emitted noise [START_REF] Bot | Noise of sliding rough contact[END_REF]. The studies of the phenomenology of roughness noise have previously shown the dependence between sound pressure level or vibrational level, surface roughness [START_REF] Nakai | A Fundamental Study on Friction Noise[END_REF][START_REF] Othman | Experimental investigation of friction noise and surface-roughness characteristics[END_REF], sliding speed [START_REF] Ben Abdelounis | An experimental study on roughness noise of dry rough flat surfaces[END_REF] and contact area [START_REF] Bot | Measurement of Friction Noise Versus Contact Area of Rough Surfaces Weakly Loaded[END_REF].

This paper reports an experimental study on vibrational level versus sliding speed and contact area for flat-flat dry contact. The aim is to establish, on one hand, the scaling laws between vibrational level Lv and sliding speed and the other hand between Lv and contact area.

The paper is organized as follows. In section 2, we present the principle, the experimental device and the procedure of the experiment. Results and discussion are presented in Section 3 while conclusions are given in Section 4.

MEASUREMENT OF VIBRATION LEVEL

Principle and experimental device

The experiment consists in rubbing the rough face of light metallic block, called slider, on the rough surface of a suspended thin plate with a very low internal dissipation (Figure 1). The metallic block is sliding at constant speed and the vibrational velocity is recorded by a laser vibrometer. The different components of the experimental setup is shown in Figure 2. The sliding motion of the slider is insured by a small brushless motor (type AKM31C class F). The thin plate, also called resonator, is attached to an aluminum frame structure using fishing wire. It's a rectangular plate made of stainless steel with size 600 mm long, 400 mm wide and 4 mm thick. The size of the rough zone of the plate is 460 × 20 mm. The slider is made of the same material as the resonator and of size 20 ×20 × 10 mm Table 2 outlines the mechanical characteristics of the resonator and the slider. The rough surfaces have been prepared by electrical discharge machining (EDM). The surface of the resonator and the slider are Gaussian (Table 1).

Experimental procedure

The experiment is run in ambient atmosphere (RH = 72 to 76% and T = 21 to 23 • C). The cleaning of the surfaces is one of the most important steps in order to obtain an acceptable reproductivity of the experiment. Before test, the rough area of the resonator and the sliders are cleaned. Each cleaning process is first done with heptane to remove all greases on the surfaces and followed respectively by acetone to ensure no residual traces of chemical remains on the surface. The surface is finally dried under a flux of nitrogen. After the cleaning procedure, the plate is attached to the aluminum frame structure using fishing lines. The slider is then carefully placed on the resonator with its rough surface in contact with that of the resonator. It is noted that the plate and the sliders are to be handled with gloves to avoid contamination of rough surfaces. Move the slider at constant speed in the range 1 to 1000 mm/s and record the vibrational velocity by the vibrometer. For each sliding speed, repeat the test five times.

RESULTS AND DISCUSSION

Vibration level versus sliding speed

The measurements are carried out in a bandwidth ranging from 10 Hz to 10 kHz at sampling frequency 51200 Hz. The vibration level is calculated with the following relationship:

L v (dB) = 20 log 10 V RMS V 0 (1)
where

V RMS = 1 T 2 -T 1 T 2 T 1 v 2 (
t)dt is the RMS velocity of the plate, V 0 = 10 -6 mm/s, v denote the vibrational velocity.

Figure 3 presents the evolution of the vibrational level L v (dB) as a function of the logarithm of sliding speed V. L v (dB) is an increasing function of the sliding speed according to the following scaling law:

L v (dB) ∝ 20 log 10 V n (2)
where n = 0.75 for V < 120 mm/s and n 2 = 0.1 for V > 120 mm/s. The evolution vibrational level versus sliding speed shows two regimes with transient speed V=120 mm/s. The creeping regime appears up to 120 mm/s, in this regime the contact rate is high. From 120 mm/s the contact is rare and the slider is in free flight. The slopes of the linear increasing function of the vibrational level versus the logarithm of the sliding speed of the two regimes are n 1 = 0.75 and n 2 = 0.1. In literature, the exponent of the scaling law (Equation 2) for flat-flat rough contact are respectively : 0.7 ≤ n ≤ 0.96 [START_REF] Ben Abdelounis | An experimental study on roughness noise of dry rough flat surfaces[END_REF][START_REF] Ben Abdelounis | Numerical simulation of friction noise[END_REF], 0.7 ≤ n ≤ 0.85 [START_REF] Stoimenov | The roughness effect on the frequency of frictional sound[END_REF], 0.71 ≤ n ≤ 0.79 [START_REF] Dang | Dynamique des interfaces multicontact[END_REF][START_REF] Dang | Direct numerical simulation of the dynamics of sliding rough surfaces[END_REF] and 0.76 ≤ n ≤ 0.77 [START_REF] Assemien | INFLUENCE OF SLIDING SPEED ON ROUGHNESS NOISE[END_REF]. The results of our study are in agreement with previous investigation but also highlight a transient speed with another exponent.

Vibration level versus contact area

The vibrational RMS velocity is measured when rubbing from 1 to 24 sliders (S = 4 -96 cm 2 ) for a constant sliding speed V=100 mm/s. Figure 4 shows the evolution of the vibrational level versus contact area. The vibrational level increase from 127,7 to 129,76 dB for 1 up to 12 sliders and remains constant from 12 sliders.

The effect of contact area on vibration level shows the existence of two regimes. In the first regime, the vibrational level Lv is proportional to the contact area while in the second one, it is constant. Concerning the relationship of Lv(dB) with contact area, Le Bot et al. [START_REF] Bot | Measurement of Friction Noise Versus Contact Area of Rough Surfaces Weakly Loaded[END_REF][START_REF] Bot | Dissipation of vibration in rough contact[END_REF] found that the dissipation in rough contact depend of the energy within the contact and also the vibration of the solids in contact. The internal damping of the solid dominates when the vibrational level increases with the number of sliders and when the vibrational remains constant, the contact damping dominates [START_REF] Bot | Dissipation of vibration in rough contact[END_REF][START_REF] Bot | Foundation of Statistical Energy Analysis in Vibroacoustics[END_REF]. The present study is in agreement with the previous observations. The vibrational level induced by the rubbing of two rough surfaces is a very complex phenomenon.

CONCLUSIONS

This paper focus on study of vibrational level versus sliding speed and contact area, it has been shown experimentally that the vibrational level is a linear increasing function of the logarithm of sliding speed with transient speed V = 120 mm/s and two slopes n 1 = 0.75 and n 2 = 0.1. In term of contact area, two regimes exist for the dependence of vibrational level Lv with contact area S . In the first regime, Lv is proportional to S while in the second one, Lv is constant.

Figure 1 :

 1 Figure 1: Principle of the experiment.

Figure 2 :

 2 Figure 2: Experimental device.

Figure 3 :

 3 Figure 3: Evolution of vibration level Lv versus sliding speed V.

Figure 4 :

 4 Figure 4: Vibrational level Lv(dB) versus contact area S (cm 2 ) for a slider 20 × 20 × 10 mm at constant V = 100 mm.

Table 1 :

 1 Characteristics of the slider and resonator

	Material properties	Slider	Resonator
	Size (mm)	20 × 20 × 10 600 × 400 × 4
	Young's modulus, E (GPa)	210	210
	Poisson's ratio, ν	0.3	0.3
	Density, ρ (kg/m 3 )	7800	7800
	Mass, (g)	31.2	7488

Table 2 :

 2 Topography parameters obtained by profilometer

	Parameters Resonator	slider
	Rq (µm)	17,10 ± 0.5 16,62 ± 0.38
	Rsk (µm)	0,37 ± 0.04 0,36 ± 0.11
	Rku (µm)	2,70 ± 0.25 2,96 ± 0.18

nguetta.assemien@ec-lyon.fr

alain.le-bot@ec-lyon.fr

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the French National Centre for Scientific Research.