Charles Alexandre
email: char.alexandre@gmail.com

Martin Bordemann
email: martin.bordemann@uha.fr

Salim Riviere
email: salim.riviere@univ-angers.fr

Friedrich Wagemann
email: wagemann@math.univ-nantes.fr

Salim Rivière

Algebraic deformation quantization of Leibniz algebras

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Introduction The algebraic structures involved in Leibniz deformation quantization

Recall that a pointed rack (see [START_REF] Fenn | Racks and links in codimension 2[END_REF]) is a pointed set (X, e) together with a binary operation ⊲ : X ×X → X such that for all x ∈ X, the map y → x⊲y is bijective and such that for all x, y, z ∈ X, the self-distributivity and unit relations x ⊲ (y ⊲ z) = (x ⊲ y) ⊲ (x ⊲ z), e ⊲ x = x, and x ⊲ e = e are satisfied. Observe that racks are not algebras over an operad, but the correct algebraic structure is that of a properad. Therefore the standard deformation theory of algebras over an operad does not apply. Imitating the notion of a Lie group, the smooth version of a pointed rack is called a Lie rack.

An important class of examples of racks are the so-called augmented racks, see [START_REF] Fenn | Racks and links in codimension 2[END_REF]. An augmented rack is the data of a group G, a G-set X and a map p : X → G such that for all x ∈ X and all g ∈ G, p(g • x) = gp(x)g -1 .

The set X becomes then a rack by setting x ⊲ y := p(x) • y. In fact, augmented racks are the Drinfeld center (or the Yetter-Drinfeld modules) in the monoidal category of G-sets over the (set-theoretical) Hopf algebra G, see for example [START_REF] Krähmer | Racks, Leibniz algebras and Yetter-Drinfel'd modules[END_REF]. Any rack may be augmented in many ways, for example by using the canonical morphism to its associated group (see [START_REF] Fenn | Racks and links in codimension 2[END_REF]) or to its group of bijections or to its group of automorphisms.

In order to formalize the notion of a rack, one needs the diagonal map diag X : X → X × X given by x → (x, x). Then the self-distributivity relation reads in terms of maps

m • (id M × m) = m • (m × m) • (id M × τ M,M × id M ) • (diag M × id M × id M ).
Axiomatizing this kind of structure, one may start with a coalgebra C and look for rack operations on this fixed coalgebra, see [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF] and [START_REF] Lebed | Objets tressés : une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles doctoral thesis Paris[END_REF]. A natural framework where this kind of structure arises is by taking point-distributions over (resp. to) the pointed manifold given by a Lie rack, see [START_REF] Serre | Lie algebras and Lie groups[END_REF], [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF], [START_REF] Mostovoy | Formal multiplications, bialgebras of distributions and nonassociative Lie theory[END_REF] or [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF]. We dub the arising structure a rack bialgebra, see Definition 1.1. We carry out some structure theory for rack bialgebras based on semigroup theory in the article [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF].

Lie racks are intimately related to Leibniz algebras h, i.e. a vector space h with a bilinear bracket [, ] : h ⊗ h → h such that for all X, Y, Z ∈ h, [X, -] acts as a derivation:

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]]. (1)
Indeed, Kinyon showed in [START_REF] Kinyon | Leibniz algebras, Lie racks, and digroups[END_REF] that the tangent space at e ∈ H of a Lie rack H carries a natural structure of a Leibniz algebra, generalizing the relation between a Lie group and its tangent Lie algebra. Conversely, every (finite dimensional real or complex) Leibniz algebra h may be integrated into a Lie rack R h (with underlying manifold h) using the rack product

X ◮ Y := e adX (Y ), (2)
noting that the exponential of the inner derivation ad X for each X ∈ h is an automorphism.

Leibniz deformation quantization

Given a finite-dimensional real Lie algebra (g, [, ]), its dual vector space g * is a smooth manifold which carries a Poisson bracket on its space of smooth functions, defined for all f, g ∈ C ∞ (g * ) and all ξ ∈ g * by the Kostant-Kirillov-Souriau formula {f, g}(ξ) := ξ, [df (ξ), dg(ξ)] .

Here df (ξ) and dg(ξ) are linear functionals on g * , identified with elements of g.

In the same way, a general finite dimensional Leibniz algebra h gives rise to a smooth manifold h * , which carries now some kind of generalized Poisson bracket, namely {f, g}(ξ) := -ξ, [df (0), dg(ξ)] ,

see [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] for an explanation why we believe that this is the correct bracket to consider. In particular, this generalized Poisson bracket need not be skewsymmetric.

The quantization procedure of this generalized Poisson bracket proposed in [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] works as follows: The cotangent lift of the above rack product

X ◮ Y = e adX (Y )
is interpreted as a symplectic micromorphism. The generating function of this micromorphism serves then as a phase function in a Fourier integral operator, whose asymptotic expansion gives rise to a star-product.

One main goal of the present article is to set up a purely algebraic framework in which one may deformation quantize the dual of a Leibniz algebra. The main feature will be to recover -in a rather explicit algebraic manner-the star-product which has been constructed in [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] by analytic methods, see Corollary 2.5. The explicit formula reads:

Let f, g ∈ C ∞ (h * ). (f ⊲ g)(α) = ∞ r=0 r r! n i1,...,ir=1 ∂ r f ∂α i1 . . . ∂α ir (0) ( ad i1 • . . . • ad ir )(g) (α), (3)
where we have chosen a basis in the finite dimensional Leibniz algebra h and the first order differential operators ad i1 are defined by

( ad i (f ))(α) = n j,k=1 α k c k ij ∂f ∂α j (α),
where c k ij are the structure constants of h w.r.t. this basis. Our first main result is thus Corollary 2.5 where we show that the above ⊲ is indeed a rack product and that it is equal (up to a sign) to the one constructed in [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] by analytic methods.

In the remaining part of the paper, we answer the natural question in which cohomology theory the first term of the above deformation quantization appears as a 2-cocycle. For this, we set up a general deformation theory framework in which the above star-product appears as a formal deformation, its infinitesimal term defining a 2-cocycle and thus a second cohomology class. The main result is here Theorem 2.12 where we show that a natural differential d on the adjoint rack bialgebra complex satisfies indeed d 2 = 0. This is a combinatorially involved computation. Observe that this cohomology theory is thus well-defined in all degrees, in contrast to the related cohomology theory in [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF] for which only degrees up to 3 exist for the moment. One main point in this part of the paper is that our deformation complex for rack bialgebras replaces the (non existing) rack cohomology complex with adjoint coefficients (cf the case of group cohomology for a group G where cohomology with adjoint coefficients in G does not exist, while the Hochschild cohomology of the group algebra K[G] with values in K[G] may play this role).

Acknowledgements: F.W. thanks Université de Haute Alsace for an invitation during which the shape of this research project was defined. Some part of the results of this paper constitute the master thesis of C.A.. We all thank the referee for useful remarks leading to an improvement of the manuscript.

Preliminaries

Rack bialgebras

In the following, let K be an associative commutative unital ring containing all the rational numbers. In the main part of this paper, we will assume K = R or C. The symbol ⊗ will always denote the tensor product of K-modules over K. For any coalgebra (C, ∆) over K, we shall use Sweedler's notation ∆(a) = (a) a (1) ⊗ a (2) for any a ∈ A. We will feel free to suppress the sum-sign in Sweedler's notation in complicated formulas for typographical reasons.

The following sections will all deal with the following type of nonassociative bialgebra: Let (B, ∆, ǫ, 1, µ) be a K-module such that (B, ∆, ǫ, 1) is a coassociative counital coaugmented coalgebra (a C 3 -coalgebra), and such that the linear map µ : B ⊗ B → B (the multiplication) is a morphism of C 3 -coalgebras (it satisfies in particular µ(1 ⊗ 1) = 1). We shall call this situation a nonassociative C 3 I-bialgebra (where I stands for 1 being an idempotent for the multiplication µ). For another nonassociative

C 3 I-bialgebra (B ′ , ∆ ′ , ǫ ′ , 1 ′ , µ ′ ) a K-linear map φ : B → B ′ will be called a morphism of nonassociative C 3 I-bialgebras iff it is a morphism of C 3 -coalgebras and is multiplicative in the usual sense φ µ(a ⊗ b) = µ ′ φ(a) ⊗ φ(b) for all a, b ∈ B. Definition 1.1. A rack bialgebra (B, ∆, ǫ, 1, µ) is a nonassociative C 3 I- bialgebra (where we write for all a, b ∈ B µ(a ⊗ b) =: a ⊲ b) such that the following identities hold for all a, b, c ∈ B 1 ⊲ a = a, (4) a
⊲ 1 = ǫ(a)1, (5)
a ⊲ (b ⊲ c) = (a) (a (1) ⊲ b) ⊲ (a (2) ⊲ c). (6)
The last condition ( 6) is called the self-distributivity condition.

Note that we do not demand that the C 3 -coalgebra B should be cocommutative nor connected. Similar definitions have been proposed in [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF] and in [START_REF] Lebed | Objets tressés : une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles doctoral thesis Paris[END_REF].

i.e. f is µ-equivariant. It is a routine check that (B, ∆, ǫ, 1, µ f ) is a rack bialgebra where for all a, b ∈ B the multiplication is defined by

µ f (a ⊗ b) := a ⊲ f b := f (a) ⊲ b. (9)
We shall call (B, ∆, ǫ, 1, µ f ) the f -gauge of (B, ∆, ǫ, 1, µ).

Example 1.2.

Let (H, ∆ H , ǫ H , µ H , 1 H , S) be a cocommutative Hopf algebra over K. Then it is easy to see (cf. also the particular case B = H and Φ = id H of Proposition 1.4 for a detailed proof) that the new multiplication µ :

H ⊗ H → H, written µ(h ⊗ h ′ ) = h ⊲ h ′ , defined by the usual adjoint representation h ⊲ h ′ := ad h (h ′ ) := (h) h (1) h ′ S(h (2) ) , (10)
equips the C 4 -coalgebra (H, ∆ H , ǫ H , 1 H ) with a rack bialgebra structure. ♦

In general, the adjoint representation does not seem to preserve the coalgebra structure if no cocommutativity is assumed.

Example 1.3. Recall that a pointed set (X, e) is a pointed rack in case there is a binary operation ⊲ : X × X → X such that for all x ∈ X, the map y → x ⊲ y is bijective and such that for all x, y, z ∈ X, the self-distributivity and unit relations

x ⊲ (y ⊲ z) = (x ⊲ y) ⊲ (x ⊲ z), e ⊲ x = x, and x ⊲ e = e are satisfied. Then there is a natural rack bialgebra structure on the vector space K[X] which has the elements of X as a basis. K[X] carries the usual coalgebra structure such that all x ∈ X are set-like: △(x) = x ⊗ x for all x ∈ X. The product µ is then induced by the rack product. By functoriality, µ is compatible with △ and e.

Observe that this construction differs slightly from the construction in [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF], Section 3.1.

♦ Remark 1.2. It is shown in Theorem 4.3 of [4] that for a C 4 -coalgebra B with a self-distributive map ⊳ = q : B ⊗ B → B which is a morphism of coalgebras, the map R q = (id B ⊗ q) • (τ ⊗ id B ) • (id B ⊗ ∆)
is a solution of the Yang-Baxter equation. We draw our reader's attention to the fact that Carter-Crans-Elhamdadi-Saito work in [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF] with right racks, while we work here with left racks. The statement of their theorem works also for left racks, but then one has to take

Rq = (id B ⊗ q) • (∆ ⊗ id B ) • τ.
In particular for any rack bialgebra, Rq is a solution of the Yang-Baxter equation.

Example 1.4.

Here we suppose K = R or C. Another general construction mechanism for rack bialgebras is exhibited in [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF]. Namely, let (M, e, ⊲) be a Lie rack. In particular, (M, e) is a pointed manifold and it makes sense to associate to it the vector space E ′ e (M ) of distributions on M which have their support in {e}. There is a corresponding functor F : Mf * → KVect from the category of pointed manifolds Mf * with values in the category of K-vector spaces KVect. Observe that J.-P. Serre [START_REF] Serre | Lie algebras and Lie groups[END_REF] used it to define the universal enveloping algebra of a Lie group, see also [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF], [START_REF] Mostovoy | Formal multiplications, bialgebras of distributions and nonassociative Lie theory[END_REF] or [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF].

In fact, this framework is a special case of a local multiplication where 0×0 = 0. Such a multiplication gives rise to a bialgebra of distributions supported at 0. If the multiplication satisfies a certain identity, the bialgebra satisfies the linearized identity. In our case, the rack identity (i.e. self-distributivity) leads to the linearized self-distributivity relation, i.e. condition [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF]. The general framework is described in [START_REF] Mostovoy | Formal multiplications, bialgebras of distributions and nonassociative Lie theory[END_REF] Section 3, see also [START_REF] Pérez-Izquierdo | An envelope for Malcev algebras[END_REF].

In our case, the pointed manifold given by a Lie rack yields a rack bialgebra, see [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF] where this functor is studied in detail. ♦

We will also need the following structure:

Definition 1.3. An augmented rack bialgebra over K is a quadruple (B, Φ, H, ℓ) consisting of a C 3 -coalgebra (B, ∆, ǫ, 1), of a cocommutative (!) Hopf algebra (H, ∆ H , ǫ H , 1 H , µ H , S), of a morphism of C 3 -coalgebras Φ : B → H, and of a left action ℓ : H ⊗ B → B of H on B which is a morphism of C 3 -coalgebras (i.e. B is a H-module-coalgebra) such that for all h ∈ H and a ∈ B h.1 = ǫ H (h)1 (11) Φ(h.a) = ad h Φ(a) . ( 12
)
where ad denotes the usual adjoint representation for Hopf algebras, see e.g. eqn [START_REF] Kinyon | Leibniz algebras, Lie racks, and digroups[END_REF].

We shall define a morphism (B, Φ, H, ℓ) → (B ′ , Φ ′ , H ′ , ℓ ′ ) of augmented rack bialgebras to be a pair (φ, ψ) of K-linear maps where φ : (B, ∆, ǫ, 1)

→ (B ′ , ∆ ′ , ǫ ′ , 1 ′
) is a morphism of C 3 -coalgebras, and ψ : H → H ′ is a morphism of Hopf algebras such that the obvious diagrams commute:

Φ ′ • φ = ψ • Φ, and ℓ ′ • (ψ ⊗ φ) = φ • ℓ (13)
An immediate consequence of this definition is the following

for all a, b ∈ B. In particular, each Hopf algebra H becomes an augmented rack bialgebra via (H, id H , H, ad). In general, for each augmented rack bialgebra the map Φ : B → H is a morphism of rack bialgebras.

Proof. We check first that ⊲ is a morphism of C 3 -coalgebras B ⊗ B → B: Let a, b ∈ B, then -thanks to the fact that the action ℓ and the maps Φ are coalgebra morphisms-

∆ µ(a ⊗ b) = ∆(a ⊲ b) = ∆ Φ(a).b = (Φ(a)), (b)
Φ(a) (1) .b (1) ⊗ Φ(a) (2) .b (2) = (a),(b)

Φ(a (1) ) .b (1) ⊗ Φ(a (2) ) .b (2) = (a),(b)

a (1) ⊲ b (1) ⊗ a (2) ⊲ b (2)
whence µ is a morphism of coalgebras. Clearly

ǫ(a ⊲ b) = ǫ Φ(a).b = ǫ H Φ(a) ǫ(b) = ǫ(a)ǫ(b)
whence µ preserves counits. We shall next compute both sides of the self-distributivity identity [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] to get an idea: For all a, b, c

∈ B a ⊲ (b ⊲ c) = Φ(a). Φ(b).c = Φ(a)Φ(b) .c,

and

(a)

(a (1) ⊲ b) ⊲ (a (2) ⊲ c) = (a)
Φ(a (1) ).b ⊲ Φ(a (2) ).c

= (a)
Φ Φ(a (1) ).b . Φ(a (2) ).c

= (a)
Φ Φ(a (1) ).b Φ(a (2) ) .c, and we compute, using the fact that Φ is a morphism of

C 3 -coalgebras, (a)
Φ Φ(a (1) ).b Φ(a

(2) ) = (a)
Φ (Φ(a) (1) ).b (Φ(a) (2) )

(12) = (a) ad Φ(a) (1) Φ(b) (Φ(a) (2) ) = (a) Φ(a) (1) Φ(b) S Φ(a) (2) (Φ(a) (3) ) = (a) Φ(a) (1) Φ(b)1 H ǫ H Φ(a) (2)
= Φ(a)Φ(b), which proves the self-distributivity identity. Moreover we have

1 B ⊲ a = Φ(1).a = 1 H .a = a,
and a ⊲ 1 = Φ(a).1

(11) = ǫ H Φ(a) 1 = ǫ B (a)1,
whence the C 3 -coalgebra becomes a rack bialgebra.

Example 1.5.

Exactly in the same way as a pointed rack gives rise to a rack bialgebra K[X], an augmented pointed rack p : X → G gives rise to an augmented rack bialgebra

p : K[X] → K[G]. ♦ Remark 1.5.
Motivated by the fact that the augmented racks p : X → G are exactly the Yetter-Drinfeld modules over the (set-theoretical) Hopf algebra G, we may ask whether augmented rack bialgebras are Yetter-Drinfeld modules.

In fact, any cocommutative augmented rack bialgebra (B, Φ, H, ℓ) gives rise to a Yetter-Drinfeld module over the Hopf algebra H. Indeed, B is a left Hmodule via ℓ, and becomes a left H-comodule via

ρ : B △B → B ⊗ B Φ⊗idB → H ⊗ B.
Now, in Sweedler notation, the coaction is denoted for all b ∈ B by

ρ(b) = (b) b (-1) ⊗ b (0) ∈ H ⊗ B.
Then the Yetter-Drinfeld compatibility relation reads

(h.b) (h.b) (-1) ⊗ (h.b) (0) = (b),(h) h (1) b (-1) S(h (3) ) ⊗ h (2) .b (0) .
This relation is true in our case, because ℓ is a morphism of coalgebras and is sent to the adjoint action via Φ.

Conversely, given a Yetter-Drinfeld module C over a Hopf algebra H, together with a linear form ǫ C :

C → K satisfying ǫ C (h.c) = ǫ H (h)ǫ C (c), then define a map Φ : C → H by Φ := (id H ⊗ ǫ C ) • ρ.
The map Φ intertwines the left action on C and the adjoint action on H thanks to the Yetter-Drinfeld condition. Now define a rack product for all x, y ∈ C by

x ⊲ y = Φ(x).y, then we obtain a Yetter-Drinfeld version of self-distributivity

x ⊲ (y ⊲ z) = (x) (x (-1) .y) ⊲ (x (0) ⊲ z),
as there is no comultiplication on C.

The fact that Φ is a morphism of coalgebras is then replaced by the identity

(id H ⊗ Φ) • ρ = ∆ H • Φ,
which one needs to demand. Finally, one needs a unit 1 C ∈ C such that for all h ∈ H, h. andΦ(1 C ) = 1 H . This is somehow the closest one can get to a rack bialgebra without having a compatible C 3 coalgebra structure on C.

1 C = ǫ H (h)1 C , ǫ C (1 C ) = 1 K , ρ(1 C ) = 1 H ⊗ 1 C ,

(Augmented) rack bialgebras for any Leibniz algebra

In this subsection, we will suppose Q ⊂ K. Let (h, [ , ]) be a Leibniz algebra over K, i.e. h is a K-module equipped with a K-linear map [ , ] : h ⊗ h → h satisfying the (left) Leibniz identity [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF].

Recall first that each Lie algebra over K is a Leibniz algebra giving rise to a functor from the category of all Lie algebras to the category of all Leibniz algebras. Furthermore, recall that each Leibniz algebra has two canonical K-submodules

Q(h) := x ∈ h | ∃ N ∈ N \ {0}, ∃ λ 1 , . . . , λ N ∈ K, ∃ x 1 , . . . , x N such that x = N r=1 λ r [x r , x r ] , (15)
z(h) := x ∈ h | ∀ y ∈ h : [x, y] = 0 . ( 16
)
It is well-known and not hard to deduce from the Leibniz identity that both Q(h) and z(h) are two-sided abelian ideals of (h, [ , ]), that Q(h) ⊂ z(h), and that the quotient Leibniz algebras

h := h/Q(h) and g(h) := h/z(h) (17)
are Lie algebras. Since the ideal Q(h) is clearly mapped into the ideal Q(h ′ ) by any morphism of Leibniz algebras h → h ′ (which is a priori not the case for z(h) !), there is an obvious functor h → h from the category of all Leibniz algebras to the category of all Lie algebras. In order to perform the following constructions of rack bialgebras for any given Leibniz algebra (h, [ , ]), choose first a two-sided ideal z ⊂ h such that

Q(h) ⊂ z ⊂ z(h), (18)
let g denote the quotient Lie algebra h/z, and let p : h → g be the natural projection. The data of z ⊂ h, i.e. of a Leibniz algebra h together with an ideal z such that Q(h) ⊂ z ⊂ z(h), could be called an augmented Leibniz algebra. Thus we are actually associating an augmented rack bialgebra to every augmented Leibniz algebra. In fact, we will see that this augmented rack bialgebra does not depend on the choice of the ideal z and therefore refrain from introducing augmented Leibniz algebras in a more formal way. The Lie algebra g naturally acts as derivations on h by means of (for all x, y ∈ h) p(x).y := [x, y] =: ad x (y) [START_REF] Rivière | On the string Lie algebra[END_REF] because z ⊂ z(h). Note that

h/z(h) ∼ = ad x ∈ Hom K (h, h) | x ∈ h . (20)
as Lie algebras. Consider now the C 5 -coalgebra (B = S(h), ∆, ǫ, 1). Here S(h) is the symmetric algebra and coalgebra on the vector space h, and S(h) is actually a commutative cocommutative Hopf algebra over K with respect to the symmetric multiplication •. The linear map p : h → g induces a unique morphism of Hopf algebras Φ = S(p) :

S(h) → S(g) (21) satisfying Φ(x 1 • • • • • x k ) = p(x 1 ) • • • • • p(x k ) (22)
for any nonnegative integer k and x 1 , . . . , x k ∈ h. In other words, the association S : V → S(V ) is a functor from the category of all K-modules to the category of all commutative unital C 5 -coalgebras. Consider now the universal enveloping algebra U(g) of the Lie algebra g. Since Q ⊂ K by assumption, the Poincaré-Birkhoff-Witt Theorem (in short: PBW) holds (see e.g. [START_REF] Quillen | Rational Homotopy Theory[END_REF]Appendix]). More precisely, the symmetrisation map ω : S(g) → U(g), defined by

ω(1 S(g) ) = 1 U(g) , and ω(ξ 1 • • • • • ξ k ) = 1 k! σ∈S k ξ σ(1) • • • ξ σ(k) , (23)
see e.g. [7, p.80, eqn (3)], is an isomorphism of C 5 -coalgebras (in general not of associative algebras). We now need an action of the Hopf algebra H = U(g) on B, and an intertwining map Φ : B → U(g). In order to get this, we first look at g-modules: The K-module h is a g-module by means of eqn [START_REF] Rivière | On the string Lie algebra[END_REF], the Lie algebra g is a g-module via its adjoint representation, and the linear map p : h → g is a morphism of g-modules since p is a morphism of Leibniz algebras. Now S(h) and S(g) are g-modules in the usual way, i.e. for all k ∈ N \ {0}, ξ, ξ 1 , . . . , ξ k ∈ g, and

x 1 . . . , x k ∈ h ξ.(x 1 • • • • • x k ) := k r=1 x 1 • • • • • (ξ.x r ) • • • • • x k , (24) ξ
.(ξ 1 • • • • • ξ k ) := k r=1 ξ 1 • • • • • [ξ.ξ r ] • • • • • ξ k , (25)
and of course ξ.1 S(h) = 0 and ξ.1 S(g) = 0. Recall that U(g) is a g-module via the adjoint representation ad ξ (u) = ξ.u = ξu -uξ (for all ξ ∈ g and all u ∈ U(g)).

It is easy to see that the map Φ ( 22) is a morphism of g-modules, and it is wellknown that the symmetrization map ω (23) is also a morphism of g-modules, see e.g. [7, p.82, Prop. 2.4.10]. Define the K-linear map Φ : S(h) → U(g) by the composition

Φ := ω • Φ. ( 26
)
Then Φ is a map of C 5 -coalgebras and a map of g-modules. Thanks to the universal property of the universal enveloping algebra, it follows that S(h) and U(g) are left U(g)-modules, via (for all ξ 1 , . . . , ξ k ∈ g, and for all a ∈ S(h))

(ξ 1 • • • ξ k ).a = ξ 1 .(ξ 2 .(• • • ξ k .a) • • • ) (27)
and the usual adjoint representation [START_REF] Kinyon | Leibniz algebras, Lie racks, and digroups[END_REF] (for all u ∈ U(g))

ad ξ1•••ξ k (u) = ad ξ1 • • • • • ad ξ k (u), (28)
and that Φ intertwines the U(g)-action on C = S(h) with the adjoint action of U(g) on itself.

Finally it is a routine check using the above identities (24) and (10) that S(h) becomes a module coalgebra. We can resume the preceding considerations in the following Theorem 1.6. Let (h, [ , ]) be a Leibniz algebra over K, let z be a two-sided ideal of h such that Q(h) ⊂ z ⊂ z(h), let g denote the quotient Lie algebra h/z by g, and let p : h → g be the canonical projection.

1. Then there is a canonical U(g)-action ℓ on the C 5 -coalgebra B := S(h) (making it into a module coalgebra leaving invariant 1) and a canonical lift of p to a map of C 5 -coalgebras, Φ : S(h) → U(g) such that eqn [START_REF] Lebed | Objets tressés : une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles doctoral thesis Paris[END_REF] holds.

Hence the quadruple (S(h), Φ, U(g), ℓ) is an augmented rack bialgebra whose associated Leibniz algebra is equal to (h, [ , ]) (independently of the choice of z).

The resulting rack multiplication µ of S(h) (written µ(a ⊗ b) = a ⊲ b) is also independent on the choice of z and is explicitly given as follows for all positive integers k, l and x 1 , . . . , x k , y 1 , . . . , y l ∈ h:

x 1 •• • ••x k )⊲ y 1 •• • ••y l ) = 1 k! σ∈S k ad s x σ(1) •• • ••ad s x σ(k) y 1 •• • ••y l ) (29)
where ad s x denotes the action of the Lie algebra h/z(h) (see eqn [START_REF] Serre | Lie algebras and Lie groups[END_REF]) on S(h) according to eqn (24).

2. In case z = Q(h), the construction mentioned in 1. is a functor h → UAR ∞ (h) from the category of all Leibniz algebras to the category of all augmented rack bialgebras associating to h the rack bialgebra UAR ∞ (h) := (S(h), Φ, U(g), ℓ)

and to each morphism f of Leibniz algebras the pair S(f ), U(f ) where f is the induced Lie algebra morphism.

3. For each nonnegative integer k, the above construction restricts to each subcoalgebra of order k, S(h

) (k) = ⊕ k r=0 S r (h), to define an augmented rack bialgebra (S(h) (k) , Φ (k) , U(g), ℓ| U(g)⊗S(h) (k) ) which in case z = Q(h) defines a functor h → UAR (k) (h) := UAR ∞ (h) (k)
from the category of all Leibniz algebras to the category of all augmented rack bialgebras.

Proof. 1. All the statements except the last two ones have already been proven. Note that for all x, y ∈ h we have by definition [x, y] = p(x).y = x ⊲ y, independently of the chosen ideal z. Moreover we compute

x 1 • • • • • x k ) ⊲ y 1 • • • • • y l ) = (ω • Φ)(x 1 • • • • • x k ) . y 1 • • • • • y l ) = 1 k! σ∈S k p(x σ(1) ) • • • p(x σ(k) ) . y 1 • • • • • y l ),
which gives the desired formula since for all x ∈ h and a ∈ S(h), we have p(x).a = ad s x (a).

2. Let f : h → h ′ be a morphism of Leibniz algebras, and let f : h → h ′ be the induced morphism of Lie algebras. Hence we get

p ′ • f = f • p ( 30
)
where

p ′ : h ′ → h ′ denotes the corresponding projection modulo Q(h ′ ). Let S(f ) : S(h) → S(h ′ ), S(f ) : S(h) → S(h ′ ),
U(f ) ξ 1 • • • ξ k = f (ξ 1 ) • • • f (ξ k )
for all positive integers k and ξ 1 , . . . , ξ k ∈ h. If ω : S(h) → U(h) and ω ′ : S(h ′ ) → U(h ′ ) denote the corresponding symmetrisation maps (23) then it is easy to see from the definitions that

ω ′ • S(f ) = U(f ) • ω. Equation (30) implies Φ′ • S(f ) = S(p ′ ) • S(f ) = S(f ) • S(p) = S(f ) • Φ,
and composing from the left with ω ′ yields the equation

Φ ′ • S(f ) = U(f ) • Φ. ( 31
)
Moreover for all x, y ∈ h we have, since f is a morphism of Leibniz algebras,

f p(x).y = f [x, y] = f (x), f (y) ′ = p ′ f (x) .f (y) = f p(x) .f (y),
hence for all ξ ∈ h f (ξ.y) = f (ξ) . f (y) , and upon using eqn (24) we get for all a ∈ S(h)

S(f ) ξ.a = f (ξ) . S(f ) a ,
showing finally for all u ∈ U(h) and all a ∈ S(h)

S(f ) u.a = U(f )(u) . S(f ) a . ( 32
)
Associating to every Leibniz algebra (h, [ , ]) the above defined augmented rack bialgebra (S(h), Φ, U(h), ℓ), and to every morphism ψ : h → h ′ of Leibniz algebras the pair of K-linear maps Ψ = S(ψ), Ψ = U(ψ) , we can easily check that Ψ is a morphism of C 5 -coalgebras, Ψ is a morphism of Hopf algebras, such that the two relevant diagrams (13) commute which easily follows from (31) and (32). The rest of the functorial properties is a routine check.

3. By definition, the U(g)-action on S(h) (cf. eqs (24) and ( 27)) leaves invariant each K-submodule S r (h) for each nonnegative integer r whence it leaves invariant each subcoalgebra of order k, S(h) (k) . It follows that the construction restricts well.

Definition 1.7. The rack bialgebra UAR ∞ (h) which is by the above theorem canonically associated to each Leibniz algebra h is called the universal augmented rack bialgebra.

Remark 1.8. This theorem should be compared to Proposition 3.5 in [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF]. In [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF], the authors work with the vector space N := K ⊕ h, while we work with the whole symmetric algebra on the Leibniz algebra. In some sense, we extend their Proposition 3.5 "to all orders", hence the name UAR ∞ (h). The subset N = K ⊕ h becomes a sub rack bialgebra denoted by UAR(h). It turns out that UAR(h) is already enough to obtain a left-adjoint to the functor of primitives and hence universality, see [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF].

The above rack bialgebra UAR ∞ (h) associated to a Leibniz algebra h can be seen as one version of an enveloping algebra of the Leibniz algebra h. The link to the universal enveloping algebra of h, h ⊗ U(h), as a dialgebra (in the sense of of Loday-Pirashvili) has been elucidated in [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF].

We shall close the subsection with a geometric explanation of some of the structures appearing here: Let h, [ , ] be a real finite-dimensional Leibniz algebra. Then for any real number , there is the following Lie rack structure on the manifold h defined by x ◮ y := e adx (y) (33)

For later use we note that on the space h[[ ]] of all formal power series the above formula makes sense if x, y are also formal power series. Moreover, pick a two-sided ideal z ⊂ h with Q(h) ⊂ z ⊂ z(h) so that the quotient algebra g := h/z is a Lie algebra. Let p : h → g be the canonical projection. Let G be the connected simply connnected Lie group having Lie algebra g. Since g acts on h as derivations, there is a unique Lie group action ℓ of G on h by automorphisms of Leibniz algebras. Consider the smooth map

φ : h → G : x → exp p(x) . (34)
Clearly φ(g.x) = gφ(x)g -1 for all x ∈ h and g ∈ G whence (h, φ, G, ℓ) is an augmented Lie rack, and it is not hard to see that the Lie rack structure coincides with (33) for = 1. The following theorem is shown in [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF]:

Theorem 1.9. The C 5 -rack bialgebra associated to the augmented Lie rack (h, φ, G, ℓ) by means of the functor F , described in Example 1.4, is isomorphic to the universal envelopping algebra of infinite order, UAR ∞ (h) (Definition 1.7).

Quantum racks

In the article [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF], the authors are interested in the generalized Poisson manifold given by the linear dual h * of a finite-dimensional Leibniz algebra h. Recall that the dual g * of a Lie algebra g is a Poisson manifold with the Kostant-Kirillov-Souriau bracket, given for f, g ∈ C ∞ (g * ) by

{f, g}(ξ

) := ξ, [df, dg] = i,j,k c k ij ∂f ∂x i (ξ) ∂g ∂x j (ξ)X k ,
where df and dg are seen as linear functions on g * , i.e. elements of g. The c k ij are the structure constants of the Lie bracket with respect to a certain basis of (X k ). In the same manner, the dual of a Leibniz algebra h * carries a bracket given for f, g

∈ C ∞ (h * ) by {f, g}(ξ) := -ξ, [df (0), dg] = - i,j,k c k ij ∂f ∂x i (0) ∂g ∂x j (ξ)X k ,
i.e. with respect to the above formula, the bracket is here evaluated in 0 ∈ h * . The bracket is neither antisymmetric, nor a biderivation, nor does it satisfy some Jacobi/Leibniz identity in general. Nevertheless, [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] shows that this is the natural bracket one encounters when following the standard deformation quantization procedure for the dual of Lie algebras. Going into details, a finite dimensional Lie algebra g admits a local Lie group G giving rise to a (local) multiplication µ : G × G → G. The cotangent lift T * µ can be interpreted as a Lagrangian submanifold of the triple product T * G × T * G × T * G. It constitutes thus a symplectic micromorphism between germs of symplectic manifolds, see [START_REF] Cattaneo | Symplectic microgeometry II: generating functions[END_REF]. When such a symplectic micromorphism is given by a generating function S, the function S can be taken as an "action" in some oscillatory integral giving the deformation quantization of the corresponding Poisson manifold by Fourier Integral Operators. In [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF], this quantization scheme is adapted to the dual of Leibniz algebras. The key ingredient is the integration of h into a Lie rack where for x, y ∈ h, the rack product is given by

X ◮ Y = e ad X (Y ).
We take then S ⊲ (X, Y, ξ) = ξ, e adX (Y ) as a generating function, and the oscillatory integral is written

Q ⊲ (f ⊗ g)(ξ) = g×g f (X) g(Y )e i S⊲(X,Y,ξ) dXdY (2π ) n .
Here n is the dimension of h and f , g are the asymptotic Fourier transforms of f and g. The stationary phase series expansion of this integral gives then the corresponding star product. Its first term is the above generalized Poisson bracket.

The main theorem of [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] reads:

Theorem 1.10. The operation

⊲ : C ∞ (h * )[[ǫ]] ⊗ C ∞ (h * )[[ǫ]] → C ∞ (h * )[[ǫ]] defined by f ⊲ g := Q ⊲ (f ⊗ g) is a quantum rack, i.e. ( 1
) ⊲ restricted to U h = {E X := e i X | X ∈ h} is a rack structure and e i X ⊲ e i Y = e i conj * (X,Y ) , (2) ⊲ restricted to ⊲ : U h × C ∞ (h * ) → C ∞ (h *
) is a rack action and

(e i X ⊲ f )(ξ) = (Ad * -X f )(ξ).
2 Deformation quantization of rack bialgebras

Algebraic deformation quantization of Leibniz algebras

In this subsection, K denotes the field of real numbers R or the field of complex numbers C. Let (h, [ , ]) be a finite dimensional Leibniz algebra of dimension n, and denote by h * its linear dual. In order to make computations more elementary we shall use a fixed basis e 1 , . . . , e n of h, but it is a routine check that all the relevant formulas are invariant under a change of basis. Let e 1 , . . . , e n be the corresponding dual basis of h * , i.e. by definition e i (e j ) = δ i j , for all i, j = 1, . . . , n. Furthermore, let c k ij for i, j, k = 1, . . . , n be the structure constants of the Leibniz algebra h with respect to the basis e 1 , . . . , e n , i.e.

c i jk = e i ([e j , e k ])
for all i, j, k = 1, . . . , n. We will denote by x, y, z, . . . elements of h, while α, β, γ, . . . will denote elements of h * . Denote by α 1 , . . . , α n the coordinates of α ∈ h * with respect to the basis e 1 , . . . , e n . For all x ∈ h, denote by x ∈ C ∞ (h * , K) the linear function given by

x(α) := α(x),
for all α ∈ h * . In the same vein, let e x be the exponential function given by e x (α) := e α(x) = e x(α) , for all α ∈ h * . For all integers i = 1, . . . , n, define a first order differential operator ad i on smooth functions f : h * → K by

( ad i (f ))(α) := n j,k=1 α k c k ij ∂f ∂α j (α).
The following star-product formula, where is a formal parameter (which may be replaced by a real number in situations where the formula is convergent), will render h * a quantum rack in the sense of Theorem 1.10, see also [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF].

Let f, g ∈ C ∞ (h * , K). (f ⊲ g)(α) := ∞ r=0 r r! n i1,...,ir =1 ∂ r f ∂α i1 . . . ∂α ir (0) ( ad i1 • . . . • ad ir )(g) (α). ( 35
)
Theorem 2.1. For all x, y ∈ h, we have

e x ⊲ e y = e x◮ y ,
where ◮ : h × h → h is the Lie rack structure (33) defined by exponentiating the adjoint action of the Leibniz algebra:

x ◮ y = e adx (y).

Proof. The proof of the theorem relies on the rack bialgebra structure on S(h) given by Definition 1.7 and is performed in the following lemmas:

Lemma 2.2. The map "hat" : h → C ∞ (h * , K) which sends x ∈ h to the linear function x extends to an injective morphism of commutative associative unital algebras Ψ :

S(h) → C ∞ (h * , K) such that Ψ(x 1 • . . . • x k ) = x 1 . . . x k
for all integers k and all x 1 , . . . , x k ∈ h.

Proof. This follows immediately from the freeness property of the algebra S(h).

Lemma 2.3. The morphism Ψ intertwines the adjoint actions ad i and ad s ei (see eqn (29)), i.e. for all i = 1, . . . , n, we have ad i (Ψ(a)) = Ψ(ad s ei (a)) for all a ∈ S(h).

Proof. Indeed, it is enough to show this for x ∈ h ⊂ S(h) as both adjoint actions are derivations. Now we have for α ∈ h * :

Ψ(ad s ei (x))(α) = [e i , x](α) = α([e i , x]) = n j,k=1 α k e k ([e i , e j ]) x j = n j,k=1 c k ij α k ∂ x ∂α j (α) = ad i (Ψ(a))(α)
Lemma 2.4. For all b ∈ S(h) and all x 1 , . . . , x r ∈ h, we have

Ψ(x 1 • . . . • x r ) ⊲ Ψ(b) = r Ψ((x 1 • . . . • x r ) ⊲ b),
where the left-hand ⊲ is the rack multiplication in the rack bialgebra S(h).

Proof. First of all, note that by linearity it is enough to show this for x 1 , . . . , x r = e i1 , . . . , e ir with i 1 , . . . , i r ∈ {1, . . . , n}. By eqn (29), we have

(e i1 • . . . • e ir ) ⊲ b = 1 k! σ∈S k (ad s i σ(1) • . . . • ad s i σ(r) )(b).
Applying Ψ gives then

Ψ((e i1 • . . . • e ir ) ⊲ b) = 1 k! σ∈S k Ψ((ad s i σ(1) • . . . • ad s i σ(r) )(b)) = 1 k! σ∈S k ad i σ(1) • . . . • ad i σ(r) (Ψ(b)),
by the previous lemma. Now compute

∂ k Ψ(e i1 • . . . • e ir ) ∂α j1 . . . ∂α j k (0).
This expression is non zero only if k = r and {i 1 , . . . , i k } = {j 1 , . . . , j k }. In this case, the result is 1. One deduces the asserted formula.

Now we come back to the proof of the theorem. The assertion of the theorem is the equality: e x ⊲ e y = e x◮ y .

Summing up the assertion of the previous lemma (taking x 1 = . . . = x r = x), we obtain:

∞ r=0 1 r! Ψ(x • . . . • x r times ) ⊲ Ψ(b) = Ψ   ∞ r=0 r r! (x • . . . • x r times ) ⊲ b   ,
and thus (as the rack product in S(h) is given by the adjoint action, using also that Ψ is multiplicative)

e x ⊲ Ψ(b) = Ψ(e adx (b)). ( 36
)
This extends then to the asserted formula using that e adx is an automorphism of S(h) (because it is the exponential of a derivation).

Corollary 2.5. The above defined star-product induces the structure of a rack with respect to the product ⊲ on the set of exponential functions U h = {E X := e i X | X ∈ h} on h * , and this star-product is equal (up to a sign) to the starproduct in Theorem 1.10, see also [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF].

Proof. Via the formula of the theorem, the self-distributivity property of the rack product ◮ in the rack bialgebra S(h) translates into the self-distributivity property of ⊲ on the set of exponential functions. Since the star-product defined in [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] is a series of bidifferential operators, and since such a series is uniquely determined by its values on exponential functions, the present starproduct coincides with the one found in [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF] thanks to the statement of the preceding theorem.

Remark 2.6. Observe that the proof of the above theorem contains also an isomorphism of commutative associative unital algebras between S(h) and the image of Ψ, i.e. the polynomial algebra generated by the x i , i = 1, . . . , n. Up to factors r , Ψ is by construction an isomorphism of rack bialgebras. This mirrors the relation between the universal enveloping algebra U(g) and the deformation quantization of S(g * ) for a Lie algebra g, i.e. between U(g) and the Gutt star product on g * , see [START_REF] Gutt | An explicit * -product on the cotangent bundle of a Lie group[END_REF], [START_REF] Dherin | Deformation quantization of Leibniz algebras[END_REF].

Remark 2.7. We would like to thank the referee for asking the interesting question of comparing the above star-product formula with the theory of Loday-Pirashvili, see [START_REF] Loday | The tensor category of linear maps and Leibniz algebras[END_REF] and [START_REF] Mostovoy | Formal multiplications, bialgebras of distributions and nonassociative Lie theory[END_REF]: recall that they work in the so-called linear category LM whose objects are diagrams M φ → N of vector spaces and linear maps with the obvious commuting squares of linear maps as morphisms. It can be seen as the category of complexes concentrated in degree 1 (M ) and 0 (N ) with differential φ. There is the obvious tensor product of complexes (truncated in degree ≥ 2) equipping LM with the structure of a symmetric monoidal category (with 0 → K as unit object and a signless flip as symmetric braiding). In this category, associative algebra objects will be dialgebras, and Lie objects can be seen as Leibniz algebras h → h = g, see [START_REF] Loday | The tensor category of linear maps and Leibniz algebras[END_REF] and [START_REF] Mostovoy | Formal multiplications, bialgebras of distributions and nonassociative Lie theory[END_REF] for more details. The symmetric algebra generated by h → h will be h⊗S(g) → S(g), and the universal enveloping algebra of h → h will be h ⊗ U(g) → U(g) where the linear maps are given by x ⊗ g → p(x) • g and x ⊗ u → p(x)u, respectively. For a possible rack-like star-product formula motivated by this category it seems to us not unreasonable to base it on the 'underlying augmented rack-bialgebras' since this worked already well for the above rack-star-product (35) on smooth functions on h * . In a slightly more general fashion, let (B, Φ B , U (g), ℓ) be an augmented rack bialgebra for a given Lie algebra g. We shall later specialize B to S(h) or its rack sub-bialgebra S(h) (1) = K1 ⊕ h mentioned in Theorem 1.6, part 3. Now, U (g) clearly acts 'diagonally' (i.e. using its comultiplication) on the K-module B ⊗ U (g) via ℓ on the first factor B and via the adjoint action on the second factor U(g). Calling this action ℓ ⊗ , and defining Φ : B ⊗U(g) → U(g) by Φ(b ⊗ u) = Φ B (b)u we easily see that (B ⊗ U(g), Φ, U(g), ℓ ⊗ ) is an augmented rack bialgebra whose rack multiplication ⊲ ′ reads

(b ⊗ u) ⊲ ′ (c ⊗ v) = (b),(u) ℓ ΦB (b (1) )u (1) (c) ⊗ ad ΦB (b (2) )u (2) (v) ( 37
)
where b, c ∈ B and u, v ∈ U(g).

The interesting cases are B = S(h) and its subcoalgebra B = S(h) (1) = K1 ⊕ h for a Leibniz algebra h. The latter is important since (K1 ⊕ h) ⊗ U(g) is the universal enveloping algebra of h as the left adjoint functor to the functor associating to any bar-unital dialgebra its 'commutator'-Leibniz algebra, see e.g. [START_REF] Alexandre | Structure theory of rack bialgebras[END_REF]Thm.2.8]. Note that the classical universal enveloping algebra of h in the sense of Loday-Pirashvili is just the submodule h ⊗ U(g) of (K1 ⊕ h) ⊗ U(g), and this is easily seen to be a rack-subalgebra (NOT a subcoalgebra) of S(h) ⊗ U(g) with respect to the multiplication ⊲ ′ in the above eqn (37). The primitive part of S(h) ⊗ U(g) is known to be Leibniz algebra isomorphic to the hemisemidirect product h⊕g, see [1, Thm.2.7,4.], hence the rack bialgebra S(h)⊗U(g) is isomorphic to the rack bialgebra S(h ⊕ g). In the finite-dimensional case over K = R or K = C we can apply formula (35) to this hemi-semidirect product situation thus giving a star-product formula on the function algebra C ∞ (h * × g * , K) reflecting the above multiplication (37), and clearly rack starproducts on the subspaces of functions (at most) linear in h * which will then be rack star-product versions corresponding to the universal enveloping algebra h ⊗ U(g) in the sense of Loday-Pirashvili. Surprisingly, there is a morphism of rack algebras Γ : S(h) → B ⊗ U(g) for B = K1 ⊕ h, cf the constructions in [START_REF] Rivière | Sur l'isomorphisme entre cohomologies de Chevalley-Eilenberg et de Hochschild[END_REF], [START_REF] Rivière | On the string Lie algebra[END_REF]: firstly, the following general statement is very easy to check: given Note that Γ does not have to be a morphism of coalgebras, and will in general NOT be a morphism of rack bialgebras. More concretely, for Γ : B = S(h) → C = (K1 ⊕ h) ⊗ U(g) we make the following ansatz:

Γ = (1ǫ + pr) ⊗ (F * (e (1) ) • Φ) • ∆. (38)
Here the maps ∆, ǫ, 1 give the usual augmented coalgebra structure of S(h), pr : S(h) → h is the canonical projection, and Φ : S(h) → U(g) is the above map ω • S(p), see eqn (26). Next, * denotes the convolution multiplication on Hom K (U(g), U(g)), and e (1) : U(g) → g ⊂ U(g) is the eulerian idempotent of the cocommutative bialgebra U(g), i.e. e (1) = ln * 1 U(g) ǫ U(g) + (id U(g) -1 U(g) ǫ U(g) ) , see e.g. [START_REF] Loday | Cyclic homology[END_REF]Ch.4,. Finally, F (s) is a formal series with rational coefficients, and F * (e (1) ) ∈ Hom K (U(g), U(g)) denotes the convolution series where s is replaced by e (1) . Observe that the bialgebra structures of S(h) and of U(g) are U(g)-module maps (recall that it is the adjoint action on U(g)) whence all convolutions of these structures are U(g)-module maps, hence also e (1) and its convolution series. It follows that Γ is a U(g)-module map. Moreover, since e (1) • ω = pr g we get e (1) • Φ = p • pr, and since e * e (1) = id U(g) , the second condition Φ C • Γ = Φ B gives -using the surjectivity of Φ-1 U(g) ǫ U(g) + e (1) * F * (e (1) ) = id U(g) = e * e (1) hence with F (s) = e s 1+s we get the morphism of rack-algebras. Moreover there is a morphism Ψ : S(h) + → h ⊗ U(g) given by a similar ansatz: in the above eqn (38) we just replace the expression (1ǫ + pr) by pr and the series F by the series G(s) = e s -1 s .

General deformation theory for rack bialgebras

In this section, (R, ∆, ǫ, µ, 1) is a cocommutative rack-bialgebra over a general commutative ring K, and we use the notation r ⊲ s to denote the rack product µ(r ⊗ s) of two elements r and s of R. In this subsection, we will often drop the symbol Σ in Sweedler's notation of (iterated) comultiplications, so that the n-iterated comultiplication of r in R reads

r (1) ⊗ • • • ⊗ r (n) := (∆ ⊗ Id ⊗n-1 ) • • • • • ∆(r) Let K = K[[ ]]
denote the K-algebra of formal power series in the indeterminate with coefficients in K. If V is a vector space over K, V stands for V[[ ]]. Recall that if W is a K-module, a K -linear morphism from V to W is the same as a power series in with coefficients in Hom K (V, W ) via the canonical map Hom K (V , W ) ∼ = Hom K (V, W ) . This identification will be used without extra mention in the following.

Definition 2.8. A formal deformation of the rack product µ is a formal power series µ

µ := n≥0 n µ n in Hom K (R ⊗ R, R) , such that 1. µ 0 = µ, 2. (R , ∆, ǫ, µ , 1) is a rack bialgebra over K . Example 2.1.
For a Leibniz algebra h, we have introduced in Definition 1.7 a cocommutative augmented rack bialgebra UAR ∞ (h). Furthermore, the rack star product defined in eqn (35), restricted to S(h) , is a deformation of the trivial rack product of S(h) given for all r, s ∈ S(h) by r ⊲ s := ǫ(r)s

The self-distributivity relation is shown in a way very similar to the proof of Theorem 2.1, see eqn (36). ♦

As in the classical setting of deformation theory of associatice products, we will relate our deformation theory of rack products to cohomology. For this, let us first examine an introductory example: To the order 0 , this is only the self-distributivity relation for ⊲. But to order 1 (neglecting order 2 and higher), we obtain: ω(a, b⊲c)+a⊲ω(b, c) = ω(a (1) ⊲b, a (2) ⊲c)+ω(a (1) , b)⊲(a (2) ⊲c)+(a (1) ⊲b)⊲ω(a (2) , c).

It will turn out that this is the cocycle condition for ω in the deformation complex which we are going to define. More precisely, we will have On the other hand, the requirement that ⊲ should be a morphism of coalgebras (with respect to the undeformed coproduct △ of R) means △ • ⊲ = (⊲ ⊗ ⊲ ) • △ [2] . This reads for a, b ∈ R to the order (neglecting higher powers of ) as ω(a, b) (1) ⊗ ω(a, b) (2) = ω(a (1) , b (1) ) ⊗ (a (2) ⊲ b (2) ) + (a (1) ⊲ b (1) ) ⊗ ω(a (2) , b (2) ). This is exactly the requirement that ω is a coderivation along ⊲ = µ, to be defined below. ♦

Recall that R being a rack bialgebra means in particular that µ : R ⊗2 → R is a morphism of coassociative coalgebras. For all positive integer n, let µ n : R ⊗n → R be the linear map defined inductively by setting • µ 1 := Id : R → R,

• µ 2 := µ : R ⊗2 → R,

• µ n := µ • (µ 1 ⊗ µ n-1 ), n ≥ 3, Definition 2.14. An infinitesimal deformation of the rack product is a deformation of the rack product over the K-algebra of dual numbers K := K / ( 2 ) , i.e. a linear map µ 1 : R ⊗2 → R such that R := R ⊗ K is a rack bialgebra over K when equipped with µ 0 + µ 1 . Two infinitesimal deformations µ 0 + µ 1 and µ 0 + µ ′ 1 are said to be equivalent if there exists an automorphism φ : R → R of the coalgebra of ( R , ∆, ǫ) of the form φ := id R + α such that φ • (µ 0 + µ 1 ) = (µ 0 + µ ′ 1 ) • φ.

As usual, being equivalent is an equivalence relation and one has the following cohomological interpretation of the set of equivalence classes of infinitesimal deformations, denoted Def (µ 0 , K ): Proposition 2.15.

Def (µ 0 , K ) = H 2 (R; R)

The identificaton is obtained by sending each equivalence class [µ 0 + µ 1 ] in Def (µ 0 , K ) to the cohomology class [µ 1 ] in H 2 (R; R).

Proof. One checks easily that the correspondence is well defined (if µ 0 + µ 1 is an infinitesimal deformation, then µ 1 is a 2-cocycle, see Example 2.2) and that it is bijective when restricted to equivalence classes.

Remark 2.16. (a) The choice of taking coderivations in the deformation complex is explained as follows: The rack product µ is a morphism of coalgebras, and we want to deform it as a morphism of coalgebras with respect to the fixed coalgebra structure we started with. Tangent vectors to µ in Hom coalg (C ⊗ C, C) are exactly coderivations along µ. This is the first step: Deformations as morphisms of coalgebras. Then as a second step, we look for 1-cocycles, meaning that we determine those morphisms of coalgebras which give rise to rack bialgebra structures. The deformation complex in [START_REF] Carter | Cohomology of the Categorical Self-Distributivity[END_REF] takes into account also the possibility of deforming the coalgebra structure, and we recover our complex by restriction.

(b) Given a Leibniz algebra h, there is a natural restriction map from the cohomology complex with adjoint coefficients of h to the deformation complex of its augmented enveloping rack bialgebra UAR(h). The induced map in cohomology is not necessarily an isomorphism, as the abelian case shows.

Observe that the deformation complex of the rack bialgebra K[R] for a rack R does not contain the complex of rack cohomology for two reasons: First, this latter complex is ill-defined for adjoint coefficients, and second, there are not enough coderivations as all elements are set-like. A way out for this last problem would be to pass to completions.

Example 1. 1 .

 1 Any C 3 coalgebra (C, ∆, ǫ, 1) carries a trivial rack bialgebra structure defined by the left-trivial multiplicaton a ⊲ 0 b := ǫ(a)b (7) which in addition is easily seen to be associative and left-unital, but in general not unital. ♦ Another method of constructing rack bialgebras is gauging: Let (B, ∆, ǫ, 1, µ) a rack bialgebra -where we write µ(a ⊗ b) = a ⊲ b for all a, b ∈ B-, and let f : B → B a morphism of C 3 -coalgebras such that for all a, b ∈ B f (a ⊲ b) = a ⊲ f (b) ,

Proposition 1 . 4 .

 14 Let (B, Φ, H, ℓ) be an augmented rack bialgebra. Then the C 3 -coalgebra (B, ǫ, 1) will become a rack bialgebra by means of the multiplication a ⊲ b := Φ(a).b

  any two augmented rack bialgebras (B, Φ B , H, ℓ) and (C, Φ C , H, ℓ ′ ) over the same cocommutative Hopf algebra H, then any Klinear map Γ : B → C intertwining the H-actions which satisfies Φ C • Γ = Φ B will give a morphism of rack-algebras, i.e. Γ(b ⊲ b ′ ) = Γ(b) ⊲ Γ(b ′ ) for all b, b ′ ∈ B.

Example 2. 2 .

 2 Let (R, ⊲) be a rack bialgebra, and suppose there exists a deformation ⊲ = ⊲ + ω of ⊲. The new rack product ⊲ should satisfy the self-distributivity identity, i.e. for all a, b, c ∈ R a ⊲ (b ⊲ c) = (a(1) ⊲ b) ⊲ (a(2) ⊲ c)

1. d 2 ,

 2 0 ω(a, b, c) = ω(a, b ⊲ c), 2. d 1,1 ω(a, b, c) = a ⊲ ω(b, c), 3. d 1,0 ω(a, b, c) = ω(a (1) ⊲ b, a (2) ⊲ c), 4. d 2 3 ω(a, b, c) = ω(a (1) , b) ⊲ (a (2) ⊲ c), 5. d 2,1 ω(a, b, c) = (a (1) ⊲ b) ⊲ ω((a(2) , c).This may perhaps help to understand the general definition of the operators d n i,µ for i = 1, . . . , n and µ ∈ {0, 1} further down.

so that

for all r 1 , . . . , r n in R. Proposition 2.9. For all n ≥ 1, the map µ n is a morphism of coalgebras satisfying

for all positive integers i and n such that 1 ≤ i < n and for all r 1 , ..., r n in R.

Proof.

• eqn (39): Let us show that the assertion of eqn (39) is true for all n and i with 1 ≤ i < n by induction over i. Suppose that the induction hypothesis is true and compute

i , r i+1 ) ⊲ µ n-1 (r

which gives, thanks to the self-distributivity relation in the rack algebra R,

where we have used the induction hypothesis. This proves the assertion.

• eqn (40): The assertion follows here again from an easy induction using the self-distributivity relation.

If (C, ∆ C ) and (D, ∆ D ) are two coassociative coalgebras and φ : C → D is a morphism of coalgebras, we denote by Coder(C, V, φ) the vector space of coderivations from C to V along φ, i.e. the vector space of linear maps f :

Let us note the following permanence property of coderivations along a map under partial convolution which will be useful in the proof of the following theorem. For a coalgebra A, maps f : A ⊗ B → V and g : A ⊗ C → V and some product ⊲ : V ⊗ V → V , the partial convolution of f and g is the map

Lemma 2.10. Let A, B, C and V be coalgebras, V carrying a product ⊲ which is supposed to be a coalgebra morphism. Let f : A ⊗ B → V be a coderivation along φ and g : A ⊗ C → V be a coalgebra morphism. Then the partial convolution f ⋆ part g is a coderivation along φ ⋆ part g.

Proof.

We compute for all a ∈ A, b ∈ B and c ∈ C

= (f (a (1) ⊗ b)) (1) ⊲ (g(a (2) ⊗ c)) (1) ⊗ (f (a (1) ⊗ b)) (2) ⊲ (g(a (2) ⊗ c)) (2) = (f (a (1) ⊗ b)) (1) ⊲ g(a (2) ⊗ c (1) ) ⊗ (f (a (1) ⊗ b)) (2) ⊲ g(a (3) ⊗ c (2) ) = φ(a (1) ⊗ b (1) ) ⊲ g(a (2) ⊗ c (1) ) ⊗ f (a (3) ⊗ b (2) ) ⊲ g(a (4) ⊗ c (2) ) + +f (a (1) ⊗ b (1) ) ⊲ g(a (2) ⊗ c (1)

Definition 2.11. The deformation complex of R is the graded vector space C * (R; R) defined in degree n by

where the maps d n i,1 and d n i,0 are defined respectively by

i-1 , r i )⊲ω(r

and

and d n n+1 by

n-1 , r n+1 ) for all ω in C n (R; R) and r 1 , . . . , r n+1 in R.

Theorem 2.12. d R is a well defined differential.

Proof. That d R is well defined means that it sends coderivations to coderivations. It suffices to show that this is already true for all maps d n i,1 , d n i,0 and d n n+1 , which is the case. For this, we use Lemma 2.10. Indeed, a cochain ω ∈ C n (R; R) is a coderivation along µ n . By Proposition 2.9, µ n is a coalgebra morphism. On the other hand, it is clear from the formula for d n i,1 that d n i,1 is a partial convolution with respect to the first i -1 tensor labels of µ i and ω. Therefore the Lemma applies to give that the result is a coderivation along the partial convolution of µ i and µ n , which is just µ n+1 again by Proposition 2.9. This shows that d n i,1 ω belongs to Coder(R ⊗n , R, µ n+1 ) as expected. The maps d n i,0 and d n n+1 can be treated in a similar way. The fact that d R squares to zero is related to the so-called cubical identities satisfied by the maps d i,1 and the maps d i,0 , namely

and auxiliary identities which express the compatibility of the maps d i,1 and d i,0 with d n n+1 , and an identity involving d n n+1 and d n+1 n+2 . One could call this kind of object an augmented cubical vector space.

We will not show the usual cubical relations, i.e. those which do not refer to the auxiliary coboundary map d n n+1 , because these are well-known to hold for rack cohomology, see [START_REF] Covez | On the conjectural Leibniz cohomology for groups[END_REF], Corollary 3.12, and our case is easily adapted from there. One possibility of adaptation (in case one works over the real or complex numbers) is to take a Lie rack, write its rack homology complex (with trivial coefficients in the real or complex numbers), and to apply the functor of point-distributions.

Let us show that the two following extra relations involving the extra face d n n+1 hold:

for all 1 ≤ i ≤ n and µ in {0, 1} and

Indeed, if ω is a n-cochain and r 1 , ..., r n+2 are elements in R, then

n , r n+2 ) By Proposition 2.9 and thanks to the self-distributivity of the rack product, this equality can be rewritten as

i+1 , ,• • •, r (1) n , r n+1 ) ⊲ ⊲ µ i (r

which proves that Relation (41) holds when µ = 1. The case µ = 0 goes as follows:

i-1 , r

i ⊲ r

(1)

i-1 , r

where we have used that the rack product is a morphism of coalgebras. Recall the following equation from Proposition 2.9:

for all s 1 , ..., s n+1 in R and 1 ≤ i ≤ n. This allows to rewrite the preceeding equality as

i-1 , r

n , r

i-1 , r

which proves that (41) holds when µ = 0. Relation (42) relies on the fact that cochains are coderivations. Indeed,

where we have used Proposition 2.9 in the last equality. By self-distributivity of ⊲ and because ω is a coderivation, this gives

n-1 , r n ) (1) ⊲ µ n (r

n-1 , r n+1 ) ⊲ ω(r

n-1 , r n ) (2) ⊲ µ n (r

n-1 , r (1) n ) ⊲ µ n (r

n-1 , r n+1 ) ⊲ µ n (r

n-1 , r (2) n ) ⊲ µ n (r

n-1 , r n+2 ) + µ n (r

n-1 , r (1) n ) ⊲ µ n (r

n-1 , r n+1 ) ⊲ ω(r

n-1 , r (2) n ) ⊲ µ n (r

n-1 , r n+2 ) Applying Proposition 2.9 again enables us to rewrite this last equality as

n-1 , r n+1 ) ⊲ µ n+1 (r

n-1 , r (2) n , r n+2 ) + µ n+1 (r

The first double sum is equal to zero thanks to the cubical relations, the second sum is zero thanks to relation (41). Relation (42) implies that the last one vanishes. This shows that d R is indeed a differential and concludes the proof of the proposition.

Definition 2.13. The cohomology of the deformation complex (C * (R; R), d R ) is called the adjoint cohomology of the rack bialgebra R and is denoted by H * (R; R).


Fuming Fang

Junichi Yamagishi

Isao Echizen
email: iechizen@nii.ac.jp

Md Sahidullah
email: md.sahidullah@inria.fr

Tomi Kinnunen

Tomi Kinnunen Transforming

Transforming acoustic characteristics to deceive playback spoofing countermeasures of speaker verification systems

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Automatic speaker verification (ASV) [START_REF] Hansen | Speaker recognition by machines and humans: A tutorial review[END_REF], a kind of biometrics authentication technology, identifies a person from a segment of speech. ASV systems typically fall into two types: text-independent and text-dependent, where the latter requests a client to speak a given phrase. Due to the convenience of ASV, it is being used in more and more applications, such as ones used in call centers and by mobile devices. However, ASV is vulnerable to several kinds of spoofing attacks (also known as presentation attacks [START_REF]Information technology Biometric presentation attack detection Part 1: Framework[END_REF]), so ASV systems need a spoofing countermeasure (CM) (also known as presentation attack detection [START_REF]Information technology Biometric presentation attack detection Part 1: Framework[END_REF]). Such attacks aim to mimic the target speaker mainly by using synthesized speech [3], converted speech [3], or playback speech [START_REF] Kinnunen | The ASVspoof 2017 challenge: Assessing the limits of replay spoofing attack detection[END_REF][START_REF] Delgado | ASVspoof 2017 Version 2.0: meta-data analysis and baseline enhancements[END_REF]. Among them, playback speech-based attacks are relatively easy to mount since an attacker who has no special knowledge can make them [START_REF] Wu | Spoofing and countermeasures for speaker verification: A survey[END_REF]. Once an attacker has collected/stolen a voice sample for the target speaker, he/she can simply play it back to an ASV system or concatenate segments of the sample to form a new utterance. Threats from this kind of attack have been confirmed by several studies [START_REF] Kinnunen | The ASVspoof 2017 challenge: Assessing the limits of replay spoofing attack detection[END_REF][START_REF] Delgado | ASVspoof 2017 Version 2.0: meta-data analysis and baseline enhancements[END_REF]7,8,9]. Here we focus on playback spoofing attacks and relevant CMs.

Four main types of CMs have been developed to protect against playback spoofing attacks. One type utilizes a text-dependent ASV system and randomly prompts for a pass-phrase [START_REF] Kinnunen | Utterance verification for text-dependent speaker recognition: a comparative assessment using the RedDots corpus[END_REF][START_REF] Zeinali | Spoken pass-phrase verification in the i-vector space[END_REF], making it difficult to mount playback attacks using phrase-fixed speech. However, it is possible to form an arbitrary utterance to spoof this type of CM if the attacker has sufficient speech data for the target speaker. The second type is based on rules describing the characteristics of genuine speech (recorded from a person). For example, Mochizuki et al. [12] distinguished genuine speech by detecting pop-noise from certain phonemes. An intractable problem related to this type of CM is that it is difficult to design suitable rules and implement them. The third type utilizes audio fingerprinting to check whether an incoming recording is similar to previously authenticated utterances that were automatically saved in the ASV system. Rodriguez et al. [13] developed such a system: if the similarity score was higher than a threshold, the recording was treated as a playback attack. A disadvantage of this type of CM is that it is sensitive to noise. In contrast, the fourth type compares the differences between genuine speech and playback speech. This type mainly utilizes a machine learning algorithm to learn the differences. An example is Wang et al.'s [START_REF] Wang | An efficient learning based smartphone playback attack detection using gmm supervector[END_REF] use of a support vector machine [START_REF] Hearst | Support vector machines[END_REF] to learn the difference in Mel-frequency cepstral coefficient (MFCC)based acoustic features.

More methods of the fourth type were presented at the second Automatic Speaker Verification Spoofing and Countermeasures Challenge (ASVspoof 2017), in which a common database was used to assess the participants' CMs. The database consists of two parts. One part contains genuine speech taken from the RedDots corpus [START_REF] Lee | The reddots data collection for speaker recognition[END_REF], which was designed for speaker verification. The other part contains recordings of the genuine speech made in various environments. For these data, the baseline [START_REF] Kinnunen | The ASVspoof 2017 challenge: Assessing the limits of replay spoofing attack detection[END_REF] with a constant Q cepstral coefficient (CQCC) [17] feature and a Gaussian mixture model (GMM) classifier had an equal error rate (EER) of 30.60%. A deep learning-based method had an EER of 6.73% [START_REF] Gałka | Playback attack detection for text-dependent speaker verification over telephone channels[END_REF], which was the best performance achieved at ASVspoof 2017 [START_REF] Kinnunen | The ASVspoof 2017 challenge: Assessing the limits of replay spoofing attack detection[END_REF]. These mainstream CMs of the fourth type are also problematic: they are based on the assumption that the attackers do not have special knowledge. Moreover, this type of CM algorithms only learn the difference from a given dataset and perhaps do not work well if the acoustic characteristics of the playback speech is transformed close to that of the genuine one. To confirm this hypothesis, we tested the effectiveness of a playback attack using speech "stolen" from the target speaker and enhanced before mounting the attack. This enhancement should remove the distortions in the stolen speech caused by the recording device and environmental noise so that they do not affect the re-recorded speech.

We evaluated the effectiveness of a playback attack using this enhanced stolen speech method against a textdependent ASV system. We used the ASVspoof 2017 scenario (Figure 1) in which the attacker is assumed to obtain from somewhere uncompressed speech for the target speaker containing the phrase used for authentication, e.g., by downloading from the web, hacking a device used by the target speaker, and talking to and surreptitiously recording the target speaker. The speech enhancement generative adversarial network (SEGAN) [19] was used to transform the acoustic characteristics of the obtained speech close to that of the genuine speech. We also investigated the effect of different types of playback loudspeakers and re-recording devices. The results showed that it is possible to fool playback spoofing CMs by transforming the acoustic characteristic of the playback speech close to that of the genuine speech.

Related work

Pioneering work on playback attacks was reported by Lindberg and Blomberg in 1999 [START_REF] Lindberg | Vulnerability in speaker verification-a study of technical impostor techniques[END_REF]. They pre-recorded the numbers one to ten of two speakers and then concatenated various combinations of them to attack a hidden Markov model (HMM) [START_REF] Rabiner | A tutorial on hidden markov models and selected applications in speech recognition[END_REF]-based text-dependent ASV system. They demonstrated considerable increase in both the EER and false acceptance rate (FAR) compared to verification without attacks. More recently, Ergunay et al. investigated the effect of playback attacks against ASV systems and also achieved a large increase in FAR [22]. Compared to these conventional playback attacks, our method further degrades the performance of playback spoofing CMs by enhancing the speech.

There are a few attack methods similar to our enhanced stolen speech method. Demiroglu et al. improved the naturalness of synthesized and converted speech before attacking a phase-based synthetic speech detector and an ASV system [START_REF] Demiroglu | Postprocessing synthetic speech with a complex cepstrum vocoder for spoofing phase-based synthetic speech detectors[END_REF]. The synthesized and converted speech signals were firstly analyzed frame by frame, and each frame was replaced with one containing the most similar natural speech selected from a dataset. A complex cepstrum vocoder was used to re-synthesize these frames so as to improve speech naturalness. Finally, the speech was directly fed into an ASV system. They reported that their method fooled four out of nine detectors. Our method can be thought of as an extension of their method as it further transforms synthesized speech close to natural speech.

Nguyen et al. reported an attack method that transforms computer-generated (CG) images into natural images before feeding them into a facial authentication system [24]. The transformation model is trained using a generative adversarial network (GAN) [START_REF] Goodfellow | Generative adversarial nets[END_REF]. The GAN discriminator, which mimics a spoofing detector, is used to distinguish CG/natural images. The discriminator is pre-trained and fixed during training of the transformation model. In contrast, we treat the authentication system as a black box, and anything regarding playback spoofing CMs and ASV systems is unknown.

Playback detectors and ASV system

Two playback spoofing CMs and a classical Gaussian mixture model with universal background model (GMM-UBM) [26]-based ASV system were used to evaluate the effectiveness of our enhanced stolen speech attack method. The two CMs were the baseline and the core method of the system with the best performance (i.e., a light convolutional neural network) of ASVspoof 2017.

Baseline of ASVspoof 2017

The baseline of ASVspoof 2017 consists of a CQCC front-end and a GMM back-end. We refer to this method as "CQCC-GMM CM". The CQCC is an acoustic feature extracted from an audio signal. CQCC extraction is performed using constant Q transform (CQT) instead of the classical short-time Fourier transformation (STFT). STFT suffers from fixed frequency resolution and fixed temporal resolution whereas CQT exhibits higher frequency resolution at lower frequencies and higher temporal resolution at higher frequencies. An audio signal is usually represented by a sequence of CQCC feature vectors.

A two-class GMM-based classifier is used for genuine/playback speech detection. One GMM is trained using genuine speech while the other one is trained using playback speech. Input to the GMMs is CQCC-based acoustic feature vectors, and the expectation maximization (EM) [27] algorithm is used for training. During prediction, the feature vectors of an audio signal are independently input into the two models, and then the joint log-likelihood for both models is calculated. Finally, the log-likelihood ratio of the genuine and playback model results is compared with a threshold to determine genuine/playback speech.

Core method of best system of ASVspoof 2017

The best system of ASVspoof 2017 was a fusion of three sub-systems: a support vector machine with i-vector features [START_REF] Dehak | Front-end factor analysis for speaker verification[END_REF], a convolutional neural network (CNN) with a recurrent neural network (RNN), and a light CNN (LCNN). The LCNN was used as the core method, which achieved an EER of 7.37%. This performance was very close to that of the fused system (6.73%). We therefore used an "LCNN CM" to evaluate our enhanced stolen speech attack method.

The LCNN consists of five convolution layers, four network-in-network (NIN) [29] layers, ten max-featuremap (MFM) layers, five max-pooling layers, and two fully connected layers. Each MFM layer acts as a maxout activation function [START_REF] Goodfellow | Maxout networks[END_REF] that splits the CNN feature maps into two groups and then performs element-wise maximization to select features. The LCNN input is a spectrum with a fixed size of 864 × 400, which is obtained by performing STFT with 1728 bins and concatenating 400 frames. Dropout [START_REF] Srivastava | Dropout: A simple way to prevent neural networks from overfitting[END_REF] is applied after the first fully connected layer. The final output layer, with a softmax activation function, is used to discriminate genuine/playback speech. This is described in more detail elsewhere [START_REF] Gałka | Playback attack detection for text-dependent speaker verification over telephone channels[END_REF].

The silence parts of the audio signal are removed, and then STFT is performed using a window length of 25 ms with a shift size of 10 ms. If a signal is shorter than 4 seconds, its content is repeated to match the length. For a longer signal, its content is repeated to match multiples of 4 seconds, and the output probabilities are averaged.

GMM-UBM-based ASV system

We use a GMM-UBM-based system for speaker verification. Even though it is a classic ASV method, GMM-UBM provides competitive performance on short-duration, textdependent ASV tasks [32]. The speaker models are created by maximum a posteriori adaptation from a UBM trained with a large amount of speech data from different speakers. Text-dependent speaker models are separately created for different passphrases following the guidelines for conducting experiments with the RedDots corpus. The recognition score is the likelihood ratio between the results of the target speaker model and those of the UBM. or (ŷ, x); the former should be classified as real data while the latter should be classified as fake data; but the latter should be treated as real data when updating the generator parameters, so adversarial training is performed.

Speech enhancement

SEGAN is a data-driven speech enhancement method that constructs a mapping from a noisy waveform to a clean waveform with the help of supervised training. More specifically, SEGAN leverages the power of the GAN composed of two adversarial networks, a discriminator D and a generator G. The discriminator predicts the probability that the input is from real data rather than from fake data generated by G. The generator learns a mapping function from a prior noise distribution p noise (z) to the distribution of the real data p data (y) to fool the discriminator. If the noise distribution is conditioned by x drawn from playback speech and y drawn from genuine speech, the output ŷ is genuinelike speech.

The objective function for training SEGAN is formulated as min

D L(D) = 1 2 E y∼p data (y) [(D(y, x) -1) 2 ] + 1 2 E z∼pnoise(z) [D(ŷ, x) 2 ] min G L(G) = E z∼pnoise(z) [(D(ŷ, x) -1) 2 ] + λ• ŷ -y 1 , (1)
where ŷ = G(z, x) is enhanced (or generated) speech and E means expectation. L 1 norm, • 1 , is used to measure the distance between the real and enhanced speech. The discriminator and generator are alternately trained by performing a min-max game.

Figure 2 shows the architecture of SEGAN. It is an endto-end model, so both the input and output of the generator are raw waveforms. The input of the discriminator is also a raw waveform combining (y, x) or (ŷ, x). The generator has an encoder-decoder structure. The encoder part is composed of 11 stacked 1-D CNN layers with a filter width of 31 and a stride of two. The decoder part is a mirror structure of the encoder part, and the corresponding layers between them are connected by a skip path. The dimension of noise z is the same as that of encoder output c and is drawn from a normal distribution. They are concatenated and input to the decoder. The architecture of the discriminator is the same as that of the encoder part of the generator except that virtual batch-normalization [START_REF] Salimans | Improved techniques for training gans[END_REF] is performed in hidden layers.

Database

In this section, we describe the speech data used for training the spoofing detectors, ASV, and SEGAN. We also describe the test data used for authentic as well as illegitimate access.

Training data for playback spoofing CM

Both the CQCC-GMM and the LCNN CM models were trained using the ASVspoof 2017 database (version 2), which was derived from the RedDots corpus. The genuine speech data in the database was taken from the Red-Dots corpus, and the playback speech data was recorded by playing the genuine speech data in various environments (including quiet and noisy places) using various recording devices and speakers with various qualities. The sampling rate was 16 kHz for both the genuine and playback speech. The database was further split into three datasets: training, development, and evaluation. Each of the datasets contained both genuine and playback speech. The GMM of the CQCC-GMM CM was trained using the training and development datasets. For the LCNN, the training dataset was used to estimate the model parameters and the development dataset was used to monitor the training process.

Training data for ASV system

We used TIMIT and RSR2015 (background subset [34]) corpora for training the UBM for the GMM-UBM-based ASV system. Only male speakers were used as the ASVspoof 2017 database was created from the male subset of the RedDots corpus. In total, we used 17,850 speech utterances from 488 speakers for UBM training. Each target speaker model was created with speech utterances from three different sessions for a fixed passphrase.

Training data for SEGAN

We used a high-quality database and two low-quality databases distorted by recording devices or environment noises to train SEGAN. The high-quality database was the voice cloning toolkit (VCTK) corpus [35]. This corpus contains data recorded in a hemi-anechoic chamber by 109 native English speakers, but we used data for only 28 speakers. One of the low-quality databases was a device-recorded VCTK (DR-VCTK) corpus [36] and the other one was a noisy VCTK (N-VCTK) corpus [37]. The DR-VCTK was created by playing the high-quality speech of the 28 speakers in office environments and recording it using relatively inexpensive consumer devices. The N-VCTK was created by adding noise to the high-quality speech of the 28 speakers. The sampling rate of these databases was 48 kHz with downsampling to 16 kHz. Two types of SEGAN were trained. One was trained using DR-VCTK and VCTK. The other was trained using N-VCTK and VCTK.

Authentication and spoofing data

We equally split the genuine speech of the evaluation dataset in the ASVspoof 2017 database into two subdatasets. One was used as authentication speech, and the other was used as "stolen speech." We enhanced the stolen speech, played it using four types of portable loudspeakers, and re-recorded it using six types of recording devices in an office room. The four loudspeakers were a high-quality speaker (BOSE Soundlink), a mediumquality speaker (SONY SRS-BTS50), a low-quality speaker (audio-technica AT-SP92), and an iPhone 6s speaker. The six recording devices were a high-quality condenser microphone (Apogee MiC 96k), a directional microphone (Sony ECM-673), a low-quality microphone (Snowball iCE), a MacBook microphone, an iPad microphone, and an iPhone 6s microphone. These devices were placed at around 30 to 50 cm from the loudspeaker. The sampling rate for the rerecording was 16 kHz. According to the used loudspeaker, four playback and re-recording sessions were performed.

Experimental setup

We evaluated 1) how our enhanced stolen speech method affects the performance of playback spoofing CMs and 2) how the enhanced speech effects an ASV system. In order to compare the difference between our method and conventional playback attacks, a paired two-tailed t-test was used.

Setup for playback spoofing CMs

Settings of the CQCC-GMM CM were the same as baseline of ASVspoof 2017 and the source code is available at [START_REF]Baseline replay attack detector[END_REF]. CQCC had 29 dimension and 0-th order cepstral coefficient was further used. Their first and second derivatives were finally used as features (90 dimensions in total). The GMM of the CQCC-GMM CM had 512 components.

The weights of the LCNN were initialized using the Xavier method [39]. The dropout rate was set to 0.5. Adam optimization [40] with momentum of 0.5 was used. The initial learning rate was 0.0001; it was reduced by 0.9 if the classification accuracy of the development dataset decreased after each epoch. There were nine epochs, and the mini-batch size was 64. The LCNN was implemented using the TensorFlow framework [START_REF] Abadi | Tensorflow: Large-scale machine learning on heterogeneous distributed systems[END_REF] and is available at [42].

To assess the performance of both spoofing CMs, we use EER, which reflects the ability of the CM to discriminate genuine speech samples from playback attacks.

Setup for ASV system

Our ASV system used MFCC-based acoustic feature extracted from a 20 ms short-term window with a 10 ms shift using 20 filters. We computed 19 MFCCs after discarding the energy coefficients. The MFCCs were further processed with RASTA filtering to suppress convolutive mismatch. The delta and double-delta coefficients were computed for a context of three frames and then augmented with static MFCCs to create a 57-dimensional feature vector. Finally, cepstral mean and variance normalization (CMVN) was performed after discarding the non-speech frames with an energy-based voice activity detector. We trained the gender-dependent UBM with 512 mixture components. The speaker models were created by adapting only the centers of UBM with a relevance factor of three.

Even though ASV spoofing evaluations have focused on standalone CM assessment, the performance of a tandem (combined) system is important for real-world deployment. Both CM and ASV can result in target speaker misses and false acceptances of impostors (either non-targets or spoofs). We therefore adopted a recently proposed tandem detection cost function (t-DCF) metric [START_REF] Kinnunen | t-DCF: a detection cost function for the tandem assessment of spoofing countermeasures and automatic speaker verification[END_REF] for evaluating the combination of two systems in a Bayes risk framework. The t-DCF is given by

C asv miss • π tar • P a + C asv fa • π non • P b + C cm fa • π spoof • P c + C cm miss • π tar • P d
, where C asv miss = 1, C asv fa = 10, C cm fa = 10, and C cm miss = 1 are unit costs related to the misses and false alarms of the two systems; π spoof = 0.0100, π non = 0.0099, and π tar = 0.9801 were the prior probabilities of the targets, non-targets, and spoofs, respectively; and P a , P b , P c , and P d are the error rates of four possible errors originating from the joint actions of the CM and ASV systems. The reported t-DCF values are minimum t-DCF values with a fixed ASV system. The higher the value, the less usable the combined (ASV and CM) system.

Setup for SEGAN

Similar to previous work [19,[START_REF] Lorenzo-Trueba | Can we steal your vocal identity from the internet?: Initial investigation of cloning obamas voice using gan, wavenet and low-quality found data[END_REF], we extracted chunks of waveforms by using a sliding window of 2 14 samples at every 2 13 samples (i.e., 50% overlap). During testing, we concatenated the results at the end of the stream without overlap. The learning rate, mini-batch size, and epoch size were set to 0.0002, 100, and 120, respectively. The λ in Equation 1 was set to 100. We used source code for improved SEGAN [START_REF] Sarfjoo | Improved SEGAN[END_REF].

Results

Tables 1 and2 show the EERs for the CQCC-GMM CM and the LCNN CM, respectively. Playback spoofing attacks using our enhanced stolen speech method had significantly higher EERs for both CMs compared to those of conventional playback attacks (without enhancement). One reason could be that the signal-to-noise ratio was higher after speech enhancement, resulting in the playing of cleaner speech. Use of the high-quality speaker with the highquality microphone and use of the low-quality speaker with the high-quality microphone when N-VCTK was used to train SEGAN resulted in the largest performance degrada- As expected, use of the high-quality speaker resulted in higher EERs because it generated more natural speech. It is interesting that the results for the iPhone 6s speaker were similar to those for the high-quality speaker. While a wide range of EERs were obtained for the recording devices, use of the high-quality microphone did not result in significantly higher EERs. The CQCC-GMM CM could not distinguish the playback speech re-recorded using the iPhone 6s. This was because features were not normalized and channel distortions greatly degraded its performance [START_REF] Delgado | ASVspoof 2017 Version 2.0: meta-data analysis and baseline enhancements[END_REF]. Enhancement based on the N-VCTK was mightier than that based on the DR-VCTK in most cases. This might be because distortion due to environmental noise has a greater effect than that due to the recording devices.

Tables 3 and4 show the t-DCF values for the combined CQCC-GMM CM and GMM-UBM-based ASV scores and for the combined LCNN CM and GMM-UBM-based ASV scores, respectively. Compared to the conventional playback attacks, an attack using our enhanced stolen speech method greatly degraded the authentication performance of both combinations. This suggests that our enhanced stolen speech method enables playback attacks to pass playback spoofing CMs and to fool ASV systems as well.

Conclusion and future work

We investigated the effectiveness of using enhanced stolen speech in playback spoofing attacks. Experimental results showed that stolen speech enhanced with SEGAN can greatly degrade the performances of baseline CQCC-GMM and advanced LCNN-based playback spoofing CMs as well as that of GMM-UBM-based ASV systems.

Since the used speech enhancement method for attack would be unknown, we plan to develop a robust playback detection method for various speech enhancement methods.

Figure 1 .

 1 Figure 1. Playback spoofing attack using enhanced stolen speech method under ASVspoof 2017 scenario. Without speech enhancement, attack is the same as a conventional playback attack.

Figure 2 .

 2 Figure2. Architecture of SEGAN. Input of discriminator is (y, x) or (ŷ, x); the former should be classified as real data while the latter should be classified as fake data; but the latter should be treated as real data when updating the generator parameters, so adversarial training is performed.

Table 1 .

 1 EERs for CQCC-GMM CM. Bold means largest degradation.

	Loudspeaker used for replay	Enhancement Directional High-quality Low-quality Mac iPad training data microphone microphone microphone book	iPhone Average 6s
		-	15.65	8.83	20.87	9.98 7.21 49.92	18.74
	High quality	DR-VCTK	28.42	18.10	29.67	14.96 8.59 50.00	24.96
		N-VCTK	35.61	23.18	33.59	16.49 9.17 50.00	28.01
		-	9.35	11.96	8.78	10.78 6.54 49.13	16.09
	Medium quality	DR-VCTK	15.71	20.16	15.76	15.27 7.43 49.92	20.71
		N-VCTK	22.56	25.62	22.03	15.61 8.36 49.92	24.02
		-	11.83	8.98	10.28	8.34 6.14 49.92	15.92
	Low quality	DR-VCTK	20.07	16.29	19.77	10.32 6.96 49.96	20.56
		N-VCTK	26.87	22.35	24.44	10.78 7.35 49.92	23.62
		-	16.28	16.54	19.83	7.19 6.40 49.53	19.30
	iPhone 6s	DR-VCTK	30.45	31.19	30.50	10.93 7.14 49.92	26.69
		N-VCTK	24.25	24.26	26.94	9.83 7.28 49.88	23.74

Table 2 .

 2 EERs for LCNN CM. Bold means largest degradation.

	Loudspeaker used for replay	Enhancement Directional High-quality Low-quality Mac training data microphone microphone microphone book	iPad	iPhone Average 6s
		-	11.19	8.00	16.14	7.71 12.95 25.04	13.51
	High quality	DR-VCTK	12.35	9.12	18.14	8.59 13.55 25.74	14.58
		N-VCTK	13.48	10.43	19.74	8.85 13.83 25.29	15.27
		-	8.78	9.98	5.92	5.47	7.09	25.25	10.42
	Medium quality	DR-VCTK	9.57	11.22	6.56	6.85	8.79	27.25	11.71
		N-VCTK	10.31	12.22	7.96	7.23	9.56	27.12	12.40
		-	7.25	6.06	5.31	9.52	7.80	16.29	8.71
	Low quality	DR-VCTK	8.44	7.10	6.07	10.08 8.76	17.05	9.58
		Noisy VCTK	10.23	8.95	7.52	10.30 9.38	17.09	10.58
		-	11.11	11.56	10.47	4.40	9.17	17.65	10.73
	iPhone 6s	DR-VCTK	13.25	14.97	13.62	5.23 11.12 18.26	12.74
		N-VCTK	11.70	12.33	11.21	4.54 10.07 18.07	11.32

Table 3 .

 3 Values of t-DCF obtained from combination of CQCC-GMM CM and ASV scores. Bold means largest degradation.

	Loudspeaker used for replay	Enhancement Directional High-quality Low-quality training data microphone microphone microphone	Mac book	iPad	iPhone Average 6s
		-	0.9361	0.9276	0.9392	0.9136 0.9118 0.9426 0.9285
	High quality	DR-VCTK	0.9412	0.9363	0.9410	0.9258 0.9210 0.9426 0.9347
		N-VCTK	0.9403	0.9373	0.9402	0.9277 0.9211 0.9431 0.9350
		-	0.9156	0.9351	0.9211	0.9222 0.9039 0.9425 0.9234
	Medium quality	DR-VCTK	0.9313	0.9386	0.9348	0.9311 0.9143 0.9428 0.9322
		N-VCTK	0.9339	0.9377	0.9362	0.9308 0.9186 0.9429 0.9334
		-	0.9225	0.9177	0.9248	0.9109 0.9020 0.9415 0.9199
	Low quality	DR-VCTK	0.9339	0.9314	0.9362	0.9220 0.9076 0.9428 0.9290
		N-VCTK	0.9353	0.9342	0.9363	0.9223 0.9105 0.9425 0.9302
		-	0.9354	0.9358	0.9379	0.9085 0.9006 0.9384 0.9261
	iPhone 6s	DR-VCTK	0.9380	0.9379	0.9388	0.9290 0.9127 0.9381 0.9324
		N-VCTK	0.9387	0.9391	0.9383	0.9247 0.9100 0.9388 0.9316

Table 4 .

 4 Values of t-DCF obtained from combination of LCNN CM and ASV scores. Bold means largest degradation.

	Loudspeaker used for replay	Enhancement Directional High-quality Low-quality training data microphone microphone microphone	Mac book	iPad	iPhone Average 6s
		-	0.9239	0.9073	0.9436	0.9063 0.9338 0.9656 0.9301
	High quality	DR-VCTK	0.9303	0.9135	0.9494	0.9098 0.9385 0.9664 0.9347
		N-VCTK	0.9346	0.9186	0.9522	0.9105 0.9390 0.9658 0.9368
		-	0.9105	0.9173	0.8990	0.8978 0.9038 0.9657 0.9157
	Medium quality	DR-VCTK	0.9159	0.9266	0.9011	0.9021 0.9107 0.9673 0.9206
		N-VCTK	0.9202	0.9293	0.9077	0.9035 0.9143 0.9675 0.9238
		-	0.9043	0.8992	0.8972	0.9122 0.9070 0.9584 0.9131
	Low quality	DR-VCTK	0.9105	0.9029	0.8999	0.9156 0.9125 0.9605 0.9170
		N-VCTK	0.9183	0.9108	0.9051	0.9156 0.9141 0.9593 0.9205
		-	0.9208	0.9238	0.9173	0.8952 0.9120 0.9530 0.9204
	iPhone 6s	DR-VCTK	0.9344	0.9396	0.9345	0.8970 0.9222 0.9540 0.9303
		N-VCTK	0.9234	0.9274	0.9214	0.8960 0.9154 0.9559 0.9233
	tion for the two CMs. The increases in EER were 2.6 and
	1.5 times, respectively.

Acknowledgement

This work was partially supported by JSPS KAK-ENHI Grant Numbers JP16H06302, 18H04120, 18H04112, 18KT0051, 17H04687, and Academy of Finland (project no. 309629). We thank Huy H. Nguyen, the Graduate University for Advanced Studies (SOKENDAI), for comments on an earlier versions of the manuscript.