Marvin Verstraete
email: marvin.verstraete@univ-lille.fr

Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic

Introduction

An important result in deformation theory asserts that every deformation problem over a field of characteristic 0 can be encoded by a differential graded Lie algebra (see [START_REF] Lurie | Derived Algebraic Geometry X: Formal Moduli Problems[END_REF] and [START_REF] Pridham | Unifying derived deformation theories[END_REF]). More precisely, any deformation problem can be described by a solution of the Maurer-Cartan equation:

d(x) + 1 2 [x, x] = 0,
in some differential graded Lie algebra. The group obtained by the integration of the differential graded Lie algebra into a Lie group, called the gauge group, moreover acts on the Maurer-Cartan set. The orbits of this action give isomorphism classes of deformation problems.

In [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF], Dotsenko-Shadrin-Vallette showed that if the differential graded Lie algebra comes from a differential graded pre-Lie algebra, then the Maurer-Cartan equation, the gauge group and its action on the Maurer-Cartan set can be described in terms of pre-Lie operations. A differential graded pre-Lie algebra is a vector space L with a bilinear operation ⋆ : L ⊗ L -→ L such that (x ⋆ y) ⋆ z -x ⋆ (y ⋆ z) = (-1) |y||z| ((x ⋆ z) ⋆ y -x ⋆ (z ⋆ y)), and which satisfies the Leibniz rule with respect to the differential. Every differential graded pre-Lie algebra is in particular a differential graded Lie algebra with the graded commutator: [x, y] = x ⋆ y -(-1) |x||y| y ⋆ x. Dotsenko-Shadrin-Vallette showed in particular that given a pre-Lie algebra L, the pre-Lie exponential map exp : L 0 -→ (1 + L 0 ) induces an isomorphism between the gauge group and the group (1 + L 0 , ⊚, 1) with ⊚ the circular product defined by

a ⊚ (1 + b) = n≥0 1 n! a{b, ..., b n },
where -{-, ..., -} denote the symmetric braces determined by the pre-Lie structure ⋆, starting with x{y} = x ⋆ y. Then, by writing the Maurer-Cartan equation as a zero-square equation, they prove that the action of the gauge group on the Maurer-Cartan set can be computed in terms of the circular product ⊚ as e λ .α = (e λ ⋆ α) ⊚ e -λ , allowing us to have an easier way to compute the Deligne groupoid associated to any differential graded pre-Lie algebra over a field of characteristic 0.

The aim of this paper is to develop a deformation theory in positive characteristic which generalizes the deformation theory controlled by pre-Lie algebras over a field of characteristic 0 developed in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF]. Our idea is to use differential graded pre-Lie algebras with divided powers.

The notion of a pre-Lie algebra with divided powers (or Γ(PreLie, -)-algebra) has been studied by Cesaro in [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]. He showed in particular that every pre-Lie algebra with divided powers comes equipped with weighted brace operations -{-, ..., -} r1,...,rn , for each collection of integers r 1 , ..., r n ≥ 0, which satisfy similar identities as the quantities x{y 1 , ..., y n } r1,...,rn = 1 i r i ! x{y 1 , ..., y 1 r1 , ..., y n , ..., y n rn } in a pre-Lie algebra over a field of characteristic 0 (see [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]Propositions 5.9-5.10] for a precise list of these identities).

Every differential graded pre-Lie algebra with divided powers L comes equipped with analogous weighted brace operations -{-, ..., -} r1,...,rn which satisfy a graded version of the identities satisfied by weighted braces in the non graded framework. In this context, we have an analogue of the Maurer-Cartan equation:

d(x) + x{x} 1 = 0.
With suitable convergence hypothesis, we also get that the circular product ⊚ can be written as

a ⊚ (1 + b) = n≥0 a{b} n
and gives rise to a group structure on 1 + L 0 . This group is called the gauge group of L. As in characteristic 0, we also show that this group acts on the Maurer-Cartan set of L.

Theorem A. Let K be a ring.

(i) In any differential graded pre-Lie algebra with divided powers L, the circular product ⊚, defined as above, endows the set 1 + L 0 with a group structure.

(ii) Suppose that µ{α, α} 1,1 = 0 for every µ ∈ L 0 and α ∈ L with odd degree. If we denote by d the differential of L, then this group acts on the Maurer-Cartan set via the formula

(1 + µ).α = (α + µ{α} 1 -d(µ)) ⊚ (1 + µ) ⊚-1 .
We prove that this new deformation theory satisfies an analogue of the Goldman-Millson theorem given in [11, §2.4]. Let Deligne(L, A) be the Deligne groupoid of the dg pre-Lie algebra with divided powers L ⊗ m A , where L is a dg pre-Lie algebra with divided powers and m A the maximal ideal of a local artinian K-algebra A. We precisely get the following result.

Theorem B. Let K be a noetherian integral domain. Let L and L be two Γ(PreLie, -)-algebras. Suppose that L and L are free as K-modules and that there is no 2-torsion. Let φ : L -→ L be a morphism of Γ(PreLie, -)-algebras such that H 0 (φ) and H 1 (φ) are isomorphisms and H 2 (φ) a monomorphism. Then for every local artinian K-algebra A, the induced functor φ * : Deligne(L, A) -→ Deligne(L, A) is an equivalence of groupoids.

Other approaches to generalize the usual deformation theory in the positive characteristic framework have been proposed recently in the literature. We have for instance a deformation theory in an associative context, via A ∞ -algebras, which is used to study deformations of group representations (see [START_REF] Milham | On the Goldman-Millson theorem for A ∞ -algebras in arbitrary characteristic[END_REF]). Another approach is given by (spectral) partition Lie algebras to get a full generalization of the Lurie-Pridham correspondence in the setting of a field with positive characteristic (see [START_REF] Brantner | PD operads and explicit partition Lie algebras[END_REF][START_REF] Brantner | Deformation theory and partition Lie algebras[END_REF]).

The main motivation for the approach developed in this paper is that operadic deformation problems are expressed in terms of pre-Lie structures. The goal is then to compute the π 0 of a mapping space Map(B c (C), P), where we take any augmented dg operad P on the target and the operad B c (C) given by the cobar of a dg coaugmented cooperad C on the source. Recall simply that B c (C) defines a cofibrant operad when C is cofibrant as a symmetric sequence (Σ * -cofibrant).

It is well known that, over a field of characteristic 0, the π 0 of this mapping space is the set of isomorphism classes of the Deligne groupoid of the Lie algebra Hom Σ (C, P). Using the pre-Lie algebra structure of Hom Σ (C, P), this can be seen as a consequence of the computations in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF]. To extend this result, we first show that Hom Σ (C, P) admits a structure of dg pre-Lie algebra with divided powers given in terms of the dg brace algebra structure of Hom(C, P). Then we get the following statement.

Theorem C. Let K be a field. Suppose that C is a Σ * -cofibrant coaugmented dg cooperad which comes with a weight decomposition and P an augmented dg operad. We then have an isomorphism:

π 0 (Map(B c (C), P)) ≃ π 0 Deligne(Hom Σ (C, P)),
where π 0 Deligne(Hom Σ (C, P)) denotes the set of isomorphism classes of the Deligne groupoid.

This theorem gives a first step for the calculation of the homotopy groups of a mapping space Map(B c (C), P) over any field.

In the first part of this paper, we recall some definitions and properties on pre-Lie algebras and pre-Lie algebras with divided powers: in §1.1 we briefly review the definition of the notion of a pre-Lie algebra and the construction of the corresponding operad; in §1.2, we review the definition of a pre-Lie algebra with divided powers and of the weighted brace operations.

In the second part, we develop the deformation theory for differential graded pre-Lie algebras with divided powers: in §2.1, we study pre-Lie algebras with divided powers in the dg framework; in §2.2, we define the circular product and prove assertion (i) of Theorem A; in §2.3, we define the Maurer-Cartan set and prove assertion (ii) of Theorem A; in §2.4, we finally prove our analogue of the Goldman-Millson theorem (Theorem B) for this new deformation theory.

We conclude this article with our application of this deformation theory for operadic deformation problems: in §3.1, we introduce some basic definitions on symmetric sequences and operads which will be useful to write our formulas; in §3.2, we study the structure of differential graded pre-Lie algebra with divided powers of the convolution operad; in §3.3, we finally give a proof of Theorem C. tree operad, of which we also recall the definition in this subsection.

In §1.2, we give recollections on the notion of a pre-Lie algebra with divided powers. These objects can be seen as pre-Lie algebras with some extra operations. We will focus on some of these operations called weighted braces that will mimic the quantities which appear in the definition of the circular product.

1.1 Pre-Lie algebras and the rooted tree operad

We will use the following basic definitions. Definition 1.1. A pre-Lie algebra on a ring K is a K-module L endowed with a bilinear morphism

⋆ : L ⊗ L -→ L such that (x ⋆ y) ⋆ z -x ⋆ (y ⋆ z) = (x ⋆ z) ⋆ y -x ⋆ (z ⋆ y).
Any pre-Lie algebra structure on L gives rise to multilinear operations denoted by -{-, ..., -}, called symmetric braces, and defined by induction on the length of the brace by

a{} = a, a{b 1 } = a ⋆ b 1 , ∀n ≥ 1, a{b 1 , ..., b n } = a{b 1 , ..., b n-1 }{b n } - n-1 i=1 a{b 1 , ..., b i-1 , b i {b n }, b i+1 ..., b n-1 }, for all a, b 1 , ..., b n ∈ L.
For our purpose, it will be more convenient to see pre-Lie algebras as algebras over an operad. This operad can be described in terms of rooted trees as follows.

Definition 1.2. (see [5, §1.5]) We call n-rooted tree a non-planar tree with n vertices equipped with a numbering from 1 to n, together with a distinguished vertex called the root. By convention, we chose to put the root at the bottom in any representation of a tree.

We let RT (n) to be the set of all trees with n vertices, and PreLie(n

) = K[RT (n)].
The collection PreLie is endowed with an operad structure. The action of Σ n on PreLie(n) for all n ≥ 1 is given by the permutation of the indices attached to the vertices. The i-th partial composition S • i T ∈ PreLie(p + q -1) of S ∈ RT (p) and T ∈ RT (q) is given by the sum of all the possible trees obtained by putting T in the vertex i of S, with the obvious choice of the numbering (see an example in [5, §1.5]). This operad is also called the rooted tree operad.

One can show that the algebras over the rooted tree operad are precisely the pre-lie algebras (see [5, §1.9]). In particular, the symmetric braces are given by

T n (x, y 1 , ..., y n ) = x{y 1 , ..., y n },
where T n is the following tree, the corolla with n leaves:

T n = 1 2 3 n + 1 ... ,
and T n (x, y 1 , ..., y n ) denotes the application of T n ∈ PreLie(n + 1) on the tensor x ⊗ y 1 ⊗ ... ⊗ y n .

Pre-Lie algebras with divided powers

In this part, we recall the notion of a pre-Lie algebra with divided powers. We obtain this definition as a particular case of a general construction, for algebras over an operad, which we briefly recall.

Every operad P on a suitable monoidal category C gives a functor S(P, -) : C -→ C, called the Schur functor, defined by

S(P, V ) = n≥0 P(n) ⊗ Σn V ⊗n ,
where we consider, in the direct sum, the coinvariants of P(n) ⊗ V ⊗n under the diagonal action of Σ n given by its action on P(n) and its action by permutation on the tensor product V ⊗n . This functor defines a monad and the category of algebras over this monad is the usual category of algebras over the operad P. In particular, pre-Lie algebras in the sense of Definition 1.1 are identified with S(PreLie, -)-algebras.

In the above definition, one can chose to take invariants instead of coinvariants. We obtain a new functor Γ(P, -) : C -→ C defined by

Γ(P, V ) = n≥0 P(n) ⊗ Σn V ⊗n .
If P(0) = 0, this functor also gives a monad (see [8, §1.1.18]). The algebras over this monad are called P-algebras with divided powers. The motivation for this terminology comes from the fact that, in the case of the commutative operad P = Com, the Γ(Com, -)-algebras are precisely the usual commutative and associative algebras over K with divided powers.

Note that if K is a field of characteristic 0, the above monads are in fact isomorphic, with an isomorphism given by the trace map T r : S(P, V ) -→ Γ(P, V ). This morphism is no longer an isomorphism in general when char(K) ̸ = 0.

In the case C = KM od and P = PreLie, if V is free, we however have an isomorphism of modules given by the orbit morphism O : S(PreLie, V ) -→ Γ(PreLie, V ) defined as follows. Let n ≥ 1 and t ∈ PreLie(n) ⊗ V ⊗n be a basis element. We set

O(t) = σ∈Σn/Stab(t) σ.t,
where Stab(t) is the stabilizer of t under the diagonal action of Σ n on PreLie(n) ⊗ V ⊗n . The map O is then extended by linearity on PreLie(n) ⊗ V ⊗n . Theorem 1.3. (A. Cesaro, [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]) Every pre-Lie algebra with divided powers L comes equipped with operations -{-, ..., -} r1,...,rn : L ×n+1 -→ L called weighted braces which satisfy the following identities:

(i) x{y σ(1) , ..., y σ(n) } r σ(1) ,...,r σ(n) = x{y 1 , ..., y n } r1,...,rn , (ii) x{y 1 , ..., y i-1 , y i , y i+1 , ..., y n } r1,...,ri-1,0,ri+1,...,rn = x{y 1 , ..., y i-1 , y i+1 , ..., y n } r1,...,ri-

for all n, m ≥ 0, r 1 , ..., r n , s 1 , ..., s m ≥ 0, 1 ≤ i ≤ n, σ ∈ Σ n and x, y 1 , ..., y n , z 1 , ..., z m ∈ L.
Note that the formula (vi) is written in a form that uses fractions for more convenience, but can be reduced to Z using the other formulas. The process works as follows. Let i such that 1 ≤ i ≤ n.

In the sum, we first fix β 1 , ..., β m and α p,q j for 1 ≤ j ≤ m, 1 ≤ q ≤ r j and p ̸ = i. We obtain a sum with (α i, 1 1 , ..., α i,1 m , ..., α i,ri 1 , ..., α i,ri m ) as variables. We identify this last tuple with a tuple of tuples of the form ((α i, 1 1 , ..., α i,1 m ); ...; (α i,ri 1 , ..., α i,ri m )). Let u be one of these tuples and suppose u = ( u 1 , ..., u 1 t1 , ..., u q , ..., u q tq ) up to permutation. Note that, if u 1 , ..., u q are given, we exactly have

r i ! t 1 !...t q !
such terms occurring in the sum. Then, by using the symmetry formula (i), the formula (iv) and by summing over all such tuples, we have in the sum:

1 j (r j )! r i ! t 1 !...t q ! t 1 !...t q ! x{y 1 {z 1 , ..., z m } α 1,1 1 ,...,α 1,1 m , ..., y 1 {z 1 , ..., z m } α 1,
y i {z 1 , ..., z m } u k = y i {z 1 , ..., z m } α1,...,αm if u k = (α 1 , ..., α m ).
Hence, it gives:

1 j̸ =i (r j )! x{y 1 {z 1 , ..., z m } α 1,1 1 ,...,α 1,1 m , ..., y 1 {z 1 , ..., z m } α 1,r 1 1 ,...,α 1,r 1 m , ..., y i {z 1 , ..., z m } u1 , ..., y i {z 1 , ..., z m } uq , ... ..., y n {z 1 , ..., z m } α n,1 1 ,...,α n,1 m , ..., y n {z 1 , ..., z m } α n,rn 1 ,...,α n,rn m , z 1 , ..., z m } 1,.
..,t1,...,tq,...,1,β1,...,βm .

By iterating this argument on the other terms, we obtain an expression over Z.

The reader can find an example of such a reduction of the formula (vi) in [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]Example 5.11], as well as a proof of the previous theorem (see [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]).

We give the explicit construction of the weighted braces.

Construction 1.4. We regard the weighted braces x{y 1 , ..., y n } r1,...,rn as the action of the corolla

F i ri on the tensor x ⊗ y 1 ⊗ ... ⊗ y 1 r1 ⊗... ⊗ y n ⊗ ... ⊗ y n rn
where we distinguish all the y i 's. If y i ̸ = y j for all i ̸ = j, then we precisely set

x{y 1 , ..., y n } r1,...,rn = γ(OF i ri (x ⊗ y 1 ⊗ ... ⊗ y 1 r1 ⊗... ⊗ y n ⊗ ... ⊗ y n rn )),
where γ is the Γ(PreLie, -)-algebra structure on L.

In order to include the case where some of the y i 's might be the same, let E n to be the free Kmodule generated by a basis e, e 1 , ..., e n . We have an obvious morphism ψ x,y1,...,yn : Γ(PreLie, E n ) -→ Γ(PreLie, L) which sends e to x and each e i to y i . We then take the orbit map at the source and apply this morphism next to have a good definition of the weighted braces.

Remark 1.5. The converse of the previous theorem is also true, provided that L is free as a Kmodule. In fact, by using the same arguments as in [3, Construction 5.14], we can more generally assert that if we have brace operations -{-, ..., -} r1,...,rn : L ×n+1 -→ M which satisfy formulas (i) -(vi) where L and M are K-modules with L free, then these operations extend to a morphism Γ(PreLie, L) -→ M .

Deformation theory of Γ(PreLie, -)-algebras

The main goal of this section is to extend the results proved by Dotsenko-Shadrin-Valette in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF] in the context of a ring of positive characteristic. The main idea is that formulas which define the circular product and the gauge action can be written in terms of weighted brace operations.

In §2.1, we revisit the definition of pre-Lie algebras with divided powers in the dg framework. In particular, we give the analogue of the weighted brace operations. We then make explicit an example of differential graded pre-Lie algebras with divided powers given by differential graded brace algebras.

In §2.2, we define the circular product ⊚ in terms of weighted brace operations that will generalize the one given in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF]. We then show that this induces a group called the gauge group associated to the Γ(PreLie, -)-algebra.

In §2.3, we define the Maurer-Cartan equation in a Γ(PreLie, -)-algebra, and then the Maurer-Cartan set. We also see that the gauge group acts on the Maurer-Cartan set by a similar formula given in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF].

In §2.4, we finally motivate this new deformation theory with an analogue of the Goldman-Millson theorem. This theorem, in particular, has the advantage to be true on integers.

Differential graded pre-Lie algebras with divided powers

As we are dealing with differential graded modules, our first goal is to define and study differential graded pre-Lie algebras with divided powers.

In the following sections, we assume that dg modules are equipped with a cohomological grading convention. We will denote by ⊗ the usual tensor product of graded modules over any ring K. This induces a symmetric monoidal category that we will denote by dgKM od. If there is no possible confusion, then we will denote by ± any sign produced by the Koszul sign rule.

Weighted braces on Γ(PreLie, -)-algebras

Our main goal here is to extend [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]Proposition 5.13] in the context of dg modules. We first begin by a basic definition. Definition 2.1. A differential graded pre-Lie algebra is an algebra over the monad S(PreLie, -) : dgKM od -→ dgKM od.

Equivalently, we can easily see that a differential graded pre-Lie algebra is a graded module L = n≥0 L n endowed with a morphism of graded modules ⋆ :

L ⊗ L -→ L such that (x ⋆ y) ⋆ z -x ⋆ (y ⋆ z) = ±((x ⋆ z) ⋆ y -x ⋆ (z ⋆ y)) and a differential d : L n -→ L n+1 , which satisfies d(x ⋆ y) = d(x) ⋆ y ± x ⋆ d(y),
where ± is the sign yielded by the permutation of x and d.

We now define the notion of a pre-Lie algebra with divided powers in the dg framework. Definition 2.2. A differential graded pre-Lie algebra with divided powers is an algebra over the monad Γ(PreLie, -) : dgKM od -→ dgKM od.

As in the non graded case, we have the orbit map

O : S(PreLie, L) -→ Γ(PreLie, L),
for every Γ(PreLie, -)-algebra L that is free as a K-module, which is defined by the same formula as in §1.2. More precisely, we use the diagonal action of Σ n on PreLie(n) ⊗ V ⊗n where Σ n acts on V ⊗n by permuting the elements of the tensor and produces a sign given by the Koszul sign rule. Proposition 2.3. Let K be a ring. A (differential) graded pre-Lie algebra with divided powers L = n≥0 L n over K comes equipped with operations, called weighted braces, which have the following form.

-If char(K) = 2, weighted braces are maps -{-, ..., -} r1,...,rn : L ×(n+1) -→ L, defined for any collection of integers r 1 , ..., r n ≥ 0, which satisfy all formulas of Theorem 1.3 and preserve the grading in the sense that L k {L k1 , ..., L kn } r1,...,rn ⊂ L k+k1r1+...+knrn .

-If char(K) ̸ = 2, by setting L ev = n≥0 L 2n and L odd = n≥0 L 2n+1 , weighted braces are maps -{-, ...,p , -, ...,q } r1,...,rp,1,...,1 : L × (L ev ) ×p × (L odd ) ×q -→ L, defined for any collection of integers p, q, r 1 , ..., r p ≥ 0, which satisfy all formulas of Theorem 1.3 with a sign and preserve the grading.

Proof. We basically do the same thing as in [ ))),

where γ is the Γ(PreLie, -)-algebra structure on L.

One can check, in both cases, that all the desired formulas are satisfied. □

As in the non graded case, the converse is also true provided that L is free as a K-module in each degree.

We also see that weighted braces satisfy the Leibniz rule with respect to the differential. Proof. This proposition follows directly from the definition of the weighted braces and the commutation of d with the monadic composition. □

We then deduce from Propositions 2.3 and 2.4 that every differential graded pre-Lie algebra with divided powers is in particular a differential graded pre-Lie algebra, with Remark 2.6. Every morphism of Γ(PreLie, -)-algebras preserves the weighted braces:

x ⋆ y = x{y} 1 . Remark 2.5. If Q ⊂ K and if L is a differential graded pre-Lie algebra,
f (x{y 1 , ..., y n } r1,...,rn ) = f (x){f (y 1 ), ..., f (y n )} r1,...,rn .

In the following sections, we deal with a convergence hypothesis to give a sense to series. There are several ways to give such hypothesis. For this paper, we will suppose that L is complete with respect to some filtration ... ⊂ F n L ⊂ F n-1 L ⊂ ... ⊂ F 1 L = L in the sense that L = lim n≥1 L/F n L in dgKM od. We also assume that the filtration is compatible with the weighted braces:

F k L{F k1 L, ..., F kn L} r1,...,rn ⊂ F k+k1r1+...+knrn L.
Moreover, we will formally extend the weighted braces to L + = K1 ⊕ L by y{1} 1 = y and, for n ≥ 1,

1{y 1 , ..., y n } = y 1 if n = 1 0 if n > 1 .
We can extend the previous filtration to L + by setting F 0 L = L + . One can easily check that weighted braces still preserve this new filtration, and we obviously have that L + is complete with respect to this filtration.

Example: differential graded brace algebras

We give an example of dg pre-Lie algebra with divided powers which are given by dg brace algebras, following the idea of the proof in the non graded framework in [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]. and such that f ⟨⟩ = f and

f ⟨g 1 , ..., g n ⟩⟨h 1 , ..., h r ⟩ = ±f ⟨Y 1 , y 1 ⟨Y 2 ⟩, ..., Y 2n-1 , y n ⟨Y 2n ⟩, Y 2n+1 ⟩,
where the sum is over all consecutive subsets Y 1 ⊔ Y 2 ⊔ ... ⊔ Y 2n+1 = {h 1 , ..., h r }, and the sign is yielded by the permutation of the y i 's with the h j 's.

The operad which governs brace algebras is denoted by Brace, and is defined, in arity n, as the K-module spanned by the planar n-trees, i.e. trees with an order on the set of inputs for each vertex (see [3, §6.1] or [4, §2] for some details on the operad Brace).

This operad allows us to represent all operations in brace algebras by the action of a planar tree, or by a planar tree labeled with the inputs. For instance, we have

f g1 h1 h2 g2 g3 h3 = 1 2 3 4 5 6 7 (f ⊗ g 1 ⊗ h 1 ⊗ h 2 ⊗ g 2 ⊗ g 3 ⊗ h 3 ) = f ⟨g 1 ⟨h 1 , h 2 ⟩, g 2 , g 3 ⟨h 3 ⟩⟩.
Remark 2.8. Because the action of the symmetric groups on Brace is free, we have that the trace map induces an isomorphism of monads S(Brace, -) -→ Γ(Brace, -).

We have an inclusion i : PreLie → Brace defined by the symmetrization of trees. Namely, i is obtained by summing over all possible ways to write a given tree t as a planar tree. For instance: .

The map i induces a morphism of monads that can be used to define a Γ(PreLie, -)-algebra structure on every dg brace algebra L, given by the following composition:

Γ(PreLie, L) Γ(Brace, L) S(Brace, L) L ≃ .
We also can explicitly compute the weighted braces.

Theorem 2.9. Every dg brace algebra L is endowed with a Γ(PreLie, -)-algebra structure. Moreover, weighted braces -{-, ..., -} r1,...,rn are explicitly given by f {g 1 , ..., g n } r1,...,rn = σ∈Sh(r1,...,rn)

±f ⟨h σ -1 (1) , ..., h σ -1 (r) ⟩,
where we set r = i r i and (h 1 , ..., h r ) = (g 1 , ..., g 1 r1 , ..., g n , ..., g n rn

) and where ± is the sign induced by the permutations of g i with g j when i ̸ = j.

Proof. Weighted braces are given by elements of the form x = OT i ri (f, g 1 , ..., g 1 r1 , ..., g n , ..., g n rn ) ∈ Γ(PreLie, L). For more convenience, we will set (h 1 , ..., h r+1 ) = (f, g 1 , ..., g 1 r1 , ..., g n , ..., g n rn ) (note that we have added f here so that these h i 's are different from the h i 's of the theorem).

We precisely have:

x = σ∈Σn/ i Σr i ±(σ.T r )(h σ -1 (1) , ..., h σ -1 (r+1) ) because Stab(T r (f, g 1 , ..., g 1 r1 , ... g n , ..., g n rn )) = i Σ ri . Now, because Σ n / i Σ ri is

in bijection with

Sh(1, r 1 , ..., r n ), we can rewrite x as

x = σ∈Sh(1,r1,...,rn) ±(σ.T r )(h σ -1 (1) , ..., h σ -1 (r+1) ).
We now embed PreLie into Brace. The tree T r can be seen in Brace as s∈Σr σ.T r where T r is the planar tree

T r = 1 2 3 r + 1
... and where we embed Σ r into Σ r+1 by fixing 1. We then have, in Γ(Brace, L),

x = σ∈Sh(1,r1,...,rn) s∈Σr ±(σs.T r )(h σ -1 (1) , ..., h σ -1 (r+1) ).

Using that every s ∈ Σ r admits a unique decomposition of the form s = ω.µ where ω ∈ Sh(r 1 , ..., r n ) and µ ∈ i Σ ri , we obtain x = σ∈Sh(1,r1,...,rn) ω∈Sh(r1,...,rn) µ∈ i Σr i ±(σµω.T r )(h σ -1 (1) , ..., h σ -1 (r+1) ).

We now need to compute y = T r -1 (x). We claim that y = ω∈Sh(r1,...,rn) (ω.T r )(f, g 1 , ..., g 1 r1 , ..., g n , ..., g n rn

)
where again we embed Σ r → Σ r+1 by fixing 1.

We compute

T r(y) = ω∈Sh(r1,...,rn) τ ∈Σr+1 ±(τ ω.T r )(h τ -1 (1) , ..., h τ -1 (r+1) ).

Similarly as before, we use that every τ ∈ Σ r+1 admits a unique decomposition of the form τ = σ.µ where σ ∈ Sh(1, r 1 , ..., r n ) and µ ∈ i Σ ri . Then: T r(y) = ω∈Sh(r1,...,rn) σ∈Sh(1,r1,...,rn) µ∈ i Σr i ±(σµω.T r )(h σ -1 (1) , ..., h σ -1 (r+1) ) = x, which gives the result. □

In particular, we can easily compute weighted braces in the case n = 1:

f {g} r = f ⟨g, ..., g r ⟩.
Remark 2.10. Let P be an operad. It is well known that n P(n) is endowed with a brace algebra structure (see [START_REF] Cesaro | On Pre-Lie algebras with divided symmetries[END_REF]). If we denote by γ : P • P -→ P the operadic composition, the brace algebra structure of n P(n) can be written as

p⟨q 1 , ..., q n ⟩ = σ γ(p ⊗ 1 ⊗ ...1 ⊗ q 1 ⊗ 1 ⊗ ... ⊗ 1 ⊗ q n ⊗ 1 ⊗ ... ⊗ 1 ⊗ σ)
where we sum on all pointed unshuffle permutations σ of type (1, ..., 1, m 1 , 1, ..., 1, m n , 1, ..., 1) with p ∈ P(r), g i ∈ P(m i ). By convention, the term in the sum is 0 if n ≥ r + 1.

Consider now the sub module n P (n) Σ . It is not preserved by the brace operations in general, however the Γ(PreLie, -) operations induced by the brace algebra structure of n P(n) preserves n P(n) Σn so that n P(n) Σn is also is Γ(PreLie, -)-algebra.

The gauge group

We can now define an analogue of the circular product given in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF] using the weighted brace operations.

Definition 2.11. Let α ∈ L + and µ ∈ L 0 . We set

α ⊚ (1 + µ) = +∞ n=0 α{µ} n .
Note that this quantity is well defined according to our convergence hypothesis.

By applying this definition in the case Q ⊂ K and using the weighted braces given by Remark 2.5, we retrieve the usual circular product given in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF].

Remark 2.12. One can see that we have 1

⊚ (1 + µ) = 1 + µ = (1 + µ) ⊚ 1 so that 1 is a unit element for ⊚. We thus have ∀µ, ν ∈ L 0 , (1 + µ) ⊚ (1 + ν) = 1 + ν + +∞ n=0 µ{ν} n ,
which shows that ⊚ preserves 1 + L 0 . Lemma 2.13. The circular product ⊚ is associative, in the sense that for all α ∈ L + and µ, ν ∈ L 0 ,

(α ⊚ (1 + µ)) ⊚ (1 + ν) = α ⊚ ((1 + µ) ⊚ (1 + ν)).
Proof. Let α ∈ L + and µ, ν ∈ L 0 . We first have

(α ⊚ (1 + µ)) ⊚ (1 + ν) = +∞ n=0 α{µ} n ⊚ (1 + ν) = +∞ n,p=0 α{µ} n {ν} p .
On the other hand, we have

α ⊚ ((1 + µ) ⊚ (1 + ν)) = α ⊚ 1 + ν + +∞ n=0 µ{ν} n = +∞ p=0 α ν + +∞ n=0 µ{ν} n p .
We thus need to prove that To prove this identity, we use formula (vi) of Theorem 1.3: In this sum, because of the symmetry, some terms occur several times. For a given p and β, we count the number of partitions of p -β = α 1 + ... + α n of the particular form r 1 α 1 + ... + r q α q . We get n( α 1 , ..., α n ) = n! r1!...rq! for this number. We then have 1 r 1 !...r q ! α{µ{ν} α 1 , ..., µ{ν} α 1 , ....., µ{ν} α q , ..., µ{ν} α q , ν} 1,...,1,β = α{µ{ν} α 1 , ..., µ{ν} α q , ν} r1,...,rq,β .

α{µ} n {ν} p = p=β+ n i=1 α i 1 n! α{µ{ν} α 1 , ...,
We conclude by formula (v) of Theorem 1.

□

We now need to find an explicit inverse for a given element 1 -µ with µ ∈ L 0 .

Definition 2.14. Let t be a non-labeled tree with n vertices and µ ∈ L 0 . We set

Ot(µ) = γ(Ot(µ ⊗n )),
for some choice of labeling of t.

Note that because O is Σ-invariant, this quantity does not depend on the choice of a labeling for t. For example, let t be the non-labeled tree t = .

Then Ot(µ) = µ{µ{µ} 2 , µ{µ} 3 , µ} 2,1,1 .

Lemma 2.15. For every µ ∈ L 0 , the element 1 -µ has an inverse in 1 + L 0 for the circular product ⊚ given by

(1 -µ) ⊚-1 = 1 + t∈rRT * Ot(µ),
where rRT * is the set of trees without any labeling and with at least one vertex.

Proof. We first see that this defines a right-inverse for 1 -µ. Indeed, we first have that

(1 -µ) ⊚ 1 + t∈rRT * Ot(µ) = 1 + t∈rRT * Ot(µ) - +∞ k=0 µ t∈rRT * Ot(µ) k .
Then, as every t ∈ rRT * can be uniquely described by its root and branches, we have that every term in the first sum at the right hand side can be uniquely described by an element from the second sum, and vice versa. Formulas from Theorem 1.3 thus give the result.

We now need to prove that it is a left inverse, which is slightly more difficult. We compute

1 + t∈rRT * Ot(µ) ⊚ (1 -µ) = 1 -µ + t∈rRT * Ot(µ) + k≥1 t∈rRT * Ot(µ) {-µ} k .
We focus on one term Ot(µ) from the first sum, for some tree t ∈ rRT * . We say that a vertex of t is extremal if it is not the root of t and if it is connected to one and only one other vertex in t. We denote by m t the number of extremal vertices.

If m t = 0, then t is the trivial tree: Ot(µ) = µ. This term does not appear in the second sum (because k ≥ 1) and vanishes with -µ.

If m t ̸ = 0, the idea is to fix a number 1 ≤ k ≤ m t , and to see which trees we can obtain if we remove k extremal vertices of t. These trees will occur in the second sum and give (-1) k Ot(µ) by adding k copies of -µ. Let X t be the set of extremal vertices of t. Let X t,k be the set of non ordered subsets of X t with k elements. When we remove k extremal vertices, we need to take into account that we can obtain the same tree by removing a different non ordered set of k extremal vertices. For example, if we take the previous tree t = and if we look at the first branch, removing the vertex at the left gives the same tree as removing the vertex at the right. Let t 1 k ,...,t p k k be all the different trees that we can get from t by removing k extremal vertices. We denote by X t i k t,k the subset of X t,k formed by all the vertices that lead to t i k when removing them from t. We then have a disjoint union

X t,k = p k i=1 X t i k t,k .
Each terms Ot i k (µ){-µ} k will then give, among other terms, (-1) k Card(X

t i k t,k )Ot(µ).
When we take the sum over i, we obtain (-1) k Card(X t,k )Ot(µ) = (-1) k m t k Ot(µ). By taking the sum over k, we therefore obtain -Ot(µ) which vanishes with Ot(µ) given by the first sum. □ From Lemma 2.13 and Lemma 2.15, we deduce assertion (i) of Theorem A:

Theorem 2.16. The triple Γ = (1 + L 0 , ⊚, 1) is a group called the gauge group of L.

Maurer-Cartan elements and the Deligne groupoid

We now aim to prove assertion (ii) of Theorem A. We first make explicit the definition of the Maurer-Cartan set. Definition 2.17. A given α ∈ L 1 is a Maurer-Cartan element if it satisfies the Maurer-Cartan equation:

d(α) + α{α} 1 = 0.
We let MC(L) to be the set of all Maurer-Cartan elements of L.

Remark 2.18. In the case Q ⊂ K, we retrieve the classical definition:

d(α) + 1 2 [α, α] = 0,
written with the dg Lie algebra structure on L.

As in the case of characteristic zero, we expect the gauge group to act on the Maurer-Cartan set. Before seeing that, we define a new operation.

Definition 2.19. Let α ∈ L + , β ∈ L and 1 + µ ∈ Γ. We set α ⊚ (1 + µ; β) = +∞ n=0 α{µ, β} n,1 .
Lemma 2.20. We have the following identities:

(α ⊚ (1 + µ)){β} 1 = α ⊚ (1 + µ; β + µ{β} 1 ), α{β} 1 ⊚ (1 + µ) = α ⊚ (1 + µ; β ⊚ (1 + µ)), d(α ⊚ (1 + µ)) = d(α) ⊚ (1 + µ) + (-1) |α| α ⊚ (1 + µ; d(µ)).
Proof. By applying formula (vi) of Theorem 1.3, we find that

(α ⊚ (1 + µ)){β} 1 = +∞ n=0 α{µ} n {β} 1 = +∞ n=0 α{µ, β} n,1 + +∞ n=1 α{µ, µ{β} 1 } n-1,1 = +∞ n=0 α{µ, β + µ{β} 1 } n,1 = α ⊚ (1 + µ; β + µ{β} 1 ),
as well as

α{β} 1 ⊚ (1 + µ) = +∞ m=0 α{β} 1 {µ} m = +∞ p,q=0 α{β{µ} p , µ} 1,q = α ⊚ (1 + µ; β ⊚ (1 + µ)).
Finally, by using the compatibility of d with weighted braces, we obtain,

d(α ⊚ (1 + µ)) = +∞ n=0 d(α){µ} n + (-1) |α| +∞ n=1 α{µ, d(µ)} n-1,1 = d(α) ⊚ (1 + µ) + (-1) |α| α ⊚ (1 + µ; d(µ)),
which concludes the proof of the lemma. □

We can now prove assertion (ii) of Theorem A.

Theorem 2.21. Suppose that µ{α, α} 1,1 = 0 for every µ ∈ L 0 and α ∈ L with odd degree. Then the gauge group Γ acts on the Maurer-Cartan set MC(L) by

(1 + µ).α = (α + µ{α} 1 -d(µ)) ⊚ (1 + µ) ⊚-1
for all (1 + µ) ∈ Γ and α ∈ MC(L).

Note that this theorem is false if we do not assume µ{α, α} 1,1 = 0 for µ ∈ L 0 and α ∈ L with odd degree. The reason is that if we set β = (1 + µ).α following the assertion of the theorem, then the proof of the theorem will exactly give the equality d(β) + β{β} 1 = (µ{α, α} 1,1 ) ⊚ (1 + µ) ⊚-1 . This hypothesis then assures that the action preserves MC(L).

In most cases, this hypothesis is satisfied. We denote three particular situations where this is satisfied:

-if char(K) = 2, because µ{α, α} 1,1 = 2µ{α} 2 = 0; -if L has no 2-torsion (e.g. if 2 ∈ K × ), because by symmetry µ{α, α} 1,1 = -µ{α, α} 1,1 ;
-if the Γ(PreLie, -)-algebra structure of L is induced by a brace algebra structure (in the sense of Theorem 2.9).

Proof. We first need to prove that β = (1 + µ).α is indeed a Maurer-Cartan element. For this, we first remark that applying d on each side of the equality d(µ) = α + µ{α} 1 -β ⊚ (1 + µ), and by using that d(α) = -α{α} 1 and the previous lemma, we have

d(β) ⊚ (1 + µ) = -α{α} 1 -µ{α{α} 1 } 1 + d(µ){α} 1 + β ⊚ (1 + µ; d(µ)).
Moreover, again by the previous lemma, we have

d(µ){α} 1 = α{α} 1 + µ{α} 1 {α} 1 -β ⊚ (1 + µ){α} 1 = α{α} 1 + µ{α{α} 1 } 1 + µ{α, α} 1,1 -β ⊚ (1 + µ; α) -β ⊚ (1 + µ; µ{α} 1 ). Then d(β) ⊚ (1 + µ) = β ⊚ (1 + µ; d(µ)) -β ⊚ (1 + µ; α) -β ⊚ (1 + µ; µ{α} 1 ),
with the remark that µ{α, α} 1,1 = 0.

Finally,

d(β) ⊚ (1 + µ) = -β ⊚ (1 + µ; β ⊚ (1 + µ)).
By the previous lemma, this gives

d(β) ⊚ (1 + µ) = -β{β} 1 ⊚ (1 + µ)
and then (d(β)+β{β} 1 )⊚(1+µ) = 0, that is to say d(β)+β{β} 1 = 0 by composing with (1+µ) ⊚-1 on the right. We thus have proved that β ∈ MC(L).

We now need to check that we have indeed an action of Γ on MC(L). We have that 1 + 0 acts trivially on MC(L), so we just need to prove that ((

1 + ν) ⊚ (1 + µ)).α = (1 + ν).((1 + µ).α).
By hypothesis, we have the following equations:

d(µ) = α + µ{α} 1 -β ⊚ (1 + µ), d(ν) = β + ν{β} 1 -γ ⊚ (1 + ν). Let 1 + λ = (1 + ν) ⊚ (1 + µ) = 1 + µ + ν ⊚ (1 + µ). We compute: α + λ{α} 1 -γ ⊚ (1 + λ) = α + µ{α} 1 + ν ⊚ (1 + µ){α} 1 + d(ν) ⊚ (1 + µ) -β ⊚ (1 + µ) -ν{β} 1 ⊚ (1 + µ) = d(µ) + d(ν) ⊚ (1 + µ) + ν ⊚ (1 + µ; α) +ν ⊚ (1 + µ; µ{α} 1 ) -ν ⊚ (1 + µ; β ⊚ (1 + µ))
by the previous lemma. We then have

α + λ{α} 1 -γ ⊚ (1 + ν) ⊚ (1 + µ) = d(µ) + d(ν) ⊚ (1 + µ) + ν ⊚ (1 + µ; d(µ)) = d(λ),
which proves the theorem. □

We can link these results to the pre-Lie deformation theory developed in [START_REF] Dotsenko | Pre-Lie deformation theory[END_REF] by Dotsenko-Shadrin-Vallette. Indeed, recall that on a differential graded Lie algebra L, we can formally add an element δ which will make the differential internal in the sense that d(µ) = [δ, µ] for all µ ∈ L. Then, when looking at L = L ⊕ Kδ, the Maurer-Cartan equation is reduced to a square-zero equation:

[α, α] = 0.
Moreover, the action of an element λ in the usual gauge group is described by the formula λ.α = (e λ ⋆ α) ⊚ e -λ , which can be written, by doing the variables substitution 1 + µ = e λ , as

(1 + µ).α = (α + µ{α} 1 ) ⊚ (1 + µ) ⊚-1 .
To retrieve our formula, do the final last variable substitution α = α+δ and use that δ{x 1 , ..., x n } = 0 as soon as n ≥ 2 to have (1 + µ).α = (α + µ{α} 1 -d(µ)) ⊚ (1 + µ) ⊚-1 , which is the precise action we have defined in the previous theorem.

We end this section with the definition of the Deligne groupoid.

Proposition-Definition 2.22. Let L be a Γ(PreLie, -)-algebra such that µ{α, α} 1,1 = 0 for every µ ∈ L 0 and α ∈ L with odd degree. We let Deligne(L) to be the category with MC(L) as set of objects and

Mor Deligne(L) (α, β) = {(1 + µ) ∈ Γ | (1 + µ).α = β}.
Then Deligne(L) is a groupoid called the Deligne groupoid of L.

Proof. It is a corollary of the previous theorem. □

An integral Goldman-Millson theorem

We conclude this part with an analogue of the Goldman-Millson theorem. This theorem allows us to give a link between two particular groupoids when changing a dg Lie algebra L to another one L which is quasi-isomorphic to L (see [11, §2.4]).

From now, we suppose that every Γ(PreLie)-algebras in this section are free as K-modules and without 2-torsion. Let A be a local artinian K-algebra with maximal ideal m A . Let L be a Γ(PreLie, -)-algebra (without any convergence hypothesis). Then L ⊗ A is also a Γ(PreLie, -)algebra with the following definitions:

(L ⊗ A) k = L k ⊗ A, γ(Ot(x 1 ⊗ a 1 , ..., x n ⊗ a n )) = γ(Ot(x 1 , ..., x n )) ⊗ a 1 ...a n , d(x ⊗ a) = dx ⊗ a.
To retrieve our convergence hypothesis, we can consider the sub Γ(PreLie, -)-algebra L ⊗ m A . This Γ(PreLie, -)-algebra has a filtration satisfying our convergence hypothesis given by

F n (L ⊗ m A ) = L ⊗ m n A
which is 0 for n big enough, because m A is nilpotent. In particular, our series will be reduced to finite sums. We can now prove Theorem B.

Theorem 2.23. Let K be a noetherian integral domain. Let L and L be two Γ(PreLie, -)-algebras. Suppose that L and L are free as K-modules and that there is no 2-torsion. Let φ : L -→ L be a morphism of Γ(PreLie, -)-algebras such that H 0 (φ) and H 1 (φ) are isomorphisms, and H 2 (φ) is a monomorphism. Then for all local artinian K-algebras A, the induced functor φ * : Deligne(L, A) -→ Deligne(L, A) is an equivalence of groupoids.

Proof. It is easy to check that all the proof given in [11, §2.5- §2.11] remains valid when changing the commutator [x, y] to x ⋆ y -±y ⋆ x.

We only note that the lemma given in [11, §2.8] can be rephrased in our context by the following assertion: for all α ∈ L 1 ⊗ m A , η ∈ L 0 ⊗ m A and u ∈ L 0 ⊗ I, we have

(1 + u + η).α = (1 + η).α -d(u).
It is a simple calculation, using the fact that I.I ⊂ I.m A = 0:

(β -d(u)) ⊚ (1 + u + η) = +∞ n=0 (β -d(u)){u + η} n = +∞ n=0 n k=0 (β -d(u)){u, η} k,n-k = β -d(u) + +∞ n=1 β{η} n = β ⊚ (1 + η) -d(u) = α + η{α} 1 -d(η) -d(u) = α + (u + η){α} 1 -d(u + η).
The other parts of the proof can be easily transposed in a Γ(PreLie, -)-algebra version and remain valid. □ Definition 2.24. Two Γ(PreLie, -)-algebras L and L are quasi-isomorphic if there exists a zig-zag of morphisms of Γ(PreLie, -)-algebras

L = L 0 -→ L 1 ←-... -→ L m-1 ←-L m = L
in which each morphism induces an isomorphism in cohomology. which is natural in A.

Application in homotopy theory for operads

The goal of this section is to establish Theorem C, which gives a computation of π 0 (Map(B c (C), P)) where C is a Σ * -cofibrant coaugmented cooperad, P an augmented operad and B c the cobar construction (see [START_REF] Benoit Fresse | Homotopy of operads & Grothendieck-Teichmüller groups[END_REF] or [START_REF] Loday | Algebraic Operads[END_REF] for a definition of this construction). In the case of a field of characteristic 0, it can be expressed in terms of the Deligne groupoid with the structure of dg Lie algebra of Hom Σ (C, P). We extend this result using a structure of Γ(PreLie, -)-algebra that underlies this dg Lie algebra structure.

In §3.1, we define infinitesimal k-composition and k-decomposition that generalize the usual infinitesimal composition and decomposition operations given in [12, §6.1]. These operations will be used in the next section to write more easily weighted brace operations of the convolution operad.

In §3.2, we recall the definition of the convolution operad Hom(C, P), as given in [12, §6.4.1], and study the Γ(PreLie, -)-algebra structure of Hom Σ (C, P). This structure will be induced by a dg brace algebra structure on Hom(C, P) given by its operadic composition. In the same way that infinitesimal composition and decomposition can be used to express the pre-Lie algebra structure of the convolution operad (see [START_REF] Loday | Algebraic Operads[END_REF]Proposition 6.4.5]), we will use infinitesimal k-composition and k-decomposition to compute weighted brace operations of the convolution operad.

In §3.3, we just use a cylinder object of B c (C) given by Fresse in [9, §5.1] to get our result: the quotient of Hom Σ (C, P) by the gauge action gives π 0 (Map(B c (C), P)).

Infinitesimal compositions and decompositions of an operad and a cooperad

We first introduce some definitions which will be useful for the computations.

Let M and N two symmetric sequences. Recall that we have a monoidal structure on the category of symmetric sequences defined by

M • N (n) = k≥0 M (k) ⊗ Σ k i1+...+i k =n Ind Σn Σi 1 ×...×Σi k (N (i 1 ) ⊗ ... ⊗ N (i k )) ,
with as unit the symmetric sequence I defined by

I(n) = K if n = 1 0 if n ̸ = 1 .
Every elements of M • N (n) can be identified as a tree of the form:

x y 1 i * i * y n i * i * ... ... ... 0 ,
where x ∈ M , y 1 , ..., y n ∈ N .

We now generalize the definition of the infinitesimal composition/decomposition defined in [START_REF] Loday | Algebraic Operads[END_REF], in order to write some formulas in a more convenient way. One can easily check that if we have morphisms of symmetric sequences f : M -→ M and g : N -→ N , then we have a morphism f

• (k) g : M • (k) N -→ M • (k) N given by f • (id I ⊕ g).
Let P an operad with composition γ : P • P -→ P and unit η : I -→ P, and let C a cooperad with coproduct ∆ : C -→ C • C and counit ε : C -→ I. We will suppose that P is augmented, i.e. the unit η : I -→ P admits a retraction π : P -→ I. Equivalently, there exists an operad P with P ≃ I ⊕ P such that the first projection on P is given by π. Similarly, we suppose that C is coaugmented, i.e. the counit ε : C -→ I admits a section s : I -→ C. Equivalently, there exists a cooperad C with C ≃ I ⊕ C such that the first projection is given by ε.

The following definition gives an extension of the usual infinitesimal composition and decomposition operations given in [12, §6.1] for k = 1.

Definition 3.2. Let k ≥ 1.
-We define the infinitesimal k-composition in P as

γ (k) : P • (k) P(n) P • P(n) P(n), γ
where the first map is the inclusion of P • (k) P in P • P.

-We define the infinitesimal k-decomposition in C as

∆ (k) : C(n) C • C(n) C • (k) C(n), ∆
where the last map is the projection of

C • C onto C • (k) C.
Because C is coaugmented, we have that the coproduct ∆ : C -→ C •C preserves the isomorphism C ≃ I ⊕ C in the following sense. We have the isomorphism

C • C ≃ I • I ⊕ C • I ⊕ I • C ⊕ k≥1 C • (k) C.
Then, we get that the restriction of ∆ on I and on C are such that ∆ :

I -→ I • I and ∆ : C -→ C • I ⊕ I • C ⊕ k≥1 C • (k) C. We can then define the infinitesimal k-decompositions on C by ∆ (0) : C C • I ⊕ I • C ⊕ k≥1 C • (k) C C • I ⊕ I • C, ∆ (k) : C C • I ⊕ I • C ⊕ k≥1 C • (k) C C • (k) C, ∆ ∆ for all k ≥ 1.
Beware that these notations are in fact abusive and have nothing to do with the infinitesimal k-decompositions ∆ (k) : C -→ C • (k) C of the cooperad C, which will not be needed in this paper.

Γ(PreLie, -)-algebra structure of the convolution operad

Let M and N be two symmetric sequences of differential graded K-modules. We define a new symmetric sequence Hom(M, N ) in dg K-modules by Hom(M, N )(n) = Hom(M (n), N (n)), the differential graded module formed by the homogeneous morphisms f : M (n) -→ N (n). The differential on Hom(M, N ) is given by

d(f ) = d M • f -(-1) deg(f ) f • d N , for all f ∈ Hom(M, N ). The action of Σ n on Hom(M (n), N (n)) is defined by ∀x ∈ M (n), f σ (x) = σ -1 f (σx),
for all σ ∈ Σ n . Proposition 3.3. (see [START_REF] Loday | Algebraic Operads[END_REF]) Let C a cooperad and P an operad. Then Hom(C, P) has the structure of a dg operad called the convolution operad of C and P.

We recall the operad structure on Hom(C, P). For f ∈ Hom(C, P)(k), g 1 ∈ Hom(C, P)(i 1 ), ..., g k ∈ Hom(C, P)(i k ) the composition γ(f ⊗ g 1 ⊗ ... ⊗ g k ⊗ id) is given by the composite

C(n) C • C(n) C(k) ⊗ C(i 1 ) ⊗ ... ⊗ C(i k ) ⊗ K[id] P(k) ⊗ P(i 1 ) ⊗ ... ⊗ P(i k ) ⊗ K[id] P • P(n) P(n) ∆ f ⊗g1⊗...⊗g k ⊗id γ where n = p i p and, for all σ ∈ Σ n , γ(f ⊗ g 1 ⊗ ... ⊗ g k ⊗ σ) = γ(f ⊗ g 1 ⊗ ... ⊗ g k ⊗ id) σ .
We now suppose that P and C are connected in the sense that P(0) = C(0) = 0 and P(1) = C(1) = K.

We have a dg brace algebra structure on Hom(C, P). Lemma 3.4. Let f ∈ Hom(C, P)(r) and g 1 ∈ Hom(C, P)(p 1 ), ..., g n ∈ Hom(C, P)(p n ). We define the (non symmetric) braces as f ⟨g 1 , ..., g n ⟩ = 1≤i1<...<in≤m σ1,...,σn

((...((f • in g n ) σn • in-1 g n-1 ) σn-1 ...) • i1 g 1 ) σ1
if n ≤ r, where each σ i is a pointed unshuffle permutation of the form (1, ..., p i , ..., 1) with p i placed at the same position as g i in f , and 0 if n > r. Then this definition endows Hom(C, P) with a differential graded brace algebra structure.

Proof. When n ≤ r, we have in fact that

f ⟨g 1 , ..., g n ⟩ = 1≤i1<...<in≤m σ γ(f ⊗ 1 ⊗ ... ⊗ 1 ⊗ g 1 ⊗ 1 ⊗ ... ⊗ 1 ⊗ g n ⊗ 1 ⊗ ... ⊗ 1 ⊗ σ),
where the second sum is taken over all pointed unshuffles σ of type (1, ..., 1, p 1 , 1, ..., 1, p n , 1, ..., 1).

The lemma follows from the structure of dg operad of the convolution operad. □

We denote, for all n ≥ 0, by Hom Σn (M (n), N (n)) the submodule of Hom(M (n), N (n)) formed by all morphisms which commute with the action of Σ n . We let Hom Σ (C, P) = n≥0 Hom Σn (C(n), P(n)).

By Theorem 2.9 and Remark 2.10, Hom Σ (C, P) is endowed with a Γ(PreLie, -)-algebra structure. Moreover, we have a unit element 1 ∈ Hom Σ (I, P) given by η : I -→ P.

We can explicitly describe the weighted braces with one input in terms of infinitesimal decompositions and compositions. We now extend the computation of π 0 (Map(B c (C), P)) on a field K with positive characteristic.

Recall first that we can give an explicit cylinder object for B c (C), where B c is the cobar construction of C, when C is Σ * -cofibrant (see for instance [START_REF] Benoit Fresse | Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads[END_REF] or [START_REF] Loday | Algebraic Operads[END_REF]). Explicitly, let K = Kσ 0 ⊕ Kσ 1 ⊕ σ 01 where deg(σ 0 ) = deg(σ 1 ) = -1, deg(σ 01 ) = 0 and d(σ 01 ) = σ 1 -σ 0 . Then there exists a derivation of operads ∂ such that the free dg operad (F(K ⊗ C), ∂) is a cylinder object for B c (C). We refer to [9, §5.1] for an explicit construction of ∂ and a proof of the previous statement. Theorem 3.7. Suppose that C is Σ * -cofibrant. We then have an isomorphism: π 0 (Map(B c (C), P)) ≃ π 0 Deligne(Hom Σ (C, P)).

Proof. It is well known (see [START_REF] Benoit Fresse | Homotopy of operads & Grothendieck-Teichmüller groups[END_REF] for instance) that π 0 (Map(B c (C), P)) ≃ (M or(B c (C), P), ∼ h ) where ∼ h is the homotopy relation. Recall also that the data of a Maurer-Cartan element α in Hom Σ (C, P) is equivalent to give a morphism of operads ϕ α from B c (C) to P (see [START_REF] Benoit Fresse | Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads[END_REF] or [START_REF] Loday | Algebraic Operads[END_REF]). We just need to show that the action of the gauge group on the Maurer-Cartan set of Hom Σ (C, P) from one Maurer-Cartan α to an other one β is equivalent to give a homotopy from ϕ α to ϕ β .

Let 1 + λ be an element of the gauge group. We define a morphism h : Cyl(B c (C)) -→ P via h : K ⊗ C -→ F(K ⊗ C) by setting h(σ 0 ⊗ γ) = α(γ),

h(σ 1 ⊗ γ) = β(γ), h(σ 01 ⊗ γ) = λ(γ),
where γ is some element of C. We claim that (1 + λ).α = β if and only if h is a homotopy from ϕ α to ϕ β . Accordingly, we must prove the equivalence

d(λ) = α + λ{α} -β ⊚ (1 + λ) ⇔ d(h) = 0
where δ is the differential of M or(B c (C), P).

Because α and β are Maurer-Cartan elements, and by definition of ∂, the second equality is always satisfied for σ ε ⊗ γ with ε = 0, 1 and γ ∈ C. We just need to check this equality on terms σ 01 ⊗ γ for any γ ∈ C: We then have the desired equivalence. □

Proposition 2 . 4 .

 24 For all differential graded pre-Lie algebras with divided powers L = n≥0 L n , the differential d is compatible with the weighted braces in the sense that d(x{y 1 , ..., y n } r1,...,rn ) = d(x){y 1 , ..., y n } r1,...,rn + n k=1 (-1) ε k x{y 1 , ..., y k , d(y k ), ..., y n } r1,...,r k -1,1,...,rn , where ε k = |x| + |y 1 | + ... + |y k-1 |.

1 i r i ! x{y 1

 11 then L is a differential graded pre-Lie algebra with divided powers whose weighted braces are explicitly given by x{y 1 , ..., y n } r1,...,rn = , ..., y 1 r1 , ..., y n , ..., y n rn } in terms of symmetric braces.

Definition 2 . 7 .

 27 A differential graded brace algebra is a differential graded module L endowed with brace operations -⟨-, ..., -⟩ : L ⊗n+1 -→ L which are compatible with the differential d: d(f ⟨g 1 , ..., g n ⟩) = d(f )⟨g 1 , ..., g n ⟩ + n k=1 ±f ⟨g 1 , ..., d(g k ), ..., g n ⟩,

Let

  Deligne(L, A) = Deligne(L ⊗ m A ) the associated Deligne groupoid. As in [11, §2.3], we remark that Deligne(-, -) defines a bifunctor such that, for all morphisms of Γ(PreLie, -)-algebras φ : L -→ L and for all morphisms of algebras ψ : A -→ A, we have the following diagram

Corollary 2 .

 2 25. If L and L are isomorphic, then for all local artinian K-algebras A, the groupoids Deligne(L, A) and Deligne(L, A) are equivalent. More precisely, we have a zig-zag of equivalence of groupoids Deligne(L, A) -→ Deligne(L 1 , A) ←-... -→ Deligne(L m-1 , A) ←-Deligne(L, A)

Definition 3 . 1 .

 31 Let M and N two symmetric sequences. Suppose that N ≃ I ⊕ N for some other symmetric sequence N . For all k ≥ 0, we define a new symmetric sequence denoted by M • (k) N called the k-infinitesimal composite of M and N defined, in each arity n, as the submodule of M • N (n) spanned by trees where exactly k elements at level 2 are in N , and the others in I.

  We now consider an augmented operad P ≃ I ⊕ P and a coaugmented cooperad C ≃ I ⊕ C. Assume that we have a decomposition by weight C ≃ k≥1 C(k) . This hypothesis is satisfied, for instance, for the coaugmentation of a Koszul cooperad P ! . We thus have an isomorphismHom Σ (C, P) ≃ k≥1 Hom Σ (C (k) , P)which is compatible with the Γ(PreLie, -)-algebra structure on Hom Σ (C, P). We then define a filtration on Hom Σ (C, P) byF n (Hom Σ (C, P)) = k≥n Hom Σ (C (k) , P)such that Hom Σ (C, P) is complete for this filtration.Because C and P are connected, the isomorphism and the filtration extend to Hom Σ (C, P) ≃ Hom Σ (I, P) × k≥1 Hom Σ (C (k) , P).

Lemma 3 . 5 .⊗ 1 ⊗⊗ 1 ⊗Theorem 3 . 6 . 3 . 3

 35113633 Let f , g ∈ Hom Σ (C, P). Then f {g} k is given by the compositeC C • (k) C P • (k) P P ∆ (k) f • (k) g γ (k).Proof. By definition, we have thatf {g} k = f ⟨g, ..., g k ⟩ = 1≤i1<...<i k ≤n σ1,...,σ k ((...((f • i k g) σ k ...) σ k-1 )• i1 g) σ1where f ∈ Hom Σ (C, P)(n). This can be written in terms of the operad structure on Hom Σ (C, P) byf {g} k = 1≤i1<...<i k ≤n σ γ(f ⊗ 1 ⊗ ... ⊗ 1 ⊗ g i1 ... ⊗ 1 ⊗ g i k ... ⊗ 1 ⊗ σ)which gives the desired identity. □In particular, we find that the pre-Lie algebra structuref ⋆ g = f ⟨g⟩ = n i=1 σ (f • i g) σis given by the composite The circular product of two elements f = 1 + f , g = 1 + g of the gauge group of Hom Σ (C, P) is given by f ⊚ g : Because f |I = g |I = 1, we have that f ⊚ g |I = 1. We thus need to show the equality on C. Recall that we have infinitesimal decompositions on C denoted by ∆ (0) and ∆ (k) for k ≥ 1 such that ∆ |C = ∆ (0) ⊕ k≥1 ∆ (k) . The map ∆ (0) will give f + g, and each ∆ (k) will give f {g} k according to the previous lemma. We thus have that the composite in the statement of the theorem gives1 + f + n≥0 f {g} n which is exactly f ⊚ g.□ Computation of π 0 (Map(B c (C), P))

  d(h)(σ 01 ⊗ γ) = d(h(σ 01 ⊗ γ)) -h(∂(σ 01 ⊗ d(γ))) = d(λ(γ)) -λ(d(γ)) -α(γ) + β(γ) -λ{α} 1 (γ) + (β ⊚ (1 + λ)(γ) -β(γ)) = d(λ)(γ) -α(γ) -λ{α} 1 (γ) + β ⊚ (1 + λ)(γ).

  1,ri+1,...,rn , (iii) x{y 1 , ..., λy i , ..., y n } r1,...,ri,...,rn = λ ri x{y 1 , ..., y i , ..., y n } r1,...,ri,...,rn , (iv) x{y 1 , ..., y i , y i , ..., y n } r1,...,ri,ri+1,...,rn = r i + r i+1 r i x{y 1 , ..., y i , ..., y n } r1,...,ri-1,ri+ri+1,ri+2,...,rn , (v) x{y 1 , ..., y i + y i , ..., y n } r1,...,ri,...,rn = , ..., z m } 1,...,1,β1,...,βm ,

			ri
			x{y 1 , ..., y i , y i , ..., y n } r1,...,s,ri-s,...,rn ,
			s=0
	(vi) x{y 1 , ..., y n } r1,...,rn {z 1 , ..., z m } s1,...,sm =
	i si=βi+ α .,.	1 j (r j )!	x{y 1 {z 1 , ..., z m } α 1,1 1 ,...,α 1,1 m , ..., y 1 {z 1 , ..., z m } α 1,r 1 1	,...,α 1,r 1 m	,
	..., y n {z 1 , ..., z m } α n,1 1 ,...,α n,1 m , ..., y n {z 1 , ..., z m } α n,rn 1	,...,α n,rn m	, z 1

  3, Proposition 5.10]. Let x, y 1 , ..., y n ∈ L. Let E n be the graded K-module generated by e, e 1 , ..., e n with matching degrees. We have an obvious morphism of graded modules from E n to L sending x to e and y i to e i . This gives rise by functoriality to a morphism ψ x,y1,...,yn : Γ(PreLie, E n ) -→ Γ(PreLie, L). Let p = , ..., y n } r1,...,rn := γ(ψ x,y1,...,yn (OF p (e ⊗ e 1 ⊗ ... ⊗ e 1

	n
	r i . We set
	i=1
	x{y 1

r1 ⊗... ⊗ e n ⊗ ... ⊗ e n rn

  µ{ν} α n , ν} 1,...,1,β , , ..., µ{ν} α n , ν} 1,...,1,β .

	which gives
	+∞ n,p=0	α{µ} n {ν} p =	+∞ n=0	+∞ p=0	p β=0 p-β= n i=1 α i	1 n!	α{µ{ν} α 1

Acknowledgements

I acknowledge support from the Labex CEMPI (ANR-11-LABX-0007-01) and from the FNS-ANR project OCHoTop (ANR-18CE93-0002-01). I also acknowledge Benoit FRESSE for his advice and support throughout the writing of this article.


Tania Alhajj
email: taniahajj@imt-atlantique.fr

Nicolas Huin
email: nicolas.huin@imt-atlantique.fr

Karine Amis
email: karine.amis@imt-atlantique.fr

Xavier Lagrange
email: xavier.lagrange@imt-atlantique.fr

Radio resource allocation in low-to medium-load regimes for energy minimization with C-RAN

Keywords: C-RAN, radio resource allocation, 5G, energy consumption

Radio resource allocation is critical in order to manage available bandwidth and optimize network behavior. In Centralized-Radio Access Networks (C-RAN), simple Radio Units (RUs) are deployed and can be controlled remotely to allow cooperation. In this paper, we consider micro-cell systems with low-to-medium load regimes. We study the resource allocation problem with minimization of the energy consumption due to radio transmission and signal processing. We propose to mix two transmission modes over a reference period allowing resource reuse: one with all RUs transmitting the same signal, the other with each RU transmitting a specific signal. We impose the same target rate for each user equipment (UE), the same transmit power for the RUs, and a maximum resource block (RB) number, which defines the constraints. The resulting optimization problem is formulated so as to be mixed integer linear programming (MILP). Simulations are run using CPLEX Python API and show the efficiency of the proposed scheme to increase the capacity while consuming the least energy. We also observe that the mode with all RUs simultaneously serving the same UE is more efficient (fewer RB and lower consumed energy) than the mode with a single RU activated at a time.

I. INTRODUCTION

Fifth Generation (5G) wireless networks are intended to deliver high data rates, serve large numbers of users, and achieve high reliability and low latency. Energy consumption is increasing due to this evolution in demands, indirectly increasing carbon footprints. Due to environmental concerns, improving the energy efficiency of systems has recently become a major issue [START_REF]The energy use implications of 5G: Reviewing whole network operational energy, embodied energy, and indirect effects[END_REF]. Some use cases have strong variations of the load (e.g., air terminal), and the network is often built to address the high regime (to simultaneously serve a high number of active users). However, periodically (e.g., at night or on weekends in business centers), only a few devices are active. One way to improve the energy efficiency of wireless systems is to make the network reactive to load variations.

Centralized-RAN (C-RAN) is an evolution of the traditional Radio Access Network (RAN) supported by 5G networks. In this centralized architecture, Base Station (BS) functions are split between two units: Radio Units (RUs) located at tower sites to achieve the desired coverage, and Centralized Units (CUs) where all processing functions can be centralized [START_REF] Larsen | A survey of the functional splits proposed for 5G mobile crosshaul networks[END_REF].

The reduction in power consumption with C-RAN has attracted the interest of researchers. The authors of [START_REF] Huang | Joint scheduling and beamforming coordination in cloud radio access networks with qos guarantees[END_REF] have proposed the scheduling and coordination of beamforming in C-RAN. Depending on their position, the User Equipments (UEs) are merged into groups, and each group is served by a set of RUs. Radio resources can be reused by RUs from the same or different groups. Cooperative beamforming between RUs of the same group is envisaged to combat interference. Their proposal proved to be highly energy-efficient. Nevertheless, in their study, they consider that the number of RUs in a cluster must be at least equal to the number of UEs. Needing as many RUs as UEs can increase the cost of deployment.

The optimization of the UE-RU association is another way to manage the interference produced in a multi-cell scenario. In [START_REF] Luo | Downlink and uplink energy minimization through user association and beamforming in c-ran[END_REF], the energy consumption on the DownLink (DL) and UpLink (UL) in a C-RAN is minimized, and joint UE-RU association and beamforming solution are used to manage interference. In this study, one UE can be connected to one RU in each direction and can be served through beamforming. More recently, a distributed UE-RU association and an RU clustering in C-RAN was proposed in [START_REF] Taleb | A fully distributed approach for joint user association and rrh clustering in cloud radio access networks[END_REF]. They divide their problem into two subproblems: UE association and RU clustering. Each UE can only be connected to one RU. The RUs connected to the same CU share the radio resources available on this CU without interference. RUs connected to different CUs interfere if they use the same resources. They showed lower energy use, increased throughput, and quick adaptation to traffic variations. Energy minimization through bandwidth allocation and UE-RU association have been considered separately. However, it is essential to treat these tasks together, as the association problem has a direct impact on resource allocation and both have an impact on energy consumption.

All mentioned studies have focused on fully loaded systems to optimize resource allocation. Few studies have proposed allocation solutions in low-load systems where some network resources are available and can be exploited to improve network performance, for example by reducing energy consumption. The trade-off of bandwidth during a low-load regime with the traditional BSs system has been proposed in [START_REF] Videv | Energy-efficient scheduling and bandwidthenergy efficiency trade-off with low load[END_REF]. The method involves allocating more bandwidth to UEs since some bandwidth is unused and therefore available. The allocation of a higher number of resources enables the use of lower-order modulation and thus decreases the total energy consumption while maintaining the UE throughput request.

The BSs consume a significant part of the total energy. The predominant source of consumption depends on the cell size. Radio power generation is critical in macro-cells, unlike in micro-cells where distances and ranges are shorter [START_REF] Gunther | Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]. Energy consumed by processing units and electronic components can therefore be predominant. So, in small-cell scenarios, C-RAN deployment can be advantageous as it centralizes the powerconsuming processing part while distributing the low-energy radio part to achieve the desired coverage.

This paper considers the reduction of energy use through optimized radio resource allocation for the DL of a micro-cell C-RAN system. We define a set of network configurations designed for the low-to-medium load regime. Given a fixed RU transmit power, a maximum Resource Block (RB) number, a target user rate, and an active users number, we search for the best UE-RU association and radio resource allocation that minimize energy consumption.

This paper is structured as follows. The system model is provided in Section II. Then, in Section III we detail the energy consumption model that we use. The considered resource allocation and the energy consumption computation are given in Sections IV and V, respectively. The optimization problem is formulated and explained in Section VI. In Section VII, we discuss the results before concluding in Section VIII.

II. SYSTEM MODEL

A. Network deployment

We consider an indoor service area where micro-cells are deployed with C-RAN. Multiple RUs are implemented to achieve indoor coverage and are connected to a CU hosted in a central site. The number of deployed RUs is given by I. Each RU i position is fixed and located by its coordinates (x i , y i ).

To keep simple mathematical expressions, we consider as our reference system a service zone that is a rectangular area of dimensions A×B m 2 . The position of UE j , given by (x j , y j ), is assumed to be uniformly distributed in the service area such that 0 ≤ x j ≤ A and 0 ≤ y j ≤ B. Note that any kind of deployment could be considered with the same model.

B. Modes of operation

In 5G New Radio (NR), Synchronization Signal Blocks (SSBs) are periodically transmitted by the BS to achieve coverage, allowing the UE to know the identity of the available cell, to synchronize, and to connect to it [START_REF] Parkvall | NR: The New 5G Radio Access Technology[END_REF].

In our work, we consider three operating modes. In lowload systems, serving one UE at a time is sufficient. Thus, in mode 1, all RUs serve simultaneously one UE at a time. In this mode, the system is one big cell and the UE is unaware of the presence of multiple RUs (see Fig. 1a). A virtual RU, denoted by RU I + 1 , operates in this mode as if all RUs were active to serve one UE simultaneously. In mode 2, one RU can be active at a time and can serve one UE on specific RBs. Each RU is in an independent cell (see Fig. 1b). In mode 2, the UE is assigned to the best-serving RU, i.e., the one that provides the highest Signal to Interference and Noise Ratio (SINR). In modes 1 and 2 there is no resource reuse. In a medium-load system, serving multiple UEs in parallel is important to flow the needed load (up to the number of RUs users served on the same RBs) but there are no specific techniques needed like beamforming (up to the number of beams users served on the same RBs). If needed, multiple RUs can transmit on the same RB in the absence of interference constraints. To this end, mode 3 is a mixture of modes 1 and 2 with possible resource reuse. Mode 3 has different possible configurations and is detailed later in Section IV.

C. Traffic model

Many previous studies have considered a full-buffer traffic model. This model considers that there is an infinite amount of data to transmit. All users are active at all times. In our study, we consider a user-requested service that needs a certain target DL data rate R T to be guaranteed [START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for cloud radio access networks[END_REF].

D. Propagation model

We consider an Indoor Factory where the heights of the receiving and transmitting antennas are lower than the average clutter height. The clutter density is less than 40% [START_REF] Jiang | 3gpp standardized 5g channel model for iiot scenarios: A survey[END_REF]. We take the path-loss model given in Table 7.4.1-1 of [START_REF]Technical specification group radio access network[END_REF] defined by the 3rd Generation Partnership Project (3GPP) in dB for an Indoor Factory Sparse clutter, Low BS height (InF-SL) with:

PL InF-SL,dB = 25.5 log(d i,j ) + 33 + 20 log(f c ), (1)
where log(u) is the logarithm with base 10 of u, f c the central frequency in GHz, d i,j the 2D distance between RU i (x i , y i ) and UE j (x j , y j ) having the same height, measured in m:

d i,j = (x j -x i ) 2 + (y j -y i ) 2 .
The path-loss parameters α and r 0 can be derived from PL dB = 10α log(d i,j ) -10 × α log(r 0 ) being defined by (1) for a given value of f c . A log-normal shadowing is considered by 3GPP. It is modeled by e σξ with ξ being a normal random variable (r.v.): ξ∼N(0, 1), σ= ln (10) 10 σ dB , and ln(u) the natural logarithm of u. We also assume the RUs to be equipped with directive antennas with the following gain in dB [START_REF]Technical Specification Group Radio Access Network; Evolved universal terrestrial radio access (E-UTRA); radio[END_REF]:

G dB (θ i,j ) = G A -min 12 θ i,j θ 3dB 2 , A m , (2)
where G A is the antenna gain in the boresight direction, θ 3dB the 3 dB-beamwidth, A m the maximum attenuation, and θ i,j the angle between RU i and UE j , measured from the antenna boresight: θ i,j = arctan xj -xi yj -yi . The UE is considered equipped with an omnidirectional antenna.

Thus, the power received by UE j (function of (x j , y j )) from RU i with transmission power P t per RB is:

P r,i,j = P t r 0 d i,j α G(θ i,j )e σξi,j . (3)
The power received by RU I + 1 is: P r,I+1,j = I i=1 P r,i,j .

III. ENERGY CONSUMPTION MODEL

Reference [START_REF] Jung | Modeling of power consumption for macro-, micro-, and rrh-based base station architectures[END_REF] gives a detailed power consumption model for C-RAN with a correlation between BS component consumption and consideration of load variations. Fig. 2 shows the BS diagram with C-RAN in the DL.

• The RU consumes energy through the radio power generation (high power amplifier consumption model taken from [START_REF] Fourikis | 2 -from array theory to shared aperture arrays[END_REF] with η PA as Power Amplifier (PA) gain) and fixed power consumption (P RU,f ) of electronic components such as the low-pass filter, mixer, digital-to-analog converter, and optical-to-electrical (O/E) converter. • In the CU, the main consumption is due to the energy consumption of Signal Processing (SP) E SP (detailed in Section III-B). A fixed power P CU,f is consumed by the E/O converter, and the SP fixed power is denoted by P SP,f . • In the RU and CU, the AC/DC power supply is characterized by its efficiency η AC/DC that describes how much actual power is delivered relative to the input power [START_REF] Calwell | Active mode power supply efficiency: key issues, measured data and the design competition opportunity[END_REF]. We consider instantaneous RU and CU switch-on and switchoff without any energy cost.

A. Reference period of study

We evaluate the energy consumption of the BS including the consumption of SP, electronic components, and radio power generation. The SP consumes energy, whose amount depends on the number of allocated RBs to serve the UE. The fixed power consumed by electronic components and the power consumed for radio power generation are evaluated over the transmission duration to compute their energy consumption. Thus, we choose a reference period T equal to the SSB transmission period, i.e., there is one SSB transmission in T . We determine the transmission duration by the number of RBs needed to guarantee the target throughput R T , divided by M RB the number of available RBs per slot, and multiplied by T .

B. Packet processing energy consumption

The energy consumption model of the SP unit is taken from [START_REF] Jung | Modeling of power consumption for macro-, micro-, and rrh-based base station architectures[END_REF] and it consists of two main parts. A fixed amount of this energy consumption comes from the SP unit components, mainly from the motherboard, peripherals, and fans. The resulting fixed power is denoted by P SP,f and evaluated over the activity period. The second contribution to the SP energy consumption is variable and used to encode packets. This energy is divided into two terms. The first one in Joules per bit, denoted by E SP,v , accounts for layer 1 functions (coding, modulation...). The second one in Joules per packet, denoted by E SP,f , corresponds to signal processing run in upper layers. The SP energy consumed for J UEs on the DL is:

E SP = T DL P SP,f Power Energy proportional to TDL + E SP,v L P,T + E SP,f N Pkt
Processing energy proportional to LP,T and NPkt ,

where L P,T =JR T T is the length of transmitted packets (N Pkt packets), and T DL the RU/CU DL activity period during T .

IV. DOWNLINK RESOURCE ALLOCATION

The I RUs are independent and share the same resources available in the system. Thus, resource reuse is possible. It consists in allocating the same resources to two different users, each served by an RU if the interference conditions allow it.

Let us refer to the jth UE by UE j where 1 ≤ j ≤ J and to the ith RU by RU i where 1 ≤ i ≤ I + 1. We introduce an activity state for each RU denoted by s i for RU i :

s i = 1 if RU i is active 0 otherwise (5)
The activation configuration ν determines which RUs are idle and which are active.

d(ν, i) = 1 if RU i is active in configuration ν 0 otherwise (6)

A. RU-UE-configuration assignment

The objective is to minimize the energy consumption while serving UEs on the available RBs. The optimization problem determines the UE's serving RU and the number of needed RBs for this service (depending on the chosen configuration by the optimization problem). To allocate the RBs, we need to determine which RU serves the UE and in which configuration. This is determined by x = (x i,j,ν ), where:

x i,j,ν = 1 if RU i is serving UE j with configuration ν 0 otherwise (7) This decision variable is the output of the optimization problem that schedules the service of UEs. The constraint on this variable ensures that each UE is served and only served once:

i,ν x i,j,ν d(ν, i) = 1 ∀ j. (8)
B. DL transmission model 1) SINR computation: On the DL, if multiple RUs transmit on the same RBs, they interfere with each other. Thus, the perceived SINR by the UE depends on the powers of the signal of interest, the interfering signal, and the receiver's noise: 2) Computation of the number of needed RBs: We aim to provide each UE with the target data rate R T . The perceived data rate per RB with configuration ν and a UE-RU pair (j, i), is computed using the modified Shannon formula [START_REF] Mogensen | Lte capacity compared to the shannon bound[END_REF]:

γ i,j,ν =      Pr,i, j i ′ ̸ =i P r,i ′ , j d(ν,i ′ )+Np if ν ̸ = 2 I , d(ν, i) = 1 ∀i ̸ = I +1 I i=1 Pr,i,j Np if ν = 2 I , d(ν, I + 1) = 1 n.a. otherwise (
R (i,j,ν) = w RB δ BW log 2 1 + γ i,j,ν δ SINR , (10)
where δ BW and δ SINR are correction factors of the bandwidth and the SINR, respectively. The number of RBs per slot to serve UE j by RU i in configuration ν with R T is:

m(i, j, ν) = R T R (i,j,ν) . (11)
Each UE requests the same service with R T , but the number of needed RBs varies for each UE. This number relies on the perceived rate (see [START_REF]Technical specification group radio access network[END_REF]) determined by the SINR γ i,j,ν in [START_REF] Jiang | 3gpp standardized 5g channel model for iiot scenarios: A survey[END_REF].

C. Occupied resources on the system level

Each UE j is served by an RU i in a configuration ν using m(i, j, ν) RBs. Each active RU in a configuration occupies a number of RBs. The number of occupied RBs at the system level y ν during configuration ν is determined by the maximum number of RBs occupied by all active RU in that configuration:

y ν = max i   j m(i, j, ν)x i,j,ν   . (12)
V. DL DATA TRANSMISSION ENERGY CONSUMPTION The optimization problem objective is to minimize the CU and RUs energy consumption. The CU's energy consumption has three parts. The first part is the fixed power P CU,f consumed by electronic components during the CU's activity period as soon as the system is active. It is independent of the number of active RUs or served UEs. The second part is the fixed SP energy E SP,f consumed per processed packet per UE. We consider one packet per slot. Thus, to evaluate the energy consumption, we count the number of occupied slots to serve all UEs. Here, each RB is counted as many times as used to serve different UEs, i.e. an RB used to serve two UEs by two RUs is counted twice. The third part is the energy consumption that depends on the packet length E SP,v . It is multiplied by the number of served UEs. Constants P CU,f , E SP,f , and E SP,v include η AC/DC . We refer the reader to Section III-B for their definition. The CU's energy consumption is thus:

E DL CU (x) = ν y ν M RB P DL CU,f T + M r +M v M RB E SP,f T T s + JE SP,v L P,T (13)
where ν y ν is the total number of RBs occupied by the selected configurations on the system level, T s the slot duration, M r = I i=1 j ν m(i, j, ν)x i,j,ν , and M v = j m(I+1, j, 2 I )x I+1,j,2 I the number of RBs occupied by the I real RUs and RU I + 1 , respectively.

RUs consume energy through electronic components and radio power generation. Each RU is active for the duration of the required RBs to meet the UE's target data rate. The energy consumed by the I RUs counts the transmission and fixed power consumption during all the RBs needed to serve the J UEs. The virtual RU assumes simultaneous activation of the I real RUs to serve a UE. The energy consumed by I real RUs and a virtual RU to serve J UEs is:

E DL RU (x) = M r + IM v M RB P RU,f + M RB P t η T T, (14)
where P RU,f includes η AC/DC , and η T = η PA η AC/DC .

VI. OPTIMIZATION PROBLEM FORMULATION This

Section is dedicated to the formulation of the optimization problem. The cost function is the BS energy consumption involved to transmit data on the DL. The first decision variable x = (x i,j,ν ) is given in Section IV-A. It determines the serving RU and configuration for each UE. We rewrite [START_REF]Technical Specification Group Radio Access Network; Evolved universal terrestrial radio access (E-UTRA); radio[END_REF] as y ν ≥ j m(i, j, ν)x i,j,ν . The second decision variable is y ν of length 2 I . It represents the system-level number of RBs. The problem is formulated with constraints on these variables: The objective function in (15a) is the minimization of the DL data transmission energy consumption (( 13) and ( 14)) and is linear. Constraints explained hereinafter are set to ensure proper system operation and service scheduling among users. Constraint C1 in (15b) is defined in Section IV-A. It ensures that all UEs are served once by one RU in one configuration. Constraint C2 in (15c) is the blocking constraint that forces the algorithm to find a solution without exceeding the number of RBs available in the system (M RB ). The number of occupied RBs on the system level in configuration ν is determined by y ν and is used to evaluate the system blocking. Constraint C3 in (15d) defines x i,j,ν as binary variables.

minimize x,yν E DL CU (x, y ν ) + E DL RU (x, y ν ) (15a) s.t. C1: i,ν x i,j,ν d(ν, i) = 1 ∀ j (15b) C2: ν y ν ≤ M RB (15c) C3: x i,j,ν ∈ {0, 1} ∀ i, j, ν (15d) C4: y ν ≥ j m(i, j, ν)x i,j,ν ∀ i, ν (15e)
In our resolution, the preliminary task is the computation of m(i, j, ν) for all triplets (i, j, ν). Then, these values are used in the optimization resolution. This step linearizes constraint C4 that imposes restrictions on the number of occupied systemlevel RBs in configuration ν. In each configuration, each active RU serves a certain number of UEs. The RBs occupied by RU i are the sum of the RBs necessary to serve all UEs served by this RU i ( j m(i, j, ν)x i,j,ν ). The RBs occupied by each RU are evaluated for all active RUs in the considered configuration (∀ i). The number of occupied RBs at the system level in configuration ν (y ν ) must be greater than the RBs occupied by each active RU in ν. This must be true for all the considered configurations (∀ ν). The formulated problem is thus Mixed Integer Linear Programming (MILP).

VII. RESULTS AND DISCUSSIONS

A. Simulations

We consider four RUs in a rectangular area fixed as in Fig. 3. The load in the network is represented by J, the number of available UEs. For each load, J UEs' positions inside the rectangular area are randomly chosen with random shadowing. We calculate m(i, j, ν) for the available UEs in all the possible configurations. The optimization problem is run using CPLEX Python API. These steps are done 10000 times for each value of J. The simulations were performed on a remote server with 792 GB of RAM and 96 cores with simultaneous multithreading. Table I lists the simulation parameters.

The running time increases with J because the number of variables and constraints in (15) increases. One simulation takes less than 0.2 s for J < 35. When the required total RB number gets close to the maximum available RB number (M RB ), the running time increases but remains acceptable: 0.5 s per simulation for J = 45 and 7.2 s for J = 50. Shadowing standard deviation 5.7 [START_REF]Technical specification group radio access network[END_REF] In [START_REF] Calwell | Active mode power supply efficiency: key issues, measured data and the design competition opportunity[END_REF], only C2 can stop the solver from finding a solution. Thus, when the solver fails, the system is considered blocked. The blocking probability estimate is the system blocking occurrences over the number of iterations. The objective function of successful trials is averaged to get the energy consumption.

B. Blocking probability

Now, let us evaluate the blocking probability of the problem in [START_REF] Calwell | Active mode power supply efficiency: key issues, measured data and the design competition opportunity[END_REF]. The blocking occurs when the system requires more than M RB RBs to serve the UEs. Fig. 4 shows the blocking probability as a function of the number of UEs present in the network for modes 1, 2, and 3, with a transmission power per RB of -13 dBm. We see that considering different configurations with four RUs and a virtual RU increases the system capacity. For a blocking probability of 10 -2 , in mode 2, 17 UEs can be served compared to 32 in mode 1, and 50 in mode 3. The better SINR in mode 1 increases the perceived data rates and reduces the number of RBs needed to achieve R T , enabling more UEs to be served before reaching M RB . The proposed solution (mode 3) increases the system capacity by 56% compared to mode 1. This increase can be further explained by examining the configurations' usage.

In Fig. 5, we evaluate the configurations' usage in mode 3. The X-axis corresponds to the set of available configurations. Each configuration is identified by the active RUs it contains. For example, configuration 8 is denoted by RU [START_REF]The energy use implications of 5G: Reviewing whole network operational energy, embodied energy, and indirect effects[END_REF][START_REF] Luo | Downlink and uplink energy minimization through user association and beamforming in c-ran[END_REF], which means that RU 1 and RU 4 are active and serve different UEs on the same RBs. On the Y-axis, we count the number of RBs where each configuration is used. For example, for J = 50, when RU 1 and RU 4 are active, they reuse 2.7 RBs to serve different UEs on these RBs. Fig. 5 has a zoom-in in the box to better see configurations used on less than 6 RBs. For low loads like J = 10, we see in Fig. 5, that the solver of the optimization problem tends to serve UEs with configurations having one active RU: first four configurations (transmission by one RU, mode 2) and mostly in ν = 16 where only RU I + 1 is active (simultaneous transmissions, mode 1). Simultaneous transmissions increase the perceived SINR and reduce the number of RBs where the BS is active to reach R T . This reduction achieves the energy minimization objective.

When the load increases and the maximum system capacity (M RB ) tends to be reached, configurations with resource reuse are used as shown in Fig. 5. This shows the importance of reusing resources to serve a larger number of UEs. For J < 40, resource reuse by two RUs is applied on less than one RB per configuration. For J ≥ 40, more configurations with resource reuse are used (on more than one RB per configuration). The configurations with three and four RUs reusing resources are increasingly used in these cases. For 45, 50, and 55 UEs, the use of the first four configurations is reduced and with 50 and 55 UEs, the use of the last configuration is also reduced.

Thus, the usage of configurations without resource reuse initially increases with J until reaching a maximum value, after which it decreases. Conversely, the usage of configurations with resource reuse consistently increases with J.

C. Energy consumption

Fig. 6 shows the energy consumed by the BS for DL data transmission. The curves end at J = 32, 17, and 50 for modes 1, 2, and 3, respectively (see Fig. 4 at 10 -2 blocking probability). Although all RUs transmit in mode 1, the energy consumed is lower than mode 2 where one RU serves a UE. Simultaneous transmissions improve the received SINR and reduce the number of RBs where the BS is active. This shows the dominance of CU's energy use at P t = -13 dBm (Fig. 7).

Energy consumption in mode 1 is comparable to mode 3 when J ≤ 32. In mode 3, energy consumption increases linearly with the number of UEs up to J = 45. However, beyond this point energy consumption escalates more rapidly due to the increased interference caused by resource reuse (see Fig. 5). This requires more RBs to serve UEs and results in higher energy consumption. Consequently, Fig. 5 shows that resource reuse is not widely used unless it becomes necessary, i.e. when M RB is reached without using resource reuse. Figures 5 and6 show how load (J) affects configuration choices. Mode 1 is more energy-efficient than mode 2. In mode 3, when there are enough RBs to serve UEs without reuse, the preferred configuration is with RU I + 1 . But with higher load, RB reuse is necessary, despite increased energy consumption.

With the proposed optimization problem, P t has a direct impact on resource allocation and energy consumption since the allocation is computed in order to minimize the energy consumption. Increasing P t has an impact on the number of needed RBs. For example, it improves the perceived SINR and reduces the number of RBs needed in the case of a single transmission without interference (first four configurations). In Fig. 7, we examine the energy consumption distribution with a variation of P t for J = 40 in mode 3. The energy consumed is divided into five parts including the fixed consumption of the CU (f (P CU,f )), signal processing in the CU (f (E SP,v ) and f (E SP,f )), the fixed consumption of the RU (f (P RU,f )), and the consumption of the PA for radio power generation (f (P t )).

We can see that increasing the transmission power from -13 dBm to 12 dBm reduces the total energy consumption of the BS. Further increasing the transmission power to 24 dBm has almost no impact on the total energy consumption. When transmitting with low power, the generation of the radio power is not considerable and the CU energy consumption is predominant. Nevertheless, for P t = 24 dBm, the part consumed for radio power generation is not to be neglected. Further increase of the transmission power, beyond 24 dBm for instance, increases the total energy consumption. When the transmission power increases, the number of RBs where the BS is active for transmission is always reduced and the BS is active for a shorter period. However, the energy consumption for data transmission increases when the transmission power is considerable, even if the BS is active for a shorter period because its generation is energy hungry.

VIII. CONCLUSION

In this paper, we have investigated DL radio resource allocation for a centralized RAN architecture under a minimum energy consumption criterion. We considered I RUs that can be active all together to serve UEs simultaneously (mode 1), or active one at a time (mode 2), or a combination of these two modes with possible resource reuse (mode 3).

We showed that using mode 1, where all RUs serve the UE simultaneously, can save energy and serve more UEs compared to mode 2, where a UE is served by one RU in low-load regimes without resource reuse. This is due to the higher SINR provided by simultaneous transmissions in mode 1. Also, the network coverage is not affected, since no RUs are switched off. Combining these modes in mode 3 with potential resource reuse can serve up to 56% more UEs while minimizing energy usage. Increasing transmission power can reduce energy consumption, but beyond a certain threshold, the power generation consumes significant energy, leading to higher total energy consumption. There is a minimum energy consumption that can be achieved by varying the transmission power.

In summary, mode 1 is effective in reducing energy consumption in low-load regimes, while combining modes 1 and 2 in mode 3 gives both low energy consumption and reasonable capacity for up to 50 users. To extend this work, multiple antenna RUs can be deployed with beamforming to carry a high load in a congested system while using minimal energy.

Fig. 1 :

 1 Fig. 1: Modes of operation (patterns illustrate cells' identities).

Fig. 2 :

 2 Fig. 2: Power-consuming units in a C-RAN architecture (DL).

  [START_REF] Ezzaouia | A dynamic transmission strategy based on network slicing for cloud radio access networks[END_REF] where N p = 10 N NF 10 K b T K w RB is the UE's noise power, with N NF being the noise figure, K b the Boltzmann constant, T K the UE's temperature, and w RB the elementary RB bandwidth.

Fig. 3 :

 3 Fig. 3: RU placement for I = 4.

Fig. 4 :

 4 Fig. 4: DL blocking probability with P t = -13 dBm per RB.

Fig. 5 :

 5 Fig. 5: Configurations usage for I = 4 (in each configuration, J increases from the left to the right).

Fig. 6 :

 6 Fig. 6: Data transmission energy consumption.

Fig. 7 :

 7 Fig. 7: Energy consumption at different P t in mode 3 (J = 40).

  I configurations that include the configurations where one or more RUs are active and exclude the configuration where all the RUs are idle. The configurations are numbered from 1 to 2 I . The first I configurations are those that have only one RU in use. Then we have the configurations where more than one RU are active and each serves one dedicated UE on the same RB. Configuration ν = 2 I -1 is the configuration where all RUs are active and each one is serving different UEs on the same RBs. Then, we have the configuration ν = 2 I where RU I + 1 is active and the I real RUs are idle. Indicator d(ν, i) determines if RU i is active in configuration ν or not:

An active RU can serve a maximum of one UE per RB. Each ν is determined by I +1 (I real and one virtual RU) states s i : ν=(s 1 , s 2 , ..., s I+1 ). For I real RUs we have 2

TABLE I :

 I Simulation parameters.

	Symbol	Parameter	Value
	A (m)	Indoor area length	100
	Am (dB)	Antenna maximum attenuation	18
	B (m)	Indoor area width	50
	E SP,f (J)	Fixed SP energy consumption per	0.0312 [13]
		packet
	fc (GHz)	Central frequency	26
	G A (dBi)	Antenna gain	7 [17]
	M RB	Available RBs within one time slot	135
	N NF (dB)	UE noise figure	7 [18]
	P CU,f (W)	CU fixed power consumption	6.25 [13]
	P RU,f (W)	RU fixed power consumption	1.22 [13] [19]
	r 0 (m)	Path-loss reference distance	0.0039
	R T (Mbps)	Target DL data rate	2
	T (ms)	Reference period	20
	Ts (ms)	Time slot duration	0.25
	T K (K)	Receiver temperature	290 [18]
	w RB (kHz)	RB bandwidth	720 [20]
	W (MHz)	System bandwidth	100
	α	Path-loss exponent	2.55
	δ BW	Bandwidth correction factor	0.56 [16]
	δ SNR	SINR correction factor	2 [16]
	η AC/DC	AC/DC power supply gain	0.8
	η PA	PA gain	0.7
	θ 3dB	Half power beamwidth	90°[17]
	µ	NR numerology	2
	σ dB (dB)

E SP,v (J/bit) Variable SP energy consumption per bit 3.75×10 -8

[START_REF] Jung | Modeling of power consumption for macro-, micro-, and rrh-based base station architectures[END_REF]

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