El-Kadri Abdaoui
email: abdaouielkadri@hotmail.com

• Sami Mabrouk

Abdenacer Makhlouf
email: abdenacer.makhlouf@uha.fr

Keong B Lee

Sami Mabrouk

Rota-Baxter Operators on Pre-Lie Superalgebras

Keywords: Rota-Baxter operator, Super O-operator, Associative superalgebra, Lie superalgebra, Pre-Lie superalgebra, L-dendriform superalgebras Mathematics Subject Classification 17A30, 17A36, 17B70

In this paper, we study Rota-Baxter operators and super O-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and L-dendriform superalgebras. Moreover, we provide all Rota-Baxter operators of weight zero on complex pre-Lie superalgebras of dimensions 2 and 3.

Introduction

Rota-Baxter operators of weight λ ∈ K fulfil the so-called Rota-Baxter relation which may be regarded as one possible generalization of the standard shuffle relation [START_REF] Guo | Baxter algebras and shuffle products[END_REF][START_REF] Rota | Ten mathematics problems I will never solve[END_REF]. They appeared for the first time in the work of the mathematician Baxter [START_REF] Baxter | An analytic problem whose solution follows from a simple algebraic identity[END_REF] in 1960 and were then intensively studied by Atkinson [START_REF] Atkinson | Some aspects of Baxter's functional equation[END_REF], Miller [START_REF] Miller | Baxter operators and endomorphisms on Banach algebras[END_REF], Rota [START_REF] Rota | Baxter operators[END_REF], Cartier [START_REF] Cartier | On the structure of free Baxter algebras[END_REF], and more recently, they reappeared in the work of Guo [START_REF] Guo | An introduction to Rota-Baxter Algebra[END_REF] and Ebrahimi-Fard [START_REF] Ebrahimi-Fard | Loday-type algebras and the Rota-Baxter relation[END_REF].

Pre-Lie algebras (called also left-symmetric algebras, Vinberg algebras, quasiassociative algebras) are a class of a natural algebraic systems appearing in many fields in mathematics and mathematical physics. They were first mentioned by Cayley in 1890 [START_REF] Cayley | On the theory of analytic forms called trees[END_REF] as a kind of rooted tree algebra and later arose again from the study of convex homogeneous cones [START_REF] Vinberg | The theory of homogeneous cones[END_REF], affine manifold and affine structures on Lie groups [START_REF] Koszul | Domaines bornés homogènes et orbites de groupes de transformations affines[END_REF], and deformation of associative algebras [START_REF] Gerstenhaber | The cohomology structure of associative ring[END_REF]. They play an important role in the study of symplectic and complex structures on Lie groups and Lie algebras [START_REF] Andrada | Complex product structure on Lie algebras[END_REF][START_REF] Chu | Symplectic homogeneous spaces[END_REF][START_REF] Dardié | Algèbres de Lie Kahlériennes et double extension[END_REF][START_REF] Dardié | Double extension symplectique d'un groupe de Lie symplectique[END_REF][START_REF] Lichnerowicz | On Lie group with left-invariant symplectic or Kählerian[END_REF], phases spaces of Lie algebras [START_REF] Bai | A further study on non-abelian phase spaces: left-symmetric algebraic approach and related geometry[END_REF][START_REF] Kupershmidt | Non-abelian phase spaces[END_REF], certain integrable systems [START_REF] Bordemann | Generalized Lax pairs, the modified classical Yang-Baxter equations, and affine geometry of Lie groups[END_REF], classical and quantum Yang-Baxter equations [START_REF] Diatta | Classical Yang-Baxter equation and left-invariant affine geometry on Lie groups[END_REF], combinatorics [START_REF] Ebrahimi-Fard | Loday-type algebras and the Rota-Baxter relation[END_REF], quantum field theory [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF] and operads [START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad[END_REF]. See [START_REF] Burde | Left-symmetric algebras, or pre-Lie algebras in geometry and physics[END_REF] for a survey. Recently, pre-Lie superalgebras, the Z 2 -graded version of pre-Lie algebras also appeared in many others fields; see, for example, [START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad[END_REF][START_REF] Gerstenhaber | The cohomology structure of associative ring[END_REF][START_REF] Vasilieva | Free left-symmetric superalgebras[END_REF]. To our knowledge, they were first introduced by Gerstenhaber in 1963 to study the cohomology structure of associative algebras [START_REF] Gerstenhaber | The cohomology structure of associative ring[END_REF]. They are a class of natural algebraic appearing in many fields in mathematics and mathematical physics, especially in supersymplectic geometry, vertex superalgebras and graded classical Yang-Baxter equation. Recently, classifications of complex pre-Lie superalgebras in dimensions two and three were given by Zhang and Bai [START_REF] Bai | On some left-symmetric superalgebras[END_REF]. See [START_REF] Aguiar | Quadri-algebras[END_REF][START_REF] Chen | Left-symmetric algebra structures on the W -algebra W (2, 2)[END_REF][START_REF] Kong | Classification of graded left-symmetric algebra structures on Witt and Virasoro algebras[END_REF][START_REF] Kong | Left-symmetric superalgebra structures on the super-Virasoro algebras[END_REF][START_REF] Wang | The classification of low-dimensional Lie superalgebras[END_REF] about further results.

It turns out that the construction of pre-Lie superalgebras from associative superalgebras uses Rota-Baxter operators. Let A be an associative superalgebra (product of x and y is denoted by x y) and R be a Rota-Baxter operator of weight λ on A, which means that it satisfies, for any homogeneous elements x, y in A, the identity R(x)R(y) = R R(x)y + x R(y) + λx y .

(0.1)

If λ = 0 (resp. λ = -1), the product

x • y = R(x)y -(-1) |x||y| y R(x), ∀ x, y ∈ H(A) (0.2)
resp.

x • y = R(x)y -(-1) |x||y| y R(x)x y, ∀ x, y ∈ H(A) (0.3) defines a pre-Lie superalgebra (see Theorem 1.2). The notion of dendriform algebras was introduced in 1995 by Loday [START_REF] Loday | Dialgebras. Dialgebras and Related Operads[END_REF]. Dendriform algebras are algebras with two operations, which dichotomize the notion of associative algebras. The motivation came from algebraic K-theory, and they have been studied quite extensively with connections to several areas in mathematics and physics, including operads, homology, Hopf algebras, Lie and Leibniz algebras, combinatorics, arithmetic and quantum field theory (see [START_REF] Ebrahimi-Fard | New identities in dendriform algebras[END_REF] and the references therein). The relationship between dendriform algebras, Rota-Baxter algebras and pre-Lie algebras was given by Aguiar and Ebrahimi-Fard [START_REF] Aguiar | Pre-Poisson algebras[END_REF][START_REF] Ebrahimi-Fard | Loday-type algebras and the Rota-Baxter relation[END_REF][START_REF] Ebrahimi-Fard | On the associative Nijenhuis relation[END_REF]. Bai, Liu, Guo and Ni generalized the concept of Rota-Baxter operator and introduced a new class of algebras, namely L-dendriform algebras, in [START_REF] Bai | O-operators on associative algebras and associative Yang-Baxter equations[END_REF][START_REF] Bai | Generalizations of the classical Yang-Baxter equation and O-operators[END_REF][START_REF] Bai | Some results on L-dendriform algebras[END_REF]. Moreover, a close relationship among associative superalgebras, Lie superalgebras, pre-Lie superalgebras and dendriform superalgebras is given as follows in the sense of commutative diagram of categories:

Lie superalgebra ←pre-Lie superalgebra ↑ ↑ associative superalgebra ←dendriform superalgebra Recently, the notion of Rota-Baxter operator on a bimodule was introduced by Aguiar [START_REF] Aguiar | Infinitesimal bialgebras, pre-Lie algebras and dendriform algebras[END_REF]. The construction of associative, Lie, pre-Lie and L-dendriform superalgebras is extended to the corresponding categories of bimodules. See [START_REF] Bai | O-operators of Loday algebras and analogues of the classical Yang-Baxter equation[END_REF][START_REF] Ebrahimi-Fard | Rota-Baxter algebras and dendriform algebras[END_REF][START_REF] Ebrahimi-Fard | Dendriform equations[END_REF][START_REF] Ebrahimi-Fard | Twisted dendriform algebras and the preLie Magnus expansions[END_REF][START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new combinatorial identities[END_REF][START_REF] Li | Rota-Baxter operators on pre-Lie algebras[END_REF][START_REF] Makhlouf | Rota-Baxter Hom-Lie-admissible algebras[END_REF] about further results and [START_REF] Bai | A unified algebraic approach to classical Yang-Baxter equation[END_REF][START_REF] Bai | Bijective 1-cocycles and classification of 3-dimensional left-symmetric algebras[END_REF][START_REF] Kupershmidt | What a classical r-matrix really is[END_REF][START_REF] Ni | Super O-operators of Jordan Superalgebras and Super Jordan Yang-Baxter Equations[END_REF][START_REF] Pei | Rota-Baxter on sl(2, C) and solution of the classical Yang-Baxter equation[END_REF] about relationships with Yang-Baxter equation.

The main purpose of this paper is to study, through Rota-Baxter operators and O-operators, the relationship between associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Moreover, we classify Rota-Baxter operators of weight zero on the complex pre-Lie superalgebras of dimensions 2 and 3.

This paper is organized as follows. In Sect. 1, we recall some definitions of associative superalgebras, Lie superalgebras and pre-Lie superalgebras and we introduce the notion of super O-operator of these superalgebras that generalizes the notion of Rota-Baxter operators. We show that every Rota-Baxter associative superalgebra of weight λ = -1 gives rise to a Rota-Baxter Lie superalgebra. Moreover, a super Ooperator on a Lie superalgebra (of weight zero) gives rise to a pre-Lie superalgebra. As an Example of computations, we provide all Rota-Baxter operators (of weight zero) on the orthosymplectic Lie superalgebra osp [START_REF] Aguiar | Infinitesimal bialgebras, pre-Lie algebras and dendriform algebras[END_REF][START_REF] Aguiar | Pre-Poisson algebras[END_REF]. In Sect. 2, we introduce the notion of L-dendriform superalgebra and then study some fundamental properties of L-dendriform superalgebras in terms of super O-operator of pre-Lie superalgebras. Their relationship with associative superalgebras is also described. Sections 3 and 4 are devoted to classification of all Rota-Baxter operators (of weight zero) on the complex pre-Lie superalgebras of dimension 2 and 3 with one-dimensional even part and with two-dimensional even part, respectively.

Throughout this paper, all superalgebras are finite-dimensional and are over a field K of characteristic zero. Let (A, •) be a superalgebra, then L • and R • denote the even left and right multiplication operators L • , R • : A → End(A) defined as L • (x)(y) = (-1) |x||y| R • (y)(x) = x • y for all homogeneous element x, y in A. In particular, when (A, [ , ]) is a Lie superalgebra, we let ad(x) denote the adjoint operator, that is, ad(x)(y) = [x, y] for all homogeneous element x, y in A.

1 Rota-Baxter Associative Superalgebras, Pre-Lie Superalgebras and Lie Superalgebras

Let (A, •) be an algebra over a field K. It is said to be a superalgebra if the underlying vector space of A is Z 2 -graded, that is, A = A 0 ⊕ A 1 , and

A i • A j ⊂ A i+ j , for i, j ∈ Z 2 .
An element of A 0 is said to be even and an element of A 1 is said to be odd. The elements of A j , j ∈ Z 2 , are said to be homogenous and of parity j. The parity of a homogeneous element x is denoted by |x|, and we refer to the set of homogeneous elements of A by H(A).

We extend to graded case the concepts of A-bimodule K-algebra, O-operator and extended O-operator introduced in [START_REF] Bai | Generalizations of the classical Yang-Baxter equation and O-operators[END_REF].

Definition 1.1 (1) An associative superalgebra is a pair (A, μ) consisting of a Z 2graded vector space A and an even bilinear map μ :

A ⊗ A -→ A, (A i A j ⊆ A i+ j , ∀ i, j ∈ Z 2 ) satisfying for all x, y, z ∈ H(A)
x(yz) = (x y)z.

(2) Let (A, μ) be an associative superalgebra and V be a Z 2 -graded vector space. Let l, r : A -→ End(V ) be two even linear maps. A triple

(V, l, r ) is called an A-bimodule if for all x, y ∈ H(A) and v ∈ H(V ) l(x y)(v) = l(x)l(y)(v), r (x y)(v) = r (y)r (x)(v), l(x)r (y)(v) = r (y)l(x)(v).
Moreover, the quadruple (V, μ V , l, r ) is said to be an A-bimodule K-superalgebra if (V, l, r ) is an A-bimodule compatible with the multiplication μ V on V , that is, for all x, y ∈ H(A) and v, w ∈ H(V ),

l(x)(μ V (v, w)) = μ V (l(x)(v), w), r (x)(μ V (v, w)) = μ V (v, r (x)(w)), μ V (r (x)(v), w) = μ V (v, l(x)(w)).
(3) Fix λ ∈ K, a pair (T, T ) of even linear maps T, T : V -→ A is called an extended super O-operator with modification T of weight λ associated with the bimodule (V, l, r

) if T satisfies λl(T (u))v = λr (T (v))u, (1.1) T (u)T (v) = T l(T (u))v + (-1) |u||v| r (T (v))u +λT (u)T (v), ∀ u, v ∈ H(V ). (1.2) (4) An even linear map T : V -→ A is called a super O-operator of weight λ associated with the bimodule K-superalgebra (V, μ V , l, r ) if it satisfies T (u)T (v) = T l(T (u))v + (-1) |u||v| r (T (v))u + λμ V (u, v) , ∀ u, v ∈ H(V ). (1.3)
Notice that the notions of super O-operator and extended super O-operator coincide when λ = 0.

In particular, a super O-operator of weight λ ∈ K associated with the bimodule K-algebra (A, μ A , L μ , R μ ) is called a Rota-Baxter operator of weight λ on A, that is, R satisfies the identity (0.1). We denote by a triple (A, μ, R) the Rota-Baxter associative superalgebra.

We define now Rota-Baxter operators on A-bimodules.

Definition 1.2 Let (A, μ, R) be a Rota-Baxter associative superalgebra of weight zero. A Rota-Baxter operator on an A-bimodule

V (relative to R) is a map R V : V -→ V such that for all x ∈ H(A) and v ∈ H(V ) R(x)R V (v) = R V R(x)v + x R V (v) , R V (v)R(x) = R V R V (v)x + v R(x) .
We have similar definitions on Lie superalgebras.

Definition 1.3 (1) A Lie superalgebra is a pair (A, [ , ]
) consisting of a Z 2 -graded vector space A, and an even bilinear map [ , ] :

A ⊗ A -→ A, ([A i , A j ] ⊆ A i+ j , ∀ i, j ∈ Z 2 ) satisfying for all x, y, z ∈ H(A), [x, y] = -(-1) |x||y| [y, x], (super-skew-symmetry) (1.4) [x, [y, z]] = [[x, y], z] + (-1) |x||y| [y, [x, z]], (super-Jacobi identity). (1.5) (2) Let (A, [ , ]
) be a Lie superalgebra, V be a Z 2 -graded vector space, and ρ :

A -→ End(V ) be an even linear map. The pair (V, ρ) is said to be an A-module or a representation of (A,

[ , ]) if for all x, y ∈ H(A) and v ∈ H(V ), ρ([x, y])(v) = ρ(x)ρ(y)v -(-1) |x||y| ρ(y)ρ(x)v. (1.6) The triple (V, [ , ] V , ρ), where [ , ] V is a super-skew-symmetric bracket, is said to be an A-module K-superalgebra if, for x ∈ H(A) and v, w ∈ H(V ), ρ(x)[v, w] V = [ρ(x)(v), w] V + (-1) |v||w| [v, ρ(x)(w)] V .
(3) Let (A, [ , ]) be a Lie superalgebra and (V, ρ) be a representation of A. An even linear map

T : V -→ A is called a super O-operator of weight λ ∈ K associated with an A-module K-superalgebra (V, [ , ] V , ρ) if T satisfies: [T (u), T (v)] = T ρ(T (u))v-(-1) |u||v| ρ(T (v))u +λ[u, v] V , ∀ u, v ∈ H(V ).
123

In particular, a super O-operator of weight λ ∈ K associated with the bimodule

(A, L • , R • ) is called a Rota-Baxter operator of weight λ ∈ K on (A, [ , ]), that is, R satisfies for all x, y, z in H(A) [R(x), R(y)] = R [R(x), y] -(-1) |x||y| [R(y), x] + λ[x, y] . (1.7)
The triple (A, [ , ], R) refers to a Rota-Baxter Lie superalgebra, see [START_REF] Wang | Operator forms of the classical Yang-Baxter equation in Lie superalgebras[END_REF].

Definition 1.4 Let (A, [ , ], R) and (A , [ , ] , R ) be two Rota-Baxter Lie super- algebras. An even homomorphism f : (A, [ , ], R) -→ (A , [ , ] , R ) is said to be a morphism of two Rota-Baxter Lie superalgebras if, for all x, y ∈ H(A), f ([x, y]) = [ f (x), f (y)] and f • R = R • f. Proposition 1.1 Let (A, μ, R) be a Rota-Baxter associative superalgebra of weight λ ∈ K. Then the triple (A, [ , ], R), where [x, y] = x y -(-1) |x||y| yx, is a Rota- Baxter Lie superalgebra of weight λ ∈ K.
We introduce the notion of super O-operators of pre-Lie superalgebras and study some properties over Lie superalgebras and pre-Lie superalgebras.

Definition 1.5

Let A be a Z 2 -graded vector space and • : A ⊗ A -→ A be an even binary operation. The pair (A, •) is called a pre-Lie superalgebra if, for x, y, z in H(A), the associator

as(x, y, z) = (x • y) • z -x • (y • z)
is super-symmetric in x and y, that is, as(x, y, z) = (-1) |x||y| as(y, x, z), or equivalently

(x • y) • z -x • (y • z) = (-1) |x||y| (y • x) • z -y • (x • z) .
(1.8)

The identity (1.8) is called pre-Lie super-identity.

Definition 1.6 Let (A,

•) be a pre-Lie superalgebra.

(1) Let V be a Z 2 -graded vector space and l, r : A -→ End(V ) be two even linear maps. The triple (V, l, r ) is said to be an A-bimodule of (A,

•) if, for x, y ∈ H(A) and v ∈ H(V ), l(x)l(y)v -l(x • y)v = (-1) |x||y| l(y)l(x)v -l(y • x)v , (1.9) l(x)r (y)v -r (y)l(x)v = (-1) |x||v| (r (x • y)v -r (y)r (x)v). (1.10) Moreover, the quadruple (V, • V , l, r ) is said to be an A-bimodule K-superalgebra if (V, l, r ) is an A-bimodule compatible with the multiplication • V on V , that is, for x, y ∈ H(A) and v, w ∈ H(V ), l(x)(v • V w) -l(x)(v) • V w = (-1) |x||v| (v • V l(x)(w)) -r (x)(v) • V w), r (x)(v • V w) -v • V r (x)(w) = (-1) |v||w| (r (x)(w • V v) -w • V r (x)(v)).

123

(2) Let (V, • V , l, r ) be an A-bimodule K-superalgebra. An even linear map T : V -→ A is called a super O-operator of weight λ ∈ K associated with (V, • V , l, r ) if it satisfies:

T (u) • T (v) = T l(T (u))v + (-1) |u||v| r (T (v))u + λu • V v , ∀ u, v ∈ H(V ). (1.11)
In particular, a super O-operator of weight λ ∈ K associated with the A-bimodule

(A, L • , R • ) is called a Rota-Baxter operator of weight λ on (A, •), that is, R satisfies R(x) • R(y) = R R(x) • y + x • R(y) + λx • y (1.12)
for all x, y, z in H(A).

Proposition 1.2 Let (A, •) be a pre-Lie superalgebra.

(1) The commutator

[x, y] = x • y -(-1) |x||y| y • x defines a Lie superalgebra (A, [ , ]
) which is called the sub-adjacent Lie superalgebra of A and A is also called a compatible pre-Lie superalgebra structure on the Lie superalgebra. [START_REF] Aguiar | Pre-Poisson algebras[END_REF] The map L • gives a representation of the Lie superalgebra

(A, [ , ]), that is, L • ([x, y]) = L • (x)L • (y) -(-1) |x||y| L • (y)L • (x).
Corollary 1.1 Let (A, •) be a pre-Lie superalgebra and (V, l, r ) be an A-bimodule. Let (A, [ , ]) be the sub-adjacent Lie superalgebra. If T is a super O-operator associated with (V, l, r ), then T is a super O-operator of (A, [ , ]) associated with (V, lr, rl). Now, we construct pre-Lie superalgebras using super O-operators on Lie superalgebras. (A,[ ,]) be a Lie superalgebra and (V, ρ) be a representation of A. Suppose that T : V -→ A is a super O-operator of weight zero associated with (V, ρ). Then, the even bilinear map

Proposition 1.3 ([54]) Let

u • v = ρ(T (u))v, ∀ u, v ∈ H(V )
defines a pre-Lie superalgebra structure on A. (A,[ ,]) be a Lie superalgebra and R be the super Ooperator (of weight zero) associated with the adjoint representation (A, ad). Then the even binary operation given by x • y = [R(x), y], for all x, y ∈ H(A), defines a pre-Lie superalgebra structure on A.

Remark 1.1 ([54]) Let

As a direct consequence, since a Rota-Baxter operator on a pre-Lie superalgebra is also a Rota-Baxter operator of its sub-adjacent Lie superalgebra, we have the following observation.

Proposition 1.4

Let A 1 = (A, •, R) be a Rota-Baxter pre-Lie superalgebra of weight zero. Then A 2 = (A, * , R) is a Rota-Baxter pre-Lie superalgebra of weight zero, where the even binary operation is defined by

x * y = R(x) • y -(-1) |x||y| y • R(x).
Example 1.1 In this example, we calculate Rota-Baxter operators of weight zero on the Lie superalgebra osp(1, 2) and give the corresponding pre-Lie superalgebras. Starting from the orthosymplectic Lie superalgebra, we consider in the sequel the matrix realization of this superalgebra.

Let osp(1, 2) = A 0 ⊕ A 1 be the Lie superalgebra where A 0 is spanned by

e 1 = ⎛ ⎝ 1 0 0 0 0 0 0 0 -1 ⎞ ⎠ , e 2 = ⎛ ⎝ 0 0 1 0 0 0 0 0 0 ⎞ ⎠ , e 3 = ⎛ ⎝ 0 0 0 0 0 0 1 0 0 ⎞ ⎠ ,
and A 1 is spanned by

e 4 = ⎛ ⎝ 0 0 0 1 0 0 0 1 0 ⎞ ⎠ , e 5 = ⎛ ⎝ 0 1 0 0 0 -1 0 0 0 ⎞ ⎠ .
The defining relations (we give only the ones with nonzero values in the right-hand side) are The Rota-Baxter operators of weight zero on the Lie superalgebra osp(1, 2) with respect to the homogeneous basis {e 1 , e 2 , e 3 , e 4 , e 5 } are:

R 1 (e 1 ) = a 1 e 1 + a 2 e 2 - 8a 2 1 a 3 (2a 3 +a 2 ) 2 e 3 , R 1 (e 2 ) = - 2a 2 a 2 1 (2a 3 +a 2 ) 2 e 1 + (2a 3 -3a 2 )a 1 2(2a 3 +a 2 ) e 2 + 2a 3 1 (2a 3 +a 2 ) 2 e 3 , R 1 (e 3 )=a 3 e 1 + (2a 3 +a 2 ) 2 8a 1 e 2 + a 1 (a 2 -6a 3 ) 2(2a 3 +a 2 ) e 3 , R 1 (e 4 )=0, R 1 (e 5 )=0, a 1 = 0, a 2 = -2a 3 . R 2 (e 1 ) = a 1 e 1 + a 2 e 2 , R 2 (e 2 ) = - 2a 2 1 a 2 e 1 -3a 1 2 e 2 + 2a 3 1 a 2 2 e 3 , R 2 (e 3 ) = a 2 2 8a 1 e 2 + a 1 2 e 3 , R 2 (e 4 ) = 0, R 2 (e 5 ) = 0, a 1 = 0, a 2 = 0. R 3 (e 1 ) = a 1 e 1 - 2a 2 1 a 3 e 3 , R 3 (e 2 ) = a 1 2 e 2 + a 3 1 2a 2 3 e 3 , R 3 (e 3 ) = a 3 e 1 + a 2 R 5 (e 1 ) = a 1 e 1 -4a 3 e 2 - 2a 2 1 a 3 e 3 , R 5 (e 2 ) = - a 2 1 4a 3 e 1 + a 1 e 2 + a 3 1 2a 2 3 e 3 , R 5 (e 3 ) = a 3 e 1 - 4a 2 3
a 1 e 2 -2a 1 e 3 , R 5 (e 4 ) = 0, R 5 (e 5 ) = 0, a 1 = 0, a 3 = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = 0, R 6 (e 3 ) = a 3 e 1 , R 6 (e 4 ) = 0, R 6 (e 5 ) = 0. R 7 (e 1 ) = a 1 e 1 + 2a 2a 1 e 3 , R 8 (e 3 ) = 0, R 8 (e 4 ) = 0, R 8 (e 5 ) = 0, a 1 = 0.

R 9 (e 1 ) = a 1 e 1 + a 2 e 2 , R 9 (e 2 ) = 0, R 9 (e 3 ) = a 2 2 e 1 + a 2 2
2a 1 e 2 , R 9 (e 4 ) = 0, R 9 (e 5 ) = 0, a 1 = 0. R 10 (e 1 ) = 0, R 10 (e 2 ) = a 7 e 1 + a 8 e 3 , R 10 (e 3 ) = 0, R 10 (e 4 ) = 0, R 10 (e 5 ) = 0. 2 e 1 + a 8 e 3 , R 12 (e 3 ) = 0, R 12 (e 4 ) = 0, R 12 (e 5 ) = 0. R 13 (e 1 ) = 0, R 13 (e 2 ) = a 7 e 1 , R 13 (e 3 ) = 0, R 13 (e 4 ) = 0, R 13 (e 5 ) = 0. The constants a i are parameters. Now, we define Rota-Baxter operators on an A-module, where A is a Rota-Baxter Lie superalgebra.

R 14 (e 1 ) = 0, R 14 (e 2 ) = 0, R 14 (e 3 ) = a 4 e 2 , R 14 (e 4 ) = 0, R 14 (e 5 ) = 0. R 15 (e 1 ) = 0, R 15 (e 2 ) = 0, R 15 (e 3 ) = 0, R 15 (e 4 ) = 0, R 15 (e 5 ) = 0. R 16 (e 1 ) = a 1 e 1 + a 2 e 2 + a 2 1 (a 2 -4a 3 ) 4a 2 3 e 3 , R 16 (e 2 ) = - a 4a 3 e 1 -a 1 a 2 4a 3 e 2 + a 3 1 (4a 3 -a 2 ) 16a 3 3 e 3 , R 16 (e 3 ) = a 3 e 1 + a 2 a 3 a 1 e 2 + a 1 (a 2 -4a 3 ) 4a 3 e 3 , R 16 (e 4 ) = 0, R 16 (e 5 ) = 0, a 1 = 0, a 3 = 0. R 17 (e 1 ) = a 1 e 1 - a 2 1 a 3 e 3 , R 17 (e 2 ) = - a 2 1 4a 3 e 1 + a 3 1 4a 2 3 e 3 , R 17 (e 3 ) = a 3 e 1 -a 1 e 3 , R 17 (e 4 ) = 0, R 17 (e 5 ) = 0, a 3 = 0. R 18 (e 1 ) = a 1 e 1 -4a 3 e 2 - 2a 2 1 a 3 e 3 , R 18 (e 2 ) = 2a 2 1 a 3 e 1 -7a 1 2 e 2 + a 3 1 2a 2 3 e 3 , R 18 (e 3 ) = a 3 e 1 + a 2 3 2a 1 e 2 + 5a 1 2 e 3 , R 18 (e 4 ) = 0, R 18 (e 5 ) = 0, a 1 = 0, a 3 = 0. R 19 (e 1 ) = a 1 e 1 + 4a 3 e 2 , R 4 (19) = - a 2 1 4a 3 e 1 -a 1 e 2 , R 19 (e 3 ) = a 3 e 1 + 4a 2 3 a 1 e 2 , R 19 (e 4 ) = 0, R 19 (e 5 ) = 0, a 1 = 0, a 3 = 0 R 20 (e 1 ) = 0, R 20 (e 2 ) = 0, R 20 (e 3 ) = a 3 e 1 + a 4 e 2 , R 20 (e 4 ) = 0, R 20 (e 5 ) = 0. R 21 (e 1 ) = -2a 3 e 2 , R 21 (e 2 ) = 0, R 21 (e 3 ) = a 3 e 1 + a 4 e 2 , R 21 (e 4 ) = 0, R 21 (e 5 ) = 0. R 22 (e 1 ) = 4a 2 5 a 6 e 2 + a 6 e 3 , R 22 (e 2 ) = -a 5 e 2 - a 2 6 4a 5 e 3 , R (e 3 ) = 4a 3 5 a 2 6 e 2 + a 5 e 3 , R 23 (e 4 ) = 0, R 23 (e 5 ) = 0, a 5 = 0, a 6 = 0. R 23 (e 1 ) = a 2 e 1 , R 23 (e 2 ) = 0, R 23 (e 3 ) = -a 2 2 e 2 + a 4 e , R 23 (e 4 ) = 0, R 23 (e 5 ) = 0. R 24 (e 1 ) = 0, R 24 (e 2 ) = a 6 e 1 + a 7 e 3 , R 24 (e 3 ) = 0, R 24 (e 4 ) = 0, R 24 (e 5 ) = 0. R 25 (e 1 ) = 0 R 25 (e 2 ) = 0, R 25 (e 3 ) = a 4 e 2 , R 25 (e 4 ) = 0, R 25 (e 5 ) = 0. R 26 (e 1 ) = 0, R 26 (e 2 ) = a 5 e 2 + a 2 5 a 4 e 3 , R 26 (e 3 ) = a 4 e 2 + a 5 e 3 , R 26 (e 4 ) = 0, R 26 (e 5 ) = 0, a 4 = 0. R 27 (e 1 ) = a 1 e 1 , R 27 (e 2 ) = 0, R 27 (e 3 ) = 0, R 27 (e 4 ) = 0, R 27 (e 5 ) = 0. 123 R 28 (e 1 ) = 0, R 28 (e 2 ) = a 7 e 3 , R 28 (e 3 ) = 0, R 28 (e 4 ) = 0, R 28 (e 5 ) = 0. R 29 (e 1 ) = a 9 e 1 -
Definition 1.7 Let (A, [ , ], R) be a Rota-Baxter Lie superalgebra of weight zero. A Rota-Baxter operator on an A-module V (relative to R) is a map R V : V -→ V such that, for all x ∈ H(A) and v ∈ H(V ), [R(x), R V (v)] = R V [R(x), v] + [x, R V (v)] , [R V (v), R(x)] = R V [R V (v), x] + [v, R(x)] ,
where the action ρ

(x)(v) is denoted by [x, v].

Proposition 1.5 Let (A, [ , ], R) be a Rota-Baxter Lie superalgebra of weight zero, V an A-module and R V a Rota-Baxter operator on V . Define new actions of A on V by

x • v = [R(x), v], v • x = [R V (v), x].
Equipped with these actions, V is a bimodule over the pre-Lie superalgebra (Remark 1.1).

Proof Let x, y be a homogeneous elements in A and v in V . We have

l(x)r (y)(v) -(-1) |x||y| r (y)l(x) -r (x • y)(v) + (-1) |x||y| r (y)r (x)(v) = (-1) |y||v| [R(x), [R V (v), y]] -(-1) |y||v| [R V ([R(x), v]), y] -(-1) |v|(|x|+|y|) [R V (v), [R(x), y]] + (-1) |v|(|x|+|y|) [R V ([R V (v), x]), y] = (-1) |y||v| [R(x), [R V (v), y]] -(-1) |y||v| [R V ([R(x), v] + [x, R V (v)]), y] -(-1) |v|(|x|+|y|) [R V (v), [R(x), y]] = (-1) |y||v| [R(x), [R V (v), y]] -(-1) |y||v| [[R(x), R V (v)], y] -(-1) |v|(|x|+|y|) [R V (v), [R(x), y]] = 0. 123 Then l(x)r (y)(v) -(-1) |x||y| r (y)l(x) = r (x • y)(v) -(-1) |x||y| r (y)r (x)(v). Similarly, we show that l(x) •l(y)v -l(x • y)v = (-1) |x||y| l(y) •l(x)v -l(y • x)v .
Now, we construct a functor from a full sub-category of the category of Rota-Baxter Lie-admissible (or associative) superalgebras to the category of pre-Lie superalgebras. The Lie-admissible algebras were studied by Albert in 1948 and Goze and Remm in 2004, they introduced the notion of G-associative algebras where G is a subgroup of the permutation group S 3 (see [START_REF] Goze | Lie-admissible algebras and operads[END_REF]). The graded case was studied by Ammar and Makhlouf in 2010 (see [START_REF] Ammar | Hom-Lie superalgebras and Hom-Lie admissible superalgebras[END_REF] for more details). Definition 1.8 (1) A Lie-admissible superalgebra is a superalgebra (A, μ) in which the supercommutator bracket, defined for all homogeneous x, y in A by

[x, y] = μ(x, y) -(-1) |x||y| μ(y, x),
satisfies the super-Jacobi identity (1.5). ( 2) Let G be a subgroup of the permutation group S 3 . A Rota-Baxter G-associative superalgebra of weight λ ∈ K is a G-associative superalgebra (A, •) together with an even linear self-map R : A -→ A that satisfies the identity

R(x) • R(y) = R(R(x) • y + x • R(y) -λx • y), ( 1.13)
for all homogeneous elements x, y, z in A.

Theorem 1.1 Let (A, •, R) be a Rota-Baxter Lie-admissible superalgebra of weight zero. Define an even binary operation " * " on any homogeneous element x, y ∈

A by x * y = R(x) • y -(-1) |x||y| y • R(x) = [R(x), y]. (1.14) Then A L = (A, * ) is a pre-Lie superalgebra.
Proof A direct consequence of Remark 1.1, since a Rota-Baxter operator on a Lieadmissible superalgebra is also a Rota-Baxter operator of its supercommutator Lie superalgebra.

Theorem 1.2 Let (A, μ, R) be a Rota-Baxter associative superalgebra of weight λ = -1. Define the even binary operation "•" on any homogeneous element x, y ∈ A by

x • y = μ(R(x), y) -(-1) |x||y| μ(y, R(x)) -μ(x, y) = R(x)y -(-1) |x||y| y R(x) -x y.
(1.15)

Then A L = (A, •) is a pre-Lie superalgebra.
123

Proof For all homogeneous elements x, y, z in A, we have

x • (y • z) = R(x)(R(y)z) -(-1) |y||z| R(x)(z R(y)) -R(x)(yz) -(-1) |x|(|y|+|z|) (R(y)z)R(x) + (-1) |x|(|y|+|z|)+|y||z| (z R(y))R(x) + (-1) |x|(|y|+|z|) (yz)R(x) -x(R(y)z) + (-1) |y||z| (z R(y))x + x(yz),
and

(x • y) • z = R(R(x)y)z -(-1) |x||y| R(y R(x))z -R(x y)z -(-1) |z|(|x|+|y|) (R(x)y)R(z) + (-1) |z|(|x|+|y|)+|x||y| (y R(x))R(z) + (-1) |z|(|x|+|y|) (x y)z -(R(x)y)z + (-1) |x||y| (y R(x))z + (x y)z.
Then, we obtain

as A L (x, y, z) -(-1) |x||y| as A L (y, x, z) = x • (y • z) -(x • y) • z -(-1) |x||y| y • (x • z) + (-1) |x||y| (y • x) • z = R(x)(R(y)z)-(-1) |y||z| R(x)(z R(y))-R(x)(yz)-(-1) |x|(|y|+|z|) (R(y)z)R(x) + (-1) |x|(|y|+|z|)+|y||z| (z R(y))R(x) + (-1) |x|(|y|+|z|) (yz)R(x) -x(R(y)z) + (-1) |x|(|y|+|z|)+|y||z| (z R(y))x + x(yz) -R(R(x)y)z + (-1) |x||y| R(y R(x))z + R(x y)z + (-1) |z|(|x|+|y|) (R(x)y)R(z) -(-1) |z|(|x|+|y|)+|x||y| (y R(x))R(z) -(-1) |z|(|x|+|y|) (x y)z + (R(x)y)z + (-1) |x||y| (y R(x))z + (x y)z -(-1) |x||y| R(y)(R(x)z) + (-1) |x|(|y|+|z|) R(y)(z R(x)) + (-1) |x||y| R(y)(xz) + (-1) |y||z| (R(x)z)R(y) -(-1) |z|(|x|+|y|) (z R(x))R(y) -(-1) |y||z| (xz)R(y) + (-1) |x||y| y(R(x)z) -(-1) |z|(|x|+|y|) (z R(x))y +(-1) |x||y| y(xz) + (-1) |x||y| R(R(y)x)z -R(x R(y))z -(-1) |x||y| R(yx)z -(-1) |z|(|x|+|y|)+|x||y| (R(y)x)R(z) +(-1) |z|(|x|+|y|) (x R(y))R(z) +(-1) |z|(|x|+|y|)+|x||y| (yx)z-(-1) |x||y| (R(y)x)z + (x R(y))z + (-1) |x||y| (yx)z.
The above sum vanishes by associativity and the Rota-Baxter identity (1.13) with λ = -1.

Corollary 1.2 Let (A, μ, R) be a Rota-Baxter associative superalgebra of weight λ = -1. Then R is still a Rota-Baxter operator of weight λ = -1 on the pre-Lie superalgebra (A, •) defined in (1.15).

As a consequence of Theorem 1.2 and Corollary 1.2, we have:

123 Proposition 1.6 Let (A, μ, R) be a Rota-Baxter associative superalgebra of weight λ = -1. Then the binary operation defined, for any homogeneous elements x, y in A, by

[x, y] = R(x)y -(-1) |x||y| y R(x) -x y + x R(y) -(-1) |x||y| R(y)x + (-1) |x||y| yx, defines a Rota-Baxter Lie superalgebra (A, [ , ], R) of weight λ = -1.

L-dendriform Superalgebras

The notion of L-dendriform algebra was introduced by Bai, Liu and Ni in 2010 (see [START_REF] Bai | Some results on L-dendriform algebras[END_REF]). In this section, we extend this notion to the graded case, and define L-dendriform superalgebra. Then we study relationships between associative superalgebras, Ldendriform superalgebras and pre-Lie superalgebras. Moreover, we introduce the notion of Rota-Baxter operator (of weight zero) on the A-bimodule and we provide a construction of associative bimodules from bimodules over L-dendriform superalgebras and a construction of L-dendriform bimodules from bimodules over pre-Lie superalgebras.

L-dendriform Superalgebras and Associative Superalgebras

Definition and Some Basic Properties

Definition 2.1 A L-dendriform superalgebra is a triple (A, , ) consisting of a Z 2graded vector space A and two even bilinear maps , : A ⊗ A -→ A satisfying, for all homogeneous elements x, y, z in A,

x (y z) = (x y) z + (x y) z + (-1) |x||y| y (x z) -(-1) |x||y| (y x) z -(-1) |x||y| (y x) z, (2.1) x (y z) = (x y) z + (-1) |x||y| y (x z) +(-1) |x||y| y (x z) -(-1) |x||y| (y x) z. (2.
2)

The associated bracket to a L-dendriform superalgebra is defined as

[x, y] = x y - (-1) |x||y| y x.
Definition 2.2 (1) Let (A, , ) be a L-dendriform superalgebra, V be a Z 2 -graded vector space, and l , r , l , r : A -→ End(V ) be four even linear maps. The tuple (V, l , r , l , r ) is an A-bimodule if for any homogeneous elements x, y ∈ A and u, v ∈ V, the following identities are satisfied (a

) [l (x), l (y)] = l ([x, y]), (b) [l (x), l (y)] = l (x • y) + (-1) |x||y| l (y)l (x), (c) r (x y)(v) = r (y)r (x)(v)+r (y)r (x)(v)+(-1) |x||v| l (x)r (y)(v)- (-1) |x||v| r (y)l (x)(v) -(-1) |x||v| r (y)l (x)(v), 123 (d) r (x y)(v) = r (y)r (x)(v)-(-1) |x||v| l (x)r (y)(v)-l (x)r (y)(v) +r (y)l (x)(v) , (e) l (x)r (y)(v) -r (y)l (x)(v) = (-1) |x||v| r (x • y)(v) -(-1) |x||v| r (y) r (x)(v). where x • y = x y -(-1) |x||y| y x, and x • y = x y + x y. Moreover, The tuple (V, V , V , l , r , l , r ) is an A-bimodule K- superalgebra if the following identities are satisfied (a) l (x)(u V v) -(-1) |x||u| u V l (x)(v) = l (x)(u) V v + l (x)(u) V v -(-1) |x||u| r (x)(u) V v -(-1) |x||u| r (x)(u) V v, (b) l (x)(u V v)-(-1) |x||u| u V l (x)(v)=l (x)(u) V v-(-1) |x||u| r (x)(u) V v + (-1) |x||u| u V l (x)v, (c) u V l (x)(v) = r (x)(u) V v -(-1) |x||u| l (x)(u V v) + (-1) |x||u| l (x)(u V v) -(-1) |x||u| l (x)(u) V v. (2) Let (A, , ) be a L-dendriform superalgebra and (V, V , V , l , r , l , r ) be an A-bimodule K-superalgebra. An even linear map T : V -→ A is called a super O-operator of weight λ ∈ K associated with (V, V , V , l , r , l , r ) if T satisfies for any homogeneous elements u, v in V T (u) T (v) = T l (T (u))v + (-1) |u||v| r (T (v))u + λu V v , T (u) T (v) = T l (T (u))v + (-1) |u||v| r (T (v))u + λu V v .
In particular, a super O-operator of weight λ ∈ K of the L-dendriform superalgebra (A, , ) associated with the bimodule (A, L , R , L , R ) is called a Rota-Baxter operator (of weight λ) on (A, , ), that is, R satisfies for any homogeneous elements x, y in A

R(x) R(y) = R R(x) y + R(x) u + λx y , R(x) R(y) = R R(x) y + R(x) y + λx y .
The following theorem provides a construction of L-dendriform superalgebras using super O-operators of associative superalgebras. (A,μ) be an associative superalgebra and (V, l, r ) be a Abimodule. If T is a super O-operator of weight zero associated with (V, l, r ), then there exists a L-dendriform superalgebra structure on V defined by

Theorem 2.1 Let

u v = l(T (u))v, u v = (-1) |u||v| r (T (v))u, ∀ u, v ∈ H(V ).
(2.3)

Proof For any homogeneous elements u, v and w in V , we have

u (v w) = l(T (u))l(T (v))w, (u v) w = l T (l(T (u))v) w, (u v) w = (-1) |u||v| l T (r (T (v))u) w, (-1) |u||v| v (u w) = (-1) |u||v| l(T (v))l(T (u))w, (-1) |u||v| (v u) w = l T (r (T (u))v) w, (-1) |u||v| (v u) w = (-1) |u||v| l T (l(T (v))u) w.
Hence,

u (v w) -(u v) w -(u v) w -(-1) |u||v| v (u w) -(v u) w -(v u) w = l(T (u))l(T (v))w -l T (l(T (u))v) w -(-1) |u||v| l T (r (T (v))u) w -(-1) |u||v| l(T (v))l(T (u))w + l T (r (T (u))v) w + (-1) |u||v| l T (l(T (v))u) w = (-1) |u||v| l(μ(T (v), T (u))) -(-1) |u||v| l(T (v))l(T (u))w -l(μ(T (u), T (v))) + l(T (u))l(T (v))w = 0,
and similarly, we have

u (v w) -(u v) w -(-1) |u||v| v (u w) -(-1) |u||v| v (u w) +(-1) |u||v| (v u) w = 0.
Therefore, (V, , ) is a L-dendriform superalgebra.

A direct consequence of Theorem 2.1 is the following construction of a Ldendriform superalgebra from a Rota-Baxter operator (of weight zero) of an associative superalgebra.

-bimodule V (relative to R) is a map R V : V -→ V such that for all homogeneous elements x in A and v in V R(x) R V (v) = R V R(x) v + x R V (v) , R V (v) R(x) = R V R V (v) x + v R(x) , R(x) R V (v) = R V R(x) v + x R V (v) , R V (v) R(x) = R V R V (v) x + v R(x) .
123 Proposition 2.1 Let (A, μ) be an associative superalgebra, R : A -→ A a Rota-Baxter operator on A, V an A-bimodule and R V a Rota-Baxter operator on V . Define a new actions of A on V by

x v = μ(R(x), v), v x = μ(R V (v), x), x v = μ(x, R V (v)), v x = μ(v, R(x)).
Equipped with these actions, V becomes an A-bimodule over the associated Ldendriform superalgebra.

Corollary 2.2 Let (V, l , r , l , r ) be an A-bimodule of a dendriform superalgebra (A, , ). Let (A, μ) be the associated associative superalgebra. If T is a super O-operator associated with (V, l , r , l , r ), then T is a super O-operator of (A, μ) associated with (V, l + l , r + r ).

L-dendriform Superalgebras and Pre-Lie Superalgebras

We have the following observation.

Proposition 2.2 Let (A, , ) be a L-dendriform superalgebra (1) The even binary operation • :

A ⊗ A -→ A given by x • y = x y -(-1) |x||y| y x, ∀ x, y ∈ H(A)
defines a pre-Lie superalgebra (A, •) which is called the associated vertical pre-Lie superalgebra of (A, , ) and (A, , ) is called a compatible L-dendriform superalgebra structure on the pre-Lie superalgebra (A, •).

(2) The even binary operation

• : A ⊗ A -→ A given by x • y = x y + x y, ∀ x, y ∈ H(A)
defines a pre-Lie superalgebra (A, •) which is called the associated horizontal pre-Lie superalgebra of (A, , ) and (A, , ) is called a compatible L-dendriform superalgebra structure on the pre-Lie superalgebra (A, •). Conversely, we can construct L-dendriform superalgebras from O-operators of pre-Lie superalgebras.

Theorem 2.2 Let (A,

•) be a pre-Lie superalgebra and (V, l, r ) be an A-bimodule. If T is a super O-operator of weight zero associated with (V, l, r ), then there exists a L-dendriform superalgebra structure on V defined by

u v = l(T (u))v, u v = -r (T (u))v, ∀ u, v ∈ H(V ).
(2.4)

Therefore, there is a pre-Lie superalgebra structure on V defined by

u • v = u v -(-1) |u||v| v u, ∀ u, v ∈ H(V ) (2.5)
as the associated vertical pre-Lie superalgebra of (V, , ) and T is a homomorphism of pre-Lie superalgebra. Furthermore, T (V ) = {T (v) / v ∈ V } ⊂ A is a pre-Lie sub-superalgebra of (A, •) and there is a L-dendriform superalgebra structure on T (V ) given by

T (u) T (v) = T (u v), T (u) T (v) = T (u v), ∀ u, v ∈ H(V ). (2.6)
Moreover, the corresponding associated vertical pre-Lie superalgebra structure on T (V ) is a pre-Lie sub-superalgebra of (A, •) and T is a homomorphism of Ldendriform superalgebra.

Proof For any homogeneous elements u, v and w in V , we have

u (v w) = l(T (u))l(T (v))w, (u v) w = l T (l(T (u))v) w, (u v) w = -l T (r (T (u))v) w, (-1) |u||v| v (u w) = (-1) |u||v| l(T (v))l(T (u))w, (-1) |u||v| (v u) w = -(-1) |u||v| l T (r (T (v))u) w, (-1) |u||v| (v u) w = (-1) |u||v| l T (l(T (v))u) w, u (v w) = -l(T (u))r (T (v))w, (u v) w = r T (l(T (u))v) w, (-1) |u||v| v (u w) = -(-1) |u||v| r (T (v))l(T (u))w, (-1) |u||v| v (u w) = (-1) |u||v| r (T (v))r (T (u)), (-1) |u||v| (v u) w = (-1) |u||v| r T (r (T (v))u) w.
Hence,

u (v w) -(u v) w -(u v) w -(-1) |u||v| v (u w) + (-1) |u||v| (v u) w + (-1) |u||v| (v u) w = l(T (u))l(T (v))w -(-1) |u||v| l(T (v))l(T (u))w -l(T (l(T (u)))v)w + l(T (r (T (u))v))w -(-1) |u||v| l(T (r (T (v))u))w + (-1) |u||v| l(T (l(T (v))u))w = l(T (u))l(T (v))w -(-1) |u||v| l(T (v))l(T (u))w -l T (u) • T (v) w 123 + (-1) |u||v| l T (v) • T (u) w = 0,
and

u (v w) -(u v) w -(-1) |u||v| v (u w) -(-1) |u||v| v (u w) + (-1) |u||v| (v u) w = -l(T (u))r (T (v))w + (-1) |u||v| r (T (v))l(T (u))w + r (T (u) • T (v))w -(-1) |u||v| r (T (v))r (T (u)) = 0.
Therefore, (V, , ) is a L-dendriform superalgebra. The other conditions follow easily.

A direct consequence of Theorem 2.2, is the following construction of a Ldendriform superalgebra from a Rota-Baxter operator (of weight zero) of a pre-Lie superalgebra. Proof Straightforward.

Corollary 2.4 Let (A, •) be a pre-Lie superalgebra and R be a Rota-Baxter operator on A (of weight zero). Then even binary operations given by

x y = R(x) • y, x y = -(-1) |x||y| y • R(x) (2.
Next, we provide a construction of a L-dendriform bimodule from a bimodule over a pre-Lie superalgebra.

Proposition 2.3 Let (A, •, R) be a Rota-Baxter pre-Lie superalgebra of weight zero, V an A-bimodule and R V a Rota-Baxter operator on V . Define new actions of

A on V by x v = R(x) • v, v x = R V (v) • x, x v = -(-1) |x||v| v • R(x), v x = -(-1) |x||v| x • R V (v).
Equipped with actions, V is a bimodule over the L-dendriform superalgebra of Corollary 2.4.

Proof Let x, y be homogeneous elements in A and v in V . Then, we have

r (x • y)(v) -(-1) |x||y| r (y)r (x)(v) -[l (x), r (y)](v) = (-1) |v|(|x|+|y|) v (x y + x y) -(-1) |v|(|x|+|y|) (v x) y -(-1) |y||v| x (v y) + (-1) |v|(|x|+|y|) (x v) y = -(R(x) • y) • R V (v) + (-1) |x||y| (y • R(x)) • R V (v) -(-1) |x||y| y • R V (x • R V (v)) -(-1) |x||y| y • R V (R(x) • v) + R(x) • (y • R V (v)) = -(R(x) • y) • R V (v) + R(x) • (y • R V (v)) -(-1) |x||y| y • (R(x) • R V (v)) + (-1) |x||y| (y • R(x)) • R V (v) = 0.
Therefore,

[l (x), r (y)](v) = r (x • y)(v) -(-1) |x||y| r (y)r (x)(v).
Similarly, we have

r (x y)(v) -(-1) |x||y| r (y)r (x)(v) -l (x)r (y)(v) -[l (x), r (y)](v) = (-1) |v|(|x|+|y|) v (x y) -(-1) |v|(|x|+|y|)+|x||y| (v x) y -(-1) |y||v| x (v y) -(-1) |y||v| x (v y) + (-1) |y||v| (x v) y = -(-1) |v|(|x|+|y|)+|x||y| R V (v) • (y • R(x)) + (-1) |x|(|y|+|v|) y • R V (R V (v) • x) + (-1) |v|(|x|+|y|)+|x||y| (R V (v) • y) • R(x) -(-1) |x|(|y|+|v|) (y • R V (v)) • R(x) + (-1) |x|(|y|+|v|) y • R V (v • R(x)) = (-1) |x|(|y|+|v|) y • (R V (v) • R(x)) -(y • R V (v)) • R(x) + (-1) |v|(|x|+|y|)+|x||y| R V (v) • y) • R(x) -R V (v) • (y • R(x)) = 0, then r (x y)(v) = (-1) |x||y| r (y)r (x)(v) + l (x)r (y)(v) + [l (x), r (y)](v).
The others axioms are similar. Therefore, (V, l , r , l , r ) is a bimodule over the L-dendriform superalgebra (A, , ).

[15]). In the following, let C be the ground field of complex numbers and {e 1 , e 2 } be a homogeneous basis of a pre-Lie superalgebra (A, •), where {e 1 } is a basis of A 0 and {e 2 } is a basis of A 1 . By direct computation and by help of a computer algebra system, we obtain the following results. Proposition 3.1 The Rota-Baxter operators (of weight zero) on two-dimensional pre-Lie superalgebras (associative or non-associative) of type B 1 , B 2 and B 3 are given as follows:

Rota-Baxter Operators on Pre-Lie Superalgebras of Type B 1

The pre-Lie superalgebra (B 1,1 , •) : e 2 • e 1 = e 2 has the Rota-Baxter operator defined as

R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = 0. The pre-Lie superalgebras of type ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (B 1,2 , •) : e 1 • e 1 = e 1 , e 2 • e 1 = e 2 (associative). (B 1,3 ) k , • : e 1 • e 1 = ke 1 , e 2 • e 1 = e 2 , k = 0, 1. (B 1,4 , •) : e 1 • e 1 = e 1 , e 1 • e 2 = e 2 (associative). (B 1,5 ) k , • : e 1 • e 1 = ke 1 , e 1 • e 2 = ke 2 , e 2 • e 1 = (k + 1)e 2 , k = 0, -1.
have only the trivial Rota-Baxter operator, that is, R 1 (e 1 ) = 0, R 1 (e 2 ) = 0.

Rota-Baxter Operators on Pre-Lie Superalgebras of Type B 2

The pre-Lie superalgebra (B 2,1 , •) (associative).

e 2 • e 2 = 1 2 e 1 .
Rota-Baxter operators RB(B 2,1 ) are:

R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = 0. R 2 (e 1 ) = a 1 e 1 , R 2 (e 2 ) = 2a 1 e 2 .
The pre-Lie superalgebra (B 2,2 , •) (associative)

e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 2 • e 1 = e 2 , e 2 • e 2 = 1 2 e 1 .
Rota-Baxter operators RB(B 2,2 ) is:

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0.
123

Rota-Baxter Operators on Pre-Lie Superalgebras of Type B 3

The pre-Lie superalgebra (B 3,1 , •) (associative).

e i • e j = 0, ∀ i, j = 1, 2, 3.

Rota-Baxter operators RB(B 3,1 ) are:

R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = a 2 e 2 .
The pre-Lie superalgebra (B 3,2 , •) (associative)

e 1 • e 1 = e 1 .
Rota-Baxter operators RB(B 3,2 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 2 .
The pre-Lie superalgebra (B 3,3 , •) (associative)

e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 2 • e 1 = e 2 .
Rota-Baxter operator RB(B 3,3 ) is:

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0.
4 Rota-Baxter Operators on Three-dimensional Pre-Lie Superalgebras

Rota-Baxter Operators on Three-dimensional Pre-Lie Superalgebras with Two-dimensional Odd Part

We still work over the ground field C of complex numbers. Using the classification of the three-dimensional pre-Lie superalgebras with one-dimensional even part was given by Zhang and Bai (see [START_REF] Bai | On some left-symmetric superalgebras[END_REF]). The purpose of this section is to provide, using Definition 1.6, all Rota-Baxter operators (of weight zero) on these pre-Lie superalgebras by direct computation. In the following, let {e 1 , e 2 , e 3 } be a homogeneous basis of a pre-Lie superalgebra (A, •), where {e 1 } is a basis of A 0 and {e 2 , e 3 } is a basis of A 1 . Rota-Baxter operators RB(C 1,1 ) are:

123

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 2 , R 2 (e 3 ) = a 1 e 2 , a 2 = 0. R 3 (e 1 ) = a 3 e 1 , R 3 (e 2 ) = 0, R 3 (e 3 ) = a 1 e 2 .
The pre-Lie superalgebra (C 1,2 ) k , • (associative)

e 1 • e 3 = ke 2 , e 3 • e 1 = (k + 1)e 2 .
Rota-Baxter operators RB((C 1,2 ) k ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 2 , R 1 (e 3 ) = a 2 e 2 , k = -1. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 2 e 2 + a 3 e 3 , k = -1. R 3 (e 1 ) = a 4 e 1 , R 3 (e 2 ) = a 1 e 2 , R 3 (e 3 ) = a 2 e 2 + a 1 a 4 a 4 -a 1 e 3 , a 1 = a 4 , k = -1. The pre-Lie superalgebra (C 1,3 , •): e 1 • e 1 = e 1 , e 3 • e 1 = e 2 .
Rota-Baxter operators RB(C 1,3 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 2 , R 2 (e 3 ) = a 1 e 2 .
The pre-Lie superalgebra (C 1,4 , •): Rota-Baxter operators RB(C 1,4 ) are:

e 1 • e 1 = e 1 ,
R 1 (e 1 ) = 0, R 1 (e 3 ) = 0, R 1 (e 2 ) = a 1 e 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0.

Rota-Baxter Operators on Pre-Lie Superalgebras of Type C 2 h

The pre-Lie superalgebra (C 2 h ,1 , •):

e 1 • e 1 = (h + 1)e 1 , e 2 • e 1 = e 2 , e 2 • e 3 = -e 1 , e 3 • e 1 = he 3 , e 3 • e 2 = e 1 , h ∈ C.
Rota-Baxter operators RB(C 2 h ,1 ) are: Case 1: If h = 0, we have The pre-Lie superalgebra (C 2 h ,2 , •):

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 3 , R 2 (e 3 ) = a 2 e 3 . R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 2 e 3 . R 4 (e 1 ) = 0, R 4 (e 2 ) =
e 1 • e 1 = (1 -h)e 1 , e 1 • e 3 = e 2 , e 2 • e 1 = e 2 , e 3 • e 1 = e 2 + he , h ∈ C. Rota-Baxter operators RB(C 2 h ,2 ) are: Case 1: If h = 0, we have R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + a 2 e 3 .

Case 2:

If h ∈ C * , we have R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 1 e 2 , a 1 = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = e 2 , R 3 (e 3 ) = 0. Case 3: If h = 1, we have R 4 (e 1 ) = a 3 e 1 , R 4 (e 2 ) = 0, R 4 (e 3 ) = a 1 e 2 , a 3 = 0. R 5 (e 1 ) = 0, R 5 (e 2 ) = 0, R 5 (e 3 ) = a 1 e 2 , a 1 = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = e 2 , R 6 (e 3 ) = 0.
The pre-Lie superalgebra (C 2 h ,3 , •):

e 1 • e 1 = (1 -h)e 1 , e 1 • e 2 = (1 -h)e 2 + e 3 , e 1 • e 3 = (1 -h)e 3 , e 2 • e 1 = (2 -h)e 2 + e 3 , e 3 • e 1 = e 3 , h = 1.
Rota-Baxter operators RB(C 2 h ,3 ) are: Case 1: If h = (-1)

1 3 or h = -(-1) 2
3 , we have

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 2 + a 3 e 3 , R 2 (e 3 ) = a 2 (h -1)e 2 + a 2 (h -1)e 3 .
Case 2: If h = 1, we have

R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = 0. R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 1 e 3 , a 1 = 0. R 5 (e 1 ) = 0, R 5 (e 2 ) = 0, R 5 (e 3 ) = a 1 e 3 , h 3 -2h 2 + 2h -1 = 0.
The pre-Lie superalgebra (C 2 h ,4 , •): 123

e 1 • e 1 = (h -1)e 1 ,
R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 , h = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 1 e 3 , R 2 (e 3 ) = 0, a 1 = 0, h = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = 0, h = 0.
The pre-Lie superalgebra (C 2 h ,5 , •): Rota-Baxter operators RB(C 2 h ,5 ) are:

e 1 • e 1 = (1 -h)e 1 ,
R 1 (e 1 ) = 0, R 1 (e 2 ) = e 2 , R 1 (e 3 ) = a 1 e 2 -a 1 e 3 , h = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0, h = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = a 2 e 2 -a 3 (h-1) 2 e 3 , R 3 (e 3 ) = a 2 (h-1) 2 h e -a 2 h e 3 , a 2 = 0, h = 0. R 4 (e 1 ) = 0, R 4 (e 2 ) = a 2 e 2 -a 2 e 3 , R 4 (e 3 ) = a 2 2 e 2 -a 2 2 e 3 , a = 0, h = 2. R 5 (e 1 ) = 0, R 5 (e 2 ) = a 3 e 3 , R 5 (e 3 ) = 0, a 3 = 0. The pre-Lie superalgebra (C 2 h ,6 ) k , • : h = 0 or 1, k = 1 associative other cases non-associative. e 1 • e 1 = ke 1 , e 2 • e 1 = e 2 , e 3 • e 1 = he 3 , h, k ∈ C. Rota-Baxter operators RB((C 2 h ,6 ) k ) are: Case 1: If h = 0, we have R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 3 , R 2 (e 3 ) = a 2 e 3 . R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 2 e 3 .
Case 2: If h ∈ C * , we have Case 5: If h = 0, 1 and k = 0, we have

R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 1 e 2 . R 5 (e 1 ) = 0, R 5 (e 2 ) = a 3 e 3 , R 5 (e 3 ) = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = a 4 e 2 + a 3 e 3 , R 6 (e 3 ) = - a 2 4 ha 3 e 2 -a 4 h e 3 , a 3 = 0. R 7 (e 1 ) = 0, R 7 (e 2 ) = 0, R 7 (e 3 ) = 0. Case 3: If h = k = 0, we have R 8 (e 1 ) = 0, R 8 (e 2 ) = 0, R 8 (e 3 ) = a 1 e 2 + a 2 e 3 . R 9 (e 1 ) = 0, R 9 (e 2 ) = a 3 e 3 , R 9 (e 3 ) = a 2 e 3 . R 10 (e 1 ) = a 5 e 1 , R 10 (e 2 ) = 0, R 10 (e 3 ) = a 2 e 3 . R 11 (e 1 ) = a 5 e 1 , R 11 (e 2 ) = 0, R 11 (e 3 ) = 0. R 12 (e 1 ) = 0, R 12 (e 2 ) = a 3 e 3 , R 12 (e 3 ) = a 2 e 3 , a 3 = 0.
R 16 (e 1 ) = 0, R 16 (e 2 ) = 0, R 16 (e 3 ) = a 1 e 2 , a 1 = 0. R 17 (e 1 ) = 0, R 17 (e 2 ) = a 3 e 3 , R 17 (e 3 ) = 0, r 2,2 = 0. R 18 (e 1 ) = 0, R 18 (e 2 ) = a 4 e 2 + a 3 e 3 , R 18 (e 3 ) = - a 2 4 ha 3 e 2 + a 4 h e 3 , a 3 = 0.
The pre-Lie superalgebra (C 2 h ,7 ) k , • : (C 2 0 ,7 ) 1 associative.

e 1 • e 1 = ke 1 , e 1 • e 2 = ke 2 , e 2 • e 1 = (k + 1)e 2 , e 3 • e 1 = he 3 , , k = 0, h = ±1. Rota-Baxter operators RB((C 2 h ,7 ) k ) are: Case 1: If h = 0, we have R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + a 2 e 3 , a 1 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 3 , R 2 (e 3 ) = a 2 e 3 , a 3 = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 2 e 3 .

Case 2:

If h ∈ C * , we have R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 1 e 2 , a 1 = 0. R 5 (e 1 ) = 0, R 5 (e 2 ) = a 3 e 3 , R 5 (e 3 ) = 0, a 3 = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = 0, R 6 (e 3 ) = 0.
The pre-Lie superalgebra (C 2 h ,8 ) k , • : h = -1 or 0, k = 1 associative other cases are non-associative

e 1 • e 1 = ke 1 , e 1 • e 3 = ke 3 , e 2 • e 1 = e 2 , e 3 • e 1 = (h + k)e 3 , h ∈ C, k = 0. Rota-Baxter operators RB((C 2 h ,8 ) k ) are: R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0. The pre-Lie superalgebra (C 2 h ) k ,9 , • : h = 0 or 1, k = -1 associative other cases are non-associative e 1 • e 1 = ke 1 , e 1 • e 2 = ke 2 , e 1 • e 3 = ke 3 , e 2 • e 1 = (k + 1)e 2 , e 3 • e 1 = (h + k)e 3 , h ∈ C, k = 0.
Rota-Baxter operators RB((C 2 h ,9 ) k ) are: Rota-Baxter operators RB(C 3,1 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 2 + a 2 e 3 , R 1 (e 3 ) = - a 2 1 a 2 e 2 -a 1 e 3 , a 2 = 0, h = 1, k = -1. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 3 , R 2 (e 3 ) = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 3 e 2 . R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = 0.
R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 1 +a 3 e 2 , R 2 (e 3 ) = - a 2 2 +a 2 a 3 2 e 2 -(a 2 +a 3 )e 3 , a 3 = 0.
The pre-Lie superalgebra (C 3,2 ) k , • :

e 1 • e 1 = ke 1 , e 2 • e 1 = e 2 , e 3 • e 1 = e 2 + e 3 .
Rota-Baxter operators RB((C 3,2 ) k ) are:

Case 1: If k = 0, we have R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 2 , a 1 = 0. R 2 (e 1 ) = a 1 e 1 , R 2 (e 2 ) = 0, R 2 (e 3 ) = a 1 e 2 , a 1 = 0. R 3 (e 1 ) = a 1 e 1 , R 3 (e 2 ) = 0, R 3 (e 3 ) = a 2 e 2 , a 1 = 0. Case 2: If k ∈ C * , we have R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 2 e 2 .
R 5 (e 1 ) = 0, R 5 (e 2 ) = a 3 e 2 +a 4 e 3 , R 5 (e 3 ) = -

a 2 3 +a 3 a 4 a 4
e 2 -(a 3 +a 4 )e 3 , a 4 = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = 0, R 6 (e 3 ) = 0.

The pre-Lie superalgebra (C 3,3 ) k , • :

e 1 • e 3 = ke 2 , e 2 • e 1 = e 2 , e 3 • e 1 = (k + 1)e 2 + e 3 , k = 0. Rota-Baxter operators RB((C 3,3 ) k ) are: R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 2 e 2 . R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = 0.
The pre-Lie superalgebra (C 3,4 ) k , • :

e 1 • e 1 = ke 1 , e 1 • e 2 = ke 2 , e 1 • e 3 = ke 3 , e 2 • e 1 = (k + 1)e 2 , e 3 • e 1 = e 2 + (k + 1)e 3 , k = 0.
Rota-Baxter operator RB((C 3,4 ) k ) is: Rota-Baxter operators RB(C 4,1 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 .
R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = a 2 e 2 + a 3 e 3 , R 1 (e 3 ) = a 4 e 2 -a 1 a 2 +a 3 a 4 a 1 -a 2 e 3 , a 1 = a 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 4 e 2 + a 5 e 3 . R 3 (e 1 ) = a 1 e 1 , R 3 (e 2 ) = a 1 e 2 + a 3 e 3 , R 3 (e 3 ) = -
e i • e j = 0, ∀ i, j = 1, 2, 3. Rota-Baxter operator RB(C 4,2 ) is: R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = a 2 e 2 + a 3 e 3 , R 1 (e 3 ) = a 4 e 2 + a 5 e 3 .
The pre-Lie superalgebra (C 4,3 , •): (associative)

e 1 • e 1 = e 1 .
Rota-Baxter operator RB(C 4,3 ) is: Rota-Baxter operators RB(C 4,4 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 2 + a 2 e 3 , R 1 (e 3 ) = a 3 e 2 + a 4 e 3 .
R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 2 , R 1 (e 3 ) = a 2 e 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 1 e 2 + a 3 e 3 , R 2 (e 3 ) = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = a 1 e 2 , R 3 (e 3 ) = 0.
The pre-Lie superalgebra (C 4,5 , •): (associative)

e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 1 • e 3 = e 3 , e 2 • e 1 = e 2 , e 3 • e 1 = e 3 .
Rota-Baxter operators RB(C 4,5 ) are: 123

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 2 + a 3 e 3 , R 2 (e 3 ) = - a 2 2 a 3 e 2 -a 2 e 3 , a 3 = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = 0.
R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 2 + a 2 e 3 . R 2 (e 1 ) = a 3 e 1 , R 2 (e 2 ) = a 4 e 2 , R 2 (e 3 ) = a 1 e 2 + a 3 a 4
a 3 -a 4 e 3 , a 3 = a 4 .

Rota-Baxter Operators on Pre-Lie Superalgebras of Type C 5

The pre-Lie superalgebra (C 5,1 ) k , • : ((C 5,1 ) 0 is associative). Rota-Baxter operators RB(C 5,2 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 2 e 3 . R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 3 e 2 .
The pre-Lie superalgebra (C 5,3 , •): Rota-Baxter operators RB(C 6,1 ) are: Rota-Baxter operators RB(C 6,4 ) are:

e 1 • e 1 = e 1 ,
R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 , a 1 = 0. R 2 (e 1 ) = a 3 e 1 , R 2 (e 2 ) = a 1 e 3 , R 2 (e 3 ) = 0.
123

R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 2 e 3 .

Classification of Rota-Baxter Operator on Three-dimensional Pre-Lie Superalgebras with Two-Dimensional Even Part

In this section, we describe all Rota-Baxter operators of weight zero on the threedimensional complex pre-Lie superalgebras with two-dimensional even part which were classified in [START_REF] Bai | On some left-symmetric superalgebras[END_REF] by Zhang and Bai. In the following, let {e 1 , e 2 , e 3 } be a homogeneous basis of a pre-Lie superalgebra (A, •), where {e 1 , e 2 } is a basis of A 0 and {e 3 } is a basis of A 1 . The computation is obtained using computer algebra system, and the operators are described with respect to the basis. The pre-Lie superalgebra ( A 1,1 , •) D 3 (associative). Rota-Baxter operators RB(( A 1,2 ) k ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 2 , R 2 (e 3 ) = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = e 1 + 1 2 e 2 , R 3 (e 3 ) = 0.
The pre-Lie superalgebra

( A 1,3 ) k 1 ,k 2 , • D 1 . e 1 • e 1 = e 1 , e 2 • e 2 = e 2 , e 3 • e 1 = k 1 e 3 , e 3 • e 2 = k 2 e 3 , k 1 , k 2 = 0, k 1 ≤ k 2 , k 1 = -k 2 .
Rota-Baxter operators RB((

A 1,3 ) k 1 ,k 2 ) are: If k 2 < 0, k 1 ≤ k 2 or (k 2 > 0, k 1 < -k 2 ) or (-k < k 1 < 0) or (0 < k 1 ≤ k 2 ) R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = 0.
The pre-Lie superalgebra ( A 1,4 , •) D 3 (associative). Rota-Baxter operators RB( A 1,4 ) are: Rota-Baxter operators RB( A 1,6 ) are:

R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 2 , R 2 (e 3 ) = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = e 1 + 1 2 e 2 , R 3 (e 3 ) = 0. The pre-Lie superalgebra ( A 1,5 ) k 1 ,k 2 , • D 1 , k 1 = k 2 = 0 or k 1 = 1, k 2 = 0 associative other cases non-associative. e 1 • e 1 = e 1 , e 2 • e 2 = e 2 , e 2 • e 3 = e 3 , e 3 • e 1 = k 1 e 3 , e 3 • e 2 = k 2 e 3 , k 1 = 0 or k = 1. Rota-Baxter operator RB(( A 1,5 ) k 1 ,k 2 ) is: R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = 0. The pre-Lie superalgebra ( A 1,6 , •) D 2 (associative).
R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 2 , R 2 (e 3 ) = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = e 1 + 1 2 e 2 , R 3 (e 3 ) = 0.

Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 2

The pre-Lie superalgebra ( A 2,1 , •) D 3 (associative).

e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 2 • e 1 = e 2 .
Rota-Baxter operators RB( A 2,1 ) are: 123

R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 1 , R 2 (e 3 ) = a 2 e 3 . The pre-Lie superalgebra ( A 2,2 ) k , • D 1 , ( A 2,2 ) 1 is associative.
R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 1 , R 2 (e 3 ) = a 2 e 3 .
The pre-Lie superalgebra Rota-Baxter operators RB(( A 2,3 ) k ) are:

( A 2,3 ) k , • D 1 e 1 • e 1 = e 1 ,
R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 1 , R 2 (e 3 ) = 0. R 3 (e 1 ) = a 1 e 2 , R 3 (e 2 ) = 0, R 3 (e 3 ) = a 2 e 3 , k = 1. R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = 0, k = 1.
The pre-Lie superalgebra ( A 2,4 , •) D 3 (associative).

Rota-Baxter operator RB(

A 2,4 ) is: R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = 0. The pre-Lie superalgebra ( A 2,5 ) k , • D 1 , ( A 2,5
) 0 is associative. Rota-Baxter operators RB(( A 2,5 ) k ) are: Rota-Baxter operators RB( A 3,4 ) are

R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 3 , k = 0. R 2 (e 1 ) = a 1 e 2 , R 2 (e 2 ) = 0, R 2 (e 3 ) = 0, k = 0. The pre-Lie superalgebra ( A 2,6 ) k , • D 1 : e 1 • e 1 = e 1 ,
R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = a 2 e 2 , R 1 (e 3 ) = 0. R 2 (e 1 ) = a 1 e 2 , R 2 (e 2 ) = a 2 e 2 , R 2 (e 3 ) = 2a 2 e 3 .
The pre-Lie superalgebras of type They have the same Rota-Baxter operators: R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = a 2 e 2 , R 1 (e 3 ) = 0.

( A 3,2 ) k , • D 1 : e 1 •e 1 = e 1 , e 3 •e 1 = ke 3 , k = 0, (( A 3,2 ) 1 is associative). ( A 3,3 ) k , •) D 1 : e 1 • e 1 =

Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 4

The pre-Lie superalgebra ( A 4,1 , •) D 3 (associative). Rota-Baxter operators RB(( A 6,1 ) k ) are:

Case 1: If k = 0, we have

R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 1 , R 2 (e 3 ) = a 2 e 3 . Case 2: If k ∈ C * , we have R 4 (e 1 ) = 0, R 4 (e 2 ) = a 3 e 1 , R 4 (e 3 ) = 0. R 5 (e 1 ) = a 1 e 2 , R 5 (e 2 ) = 0, R 5 (e 3 ) = 0. R 5 (e 1 ) = a 1 e 2 , R 5 (e 2 ) = 0, R 5 (e 3 ) = 0, a 1 = 0.
The pre-Lie superalgebra ( A 6,2 , •) D 5 :

e 1 • e 2 = -

Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 7 h

The pre-Lie superalgebra ( A They have the same Rota-Baxter operators, that is, R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 1 , R 1 (e 3 ) = 0, a 1 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0. They have the same Rota-Baxter operators, that is, R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 1 , R 1 (e 3 ) = 0, a 1 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0. Remark 4.1 Using the above classification and Corollary 2.4, one may construct the two-and three-dimensional L-dendriform superalgebras associated with the Rota-Baxter pre-Lie superalgebras of dimension 2 and 3 (of weight zero) described above.

[e 1 ,

 1 e 2 ] = 2e 2 , [e 1 , e 3 ] = -2e 3 , [e 2 , e 3 ] = e 1 , [e 3 , e 5 ] = e 4 , [e 2 , e 4 ] = e 5 , [e 1 , e 4 ] = -e 4 , [e 1 , e 5 ] = e 5 , [e 5 , e 4 ] = e 1 , [e 5 , e 5 ] = -2e 2 , [e 4 , e 4 ] = 2e 3 .

R 11 (e 1 ) 2 5a 8

 11128 = 0, R 11 (e 2 ) = a 5 e 2 + a 8 e 3 , R 11 (e 3 ) = a e 2 + a 5 e 3 , R 11 (e 4 ) = 0, R 11 (e 5 ) = 0, a 8 = 0. R 12 (e 1 ) = a 6 e 3 , R 12 (e 2 ) = -a 6

2a 2 9 ae 3 , R 29 (e 3 ) = a 10 e 1 + a 2 5 2a 9 e 2 -3a 9 2 e 3 , 2 9a

 932935232 10 e 3 , R 29 (e 2 ) R 29 (e 4 ) = -a 9 e 4 +a 10 e 5 , R 29 (e 5 ) = -a 10 e 4 +a 9 e 5 , a 9 = 0, a 10 = 0. R 30 (e 1 ) = -a 10 e 2 , R 30 (e 2 ) = 0, R 30 (e 3 ) = a 10 2 e 1 + a 4 e 2 , R 30 (e 4 ) = a 10 e 5 , R 30 (e 5 ) = 0. R 31 (e 1 ) = a 11 e 3 , R 31 (e 2 ) = -a 11 2 e 1 + a 8 e 3 , R 31 (e 3 ) = 0, R 31 (e 4 ) = 0, R 31 (e 5 ) = a 11 e 4 .

Corollary 2 . 1 Definition 2 . 3

 2123 Let (A, μ, R) be a Rota-Baxter associative superalgebra of weight zero. Then, the even binary operations given byx y = μ(R(x), y), x y = μ(x, R(y)), ∀ x, y ∈ H(A)defines a L-dendriform superalgebra structure on A. Let (A, , ) be a L-dendriform superalgebra and R : A -→ A be a Rota-Baxter operator of weight zero. A Rota-Baxter operator on A

( 3 )Corollary 2 . 3

 323 Both (A, •) and (A, •) have the same sub-adjacent Lie superalgebra g(A) defined by[x, y] = x y + x y -(-1) |x||y| y x -(-1) |x||y| y x, ∀ x, y ∈ H(A).Proof Straightforward. Let (V, l , r , l , r ) be a bimodule of a L-dendriform superalgebra (A, , ). Let (A, •) be the associated pre-Lie superalgebra. If T is a super Ooperator associated with (V, l , r , l , r ), then T is a super O-operator of (A, •) associated with (V, l , r ), where l = l + (-1) |u||v| r and r = l + r .

Proposition 4 . 1

 41 The Rota-Baxter operators (of weight zero) on three-dimensional pre-Lie superalgebras (associative or non-associative) with two-dimensional odd part of type C 1 , C 2 h , C 3 , C 4 , C 5 and C 6 are given as follows: 4.1.1 Rota-Baxter Operators on Pre-Lie Superalgebras of Type C 1 The pre-Lie superalgebra (C 1,1 , •) e 2 • e 3 = -e 1 , e 3 • e 1 = e 2 , e 3 • e 2 = e 1 .

e 1 •

 1 e 2 = e 2 , e 1 • e 3 = e 3 , e 2 • e 1 = e 2 , e 3 • e 1 = e 2 + e 3 .

a 3 e 3 ,Case 2 : 123 R 6 (e 1 ) 2 5ha 3 e 2 -a 5 h e 3 ,Case 3 : 2 5a 3

 321236123323 R 4 (e 3 ) = a 2 e 3 , a 3 = 0. If h ∈ C * , we haveR 5 (e 1 ) = 0, R 5 (e 2 ) = 0, R 5 (e 3 ) = a 1 e 2 . = 0, R 6 (e 2 ) = a 3 e 3 , R 6 (e 3 ) = 0, a 3 = 0. R 7 (e 1 ) = 0, R 7 (e 2 ) = a 5 e 2 + a 3 e 3 , R 7 (e 3 ) = -a a 3 = 0, a 5 = 0. R 8 (e 1 ) = 0, R 8 (e 2 ) = 0, R 8 (e 3 ) = 0. If h = -1, we have R 9 (e 1 ) = 0, R 9 (e 2 ) = a 3 e 3 , R 9 (e 3 ) = 0.R 10 (e 1 ) = 0, R 10 (e 2 ) = a 5 e 2 + a 3 e 3 , R 10 (e 3 ) = a e 3 + a 5 e 3 , a = 0. R 11 (e 1 ) = a 4 e 1 , R 11 (e 2 ) = 0, R 11 (e 3 ) = a 1 e 2 , a 4 = 0. R 12 (e 1 ) = a 4 e 1 , R 12 (e 2 ) = 0, R 12 (e 3 ) = 0, a 4 = 0.

e 1 •

 1 e 2 = e 3 , e 2 • e 1 = e 2 + e 3 , e 3 • e 1 = he , h = ±1. Rota-Baxter operators RB(C 2 h ,4 ) are:

e 1 • e 2 =

 12 (1h)e 2 + e 3 , e 1 • e 3 = (1h)e 3 , e 2 • e 1 = (2h)e 2 + e 3 , e 3 • e 1 = e 3 , h = ±1.

Case 4 :

 4 If h = 1 and k = 0, we have R 13 (e 1 ) = a 5 e 1 , R 13 (e 2 ) = 0, R 13 (e 3 ) = a 1 e 2 . R 14 (e 1 ) = a 5 e 1 , R 14 (e 2 ) = a 3 e 3 , R 14 (e 3 ) = 0. R 15 (e 1 ) = a 5 e 1 , R 15 (e 2 ) = a 4 e 2 + a 3 e 3 , R 15 (e 3 ) = -a 2

123 4 . 1 . 3

 413 Rota-Baxter Operators on Pre-Lie Superalgebras of Type C 3 The pre-Lie superalgebra (C 3,1 , •): e 1 • e 1 = 2e 1 , e 2 • e 1 = e 2 , e 2 • e 3 = -e 1 , e 3 • e 1 = e 2 + e 3 , e 3 • e 2 = e 1 .

123 4 . 1 . 4

 414 Rota-Baxter Operators on Pre-Lie Superalgebras of Type C 4 The pre-Lie superalgebra (C 4,1 , •): (associative) e 2 • e 3 = -e 1 , e 3 • e 2 = e 1 .

a 2 1 a 3

 13 e 2 + a 5 e 3 , a 3 = 0. The pre-Lie superalgebra (C 4,2 , •): (associative).

  The pre-Lie superalgebra (C 4,4 , •): (associative) e 1 • e 1 = e 1 , e 1 • e 3 = e 3 , e 3 • e 1 = e 3 .

  The pre-Lie superalgebra (C 4,6 , •): (associative)e 1 • e 3 = e 2 , e 3 • e 1 = e 2 .Rota-Baxter operators RB(C 4,6 ) are:

e 1 • e 2 = 1 : 2 :

 1212 ke 3 , e 2 • e 1 = ke 3 , e 3 • e 3 = e 1 , k = or 1. Rota-Baxter operators RB((C 5,1 ) k ) are:Case If k = 0, we haveR 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 1 e 3 , R 2 (e 3 ) = a 3 e 2 . R 3 (e 1 ) = 0, R 3 (e 2 ) = a 4 e 2 , R 3 (e 3 ) = a 3 e 2 . R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 2 e 3 . R 5 (e 1 ) = a 5 e 1 , R 5 (e 2 ) = a 4 e 2 , R 5 (e 3 ) = a 5 a 4 a 5 -a 4 e 3 , a 4 = a . Case If k = 1, we have R 6 (e 1 ) = 0, R 6 (e 2 ) = a 1 e 3 , R 6 (e 3 ) = a 2 e 3 . R 7 (e 1 ) = 0, R 7 (e 2 ) = 0, R 7 (e 3 ) = a 2 e 3 . R 8 (e 1 ) = 0, R 8 (e 2 ) = a 1 e 3 , R 8 (e 3 ) = e 3 . R 9 (e 1 ) = a 5 e 1 , R 9 (e 2 ) = a 5 e 2 , R 9 (e 3 ) =a 5 2 e 3 , a 5 = 0. R 10 (e 1 ) = a 5 e 1 , R 10 (e 2 ) = a 5 a 5 -1 e 2 , R 10 (e 3 ) = e 3 , a 5 = 1. The pre-Lie superalgebra (C 5,2 , •): e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 2 • e 1 = e 2 , e 2 • e 3 = e 1 .

e 1 • 2 . 2 5a 1 e 2 +

 12212 e 2 = e 2 + e 3 , e 2 • e 1 = e 2 + e 3 , e • e 3 = e 1 .Rota-Baxter operators RB(C 5,3 ) are:R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 2 e 3 . R 3 (e 1 ) = 0, R 3 (e 2 ) = a 3 e 2 , R 3 (e 3 ) = -a 3 e 2 .The pre-Lie superalgebra (C 5,4 ) k , • : (associative)e 2 • e 3 = 1 2 + k e 1 , e 3 • e 2 = 1 2 k e 1 , k ≥ 0, k =1Rota-Baxter operators RB((C 5,4 ) k ) are:R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 3 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 2 , R 2 (e 3 ) = a 4 e 2 . R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 2 e 3 . R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 4 e 2 . R 5 (e 1 ) = 0, R 5 (e 2 ) = a 1 e 3 , R 5 (e 3 ) = 0, a 1 = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = a 3 e 2 , R 6 (e 3 ) = 0, a 3 = 0. R 7 (e 1 ) = a 5 e 1 , R 7 (e 2 ) = 0, R 7 (e 3 ) = 0.R 8 (e 1 ) = a 5 e 1 , R 8 (e 2 ) = a 5 e 2 + a 1 e 3 , R 8 (e 3 ) = a a 5 e , a 1 = 0, a 5 = 0, k = 0. R 9 (e 1 ) = a 5 e 1 , R 9 (e 2 ) = a 3 e 2 , R 9 (e 3 ) = -a 5 a 3 a 5 -a 3 e 3 , a 3 = a . 4.1.6 Rota-Baxter Operators on Pre-Lie Superalgebras of TypeC The pre-Lie superalgebra (C 6,1 , •) (associative) e 2 • e 2 = 1 2 e 1 , e 2 • e 3 = -e 1 , e 3 • e 2 = e 1 .

2 • e 2 = 1 2 e 1 . 1 • e 2 = e 3 , e 2 • e 1 = e 3 , e 2 • e 2 = 1 2 e 1 . 3 (e 1 ) = a 3 e 1 , R 3 (e 2 ) = 2a 3 e 2 + a 1 e 3 , R 3 (e 3 ) = 2a 3 3 e 3 .

 21123213213113223333 The pre-Lie superalgebra (C 6,2 , •): (associative) e Rota-Baxter operators RB(C 6,2 ) are:R 1 (e 1 ) = a 1 e 1 , R 1 (e 2 ) = a 2 e 3 , R 1 (e 3 ) = a 3 e 3 . R 2 (e 1 ) = a 1 e 1 , R 2 (e 2 ) = 2a 1 e 2 + a 2 e 3 , R 2 (e 3 ) = a 3 e 3 .The pre-Lie superalgebra (C 6,3 , •) (associative) e Rota-Baxter operators RB(C 6,3 ) are:R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 2 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = a 3 e 1 , R 2 (e 2 ) = a 1 e 3 , R 2 (e 3 ) =0, a 3 = 0. R The pre-Lie superalgebra (C 6,4 , •): (associative) e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 2 • e 1 = e 2 , e 2 • e 2 = 1 2 e 1 .

e 1 • e 1 = e 1 , e 2 • e 2 = e 2 .

 111222 Rota-Baxter operators RB( A 1,1 ) are:R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 1 2 e 2 , R 2 (e 3 ) = a 1 e 3 . R 3 (e 1 ) = 0, R 3 (e 2 ) = e 1 + 1 2 e 2 , R 3 (e 3 ) = a 1 e 3 .The pre-Lie superalgebra( A 1,2 ) k , • D 1 , ( A 1,2 ) 1 associative.e 1 • e 1 = e 1 , e 2 • e 2 = e 2 , e 3 • e 1 = ke 3 , k ∈ C * .

e 1 •

 1 e 1 = e 1 , e 2 • e 2 = e 2 , e 2 • e 3 = e 3 , e 3 • e = e 3 .

e 1 •

 1 e 1 = e 1 , e 2 • e 2 = e 2 , ,e 2 • e 3 = e 3 , e 3 • e 2 = e 3 , e 3 • e 3 = e 2 .

e 1 •

 1 e 1 = e 1 , e 1 • e 2 = e 2 , e 2 • e 1 = e 1 , e 3 • e 1 = ke 3 , k = 0. Rota-Baxter operators RB(( A 2,2 ) k ) are:

e 1 •

 1 e 2 = e 2 , e 2 • e 1 = e 2 , e 3 • e 1 = ke 3 , e 3 • e 2 = e 3 .

e 1 •

 1 e 1 = e 1 , e 1 • e 2 = e 2 , e 1 • e 3 = e 3 , e 2 • e 1 = e 2 , e 3 • e 1 = e 3 .

e 1 •

 1 e 1 = e 1 , e 1 • e 2 = e 2 , e 1 • e 3 = e 3 , e 2 • e 1 = e 2 , e 3 • e 1 = ke 3 , k = 1.

e 1 • e 2 = e 2 , e 1 • e 3 = e 3 , e 2 • e 1 = e 2 , e 3 • e 1 = ke 3 , e 3 • e 2 = e 3 .

 22133212313323 Rota-Baxter operators RB((A 2,6 ) k ) are: R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = 0.The pre-Lie superalgebra ( A 2,7 , •) D 2 (associative).

e 1 • e 1 = e 1 , e 1 • e 2 = e 2 , e 1 • e 3 = e 3 , e 2 • e 1 = e 2 , e 3 • e 1 = e 3 , e 3 • e 3 = e 2 .

 111122133212313332 Rota-Baxter operators RB( A 2,7 ) are: R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = 0.

123 4 . 2 . 3 e 1 • e 1 = e 1 .R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = a 2 e 2 , R 1 (e 3 ) = a 3 e 3 .

 423111112122133 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 3The pre-Lie superalgebra ( A 3,1 , •) D 3 (associative).Rota-Baxter operators RB( A 3,1 ) are:The pre-Lie superalgebra ( A 3,4 , •) D 2 : (associative). e 1 • e 1 = e 1 , e 3 • e 3 = e 2 .

e 1 , e 3 • e 1 =

 131 ke 3 , e 3 • e 2 = e 3 . ( A 3,5 , •) D 3 : e 1 • e 1 = e 1 , e 1 • e 3 = e 3 , e 3 • e 1 = e 3 , (associative).( A 3,6 ) k , • D 1 : e 1 • e 1 = e 1 , e 1 • e 3 = e 3 , e 3 • e 1 = ke 3 , k = 1, (( A 3,6 ) 0 is associative). ( A 3,7 ) k , •D1 : e 1 • e 1 = e 1 , e 1 • e 3 = e 3 , e 3 • e 1 = ke 3 , e 3 • e 2 = e 3 . ( A 3,8 , •) D 2 : e 1 • e 1 = e 1 , e 1 • e 3 = e 3 , e 3 • e 1 = e 3 , e 3 • e 3 = e 1 , (associative).

  e i • e j = 0, ∀ i, j = 1, 2, 3.Rota-Baxter operators RB( A 4,1 ) are:R 1 (e 1 ) = a 1 e 1 + a 2 e 2 , R 1 (e 2 ) = a 3 e 1 + a 4 e 2 , R 1 (e 3 ) = a 5 e 3 .The pre-Lie superalgebra ( A 4,2 , •) D 1 :e 3 • e 1 = e 3 .Rota-Baxter operators RB( A 4,2 ) are: 123 4.2.6 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 6 The pre-Lie superalgebra ( A 6,1 ) k , • (D 4 ) μ : k = 0 or -1 associative other cases non-associative e 1 • e 2 = -e 1 , e 2 • e 2 = -e 2 , e 3 • e 2 = ke 3 .

e 1 , e 2 • e 2 = -e 2 , e 3 • e 2 = - 1 2 e 3 , e 3 • e 3 = e 1 .R 1 (e 1 ) = a 1 e 2 ,

 122233331112 Rota-Baxter operators RB( A 6,2 ) are:R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = 0, R 1 (e 3 ) = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 1 , R 2 (e 3 ) = 0.The pre-Lie superalgebra ( A 6,3 ) k , • (D 4 ) μ : k = 0 or -1 associative other cases are non-associative e 1 • e 2 = -e 1 , e 2 • e 2 = -e 2 , e 2 • e 3 = -e 3 , e 3 • e 2 = ke 3 .Rota-Baxter operators RB(( A 6,3 ) k ) are: R 1 (e 2 ) = 0, R 1 (e 3 ) = 0, a 1 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 1 , R 2 (e 3 ) = 0.

7 h , 1 , •) D 5 :e 1 • e 2 = -e 1 , e 2 • e 2 = 2 e 3 ,R 3 (e 1 ) 1 : 2 1a 2 e 1 -a 1 e 2 , 3 . 2 : 2 1a 2 ( A 9 , 2 )e 2 • e 1 = e 1 e 2 • e 2 = e 2 e 2 •e 2 • e 2 = e 1 e 2 • e 2 = e 2 e 2 • 2 1a 2 1 : 3 ) = a 2 e 3 . 2 :

 151212223311212322292222222221332 he 2 , e 3 • e 2 = -1 e 3 • e 3 = e 1 , h = -1.Rota-Baxter operators RB( A 7 h ,1 ) are: R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 1 , R 1 (e 3 ) = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 2 e 2 , R 2 (e 3 ) = 0, h = 0. 123 = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = 0, h = 0.The pre-Lie superalgebra ( A 7 h ,2 ) k , • (D 4 ) μ :e 1 • e 2 = -e 1 , e 2 • e 2 = he 2 , e 2 • e 3 = he 3 , e 3 • e 2 = ke 3 , h = 0, -1.Rota-Baxter operators RB((A 7 h ,2 ) k ) are: R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 1 , R 1 (e 3 ) = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0.The pre-Lie superalgebra( A 7 h ,3 ) k , • (D 4 ) μ : e 1 • e 2 = -e 1 , e 2 • e 2 = he 2 , e 3 • e 2 = ke 3 , h = -1.Rota-Baxter operators RB((A 7 h ,3 ) k ) are: R 1 (e 1 ) = a 1 e 2 , R 1 (e 2 ) = a 2 e 1 , R 1 (e 3 ) = a 3 e 3 , h = -1 2 , k = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 4 e 2 , R 2 (e 3 ) = 0, h = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = a 4 e 2 , R 3 (e 3 ) = a 3 e 3 , h = k = 0. R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 3 e 3 , h = 0, k = 0. R 5 (e 1 ) = 0, R 5 (e 2 ) = a 2 e 1 , R 5 (e 3 ) = 0, h = -1 2 . R 6 (e 1 ) = 0, R 6 (e 2 ) = a 2 e 1 , R 6 (e 3 ) = a 3 e 3 , h = -1 2 , k = 0. R 7 (e 1 ) = 0, R 7 (e 2 ) = 0, R 7 (e 3 ) = 0, h = -1 2 , 0.4.2.8 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 8The pre-Lie superalgebra ( A 8,1 ) k , • (D 4 ) μ :e 1 • e 1 = 2e 1 , e 2 • e 1 = e 2 , e 2 • e 2 = e 1 , e 3 • e 1 = ke 3 .Rota-Baxter operators RB(( A 8,1 ) k ) are:R 1 (e 1 ) = 0, R 1 (e 2 ) = 0, R 1 (e 3 ) = a 1 e 3 , k = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = 0.4.2.9 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 9The pre-Lie superalgebra( A 9,1 ) k , •(D 4 ) μ : k = 0 or 1 associative others cases are non-associative. e 2 • e 1 = e 1 , e 2 • e 2 = e 2 , e 3 • e 2 = ke 3 .Rota-Baxter operators RB(( A 9,1 ) k ) are:Case If k = 0, we have R 1 (e 1 ) = a 1 e 1 + a 2 e 2 , R 1 (e 2 ) = -a R 1 (e 3 ) = a 3 e 3 , a 2 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 4 e 1 , R 2 (e 3 ) = a 3 e 3 . R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = a 3 e Case If k ∈ C * , we have R 4 (e 1 ) = 0, R 4 (e 2 ) = a 4 e 1 , R 4 (e 3 ) = 0. R 5 (e 1 ) = a 1 e 1 + a 2 e 2 , R 5 (e 2 ) = -a e 1a 1 e 2 , R 5 (e 3 ) = 0, a 2 = 0. R 6 (e 1 ) = 0, R 6 (e 2 ) = 0, R 6 (e 3 ) = 0.The pre-Lie superalgebras ( A 9,2 ) k , • (D 4 ) μ and ( A 9,3 , •) D 5 : e 3 = e 3 e 3 • e 1 = ke 3 . e 3 = e 3 e 3 • e 3 = 1 2 e 1 e 3 • e 3 = e 1 .k = 0 or 1 associative other cases are non associative. They have the same Rota-Baxter operators RB(A), that is,R 1 (e 1 ) = a 1 e 1 + a 2 e 2 , R 1 (e 2 ) = -a e 1a 1 e 2 , R 1 (e 3 ) = 0, a 2 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = a 3 e 1 , R 2 (e 3 ) = 0. R 3 (e 1 ) = 0, R 3 (e 2 ) = 0, R 3 (e 3 ) = 0.4.2.10 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 10 hThe pre-Lie superalgebra ( A 10 h ,1 ) k , • (D 4 ) μ :e 1 • e 2 = (h -1)e 1 , e 2 • e 1 = he 1 , e 2 • e 2 = e 1 + he 2 , e 3 • e 2 = ke 3 , h = 0. Rota-Baxter operators RB(( A 10 h ,1 ) k ) are:Case If k = 0, we have R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 1 , R 1 (e 3 ) = a 2 e 3 . R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e Case If k ∈ C * , we have R 3 (e 1 ) = 0, R 3 (e 2 ) = a 1 e 1 , R 3 (e 3 ) = 0, a 1 = 0. R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = 0.The pre-Lie superalgebras ( A 10 h ,2 ) k , • (D 4 ) μ and ( A 10 h ,3 , •) D 5 , where( A 10 h ,2 ) k , • ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩e 1 • e 2 = (h -1)e 1 e 2 • e 1 = he 1 e 2 • e 2 = e 1 + he 2 , h = 0 e 2 • e 3 = he 3 e 3 • e 2 = ke 3 ( A 10 h ,3 , •) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e 1 • e 2 = (h -1)e 1 e 2 • e 1 = he 1 e 2 • e 2 = e 1 + he 2 , h = 0 e 2 • e 3 = he 3 e 3 • e 2 = h -1 2 e 3 e 3 • e 3 = e 1 .

4. 2 . 1 :Case 2 :

 212 11 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 11 The pre-Lie superalgebra ( A 11,1 ) k , • (D 4 ) μ : e 1 • e 2 = -e 1 , e 2 • e 2 = e 1e 2 , e 3 • e 2 = ke 3 .Rota-Baxter operators RB((A 11,1 ) k ) are: Case If k = 0, we have R 1 (e 1 ) = 0, R 1 (e 2 ) = a 1 e 1 , R 1 (e 3 ) = a 2 e 3 , a 1 = 0. R 2 (e 1 ) = 0, R 2 (e 2 ) = 0, R 2 (e 3 ) = a 2 e 3 . If k ∈ C * , we have R 3 (e 1 ) = 0, R 3 (e 2 ) = a 1 e 1 , R 3 (e 3 ) = 0, a 1 = 0. R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = 0.The pre-Lie superalgebras ( A 11,2 , •) D 5 and ( A 11,3 ) k , • (D 4 ) μ , where ( A 11,2 , •) : e 1 • e 2 = -e 1 , e 2 • e 2 = e 1e 2 , e 3 • e 2 = -1 2 e 3 , e 3 • e 3 = e 1 . ( A 11,3 ) k , • : e 1 • e 2 = -e 1 , e 2 • e 2 = e 1e 2 , e 2 • e 3 = -e 3 , e 3 • e 2 = ke 3 .

  1 a 5 a 6 e 2 + a 6 e 3 , R 7 (e 2 ) = a6 2 e 1 + a 5 e 2 +

	a 2 6 2a 1 e 3 , R 7 (e 3 ) = e 2 + a 5 e 3 , R 7 (e 4 ) = 0, R 7 (e 5 ) = 0, a 1 = 0, a 6 = 0. R 8 (e 1 )=a 1 e 1 + a 6 e 3 , R 8 (e 2 ) = a 6 a 1 a 5 a 6 e 1 + 2a 1 a 2 5 a 2 6 2 e 1 + a 2 6

  Proposition 4.2 The Rota-Baxter operators (of weight zero) on three-dimensional pre-Lie superalgebras (associative or non-associative) with two-dimensional even part of type A1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 h , A 8 ,A 9 , A 10 h and A 11 are given as follows: 4.2.1 Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 1

2a 1 e 2 -3a[START_REF] Aguiar | Infinitesimal bialgebras, pre-Lie algebras and dendriform algebras[END_REF] 2 e 3 , R 3 (e

) = 0, R 3 (e

) = 0, a 1 = 0, a 3 = 0. R 4 (e 1 ) = 0, R 4 (e 2 ) = 0, R 4 (e 3 ) = a 3 e 1 + a 4 e 2 , R 4 (e 4 ) = 0, R 4 (e 5 ) = 0.

Rota-Baxter Operators on Two-dimensional Pre-Lie SuperalgebrasThe purpose of this section is to compute all Rota-Baxter operators (of weight zero) on the two-dimensional complex pre-Lie superalgebras given by Zhang and Bai (see

a 3 e 2a e 3 , a 3 = 0.

Acknowledgements

We would like to thank Chengming Bai for his valuable remarks and suggestions. The study was funded by Laboratoire de Mathématiques Appliquées et Analyse Harmonique with Grant No. LR11ES52.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

R 1 (e 1 ) = a 1 e 1 + a 2 e 2 , R 1 (e 2 ) = a 3 e 1 + a 4 e 2 , R 1 (e 3 ) = 0.

The pre-Lie superalgebra ( A 4,3 , •) D 2 : (associative)

Rota-Baxter operators RB( A 4,3 ) are:

Rota-Baxter Operators on Pre-Lie Superalgebras of Type A 5

The pre-Lie superalgebra

Rota-Baxter operators RB( A 5,1 ) are:

Rota-Baxter operators RB( A 5,2 ) are:

The pre-Lie superalgebra ( A 5,3 , •) D 1 :

Rota-Baxter operators RB( A 5,3 ) are:

The pre-Lie superalgebra ( A 5,4 , •) D 1 : e 1 • e 1 = e 2 , e 3 • e 2 = e 3 . Rota-Baxter operators RB( A 5,4 ) are: R 1 (e 1 ) = a 1 e 1 + a 2 e 2 , R 1 (e 2 ) = a 1 2 e 2 , R 1 (e 3 ) = 0. R 2 (e 1 ) = a 2 e 2 , R 2 (e 2 ) = a 3 e 2 , R 2 (e 3 ) = 0.


Baidar Lotfi

Mihoubi Miloud
email: miloud.mihoubi@univ-sba.dz

Rahmoun Abdellatif

Lorenz Pascal
email: lorenz@ieee.org

PSO Variants for Localization Challenge in Wireless Sensor Network and Internet Of Thing

Keywords: Particle Swarm Optimisation Wireless sensor network, Optimisation problem, node localization

Wireless Sensor Network (WSN) has recently been extensively investigated due to their numerous applications where processes have to be spread over a large area. One of several technical aspects of WSNs is the node localization. Most approaches in the recent literature rely on algorithms that maximize the localization rate with a minimum runtime. In this paper, we introduce a comparative study about the PSO (particle swarm optimization) and its variants, The node localization problem is seen as an optimization problem in a multi-dimensional space, PSO computes iteratively (through evolution) the nodes positions using the Euclidian distance as fitness. Deploying this algorithm on a large WSN with hundreds of sensors shows pretty good performance in terms of node localization, two network topology are treated in this paper for contributing the advantages of range communication as an influencing factor.

I.

I NTRODUCTION WSN (Wireless Sensor Network) consists of a large number of densely deployed nodes which are tiny, low power, inexpensive, multi-functional connected by a wireless medium. Its main task is to provide information of environment at each moment, the sensors are installed for tracking and monitoring the environmental requirements and physical phenomenon such as climate prediction, analysis of sane, atmospheric pressure, etc. WSN is employed are several application domains such as security and surveillance, data aggregation, environment sensing, industrial process control, structural health monitoring [START_REF] Adnan | Bio-Mimic Optimization Strategies in Wireless Sensor Networks: A Survey[END_REF].

The gathered information needs to be associated with sensor nodes rental to provide an accurate view of the sensor field.

Each sensor can monitor its region and proceed to send the collected data to sink node, the effectiveness of WSN is bound to its ability to collect data and transmit them in a minimum time and with greater precision [START_REF] Akyildiz | Wireless Sensor Networks: A Survey[END_REF].Numerous challenges are addressed in WSN and have been largely browsed: energy minimization (optimization), energy harvesting, compression schemes, self-organizing network algorithms, routing protocols, security, quality of service management, etc…

The three most essential problems are energy efficiency, quality of service and security management [START_REF] Sohraby | Wireless sensor networks: Technology, protocols, and applications[END_REF], moreover, one of the most issues in WSN is node localization, the information location plays a vital role in coverage, deployment purpose, routing information, location service, target tracking, and rescue operations.

The intuitive solution is to equip every node with a Global Positioning System (GPS) receiver that can accurately provide the sensors with their exact location, localization by equipping each node with GPS is not suitable, because it is less energy efficient and expensive, it needs large size of hardware and has a line of sight problem. On the other hand, If GPS is installed on every node, then it increases the node size and deployment cost. Furthermore, GPS is not energy efficient as it consumes a lot of energy and not suitable for a network like WSN, where for instance, it does not work at all in indoor environments.

Numerous localization methods have been proposed for solving the problem, in place of equipping each node by a GPS, the majority of methods allowed to use a certain number of nodes in the network equipped with GPS. These nodes are generally known by anchor nodes, beacons or landmark which their position is known, the remain sensors communicate with the anchor nodes for determinate its positions by using some localization algorithm. This paper is divided into eight sections, section two gives an overview treatment of node localization problem by metaheuristic algorithm where related work is presented and analysed in this section, the node localization problem is formulated in section three, In the fourth section a comparative study is presented and analysed to evaluate the performance of the proposed algorithm, Our conclusion is drawn in the final section.

II.

RELATED WORK Fig. 1:Localization scheme Several surveys on localization are detailed in [START_REF] Alrajeh | Localization Techniques in Wireless Sensor Networks[END_REF][START_REF] Chowdhurya | Advances on Localization Techniques forWireless Sensor Networks: A Survey[END_REF][START_REF]Localization Algorithms in Wireless Sensor Networks: Current Approaches and Future Challenges[END_REF][START_REF] Han | Localization algorithms of Wireless Sensor Networks: a survey[END_REF][START_REF] Hoang | An RSS comparison based Localization in Wireless Sensor Networks[END_REF] and node localization can be divided via much consideration and according to several criteria: the mobility of the nodes (static vs. mobile, mixed), the execution environment (centralized vs. distributed, mixed), distance estimation (range-free and range-based). Localization scheme can be divided into three components combined show Fig. 1, where several methods are proposed the first phase (distance estimation). The main task of the first phase is evaluating the distance between beacons nodes and target nodes, where many techniques are used , we found Radio Signal Strength Indicator ( RSSI ) [START_REF] Cheon | IEEE 802.15.4 ZigBee-Based Time-of-Arrival Estimation for Wireless Sensor Networks[END_REF], Time of Arrival ( TOA ) and Time Difference of Arrival ( TDOA ) [START_REF] Zhu | A distributed continuous-time algorithm for network localization using angle-of-arrival information[END_REF], Angle of Arrival (AOA) [START_REF] Pk | node localization in wirless sensor network[END_REF] are applied to calculate the angle or distance [START_REF] Singh | A Hybrid Localization Algorithm for Wireless Sensor Networks[END_REF]. The selection of distance estimation in localization scheme is an important factor that influences the final performance of the system and that each one is specified according to the field of application.

The second phase named position computation, by exploiting the data from distance estimation phase, it is possible to calculate the coordinates of the target nodes, wherein the several approaches are proposed in this phase: such methods include trilateration (Fig. 2.a), multilateration(Fig. 2.a), triangulation, probabilistic approaches, bounding box, the central position, fingerprinting [START_REF] Boukerche | Localization systems for wireless sensor networks[END_REF], there is other work concentrates all information about distances computing that uses mathematical optimization techniques to compute the positions of the nodes. The third phase of localization scheme is localization algorithm, the combining operation schematized in Fig. 1, localization algorithm determines how the information concerning distances and positions is manipulated in order to allow nodes of a WSN to estimate their positions.

Localization algorithm has received much attention in the last two years, recently, metaheuristic methods have been seen as important tools to solve the problem of optimization, and node localization is treated as a multidimensional optimization problem, where a substantial number of methods are addressed through population-based stochastic techniques.

PSO (particle swarm optimization) is proposed [START_REF] Gopakumar | Localization in wireless sensor networks using particle swarm optimization[END_REF] for node localization problem. After getting the estimated distances and calculated coordinates in the network by the estimation and calculation phases.

The basic idea of this approach that all unknown nodes send the coordinates and distance to a base station, the base station runs PSO to minimize the tracking error defined. The approach does not take into account the flip ambiguity problems (Fig. 3), and location of nodes that do not have at least three tags in their neighborhood, The system only works well if either beacon have sufficient range or there is a large number of beacons. In addition, the base station requires estimates the range of all target nodes located in the surrounding of beacons nodes. This requires a lot of communication that can lead to congestion, delays and the depletion of energy. More than, the proposed scheme has limited scalability because the dimensionality PSO is twice the number of target nodes. Fig. 3:Flip ambiguity phenomenon PSO approach BFA are proposed [START_REF] Kulkarni | Bio-inspired node localization in wireless sensor networks[END_REF] to deal with the localization issue, the distributed architecture is utilized for both approaches, iterative fashion to locate the nodes, this means that each target node that has three or more beacons in its hearing range runs localization Algorithm to minimize the localization error, and each node localized in the iteration i is considered as an anchor nodes at iteration i+1 .

The process continues iteratively until either all the nodes get localized or no more nodes can be localized, the proposed method does not require that each node transmit it range measurement to a central node. Besides, it can localize all nodes that have three beacons nodes in their range. As the localization iterations pass by, a node may get the number of references for localization, which mitigates the a b flip ambiguity problem, the situation that results in large localization error when the references are near collinear. However, the proposed method is prone to error accumulation.

A comparison of the performance of PSO and BFA in the number of nodes localized, localization accuracy and computation time is presented according to the results, it has concluded that the two approaches have a trade problem where simulations obtained showed that PSO is convergent quickly by contribution to BFA, on other hand BFA gives many error accuracies than PSO .

Binary PSO is proposed to minimize localization time and energy computing and Improving the original approach, where the same strategy of localization phase utilized in [START_REF] Zain | Distributed Localization for Wireless Sensor Networks using Binary Particle Swarm Optimization (BPSO)[END_REF], this approach has proven effective in comparison with the original approach, although it is efficient in terms of convergent but localization error remains low than the original PSO.

h-best PSO is proposed for enhancement of PSO for node localization problem [START_REF] Kumar | Meta-Heuristic Range Based Node LocalizationAlgorithm for Wireless Sensor Networks[END_REF], the objective of the proposed approach is to have a fast converging for localization and have a significant localization rate in terms of nodes and minimizing the location error, the results obtained demonstrated that the proposed approach is efficient when compared with PSO original and BBO, Although The HPSO-based localization algorithm determined the accurate coordinates Quickly, whereas BBObased localization algorithm find the coordinate of the nodes more accurately. The proposed algorithms also reduce the number of transmissions to the base station, that helps the node to retain more energy, and so, the node can perform for a long period.

in [START_REF] Chandirasekaran | A Case Study of Bio-Optimization Techniques forWireless Sensor Network in Node Location Awareness[END_REF], a comparative study is presented in this paper, and three famous metaheuristics: PSO, SFLA, FFA are proposed for node localization in distributed scheme range based, several parameters are combined in comparison as execution time, localization error, number of the nonlocalized node. For understanding the performance of the proposed algorithms, LabVIEW platform is used. The obtained results show that the PSO converge rapidly then FFA and SFLA algorithms but it gives more average error then the two other. The point common is that there is not the grand difference in the errors offered by both, the selection of the algorithms to use for localization depends entirely on the hardware available to the user and the time constraints involved. The using of short transmission radius is not suitable for these algorithms.

III. N ODE LOCALIZATION PROBLEM FORMULATION AS OPTIMIZATION PROBLEM

The main objective of localization is to find out the coordinate of target nodes by using anchor nodes with single hop range-based distributed technique.

The main steps of node localization scheme are:

1/ M target nodes and N anchor nodes are randomly deployed in the sensor field. Each target node and anchor node has transmission range R. Anchor nodes compute their location awareness and transmit their coordinates. The nodes which get settled at the end of iteration serve as reference nodes during the next iteration and behave like anchors.

2/ the node that falls within transmission range of three or more anchors is considered as localized node.

3/ each localized node measures its distance from each of its neighbouring anchors. The distance measurements are corrupted with Gaussian noise , due to environment consideration. A node estimates its distance from i th anchor as = [ + ],where d is actual distance given by: .. Whereas (x, y) is the location of the target node and ( , ) is the location of the ith anchor node in the neighborhood . 4/ each localizable node runs meta-heuristic algorithmindependently to localize itself by finding its coordinates (x, y). The objective function (mean of square of error between actual and estimated distances of computed node coordinates and the actual node coordinates), is minimized by:

(

Where N ≥ 3 ( 2D location of a node needs minimum 3 anchors) is the number of anchors with in transmission range, R, of the target node.. 5/ evolves the optimal location of target nodes, i.e. (x, y) by minimizing the error function 6/ after coordinates of all localizable nodes (say, ML) are determined, the mean localization error is computed as the mean of square of distance so of computed nodes coordinates ( ) and the actual node coordinates ( , for i = 1, 2, ML determined for metaheuristic algorithm, as shown below:

…………….(3)

Where "L=M-N " 7/ Steps 2 to 6 are repeated by number of iteration defined or until all target nodes get localized or no more nodes can be localized. The performance of the localization algorithm is based on EL and MML, where MML = [M -ML] is the number of nodes that could not be localized. Lesser the value of ML and L, better the performance is.

As the iterations progress, the number of localized nodes increases. This increases the number of references available for already localized nodes. A node that localizes using just three references in an iteration k may have more references in iteration k +1. This decreases the probability of the flip ambiguity.

SIMULATION EXPERIMENTS AND PERFORMANCE EVALUATION

A. Simulation Experiments Two major metrics [START_REF] Abdi | a hybrid rssi based localization algorithm dor wsn using a mobile anchor node[END_REF] are proposed for performance evaluation:

1) Average Location Errors (ALE): This is equal to average distance between the estimated location (

) and actual location ( , of all sensor nodes.

2) Average Execution Time (AET): which is the average time required for localization of all sensor nodes

The total nodes example are deployed in Fig. 5 , the Cshape of the area is selected for the first deployment , where in the first picture 400 nodes are deployed randomly , 0,2 % are anchors nodes and 0,8 are the target nodes .

The main objective is to find out the location of the 0.8 % unknownnodes by the meta-heuristic localization algorithm

The second example are deployed in Fig. 6 , the square shape of the area is selected for the first deployment , where in the first picture 400 nodes are deployed randomly , 0,2 % are anchors nodes and 0,8 are the target nodes .

B. Performance Evaluation

The same configuration of the network is simulated for the three algorithms PSO, HPSO,BPSO. From Fig. 7, we show that HPSO converge quickly than the two others, but in the fig. 8, we found that the localization error of original PSO gives more accuracy then the two others algorithms. The interpretation of this difference focuses on the variation between the exploration and exploitation, where the exploitation consists of probing a limited region of the search space with the hope of improving a promising solution K that we already have at hand. This operation amounts then to intensifying (refining) the search in the vicinity of K, on the other hand, the exploration consists of probing a much larger portion of the search space with the hope of finding other promising solutions that are yet to be refined. This operation amounts then to diversifying the search in order to avoid getting trapped in a local optimum.

VI. C ONCLUSION

Wireless sensor networks are extensively used in most of the areas of modern world intelligent applications. Most of these applications demand location information of the sensor nodes.

In this paper node localization challenge is treated as multidimensional optimization, and three variant of Particle Swarm Optimization are evaluated under two important criteria the performance evaluation (localization error and computing time) are addressed in this paper among the three variants approach of PSO.

To balance between the exploration and exploitation phases is was a delicate task where the first one converge and the other diverge at the same time. This task makes the operation very difficult to be realized.

The future objective is focalized to add energy consumption as third criteria for performance evaluation of node localization in a wireless sensor network, on the other hand ,hybridization with other algorithms may also prove to be fruitful. , we will extend the scheme to counter different types of malicious attacks in sensor localization without incurring too much additional computational cost and communication overhead and apply it to different network environments to further verify and improve the accuracy of sensor localization in hostile or untrusted environments.

Fig. 2 :

 2 Fig. 2: (a) trilateration, (b) Multilateration

  ................................(1)

1 -Fig. 4 :

 14 Fig.4: Flowchart of node localization

Fig. 5 :

 5 Fig.5: network deployment in C-shape

  Fig.8: localization Error obtained during localization