Rota–Baxter Operators on Pre-Lie Superalgebras

Bulletin of the Malaysian Mathematical Sciences Society, Dec 2017

In this paper, we study Rota–Baxter operators and super \(\mathcal {O}\)-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.

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Rota–Baxter Operators on Pre-Lie Superalgebras

Rota-Baxter Operators on Pre-Lie Superalgebras El-Kadri Abdaoui 0 1 2 Sami Mabrouk 0 1 2 Abdenacer Makhlouf 0 1 2 Communicated by See Keong Lee. 0 1 2 B Sami Mabrouk Mabrouksami 0 1 2 @yahoo.fr 0 1 2 El-Kadri Abdaoui 0 1 2 Abdenacer Makhlouf 0 1 2 0 Université de Haute Alsace , 4 rue des frères Lumière, 68093 Mulhouse , France 1 Faculté des Sciences, Université de Gafsa , Gafsa , Tunisia 2 Faculté des Sciences Sfax, Université de Sfax , BP 1171, 3038 Sfax , Tunisia 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. Mathematics Subject Classification 17A30 · 17A36 · 17B70 Rota-Baxter operator; Super O-operator; Associative superalgebra; Lie superalgebra; Pre-Lie superalgebra; L -dendriform superalgebras 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 [ 36,51 ]. They appeared for the first time in the work of the mathematician Baxter [ 7 ] in 1960 and were then intensively studied by Atkinson [ 6 ], Miller [47], Rota [50], Cartier [ 18 ], and more recently, they reappeared in the work of Guo [37] and Ebrahimi-Fard [ 27 ]. 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 [ 20 ] as a kind of rooted tree algebra and later arose again from the study of convex homogeneous cones [53], affine manifold and affine structures on Lie groups [40], and deformation of associative algebras [ 34 ]. They play an important role in the study of symplectic and complex structures on Lie groups and Lie algebras [ 5,22,24,25,44 ], phases spaces of Lie algebras [8,42], certain integrable systems [ 16 ], classical and quantum Yang–Baxter equations [ 26 ], combinatorics [ 27 ], quantum field theory [ 23 ] and operads [ 19 ]. See [ 17 ] for a survey. Recently, pre-Lie superalgebras, the Z2-graded version of pre-Lie algebras also appeared in many others fields; see, for example, [ 19,34,52 ]. To our knowledge, they were first introduced by Gerstenhaber in 1963 to study the cohomology structure of associative algebras [34]. 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 [ 15 ]. See [ 3,21,38,39,55 ] 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 (0.1) (0.2) (0.3) R(x )R(y) = R R(x )y + x R(y) + λx y . If λ = 0 (resp. λ = −1), the product resp. x ◦ y = R(x )y − (−1)|x||y| y R(x ) − x y, ∀ x , y ∈ H(A) defines a pre-Lie superalgebra (see Theorem 1.2). The notion of dendriform algebras was introduced in 1995 by Loday [45]. 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 [ 30 ] and the references therein). The relationship between dendriform algebras, Rota–Baxter algebras and pre-Lie algebras was given by Aguiar and Ebrahimi-Fard [ 2,27,28 ]. Bai, Liu, Guo and Ni generalized the concept of Rota–Baxter operator and introduced a new class of algebras, namely L-dendriform algebras, in [ 12–14 ]. 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 [1]. The construction of associative, Lie, pre-Lie and L-dendriform superalgebras is extended to the corresponding categories of bimodules. See [ 9,29,31–33,43,46 ] about further results and [ 10,11,41,48,49 ] 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(1, 2). 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 → E nd(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. Let (A, ◦) be an algebra over a field K. It is said to be a superalgebra if the underlying vector space of A is Z2-graded, that is, A = A0 ⊕ A1, and Ai ◦ A j ⊂ Ai+ j , for i, j ∈ Z2. An element of A0 is said to be even and an element of A1 is said to be odd. The elements of A j , j ∈ Z2, 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 [ 13 ]. Definition 1.1 (1) An associative superalgebra is a pair (A, μ) consisting of a Z2graded vector space A and an even bilinear map μ : A ⊗ A −→ A, (Ai A j ⊆ Ai+ j , ∀ i, j ∈ Z2) 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 Z2-graded vector space. Let l, r : A −→ E nd(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 , μV , l, r ) is said to be an A-bimodule K-superalgebra if (V , l, r ) is an A-bimodule compatible with the multiplication μV on V , that is, for all x , y ∈ H(A) and v, w ∈ H(V ), l(x )(μV (v, w)) = μV (l(x )(v), w), r (x )(μV (v, w)) = μV (v, r (x )(w)), μV (r (x )(v), w) = μV (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, T (u)T (v) = T l(T (u))v + (−1)|u||v|r (T (v))u +λT (u)T (v), ∀ u, v ∈ H(V ). (1.1) (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 , μV , l, r ) if it satisfies T (u)T (v) = T l(T (u))v + (−1)|u||v|r (T (v))u + λμV (u, v) , ∀ u, v ∈ H(V ). 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, μA, Lμ, Rμ) 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 RV : V −→ V such that for all x ∈ H(A) and v ∈ H(V ) R(x )RV (v) = RV R(x )v + x RV (v) , RV (v)R(x ) = RV RV (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 Z2-graded vector space A, and an even bilinear map [ , ] : A ⊗ A −→ A, ([Ai , A j ] ⊆ Ai+ j , ∀ i, j ∈ Z2) satisfying for all x , y, z ∈ H(A), [x , y] = −(−1)|x||y|[y, x ], (super-skew-symmetry) [x , [y, z]] = [[x , y], z] + (−1)|x||y|[y, [x , z]], (super-Jacobi identity). (2) Let (A, [ , ]) be a Lie superalgebra, V be a Z2-graded vector space, and ρ : A −→ E nd(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. 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 ). (1.3) (1.4) (1.5) (1.6) 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 [54]. Definition 1.4 Let (A, [ , ], R) and (A , [ , ] , R ) be two Rota–Baxter Lie superalgebras. 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 Z2-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 Z2-graded vector space and l, r : A −→ E nd(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 , l(x )r (y)v − r (y)l(x )v = (−1)|x||v|(r (x ◦ y)v − r (y)r (x )v). (1.9) (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)). (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. (2) 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 , l − r, r − l). Now, we construct pre-Lie superalgebras using super O-operators on Lie superalgebras. Proposition 1.3 ([54]) Let (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 u ◦ v = ρ(T (u))v, ∀ u, v ∈ H(V ) defines a pre-Lie superalgebra structure on A. Remark 1.1 ([54]) Let (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. 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 A1 = (A, ◦, R) be a Rota–Baxter pre-Lie superalgebra of weight zero. Then A2 = (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) = A0 ⊕ A1 be the Lie superalgebra where A0 is spanned by The defining relations (we give only the ones with nonzero values in the right-hand side) are [e1, e2] = 2e2, [e1, e3] = −2e3, [e2, e3] = e1, [e3, e5] = e4, [e2, e4] = e5, [e1, e4] = −e4, [e1, e5] = e5, [e5, e4] = e1, [e5, e5] = −2e2, [e4, e4] = 2e3. The Rota–Baxter operators of weight zero on the Lie superalgebra osp(1, 2) with respect to the homogeneous basis {e1, e2, e3, e4, e5} are: R1(e1) = a1e1 + a2e2 − (2a83a+12aa32)2 e3, R1(e2) = − (2a23a+2aa122)2 e1 + (22a(23−a33+aa2)2a)1 e2 + (2a32+a13a2)2 e3, R1(e3)=a3e1 + (2a38+a1a2)2 e2 + a21((2aa23−+6aa23)) e3, R1(e4)=0, R1(e5)=0, a1 = 0, a2 = −2a3. R2(e1) = a1e1 + a2e2, R2(e2) = − 2aa212 e1 − 32a1 e2 + a 2a213 e3, R2(e3) = 8aa21 e2 2 2 + a21 e3, R2(e4) = 0, R2(e5) = 0, a1 = 0, a2 = 0. R3(e1) = a1e1 − 2aa312 e3, R3(e2) = a21 e2 + 2a132 e3, R3(e3) = a3e1 + 2aa31 e2 a3 2 − 32a1 e3, R3(e4) = 0, R3(e5) = 0, a1 = 0, a3 = 0. R4(e1) = 0, R4(e2) = 0, R4(e3) = a3e1 + a4e2, R4(e4) = 0, R4(e5) = 0. − a1e3, R17(e4) = 0, R17(e5) = 0, a3 = 0. R5(e1) = a1e1 − 4a3e2 − 2aa312 e3, R5(e2) = − 4a13 e1 + a1e2 + 2a132 e3, R5(e3) = a2 a3 a13(41a63a−33a2) e3, R16(e3) = a3e1 + aa2a13 e2 + a1(a42a−34a3) e3, R16(e4) = 0, R16(e5) = 0, a1 = 0, a3 = 0. 2 a2 a3 R17(e1) = a1e1 − aa13 e3, R17(e2) = − 4a13 e1 + 4a132 e3, R17(e3) = a3e1 R18(e1) = a1e1 − 4a3e2 − 2aa312 e3, R18(e2) = 2aa312 e1 − 72a1 e2 + 2a132 e3, R18(e3) a3 2 = a3e1 + 2aa31 e2 + 52a1 e3, R18(e4) = 0, R18(e5) = 0, a1 = 0, a3 = 0. a2 R19(e1) = a1e1 + 4a3e2, R4(19) = − 4a13 e1 − a1e2, R19(e3) = a3e1 + R22(e1) = 4aa652 e2 + a6e3, R22(e2) = −a5e2 − 4aa65 e3, R22(e3) = a 4a253 e2 + 2 6 aR52e33(,e1R) 23=(e4a)2=e1,0,RR2323(e(e25)) == 00,, aR52=3(e03,) a=6 =−0a22. e2 + a4e2, R23(e4) = 0, The constants ai are parameters. Now, we define Rota–Baxter operators on an A-module, where A is a Rota–Baxter Lie superalgebra. 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 RV : V −→ V such that, for all x ∈ H(A) and v ∈ H(V ), [R(x ), RV (v)] = RV [R(x ), v] + [x , RV (v)] , [RV (v), R(x )] = RV [RV (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 RV a Rota–Baxter operator on V . Define new actions of A on V by x ◦ v = [R(x ), v], v ◦ x = [RV (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 ), [RV (v), y]] − (−1)|y||v|[RV ([R(x ), v]), y] − (−1)|v|(|x|+|y|)[RV (v), [R(x ), y]] + (−1)|v|(|x|+|y|)[RV ([RV (v), x ]), y] = (−1)|y||v|[R(x ), [RV (v), y]] − (−1)|y||v| [RV ([R(x ), v] + [x , RV (v)]), y] − (−1)|v|(|x|+|y|)[RV (v), [R(x ), y]] = (−1)|y||v|[R(x ), [RV (v), y]] − (−1)|y||v|[[R(x ), RV (v)], y] − (−1)|v|(|x|+|y|)[RV (v), [R(x ), y]] = 0. 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 S3 (see [ 35 ]). The graded case was studied by Ammar and Makhlouf in 2010 (see [ 4 ] 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 S3. 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 AL = (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 = R(x )y − (−1)|x||y| y R(x ) − x y. x ◦ y = μ(R(x ), y) − (−1)|x||y|μ(y, R(x )) − μ(x , y) Then AL = (A, ◦) is a pre-Lie superalgebra. (1.15) 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) − (R(x)y)z + (−1)|x||y|(y R(x))z + (x y)z. + (−1)|z|(|x|+|y|)+|x||y|(y R(x))R(z) + (−1)|z|(|x|+|y|)(x y)z Then, we obtain asAL (x, y, z) − (−1)|x||y|asAL (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|)(x y)z + (−1)|z|(|x|+|y|)(R(x)y)R(z) − (−1)|z|(|x|+|y|)+|x||y|(y R(x))R(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)(x z) + (−1)|y||z|(R(x)z)R(y) − (−1)|z|(|x|+|y|)(z R(x))R(y) − (−1)|y||z|(x z)R(y) + (−1)|x||y| y(R(x)z) − (−1)|z|(|x|+|y|)(z R(x))y +(−1)|z|(|x|+|y|)(x R(y))R(z) +(−1)|x||y| y(x z) + (−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||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: 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. 2 L-dendriform Superalgebras The notion of L-dendriform algebra was introduced by Bai, Liu and Ni in 2010 (see [ 14 ]). 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. 2.1 L-dendriform Superalgebras and Associative Superalgebras 2.1.1 Definition and Some Basic Properties Definition 2.1 A L-dendriform superalgebra is a triple (A, , ) consisting of a Z2graded vector space A and two even bilinear maps , : A ⊗ A −→ A satisfying, for all homogeneous elements x , y, z in A, x x (y (y z) = (x z) = (x y) y) −(−1)|x||y|(y +(−1)|x||y| y z + (x y) x ) (x z + (−1)|x||y| y z − (−1)|x||y|(y z) − (−1)|x||y|(y x ) x ) (x z) z, The associated bracket to a L-dendriform superalgebra is defined as [x , y] = x (−1)|x||y| y x . Definition 2.2 (1) Let (A, , ) be a L-dendriform superalgebra, V be a Z2-graded vector space, and l , r , l , r : A −→ E nd(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), (2.1) (2.2) y − (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 Ksuperalgebra 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 (u) T (v) = T l (T (u))v + (−1)|u||v|r (T (v))u + λu T (v) = T l (T (u))v + (−1)|u||v|r (T (v))u + λu V v , 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(x ) R(y) = R R(x ) R(y) = R R(x ) y + R(x ) y + R(x ) u + λx y + λx y , y . The following theorem provides a construction of L-dendriform superalgebras using super O-operators of associative superalgebras. Theorem 2.1 Let (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 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 = (−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. 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. Corollary 2.1 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight zero. Then, the even binary operations given by x y = μ(R(x ), y), x y = μ(x , R(y)), ∀ x , y ∈ H(A) defines a L-dendriform superalgebra structure on A. Definition 2.3 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-bimodule V (relative to R) is a map RV : V −→ V such that for all homogeneous elements x in A and v in V R(x ) RV (v) R(x ) RV (v) RV (v) = RV R(x ) v + x R(x ) = RV RV (v) x + v RV (v) = RV R(x ) v + x R(x ) = RV RV (v) x + v RV (v) , R(x ) , RV (v) , R(x ) . Proposition 2.1 Let (A, μ) be an associative superalgebra, R : A −→ A a Rota– Baxter operator on A, V an A-bimodule and RV a Rota–Baxter operator on V . Define a new actions of A on V by x x v = μ(R(x ), v), v v = μ(x , RV (v)), v x = μ(RV (v), x ), 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 ). 2.2 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 preLie 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 preLie superalgebra of (A, , ) and (A, , ) is called a compatible L-dendriform superalgebra structure on the pre-Lie superalgebra (A, •). (3) Both (A, ◦) and (A, •) have the same sub-adjacent Lie superalgebra g(A) defined by Corollary 2.3 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 . u 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 ) 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 (r(T (u))v) 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|r(T (v))l(T (u))w, (−1)|u||v|(v u) w = (−1)|u||v|r T (r(T (v))u) w. Hence, 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 + (−1)|u||v|(v v) u) v) w + (−1)|u||v|(v u) w (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 (2.4) (2.5) + (−1)|u||v|l T (v) ◦ T (u) w and = 0, u (v = −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. 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 ) (2.7) defines a L-dendriform superalgebra structure on A. Lemma 2.1 Let {R1, R2} be a pair of commuting Rota–Baxter operators (of weight zero) on a pre-Lie superalgebra (A, ◦). Then R2 is a Rota–Baxter operator (of weight zero) on the L-dendriform superalgebra (A, , ) defined in (2.7) with R = R1. Theorem 2.3 Let (A, ◦) be a pre-Lie superalgebra. Then there exists a compatible L-dendriform superalgebra structure on (A, ◦) such that (A, ◦) is the associated vertical pre-Lie superalgebra if and only if there exists an invertible super O-operator (of weight zero) of (A, ◦). 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 RV a Rota–Baxter operator on V . Define new actions of A on V by x v = R(x ) ◦ v, v v = −(−1)|x||v|v ◦ R(x ), v x = RV (v) ◦ x , x x = −(−1)|x||v|x ◦ RV (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 = (−1)|v|(|x|+|y|)v y) − (−1)|v|(|x|+|y|)(v y + x x ) y − (−1)|y||v|x (v y) + (−1)|v|(|x|+|y|)(x v) y = −(R(x ) ◦ y) ◦ RV (v) + (−1)|x||y|(y ◦ R(x )) ◦ RV (v) − (−1)|x||y| y ◦ RV (x ◦ RV (v)) − (−1)|x||y| y ◦ RV (R(x ) ◦ v) + R(x ) ◦ (y ◦ RV (v)) = −(R(x ) ◦ y) ◦ RV (v) + R(x ) ◦ (y ◦ RV (v)) − (−1)|x||y| y ◦ (R(x ) ◦ RV (v)) + (−1)|x||y|(y ◦ R(x )) ◦ RV (v) = 0. Therefore, Similarly, we have [l (x ), r (y)](v) = r (x • y)(v) − (−1)|x||y|r (y)r (x )(v). r (x = (−1)|v|(|x|+|y|)v = −(−1)|v|(|x|+|y|)+|x||y| RV (v) ◦ (y ◦ R(x )) + (−1)|x|(|y|+|v|) y ◦ RV (RV (v) ◦ x ) + (−1)|v|(|x|+|y|)+|x||y|(RV (v) ◦ y) ◦ R(x ) − (−1)|x|(|y|+|v|)(y ◦ RV (v)) ◦ R(x ) + (−1)|x|(|y|+|v|) y ◦ RV (v ◦ R(x )) = (−1)|x|(|y|+|v|) y ◦ (RV (v) ◦ R(x )) − (y ◦ RV (v)) ◦ R(x ) + (−1)|v|(|x|+|y|)+|x||y| RV (v) ◦ y) ◦ R(x ) − RV (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, , ). 3 Rota–Baxter Operators on Two-dimensional Pre-Lie Superalgebras The 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 [ 15 ]). In the following, let C be the ground field of complex numbers and {e1, e2} be a homogeneous basis of a pre-Lie superalgebra (A, ◦), where {e1} is a basis of A0 and {e2} is a basis of A1. 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 preLie superalgebras (associative or non-associative) of type B1, B2 and B3 are given as follows: 3.1 Rota–Baxter Operators on Pre-Lie Superalgebras of Type B1 The pre-Lie superalgebra (B1,1, ◦) : e2 ◦ e1 = e2 has the Rota–Baxter operator defined as R1(e1) = a1e1, R1(e2) = 0. ⎧ ⎪⎪⎪⎪⎨ The pre-Lie superalgebras of type (B1,2, ◦) : e1 ◦ e1 = e1, e2 ◦ e1 = e2 (associative). (B1,3)k , ◦ : (B1,4, ◦) : e1 ◦ e1 = ke1, e1 ◦ e1 = e1, e2 ◦ e1 = e2, k = 0, 1. e1 ◦ e2 = e2 (associative). (B1,5)k , ◦ : e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e2 ◦ e1 = (k + 1)e2, k = 0, −1. have only the trivial Rota–Baxter operator, that is, R1(e1) = 0, R1(e2) = 0. 3.2 Rota–Baxter Operators on Pre-Lie Superalgebras of Type B2 The pre-Lie superalgebra (B2,1, ◦) (associative). R1(e1) = a1e1, R1(e2) = 0. R2(e1) = a1e1, R2(e2) = 2a1e2. The pre-Lie superalgebra (B2,2, ◦) (associative) Rota–Baxter operators RB(B2,2) is: 1 e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e2 ◦ e2 = 2 e1. R1(e1) = 0, R1(e2) = 0. 3.3 Rota–Baxter Operators on Pre-Lie Superalgebras of Type B3 The pre-Lie superalgebra (B3,1, ◦) (associative). The pre-Lie superalgebra (B3,2, ◦) (associative) Rota–Baxter operators RB(B3,2) are: R1(e1) = 0, R1(e2) = a1e2. The pre-Lie superalgebra (B3,3, ◦) (associative) R1(e1) = 0, R1(e2) = 0. e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2. 4 Rota–Baxter Operators on Three-dimensional Pre-Lie Superalgebras 4.1 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 [ 15 ]). 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 {e1, e2, e3} be a homogeneous basis of a pre-Lie superalgebra (A, ◦), where {e1} is a basis of A0 and {e2, e3} is a basis of A1. Proposition 4.1 The Rota–Baxter operators (of weight zero) on three-dimensional pre-Lie superalgebras (associative or non-associative) with two-dimensional odd part of type C1, C2h , C3, C4, C5 and C6 are given as follows: 4.1.1 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C1 The pre-Lie superalgebra (C1,1, ◦) e2 ◦ e3 = −e1, e3 ◦ e1 = e2, e3 ◦ e2 = e1. Rota–Baxter operators RB(C1,1) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + e3. R2(e1) = 0, R2(e2) = a2e2, R2(e3) = a1e2, a2 = 0. R3(e1) = a3e1, R3(e2) = 0, R3(e3) = a1e2. The pre-Lie superalgebra (C1,2)k, ◦ (associative) e1 ◦ e3 = ke2, e3 ◦ e1 = (k + 1)e2. Rota–Baxter operators RB((C1,2)k) are: R1(e1) = 0, R1(e2) = a1e2, R1(e3) = a2e2, k = −1. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e2 + a3e3, k = −1. R3(e1) = a4e1, R3(e2) = a1e2, R3(e3) = a2e2 + aa41−aa41 e3, a1 = a4, k = −1. The pre-Lie superalgebra (C1,3, ◦): e1 ◦ e1 = e1, e3 ◦ e1 = e2. Rota–Baxter operators RB(C1,3) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3. R2(e1) = 0, R2(e2) = a3e2, R2(e3) = a1e2. The pre-Lie superalgebra (C1,4, ◦): e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e2 + e3. Rota–Baxter operators RB(C1,4) are: R1(e1) = 0, R1(e3) = 0, R1(e2) = a1e2. R2(e1) = 0, R2(e2) = 0, R2(e3) = 0. 4.1.2 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C2h The pre-Lie superalgebra (C2h,1, ◦): e1 ◦ e1 = (h + 1)e1, e2 ◦ e1 = e2, e2 ◦ e3 = −e1, e3 ◦ e1 = he3, e3 ◦ e2 = e1, h ∈ C. Rota–Baxter operators RB(C2h,1) are: Case 1: If h = 0, we have R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3. R2(e1) = 0, R2(e2) = a3e3, R2(e3) = a2e3. R3(e1) = 0, R3(e2) = 0, R3(e3) = a2e3. R4(e1) = 0, R4(e2) = a3e3, R4(e3) = a2e3, a3 = 0. Case 2: If h ∈ C∗, we have R5(e1) = 0, R5(e2) = 0, R5(e3) = a1e2. R6(e1) = 0, R6(e2) = a3e3, R6(e3) = 0, a3 = 02. a R7(e1) = 0, R7(e2) = a5e2 + a3e3, R7(e3) = − ha53 e2 − ah5 e3, a3 = 0, a5 = 0. R8(e1) = 0, R8(e2) = 0, R8(e3) = 0. Case 3: If h = −1, we have R9(e1) = 0, R9(e2) = a3e3, R9(e3) = 0. The pre-Lie superalgebra (C2h,2, ◦): e1 ◦ e1 = (1 − h)e1, e1 ◦ e3 = e2, e2 ◦ e1 = e2, e3 ◦ e1 = e2 + he3, h ∈ C. R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3. R2(e1) = 0, R2(e2) = 0, R2(e3) = a1e2, a1 = 0. R3(e1) = 0, R3(e2) = e2, R3(e3) = 0. Case 3: If h = 1, we have R4(e1) = a3e1, R4(e2) = 0, R4(e3) = a1e2, a3 = 0. R5(e1) = 0, R5(e2) = 0, R5(e3) = a1e2, a1 = 0. R6(e1) = 0, R6(e2) = e2, R6(e3) = 0. The pre-Lie superalgebra (C2h,3, ◦): e1 ◦ e1 = (1 − h)e1, e1 ◦ e2 = (1 − h)e2 + e3, e1 ◦ e3 = (1 − h)e3, e2 ◦ e1 = (2 − h)e2 + e3, e3 ◦ e1 = e3, h = 1. R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e3. R2(e1) = 0, R2(e2) = a2e2 + a3e3, R2(e3) = a2(h − 1)e2 + a2(h − 1)e3. Case 2: If h = 1, we have R3(e1) = 0, R3(e2) = 0, R3(e3) = 0. R4(e1) = 0, R4(e2) = 0, R4(e3) = a1e3, a1 = 0. R5(e1) = 0, R5(e2) = 0, R5(e3) = a1e3, h3 − 2h2 + 2h − 1 = 0. The pre-Lie superalgebra (C2h,4, ◦): e1 ◦ e1 = (h − 1)e1, e1 ◦ e2 = e3, e2 ◦ e1 = e2 + e3, e3 ◦ e1 = he3, h = ±1. Rota–Baxter operators RB(C2h,4) are: R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3, h = 0. R2(e1) = 0, R2(e2) = a1e3, R2(e3) = 0, a1 = 0, h = 0. R3(e1) = 0, R3(e2) = 0, R3(e3) = 0, h = 0. The pre-Lie superalgebra (C2h,5, ◦): e1 ◦ e1 = (1 − h)e1, e1 ◦ e2 = (1 − h)e2 + e3, e1 ◦ e3 = (1 − h)e3, e2 ◦ e1 = (2 − h)e2 + e3, e3 ◦ e1 = e3, h = ±1. Rota–Baxter operators RB(C2h,5) are: R1(e1) = 0, R1(e2) = e2, R1(e3) = a1e2 − a1e3, h = 0. R2(e1) = 0, R2(e2) = 0, R2(e3) = 0, h = 0. R3(e1) = 0, R3(e2) = a2e2 − (h −a31)2 e3, R3(e3) = a2(hh−1)2 e2 − ah2 e3, a2 = 0, h = 0. R4(e1) = 0, R4(e2) = a2e2 − a2e3, R4(e3) = a22 e2 − a22 e3, a2 = 0, h = 2. R5(e1) = 0, R5(e2) = a3e3, R5(e3) = 0, a3 = 0. The pre-Lie superalgebra (C2h,6)k, ◦ : h = 0 or 1, k = 1 associative other cases non-associative. e1 ◦ e1 = ke1, e2 ◦ e1 = e2, e3 ◦ e1 = he3, h, k ∈ C. Rota–Baxter operators RB((C2h,6)k) are: Case 1: If h = 0, we have R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3. R2(e1) = 0, R2(e2) = a3e3, R2(e3) = a2e3. R3(e1) = 0, R3(e2) = 0, R3(e3) = a2e3. Case 2: If h ∈ C∗, we have R4(e1) = 0, R4(e2) = 0, R4(e3) = a1e2. R5(e1) = 0, R5(e2) = a3e3, R5(e3) = 0. a2 R6(e1) = 0, R6(e2) = a4e2 + a3e3, R6(e3) = − ha43 e2 − ah4 e3, a3 = 0. R7(e1) = 0, R7(e2) = 0, R7(e3) = 0. Case 3: If h = k = 0, we have R8(e1) = 0, R8(e2) = 0, R8(e3) = a1e2 + a2e3. R9(e1) = 0, R9(e2) = a3e3, R9(e3) = a2e3. R10(e1) = a5e1, R10(e2) = 0, R10(e3) = a2e3. R11(e1) = a5e1, R11(e2) = 0, R11(e3) = 0. R12(e1) = 0, R12(e2) = a3e3, R12(e3) = a2e3, a3 = 0. Case 4: If h = 1 and k = 0, we have R13(e1) = a5e1, R13(e2) = 0, R13(e3) = a1e2. R14(e1) = a5e1, R14(e2) = a3e3, R14(e3) = 0. 2 R15(e1) = a5e1, R15(e2) = a4e2 + a3e3, R15(e3) = − aa43 e2 − a4e3, a3 = 0. Case 5: If h = 0, 1 and k = 0, we have R16(e1) = 0, R16(e2) = 0, R16(e3) = a1e2, a1 = 0. R17(e1) = 0, R17(e2) = a3e3, R17(e3) = 0, r2,2 = 02. a R18(e1) = 0, R18(e2) = a4e2 + a3e3, R18(e3) = − ha43 e2 + ah4 e3, a3 = 0. The pre-Lie superalgebra (C2h,7)k, ◦ : (C20,7)1 associative. e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e2 ◦ e1 = (k + 1)e2, e3 ◦ e1 = he3, , k = 0, h = ±1. R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3, a1 = 0. R2(e1) = 0, R2(e2) = a3e3, R2(e3) = a2e3, a3 = 0. R3(e1) = 0, R3(e2) = 0, R3(e3) = a2e3. R4(e1) = 0, R4(e2) = 0, R4(e3) = a1e2, a1 = 0. R5(e1) = 0, R5(e2) = a3e3, R5(e3) = 0, a3 = 0. R6(e1) = 0, R6(e2) = 0, R6(e3) = 0. The pre-Lie superalgebra (C2h,8)k, ◦ : h = −1 or 0, k = 1 associative other cases are non-associative e1 ◦ e1 = ke1, e1 ◦ e3 = ke3, e2 ◦ e1 = e2, e3 ◦ e1 = (h + k)e3, h ∈ C, k = 0. Rota–Baxter operators RB((C2h,8)k) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2. R2(e1) = 0, R2(e2) = 0, R2(e3) = 0. The pre-Lie superalgebra (C2h)k,9, ◦ : h = 0 or 1, k = −1 associative other cases are non-associative e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e1 ◦ e3 = ke3, e2 ◦ e1 = (k + 1)e2, e3 ◦ e1 = (h + k)e3, h ∈ C, k = 0. Rota–Baxter operators RB((C2h,9)k) are: 2 R1(e1) = 0, R1(e2) = a1e2 + a2e3, R1(e3) = − aa12 e2 − a1e3, a2 = 0, h = 1, k = −1. R2(e1) = 0, R2(e2) = a2e3, R2(e3) = 0. R3(e1) = 0, R3(e2) = 0, R3(e3) = a3e2. R4(e1) = 0, R4(e2) = 0, R4(e3) = 0. 4.1.3 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C3 The pre-Lie superalgebra (C3,1, ◦): e1 ◦ e1 = 2e1, e2 ◦ e1 = e2, e2 ◦ e3 = −e1, e3 ◦ e1 = e2 + e3, e3 ◦ e2 = e1. Rota–Baxter operators RB(C3,1) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2. R2(e1) = 0, R2(e2) = a2e1+a3e2, R2(e3) = − a22+a2a3 e2−(a2+a3)e3, a3 = 0. 2 The pre-Lie superalgebra (C3,2)k , ◦ : e1 ◦ e1 = ke1, e2 ◦ e1 = e2, e3 ◦ e1 = e2 + e3. Rota–Baxter operators RB((C3,2)k ) are: Case 1: If k = 0, we have R1(e1) = a1e1, R1(e2) = 0, R1(e3) = a2e2, a1 = 0. R2(e1) = a1e1, R2(e2) = 0, R2(e3) = a1e2, a1 = 0. R3(e1) = a1e1, R3(e2) = 0, R3(e3) = a2e2, a1 = 0. Case 2: If k ∈ C∗, we have R4(e1) = 0, R4(e2) = 0, R4(e3) = a2e2. R5(e1) = 0, R5(e2) = a3e2+a4e3, R5(e3) = − a32+a3a4 e2−(a3+a4)e3, a4 = 0. a4 R6(e1) = 0, R6(e2) = 0, R6(e3) = 0. The pre-Lie superalgebra (C3,3)k , ◦ : e1 ◦ e3 = ke2, e2 ◦ e1 = e2, e3 ◦ e1 = (k + 1)e2 + e3, k = 0. Rota–Baxter operators RB((C3,3)k ) are: R1(e1) = a1e1, R1(e2) = 0, R1(e3) = a2e2. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e2. R3(e1) = 0, R3(e2) = 0, R3(e3) = 0. The pre-Lie superalgebra (C3,4)k , ◦ : Rota–Baxter operator RB((C3,4)k ) is: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2. e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e1 ◦ e3 = ke3, e2 ◦ e1 = (k + 1)e2, e3 ◦ e1 = e2 + (k + 1)e3, k = 0. 4.1.4 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C4 The pre-Lie superalgebra (C4,1, ◦): (associative) e2 ◦ e3 = −e1, e3 ◦ e2 = e1. Rota–Baxter operators RB(C4,1) are: R1(e1) = a1e1, R1(e2) = a2e2 + a3e3, R1(e3) = a4e2 − a1aa21−+aa23a4 e3, a1 = a2. R2(e1) = 0, R2(e2) = 0, R2(e3) = a4e2 + a5e3. 2 R3(e1) = a1e1, R3(e2) = a1e2 + a3e3, R3(e3) = − aa13 e2 + a5e3, a3 = 0. The pre-Lie superalgebra (C4,2, ◦): (associative). R1(e1) = a1e1, R1(e2) = a2e2 + a3e3, R1(e3) = a4e2 + a5e3. The pre-Lie superalgebra (C4,3, ◦): (associative) R1(e1) = 0, R1(e2) = a1e2 + a2e3, R1(e3) = a3e2 + a4e3. The pre-Lie superalgebra (C4,4, ◦): (associative) e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = e3. Rota–Baxter operators RB(C4,4) are: R1(e1) = 0, R1(e2) = a1e2, R1(e3) = a2e2. R2(e1) = 0, R2(e2) = a1e2 + a3e3, R2(e3) = 0. R3(e1) = 0, R3(e2) = a1e2, R3(e3) = 0. The pre-Lie superalgebra (C4,5, ◦): (associative) e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e3. Rota–Baxter operators RB(C4,5) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2. 2 R2(e1) = 0, R2(e2) = a2e2 + a3e3, R2(e3) = − aa23 e2 − a2e3, a3 = 0. R3(e1) = 0, R3(e2) = 0, R3(e3) = 0. The pre-Lie superalgebra (C4,6, ◦): (associative) e1 ◦ e3 = e2, e3 ◦ e1 = e2. Rota–Baxter operators RB(C4,6) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3. R2(e1) = a3e1, R2(e2) = a4e2, R2(e3) = a1e2 + aa33−aa44 e3, a3 = a4. 4.1.5 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C5 The pre-Lie superalgebra (C5,1)k , ◦ : ((C5,1)0 is associative). e1 ◦ e2 = ke3, e2 ◦ e1 = ke3, e3 ◦ e3 = e1, k = 0 or 1. Rota–Baxter operators RB((C5,1)k ) are: Case 1: If k = 0, we have R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3. R2(e1) = 0, R2(e2) = a1e3, R2(e3) = a3e2. R3(e1) = 0, R3(e2) = a4e2, R3(e3) = a3e2. R4(e1) = 0, R4(e2) = 0, R4(e3) = a2e3. R5(e1) = a5e1, R5(e2) = a4e2, R5(e3) = aa55−aa44 e3, a4 = a5. Case 2: If k = 1, we have R6(e1) = 0, R6(e2) = a1e3, R6(e3) = a2e3. R7(e1) = 0, R7(e2) = 0, R7(e3) = a2e3. R8(e1) = 0, R8(e2) = a1e3, R8(e3) = e3. R9(e1) = a5e1, R9(e2) = a5e2, R9(e3) = a25 e3, a5 = 0. R10(e1) = a5e1, R10(e2) = a5a−51 e2, R10(e3) = e3, a5 = 1. The pre-Lie superalgebra (C5,2, ◦): e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e2 ◦ e3 = e1. Rota–Baxter operators RB(C5,2) are: R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3. R3(e1) = 0, R3(e2) = 0, R3(e3) = a3e2. The pre-Lie superalgebra (C5,3, ◦): e1 ◦ e1 = e1, e1 ◦ e2 = e2 + e3, e2 ◦ e1 = e2 + e3, e2 ◦ e3 = e1. Rota–Baxter operators RB(C5,3) are: R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3. R3(e1) = 0, R3(e2) = a3e2, R3(e3) = −a3e2. The pre-Lie superalgebra (C5,4)k , ◦ : (associative) e2 ◦ e3 = 1 2 + k e1, e3 ◦ e2 = 1 1 2 − k e1, k ≥ 0, k = 2 . Rota–Baxter operators RB((C5,4)k) are: 2 R8(e1) = a5e1, R8(e2) = a5e2 + a1e3, R8(e3) = aa51 e2 + a5e3, a1 = 0, a5 = 0, k = 0. R9(e1) = a5e1, R9(e2) = a3e2, R9(e3) = − aa55−aa33 e3, a3 = a5. 4.1.6 Rota–Baxter Operators on Pre-Lie Superalgebras of TypeC6 The pre-Lie superalgebra (C6,1, ◦) (associative) 1 e2 ◦ e2 = 2 e1, e2 ◦ e3 = −e1, e3 ◦ e2 = e1. Rota–Baxter operators RB(C6,1) are: R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3, a1 = 0. R2(e1) = a3e1, R2(e2) = a1e3, R2(e3) = 0. The pre-Lie superalgebra (C6,2, ◦): (associative) R1(e1) = a1e1, R1(e2) = a2e3, R1(e3) = a3e3. R2(e1) = a1e1, R2(e2) = 2a1e2 + a2e3, R2(e3) = a3e3. The pre-Lie superalgebra (C6,3, ◦) (associative) 1 e1 ◦ e2 = e3, e2 ◦ e1 = e3, e2 ◦ e2 = 2 e1. Rota–Baxter operators RB(C6,3) are: R1(e1) = 0, R1(e2) = a1e2, R1(e3) = a2e3. RR23((ee11)) == aa33ee11,, RR32((ee22)) == 2aa1e33e,2 +R2a(1ee33),=R03,(e3a)3== 203a.3 e3. The pre-Lie superalgebra (C6,4, ◦): (associative) 1 e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e2 ◦ e2 = 2 e1. Rota–Baxter operators RB(C6,4) are: R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3. 4.2 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 [ 15 ] by Zhang and Bai. In the following, let {e1, e2, e3} be a homogeneous basis of a pre-Lie superalgebra (A, ◦), where {e1, e2} is a basis of A0 and {e3} is a basis of A1. The computation is obtained using computer algebra system, and the operators are described with respect to the basis. 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, A2, A3, A4, A5, A6, A7h , A8, A9, A10h and A11 are given as follows: 4.2.1 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A1 The pre-Lie superalgebra ( A1,1, ◦) e1 ◦ e1 = e1, e2 ◦ e2 = e2. Rota–Baxter operators RB( A1,1) are: R1(e1) = 0, R1(e2) = 01, R1(e3) = a1e3. R2(e1) = 0, R2(e2) = 2 e2, R2(e3) = a1e3. 1 R3(e1) = 0, R3(e2) = e1 + 2 e2, R3(e3) = a1e3. The pre-Lie superalgebra ( A1,2)k , ◦ D1, ( A1,2)1 associative. e1 ◦ e1 = e1, e2 ◦ e2 = e2, e3 ◦ e1 = ke3, k ∈ C∗. Rota–Baxter operators RB(( A1,2)k ) are: R1(e1) = 0, R1(e2) = 01, R1(e3) = 0. R2(e1) = 0, R2(e2) = 2 e2, R2(e3) = 0. 1 R3(e1) = 0, R3(e2) = e1 + 2 e2, R3(e3) = 0. The pre-Lie superalgebra ( A1,3)k1,k2 , ◦ D1. e1 ◦ e1 = e1, e2 ◦ e2 = e2, e3 ◦ e1 = k1e3, e3 ◦ e2 = k2e3, k1, k2 = 0, k1 ≤ k2, k1 = −k2. Rota–Baxter operators RB(( A1,3)k1,k2 ) are: If k2 < 0, k1 ≤ k2 or (k2 > 0, k1 < −k2) or (−k2 < k1 < 0) or (0 < R1(e1) = 0, R1(e2) = 0, R1(e3) = 0. The pre-Lie superalgebra ( A1,4, ◦) e1 ◦ e1 = e1, e2 ◦ e2 = e2, e2 ◦ e3 = e3, e3 ◦ e2 = e3. Rota–Baxter operators RB( A1,4) are: D1, k1 = k2 = 0 or k1 = 1, k2 = e1 ◦ e1 = e1, e2 ◦ e2 = e2, e2 ◦ e3 = e3, e3 ◦ e1 = k1e3, e3 ◦ e2 = k2e3, k1 = 0 or k2 = 1. Rota–Baxter operator RB(( A1,5)k1,k2 ) is: R1(e1) = 0, R1(e2) = 0, R1(e3) = 0. The pre-Lie superalgebra ( A1,6, ◦) e1 ◦ e1 = e1, e2 ◦ e2 = e2, , e2 ◦ e3 = e3, e3 ◦ e2 = e3, e3 ◦ e3 = e2. Rota–Baxter operators RB( A1,6) are: R1(e1) = 0, R1(e2) = 01, R1(e3) = 0. R2(e1) = 0, R2(e2) = 2 e2, R2(e3) = 0. 1 R3(e1) = 0, R3(e2) = e1 + 2 e2, R3(e3) = 0. 4.2.2 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A2 The pre-Lie superalgebra ( A2,1, ◦) Rota–Baxter operators RB( A2,1) are: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = a2e3. 1 R2(e1) = 0, R2(e2) = 2 e1, R2(e3) = a2e3. The pre-Lie superalgebra ( A2,2)k , ◦ D1, ( A2,2)1 is associative. e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e1, e3 ◦ e1 = ke3, k = 0. Rota–Baxter operators RB(( A2,2)k ) are: e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e3 ◦ e1 = ke3, e3 ◦ e2 = e3. Rota–Baxter operators RB(( A2,3)k ) are: The pre-Lie superalgebra ( A2,4, ◦) e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e3. Rota–Baxter operator RB( A2,4) is: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0. The pre-Lie superalgebra ( A2,5)k , ◦ D1, ( A2,5)0 is associative. e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = ke3, k = 1. Rota–Baxter operators RB(( A2,5)k ) are: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = a2e3, k = 0. R2(e1) = a1e2, R2(e2) = 0, R2(e3) = 0, k = 0. The pre-Lie superalgebra ( A2,6)k , ◦ D1: e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = ke3, e3 ◦ e2 = e3. Rota–Baxter operators RB(( A2,6)k ) are: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0. The pre-Lie superalgebra ( A2,7, ◦) e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e3, e3 ◦ e3 = e2. Rota–Baxter operators RB( A2,7) are: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0. 4.2.3 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A3 The pre-Lie superalgebra ( A3,1, ◦) Rota–Baxter operators RB( A3,1) are: R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = a3e3. The pre-Lie superalgebra ( A3,4, ◦) D2: (associative). e1 ◦ e1 = e1, e3 ◦ e3 = e2. Rota–Baxter operators RB( A3,4) are R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = 0. R2(e1) = a1e2, R2(e2) = a2e2, R2(e3) = 2a2e3. The pre-Lie superalgebras of type ( A3,2)k , ◦ D1 : e1◦e1 = e1, e3◦e1 = ke3, k = 0, (( A3,2)1 is associative). D3 D1 : : D1 e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = ke3, k = 1, D1 : : e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = ke3, e3 ◦ e2 = e3. e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = e3, e3 ◦ e3 = e1, They have the same Rota–Baxter operators: R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = 0. 4.2.4 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A4 The pre-Lie superalgebra ( A4,1, ◦) ei ◦ e j = 0, ∀ i, j = 1, 2, 3. Rota–Baxter operators RB( A4,1) are: R1(e1) = a1e1 + a2e2, R1(e2) = a3e1 + a4e2, R1(e3) = a5e3. The pre-Lie superalgebra ( A4,2, ◦) D1: e3 ◦ e1 = e3. Rota–Baxter operators RB( A4,2) are: R1(e1) = a1e1 + a2e2, R1(e2) = a3e1 + a4e2, R1(e3) = 0. The pre-Lie superalgebra ( A4,3, ◦) D2: (associative) e3 ◦ e3 = e2. Rota–Baxter operators RB( A4,3) are: R1(e1) = a1e1 + a2e2, R1(e2) = a3e2, R1(e3) = 2a3e3. R2(e1) = a1e1 + a2e2, R2(e2) = a4e1 + a3e2, R2(e3) = 0. 4.2.5 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A5 The pre-Lie superalgebra ( A5,1, ◦) D2 (associative) e1 ◦ e1 = e2, e3 ◦ e3 = e2. Rota–Baxter operators RB( A5,1) are: R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = 0. R2(e1) = a1e2, R2(e2) = a2e2, R2(e3) = 2a2e3. a3 e2, R3(e3) = 0. RR34((ee11)) == aa33ee11 ++ aa11ee22,, RR34((ee22)) == a223 e2, R4(e3) = a3e3. The pre-Lie superalgebra ( A5,2, ◦) Rota–Baxter operators RB( A5,2) are: a1 e2, R1(e3) = a3e3. R1(e1) = a1e1 + a2e2, R1(e2) = 2 R2(e1) = a2e2, R2(e2) = a4e2, R2(e3) = 0. The pre-Lie superalgebra ( A5,3, ◦) D1: e1 ◦ e1 = e2, e3 ◦ e1 = e3. Rota–Baxter operators RB( A5,3) are: a1 e2, R1(e3) = 0. R1(e1) = a1e1 + a2e2, R1(e2) = 2 R2(e1) = a2e2, R2(e2) = a3e2, R2(e3) = 0. The pre-Lie superalgebra ( A5,4, ◦) D1: Rota–Baxter operators RB( A5,4) are: a1 e2, R1(e3) = 0. R1(e1) = a1e1 + a2e2, R1(e2) = 2 R2(e1) = a2e2, R2(e2) = a3e2, R2(e3) = 0. 4.2.6 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A6 The pre-Lie superalgebra ( A6,1)k , ◦ other cases non-associative (D4)μ : k = 0 or − 1 associative e1 ◦ e2 = −e1, e2 ◦ e2 = −e2, e3 ◦ e2 = ke3. Rota–Baxter operators RB(( A6,1)k ) are: Case 1: If k = 0, we have R1(e1) = a1e2, R1(e2) = 0, R1(e3) = a2e3. R2(e1) = 0, R2(e2) = a3e1, R2(e3) = a2e3. Case 2: If k ∈ C∗, we have R4(e1) = 0, R4(e2) = a3e1, R4(e3) = 0. R5(e1) = a1e2, R5(e2) = 0, R5(e3) = 0. R5(e1) = a1e2, R5(e2) = 0, R5(e3) = 0, a1 = 0. The pre-Lie superalgebra ( A6,2, ◦) D5: 1 e1 ◦ e2 = −e1, e2 ◦ e2 = −e2, e3 ◦ e2 = − 2 e3, e3 ◦ e3 = e1. Rota–Baxter operators RB( A6,2) are: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0. R2(e1) = 0, R2(e2) = a2e1, R2(e3) = 0. The pre-Lie superalgebra ( A6,3)k , ◦ other cases are non-associative e1 ◦ e2 = −e1, e2 ◦ e2 = −e2, e2 ◦ e3 = −e3, e3 ◦ e2 = ke3. Rota–Baxter operators RB(( A6,3)k ) are: R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0, a1 = 0. R2(e1) = 0, R2(e2) = a2e1, R2(e3) = 0. 4.2.7 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A7h The pre-Lie superalgebra ( A7h,1, ◦) D5: 1 e1 ◦ e2 = −e1, e2 ◦ e2 = he2, e3 ◦ e2 = − 2 e3, e3 ◦ e3 = e1, h = −1. Rota–Baxter operators RB( A7h,1) are: R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0. R2(e1) = 0, R2(e2) = a2e2, R2(e3) = 0, h = 0. (D4)μ : k = 0 or − 1 associative R3(e1) = 0, R3(e2) = 0, R3(e3) = 0, h = 0. The pre-Lie superalgebra ( A7h,2)k, ◦ R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0. R2(e1) = 0, R2(e2) = 0, R2(e3) = 0. The pre-Lie superalgebra ( A7h,3)k, ◦ e1 ◦ e2 = −e1, e2 ◦ e2 = he2, e3 ◦ e2 = ke3, h = −1. Rota–Baxter operators RB(( A7h,3)k) are: 4.2.8 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A8 The pre-Lie superalgebra ( A8,1)k, ◦ (D4)μ: e1 ◦ e1 = 2e1, e2 ◦ e1 = e2, e2 ◦ e2 = e1, e3 ◦ e1 = ke3. Rota–Baxter operators RB(( A8,1)k) are: R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e3, k = 0. R2(e1) = 0, R2(e2) = 0, R2(e3) = 0. 4.2.9 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A9 The pre-Lie superalgebra ( A9,1)k , ◦ others cases are non-associative. (D4)μ: k = 0 or 1 associative e2 ◦ e1 = e1, e2 ◦ e2 = e2, e3 ◦ e2 = ke3. Rota–Baxter operators RB(( A9,1)k) are: Case 1: If k = 0, we have 2 R1(e1) = a1e1 + a2e2, R1(e2) = − aa12 e1 − a1e2, R1(e3) = a3e3, a2 = 0. R2(e1) = 0, R2(e2) = a4e1, R2(e3) = a3e3. R3(e1) = 0, R3(e2) = 0, R3(e3) = a3e3. R4(e1) = 0, R4(e2) = a4e1, R4(e3) =2 0. a R5(e1) = a1e1 + a2e2, R5(e2) = − a12 e1 − a1e2, R5(e3) = 0, a2 = 0. R6(e1) = 0, R6(e2) = 0, R6(e3) = 0. The pre-Lie superalgebras ( A9,2)k, ◦ (D4)μ and ( A9,3, ◦) D5: ( A9,2)k , ◦ ⎧ e2 ◦ e1 = e1 ⎪⎪⎨ e2 ◦ e2 = e2 e2 ◦ e3 = e3 ⎪⎪⎩ e3 ◦ e1 = ke3 . ⎧ e2 ◦ e2 = e1 ⎪⎪⎪⎪ e2 ◦ e2 = e2 ( A9,3, ◦) ⎨ e2 ◦ e3 = e3 1 ⎪⎪ e3 ◦ e3 = 2 e1 ⎪⎪⎩ e3 ◦ e3 = e1 . 4.2.10 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A10h The pre-Lie superalgebra ( A10h,1)k, ◦ (D4)μ: e1 ◦ e2 = (h − 1)e1, e2 ◦ e1 = he1, e2 ◦ e2 = e1 + he2, e3 ◦ e2 = ke3, h = 0. Rota–Baxter operators RB(( A10h,1)k) are: Case 1: If k = 0, we have R1(e1) = 0, R1(e2) = a1e1, R1(e3) = a2e3. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3. Case 2: If k ∈ C∗, we have R3(e1) = 0, R3(e2) = a1e1, R3(e3) = 0, a1 = 0. R4(e1) = 0, R4(e2) = 0, R4(e3) = 0. The pre-Lie superalgebras ( A10h,2)k, ◦ (D4)μ and ( A10h,3, ◦) D5, where ⎧ e1 ◦ e2 = (h − 1)e1 (A10h,2)k, ◦ ⎪⎪⎪⎨⎪⎧ eee221 ◦◦◦ eee221 === (ehh1e1+− 1h)ee21, h = 0 (A10h,3, ◦) ⎨⎪⎪⎪⎪⎪⎪⎪ eee222 ◦◦◦ eee231 === ehh1ee31+ he2, h = 0 ⎪⎪ e2 ◦ e3 = he3 ⎪⎪⎩ e3 ◦ e2 = ke3 ⎪⎪⎪⎪⎪ e3 ◦ e2 = h − 21 e3 ⎪⎩⎪ e3 ◦ e3 = e1 They have the same Rota–Baxter operators, that is, R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0, a1 = 0. R2(e1) = 0, R2(e2) = 0, R2(e3) = 0. 4.2.11 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A11 The pre-Lie superalgebra ( A11,1)k , ◦ ( D4)μ: e1 ◦ e2 = −e1, e2 ◦ e2 = e1 − e2, e3 ◦ e2 = ke3. Rota–Baxter operators RB(( A11,1)k ) are: Case 1: If k = 0, we have R1(e1) = 0, R1(e2) = a1e1, R1(e3) = a2e3, a1 = 0. R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3. Case 2: If k ∈ C∗, we have R3(e1) = 0, R3(e2) = a1e1, R3(e3) = 0, a1 = 0. R4(e1) = 0, R4(e2) = 0, R4(e3) = 0. The pre-Lie superalgebras ( A11,2, ◦) D5 and ( A11,3)k , ◦ ( D4)μ, where ( A11,2, ◦) : ( A11,3)k , ◦ : 1 e1 ◦ e2 = −e1, e2 ◦ e2 = e1 − e2, e3 ◦ e2 = − 2 e3, e3 ◦ e3 = e1. e1 ◦ e2 = −e1, e2 ◦ e2 = e1 − e2, e2 ◦ e3 = −e3, e3 ◦ e2 = ke3. They have the same Rota–Baxter operators, that is, R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0, a1 = 0. R2(e1) = 0, R2(e2) = 0, R2(e3) = 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. 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. 1. Aguiar, M.: Infinitesimal bialgebras, pre-Lie algebras and dendriform algebras. In: Hopf Algebras. Lecture Notes in Pure and Applied Mathematics, vol. 237, pp. 1–33 (2004) 2. Aguiar, M.: Pre-Poisson algebras. Lett. Math. Phys. 54, 263–277 (2000) 3. Aguiar, M., Loday, J.L.: Quadri-algebras. J. Pure Appl. Algebra 191, 221–251 (2004) 37. Guo, L.: An introduction to Rota–Baxter Algebra, Surveys of Modern Mathematics, vol. 4. International Press, Higher Education Press, Somerville, Beijing (2012) 38. Kong, X., Chen, H., Bai, C.: Classification of graded left-symmetric algebra structures on Witt and Virasoro algebras. Int. J. Math. 22(2), 201–202 (2011) 39. Kong, X.L., Bai, C.M.: Left-symmetric superalgebra structures on the super-Virasoro algebras. Pac. J. Math. 235(1), 43–55 (2008) 40. Koszul, J.-L.: Domaines bornés homogènes et orbites de groupes de transformations affines. Bull. Soc. Math. Fr. 89, 515–533 (1961) 41. Kupershmidt, B.A.: What a classical r-matrix really is. J. Nonlinear Math. Phys. (6), 448–488 (1999) 42. Kupershmidt, B.A.: Non-abelian phase spaces. J. Phys. A Math. Gen. 27, 2801–2809 (1994) 43. Li, X., Hou, D., Bai, C.: Rota–Baxter operators on pre-Lie algebras. J. Nonlinear Math. Phys. 14(2), 269–289 (2007) 44. Lichnerowicz, A., Medina, A.: On Lie group with left-invariant symplectic or Kählerian. Lett. Math. Phys. 16(3), 225–235 (1988) 45. Loday, J.-L.: Dialgebras. Dialgebras and Related Operads. Lecture Notes in Mathematics, vol. 1763, pp. 7–66. Springer, New York (2001) 46. Makhlouf, A., Yau, D.: Rota–Baxter Hom-Lie-admissible algebras. Commun. Algebra 42(37), 1231– 1257 (2013) 47. Miller, J.B.: Baxter operators and endomorphisms on Banach algebras. J. Math. Anal. Appl. 25, 503– 520 (1969) 48. Ni, J., Wang, Y., Hou, D.: Super O-operators of Jordan Superalgebras and Super Jordan Yang–Baxter Equations. Frontiers Mathematics in China. Higher Education Press, Springer, Berlin Heidelberg (2014). https://doi.org/10.1007/s11464-014-0339-9 49. Pei, J., Bai, C., Guo, L.: Rota–Baxter on sl(2, C) and solution of the classical Yang–Baxter equation. J. Math. Phys. 55, 021701 (2014). https://doi.org/10.1063/1.4863898 50. Rota, G.-C.: Baxter operators. In: Kung, J.P.S. (ed.) Gian-Carlo Rota on Combinatorics, Introductory Paper and commentaries. Birkhauser, Boston (1995) 51. Rota, G.-C.: Ten mathematics problems I will never solve. Mitt. Dtsch.-Ver 2, 45–52 (1998) 52. Vasilieva, E.A., Mikhalev, A.A.: Free left-symmetric superalgebras. Fund. Appl. Math. 2, 611–613 (1996) 53. Vinberg, E.B.: The theory of homogeneous cones. Trudy Moskov. Mat. Obsc. 12, 303–358 (1963) 54. Wang, Y., Hou, D., Bai, C.: Operator forms of the classical Yang–Baxter equation in Lie superalgebras. Int. J. Geom. Methods Mod. Phys. 7(4), 583–597 (2010) 55. Wang, Z.G.: The classification of low-dimensional Lie superalgebras. East China Normal University, Dissertation (2006). (in Chinese) 4. Ammar , F. , Makhlouf , A. : Hom-Lie superalgebras and Hom-Lie admissible superalgebras . J. Algebra 324 , 1513 - 1528 ( 2010 ) 5. Andrada , A. , Salamon , S. : Complex product structure on Lie algebras . Forum Math . 17 , 261 - 295 ( 2005 ) 6. Atkinson , F.V.: Some aspects of Baxter's functional equation . J. Math. Anal. Appl . 7 , 1 - 30 ( 1967 ) 7. Baxter , G. : An analytic problem whose solution follows from a simple algebraic identity . Pac. J. Math. 10 , 731 - 742 ( 1960 ) 8. Bai , C.M.: A further study on non-abelian phase spaces: left-symmetric algebraic approach and related geometry . Rev. Math. Phys. 18 , 545 - 564 ( 2006 ) 9. Bai , C.M.: O-operators of Loday algebras and analogues of the classical Yang-Baxter equation . Commun. Algebra 38 , 4277 - 4321 ( 2010 ) 10. Bai , C.M.: A unified algebraic approach to classical Yang-Baxter equation . J. Phys. A Math. Theor . 40 , 11073 - 11082 ( 2007 ) 11. Bai , C.M.: Bijective 1-cocycles and classification of 3-dimensional left-symmetric algebras . Commun. Algebra 37 , 1016 - 1057 ( 2009 ) 12. Bai , C.M. , Guo , L. , Ni , X.: O-operators on associative algebras and associative Yang-Baxter equations . Pac. J. Math. 256 , 257 - 289 ( 2012 ) 13. Bai , C.M. , Guo , L. , Ni , X. : Generalizations of the classical Yang-Baxter equation and O-operators . J. Math. Phys. 52 , 063515 ( 2011 ) 14. Bai , C.M. , Liu , L.G. , Ni , X.: Some results on L-dendriform algebras . J. Geom. Phys . 60 , 940 - 950 ( 2010 ) 15. Bai , C.M. , Zhang , R.: On some left-symmetric superalgebras . J. Algebra Appl . 11 ( 5 ), 1250097 ( 2012 ) 16. Bordemann , M. : Generalized Lax pairs, the modified classical Yang-Baxter equations, and affine geometry of Lie groups . Commun. Math. Phys . 135 ( 1 ), 201 - 216 ( 1990 ) 17. Burde , D. : Left-symmetric algebras, or pre-Lie algebras in geometry and physics . Cent. Eur. J. Math. 4 ( 3 ), 323 - 357 ( 2006 ) 18. Cartier , P. : On the structure of free Baxter algebras . Adv. Math. 9 , 253 - 265 ( 1972 ) 19. Chapoton , F. , Livernet , M. : Pre-Lie algebras and the rooted trees operad . Int. Math. Res. Notices 8 , 395 - 408 ( 2001 ) 20. Cayley , A. : On the theory of analytic forms called trees . In: Cayley, A . (ed.) Collected Mathematical Papers of Arthur Cayley. Notices , vol. 3 , pp. 242 - 246 . Cambridge University Press, Cambridge ( 1890 ) 21. Chen , H. , Li , J. : Left-symmetric algebra structures on the W -algebra W (2, 2 ). Linear Algebra Appl . 437 , 1821 - 1834 ( 2012 ) 22. Chu , B.Y. : Symplectic homogeneous spaces . Trans. Am. Math. Soc . 197 , 145 - 159 ( 1974 ) 23. Connes , A. , Kreimer , D. : Hopf algebras, renormalization and noncommutative geometry . Commun. Math. Phys. 199 , 203 - 242 ( 1998 ) 24. Dardié , J.M. , Médina , A. : Algèbres de Lie Kahlériennes et double extension. J. Algebra 185 , 744 - 795 ( 1996 ) 25. Dardié , J.M. , Médina , A. : Double extension symplectique d'un groupe de Lie symplectique . Adv. Math . 117 , 208 - 227 ( 1996 ) 26. Diatta , A. , Medina , A. : Classical Yang-Baxter equation and left-invariant affine geometry on Lie groups . Manuscipta Math . 114 , 477 - 486 ( 2004 ) 27. Ebrahimi-Fard , K. : Loday-type algebras and the Rota-Baxter relation . Lett. Math. Phys. 61 , 139 - 147 ( 2002 ) 28. Ebrahimi-Fard , K. : On the associative Nijenhuis relation . Elect. J. Comb . 11 ( 1 ), 38 ( 2004 ) 29. Ebrahimi-Fard , K. , Guo , L. : Rota-Baxter algebras and dendriform algebras . J. Pure Appl. Algebra 212 , 320 - 339 ( 2008 ) 30. Ebrahimi-Fard , K. , Manchon , D. , Patras , F. : New identities in dendriform algebras . J. Algebra 320 , 708 - 727 ( 2008 ) 31. Ebrahimi-Fard , K. , Manchon , D. : Dendriform equations . J. Algebra 322 , 4053 - 4079 ( 2009 ) 32. Ebrahimi-Fard , K. , Manchon , D. : Twisted dendriform algebras and the preLie Magnus expansions , ( 2009 ). arXiv: 0910 . 2166 33. Ebrahimi-Fard , K. , Gracia-Bondia , J.M. , Patras , F. : Rota-Baxter algebras and new combinatorial identities . Lett. Math. Phys. 81 , 61 - 75 ( 2007 ) 34. Gerstenhaber , M.: The cohomology structure of associative ring . Ann. Math. 78 , 267 - 288 ( 1963 ) 35. Goze , M. , Remm , E.: Lie-admissible algebras and operads . J. Algebra 273 , 129 - 152 ( 2004 ) 36. Guo , L. , Keigher , W. : Baxter algebras and shuffle products . Adv. Math . 150 ( 1 ), 117 - 149 ( 2000 )


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El-Kadri Abdaoui, Sami Mabrouk, Abdenacer Makhlouf. Rota–Baxter Operators on Pre-Lie Superalgebras, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-40, DOI: 10.1007/s40840-017-0565-x