Rota–Baxter Operators on Pre-Lie Superalgebras

In this paper, we study Rota–Baxter operators and super O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}$$\end{document}-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 [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 Z 2 -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 R(x)R(y) = R R(x)y + x R(y) + λx y . (0.1) If λ = 0 (resp. λ = −1), the product 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][13][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][32][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 → 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.

Rota-Baxter Associative Superalgebras, Pre-Lie Superalgebras and Lie Superalgebras
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.
We have similar definitions on Lie superalgebras.
consisting of a Z 2 -graded vector space A, and an even bilinear map In particular, a super O-operator of weight λ ∈ K associated with the bimodule The triple (A, [ , ], R) refers to a Rota-Baxter Lie superalgebra, see [54].
We introduce the notion of super O-operators of pre-Lie superalgebras and study some properties over Lie superalgebras and pre-Lie superalgebras.
(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, (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: In particular, a super O-operator of weight λ ∈ K associated with the A-bimodule for all x, y, z in H(A).
(1) The commutator ) 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, 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
•, 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 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.
The constants a i are parameters. Now, we define Rota-Baxter operators on an A-module, where A is a Rota-Baxter Lie superalgebra.

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
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 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 [35]). The graded case was studied by Ammar and Makhlouf in 2010 (see [4] for more details).
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 (1.14) 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
Define the even binary operation "•" on any homogeneous element x, y ∈ A by Proof For all homogeneous elements x, y, z in A, we have and Then, we obtain The above sum vanishes by associativity and the Rota-Baxter identity (1.13) with λ = −1.

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.

Definition and Some Basic Properties
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, The associated bracket to a L-dendriform superalgebra is defined as 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 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
Proof For any homogeneous elements u, v and w in V , we have Hence, and similarly, we have 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 defines a L-dendriform superalgebra structure on A.

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
Equipped with these actions, V becomes an A-bimodule over the associated Ldendriform superalgebra. (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.  Conversely, we can construct L-dendriform superalgebras from O-operators of pre-Lie superalgebras. (V, l, r ), then there exists a L-dendriform superalgebra structure on V defined by

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
Therefore, there is a pre-Lie superalgebra structure on V defined by as the associated vertical pre-Lie superalgebra of (V, , ) and T is a homomorphism of pre-Lie superalgebra.
and there is a L-dendriform superalgebra structure on T (V ) given by 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 Hence, 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
defines a L-dendriform superalgebra structure on A. Proof Straightforward.
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
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 Therefore, Similarly, we have 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 have only the trivial Rota-Baxter operator, that is,

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 {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 .

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 C 1 , C 2 h , C 3 , C 4 , C 5 and C 6 are given as follows:

Case 2:
If h ∈ C * , we have

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 {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.

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
, A 10 h and A 11 are given as follows:
Rota-Baxter operators RB( A 3,4 ) are The pre-Lie superalgebras of type

Case 2:
If k ∈ C * , we have