Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, we construct solutions of the classical Leibniz Yang-Baxter equation using relative Rota-Baxter operators and Leibniz-dendriform algebras.

1.1. Leibniz algebras and Leibniz bialgebras. The notion of a Leibniz algebra was introduced by Loday [27,28] with the motivation in the study of the periodicity in algebraic K-theory. Recently Leibniz algebras were studied from different aspects due to applications in both mathematics and physics. In particular, integration of Leibniz algebras were studied in [10,13] and deformation quantization of Leibniz algebras was studied in [14]. As the underlying structure of embedding tensor, Leibniz algebras also have application in higher gauge theories, see [24,30] for more details.
For a given algebraic structure, a bialgebra structure on this algebra is obtained by a comultiplication together with some compatibility conditions between the multiplication and the comultiplication.A good compatibility condition is prescribed by a rich structure theory and effective constructions. The most famous examples of bialgebras are associative bialgebras and Lie bialgebras, which have important applications in both mathematics and mathematical physics, e.g. a Lie bialgebra is the algebraic structure corresponding to a Poisson-Lie group and the classical structure of a quantized universal enveloping algebra [12,15].
The purpose of this paper is to study the bialgebra theory for Leibniz algebras with the motivation from the great importance of Lie bialgebras. It is well known that a Lie bialgebra is equivalent to a Manin tripe of Lie algebras. In the definition of a Manin triple, one needs to use a quadratic Lie algebra, which is a Lie algebra equipped with a symmetric nondegenerate invariant bilinear form. However, to define a quadratic Leibniz algebra, we need to use a skew-symmetric bilinear form. This is supported by the fact that the operad of Lie algebras is a cyclic operad, but the operad of Leibniz algebras is an anticyclic operad. Actually, it is observed by Chapoton in [11] using the operad theory that one should use the aforementioned skew-symmetric invariant bilinear form on a Leibniz algebra. As soon as we have the correct notion of a quadratic Leibniz algebra, we can define Manin triples and dual representations of Leibniz algebras. We introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Even though we obtain some nice results totally parallel to the context of Lie bialgebras, we need to emphasize that our bialgebra theory are not generalization of Lie bialgebras, namely the restriction of our theory on Lie algebras is independent of Lie bialgebras.
1.2. Triangular Leibniz bialgebras: relative Rota-Baxter operator approach. Due to the importance of the classical Yang-Baxter equation and triangular Lie bialgebras, it is natural to define the Leibniz analogue of the classical Yang-Baxter equation and triangular Leibniz bialgebras. This is a very hard problem due to that the representation theory of Leibniz algebras is not good, e.g. there is no tensor product in the module category of Leibniz algebras. We solve this problem using relative Rota-Baxter operators and the twisting theory of twilled Leibniz algebras.
A Rota-Baxter operator on a Lie algebra was introduced in the 1980s as the operator form of the classical Yang-Baxter equation. To better understand the classical Yang-Baxter equation and the related integrable systems, the more general notion of an O-operator on a Lie algebra was introduced by Kupershmidt [25], which can be traced back to Bordemann [9]. An O-operator gives rise to a skew-symmetric r-matrix in a larger Lie algebra [4]. In the context of associative algebras, O-operators give rise to dendriform algebras [29], play important role in the bialgebra theory [5] and lead to the splitting of operads [6]. See the book [20] for more details.
The twisting theory was introduced by Drinfeld in [16] motivated by the study of quasi-Lie bialgebras and quasi-Hopf algebras. As a useful tool in the study of bialgebras, the twisting theory was further applied to associative algebras and Poisson geometry, see [22,23,31,34] for more details.
A representation of a Leibniz algebra (g, [·, ·] g ) is a triple (V; ρ L , ρ R ), where V is a vector space, ρ L , ρ R : g → gl(V) are linear maps such that the following equalities hold for all x, y ∈ g, Here [·, ·] : ∧ 2 gl(V) → gl(V) is the commutator Lie bracket on gl(V), the vector space of linear transformations on V.
Define the left multiplication L : g −→ gl(g) and the right multiplication R : g −→ gl(g) by L x y = [x, y] g and R x y = [y, x] g respectively for all x, y ∈ g. Then (g; L, R) is a representation of (g, [·, ·] g ), which is called the regular representation. Define two linear maps L * , R * : g −→ gl(g * ) with x −→ L * x and x −→ R * x respectively by If there is a Leibniz algebra structure on the dual space g * , we denote the left multiplication and the right multiplication by L and R respectively.
2.1. Quadratic Leibniz algebras and the Leibniz analogue of the string Lie 2-algebra. It is observed by Chapoton in [11] using the operad theory that one need to use skew-symmetric bilinear forms instead of symmetric bilinear forms on a Leibniz algebra. This is the key ingredient in our study of Leibniz bialgebras. Definition 2.2. ( [11]) A quadratic Leibniz algebra is a Leibniz algebra (g, [·, ·] g ) equipped with a nondegenerate skew-symmetric bilinear form ω ∈ ∧ 2 g * such that the following invariant condition holds: Remark 2.3. In the original definition of a nondegenerate skew-symmetric invariant bilinear form on a Leibniz algebra (g, [·, ·] g ) given in [11], there is a superfluous condition In fact, by (5), we have Remark 2.4. Note that we use skew-symmetric bilinear forms instead of symmetric bilinear forms and use the invariant condition (5) instead of the invariant condition B([x, y] g , z) = B(x, [y, z] g ) and this is the main ingredient in our study of Leibniz bialgebras. In [8], the author use symmetric bilinear form and invariant condition B([x, y] g , z) = B(x, [y, z] g ) to study Leibniz bialgebras so that one has to add some strong conditions. As we will see, everything in the following study is natural in the sense that we do not need to add any extra conditions on the Leibniz algebra.
Recall that a quadratic Lie algebra is a Lie algebra (k, [·, ·] k ) equipped with a nondegenerate symmetric bilinear form B ∈ Sym 2 (k * ), which is invariant in the sense that Associated to a quadratic Lie algebra (k, [·, ·] k , B), we have a closed 3-formΘ ∈ ∧ 3 k * given bȳ which is known as the Cartan 3-form.
Let (g, [·, ·] g , ω) be a quadratic Leibniz algebra. Define Θ ∈ ⊗ 3 g * by This 3-tensor can be viewed as the Leibniz analogue of the Cartan 3-form on a quadratic Lie algebra as the following lemma shows. Proof. For all x, y, z, w ∈ g, by the fact that [g, g] g is a left center, we have +ω(x, [[y, z] g + [z, y] g , w] g ) = 0, which finishes the proof.
Proof. It follows from Lemma 2.5 and we omit details.
Remark 2.7. A semisimple Lie algebra with the Killing form is naturally a quadratic Lie algebra. How to construct a skew-symmetric bilinear form associated to a Leibniz algebra such that it is invariant in the sense of (5) is not known yet.

2.2.
Matched pairs, Manin triples of Leibniz algebras and Leibniz bialgebras. In this subsection, first we recall the notion of a matched pair of Leibniz algebras. Then we introduce the notions of a Manin triple of Leibniz algebras and a Leibniz bialgebra. Finally, we prove the equivalence between matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras.
hold for all x, y ∈ g 1 and u, v ∈ g 2 , then we call (g 1 , ) a matched pair of Leibniz algebras.
) be a matched pair of Leibniz algebras. Then there is a Leibniz algebra structure on g 1 ⊕ g 2 defined by In the Lie algebra context, to relate matched pairs of Lie algebras to Lie bialgebras and Manin triples for Lie algebras, we need the notions of the coadjoint representation, which is the dual representation of the adjoint representation. Now we investigate the dual representation in the Leibniz algebra context. Lemma 2.10. Let (V; ρ L , ρ R ) be a representation of a Leibniz algebra (g, [·, ·] g ). Then is a representation of (g, [·, ·] g ), which is called the dual representation of (V; ρ L , ρ R ).
Example 2.12. Let (g, [·, ·] g ) be a Leibniz algebra. Then (g ⋉ L * ,−L * −R * g * , g, g * ) is a Manin triple of Leibniz algebras, where the natural nondegenerate skew-symmetric bilinear form ω on g ⊕ g * is given by: For a Leibniz algebra (g * , [·, ·] g * ), let △ : g −→ ⊗ 2 g be the dual map of [·, ·] g * : Definition 2.13. Let (g, [·, ·] g ) and (g * , [·, ·] g * ) be Leibniz algebras. Then (g, g * ) is called a Leibniz bialgebra if the following conditions hold: (a) For all x, y ∈ g, we have Until now, we have recalled the notion of a matched pair, introduced the notions of a Manin triple of Leibniz algebras and a Leibniz bialgebra. Similar to the case of Lie algebras, these objects are equivalent when we consider the dual representation of the regular representation in a matched pair of Leibniz algebras. The following theorem is the main result in this section.
(iii) (g ⊕ g * , g, g * ) is a Manin triple of Leibniz algebras, where the invariant skew-symmetric bilinear form on g ⊕ g * is given by (15).

(Relative) Rota-Baxter operators and twisting theory
In this section, first we recall the graded Lie algebra whose Maurer-Cartan elements are Leibniz algebra structures, and define the bidegree of a multilinear map which is the technical tool in our later study. Then we introduce the notion of a relative Rota-Baxter operator on a Leibniz algebra, and construct the graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter operators. Finally, we give the twisting theory of twilled Leibniz algebras. These structures and theories are the main ingredient in our later study of Leibniz bialgebras.
Let g be a vector space. We consider the graded vector space C * (g, g) = ⊕ n≥1 C n (g, g) = ⊕ n≥1 Hom(⊗ n g, g). The Balavoine bracket on the graded vector space C * (g, g) is given by: where P•Q ∈ C p+q+1 (g, g) is defined by and • k is defined by It is well known that Let g 1 and g 2 be vector spaces and elements in g 1 will be denoted by x, y, z, x i and elements in g 2 will be denoted by u, v, v i . Let c : g ⊗n 2 → g 1 be a linear map. We can construct a linear map c ∈ C n (g 1 ⊕ g 2 , g 1 ⊕ g 2 ) bŷ In general, for a given linear map f : We call the linear mapf a horizontal lift of f , or simply a lift. Let H : g 2 → g 1 be a linear map. Its lift is given byĤ(x, v) = (H(v), 0). Obviously we haveĤ •Ĥ = 0.
We denote by g l,k the direct sum of all (l + k)-tensor powers of g 1 and g 2 , where l (resp. k) is the number of g 1 (resp. g 2 ). By the properties of the Hom-functor, we have where the isomorphism is the horizontal lift.
It is obvious that we have the following lemmas: Proof. Assume that || f || = −1|l and ||g|| = −1|k. Then f and g are both horizontal lift of linear maps in C * (g 2 , g 1 ). By the definition of the lift, we have f and g ∈ C m (g 1 ⊕ g 2 , g 1 ⊕ g 2 ) be homogeneous linear maps with bidegrees l f |k f and l g |k g respectively. Then the composition f is a homogeneous linear map of the bidegree l f + l g |k f + k g .
Lemma 3.6. If || f || = l f |k f and ||g|| = l g |k g , then [ f, g] B has the bidegree l f + l g |k f + k g .

3.2.
(Relative) Rota-Baxter operators. First we introduce the notion of a (relative) Rota-Baxter operator and give some examples.
Remark 3.8. When (g, [·, ·] g ) is a Lie algebra and ρ R = −ρ L , we obtain the notion of a relative Rota-Baxter operator (an O-operator) on a Lie algebra with respect to a representation.  with respect to the representation (g * ; L * , −L * − R * ) 1 if and only if It is straightforward to deduce that L e 1 (e 1 , e 2 ) = (e 1 , e 2 ) 0 0 0 0 , L e 2 (e 1 , e 2 ) = (e 1 , e 2 ) 1 1 0 0 , = a 21 (a 11 + a 21 )e 1 , and In the sequel, we construct the graded Lie algebra that characterize relative Rota-Baxter operators as Maurer-Cartan elements.
Let (V; ρ L , ρ R ) be a representation of a Leibniz algebra (g, [·, ·] g ). Then there is a Leibniz algebra structure on g ⊕ V given by This Leibniz algebra is called the semidirect product of g and (V; ρ L , ρ R ), and denoted by g⋉ ρ L ,ρ R V. We denote the above semidirect product Leibniz multiplication byμ 1 .
Consider the graded vector space More precisely, we have Moreover, its Maurer-Cartan elements are relative Rota-Baxter operators on the Leibniz algebra (g, [·, ·] g ) with respect to the representation (V; ρ L , ρ R ).
Proof. The graded Lie algebra (C * (V, g), {·, ·}) is obtained via the derived bracket [21,35]. In fact, the Balavoine bracket [·, ·] B associated to the direct sum vector space g ⊕ V gives rise to a graded Lie algebra (C * (g ⊕ V, g ⊕ V), [·, ·] B ). Sinceμ 1 is the semidirect product Leibniz algebra structure on the vector space g ⊕ V. By Theorem 3.1, we deduce that ( is a differential graded Lie algebra. Obviously C * (V, g) is an abelian subalgebra. Further, we define the derived bracket on the graded vector space C * (V, g) by By Lemma 3.6, the derived bracket {·, ·} is closed on C * (V, g), which implies that (C * (V, g), {·, ·}) is a graded Lie algebra. Moreover, it is straightforward to obtain the above concrete graded Lie bracket {·, ·} on C * (V, g) = ⊕ +∞ k=1 C k (V, g).
Thus, Maurer-Cartan elements are precisely relative Rota-Baxter operators on g with respect to the representation (V; ρ L , ρ R ). The proof is finished.
This is the main ingredient in our later study of the classical Leibniz Yang-Baxter equation and the classical Leibniz r-matrix.
where P g 1 and P g 2 are the natural projections from G to g 1 and g 2 respectively. The multiplication [(x, u), (y, v)] G of G is uniquely decomposed by the canonical projections P g 1 and P g 2 into eight multiplications: Write Ω =φ 1 +μ 1 +μ 2 +φ 2 as in Lemma 3.11. Then we obtain [μ 1 ,μ 2 ] B = 0, Proof. By Lemma 3.6 and Lemma 3.3, the proof is straightforward.
Proposition 3.17. The twisting Ω H is a Leibniz algebra structure on G.

Proof.
By which implies that Ω H is a Leibniz algebra structure on G by Theorem 3.1. Obviously, Ω H is also decomposed into the unique four substructures. The twisting operations are completely determined by the following result.
Proof. By (33)  In the sequel, we consider a special case of the above twisting theory. Let (V; ρ L , ρ R ) be a representation of a Leibniz algebra (g, [·, ·] g ). Consider the twilled Leibniz algebra (g⋉ ρ L ,ρ R V, g, V). Denote the Leibniz bracket [·, ·] ⋉ by Ω. Write Ω =μ 1 +μ 2 . Thenμ 2 = 0.  V, Ω H ), g, V) is a twilled Leibniz algebra if and only if H is a relative Rota-Baxter operator on the Leibniz algebra (g, [·, ·] g ) with respect to the representation (V; ρ L , ρ R ). Moreover, the Leibniz algebra structure on V is given by [u, v] Proof. By Proposition 3.19, the twisting have the form: Thus, the twisting ((g ⊕ V, Ω H ), g, V) is a twilled Leibniz algebra if and only ifφ H 2 = 0, which implies that H is a relative Rota-Baxter operator by Theorem 3.10.
By Lemma 3.13, we deduce thatμ H 2 is a Leibniz algebra multiplication on V. It is straightforward to deduce that the multiplication on V is given by (38).

The classical Leibniz Yang-Baxter equation and triangular Leibniz bialgebras
In this section, first we construct a Leibniz bialgebra using a symmetric relative Rota-Baxter operator. Then we define the classical Leibniz Yang-Baxter equation using the graded Lie algebra obtained in Theorem 3.10. Its solutions are called classical Leibniz r-matrices. Using the twisting theory given in Section 3, we define a triangular Leibniz bialgebra successfully. Finally, we generalize a Semonov-Tian-Shansky's result in [32] about the relation between the operator form and the tensor form of a classical r-matrix to the context of Leibniz algebras.
First by Theorem 3.20, we have Corollary 4.1. Let K : g * → g be a relative Rota-Baxter operator on g with respect to the representation (g * ; L * , −L * − R * ). Then g * Let K : g * → g be a relative Rota-Baxter operator on a Leibniz algebra (g, [·, ·] g ) with respect to the representation (g * ; L * , −L * − R * ). Then eK preserves the bilinear form ω given by (15) if and only if K * = K. Here K * is the dual map of K, i.e. Kξ, η = ξ, K * η , for all ξ, η ∈ g * .
Proof. Since eK is a Leibniz algebra isomorphism and preserves the bilinear form ω, for all X, Y, Z ∈ g ⊕ g * , we have which implies that (g ⊕ g * , Ω K ) is a quadratic Leibniz algebra. It is obvious that eK is an isomorphism from the quadratic Leibniz algebra (g ⊕ g * , Ω K ) to (g ⊕ g * , Ω).
In the sequel, to define the classical Leibniz Yang-Baxter equation, we transfer the above graded Lie algebra structure to the tensor space.
The general formula of [[P, Q]] is very sophisticated. But for P = x ⊗ y and Q = z ⊗ w, there is an explicit expression, which is enough for our application.
Moreover, we can obtain the tensor form of a relative Rota-Baxter operator on g with respect to the representation (g * ; L * , −L * − R * ).   Therefor, for all r = ae 1 ⊗ e 1 + be 1 ⊗ e 2 + be 2 ⊗ e 1 + ce 2 ⊗ e 2 ∈ Sym 2 (g), we have   The above classical Leibniz r-matrices actually correspond to symmetric relative Rota-Baxter operators given in Example 3.9.
Proof. By r ∈ Sym 2 (g) and [[r, r]] = 0, we deduce that r ♯ : g * → g is a relative Rota-Baxter operator on g with respect to the representation (g * ; L * , −L * − R * ) and (r ♯ ) * = r ♯ . By Theorem 4.4, we obtain that (g, g * r ♯ ) is a Leibniz bialgebra. Definition 4.13. Let (g, [·, ·] g ) be a Leibniz algebra and r ∈ Sym 2 (g) a solution of the classical Leibniz Yang-Baxter equation in g. We call the Leibniz bialgebra (g, g * r ♯ ) the triangular Leibniz bialgebra associated to the classical Leibniz r-matrix r.
Remark 4.14. In Section 2, we define a Leibniz bialgebra, which is equivalent to a Manin triple of Leibniz algebras. Note that there is no cohomology theory can be used in the theory of Leibniz bialgebras. Thus, there is not an obvious way to define a "coboundary Leibniz bialgebra". Nevertheless, using the twisting method in the theory of twilled Leibniz algebras, we define triangular Leibniz bialgebras successfully.
In the Lie algebra context, we know that the dual description of a classical r-matrix is a symplectic structure on a Lie algebra. Now we investigate the dual description of a classical Leibniz r-matrix. A symmetric 2-form B ∈ Sym 2 (g * ) on a Leibniz algebra (g, [·, ·] g ) induces a linear map B ♮ : g → g * by B ♮ (x), y := B(x, y), ∀x, y ∈ g.
B is said to be nondegenerate if B ♮ : g → g * is an isomorphism. Similarly, r ∈ Sym 2 (g) is said to be nondegenerate if r ♯ : g * → g is an isomorphism.
satisfies the following "closed" condition Proof. Let r ∈ Sym 2 (g) be nondegenerate. It is obvious that B is symmetric and nondegenerate.
Proof. For all x, y, z ∈ g, by (6) we have •ω ♮ . Therefore, ω ♮ is an isomorphism between representations. The proof is finished.
Proof. For all x, y ∈ g, by Lemma 4.16, we have , which implies the conclusion.
Corollary 4.18. Let (g, [·, ·] g , ω) be a quadratic Leibniz algebra. Then r ∈ Sym 2 (g) is a solution of the classical Leibniz Yang-Baxter equation in g if and only if r ♯ • ω ♮ is a Rota-Baxter operator on (g, [·, ·] g ), that is, Remark 4.19. In [18], the authors defined R ± -matrix for Leibniz algebras as a direct generalization of Semonov-Tian-Shansky's approach in [32], without any bialgebra theory for Leibniz algebras. It is straightforward to see that their R + -matrices in a Leibniz algebra are simply Rota-Baxter operator on the Leibniz algebra. By the above corollary, if r is a classical Leibniz r-matrix in a quadratic Leibniz algebra (g, [·, ·] g , ω), then r ♯ • ω ♮ is an R + -matrix introduced in [18].
Our bialgebra theory for Leibniz algebras enjoys many good properties parallelling to that for Lie algebras. This justifies its correctness.

Solutions of the classical Leibniz Yang-Baxter equations
In this section, first we show that a relative Rota-Baxter operator on a Leibniz (g, [·, ·] g ) with respect to a general representation (V; ρ L , ρ R ) gives rise to a solution of the classical Leibniz Yang-Baxter equation in a larger Leibniz algebra. Then we introduce the notion of a Leibnizdendriform algebra, which is the underlying algebraic structure of a relative Rota-Baxter operator on a Leibniz algebra. This type of algebras play important role in our study of the classical Leibniz Yang-Baxter equation. There is a natural solution of the classical Leibniz Yang-Baxter equation in the semidirect product Leibniz algebra A ⋉ L * ⊳ ,−L * ⊳ −R * ⊲ A * associated to a Leibniz-dendriform algebra (A, ⊲, ⊳).
In the sequel, we introduce the notion of a Leibniz-dendriform algebra as the underlying algebraic structure of a relative Rota-Baxter operator on a Leibniz algebra. Definition 5.3. A Leibniz-dendriform algebra is a vector space A equipped with two binary operations ⊲ and ⊳ : A ⊗ A → A such that for all x, y, z ∈ A, we have Proposition 5.4. Let (A, ⊲, ⊳) be a Leibniz-dendriform algebra. Then the binary operation [·, ·] ⊲,⊳ : A ⊗ A → A given by defines a Leibniz algebra, which is called the sub-adjacent Leibniz algebra of (A, ⊲, ⊳) and (A, ⊲, ⊳) is called a compatible Leibniz-dendriform algebra structure on (A, [·, ·] ⊲,⊳ ).
Thus, we obtain that Id : A → A is a relative Rota-Baxter operator on the Leibniz algebra (A, [·, ·] ⊲,⊳ ) with respect to the representation (A; L ⊳ , R ⊲ ). The proof is finished.
Proposition 5.7. Let K : V → g be a relative Rota-Baxter operator on a Leibniz algebra (g, [·, ·] g ) with respect to a representation (V; ρ L , ρ R ). Then there is a Leibniz-dendriform algebra structure on V given by Proof. By (1) and (24), we have which implies that (49) in Definition 5.3 holds.
We give a sufficient and necessary condition for the existing of a compatible Leibniz-dendriform algebra structure on a Leibniz algebra.
Proposition 5.8. Let (g, [·, ·] g ) be a Leibniz algebra. Then there is a compatible Leibniz-dendriform algebra on g if and only if there exists an invertible relative Rota-Baxter operator K : V → g on g with respect to a representation (V; ρ L , ρ R ). Furthermore, the compatible Leibniz-dendriform algebra structure on g is given by x ⊲ y := K(ρ R (y)K −1 x), x ⊳ y := K(ρ L (x)K −1 y), ∀x, y ∈ g. (55) Proof. Let K : V → g be an invertible relative Rota-Baxter operator on g with respect to a representation (V; ρ L , ρ R ). By Proposition 5.7, there is a Leibniz-dendriform algebra on V given by u ⊲ v := ρ R (Kv)u, u ⊳ v := ρ L (Ku)v, ∀u, v ∈ V.
Proof. Since (A, ⊲, ⊳) is a Leibniz-dendriform algebra, the identity map Id : A → A is a relative Rota-Baxter operator on the sub-adjacent Leibniz algebra (A, [·, ·] ⊲,⊳ ) with respect to the representation (A; L ⊳ , R ⊲ ). By Theorem 5.2, r = n i=1 (e * i ⊗ e i + e i ⊗ e * i ) is a symmetric solution of the classical Leibniz Yang-Baxter equation in A ⋉ L * ⊳ ,−L * ⊳ −R * ⊲ A * . It is obvious that the corresponding bilinear form B ∈ Sym 2 (A ⊕ A * ) is given by (57). The proof is finished.
The above results can be viewed as the Leibniz analogue of the results given in [4].