The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras

Abstract

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator.

1. Introduction

Hom-algebras were first introduced in the Lie algebra setting [1] with motivation from physics, though their origin can be traced back to earlier literature, such as [2]. In [3], Makhlouf and Silvestrov introduced the definition of Hom-algebras, where the associativity of a Hom-algebra is twisted by an endomorphism. Later, Makhlouf and Silvestrov [4,5] extended the notions of bialgebras and Hopf algebras to Hom-bialgebras and Hom–Hopf algebras. In [6], Yau extended the notion of Lie bialgebras to Hom–Lie bialgebras and showed solutions for the classical Hom–Yang–Baxter equation using the twisted map. In [7], Sheng and Bai defined a new kind of Hom-Lie bialgebra, which was equivalent to Manin triples of Hom-Lie algebras, and constructed solutions of the classical Hom–Yang–Baxter equation in terms of $\mathcal{O}$-operators.

Leibniz algebras were first introduced by Loday [8,9] with the motivation from the study of periodicity in algebraic K-theory, which played an important role in mathematics and physics. It is well known that both Lie algebras and commutative, associative algebras have a theory of bialgebras applied to many fields. For Leibniz algebras, Barreiro and Benayadi [10] introduced the notion of Leibniz bialgebras via the double construction of Leibniz algebras and studied the classical Yang–Baxter equation, which is different to Barreiro and Benayadi’s Leibniz bialgebras. Tang and Sheng [11] defined a new kind of Leibniz bialgebras and proved they were equivalent with matched pairs of Leibniz algebras and Manin triples of Leibniz algebras. Now it is natural to introduce the notion of Hom–Leibniz bialgebras and explore their equivalent characterizations. This was our first motivation for writing the present paper.

The classical Yang–Baxter equation was investigated by Sklyanin [12] in the context of the quantum inverse scattering method, which has a close connection with many branches of mathematical physics and pure mathematics, including quantum groups, quantum integrable systems, Hamiltonian structures, braided categories and invariants of knots and links. The generalization of the classical Yang–Baxter equation in the Hom-type case, namely, the classical Hom–Yang–Baxter equation in Hom–Lie algebras, has been studied widely in [6,13,14,15,16]. In [11], Tang and Sheng generalized the classical Yang–Baxter equation to the classical Leibniz Yang–Baxter equation and characterized relative Rota–Baxter operators on Leibniz algebras as Maurer–Cartan elements. Motivated by the recent work on the classical Hom–Yang–Baxter equation and the relative Rota–Baxter operator on Leibniz algebras, in this paper, we introduce the notions of the classical Hom–Leibniz Yang–Baxter equation and the relative Rota–Baxter operator on Hom–Leibniz algebras. This was another motivation for writing the present paper.

The paper is organized as follows. In Section 3, we present the notion of Hom–Leibniz bialgebras and show that matched pairs of Hom–Leibniz algebras, Manin triples of Hom–Leibniz algebras, and Hom–Leibniz bialgebras are equivalent. In Section 4, we introduce the notion of a relative Rota–Baxter operator on a Hom–Leibniz algebra and show that a Hom–Leibniz algebra together with a relative Rota–Baxter operator yields a Hom–pre-Leibniz algebra. In Section 5, we recall the notion of the classical Hom–Leibniz Yang–Baxter equation and obtain its connection with the relative Rota–Baxter operator on Hom–Leibniz algebras.

2. Preliminaries

Throughout this paper, we work over the complex field $\mathbb{C}$, and all the vector spaces are finite-dimensional. We now recall some useful definitions in [17].

Definition 1.

A Hom–Leibniz algebra is a triple $(g,{[·,·]}_g,{\varphi }_g)$ consisting of a linear space g, a bilinear operation ${[·,·]}_g:g\otimes g\to g$ and an algebra homomorphism ${\varphi }_g:g\to g$ satisfying

$$\begin{array}{c}\hfill {[{\varphi }_g\left(x\right),{[y,z]}_g]}_g={[{[x,y]}_g,{\varphi }_g\left(z\right)]}_g+{[{\varphi }_g\left(y\right),{[x,z]}_g]}_g,\forall x,y,z\in g.\end{array}$$

A Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ is said to be regular (involutive), if ${\varphi }_g$ is nondegenerate (satisfies ${\varphi }_g^2=Id$). Define two linear maps $L^{*},R^{*}:g\to gl\left(g^{*}\right)$ with $x\mapsto L_x^{*}$ and $x\mapsto R_x^{*}$, respectively, by

$$\begin{array}{c}\hfill \langle L_x^{*}\xi ,y\rangle =-\langle \xi ,{[x,y]}_g\rangle ,\langle R_x^{*}\xi ,y\rangle =-\langle \xi ,{[y,x]}_g\rangle ,\forall x,y\in g,\xi \in g^{*}.\end{array}$$

If there is a Hom–Leibniz algebra structure on the dual space $g^{*}$, we denote the left multiplication and the right multiplication by $\mathcal{L}$ and $\mathcal{R}$, respectively.

Definition 2.

A representation of a Hom–Leibniz algebra $(g,{[·,·]}_g,,{\varphi }_g)$ is a triple $(V,{\varphi }_V,$

${\rho }^L,{\rho }^R)$, where V is a vector space and $gl\left(V\right)$ denotes the linear endomorphisms of V; ${\varphi }_V\in gl\left(V\right)$, ${\rho }^L,{\rho }^R:g\to gl\left(V\right)$ are three linear maps such that the following equalities hold for all $x,y\in g$:

(1) 

${\rho }^L\left({\varphi }_g\left(x\right)\right)\circ {\varphi }_V={\varphi }_V\circ {\rho }^L\left(x\right),{\rho }^R\left({\varphi }_g\left(x\right)\right)\circ {\varphi }_V={\varphi }_V\circ {\rho }^R\left(x\right);$

(2) 

${\rho }^L\left({[x,y]}_g\right)\circ {\varphi }_V={\rho }^L\left({\varphi }_g\left(x\right)\right)\circ {\rho }^L\left(y\right)-{\rho }^L\left({\varphi }_g\left(y\right)\right)\circ {\rho }^L\left(x\right);$

(3) 

${\rho }^R\left({[x,y]}_g\right)\circ {\varphi }_V={\rho }^L\left({\varphi }_g\left(x\right)\right)\circ {\rho }^R\left(y\right)-{\rho }^R\left({\varphi }_g\left(y\right)\right)\circ {\rho }^L\left(x\right);$

(4) 

${\rho }^R\left({\varphi }_V\left(y\right)\right)\circ {\rho }^L\left(x\right)=-{\rho }^R({\varphi }_V\left(y\right)\circ {\rho }^R\left(x\right).$

Define the left multiplication $L:g\to gl\left(g\right)$ and the right multiplication $R:g\to gl\left(g\right)$ by $L_{x}y={[x,y]}_g$ and $R_{x}y={[y,x]}_g$, respectively, for all $x,y\in g$. Then $(g,{\varphi }_g,L,R)$ is a representation of $(g,{[·,·]}_g,{\varphi }_g)$, which is called a regular representation.

3. Hom–Leibniz Bialgebras

In this section, we will introduce the notion of a Hom–Leibniz bialgebra and prove that matched pairs of Hom–Leibniz algebras, Manin triples of Hom–Leibniz algebras and Hom–Leibniz bialgebras are equivalent.

Definition 3.

Let $(g_1,{[·,·]}_{g_1},{\varphi }_{g_1})$ and $(g_2,{[·,·]}_{g_2},{\varphi }_{g_2})$ be two Hom–Leibniz algebras. If there exists a representation $({\rho }_1^L,{\rho }_1^R)$ of $g_1$ on $g_2$ and a representation $({\rho }_2^L,{\rho }_2^R)$ of $g_2$ on $g_1$ such that the following identities hold:

$$\begin{array}{ccc}& & {\rho }_1^R\left({\varphi }_{g_1}\left(x\right)\right){[u,v]}_{g_2}-{[{\varphi }_{g_2}\left(u\right),{\rho }_1^R\left(x\right)v]}_{g_2}+{[{\varphi }_{g_2}\left(v\right),{\rho }_1^R\left(x\right)u]}_{g_2}\hfill \\ & & -{\rho }_1^R\left({\rho }_2^L\left(v\right)x\right){\varphi }_{g_2}\left(u\right)+{\rho }_1^R\left({\rho }_2^L\left(u\right)x\right){\varphi }_{g_2}\left(v\right)=0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & {\rho }_1^L\left({\varphi }_{g_1}\left(x\right)\right){[u,v]}_{g_2}-{[{\rho }_1^L\left(x\right)u,{\varphi }_{g_2}\left(v\right)]}_{g_2}-{[{\varphi }_{g_2}\left(u\right),{\rho }_1^L\left(x\right)v]}_{g_2}\hfill \\ & & -{\rho }_1^L\left({\rho }_2^R\left(u\right)x\right){\varphi }_{g_2}\left(v\right)-{\rho }_1^R\left({\rho }_2^R\left(v\right)x\right){\varphi }_{g_2}\left(u\right)=0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & {[{\rho }_1^L\left(x\right)u,{\varphi }_{g_2}\left(v\right)]}_{g_2}+{\rho }_1^L\left({\rho }_2^R\left(u\right)x\right){\varphi }_{g_2}\left(v\right)+{[{\rho }_1^R\left(x\right)u,{\varphi }_{g_2}\left(v\right)]}_{g_2}+{\rho }_1^L\left({\rho }_2^L\left(u\right)x\right){\varphi }_{g_2}\left(v\right)=0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & {\rho }_2^R\left({\varphi }_{g_2}\left(u\right)\right){[x,y]}_{g_1}-{[{\varphi }_{g_1}\left(x\right),{\rho }_2^R\left(u\right)y]}_{g_1}+{[{\varphi }_{g_1}\left(y\right),{\rho }_2^R\left(u\right)x]}_{g_1}\hfill \\ & & -{\rho }_2^R\left({\rho }_1^L\left(y\right)u\right){\varphi }_{g_1}\left(x\right)+{\rho }_2^R\left({\rho }_1^L\left(x\right)u\right){\varphi }_{g_1}\left(y\right)=0,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & {\rho }_2^L\left({\varphi }_{g_2}\left(u\right)\right){[x,y]}_{g_1}-{[{\rho }_2^L\left(u\right)x,{\varphi }_{g_1}\left(y\right)]}_{g_1}-{[{\varphi }_{g_1}\left(x\right),{\rho }_2^L\left(u\right)y]}_{g_1}\hfill \\ & & -{\rho }_2^L\left({\rho }_1^R\left(x\right)u\right){\varphi }_{g_1}\left(y\right)-{\rho }_2^R\left({\rho }_1^R\left(y\right)u\right){\varphi }_{g_1}\left(x\right)=0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & {[{\rho }_2^L\left(u\right)x,{\varphi }_{g_1}\left(y\right)]}_{g_1}+{\rho }_2^L\left({\rho }_1^R\left(x\right)u\right){\varphi }_{g_1}\left(y\right)+{[{\rho }_2^R\left(u\right)x,{\varphi }_{g_1}\left(y\right)]}_{g_1}+{\rho }_2^L\left({\rho }_1^L\left(x\right)u\right){\varphi }_{g_1}\left(y\right)=0,\hfill \end{array}$$

(6)

for all $x,y\in g_1$ and $u,vs.\in g_2$, then we call $(g_1,{\varphi }_{g_1},g_2,{\varphi }_{g_2},({\rho }_1^L,{\rho }_1^R),({\rho }_2^L,{\rho }_2^R))$ a matched pair of Hom–Leibniz algebras.

Proposition 1.

Let $(g_1,{\varphi }_{g_1},g_2,{\varphi }_{g_2},({\rho }_1^L,{\rho }_1^R),({\rho }_2^L,{\rho }_2^R))$ be a matched pair of Hom–Leibniz algebras. Then there is a Hom–Leibniz algebra structure on $g_1\oplus g_2$ defined by

$$\begin{array}{ccc}& & {\varphi }_{\bowtie }(x+u)={\varphi }_{g_1}\left(x\right)+{\varphi }_{g_2}\left(u\right),\hfill \\ & & {[x+u,y+v]}_{\bowtie }={[x,y]}_{g_1}+{\rho }_2^R\left(v\right)x+{\rho }_2^L\left(u\right)y+{[u,v]}_{g_2}+{\rho }_1^L\left(x\right)v+{\rho }_1^R\left(y\right)u,\hfill \end{array}$$

for all $x,y\in g_1$ and $u,vs.\in g_2$.

Proof. 

For any $x,y,z\in g_1$ and $u,v,w\in g_2$, we have

$$\begin{array}{ccc}& & {[{[x+u,y+v]}_{\bowtie },{\varphi }_{\bowtie }(z+w)]}_{\bowtie }+{[{\varphi }_{\bowtie }(y+v),{[x+u,z+w]}_{\bowtie }]}_{\bowtie }\hfill \\ & =& {[{[x,y]}_{g_1}+{\rho }_2^R\left(v\right)x+{\rho }_2^L\left(u\right)y+{[u,v]}_{g_2}+{\rho }_1^L\left(x\right)v+{\rho }_1^R\left(y\right)u,{\varphi }_{g_1}\left(z\right)+{\varphi }_{g_2}\left(w\right)]}_{\bowtie }\hfill \\ & & +{[{\varphi }_{g_1}\left(y\right)+{\varphi }_{g_2}\left(v\right),{[x,z]}_{g_1}+{\rho }_2^R\left(w\right)x+{\rho }_2^L\left(u\right)z+{[u,w]}_{g_2}+{\rho }_1^L\left(x\right)w+{\rho }_1^R\left(z\right)u]}_{\bowtie }\hfill \\ & =& {[{[x,y]}_{g_1},{\varphi }_{g_1}\left(z\right)]}_{g_1}+{[{\rho }_2^R\left(v\right)x,{\varphi }_{g_1}\left(z\right)]}_{g_1}+{[{\rho }_2^R\left(u\right)y,{\varphi }_{g_1}\left(z\right)]}_{g_1}+{\rho }_2^R\left({\varphi }_{g_2}\left(w\right)\right){[x,y]}_{g_1}\hfill \\ & & +{\rho }_2^R\left({\varphi }_{g_2}\left(w\right)\right){\rho }_2^R\left(v\right)x+{\rho }_2^R\left({\varphi }_{g_2}\left(w\right)\right){\rho }_2^L\left(u\right)y+{\rho }_2^L\left({[u,v]}_{g_2}\right){\varphi }_{g_1}\left(z\right)+{\rho }_2^L\left({\rho }_1^L\left(x\right)v\right){\varphi }_{g_1}\left(z\right)\hfill \\ & & +{\rho }_2^L\left({\rho }_1^R\left(y\right)u\right){\varphi }_{g_1}\left(z\right)+{[{[u,v]}_{g_2},{\varphi }_{g_2}\left(w\right)]}_{g_2}+{[{\rho }_1^L\left(x\right)v,{\varphi }_{g_2}\left(w\right)]}_{g_2}+{[{\rho }_1^R\left(y\right)u,{\varphi }_{g_2}\left(w\right)]}_{g_2}\hfill \\ & & +{\rho }_1^L\left({[x,y]}_{g_1}\right){\varphi }_{g_2}\left(w\right)+{\rho }_1^L\left({\rho }_2^R\left(v\right)x\right){\varphi }_{g_2}\left(w\right)+{\rho }_1^L\left({\rho }_2^L\left(u\right)y\right){\varphi }_{g_2}\left(w\right)+{\rho }_1^R\left({\varphi }_{g_1}\left(z\right)\right){[u,v]}_{g_2}\hfill \\ & & +{\rho }_1^R\left({\varphi }_{g_1}\left(z\right)\right){\rho }_1^L\left(x\right)v+{\rho }_1^R\left({\varphi }_{g_1}\left(z\right)\right){\rho }_1^R\left(y\right)u+{[{\varphi }_{g_1}\left(y\right),{[x,z]}_{g_1}]}_{g_1}+{[{\varphi }_{g_1}\left(y\right),{\rho }_2^R\left(w\right)x]}_{g_1}\hfill \\ & & +{[{\varphi }_{g_1}\left(y\right),{\rho }_2^L\left(u\right)z]}_{g_1}+{\rho }_2^R\left({[u,w]}_{g_2}\right){\varphi }_{g_1}\left(y\right)+{\rho }_2^R\left({\rho }_1^L\left(x\right)w\right){\varphi }_{g_1}\left(y\right)+{\rho }_2^R\left({\rho }_1^R\left(z\right)u\right){\varphi }_{g_1}\left(y\right)\hfill \\ & & +{\rho }_2^L\left({\varphi }_{g_2}\left(v\right)\right){[x,z]}_{g_1}+{\rho }_2^L\left({\varphi }_{g_2}\left(v\right)\right){\rho }_2^R\left(w\right)x+{\rho }_2^L\left({\varphi }_{g_2}\left(v\right)\right){\rho }_2^L\left(u\right)z+{[{\varphi }_{g_2}\left(v\right),{[u,w]}_{g_2}]}_{g_2}\hfill \\ & & +{[{\varphi }_{g_2}\left(v\right),{\rho }_1^L\left(x\right)w]}_{g_2}+{[{\varphi }_{g_2}\left(v\right),{\rho }_1^R\left(z\right)u]}_{g_2}+{\rho }_1^L\left({\varphi }_{g_1}\left(y\right)\right){[u,w]}_{g_2}+{\rho }_1^L\left({\varphi }_{g_1}\left(y\right)\right){\rho }_1^L\left(x\right)w\hfill \\ & & +{\rho }_1^L\left({\varphi }_{g_1}\left(y\right)\right){\rho }_1^R\left(z\right)u+{\rho }_1^R\left({[x,z]}_{g_1}\right){\varphi }_{g_2}\left(v\right)+{\rho }_1^R\left({\rho }_2^R\left(w\right)x\right){\varphi }_{g_2}\left(v\right)+{\rho }_1^R\left({\rho }_2^L\left(u\right)z\right){\varphi }_{g_2}\left(v\right).\hfill \end{array}$$

Further, we have

$$\begin{array}{ccc}& & {[{\varphi }_{{}_{\bowtie }}(x+u),{[y+v,z+w]}_{\bowtie }]}_{\bowtie }\hfill \\ & =& {[{\varphi }_{g_1}\left(x\right)+{\varphi }_{g_2}\left(u\right),{[y,z]}_{g_1}+{\rho }_2^R\left(w\right)y+{\rho }_2^L\left(v\right)z+{[v,w]}_{g_2}+{\rho }_1^L\left(y\right)w+{\rho }_1^R\left(z\right)v]}_{\bowtie }\hfill \\ & =& {[{\varphi }_{g_1}\left(x\right),{[y,z]}_{g_1}]}_{g_1}+{[{\varphi }_{g_1}\left(x\right),{\rho }_2^R\left(w\right)y]}_{g_1}+{[{\varphi }_{g_1}\left(x\right),{\rho }_2^L\left(v\right)z]}_{g_1}+{\rho }_2^R\left({[v,w]}_{g_2}\right){\varphi }_{g_1}\left(x\right)\hfill \\ & & +{\rho }_2^R\left({\rho }_1^L\left(y\right)w\right){\varphi }_{g_1}\left(x\right)+{\rho }_2^R\left({\rho }_1^R\left(z\right)v\right)){\varphi }_{g_1}\left(x\right)+{\rho }_2^L\left({\varphi }_{g_2}\left(u\right)\right){[y,z]}_{g_1}+{\rho }_2^L\left({\varphi }_{g_2}\left(u\right)\right){\rho }_2^R\left(w\right)y\hfill \\ & & +{\rho }_2^L\left({\varphi }_{g_2}\left(u\right)\right){\rho }_2^L\left(v\right)z+{[{\varphi }_{g_2}\left(u\right),{[v,w]}_{g_2}]}_{g_2}+{[{\varphi }_{g_2}\left(u\right),{\rho }_1^L\left(y\right)w]}_{g_2}+{[{\varphi }_{g_2}\left(u\right),{\rho }_1^R\left(z\right)v]}_{g_2}\hfill \\ & & +{\rho }_1^L\left({\varphi }_{g_1}\left(x\right)\right){[v,w]}_{g_2}+{\rho }_1^L\left({\varphi }_{g_1}\left(x\right)\right){\rho }_1^L\left(y\right)w+{\rho }_1^L\left({\varphi }_{g_1}\left(x\right)\right){\rho }_1^R\left(z\right)v+{\rho }_1^R\left({[y,z]}_{g_1}\right){\varphi }_{g_2}\left(u\right)\hfill \\ & & +{\rho }_1^R\left({\rho }_2^R\left(w\right)y\right){\varphi }_{g_2}\left(u\right)+{\rho }_1^R\left({\rho }_2^L\left(v\right)z\right){\varphi }_{g_2}\left(u\right).\hfill \end{array}$$

By Equation (1), we have

$$\begin{array}{ccc}& & {\rho }_1^R\left({\varphi }_{g_1}\left(z\right)\right){[u,v]}_{g_2}+{[{\varphi }_{g_2}\left(v\right),{\rho }_1^R\left(z\right)u]}_{g_2}+{\rho }_1^R\left({\rho }_2^L\left(u\right)z\right){\varphi }_{g_2}\left(v\right)\hfill \\ & =& {\rho }_1^R\left({\rho }_2^L\left(v\right)z\right){\varphi }_{g_2}\left(u\right)+{[{\varphi }_{g_2}\left(u\right),{\rho }_1^R\left(z\right)v]}_{g_2}.\hfill \end{array}$$

By Equation (2), we have

$$\begin{array}{ccc}& & {[{\rho }_1^L\left(x\right)v,{\varphi }_{g_2}\left(w\right)]}_{g_2}+{[{\varphi }_{g_2}\left(v\right),{\rho }_1^L\left(x\right)w]}_{g_2}+{\rho }_1^L\left({\rho }_2^R\left(v\right)x\right){\varphi }_{g_2}\left(w\right)+{\rho }_1^R\left({\rho }_2^R\left(w\right)x\right){\varphi }_{g_2}\left(v\right)\hfill \\ & =& {\rho }_1^L\left({\varphi }_{g_1}\left(x\right)\right){[v,w]}_{g_2}.\hfill \end{array}$$

By Equation (3), we have

$$\begin{array}{ccc}& & {[{\rho }_1^L\left(y\right)u,{\varphi }_{g_2}\left(w\right)]}_{g_2}+{\rho }_1^L\left({\rho }_2^R\left(u\right)y\right){\varphi }_{g_2}\left(w\right)+{[{\rho }_1^R\left(y\right)u,{\varphi }_{g_2}\left(w\right)]}_{g_2}+{\rho }_1^L\left({\rho }_2^L\left(u\right)y\right){\varphi }_{g_2}\left(w\right)=0.\hfill \end{array}$$

By Equation (4), we have

$$\begin{array}{ccc}& & {\rho }_2^R\left({\varphi }_{g_2}\left(w\right)\right){[x,y]}_{g_1}+{[{\varphi }_{g_1}\left(y\right),{\rho }_2^R\left(w\right)x]}_{g_1}+{\rho }_2^R\left({\rho }_1^L\left(x\right)w\right){\varphi }_{g_1}\left(y\right)\hfill \\ & =& {[{\varphi }_{g_1}\left(x\right),{\rho }_2^R\left(w\right)y]}_{g_1}+{\rho }_2^R\left({\rho }_1^L\left(y\right)w\right){\varphi }_{g_1}\left(x\right).\hfill \end{array}$$

By Equation (5), we have

$$\begin{array}{ccc}& & {\rho }_2^L\left({\varphi }_{g_2}\left(v\right)\right){[x,z]}_{g_1}\hfill \\ & & {[{\rho }_2^L\left(v\right)x,{\varphi }_{g_1}\left(z\right)]}_{g_1}+{[{\varphi }_{g_1}\left(x\right),{\rho }_2^L\left(v\right)z]}_{g_1}+{\rho }_2^L\left({\rho }_1^R\left(x\right)v\right){\varphi }_{g_1}\left(z\right)-{\rho }_2^R\left({\rho }_1^R\left(z\right)v\right){\varphi }_{g_1}\left(x\right).\hfill \end{array}$$

By Equation (6), we have

$$\begin{array}{ccc}& & {[{\rho }_2^L\left(v\right)x,{\varphi }_{g_1}\left(z\right)]}_{g_1}+{\rho }_2^L\left({\rho }_1^R\left(x\right)v\right){\varphi }_{g_1}\left(z\right)+{[{\rho }_2^R\left(v\right)x,{\varphi }_{g_1}\left(z\right)]}_{g_1}+{\rho }_2^L\left({\rho }_1^L\left(x\right)v\right){\varphi }_{g_1}\left(z\right)=0.\hfill \end{array}$$

As $({\rho }_1^L,{\rho }_1^R)$ is a representation of $g_1$ on $g_2$ and $({\rho }_2^L,{\rho }_2^R)$ is a representation of $g_2$ on $g_1$, $(g_1\oplus g_2,{[·,·]}_{\bowtie },{\varphi }_{g_1}+{\varphi }_{g_2})$ is a Hom–Leibniz algebra. □

According to [7], it is easy to obtain the following result by Definition 3.

Lemma 1.

Let $(g,{[·,·]}_g,{\varphi }_g)$ be a Hom–Leibniz algebra and $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$ be a representation. Then $(V^{*},{\varphi }_V^{*},{\left({\rho }^L\right)}^{*},-{\left({\rho }^L\right)}^{*}-{\left({\rho }^R\right)}^{*})$ is a representation of $(g,{[·,·]}_g,{\varphi }_g)$ if the following equalities hold for all $x,y\in g$:

(i) 

${\varphi }_V\circ {\rho }^L\left({\varphi }_g\left(x\right)\right)={\rho }^L\left(x\right)\circ {\varphi }_V,{\varphi }_V\circ {\rho }^R\left({\varphi }_g\left(x\right)\right)={\rho }^R\left(x\right)\circ {\varphi }_V,$

(ii) 

${\varphi }_V\circ {\rho }^L\left({[x,y]}_g\right)={\rho }^L\left(x\right)\circ {\rho }^L\left({\varphi }_g\left(y\right)\right)-{\rho }^L\left(y\right)\circ {\rho }^L\left({\varphi }_g\left(x\right)\right),$

(iii) 

${\varphi }_V\circ {\rho }^R\left({[x,y]}_g\right)={\rho }^L\left(x\right)\circ {\rho }^R\left({\varphi }_g\left(y\right)\right)-{\rho }^R\left(y\right)\circ {\rho }^L\left({\varphi }_g\left(x\right)\right),$

(iv) 

${\rho }^R\left({\varphi }_V\left(y\right)\right)\circ {\rho }^L\left(x\right)=-{\rho }^R\left({\varphi }_V\left(y\right)\right)\circ {\rho }^R\left(x\right).$

A representation $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$ is called admissible if $(V^{*},{\varphi }_V^{*},{\left({\rho }^L\right)}^{*},-{\left({\rho }^L\right)}^{*}-{\left({\rho }^R\right)}^{*})$ is also a representation—i.e., conditions (i)–(iv) in the above lemma are satisfied.

Definition 4.

A Manin triple of Hom–Leibniz algebras is a quintuple $(\mathcal{G},g_1,g_2,{\varphi }_{g_1},{\varphi }_{g_2})$, where

(1) 

$(\mathcal{G},{[·,·]}_{\mathcal{G}},{\varphi }_{\mathcal{G}},\omega )$ is a quadratic Hom–Leibniz algebra,

(2) 

Both $(g_1,{\varphi }_{g_1})$ and $(g_2,{\varphi }_{g_2})$ are isotropic subalgebras of $(\mathcal{G},{[·,·]}_{\mathcal{G}},{\varphi }_{\mathcal{G}})$,

(3) 

$\mathcal{G}=g_1\oplus g_2$ as vector spaces.

For a Hom–Leibniz algebra $(g^{*},{[·,·]}_{g^{*}},{\varphi }_{g^{*}})$, let $\Delta :g\to g\otimes g$ be the dual map of ${[·,·]}_{g^{*}}:g^{*}\otimes g^{*}\to g^{*}$; i.e.,

$$\begin{array}{c}\hfill \langle \Delta x,\xi \otimes \eta \rangle =\langle x,{[\xi ,\eta ]}_{g^{*}}\rangle .\end{array}$$

Definition 5.

Let $(g,{[·,·]}_g,{\varphi }_g)$ and $(g^{*},{[·,·]}_{g^{*}},{\varphi }_g^{*})$ be admissible Hom–Leibniz algebras. Then $(g,g^{*})$ is called a Hom–Leibniz bialgebra if the following conditions hold:

(1) 

For any $x,y\in g$,

$$\begin{array}{c}\hfill (R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)(\Delta y)={\tau }_{12}\left((R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)(\Delta x)\right),\end{array}$$

where ${\tau }_{12}:g\otimes g\to g\otimes g$ is the exchange operator defined by ${\tau }_{12}(x\otimes y)=y\otimes x$.

(2) 

For any $x,y\in g$,

$$\begin{array}{ccc}& & \Delta {[x,y]}_g\hfill \\ & =& ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)}-L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g-R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g-{\tau }_{12}\circ ({\varphi }_g\otimes L_{{\varphi }_g\left(y\right)})-{\tau }_{12}\circ ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)}))\Delta x\hfill \\ & & +({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)}+L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g+R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y.\hfill \end{array}$$

Theorem 1.

Let $(g,{[·,·]}_g,{\varphi }_g)$ and $(g^{*},{[·,·]}_{g^{*}},{\varphi }_g^{*})$ be two admissible Hom–Leibniz algebras. Then the following conditions are equivalent:

(i) 

$(g,g^{*})$ is a Hom–Leibniz bialgebra.

(ii) 

$(g,{\varphi }_g,g^{*},{\varphi }_g^{*},(L^{*},-L^{*}-R^{*}),({\mathcal{L}}^{*},-{\mathcal{L}}^{*}-{\mathcal{R}}^{*}))$ is a matched pair of Hom–Leibniz algebras.

(iii) 

$(g\oplus g^{*},g,g^{*},{\varphi }_g,{\varphi }_g^{*})$ is a Manin triple of Hom–Leibniz algebras, where the invariant skew-symmetric bilinear form ω on $g\oplus g^{*}$ is given by

$$\begin{array}{c}\hfill \omega (x+\xi ,y+\eta )=\langle \xi ,y\rangle -\langle \eta ,x\rangle ,\forall x,y\in g,\xi ,\eta \in g^{*}.\end{array}$$

Proof. 

We only prove that (i) is equivalent to (ii). Others are similar to [11]. For any $x,y\in g$ and $\xi ,\eta \in g^{*}$, consider the left hand side of Equation (6). We have

$$\begin{array}{ccc}& & {[{\mathcal{L}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g+{\mathcal{L}}_{(-L_x^{*}-R_x^{*})\xi }^{*}{\varphi }_g\left(y\right)+{[(-{\mathcal{L}}_{\xi }^{*}-{\mathcal{R}}_{\xi }^{*})x,{\varphi }_g\left(y\right)]}_g+{\mathcal{L}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right)\hfill \\ & =& -{\mathcal{L}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(y\right)-{[{\mathcal{R}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g.\hfill \end{array}$$

Furthermore, by straightforward computations, we have

$$\begin{array}{ccc}& & \langle -{\mathcal{L}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(y\right)-{[{\mathcal{R}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g,\eta \rangle \hfill \\ & =& \langle {\varphi }_g\left(y\right),{[R_x^{*}\xi ,\eta ]}_{g^{*}}\rangle -\langle R_{{\varphi }_g\left(y\right)}{\mathcal{R}}_{\xi }^{*}x,\eta \rangle \hfill \\ & =& \langle \Delta y,R_{{\varphi }_g\left(x\right)}^{*}{\varphi }_g^{*}\left(\xi \right)\otimes {\varphi }_g^{*}\left(\eta \right)\rangle -\langle \Delta x,R_{{\varphi }_g\left(y\right)}^{*}\eta \otimes {\varphi }_g^{*}\left(\xi \right)\rangle \hfill \\ & =& -\langle (R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)(\Delta y),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle (R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)(\Delta x),\eta \otimes {\varphi }_g^{*}\left(\xi \right)\rangle \hfill \\ & =& -\langle (R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)(\Delta y),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle {\tau }_{12}\left[(R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)(\Delta x)\right],{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle .\hfill \end{array}$$

Therefore, Equation (6) is equivalent to

$$\begin{array}{c}\hfill (R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)(\Delta y)={\tau }_{12}\left((R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)(\Delta x)\right).\end{array}$$

(7)

The left hand side of Equation (5) is equal to

$$\begin{array}{ccc}& & {\mathcal{L}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}{[x,y]}_g-{[{\mathcal{L}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g-{[{\varphi }_g\left(x\right),{\mathcal{L}}_{\xi }^{*}y]}_g\hfill \\ & & -{\mathcal{L}}_{(-L_x^{*}-R_x^{*})\xi }^{*}{\varphi }_g\left(y\right)-(-{\mathcal{L}}_{(-L_y^{*}-R_y^{*})\xi }^{*}-{\mathcal{R}}_{(-L_x^{*}-R_x^{*})\xi }^{*}){\varphi }_g\left(x\right)\hfill \\ & =& {\mathcal{L}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}{[x,y]}_g-{[{\mathcal{L}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g-{[{\varphi }_g\left(x\right),{\mathcal{L}}_{\xi }^{*}y]}_g+{\mathcal{L}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right)+{\mathcal{L}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(y\right)-{\mathcal{L}}_{L_y^{*}\xi }^{*}{\varphi }_g\left(x\right)\hfill \\ & & -{\mathcal{L}}_{R_y^{*}\xi }^{*}{\varphi }_g\left(x\right)-{\mathcal{R}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(x\right)-{\mathcal{R}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(x\right).\hfill \end{array}$$

Furthermore, we have

$$\begin{array}{ccc}& & \langle {\mathcal{L}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}{[x,y]}_g-{[{\mathcal{L}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g-{[{\varphi }_g\left(x\right),{\mathcal{L}}_{\xi }^{*}y]}_g+{\mathcal{L}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right)+{\mathcal{L}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(y\right)-{\mathcal{L}}_{L_y^{*}\xi }^{*}{\varphi }_g\left(x\right)\hfill \\ & & -{\mathcal{L}}_{R_y^{*}\xi }^{*}{\varphi }_g\left(x\right)-{\mathcal{R}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(x\right)-{\mathcal{R}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(x\right),\eta \rangle \hfill \\ & =& -\langle {[x,y]}_g,{[{\varphi }_g^{*}\left(\xi \right),\eta ]}_{g^{*}}\rangle -\langle x,[\xi ,R_{{\varphi }_g\left(y\right)}^{*}\eta ]\rangle -\langle y,[\xi ,L_{{\varphi }_g\left(x\right)}^{*}\eta ]\rangle -\langle {\varphi }_g\left(y\right),[L_x^{*}\xi ,\eta ]\rangle \hfill \\ & & -\langle {\varphi }_g\left(y\right),[R_x^{*}\xi ,\eta ]\rangle +\langle {\varphi }_g\left(x\right),{[L_y^{*}\xi ,\eta ]}_{g^{*}}\rangle +\langle {\varphi }_g\left(x\right),{[R_y^{*}\xi ,\eta ]}_{g^{*}}\rangle +\langle {\varphi }_g\left(x\right),{[\eta ,L_y^{*}\xi ]}_{g^{*}}\rangle \hfill \\ & & +\langle {\varphi }_g\left(x\right),{[\eta ,R_y^{*}\xi ]}_{g^{*}}\rangle \hfill \\ & =& -\langle \Delta {[x,y]}_g,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle -\langle \Delta x,\xi \otimes R_{{\varphi }_g\left(y\right)}^{*}\eta \rangle -\langle \Delta y,\xi \otimes L_{{\varphi }_g\left(x\right)}^{*}\eta \rangle -\langle \Delta y,L_{{\varphi }_g\left(x\right)}^{*}{\varphi }_g^{*}\left(\xi \right)\otimes {\varphi }_g^{*}\left(\eta \right)\rangle \hfill \\ & & -\langle \Delta {\varphi }_g\left(y\right),R_x^{*}\xi \otimes \eta \rangle +\langle \Delta x,L_{{\varphi }_g\left(y\right)}^{*}{\varphi }_g^{*}\left(\xi \right)\otimes {\varphi }_g^{*}\left(\eta \right)\rangle +\langle \Delta x,R_{{\varphi }_g\left(y\right)}^{*}{\varphi }_g^{*}\left(\xi \right)\otimes {\varphi }_g^{*}\left(\eta \right)\rangle \hfill \\ & & +\langle \Delta x,{\varphi }_g^{*}\left(\eta \right)\otimes L_{{\varphi }_g\left(y\right)}^{*}{\varphi }_g^{*}\left(\xi \right)\rangle +\langle \Delta x,{\varphi }_g^{*}\left(\eta \right)\otimes R_{{\varphi }_g\left(y\right)}^{*}{\varphi }_g^{*}\left(\xi \right)\rangle \hfill \\ & =& -\langle \Delta {[x,y]}_g,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)})\Delta x,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle ({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)})\Delta y,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle \hfill \\ & & +\langle (L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle (R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle \hfill \\ & & -\langle (L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)\Delta x,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle -\langle (R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)\Delta x,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle \hfill \\ & & -\langle {\tau }_{12}(({\varphi }_g\otimes L_{{\varphi }_g\left(y\right)})\Delta x),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle -\langle {\tau }_{12}(({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)})\Delta x),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle .\hfill \end{array}$$

Therefore, Equation (5) is equivalent to

$$\begin{array}{ccc}& & \Delta {[x,y]}_g\hfill \\ & =& ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)}-L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g-R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g-{\tau }_{12}\circ ({\varphi }_g\otimes L_{{\varphi }_g\left(y\right)})-{\tau }_{12}\circ ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)}))\Delta x\hfill \\ & & +({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)}+L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g+R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y.\hfill \end{array}$$

(8)

The left hand side of Equation (4) is equal to

$$\begin{array}{ccc}& & (-{\mathcal{L}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}-{\mathcal{R}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}){[x,y]}_g+{[{\varphi }_g\left(x\right),{\mathcal{L}}_{\xi }^{*}y+{\mathcal{R}}_{\xi }^{*}y]}_g-{[{\varphi }_g\left(y\right),{\mathcal{L}}_{\xi }^{*}x+{\mathcal{R}}_{\xi }^{*}x]}_g\hfill \\ & & +{\mathcal{L}}_{L_y^{*}\xi }^{*}{\varphi }_g\left(x\right)+{\mathcal{R}}_{L_y^{*}\xi }^{*}{\varphi }_g\left(x\right)-{\mathcal{L}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right)-{\mathcal{R}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right).\hfill \end{array}$$

Furthermore, we have

$$\begin{array}{ccc}& & \langle (-{\mathcal{L}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}-{\mathcal{R}}_{{\varphi }_g^{*}\left(\xi \right)}^{*}){[x,y]}_g+{[{\varphi }_g\left(x\right),{\mathcal{L}}_{\xi }^{*}y+{\mathcal{R}}_{\xi }^{*}y]}_g-{[{\varphi }_g\left(y\right),{\mathcal{L}}_{\xi }^{*}x+{\mathcal{R}}_{\xi }^{*}x]}_g\hfill \\ & & +{\mathcal{L}}_{L_y^{*}\xi }^{*}{\varphi }_g\left(x\right)+{\mathcal{R}}_{L_y^{*}\xi }^{*}{\varphi }_g\left(x\right)-{\mathcal{L}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right)-{\mathcal{R}}_{L_x^{*}\xi }^{*}{\varphi }_g\left(y\right),\eta \rangle \hfill \\ & =& \langle \Delta {[x,y]}_g,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle {\tau }_{12}(\Delta {[x,y]}_g),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle -\langle ({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)})\Delta y,\xi \otimes \eta \rangle \hfill \\ & & -\langle ({\tau }_{12}(L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle ({\varphi }_g\otimes L_{{\varphi }_g\left(y\right)})\Delta x,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle \hfill \\ & & +\langle {\tau }_{12}((L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)\Delta x),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle +\langle (L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)\Delta x,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle \hfill \\ & & +\langle {\tau }_{12}(({\varphi }_g\otimes L_{{\varphi }_g\left(y\right)})\Delta x),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle -\langle (L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y,{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle \hfill \\ & & -\langle {\tau }_{12}(({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)})\Delta y),{\varphi }_g^{*}\left(\xi \right)\otimes \eta \rangle .\hfill \end{array}$$

Therefore, Equation (4) is equivalent to

$$\begin{array}{ccc}& & \Delta {[x,y]}_g+{\tau }_{12}(\Delta {[x,y]}_g)\hfill \\ & =& (-{\varphi }_g\otimes L_{{\varphi }_g\left(y\right)}-{\tau }_{12}\circ (L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)-L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g-{\tau }_{12}\circ (L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g))\Delta x\hfill \\ & & +({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)}+L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g+{\tau }_{12}\circ (L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)+{\tau }_{12}\circ ({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)}))\Delta y.\hfill \end{array}$$

(9)

By Equations (8) and (9), we deduce that

$$\begin{array}{ccc}& & \Delta {[x,y]}_g+{\tau }_{12}\circ \Delta {[x,y]}_g)\hfill \\ & =& ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)}-L_{{\varphi }_g\left(y\right)}\otimes Id-R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g-{\tau }_{12}\circ ({\varphi }_g\otimes L_{{\varphi }_g\left(y\right)})-{\tau }_{12}\circ ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)}))\Delta x\hfill \\ & & +({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)}+L_{{\varphi }_g\left(x\right)}\otimes Id+R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)\Delta y+({\tau }_{12}\circ ({\varphi }_g\otimes R_{{\varphi }_g\left(y\right)})\hfill \\ & & -{\tau }_{12}\circ (L_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)-{\tau }_{12}\circ (R_{{\varphi }_g\left(y\right)}\otimes {\varphi }_g)-Id\otimes L_{{\varphi }_g\left(y\right)}-{\varphi }_g\otimes R_{{\varphi }_g\left(y\right)})\Delta x\hfill \\ & & +({\tau }_{12}\circ ({\varphi }_g\otimes L_{{\varphi }_g\left(x\right)})+{\tau }_{12}\circ (L_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g)+{\tau }_{12}\circ (R_{{\varphi }_g\left(x\right)}\otimes {\varphi }_g))\Delta y\hfill \\ & =& therighthandsideofEquation\left(7\right).\hfill \end{array}$$

Thus, by Equations (5) and (6), we can deduce that Equation (4) holds.

Consider the left hand side of Equation (3). It is equal to $-L_{{\mathcal{R}}_{\xi }^{*}x}^{*}{\varphi }_g^{*}\left(\eta \right)-{[R_x^{*}\xi ,{\varphi }_g^{*}\left(\eta \right)]}_{g^{*}}$. For any $y\in g$, we have

$$\begin{array}{ccc}& & \langle -L_{R_{\xi }^{*}x}^{*}{\varphi }_g^{*}\left(\eta \right)-[R_x^{*}\xi ,{\varphi }_g^{*}\left(\eta \right)],y\rangle \hfill \\ & =& \langle {\varphi }_g^{*}\left(\eta \right),[R_{\xi }^{*}x,y]\rangle +\langle {\varphi }_g^{*}\left(\eta \right),{\mathcal{L}}_{R_x^{*}\xi }^{*}y\rangle \hfill \\ & =& \langle \eta ,{\mathcal{L}}_{R_x^{*}\xi }^{*}{\varphi }_g\left(y\right)+{[{\mathcal{R}}_{\xi }^{*}x,{\varphi }_g\left(y\right)]}_g\rangle ,\hfill \end{array}$$

which implies that Equation (3) is equivalent to Equation (6). Similarly, if Equation (3) holds, we can deduce that Equation (2) is equivalent to Equation (5). Furthermore, by Equations (2) and (3), we can deduce that Equation (1) holds naturally. Therefore, $(g,{\varphi }_g,g^{*},{\varphi }_g^{*},(L^{*},-L^{*}-R^{*}),({\mathcal{L}}^{*},-{\mathcal{L}}^{*}-{\mathcal{R}}^{*}))$ is a matched pair of Hom–Leibniz algebras if and only if Equations (8) and (9) hold. Thus, $(g,g^{*})$ is a Hom–Leibniz bialgebra if and only if $(g,{\varphi }_g,g^{*},{\varphi }_g^{*},(L^{*},-L^{*}-R^{*}),({\mathcal{L}}^{*},-{\mathcal{L}}^{*}-{\mathcal{R}}^{*}))$ is a matched pair of Hom–Leibniz algebras. The proof is finished. □

Corollary 1.

Let $(g,g^{*})$ be a Hom–Leibniz bialgebra. Then $(g^{*},g)$ is also a Hom–Leibniz bialgebra.

4. Relative Rota–Baxter Operators on Hom–Leibniz Algebras

In this section, we will introduce the notion of relative Rota–Baxter operators on Hom–Leibniz algebras and prove that any Hom–Leibniz algebra together with a relative Rota–Baxter operator naturally gives rise to a Hom–pre-Leibniz algebra.

Definition 6.

Let $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$ be a representation of a Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$. A relative Rota–Baxter operator on $(g,{[·,·]}_g,{\varphi }_g)$ with respect to the representation $(V,{\varphi }_V,{\rho }^L$, ${\rho }^R)$ is a linear map $K:V\to g$ such that

$$\begin{array}{ccc}& & K\circ {\varphi }_V={\varphi }_g\circ K,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & {[K\left(u\right),K\left(v\right)]}_g=K({\rho }^L\left(Ku\right)v+{\rho }^R\left(Kv\right)u),\forall u,v\in V.\hfill \end{array}$$

(11)

Obviously, the notion of relative Rota–Baxter operators on Hom–Leibniz algebras is a generalization of relative Rota–Baxter operators on Leibniz algebras in [11].

Example 1.

Consider the 2-dimensional Hom–Leibniz algebra $(g,[·,·],{\varphi }_g)$ given with respect to a basis $\{e_1,e_2\}$ by

$$\begin{array}{ccc}& & [e_1,e_1]=0,[e_1,e_2]=0,[e_2,e_1]=e_1,[e_2,e_2]=e_1,\hfill \\ & & {\varphi }_g\left(e_1\right)=e_1,{\varphi }_g\left(e_2\right)=-e_2.\hfill \end{array}$$

Let $\{e_1^{*},e_2^{*}\}$ be the dual basis. Then $K=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$ is a relative Rota–Baxter operator on $(g,[·,·],{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},L^{*}-R^{*})$ if and only if the following equalities hold:

$$\begin{array}{c}\hfill [K\left(e_i^{*}\right),K\left(e_j^{*}\right)]=K(L_{K{e}_i^{*}}^{*}\left(e_j^{*}\right)-L_{K{e}_j^{*}}^{*}\left(e_i^{*}\right)-R_{K{e}_j^{*}}^{*}\left(e_i^{*}\right)),i,j=1,2.\end{array}$$

It is straightforward to deduce that

$$\begin{array}{c}{L}_{e_1}(e_1,e_2)=(e_1,e_2)\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),L_{e_2}(e_1,e_2)=(e_1,e_2)\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right),\hfill \\ R_{e_1}(e_1,e_2)=(e_1,e_2)\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),R_{e_2}(e_1,e_2)=(e_1,e_2)\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & L_{e_1}^{*}(e_1^{*},e_2^{*})=(e_1^{*},e_2^{*})\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),L_{e_2}^{*}(e_1^{*},e_2^{*})=(e_1^{*},e_2^{*})\left(\begin{array}{cc}-1& 0\\ -1& 0\end{array}\right),\hfill \\ & & R_{e_1}^{*}(e_1^{*},e_2^{*})=(e_1^{*},e_2^{*})\left(\begin{array}{cc}0& 0\\ -1& 0\end{array}\right),R_{e_2}^{*}(e_1^{*},e_2^{*})=(e_1^{*},e_2^{*})\left(\begin{array}{cc}0& 0\\ -1& 0\end{array}\right).\hfill \end{array}$$

We have

$$\begin{array}{c}\hfill [K{e}_1^{*},K{e}_1^{*}]=[a_{11}{e}_1+a_{21}{e}_2,a_{11}{e}_1+a_{21}{e}_2]=a_{21}(a_{11}+a_{21})e_1,\end{array}$$

and

$$\begin{array}{ccc}\hfill K(L_{K{e}_1^{*}}^{*}{e}_1^{*}-L_{K{e}_1^{*}}^{*}{e}_1^{*}-R_{K{e}_1^{*}}^{*}{e}_1^{*})& =& (a_{11}+a_{21})K\left(e_2^{*}\right)\hfill \\ & =& (a_{11}+a_{21})(a_{12}{e}_1+a_{22}{e}_2)\hfill \\ & =& (a_{11}+a_{21})a_{12}{e}_1+(a_{11}+a_{21})a_{22}{e}_2.\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}& & a_{21}(a_{11}+a_{21})=(a_{11}+a_{21})a_{12},(a_{21}+a_{22})a_{22}=0.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{ccc}& & a_{21}(a_{12}+a_{22})=a_{22}{a}_{11}+(a_{12}+2a_{22})a_{12},a_{22}(a_{21}+a_{12}+2a_{22})=0;\hfill \\ & & a_{22}(a_{11}+a_{21})=-a_{22}(a_{11}+a_{12}),-a_{22}(a_{21}+a_{22})=0,a_{22}(a_{12}+a_{22})=0.\hfill \end{array}$$

To summarize the above discussion, we have

(1) 

If $a_{22}=0$, then $K=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& 0\end{array}\right)$ is a relative Rota–Baxter operator on $(g,[·,·],{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},L^{*}-R^{*})$ if and only if

$$\begin{array}{c}\hfill (a_{12}-a_{21})a_{12}=(a_{12}-a_{21})(a_{11}+a_{21})=0.\end{array}$$

(2) 

If $a_{22}\ne 0$, then $K=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$ is a relative Rota–Baxter operator on $(g,[·,·],{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},L^{*}-R^{*})$ if and only if

$$\begin{array}{c}\hfill a_{11}=-a_{12}=-a_{21}=a_{22}.\end{array}$$

Definition 7.

A Hom–pre-Leibniz algebra is a quadruple $(A,{\varphi }_A,▹,◃)$ consisting of a vector space A, two binary operations ▹ and $◃:A\otimes A\to A$ and a linear map ${\varphi }_A:A\to A$ satisfying the following conditions:

(A1) 

$(x◃y)◃{\varphi }_A\left(z\right)={\varphi }_A\left(x\right)◃(y◃z)-{\varphi }_A\left(y\right)◃(x◃z)-(x▹y)◃{\varphi }_A\left(z\right),$

(A2) 

${\varphi }_A\left(x\right)◃(y▹z)=(x◃y)▹{\varphi }_A\left(z\right)+{\varphi }_A\left(y\right)▹(x◃z)+{\varphi }_A\left(y\right)▹(x▹z),$

(A3) 

${\varphi }_A\left(x\right)▹(y▹z)=(x▹y)▹{\varphi }_A\left(z\right)+{\varphi }_A\left(y\right)◃(x▹z)-{\varphi }_A\left(x\right)▹(y◃z),$

for all $x,y,z\in A$.

Proposition 2.

Let $(A,{\varphi }_A,▹,◃)$ be a Hom–pre-Leibniz algebra. Define ${[·,·]}_{▹,◃}:A\otimes A\to A$ by

$$\begin{array}{c}\hfill {[x,y]}_{▹,◃}=x▹y+x◃y,\forall x,y\in A.\end{array}$$

Then, ${[·,·]}_{▹,◃}$ defines a Hom–Leibniz algebra, which is called the sub-adjacent Hom–Leibniz algebra of $(A,{\varphi }_A,▹,◃)$ and $(A,{\varphi }_A,▹,◃)$ is called a compatible Hom–pre-Leibniz algebra structure on $(A,{[·,·]}_{▹,◃},{\varphi }_A)$.

Proof. 

For any $x,y,z\in A$, we have

$$\begin{array}{ccc}& & {[{\varphi }_A\left(x\right),{[y,z]}_{▹,◃}]}_{▹,◃}\hfill \\ & =& {[{\varphi }_A\left(x\right),y▹z+y◃z]}_{▹,◃}\hfill \\ & =& {\varphi }_A\left(x\right)◃(y◃z)+{\varphi }_A\left(x\right)▹(y◃z)+{\varphi }_A\left(x\right)◃(y▹z)+{\varphi }_A\left(x\right)▹(y▹z).\hfill \end{array}$$

Further, we have

$$\begin{array}{ccc}& & {[{[x,y]}_{▹,◃},{\varphi }_A\left(z\right)]}_{▹,◃}+{[{\varphi }_A\left(y\right),{[x,z]}_{▹,◃}]}_{▹,◃}\hfill \\ & =& {[x◃y+x▹y,{\varphi }_A\left(z\right)]}_{▹,◃}+{[{\varphi }_A\left(y\right),x◃z+x▹z]}_{▹,◃}\hfill \\ & =& (x◃y)◃{\varphi }_A\left(z\right)+(x◃y)▹{\varphi }_A\left(z\right)+(x▹y)◃{\varphi }_A\left(z\right)+(x▹y)▹{\varphi }_A\left(z\right)\hfill \\ & & +{\varphi }_A\left(y\right)◃(x◃z)+{\varphi }_A\left(y\right)▹(x◃z)+{\varphi }_A\left(y\right)◃(x▹z)+{\varphi }_A\left(y\right)▹(x▹z).\hfill \end{array}$$

Then, $(A,{[·,·]}_{▹,◃},{\varphi }_A)$ is a Hom–Leibniz algebra. □

Let $(A,{\varphi }_A,L_{◃},R_{▹})$ be a Hom–pre-Leibniz algebra. Define two linear maps $L_{◃}:A\to gl\left(A\right)$ and $R_{▹}:A\to gl\left(A\right)$ by

$$\begin{array}{c}\hfill L_{◃}\left(x\right)y=x◃y,R_{▹}\left(x\right)y=y▹x,\forall x,y\in A.\end{array}$$

Proposition 3.

Let $(A,▹,◃,{\varphi }_A)$ be a Hom–pre-Leibniz algebra. Then $(A,{\varphi }_A,L_{◃}$, $R_{▹})$ is a representation of the sub-adjacent Hom–Leibniz algebra $(A,{[·,·]}_{▹,◃},{\varphi }_A)$. Moreover, the identity map $Id:A\to A$ is a relative Rota–Baxter operator on the Hom–Leibniz algebra $(A,{[·,·]}_{▹,◃},{\varphi }_A)$ with respect to the representation $(A,{\varphi }_A,L_{◃},R_{▹})$.

Proof. 

For any $x,y,z\in A$, we have

$$\begin{array}{ccc}\hfill L_{◃}\left({[x,y]}_{▹,◃}\right){\varphi }_A\left(z\right)& =& {[x,y]}_{▹,◃}◃{\varphi }_A\left(z\right)\hfill \\ & \stackrel{\left(A1\right)}{=}& {\varphi }_A\left(x\right)◃(y◃z)-{\varphi }_A\left(y\right)◃(x◃z)\hfill \\ & =& L_{◃}\left({\varphi }_A\left(x\right)\right)\circ L_{◃}\left(y\right))z-L_{◃}\left({\varphi }_A\left(y\right)\right)\circ L_{◃}\left(x\right)z,\hfill \\ \hfill R_{▹}\left({[x,y]}_{▹,◃}\right){\varphi }_A\left(z\right)& =& {\varphi }_A\left(z\right)▹{[x,y]}_{▹,◃}\hfill \\ & \stackrel{\left(A2\right)}{=}& {\varphi }_A\left(x\right)◃(z▹y)-(x◃z)▹{\varphi }_A\left(y\right)\hfill \\ & =& L_{◃}\left({\varphi }_A\left(x\right)\right)\circ R_{▹}\left(y\right))z-R_{▹}\left({\varphi }_A\left(y\right)\right)\circ L_{◃}\left(x\right)z.\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}& & (R_{▹}\left({\varphi }_A\left(y\right)\right)L_{◃}\left(x\right)+R_{▹}\left({\varphi }_A\left(y\right)\right)R_{▹}\left(x\right))z\hfill \\ & =& (x◃z)◃{\varphi }_A\left(y\right)+(z▹x)▹{\varphi }_A\left(y\right)\hfill \\ & \stackrel{(A2,A3)}{=}& {\varphi }_A\left(x\right)◃(z▹y)-{\varphi }_A\left(z\right)▹(x◃y)-{\varphi }_A\left(z\right)▹(x▹y)\hfill \\ & & +{\varphi }_A\left(z\right)▹(x▹y)-{\varphi }_A\left(x\right)◃(z▹y)+{\varphi }_A\left(z\right)▹(x◃y)\hfill \\ & =& 0.\hfill \end{array}$$

Therefore, $(A,{\varphi }_A,L_{◃},R_{▹})$ is a representation of the sub-adjacent Hom–Leibniz algebra $(A,{[·,·]}_{▹,◃}$, ${\varphi }_A)$. Moreover, we have

$$\begin{array}{c}\hfill Id(L_{◃}\left(Idx\right)y+R_{▹}\left(Idy\right)x)=x◃y+x▹y={[Idx,Idy]}_{▹,◃}.\end{array}$$

Thus, $Id:A\to A$ is a relative Rota–Baxter operator on the Hom–Leibniz algebra $(A,▹,◃,{\varphi }_A)$ with respect to the representation $(A,{\varphi }_A,L_{◃},R_{▹})$. □

Proposition 4.

Let K be a relative Rota–Baxter operator on a Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ with respect to a representation $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$. Then there is a Hom–pre-Leibniz algebra structure on V given by

$$\begin{array}{c}\hfill u▹v:={\rho }^R\left(Kv\right)u,u◃v:={\rho }^L\left(Ku\right)v,\forall u,v\in V.\end{array}$$

Proof. 

For any $u,v,w\in V$, we have

$$\begin{array}{ccc}& & {\varphi }_V\left(u\right)◃(v◃w)-{\varphi }_V\left(v\right)◃(u◃w)-(u▹v)◃{\varphi }_V\left(w\right)-(u◃v)◃{\varphi }_V\left(w\right)\hfill \\ & =& {\varphi }_V\left(u\right)◃\left({\rho }^L\left(Kv\right)w\right)-{\varphi }_V\left(v\right)◃\left({\rho }^L\left(Ku\right)w\right)-\left({\rho }^R\left(Kv\right)u\right)◃{\varphi }_V\left(w\right)-\left({\rho }^L\left(Ku\right)v\right)◃{\varphi }_V\left(w\right)\hfill \\ & =& {\rho }^L\left(K{\varphi }_V\left(u\right)\right){\rho }^L\left(Kv\right)w-{\rho }^L\left(K{\varphi }_V\left(v\right)\right){\rho }^L\left(Ku\right)w-{\rho }^L\left(K\left({\rho }^R\left(Ku\right)v\right)\right){\varphi }_V\left(w\right)-{\rho }^L\left(K\left({\rho }^L\left(Ku\right)v\right)\right){\varphi }_V\left(w\right)\hfill \\ & =& {\rho }^L\left({[Ku,Kv]}_g\right){\varphi }_V\left(w\right)-{\rho }^L\left(K\left({\rho }^R\left(Ku\right)v\right)\right){\varphi }_V\left(w\right)-{\rho }^L\left(K\left({\rho }^L\left(Ku\right)v\right)\right){\varphi }_V\left(w\right)\hfill \\ & =& 0,\hfill \end{array}$$

which implies that (A1) in Definition 7 holds.

Similarly, we can show that (A2) and (A3) also hold. Thus, $(V,▹,◃,{\varphi }_V)$ is a Hom–pre-Leibniz algebra. □

Next we will give a sufficient and necessary condition for the existence of a compatible Hom–pre-Leibniz algebra structure on a Hom–Leibniz algebra.

Proposition 5.

Let $(g,{[·,·]}_g,{\varphi }_g)$ be a Hom–Leibniz algebra. Then, there is a compatible Hom–pre-Leibniz algebra on g if and only if there exists an invertible relative Rota–Baxter operator $K:V\to g$ on g with respect to a representation $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$. Furthermore, the compatible Hom–pre-Leibniz algebra structure on g is given by

$$\begin{array}{c}\hfill x▹y:=K\left({\rho }^R\left(y\right)K^{-1}x\right),x◃y:=K\left({\rho }^L\left(x\right)K^{-1}y\right),\forall x,y\in g.\end{array}$$

Proof. 

Let $K:V\to g$ be an invertible relative Rota–Baxter operator on g with respect to a representation $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$. By Proposition 4, there is a Hom–pre-Leibniz algebra on V given by

$$\begin{array}{c}\hfill u▹v:={\rho }^R\left(Kv\right)u,u◃v:={\rho }^L\left(Ku\right)v,\forall u,v\in V.\end{array}$$

Since K is an invertible relative Rota–Baxter operator, we obtain that

$$\begin{array}{c}\hfill x▹y:=K(K^{-1}x▹K^{-1}y)=K\left({\rho }^R\left(y\right)K^{-1}x\right),x◃y:=K(K^{-1}x◃K^{-1}y)=K\left({\rho }^L\left(x\right)K^{-1}y\right),\end{array}$$

is a Hom–pre-Leibniz algebra on $(g,{[·,·]}_g,{\varphi }_g)$. Furthermore, we have

$$\begin{array}{ccc}& & x▹y+x◃y\hfill \\ & =& K\left({\rho }^R\left(y\right)K^{-1}x\right)+K\left({\rho }^L\left(x\right)K^{-1}y\right)\hfill \\ & =& K\left({\rho }^R(K\circ K^{-1}y)K^{-1}x\right)+K\left({\rho }^L(K\circ K^{-1}x)K^{-1}y\right)\hfill \\ & =& {[x,y]}_g.\hfill \end{array}$$

On the other hand, by Proposition 4, $(g,{\varphi }_g,L_{◃},R_{▹})$ is a representation of the Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$. Moreover, $Id:g\to g$ is a relative Rota–Baxter operator on the Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ with respect to the representation $(g,{\varphi }_g,L_{◃},R_{▹})$. □

Proposition 6.

Let K be a relative Rota–Baxter operator on a Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ with respect to $(V,{\varphi }_V,{\rho }^L,{\rho }^R)$. Define

$$\begin{array}{c}\hfill {[u,v]}_K:={\rho }^L\left(Ku\right)v+{\rho }^R\left(Kv\right)u,\forall u,v\in V.\end{array}$$

Then, $(V,{[·,·]}_K,{\varphi }_V)$ is a Hom–Leibniz algebra.

Proof. 

It follows from Propositions 3 and 5. □

Corollary 2.

Let $(g,{[·,·]}_g,{\varphi }_g)$ be an admissible Hom–Leibniz algebra and $K:g^{*}\to g$ be a relative Rota–Baxter operator on g with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},-L^{*}-R^{*})$. Then, $g_K^{*}:=(g^{*},{[·,·]}_K,{\varphi }_g^{*})$ is a Hom–Leibniz algebra, where ${[·,·]}_K$ is given by

$$\begin{array}{c}\hfill {[\xi ,\eta ]}_K=L_{K\xi }^{*}\eta -L_{K\eta }^{*}\xi -R_{K\eta }^{*}\xi ,\forall \xi ,\eta \in g^{*}.\end{array}$$

Definition 8.

A quadratic Hom–Leibniz algebra is a Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ equipped with a nondegenerate skew-symmetric bilinear form $\omega \in g^{*}\otimes g^{*}$ such that the following invariant condition holds:

$$\begin{array}{ccc}& & \omega ({\varphi }_g\left(x\right),y)=\omega (x,{\varphi }_g\left(y\right)),\hfill \\ & & \omega ({\varphi }_g\left(x\right),{[y,z]}_g)=\omega ({[x,z]}_g+{[z,x]}_g,{\varphi }_g\left(y\right)),\mathrm{for}x,y,z\in g.\hfill \end{array}$$

Remark 1.

In [18], Ammar, Mabrouk and Makhlouf introduced the notion of quadratic Hom–Leibniz algebras as follows:

$$\begin{array}{ccc}& & \omega ({\varphi }_g\left(x\right),y)=\omega (x,{\varphi }_g\left(y\right)),\hfill \\ & & \omega ({\varphi }_g\left(x\right),{[y,z]}_g)=-\omega ({[y,x]}_g,{\varphi }_g\left(z\right)).\hfill \end{array}$$

Combined with [11], we give the notion of quadratic Hom–Leibniz algebras different from [18]. It is worth mentioning that our notion of quadratic Hom–Leibniz algebras can include the notion of quadratic Hom–Leibniz algebras in [18].

Proposition 7.

Let $K:g^{*}\to g$ be a relative Rota–Baxter operator on a Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},-L^{*}-R^{*})$ and $K^{*}=K$, where $K^{*}$ is the dual map of K. Then $(g\oplus g^{*},g,g^{*},{\varphi }_g,{\varphi }_g^{*})$ is a quadratic Hom–Leibniz algebra with the invariant bilinear form ω given by Equation (7).

Proof. 

For any $x,y,z\in g$ and $\xi ,\eta ,\zeta \in g^{*}$, we have

$$\begin{array}{ccc}& & \omega (x+\xi +K\xi ,y+\eta +K\eta )\hfill \\ & =& \omega (x+\xi ,y+\eta )+\omega (x+\xi ,K\eta )+\omega (K\xi ,y+\eta )+\omega (K\xi ,K\eta )\hfill \\ & =& \omega (x+\xi ,y+\eta )+\omega (\xi ,K\eta )+\omega (K\xi ,\eta )\hfill \\ & =& \omega (x+\xi ,y+\eta )+\langle \xi ,K\eta \rangle +\langle K\xi ,\eta \rangle \hfill \\ & =& \omega (x+\xi ,y+\eta )+\langle (K^{*}-K)\xi ,\eta \rangle .\hfill \end{array}$$

Thus, $\omega (x+\xi +K\xi ,y+\eta +K\eta )=\omega (x+\xi ,y+\eta )$ if and only if $K^{*}=K$. Further, we have

$$\begin{array}{ccc}& & \omega ({\varphi }_g\left(x\right)+{\varphi }_g^{*}\left(\xi \right)+{\varphi }_g\left(K\xi \right),{[y+\eta +K\eta ,z+\zeta +K\zeta ]}_{\bowtie })\hfill \\ & =& \omega ({[x+\xi +K\xi ,z+\zeta +K\zeta ]}_{\bowtie }+{[z+\zeta +K\zeta ,x+\xi +K\xi ]}_{\bowtie },{\varphi }_g\left(y\right)+{\varphi }_g^{*}\left(\eta \right)+{\varphi }_g\left(K\eta \right)).\hfill \end{array}$$

Hence, $(g\oplus g^{*},g,g^{*},{\varphi }_g,{\varphi }_g^{*})$ is a quadratic Hom–Leibniz algebra. □

By Corollary 2, Proposition 7 and Theorem 1, we obtain

Theorem 2.

Let $K:g^{*}\to g$ be a relative Rota–Baxter operator on a Hom–Leibniz algebra $(g,{[·,·]}_g,{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},-L^{*}-R^{*})$ and $K^{*}=K$. Then $(g,g_K^{*})$ is a Hom–Leibniz bialgebra.

5. The Classical Hom–Leibniz Yang–Baxter Equation

In this section, we will recall the notion of the classical Hom–Leibniz Yang–Baxter equation according to [10,11] and study its properties.

Definition 9.

Let $(g,{[·,·]}_g,{\varphi }_g)$ be a Hom–Leibniz algebra and $r=\sum r_1\otimes r_2\in g\otimes g$ be symmetric. Then the equation

$$\begin{array}{c}\hfill \left[[r,r]\right]={[r_{12},r_{13}]}_g+{[r_{13},r_{12}]}_g-{[r_{12},r_{23}]}_g-{[r_{13},r_{23}]}_g=0,\end{array}$$

where

$$\begin{array}{c}\hfill r_{12}:=r_1\otimes r_2\otimes 1,r_{13}:=r_1\otimes 1\otimes r_2,r_{23}:=1\otimes r_1\otimes r_2\end{array}$$

is called the classical Hom–Leibniz Yang–Baxter equation in g, and r is called a solution of the classical Hom–Leibniz Yang–Baxter equation.

Example 2.

Consider the 2-dimensional Hom–Leibniz algebra $(g,[·,·],{\varphi }_g)$ given with respect to a basis $\{e_1,e_2\}$ by

$$\begin{array}{ccc}& & [e_1,e_1]=0,[e_1,e_2]=0,[e_2,e_1]=e_1,[e_2,e_2]=e_1,\hfill \\ & & {\varphi }_g\left(e_1\right)=e_1,{\varphi }_g\left(e_2\right)=-e_2.\hfill \end{array}$$

For any $r=a{e}_1\otimes e_1+b{e}_1\otimes e_2+b{e}_2\otimes e_1+c{e}_2\otimes e_2$. By direct calculation, we have

$$\begin{array}{ccc}\hfill \left[\right[r,r\left]\right]& =& 4ca{e}_2\otimes e_1\otimes e_1-2ca{e}_1\otimes e_2\otimes e_1-2ca{e}_1\otimes e_1\otimes e_2\hfill \\ & & +2c^2{e}_2\otimes e_1\otimes e_2-4c^2{e}_1\otimes e_2\otimes e_2+2c^2{e}_2\otimes e_2\otimes e_1=0.\hfill \end{array}$$

Thus,

(1) 

If $c=0$, any $r=a{e}_1\otimes e_1+b{e}_1\otimes e_2+b{e}_2\otimes e_1$ is a solution of the classical Hom–Leibniz Yang–Baxter equation.

(2) 

If $c\ne 0$, any $r=a{e}_1\otimes e_1+b{e}_1\otimes e_2+b{e}_2\otimes e_1+c{e}_2\otimes e_2$ is a solution of the classical Hom–Leibniz Yang–Baxter equation if and only if

$$\begin{array}{c}\hfill a=-b=c.\end{array}$$

Let r be a solution of the classical Hom–Leibniz Yang–Baxter equation in g, and define the linear map $r^{♮}:g^{*}\to g$ such that ${\varphi }_g\circ r^{♮}=r^{♮}\circ {\varphi }_g^{*}$ by

$$\begin{array}{c}\hfill \langle r^{♮}\left(\xi \right),\eta \rangle =\langle r,\xi \otimes \eta \rangle ,\forall \xi ,\eta \in g^{*};\end{array}$$

then, $(g,g_{r^{♮}}^{*})$ is a Hom–Leibniz bialgebra.

Let $(g,{[·,·]}_g,{\varphi }_g)$ be a regular admissible Hom–Leibniz algebra and $r\in g\otimes g$ be invertible (that is, $r^{♮}$ is invertible). Define $B\in g^{*}\otimes g^{*}$ by

$$\begin{array}{c}\hfill B(x,y)=\langle r^{{♮}^{-1}}\left(x\right),y\rangle .\end{array}$$

Proposition 8.

With the notation above, r is a nondegenerate solution of the classical Hom–Leibniz Yang–Baxter equation satisfying ${\varphi }_g^{2}r=r$ in a Hom–Leibniz algebra g if and only if the symmetric nondegenerate bilinear form B satisfies

$$\begin{array}{c}\hfill B({\varphi }_g\left(z\right),{[x,y]}_g))=-B({\varphi }_g\left(y\right),{[x,z]}_g)+B({\varphi }_g\left(x\right),{[y,z]}_g)+B({\varphi }_g\left(x\right),{[z,y]}_g).\end{array}$$

Proof. 

B is symmetric nondegenerate bilinear since r is nondegenerate. For any $x,y,z\in g$, there exist ${\eta }_1,{\eta }_2,{\eta }_3\in g^{*}$ such that $r^{♮}\left({\eta }_1\right)=x,r^{♮}\left({\eta }_2\right)=y,r^{♮}\left({\eta }_3\right)=z$, so we have

$$\begin{array}{ccc}& & B({\varphi }_g\left(z\right),{[x,y]}_g))\hfill \\ & =& \langle r^{{♮}^{-1}}\left({\varphi }_g\left(z\right)\right),{[x,y]}_g\rangle \hfill \\ & =& \langle {\varphi }_g^{*}\left({\eta }_3\right),{[r^{♮}\left({\eta }_1\right),r^{♮}\left({\eta }_2\right)]}_g\rangle \hfill \\ & =& \langle {\varphi }_g^{*}\left({\eta }_3\right),r^{♮}\left(L_{r^{♮}\left({\eta }_1\right)}^{*}{\eta }_2\right)\rangle -\langle {\varphi }_g^{*}\left({\eta }_3\right),r^{♮}\left(L_{r^{♮}\left({\eta }_2\right)}^{*}{\eta }_1\right)\rangle -\langle {\varphi }_g^{*}\left({\eta }_3\right),r^{♮}\left(R_{r^{♮}\left({\eta }_2\right)}^{*}{\eta }_1\right)\rangle \hfill \\ & =& \langle r^{♮}\left({\varphi }_g^{*}\left({\eta }_3\right)\right),L_{r^{♮}\left({\eta }_1\right)}^{*}{\eta }_2\rangle -\langle r^{♮}\left({\varphi }_g^{*}\left({\eta }_3\right)\right),L_{r^{♮}\left({\eta }_2\right)}^{*}{\eta }_1\rangle -\langle r^{♮}\left({\varphi }_g^{*}\left({\eta }_3\right)\right),R_{r^{♮}\left({\eta }_2\right)}^{*}{\eta }_1\rangle \hfill \\ & =& -\langle {[r^{♮}\left({\eta }_1\right),r^{♮}\left({\eta }_3\right)]}_g,{\varphi }_g^{*}\left({\eta }_2\right)\rangle +\langle {[r^{♮}\left({\eta }_2\right),r^{♮}\left({\eta }_3\right)]}_g,{\varphi }_g^{*}\left({\eta }_1\right)\rangle +\langle {[r^{♮}\left({\eta }_3\right),r^{♮}\left({\eta }_2\right)]}_g,{\varphi }_g^{*}\left({\eta }_1\right)\rangle \hfill \\ & =& -B({\varphi }_g\left(y\right),{[x,z]}_g)+B({\varphi }_g\left(x\right),{[y,z]}_g)+B({\varphi }_g\left(x\right),{[z,y]}_g).\hfill \end{array}$$

The proof is finished. □

Lemma 2.

Let $(g,{[·,·]}_g,{\varphi }_g,\omega )$ be a quadratic Hom–Leibniz algebra. Then the map

$${\omega }^{♮}:g\to g^{*},{\omega }^{♮}\left(x\right)\left(y\right)=\omega (x,y),\mathrm{for}x,y\in g$$

is an isomorphism from the regular representation $(g,{\varphi }_g,L,R)$ to its dual representation $(g^{*},{\varphi }_g^{*},L^{*},$ $-L^{*}-R^{*})$.

Proof. 

Straightforward. □

Proposition 9.

Let $(g,{[·,·]}_g,{\varphi }_g,\omega )$ be a quadratic Hom–Leibniz algebra and $K:g^{*}\to g$ be two linear maps. Then K is a relative Rota–Baxter operator on $(g,{[·,·]}_g,{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},-L^{*}-R^{*})$ if and only if $K\circ {\omega }^{♮}$ is a Rota–Baxter operator on $(g,{[·,·]}_g,{\varphi }_g)$.

Proof. 

For any $x,y\in g$, we have

$$\begin{array}{ccc}& & K\circ {\omega }^{♮}({[K\circ {\omega }^{♮}\left(x\right),y]}_g+{[x,K\circ {\omega }^{♮}\left(y\right)]}_g)\hfill \\ & =& K({\omega }^{♮}\left(L_{K\circ {\omega }^{♮}\left(x\right)}y\right)+{\omega }^{♮}\left(R_{K\circ {\omega }^{♮}\left(y\right)}x\right))\hfill \\ & =& K(L_{K\circ {\omega }^{♮}\left(x\right)}^{*}{\omega }^{♮}\left(y\right)-L_{S\circ {\omega }^{♮}\left(y\right)}^{*}{\omega }^{♮}\left(x\right)-R_{K\circ {\omega }^{♮}\left(y\right)}^{*}{\omega }^{♮}\left(x\right)).\hfill \end{array}$$

Thus, it follows that $K\circ {\omega }^{♮}$ is a Rota–Baxter operator on $(g,{[·,·]}_g,{\varphi }_g)$ if and only if

$$\begin{array}{ccc}& & {[K\circ {\omega }^{♮}\left(x\right),K\circ {\omega }^{♮}\left(y\right)]}_g=K(L_{K\circ {\omega }^{♮}\left(x\right)}^{*}{\omega }^{♮}\left(y\right)-L_{K\circ {\omega }^{♮}\left(y\right)}^{*}{\omega }^{♮}\left(x\right)-R_{K\circ {\omega }^{♮}\left(y\right)}^{*}{\omega }^{♮}\left(x\right)).\hfill \end{array}$$

Since ${\omega }^{♮}$ is an isomorphism, these identities hold if and only if K is a relative Rota–Baxter operator on $(g,{[·,·]}_g,{\varphi }_g)$ with respect to the representation $(g^{*},{\varphi }_g^{*},L^{*},-L^{*}-R^{*})$. □

Corollary 3.

Let $(g,{[·,·]}_g,{\varphi }_g,\omega )$ be a quadratic Hom–Leibniz algebra. Then $r\in g\otimes g$ is a solution of the classical Hom–Leibniz Yang–Baxter equation in g if and only if $r^{♮}\circ {\omega }^{♮}$ is a relative Rota–Baxter operator on $(g,{[·,·]}_g,{\varphi }_g)$, where $r^{♮}:g^{*}\to g$ is defined by $\langle r^{♮}\left(\xi \right),\eta \rangle =\langle r,\xi \otimes \eta \rangle$ for all $\xi ,\eta \in g^{*}$; that is,

$$\begin{array}{ccc}& & {[r^{♮}\circ {\omega }^{♮}\left(x\right),r^{♮}\circ {\omega }^{♮}\left(y\right)]}_g=r^{♮}\circ {\omega }^{♮}({[r^{♮}\circ {\omega }^{♮}\left(x\right),y]}_g+{[x,r^{♮}\circ {\omega }^{♮}\left(y\right)]}_g).\hfill \end{array}$$