3-Hom–Lie Yang–Baxter Equation and 3-Hom–Lie Bialgebras

Abstract

In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define $\mathcal{O}$-operators of 3-Hom–Lie algebras and construct solutions of the 3-Hom–Lie Yang–Baxter equation in terms of $\mathcal{O}$-operators and 3-Hom–pre-Lie algebras. Finally, we show that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra.

1. Introduction

Hom–algebras were first introduced in the Lie algebra setting [1] with motivation from physics though the origin can be traced back in earlier literature such as [2], where the Jacobi identity was twisted by an endomorphism, namely $\left[\alpha \right(x),[y,z\left]\right]+\left[\alpha \right(y),[z,x\left]\right]+\left[\alpha \right(z),[x,y\left]\right]=0.$ In [3], Yau extended the notion of Lie bialgebras to Hom–Lie bialgebras and studied the classical Hom–Yang–Baxter equation using the twisted map, namely

$$CH\left(r\right)=[r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13},r^{23}]=0.$$

In [4], 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. Later, in [5], Tao, Bai and Guo introduced the notion of a Hom–Lie bialgebra with emphasis on its compatibility with a Manin triple of Hom–Lie algebras associated to a nondegenerate symmetric bilinear form satisfying a new invariance condition.

3-Lie algebras were special types of n-Lie algebras and played an important role in string theory [6,7]. In [8], Sheng and Tang proved that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. In [9], Ataguema, Makhlouf and Silvestrov extended the notion of 3-Lie algebras to 3-Hom–Lie algebras and presented constructions from 3-Lie algebras. Because of close relation to discrete and conformal vector fields, 3-Lie algebras and 3-Hom–Lie algebras were widely studied in the following aspects. In [10], Liu, Chen and Ma described the representations and module-extensions of 3-Hom–Lie algebras. In [11], Abdaoui, Mabrouk, Makhlouf and Massoud introduced and studied 3-Hom–Lie bialgebras, which are a ternary version of Hom–Lie bialgebras introduced by Yau. In [12], Ben Hassine, Chtioui and Mabrouk introduced the notion of 3-Hom–L-dendriform algebras which is the dendriform version of 3-Hom–Lie-algebras and studied their properties, the authors introduced the classical Yang–Baxter equation and Manin triples for 3-Lie algebras in [13,14]. Recently, we introduced the notion of 3-Hom–Lie-Rinehart algebras and systematically described a cohomology complex by considering coefficient modules in [15]. Motivated by the work of [4,8], it is natural and meaningful to study 3-Hom–Lie bialgebras and the phase space on 3-Hom–Lie algebras. This becomes our first motivation for writing the present paper.

The classical Yang–Baxter equation was investigated by Sklyanin [16] in the context of quantum inverse scattering method, which has a close connection with many branches of mathematical physics and pure mathematics. In [3], Yau extended the notion of classical Yang–Baxter equation to classical Hom–Yang–Baxter equation and presented some solutions using the twisting method. In [17], Wang, Wu and Cheng studied the 3-Lie classical Hom–Yang–Baxter equation on coboundary local cocycle 3-Hom–Lie bialgebras. Recently, the classical Hom–Yang–Baxter equation in Hom–Lie algebras has been studied widely in terms of Hom–$\mathcal{O}$-operators [18] and quasitriangular structures [3]. Motivated by the recent work on the classical Hom–Yang–Baxter equation, in this paper, we will study 3-Lie classical Hom–Yang–Baxter equation in terms of $\mathcal{O}$-operators. This becomes another motivation for writing the present paper.

In this paper, we continue the study of 3-Hom–Lie algebras and give a new description of 3-Hom–Lie bialgebras. It needs to be emphasized that there are results on 3-Hom–Lie algebras in this paper which are not “parallel” to the case of Hom–Lie algebras given in [4]. Because of the complexity of 3-Hom–Lie algebras, we need some technique to complete this paper. Now given a 3-Hom–Lie bialgebra $(L,L^{*})$, $L\oplus L^{*}$ is a 3-Hom–Lie algebra such that $(L\oplus L^{*},L,L^{*})$ is a Manin triple of 3-Hom–Lie algebras. We also study the 3-Lie classical Hom–Yang–Baxter equation in detail, and construct a solution in the semidirect 3-Hom–Lie algebra by introducing a notion of an $\mathcal{O}$-operator for a 3-Hom–Lie algebra. Finally, we describe symplectic structures and phase spaces of 3-Hom–Lie algebras from 3-Hom–pre-Lie algebra structures.

This paper is organized as follows. In Section 2, we recall some concepts and results, and introduce the notions of the matched pairs of 3-Hom–Lie algebras, the 3-Hom–Lie bialgebras and the Manin triples of 3-Hom–Lie algebras. In Section 3, we introduce the notion of the $\mathcal{O}$-operator and construct solutions of the 3-Lie classical Hom–Yang–Baxter equation in terms of $\mathcal{O}$-operators and 3-Hom–pre-Lie algebras. In Section 4, we introduce the notion of the phase space of a 3-Hom–Lie algebra and show that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra.

2. 3-Hom–Lie Bialgebras

In this section, we will recall some basic notions and facts about 3-Hom–Lie algebras and present some examples. Then we give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras.

Definition 1

([19]). A 3-Hom–Lie algebra is a triple $(L,[·,·,·],\alpha )$ consisting of a vector space L, a 3-ary skew-symmetric operation $[·,·,·]:{\wedge }^{3}L\to L$ and an algebra morphism $\alpha :L\to L$ satisfying the following 3-Hom–Jacobi identity

$$\begin{array}{ccc}\hfill \left[\alpha \right(x),\alpha (y),[u,v,w\left]\right]& =& \left[\right[x,y,u],\alpha (v),\alpha (w\left)\right]+\left[\alpha \right(u),[x,y,v],\alpha (w\left)\right]\hfill \\ & & +\left[\alpha \right(u),\alpha (v),[x,y,w\left]\right],\hfill \end{array}$$

for any $x,y,u,v,w\in L$.

A 3-Hom–Lie algebra is called regular if $\alpha$ is an algebra automorphism.

Example 1.

Let $(L,[·,·,·\left]\right)$ be a 3-Lie algebra and $\alpha :L\to L$ an algebra morphism, then the algebra $(L,{[·,·,·]}_{\alpha },\alpha )$ is a 3-Hom–Lie algebra, where ${[·,·,·]}_{\alpha }$ is defined by

$$\begin{array}{c}\hfill {[x_1,x_2,x_3]}_{\alpha }=[\alpha \left(x_1\right),\alpha \left(x_2\right),\alpha \left(x_3\right)].\end{array}$$

Example 2.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $\beta :L\to L$ an algebra morphism such that $\alpha \beta =\beta \alpha$, then $(L,{[·,·,·]}_{\alpha \beta }=[·,·,·]\circ (\beta \otimes \beta \otimes \alpha ),\alpha \circ \beta )$ is a 3-Hom–Lie algebra.

Example 3.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra over a filed F and t an indeterminate, define $\overline{L}:\{\sum (x\otimes t+y\otimes t^n)\subset L\otimes (F\left[t\right]/t^{n+1})|x,y\in L)\}$, $\overline{\alpha }\left(\overline{L}\right)=\{\sum (\alpha \left(x\right)\otimes t+\alpha \left(y\right)\otimes t^n):x,y\in L\}.$ Then $(\overline{L},\overline{\alpha })$ is a 3-Hom–Lie algebra with the operation $[x_1\otimes t^{i_1},x_2\otimes t^{i_2},x_3\otimes t^{i_3}]=[x_1,x_2,x_3]\otimes t^{\sum i_j}$ for all $x_1,x_2,x_3\in L$ and $i_1,i_2,i_3\in \{1,2,3\}$.

Definition 2

([10]). A representation of a 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ on the vector space V with respect to $A\in gl\left(V\right)$ is a bilinear map $\rho :L\wedge L\to gl\left(V\right)$, such that for any $x,y,z,u,v\in L$, the following equalities are satisfied:

$$\begin{array}{ccc}& & \rho \left(\alpha \right(u),\alpha (v\left)\right)\circ A=A\circ \rho (u,v),\hfill \\ & & \rho \left(\right[x,y,z],\alpha (u\left)\right)\circ A=\rho \left(\alpha \right(y),\alpha (z\left)\right)\rho (x,u)+\rho \left(\alpha \right(z),\alpha (x\left)\right)\rho (y,u)\hfill \\ & & +\rho \left(\alpha \right(x),\alpha (y\left)\right)\rho (z,u),\hfill \\ & & \rho \left(\alpha \right(x),\alpha (y\left)\right)\rho (z,u)=\rho \left(\alpha \right(z),\alpha (u\left)\right)\rho (x,y)+\rho \left(\right[x,y,z],\alpha (u\left)\right)\circ A\hfill \\ & & +\rho \left(\alpha \right(z),[x,y,u\left]\right)\circ A.\hfill \end{array}$$

Then $(V,A,\rho )$ is called a representation of L.

Lemma 1

([10]). Let $(V,A,\rho )$ be a representation of a 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$. Then there is a 3-Hom–Lie algebra structure on the direct sum of vector spaces $L\oplus V$, defined by

$$\begin{array}{ccc}\hfill [x_1+v_1,x_2+v_2,x_3+v_3]& =& [x_1,x_2,x_3]+\rho (x_1,x_2)v_3+\rho (x_2,x_3)v_1+\rho (x_3,x_1)v_2,\hfill \\ \hfill (\alpha \oplus A)(x_1+v_1)& =& \alpha \left(x_1\right)+A{v}_1,\hfill \end{array}$$

for any $x_1,x_2,x_3\in L$ and $v_1,v_2,v_3\in V$.

Example 4.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $ad(x,y)z=[x,y,z]$, for all $x,y,z\in L$. Then, $(L,\alpha ,ad)$ is called a regular representation of L.

Definition 3.

Let $(L,[·,·,·],\alpha )$ and $(L^{\prime },{[·,·,·]}^{\prime },{\alpha }^{\prime })$ be two 3-Hom–Lie algebras. A morphism from $(L,[·,·,·],\alpha )$ to $(L^{\prime },{[·,·,·]}^{\prime },{\alpha }^{\prime })$ is a 3-Lie algebra morphism $f:L\to L^{\prime }$ satisfying $f\circ \alpha ={\alpha }^{\prime }\circ f$.

Proposition 1.

If $f:(L,[·,·,·],\alpha )⟶(L^{\prime },{[·,·,·]}^{\prime },{\alpha }^{\prime })$ is a 3-Hom–Lie algebras morphism, then $(L^{\prime },\rho ,{\alpha }^{\prime })$ becomes a representation of L via f, that is, for all $(x,y,z)\in L^2\times L^{\prime }$, $\rho (x,y)z={[f\left(x\right),f\left(y\right),z]}^{\prime }.$

Proof. 

First, for any $x,y\in L,z\in L^{\prime }$ we have

$$\begin{array}{cc}\hfill \rho (\alpha \left(x\right),\alpha \left(y\right)){\alpha }^{\prime }\left(z\right)& ={[f\left(\alpha \left(x\right)\right),f\left(\alpha \left(y\right)\right),{\alpha }^{\prime }\left(z\right)]}^{\prime }\hfill \\ \hfill & ={[{\alpha }^{\prime }\left(f\left(x\right)\right),{\alpha }^{\prime }\left(f\left(y\right)\right),{\alpha }^{\prime }\left(z\right)]}^{\prime }\hfill \\ \hfill & ={\alpha }^{\prime }{[f\left(x\right),f\left(y\right),z]}^{\prime }\hfill \\ \hfill & ={\alpha }^{\prime }\rho (x,y)z.\hfill \end{array}$$

Next, for all $x,y,z,u\in L,z\in L^{\prime }$ we have

$$\begin{array}{cc}\hfill & \rho ([x,y,z],\alpha \left(u\right))\circ {\alpha }^{\prime }-\rho (\alpha \left(y\right),\alpha \left(z\right))\rho (x,u)-\rho (\alpha \left(z\right),\alpha \left(x\right))\rho (y,x)-\rho (\alpha \left(x\right),\alpha \left(y\right))\rho (z,u)\hfill \\ \hfill & =[f{([x,y,z],f\left(\alpha \left(x\right)\right),{\alpha }^{\prime }\left(v\right)]}^{\prime }-{[f\left(\alpha \left(y\right)\right),f\left(\alpha \left(z\right)\right),\rho (x,u)v]}^{\prime }\hfill \\ & -{[f\left(\alpha \left(z\right)\right),f\left(\alpha \left(x\right)\right),\rho (y,u)v]}^{\prime }-{[f\left(\alpha \left(x\right)\right),f\left(\alpha \left(y\right)\right),\rho (z,u)v]}^{\prime }\hfill \\ \hfill & =\left[{[f\left(x\right),f\left(y\right),f\left(z\right)]}^{\prime }{\alpha }^{\prime }f\left(u\right)\right),{\alpha }^{\prime }\left(v\right){{]}^{\prime }-\left[{\alpha }^{\prime }f\left(y\right)\right),{\alpha }^{\prime }\left(f\left(z\right)\right),{[f\left(x\right),f\left(u\right),v]}^{\prime }]}^{\prime }\hfill \\ & -\left[{\alpha }^{\prime }f\left(z\right)\right),{\alpha }^{\prime }\left(f\left(x\right)\right),{[f\left(y\right),f\left(u\right),v]}^{\prime }{{]}^{\prime }-\left[{\alpha }^{\prime }f\left(x\right)\right),{\alpha }^{\prime }\left(f\left(y\right)\right),{[f\left(z\right),f\left(u\right),v]}^{\prime }]}^{\prime }\hfill \\ \hfill & =0(by3-HomJacobiidentity),\hfill \\ \hfill & \rho (\alpha \left(x\right),\alpha \left(y\right))\rho (z,u)-\rho (\alpha \left(z\right),\alpha \left(u\right))\rho (x,y)-\rho ([x,y,z],\alpha \left(u\right)){\alpha }^{\prime }\left(v\right)-\rho (\alpha \left(z\right),[x,y,u]){\alpha }^{\prime }\left(v\right)\hfill \\ \hfill & ={\left[f\right(\alpha \left(x\right),f(\alpha \left(y\right),\rho (z,u)v]}^{\prime }-\left[f\right(\alpha \left(z\right),f{(\alpha \left(u\right),\rho (x,y)v]}^{\prime }\hfill \\ \hfill & -[f{([x,y,z],f\left(\alpha \left(u\right)\right),{\alpha }^{\prime }\left(v\right)]}^{\prime }-{[f\left(\alpha \left(z\right)\right),f\left([x,y,u]\right),{\alpha }^{\prime }\left(v\right)]}^{\prime }\hfill \\ \hfill & ={[{\alpha }^{\prime }\left(f\left(x\right)\right),{\alpha }^{\prime }\left(f\left(y\right)\right),{[f\left(z\right),f\left(u\right),v]}^{\prime }]}^{\prime }-{[{\alpha }^{\prime }\left(f\left(u\right)\right),{[f\left(x\right),f\left(y\right),v]}^{\prime }]}^{\prime }\hfill \\ \hfill & -{[{[f\left(x\right),f\left(y\right),f\left(z\right)]}^{\prime },{\alpha }^{\prime }\left(f\left(u\right)\right),{\alpha }^{\prime }\left(v\right)]}^{\prime }-{[{\alpha }^{\prime }\left(f\left(z\right)\right),{[f\left(x\right),f\left(y\right),f\left(u\right)]}^{\prime },{\alpha }^{\prime }\left(v\right)]}^{\prime }\hfill \\ \hfill & =0(by3-HomJacobiidentity).\hfill \end{array}$$

This finishes the proof. □

Proposition 2.

Let $(L,[·,·,·],\alpha )$ and $(L^{\prime },{[·,·,·]}^{\prime },{\alpha }^{\prime })$ be two 3-Hom–Lie algebras. Suppose that there are two skew-symmetric linear maps $\rho :L\otimes L\to gl\left(L^{\prime }\right)$ and $\mu :L^{\prime }\otimes L^{\prime }\to gl\left(L\right)$ which are representations of L and $L^{\prime }$ respectively, satisfying the following equations:

$$\begin{array}{ccc}& & \mu ({\alpha }^{\prime }\left(a_4\right),{\alpha }^{\prime }\left(a_5\right))[x_1,x_2,x_3]-[\mu (a_4,a_5)x_1,\alpha \left(x_2\right),\alpha \left(x_3\right)]\hfill \\ & & -[\alpha \left(x_1\right),\mu (a_4,a_5)x_2,\alpha \left(x_3\right)]-[\alpha \left(x_1\right),\alpha \left(x_2\right),\mu (a_4,a_5)x_3]=0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & \mu (\rho (x_1,x_4)a_5,{\alpha }^{\prime }\left(a_3\right))\alpha \left(x_2\right)-\mu (\rho (x_2,x_4)a_5,{\alpha }^{\prime }\left(a_3\right))\alpha \left(x_1\right)\hfill \\ & & -\mu (\rho (x_1,x_2)a_3,{\alpha }^{\prime }\left(a_3\right))\alpha \left(x_4\right)+[\alpha \left(x_1\right),\alpha \left(x_2\right),\mu (a_3,a_5)x_4]=0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & [\mu (a_2,a_3)x_1,\alpha \left(x_4\right),\alpha \left(x_5\right)]-\mu ({\alpha }^{\prime }\left(a_2\right),{\alpha }^{\prime }\left(a_3\right))[x_1,x_4,x_5]\hfill \\ & & -\mu (\rho (x_4,x_5)a_2,{\alpha }^{\prime }\left(a_3\right))\alpha \left(x_1\right)-\mu ({\alpha }^{\prime }\left(a_2\right),\rho (x_4,x_5)a_3)\alpha \left(x_1\right)=0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & \rho (\alpha \left(x_4\right),\alpha \left(x_5\right)){[a_1,a_2,a_3]}^{\prime }-{[\rho (x_4,x_5)a_1,{\alpha }^{\prime }\left(a_2\right),{\alpha }^{\prime }\left(a_3\right)]}^{\prime }\hfill \\ & & -{[{\alpha }^{\prime }\left(a_1\right),\rho (x_4,x_5)a_2,{\alpha }^{\prime }\left(a_3\right)]}^{\prime }-{[{\alpha }^{\prime }\left(a_1\right),{\alpha }^{\prime }\left(a_2\right),\rho (x_4,x_5)a_3]}^{\prime }=0,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \rho (\mu (a_1,a_4)x_5,\alpha \left(x_3\right)){\alpha }^{\prime }\left(a_2\right)-\rho (\mu (a_2,a_4)x_5,\alpha \left(x_3\right)){\alpha }^{\prime }\left(a_1\right)\hfill \\ & & -\rho (\mu (a_1,a_2)x_3,\alpha \left(x_5\right)){\alpha }^{\prime }\left(a_4\right)+{[{\alpha }^{\prime }\left(a_1\right),{\alpha }^{\prime }\left(a_2\right),\rho (x_3,x_5)a_4]}^{\prime }=0,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & [\rho (x_2,x_3)a_1,{\alpha }^{\prime }\left(a_4\right),{\alpha }^{\prime }\left(a_5\right)]-\rho (\alpha \left(x_2\right),\alpha \left(x_3\right)){[a_1,a_4,a_5]}^{\prime }\hfill \\ & & -\rho (\mu (a_4,a_5)x_2,\alpha \left(x_3\right)){\alpha }^{\prime }\left(a_1\right)-\rho (\alpha \left(x_2\right),\mu (a_4,a_5)x_3){\alpha }^{\prime }\left(a_1\right)=0,\hfill \end{array}$$

(6)

for any $x_i\in L$ and $a_i\in L^{\prime },1\le i\le 5$. Then, there is a 3-Hom–Lie algebra structure on $L\oplus L^{\prime }$ defined by

$$\begin{array}{ccc}& & (\alpha \oplus {\alpha }^{\prime })(x_1+a_1)=\alpha \left(x_1\right)+{\alpha }^{\prime }\left(a_1\right),\hfill \\ & & {[x_1+a_1,x_2+a_2,x_3+a_3]}_{L\oplus L^{\prime }}=[x_1,x_2,x_3]+\rho (x_1,x_2)a_3+\rho (x_3,x_1)a_2+\rho (x_2,x_3)a_1\hfill \\ & & +{[a_1,a_2,a_3]}^{\prime }+\mu (a_1,a_2)x_3+\mu (a_3,a_1)x_2+\mu (a_2,a_3)x_1.\hfill \end{array}$$

Moreover, $(L,L^{\prime },[·,·,·],{[·,·,·]}^{\prime },\rho ,\mu ,\alpha ,{\alpha }^{\prime })$ satisfying the above conditions is called a matched pair of 3-Hom–Lie algebras.

Proof. 

Straightforward. □

Definition 4.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra. A bilinear form $\langle ·,·\rangle$ on L is called invariant if it satisfies

$$\begin{array}{c}\hfill \langle [x,y,z],\alpha \left(u\right)\rangle +\langle [x,y,u],\alpha \left(z\right)\rangle =0,\forall x,y,z,u\in L.\end{array}$$

A 3-Hom–Lie algebra L is called pseudo-metric if there is a non-degenerate symmetric invariant bilinear form on L.

Definition 5.

A Manin triple of 3-Hom–Lie algebras consists of a pseudo-metric 3-Hom–Lie algebra $(L,[·,·,·],\langle ·,·\rangle ,\alpha )$ and 3-Hom–Lie algebras $L_1$ and $L_2$ such that

(1) $L_1,L_2$ are isotropic 3-Hom–Lie subalgebras of L;

(2) $L=L_1\oplus L_2$ as the direct sum of vector spaces;

(3) For all $x_1,y_1\in L_1$ and $x_2,y_2\in L_2$, we have pr${}_1[x_1,y_1,x_2]=0$ and pr${}_2[x_2,y_2,x_1]=0$, where pr${}_1$ and pr${}_2$ denote the projections from $L_1\oplus L_2$ to $L_1,L_2$, respectively.

Given a representation $(V,A,\rho )$, define ${\rho }^{*}:L\wedge L\to gl\left(V^{*}\right)$ by

$$\begin{array}{c}\hfill \langle {\rho }^{*}(x,y)\left(f\right),v\rangle =-\langle f,\rho (x,y)\left(v\right)\rangle ,\forall x,y\in L,f\in V^{*},v\in V.\end{array}$$

As observed in [4], $(V^{*},A^{*},{\rho }^{*})$ is not a representation of L on $V^{*}$ with respect to $A^{*}$ in general. It is easy to obtain the following result by Proposition 2.

Proposition 3.

Let $(V,A,\rho )$ be a representation of a 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$. Then $(V^{*},A^{*},{\rho }^{*})$ is a representation of the 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ if the following conditions hold:

$$\begin{array}{ccc}\hfill \left(i\right)& & A\circ \rho \left(\alpha \right(u),\alpha (v\left)\right)=\rho (u,v)\circ A,\hfill \\ \hfill \left(ii\right)& & A\circ \rho \left(\right[x,y,z],\alpha (u\left)\right)=\rho (y,z)\rho \left(\alpha \right(x),\alpha (u\left)\right)+\rho (z,x)\rho \left(\alpha \right(y),\alpha (u\left)\right)\hfill \\ & & +\rho (x,y)\rho \left(\alpha \right(z),\alpha (u\left)\right),\hfill \\ \hfill \left(iii\right)& & \rho (x,y)\rho \left(\alpha \right(z),\alpha (u\left)\right)=\rho (z,u)\rho \left(\alpha \right(x),\alpha (y\left)\right)+A\circ \rho \left(\right[x,y,z],\alpha (u\left)\right)\hfill \\ & & +A\circ \rho \left(\alpha \right(z),[x,y,u\left]\right),\hfill \end{array}$$

for all $x,y,z,u,v\in L$.

A representation $(V,A,\rho )$ is called admissible if $(V^{*},A^{*},{\rho }^{*})$ is also a representation, i.e., conditions (i),(ii) and (iii) in Proposition 3 are satisfied. When we focus on the adjoint representation, we have the following corollary:

Corollary 1.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra. The adjoint representation $(L,\alpha ,ad)$ is admissible if the following three equations hold:

$$\begin{array}{c}[(id-{\alpha }^2)\left(u\right),(id-{\alpha }^2)\left(v\right),\alpha \left(w\right)]=0,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & [[\alpha \left(x\right),\alpha \left(y\right),\alpha \left(z\right)],{\alpha }^2\left(u\right),\alpha \left(w\right)]=[y,z,[\alpha \left(x\right),\alpha \left(u\right),w]]+[z,x,[\alpha \left(y\right),\alpha \left(u\right),w]]\hfill \\ & & +[x,y,[\alpha \left(z\right),\alpha (u,w]],\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & [x,y,[\alpha \left(z\right),\alpha \left(u\right),w]]=[z,u,[\alpha \left(x\right),\alpha \left(y\right)]]+[[\alpha \left(x\right),\alpha \left(y\right),\alpha \left(z\right)],{\alpha }^2\left(u\right),\alpha \left(w\right)]\hfill \\ & & +[{\alpha }^2\left(z\right),[\alpha \left(x\right),\alpha \left(y\right),\alpha \left(u\right)],\alpha \left(w\right)],\hfill \end{array}$$

(9)

for all $x,y,z,u,v,w\in L.$

Definition 6.

A 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ is called admissible if its adjoint representation is admissible, i.e., Equations (7)–(9) are satisfied.

In the following, we concentrate on the case that $L^{\prime }$ is $L^{*}$, the dual space of L, and ${\alpha }^{\prime }$ = ${\alpha }^{*}$, $\rho =a{d}^{*},\mu =a{\partial }^{*}$, where $a{\partial }^{*}$ is the dual map of $a\partial$.

Let $(L,[·,·,·],\alpha )$ be an admissible 3-Hom–Lie algebra. Then, we have a natural nondegenerate symmetric bilinear form $\langle ·,·\rangle$ on $L\oplus L^{*}$ given by

$$\begin{array}{c}\hfill \langle x+\xi ,y+\eta \rangle =\langle x,\eta \rangle +\langle y,\xi \rangle ,\forall x,y\in L,\xi ,\eta \in L^{*}.\end{array}$$

(10)

There is also a twist map $\alpha \oplus {\alpha }^{*}$ and a bracket operation ${[·,·,·]}_{L\oplus L^{*}}$ on $L\oplus L^{*}$ given by

$$\begin{array}{ccc}& & (\alpha \oplus {\alpha }^{*})(x+\xi )=\alpha \left(x\right)+{\alpha }^{*}\left(\xi \right),\hfill \\ & & {[x+\xi ,y+\eta ,z+\gamma ]}_{L\oplus L^{*}}=[x,y,z]+a{d}_{x,y}^{*}\gamma +a{d}_{y,z}^{*}\xi +a{d}_{z,x}^{*}\eta \hfill \\ & & +a{\partial }_{\xi ,\eta }^{*}z+a{\partial }_{\eta ,\gamma }^{*}x+a{\partial }_{\gamma ,\xi }^{*}y+{[\xi ,\eta ,\gamma ]}^{*}.\hfill \end{array}$$

(11)

Note that the bracket operation ${[·,·,·]}_{L\oplus L^{*}}$ is naturally invariant with respect to the symmetric bilinear form $\langle ·,·\rangle$ and satisfies the condition (10). Assume that $(L\oplus L^{*},{[·,·,·]}_{L\oplus L^{*}},\alpha \oplus {\alpha }^{*})$ is a 3-Hom–Lie algebra, then obviously L and $L^{*}$ are isotropic subalgebras. Consequently, $((L\oplus L^{*},\langle ·,·\rangle ,\alpha \oplus {\alpha }^{*}),L,L^{*})$ is a Manin triple, which is called the standard Manin triple of 3-Hom–Lie algebras.

Next we will show a close relation between the matched pair and the Manin triple of admissible 3-Hom–Lie algebras.

Lemma 2.

Let $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ be two admissible 3-Hom–Lie algebras. If Equations (1)–(3) hold. Then, $(L,L^{*},a{d}^{*},a{\partial }^{*},\alpha ,{\alpha }^{*})$ is a matched pair.

Proof. 

For any $x_1,x_2,x_4\in L$ and $a_3,a_5,a_6\in L^{*}$, we have

$$\begin{array}{ccc}& & \langle -a{\partial }_{a{d}_{x_1,x_2}^{*}{a}_3,a_5}^{*}\alpha \left(x_4\right)+a{\partial }_{a{d}_{x_1,x_4}^{*}{a}_5,a_3}^{*}\alpha \left(x_2\right)-a{\partial }_{a{d}_{x_2,x_4}^{*}{a}_5,a_3}^{*}\alpha \left(x_1\right)\hfill \\ & & +[\alpha \left(x_1\right),\alpha \left(x_2\right),a{\partial }_{a_3,a_5}^{*}{x}_4],a_6\rangle \hfill \\ & =& \langle {[a{d}_{x_1,x_2}^{*}{a}_3,a_5,a_6]}^{*},\alpha \left(x_4\right)>-<{[a{d}_{x_1,x_4}^{*}{a}_5,a_3,a_6]}^{*},\alpha \left(x_2\right)\rangle \hfill \\ & & +\langle {[a{d}_{x_2,x_4}^{*}{a}_5,a_3,a_6]}^{*}-a{d}_{\alpha \left(x_2\right),a{\partial }_{a_3,a_5}^{*}{x}_4}^{*}{a}_6,\alpha \left(x_1\right)\rangle \hfill \\ & =& \langle x_1,a{d}_{x_2,a{\partial }_{a_5,a_6}^{*}{x}_4}^{*}{\alpha }^{*}\left(a_3\right)+a{d}_{x_4,a{\partial }_{a_3,a_6}^{*}{x}_2}^{*}{\alpha }^{*}\left(a_5\right)\hfill \\ & & +a{d}_{x_2,x_4}^{*}{a}_5,{\alpha }^{*}\left(a_3\right),{\alpha }^{*}\left(a_6\right){]}^{*}-a{d}_{x_2,a{\partial }_{a_3,a_5}^{*}{x}_4}^{*}{\alpha }^{*}\left(a_6\right)\rangle ,\hfill \end{array}$$

which implies the equivalence between Equations (2) and (5). The proofs of Equation (1) ⟺ Equation (4), Equation (3) ⟺ Equation (6) are similar. □

Proposition 4.

Let $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ be two admissible 3-Hom–Lie algebras. Then $(L\oplus L^{*},\langle ·,·\rangle ,\alpha \oplus {\alpha }^{*},L,L^{*})$ under the nondegenerate symmetric bilinear form (10) and the bracket operation (11) is a standard Manin triple if and only if $(L,L^{*},a{d}^{*},a{\partial }^{*},\alpha ,{\alpha }^{*})$ is a matched pair.

Proof. 

Straightforward from Lemma 2. □

Theorem 1.

Let $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ be two admissible 3-Hom–Lie algebras, $\Delta :L\to L\otimes L\otimes L$ a linear map. Suppose that ${\Delta }^{*}:L^{*}\otimes L^{*}\otimes L^{*}\to L^{*}$ defines a 3-Hom–Lie algebra structure ${[·,·,·]}^{*}$ on $L^{*}$. Then, $(L,L^{*},a{d}^{*},a{\partial }^{*},\alpha ,{\alpha }^{*})$ is a matched pair if and only if the following equations are satisfied:

$$\begin{array}{ccc}& & \Delta \left([x,y,z]\right)=(\alpha \otimes \alpha \otimes a{d}_{y,z})\Delta \left(x\right)+(\alpha \otimes \alpha \otimes a{d}_{z,x})\Delta \left(y\right)+(\alpha \otimes \alpha \otimes a{d}_{x,y})\Delta \left(z\right)\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & \Delta \left([x,y,z]\right)=(\alpha \otimes \alpha \otimes a{d}_{y,z})\Delta \left(x\right)+(\alpha \otimes a{d}_{y,z}\otimes \alpha )\Delta \left(x\right)+(a{d}_{y,z}\otimes \alpha \otimes \alpha )\Delta \left(x\right)\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & (a{d}_{x,y}\otimes \alpha \otimes \alpha +\alpha \otimes \alpha \otimes a{d}_{x,y})\Delta \left(z\right)=(\alpha \otimes a{d}_{z,x}\otimes \alpha )\Delta \left(y\right)+(\alpha \otimes a{d}_{y,z}\otimes \alpha )\Delta \left(x\right)\hfill \end{array}$$

(14)

for any $x,y,z\in L$.

Proof. 

Let $\{e_1,e_2,...,e_n\}$ be a basis of L and $\{e_1^{*},e_2^{*},...,e_n^{*}\}$ the dual basis. Suppose

$$\begin{array}{c}\hfill [e_i,e_j,e_k]=\sum _{l=1}^n{c}_{ijk}^l{e}_l,{[e_i^{*},e_j^{*},e_k^{*}]}^{*}=\sum _{l=1}^n{d}_{ijk}^l{e}_l^{*}.\end{array}$$

Let

$$\begin{array}{ccc}& & \alpha \left(e_i\right)=\sum _s{f}_s{e}_s,\alpha \left(e_j\right)=\sum _n{g}_n{e}_n,\alpha \left(e_k\right)=\sum _n{h}_m{e}_m,\hfill \\ & & {\alpha }^{*}\left(e_{\xi }^{*}\right)=\sum _s{f}_s^{*}{e}_s^{*},{\alpha }^{*}\left(e_{\eta }^{*}\right)=\sum _n{g}_n^{*}{e}_n^{*},{\alpha }^{*}\left(e_k^{*}\right)=\sum _m{h}_m^{*}{e}_m^{*}.\hfill \end{array}$$

Then we have

$$\begin{array}{c}\hfill a{d}_{e_i,e_j}^{*}{e}_k^{*}=-\sum _{l=1}^n{c}_{ijk}^l{e}_l^{*},a{\partial }_{e_i^{*},e_j^{*}}^{*}{e}_k=-\sum _{l=1}^n{d}_{ijk}^l{e}_l,\Delta \left(e_k\right)=\sum _{i,j,l=1}^n{d}_{ijl}^k{e}_i\otimes e_j\otimes e_k.\end{array}$$

By Equation (1), we have

$$\begin{array}{ccc}& & a{\partial }_{{\alpha }^{*}\left(e_{\xi }^{*}\right),{\alpha }^{*}\left(e_{\eta }^{*}\right)}^{*}[e_i,e_j,e_k]-[a{\partial }_{e_{\xi }^{*},e_{\eta }^{*}}^{*}{e}_i,\alpha \left(e_j\right),\alpha \left(e_k\right)]\hfill \\ & & -[\alpha \left(e_i\right),a{\partial }_{e_{\xi }^{*},e_{\eta }^{*}}^{*}{e}_j,\alpha \left(e_k\right)]-[\alpha \left(e_i\right),\alpha \left(e_j\right),a{\partial }_{e_{\xi }^{*},e_{\eta }^{*}}^{*}{e}_k]=0.\hfill \end{array}$$

It follows that

$$\begin{array}{c}\hfill \sum _{l=1}^n(-f_s^{*}{g}_n^{*}{d}_{snm}^l{c}_{ijk}^l+g_n{h}_m{d}_{\xi \eta l}^i{c}_{lnm}^m+f_s{h}_m{d}_{\xi \eta l}^j{c}_{slm}^m+f_s{g}_n{d}_{\xi \eta l}^k{c}_{snl}^m)=0,\end{array}$$

as the coefficient of $e_m$. On the other hand, the left hand side of the above equation is also the coefficient of $e_{\xi }\otimes e_{\eta }\otimes e_m$ in Equation (12). Thus, we deduce that Equation (1) is equivalent to Equation (12). The proofs of the other case are similar. □

Definition 7.

Let $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ be two admissible 3-Hom–Lie algebras, $\Delta :L\to L\otimes L\otimes L$ be a linear map. Suppose that ${\Delta }^{*}:L^{*}\otimes L^{*}\otimes L^{*}\to L^{*}$ defines a 3-Hom–Lie algebra structure ${[·,·,·]}^{*}$ on $L^{*}$. If Δ satisfies Equations (12)–(14), then we call $(L,L^{*},\alpha ,\Delta )$ a double construction 3-Hom–Lie bialgebra.

Example 5.

Consider the 4-dimensional 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ with respect to a basis $\{e_1,e_2,e_3,e_4\}$ given by

$$\begin{array}{c}\hfill [e_2,e_3,e_4]=e_1,\alpha \left(e_1\right)=-e_1,\alpha \left(e_2\right)=e_2,\alpha \left(e_3\right)=e_3,\alpha \left(e_4\right)=e_4.\end{array}$$

Define the skew-symmetric linear map $\Delta :L\to L\otimes L\otimes L$ satisfying Equation (12) is given as follows

$$\begin{array}{ccc}& & \Delta \left(e_1\right)=0,\Delta \left(e_2\right)=e_1\wedge e_2\wedge e_3+e_1\wedge e_2\wedge e_4+e_1\wedge e_3\wedge e_4,\hfill \\ & & \Delta \left(e_3\right)=e_1\wedge e_2\wedge e_3-e_1\wedge e_3\wedge e_4+e_1\wedge e_2\wedge e_4,\hfill \\ & & \Delta \left(e_4\right)=-e_1\wedge e_2\wedge e_4+e_1\wedge e_3\wedge e_4+e_1\wedge e_2\wedge e_3,\hfill \end{array}$$

then $(L,\Delta )$ is a double construction 3-Hom–Lie bialgebra.

Combining Lemma 2, Proposition 6, Theorem 1 and Definition 7, we have

Theorem 2.

Let $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ be two admissible 3-Hom–Lie algebras, $\Delta :L\to L\otimes L\otimes L$ be a linear map. Suppose that ${\Delta }^{*}:L^{*}\otimes L^{*}\otimes L^{*}\to L^{*}$ defines a 3-Hom–Lie algebra structure ${[·,·,·]}^{*}$ on $L^{*}$. Then, the following statements are equivalent:

(1) $(L,L^{*},\alpha ,\Delta )$ is a double construction 3-Hom–Lie bialgebra.

(2) $(L\oplus L^{*},\langle ·,·\rangle ,\alpha \oplus {\alpha }^{*})$ is a standard Manin triple of admissible 3-Hom–Lie algebras.

(3) $(L,L^{*},a{d}^{*},a{\partial }^{*},\alpha ,{\alpha }^{*})$ is a matched pair of admissible 3-Hom–Lie algebras.

Example 6.

Consider the 4-dimensional 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ in Example 5 and $\{e_1^{*},e_2^{*},e_3^{*},e_4^{*}\}$ is the dual basis. On the vector space $L\oplus L^{*}$ define a bilinear form $\langle ·,·\rangle$ by Equation (10), the non-zero product of 3-Hom–Lie algebra structure on $L\oplus L^{*}$ is given by

$$\begin{array}{ccc}& & [e_2,e_3,e_4]=e_1,\alpha \left(e_1\right)=-e_1,\alpha \left(e_2\right)=e_2,\alpha \left(e_3\right)=e_3,\alpha \left(e_4\right)=e_4,\hfill \\ & & {[e_1^{*},e_2^{*},e_3^{*}]}^{*}=e_2^{*}+e_3^{*}+e_4^{*},{[e_1^{*},e_2^{*},e_4^{*}]}^{*}=e_2^{*}+e_3^{*}-e_4^{*},\hfill \\ & & {[e_1^{*},e_3^{*},e_4^{*}]}^{*}=e_2^{*}-e_3^{*}-e_4^{*},[e_1,e_2,e_1^{*}]=-e_3^{*},[e_1,e_3,e_1^{*}]=e_2^{*},\hfill \\ & & [e_2,e_3,e_1^{*}]=-e_1^{*},[e_2,e_1^{*},e_2^{*}]=-e_3-e_4,[e_2,e_2^{*},e_3^{*}]=-e_1,\hfill \\ & & [e_2,e_1^{*},e_3^{*}]=e_2-e_4,[e_2,e_2^{*},e_4^{*}]=-e_1,[e_2,e_1^{*},e_4^{*}]=e_2+e_3,\hfill \\ & & [e_2,e_3^{*},e_4^{*}]=-e_1,[e_3,e_1^{*},e_2^{*}]=-e_3-e_4,[e_3,e_2^{*},e_3^{*}]=-e_1,\hfill \\ & & [e_3,e_2^{*},e_4^{*}]=-e_1,[e_3,e_1^{*},e_4^{*}]=e_2-e_3,[e_3,e_3^{*},e_4^{*}]=e_1,\hfill \\ & & [e_3,e_3^{*},e_4^{*}]=e_1,[e_4,e_1^{*},e_2^{*}]=-e_3+e_4,[e_4,e_2^{*},e_3^{*}]=-e_1,\hfill \\ & & [e_4,e_1^{*},e_3^{*}]=e_2-e_4,[e_3,e_1^{*},e_3^{*}]=e_2+e_4.\hfill \end{array}$$

They correspond to the double construction 3-Hom–Lie bialgebra $(L,\Delta )$ given in Example 5.

3. $\mathcal{O}$-Operators and 3-Hom–pre-Lie Algebras

In this section, we mainly study the $\mathcal{O}$-operator of a 3-Hom–Lie algebra and present a class of solutions of 3-Hom–Lie Yang–Baxter equations.

Definition 8.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $(V,A,\rho )$ a representation. A linear operator $T:V\to L$ is called an $\mathcal{O}$-operator associated to $(V,A,\rho )$ if T satisfies: for any $u,v,w\in L$,

$$\begin{array}{ccc}& & \alpha \circ T=T\circ A,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & [Tu,Tv,Tw]=T\left(\rho \right(Tu,Tv)w+\rho (Tv,Tw)u+\rho (Tw,Tu\left)v\right).\hfill \end{array}$$

(16)

Example 7.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra. An $\mathcal{O}$-operator of L associated to the adjoint representation $(L,ad,\alpha )$ is nothing but the Rota-Baxter operator of weight zero introduced in [17].

Definition 9.

A 3-Hom–pre-Lie algebra is a triple $(L,\{·,·,·\},\alpha )$ consisting of a vector space L, with a trilinear map $\{·,·,·\}:L\otimes L\otimes L\to L$ and an algebra morphism $\alpha :L\to L$ satisfying

$$\begin{array}{ccc}\hfill \{x,y,z\}& =& -\{y,x,z\},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill \left\{\alpha \right(x),\alpha (y),\{z,u,v\left\}\right\}& =& \{{[x,y,z]}_C,\alpha \left(u\right),\alpha \left(v\right)\}+\{\alpha \left(z\right),{[x,y,u]}_C,\alpha \left(v\right)\}\hfill \\ & & +\{\alpha \left(z\right),\alpha \left(u\right),{[x,y,v]}_C\},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill \left\{\right[x,y,z],\alpha (u),\alpha (v\left)\right\}& =& \{\alpha \left(x\right),\alpha \left(y\right),{[z,u,v]}_C\}+\{\alpha \left(y\right),\alpha \left(z\right),{[x,u,v]}_C\}\hfill \\ & & +\{\alpha \left(z\right),\alpha \left(x\right),{[y,u,v]}_C\},\hfill \end{array}$$

(19)

for any $x,y,z,u,v\in L$.

Proposition 5.

Let $(L,\{·,·,·\},\alpha )$ be a 3-Hom–pre-Lie algebra. Then, the induced 3-commutator

$$\begin{array}{c}\hfill {[x,y,z]}_C=\{x,y,z\}+\{y,z,x\}+\{z,x,y\},\end{array}$$

(20)

defines a 3-Hom–Lie algebra $(L^c,{\{·,·,·\}}_C,\alpha )$.

Proof. 

It is easy to check that ${[·,·,·]}_C$ is skew-symmetric. For any $x_1,x_2,x_3,x_4,x_5\in L$, we have

$$\begin{array}{ccc}& & {[\alpha \left(x_1\right),\alpha \left(x_2\right),{[x_3,x_4,x_5]}_C]}_C-{[{[x_1,x_2,x_3]}_C,\alpha \left(x_4\right),\alpha \left(x_5\right)]}_C-{[\alpha \left(x_3\right),{[x_1,x_2,x_4]}_C,\alpha \left(x_5\right)]}_C\hfill \\ & & -{[\alpha \left(x_3\right),\alpha \left(x_4\right),{[x_1,x_2,x_5]}_C,]}_C\hfill \\ & =& \{\alpha \left(x_1\right),\alpha \left(x_2\right),\{x_3,x_4,x_5\}\}+\{\alpha \left(x_1\right),\alpha \left(x_2\right),\{x_4,x_5,x_3\}\}+\{\alpha \left(x_1\right),\alpha \left(x_2\right),\{x_5,x_3,x_4\}\}\hfill \\ & & +\{\alpha \left(x_2\right),{[x_3,x_4,x_5]}_C,\alpha \left(x_1\right)\}+\{[x_3,x_4,x_5],\alpha \left(x_1\right),\alpha \left(x_2\right)\}\hfill \\ & & -\{\alpha \left(x_4\right),\alpha \left(x_5\right),\{x_1,x_2,x_3\}\}-\{\alpha \left(x_4\right),\alpha \left(x_5\right),\{x_2,x_3,x_1\}\}-\{\alpha \left(x_4\right),\alpha \left(x_5\right),\{x_3,x_1,x_2\}\}\hfill \\ & & -\{[x_1,x_2,x_3],\alpha \left(x_4\right),\alpha \left(x_5\right)\}-\{\alpha \left(x_5\right),{[x_1,x_2,x_3]}_C,\alpha \left(x_4\right)\}\hfill \\ & & -\{\alpha \left(x_5\right),\alpha \left(x_3\right),\{x_1,x_2,x_4\}\}-\{\alpha \left(x_5\right),\alpha \left(x_3\right),\{x_2,x_4,x_1\}\}-\{\alpha \left(x_5\right),\alpha \left(x_3\right),\{x_2,x_4,x_1\}\}\hfill \\ & & -\{[x_1,x_2,x_4],\alpha \left(x_5\right),\alpha \left(x_3\right)\}-\{\alpha \left(x_3\right),{[x_1,x_2,x_4]}_C,\alpha \left(x_5\right)\}\hfill \\ & & -\{\alpha \left(x_3\right),\alpha \left(x_4\right),\{x_1,x_2,x_5\}\}-\{\alpha \left(x_3\right),\alpha \left(x_4\right),\{x_2,x_5,x_1\}\}-\{\alpha \left(x_3\right),\alpha \left(x_4\right),\{x_5,x_1,x_2\}\}\hfill \\ & & -\{{[x_1,x_2,x_5]}_C,\alpha \left(x_3\right),\alpha \left(x_4\right)\}-\{\alpha \left(x_4\right),{[x_1,x_2,x_5]}_C,\alpha \left(x_3\right)\}\hfill \\ & =& 0.\hfill \end{array}$$

Thus the proof is finished. □

Definition 10.

Let $(L,\{·,·,·\},\alpha )$ be a 3-Hom–pre-Lie algebra. The 3-Hom–Lie algebra $(L^c,{[·,·,·]}_C,\alpha )$ is called the sub-adjacent 3-Hom–Lie algebra of $(L,\{·,·,·\},\alpha )$ and $(L,\{·,·,·\},\alpha )$ is called a compatible 3-Hom–pre-Lie algebra of the 3-Hom–Lie algebra $(L^c,{[·,·,·]}_C,\alpha )$.

Definition 11.

Let $(L,\{·,·,·\},\alpha )$ and $(L^{\prime },{\{·,·,·\}}^{\prime },{\alpha }^{\prime })$ be two 3-Hom–pre-Lie algebras. A morphism from $(L,\{·,·,·\},\alpha )$ to $(L^{\prime },{\{·,·,·\}}^{\prime },{\alpha }^{\prime })$ is a 3-pre-Lie algebra morphism $f:L\to L^{\prime }$ satisfying $f\circ \alpha ={\alpha }^{\prime }\circ f$.

Theorem 3.

Let $\mathcal{L}=(L,\{·,·,·\},\alpha )$ be a 3-Hom–pre-Lie algebra and ${\alpha }^{\prime }:\mathcal{L}\to \mathcal{L}$ be a 3-pre-Lie algebras morphism such that α and ${\alpha }^{\prime }$ commute. Define

$$\begin{array}{c}\hfill {\{·,·,·\}}_{{\alpha }^{\prime }}:L\times L\to L,{\{x,y,z\}}_{{\alpha }^{\prime }}={\alpha }^{\prime }\left(\{x,y,z\}\right),\forall x,y,z\in L.\end{array}$$

Then ${\mathcal{L}}_{\alpha }^{\prime }=(L_{\alpha }^{\prime }=L,{\{x,y,z\}}_{{\alpha }^{\prime }},{\alpha }^{\prime })$ is a 3-Hom–pre-Lie algebra, called ${\alpha }^{\prime }$-twist or Yau twist of $\mathcal{L}$. Moreover, assume that ${\mathcal{L}}^{\prime }=(L^{\prime },{\{·,·,·\}}^{\prime },\beta )$ is another 3-Hom–pre-Lie algebra, and ${\beta }^{\prime }:{\mathcal{L}}^{\prime }\to {\mathcal{L}}^{\prime }$ is a 3-Hom–pre-pre-Lie algebras morphism such that α and ${\alpha }^{\prime }$ commute. Let $f:\mathcal{L}\to {\mathcal{L}}^{\prime }$ be a 3-Hom–pre-Lie algebras morphism satisfying $f\circ {\alpha }^{\prime }={\beta }^{\prime }\circ f$. Then, $f:{\mathcal{L}}_{{\alpha }^{\prime }}\to {\mathcal{L}}_{{\beta }^{\prime }}^{\prime }$ is a 3-Hom–pre-Lie algebras morphism.

Proof. 

Let $x,y,z\in L$,

$$\begin{array}{cc}\hfill {\{x,y,z\}}_{{\alpha }^{\prime }}& ={\alpha }^{\prime }\left(\{x,y,z\}\right)\hfill \\ & =\{{\alpha }^{\prime }\left(x\right),{\alpha }^{\prime }\left(y\right),{\alpha }^{\prime }\left(z\right)\}\hfill \\ & =-\{{\alpha }^{\prime }\left(y\right),{\alpha }^{\prime }\left(x\right),{\alpha }^{\prime }\left(z\right)\}\hfill \\ & =-{\alpha }^{\prime }\left(\{y,x,z\}\right)\hfill \\ & =-{\{y,x,z\}}_{{\alpha }^{\prime }},\hfill \\ \hfill {\{\alpha {\alpha }^{\prime }\left(x\right),\alpha {\alpha }^{\prime }\left(y\right),{\{z,u,v\}}_{{\alpha }^{\prime }}\}}_{{\alpha }^{\prime }}& =\{\alpha {\alpha }^{\prime 2}\left(x\right),\alpha {\alpha }^{\prime 2}\left(y\right),{\alpha }^{\prime }\{{\alpha }^{\prime }\left(z\right),{\alpha }^{\prime }\left(u\right),{\alpha }^{\prime }\left(v\right)\}\}\hfill \\ \hfill & =\{\alpha {\alpha }^{\prime 2}\left(x\right),\alpha {\alpha }^{\prime 2}\left(y\right),\{{\alpha }^{\prime 2}\left(z\right),{\alpha }^{\prime 2}\left(u\right),{\alpha }^{\prime 2}\left(v\right)\}\}\hfill \\ \hfill & =\{{[{\alpha }^{\prime 2}\left(x\right),{\alpha }^{\prime 2}\left(y\right),{\alpha }^{\prime 2}\left(z\right)]}_C,\alpha {\alpha }^{\prime 2}\left(u\right),\alpha {\alpha }^{\prime 2}\left(v\right)\}\hfill \\ \hfill & +\{\alpha {\alpha }^{\prime 2}\left(z\right),[{\alpha }^{\prime 2}\left(x\right),{\alpha }^{\prime 2}\left(y\right),{\alpha }^{\prime 2}\left(u\right)],\alpha {\alpha }^{\prime 2}\left(v\right)\}\hfill \\ \hfill & +\{\alpha {\alpha }^{\prime 2}\left(z\right),\alpha {\alpha }^{\prime 2}\left(u\right),[{\alpha }^{\prime 2}\left(x\right),{\alpha }^{\prime 2}\left(y\right),{\alpha }^{\prime 2}\left(v\right)]\}\hfill \\ \hfill & ={\{{\left({[x,y,z]}_{{\alpha }^{\prime }}\right)}_C,\alpha {\alpha }^{\prime }\left(u\right),\alpha {\alpha }^{\prime }\left(v\right)\}}_{{\alpha }^{\prime }}\hfill \\ \hfill & +{\{\alpha {\alpha }^{\prime }\left(z\right),{\left({[x,y,u]}_{{\alpha }^{\prime }}\right)}_C,\alpha {\alpha }^{\prime }\left(v\right)\}}_{{\alpha }^{\prime }}\hfill \\ \hfill & +{\{\alpha {\alpha }^{\prime }\left(z\right),\alpha {\alpha }^{\prime }\left(u\right)\}}_{{\alpha }^{\prime }},{\left({[x,y,v]}_{{\alpha }^{\prime }}\right)}_C.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill \{[x,y,z],\alpha {\alpha }^{\prime }\left(u\right),\alpha {\alpha }^{\prime }\left(v\right)\}& =\{\alpha {\alpha }^{\prime }\left(x\right),\alpha {\alpha }^{\prime }\left(y\right),[z,u,v]\}\hfill \\ & +\{\alpha {\alpha }^{\prime }\left(y\right),\alpha {\alpha }^{\prime }\left(z\right),{[x,u,v]}_C\}\hfill \\ & +\{\alpha {\alpha }^{\prime }\left(z\right),\alpha {\alpha }^{\prime }\left(x\right),{[y,u,v]}_C\}.\hfill \end{array}$$

For the second assertion, we have

$$\begin{array}{cc}\hfill f\left({\{x,y,z\}}_{{\alpha }^{\prime }}\right)& =f\left(\{{\alpha }^{\prime }\left(x\right),{\alpha }^{\prime }\left(y\right),{\alpha }^{\prime }\left(z\right)\}\right)\hfill \\ \hfill & ={\{f\left({\alpha }^{\prime }\left(x\right)\right),f\left({\alpha }^{\prime }\left(y\right)\right),f\left({\alpha }^{\prime }\left(z\right)\right)\}}^{\prime })\hfill \\ \hfill & ={\{{\beta }^{\prime }\left(f\left(x\right)\right),{\beta }^{\prime }\left(f\left(y\right)\right),{\beta }^{\prime }\left(f\left(z\right)\right)\}}^{\prime }).\hfill \end{array}$$

□

Corollary 2.

If $\mathcal{A}=(A,\{·,·,·\},\alpha )$ is a 3-Hom–pre-Lie algebra, for any $n\in {\mathbb{N}}^{*}$, the following results hold:

1. 

The $n\mathrm{th}$ derived 3-Hom–pre-Lie algebra of type 1 of $\mathcal{A}$ is defined by ${\mathcal{A}}_1^n=(A,{\{·,·,·\}}^{\left(n\right)}={\alpha }^n\circ \{·,·,·\},{\alpha }^{n+1}).$

2. 

The $n\mathrm{th}$ derived 3-Hom–pre-Lie algebra of type 2 of A is defined by ${\mathcal{A}}_2^n=(A,{\{·,·,·\}}^{(2^n-1)}={\alpha }^{2^n-1}\circ \{·,·,·\},{\alpha }^{2^n}).$

Proof. 

Apply Theorem 3 with ${\alpha }^{\prime }={\alpha }^n$ and ${\alpha }^{\prime }={\alpha }^{2^n-1}$ respectively. □

Define the left multiplication $\mathcal{L}:{\wedge }^{2}L\to gl\left(L\right)$ by $\mathcal{L}(x,y)z=\{x,y,z\}$ for all $x,y,z\in L$. Then $(L,\mathcal{L},\alpha )$ is a representation of the 3-Hom–Lie algebra L. Similarly, we define the right multiplication $\mathcal{R}:{\wedge }^{2}L\to gl\left(L\right)$ by $\mathcal{R}(x,y)z=\{z,x,y\}$. If there is an admissible 3-Hom–pre-Lie algebra structure on its dual space $L^{*}$, we denote the left multiplication and right multiplication by ${\mathcal{L}}^{*}$ and ${\mathcal{R}}^{*}$ respectively.

Proposition 6.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $(V,A,\rho )$ a representation. Suppose that the linear map $T:V\to L$ is an $\mathcal{O}$-operator associated to $(V,A,\rho )$. Then, there exists a 3-Hom–pre-Lie algebra structure on V given by

$$\begin{array}{c}\hfill \{u,v,w\}=\rho (Tu,Tv)w,\forall u,v,w\in V.\end{array}$$

Proof. 

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

$$\begin{array}{c}\hfill \{u,v,w\}=\rho (Tu,Tv)w=-\rho (Tv,Tu)w=-\{v,u,w\}.\end{array}$$

Since ${[u,v,w]}_C=\{u,v,w\}+\{v,w,u\}+\{w,u,v\},$ we have

$$\begin{array}{c}\hfill {[u,v,w]}_C=\rho (Tu,Tv)w+\rho (Tv,Tw)u+\rho (Tw,Tu)v.\end{array}$$

Because T is an $\mathcal{O}$-operator, we have

$$\begin{array}{c}\hfill T{[u,v,w]}_C=[Tu,Tv,Tw].\end{array}$$

For any $v_1,v_2,v_3,v_4,v_5\in V$, we have

$$\begin{array}{ccc}& & \{\beta \left(v_1\right),\beta \left(v_2\right),\{v_3,v_4,v_5\}\}=\rho (T\circ A\left(v_1\right),T\circ A\left(v_2\right))\rho (T{v}_3,T{v}_4)v_5,\hfill \\ & & \{[v_1,v_2,v_3],\beta \left(v_4\right),\beta \left(v_5\right)\}\hfill \\ & =& \rho (T[v_1,v_2,v_3],TA\left(v_4\right))A\left(v_5\right)=\rho ([T{v}_1,T{v}_2,T{v}_3],T\circ A\left(v_4\right))A\left(v_5\right),\hfill \\ & & \{\beta \left(v_3\right),[v_1,v_2,v_4],\beta \left(v_5\right)\}\hfill \\ & =& \rho (T\circ A\left(v_3\right),T[v_1,v_2,v_4])A\left(v_5\right)=\rho (T\circ A\left(v_3\right),[T{v}_1,T{v}_2,T{v}_4])A\left(v_5\right),\hfill \\ & & \{\beta \left(v_3\right),\beta \left(v_4\right),\{v_1,v_2,v_5\}\}=\rho (T\circ A\left(v_3\right),T\circ A\left(v_4\right))\rho (T{v}_1,T{v}_2)v_5.\hfill \end{array}$$

Since $(V,A,\rho )$ is a representation, we can check that Equations (18) and (19) hold. This finishes the proof. □

Corollary 3.

Let $T:V\to L$ be an $\mathcal{O}$-operator on a 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ associated to the representation $(V,A,\rho )$. Then, T is a morphism from the 3-Hom–Lie algebra $(V,{[·,·]}_C,A)$ to $(A,[·,·],\alpha )$.

Proof. 

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

$$\begin{array}{cc}\hfill T\left({[u,v,w]}_C\right)& =T\left(\right\{u,v,w\}+\{w,u,v\}+\{v,w,u\left\}\right)\hfill \\ \hfill & =T\left(\rho \right(Tu,Tv)w+\rho (Tw,Tu)v+\rho (Tv,Tw\left)u\right)\hfill \\ \hfill & =[Tu,Tv,Tw],\hfill \end{array}$$

as desired. □

Example 8.

Let $(A,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $R:A⟶A$ a Rota-Baxter operator. Define a new operation on A by $\{x,y,z\}=\left[R\right(x),R(y),z].$ Then, $(A,\{·,·,·\},\alpha )$ is a 3-Hom–pre-Lie algebra and R is a homomorphism from the sub-adjacent 3-Hom–Lie algebra $(A,{[·,·,·]}_C,\alpha )$ to $(A,[·,·,·],\alpha )$.

Proposition 7.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra. Then there exists a compatible 3-Hom–pre-Lie algebra if and only if there exists an invertible $\mathcal{O}$-operator of L.

Proof. 

Let T be an invertible $\mathcal{O}$-operator of L associated to a representation $(V,A,\rho )$. Then there exists a 3-Hom–pre-Lie algebra structure on $(V,A,\rho )$ defined by

$$\begin{array}{c}\hfill \{u,v,w\}=\rho (Tu,Tv)\left(w\right),\forall u,v,w\in V.\end{array}$$

Moreover, there is an induced 3-Hom–pre-Lie algebra structure $\{·,·,·\}$ on $L=T\left(V\right)$ given by

$$\begin{array}{c}\hfill \{x,y,z\}=T\{T^{-1}x,T^{-1}y,T^{-1}z\}=T\rho (x,y)T^{-1}z.\end{array}$$

Since T is an $\mathcal{O}$-operator, we have

$$\begin{array}{ccc}\hfill [x,y,z]& =& T\rho (y,z)T^{-1}x+T\rho (z,x)T^{-1}y+T\rho (x,y)T^{-1}z\hfill \\ & =& \{x,y,z\}+\{y,z,x\}+\{z,x,y\}.\hfill \end{array}$$

Therefore, $(L,\{·,·,·\},\alpha )$ is a compatible 3-Hom–pre-Lie algebra.

Conversely, the identity map $id$ is an $\mathcal{O}$-operator of L. □

Definition 12

([17]). Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $r\in L\otimes L$. The equation

$$\begin{array}{c}\hfill {\left[[r,r,r]\right]}^{\alpha }=0\end{array}$$

is called the 3-Hom–Lie Yang–Baxter equation.

Let $(L,[·,·,·],\alpha )$ be an admissible 3-Hom–Lie algebra. For any $r\in L\otimes L$, the induced skew-symmetric linear map $r:L^{*}\to L$ is defined by

$$\begin{array}{c}\hfill \langle r\left(\xi \right),\eta \rangle =\langle r,\xi \wedge \eta \rangle .\end{array}$$

We denote the ternary operation ${\Delta }^{*}:L^{*}\otimes L^{*}\otimes L^{*}\to L^{*}$ by ${[·,·,·]}^{*}$. According to [17], for any $r={\sum }_i{x}_i\otimes y_i\in L\otimes L$ and $x\in L$, one can define

$$\begin{array}{ccc}& & {\Delta }_1\left(x\right)=\sum _{i,j}[x,x_i,x_j]\otimes \alpha \left(y_j\right)\otimes \alpha \left(y_i\right),\hfill \\ & & {\Delta }_2\left(x\right)=\sum _{i,j}\alpha \left(y_i\right)\otimes [x,x_i,x_j]\otimes \alpha \left(y_j\right),\hfill \\ & & {\Delta }_3\left(x\right)=\sum _{i,j}\alpha \left(y_j\right)\otimes \alpha \left(y_i\right)\otimes [x,x_i,x_j].\hfill \end{array}$$

Proposition 8.

Let $(L,[·,·,·],\alpha )$ be an admissible 3-Hom–Lie algebra and $r\in L\otimes L$ such that ${\alpha }^{{\otimes }^2}\left(r\right)=r$. Suppose that r is skew-symmetric and $\Delta ={\Delta }_1+{\Delta }_2+{\Delta }_3:L\to L\otimes L\otimes L$. Then

$$\begin{array}{c}\hfill {[\xi ,\eta ,\gamma ]}^{*}=a{d}_{r\left(\xi \right),r\left(\eta \right)}^{*}\gamma +a{d}_{r\left(\eta \right),r\left(\gamma \right)}^{*}\xi +a{d}_{r\left(\gamma \right),r\left(\xi \right)}^{*}\eta .\end{array}$$

(21)

Furthermore, we have

$$\begin{array}{c}\hfill [r\left(\xi \right),r\left(\eta \right),r\left(\gamma \right)]-r\left({[\xi ,\eta ,\gamma ]}^{*}\right)=\left[[r,r,r]\right](\xi ,\eta ,\gamma ),\end{array}$$

(22)

for any $\xi ,\eta ,\gamma \in L^{*}$.

Proof. 

Let $r={\sum }_i{x}_i\otimes y_i$, then for any $x,y\in L$ and $\xi ,\eta ,\gamma \in L^{*}$, we have

$$\begin{array}{ccc}\hfill \langle x,a{d}_{r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right)}^{*}\gamma \rangle & =& \langle -[r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right),x],\gamma \rangle \hfill \\ & =& -\langle r,{\alpha }^{*}\left(\eta \right)\otimes a{d}_{r\circ {\alpha }^{*}\left(\xi \right),x}^{*}\gamma \rangle \hfill \\ & =& \sum _i\langle y_i,{\alpha }^{*}\left(\eta \right)\rangle \langle r,{\alpha }^{*}\left(\xi \right)\otimes a{d}_{x,x_i}^{*}\gamma \rangle \hfill \\ & =& \sum _i\langle y_i,{\alpha }^{*}\left(\eta \right)\rangle \langle y_j,{\alpha }^{*}\left(\xi \right)\rangle \langle [x,x_i,x_j],\gamma \rangle \hfill \\ & =& \sum _{i,j}\langle \alpha \left(y_i\right),\eta \rangle \langle \alpha \left(y_j\right),\xi \rangle \langle [x,x_i,x_j],\gamma \rangle \hfill \\ & =& \langle \sum _{i,j}\alpha \left(y_j\right)\otimes \alpha \left(y_i\right)\otimes [x,x_i,x_j],\xi \otimes \eta \otimes \gamma \rangle \hfill \\ & =& \langle {\Delta }_3\left(x\right),\xi \otimes \eta \otimes \gamma \rangle .\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}\hfill \langle x,a{d}_{r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right)}^{*}\left(\xi \right)\rangle & =& \langle {\Delta }_1\left(x\right),\xi \otimes \eta \otimes \gamma \rangle ,\langle x,a{d}_{r\circ {\alpha }^{*}\left(\gamma \right),r\circ {\alpha }^{*}\left(\xi \right)}^{*}\left(\eta \right)\rangle \hfill \\ & =& \langle {\Delta }_1\left(x\right),\xi \otimes \eta \otimes \gamma \rangle .\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}& & \langle \Delta \left(x\right),\xi \otimes \eta \otimes \gamma \rangle \hfill \\ & =& \langle {\Delta }_1\left(x\right)+{\Delta }_2\left(x\right)+{\Delta }_3\left(x\right),\xi \otimes \eta \otimes \gamma \rangle \hfill \\ & =& \langle x,a{d}_{r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right)}^{*}\xi \rangle +\langle x,a{d}_{r\circ {\alpha }^{*}\left(\gamma \right),r\circ {\alpha }^{*}\left(\xi \right)}^{*}\eta \rangle +\langle x,a{d}_{r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right)}^{*}\gamma \rangle \hfill \\ & =& \langle x,{[\xi ,\eta ,\gamma ]}^{*}\rangle .\hfill \end{array}$$

So Equation (21) holds as required. For Equation (22) we take any $\kappa \in L^{*}$ and compute

$$\begin{array}{ccc}& & \left[\right[r,r,r\left]\right](\xi ,\eta ,\gamma ,\kappa )\hfill \\ & =& \sum _{i,j,k}([x_i,x_j,x_k]\otimes \alpha \left(y_i\right)\otimes \alpha \left(y_j\right)\otimes \alpha \left(y_k\right)(\xi ,\eta ,\gamma ,\kappa )\hfill \\ & & +\alpha \left(x_i\right)\otimes [y_i,x_j,x_k]\otimes \alpha \left(y_j\right)\otimes \alpha \left(y_k\right)(\xi ,\eta ,\gamma ,\kappa )\hfill \\ & & \alpha \left(x_i\right)\otimes \alpha \left(x_j\right)\otimes [y_i,y_j,x_k]\otimes \alpha \left(y_k\right)(\xi ,\eta ,\gamma ,\kappa )+\hfill \\ & & \alpha \left(x_i\right)\otimes \alpha \left(x_j\right)\otimes \alpha \left(x_k\right)\otimes [y_i,y_j,y_k](\xi ,\eta ,\gamma ,\kappa ))\hfill \\ & =& \sum _{i,j,k}\langle \xi ,[x_i,x_j,x_k]\rangle \langle \eta ,\alpha \left(y_i\right)\rangle \langle \gamma ,\alpha \left(y_j\right)\rangle \langle \kappa ,\alpha \left(y_k\right)\rangle +\langle \eta ,[y_i,x_j,x_k]\rangle \langle \xi ,\alpha \left(x_i\right)\rangle \hfill \\ & & \langle \gamma ,\alpha \left(y_j\right)\rangle \langle \kappa ,\alpha \left(y_k\right)\rangle \langle \gamma ,[y_i,y_j,x_k]\rangle \langle \xi ,\alpha \left(x_i\right)\rangle \langle \eta ,\alpha \left(x_j\right)\rangle \langle \kappa ,\alpha \left(y_k\right)\rangle +\hfill \end{array}$$

$$\begin{array}{ccc}& & \langle \kappa ,[y_i,y_j,y_k]\rangle \langle \xi ,\alpha \left(x_i\right)\rangle \langle \eta ,\alpha \left(x_j\right)\rangle \langle \gamma ,\alpha \left(x_k\right)\rangle \hfill \\ & =& -\langle \xi ,[r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right),r\circ {\alpha }^{*}\left(\kappa \right)\rangle -\langle \eta ,[r\circ {\alpha }^{*}\left(\gamma \right),r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\kappa \right)\rangle \hfill \\ & & -\langle \gamma ,[r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\kappa \right)\rangle +\langle \kappa ,[r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right)\rangle \hfill \\ & =& \langle [r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right)]-r\circ {\alpha }^{*}\left({[\xi ,\eta ,\gamma ]}^{*}\right),\kappa \rangle .\hfill \end{array}$$

So Equation (22) holds and this finishes the proof. □

Proposition 9.

Let $(L,[·,·,·],\alpha )$ be a regular 3-Hom–Lie algebra and $r\in L\otimes L$ such that ${\alpha }^{{\otimes }^2}\left(r\right)=r$. Suppose r is skew-symmetric and nondegenerate. Then, r is a solution of the 3-Hom–Lie Yang–Baxter equation if and only if the nondegenerate skew-symmetric bilinear form B on L defined by $B(x,y)=\langle r^{-1}\left(x\right),y\rangle$ satisfies

$$\begin{array}{c}\hfill B\left(\alpha \right[x,y,z],w)-B\left(\alpha \right[x,y,w],z)+B\left(\alpha \right[x,z,w],y)-B\left(\alpha \right[y,z,w],x)=0,\end{array}$$

for any $x,y,z,w\in L.$

Proof. 

For any $x,y,z,w\in L$, there exists $\xi ,\eta ,\gamma ,\kappa \in L^{*}$ such that $r\left(\xi \right)=x,r\left(\eta \right)=y,$$r\left(\gamma \right)=z,$$r\left(\kappa \right)=w$. If ${\left[[r,r,r]\right]}^{\alpha }=0$, we have

$$\begin{array}{ccc}& & B\left(\alpha \right[x,y,z],w)\hfill \\ & =& \langle \alpha \left[r\right(\xi ),r(\eta ),r(\gamma \left)\right],\kappa \rangle \hfill \\ & =& \langle r\circ {\alpha }^{*}(a{d}_{r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right)}^{*}\gamma +a{d}_{r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right)}^{*}\xi +a{d}_{r\circ {\alpha }^{*}\left(\gamma \right),r\circ {\alpha }^{*}\left(\xi \right)}^{*}\eta ),\kappa \rangle \hfill \\ & =& \langle a{d}_{r\circ {\alpha }^{*}\left(\xi \right),r\circ {\alpha }^{*}\left(\eta \right)}^{*}\gamma +a{d}_{r\circ {\alpha }^{*}\left(\eta \right),r\circ {\alpha }^{*}\left(\gamma \right)}^{*}\xi +a{d}_{r\circ {\alpha }^{*}\left(\gamma \right),r\circ {\alpha }^{*}\left(\xi \right)}^{*}\eta ,-\alpha \circ r\left(\kappa \right)\rangle \hfill \\ & =& \langle \gamma ,\alpha [x,y,w]\rangle -\langle -\xi ,\alpha [y,z,w]\rangle -\langle -\eta ,\alpha [z,x,w]\rangle \hfill \\ & =& B\left(\alpha \right[x,y,w],z)-B\left(\alpha \right[x,z,w],y)+B\left(\alpha \right[y,z,w],x).\hfill \end{array}$$

Thus the proof is finished. □

4. Symplectic Structures and Phase Spaces of 3-Hom–Lie Algebras

In this section, we introduce the notions of symplectic structures and phase spaces of 3-Hom–Lie algebras, and prove that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra.

Definition 13.

A symplectic structure on a regular 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ is a nondegenerate skew-symmetric bilinear form $\omega \in L^{*}\wedge L^{*}$ satisfying the following equality

$$\begin{array}{c}\hfill \omega \left(\right[x,y,z],\alpha (w\left)\right)-\omega \left(\right[y,z,w],\alpha (x\left)\right)+\omega \left(\right[z,w,x],\alpha (y\left)\right)-\omega \left(\right[w,x,y],\alpha (z\left)\right)=0,\end{array}$$

(23)

for any $x,y,z,w\in L$.

Definition 14

([20]). Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and $B:L\times L\to F$ be a non-degenerate symmetric bilinear form on L. If B satisfies

$$\begin{array}{c}\hfill B\left(\right[x,y,z],w)+B(z,[x,y,w\left]\right)=0,\forall x,y,z,w\in L.\end{array}$$

(24)

Then B is called a metric on 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$ and $(L,[·,·,·],\alpha ,B)$ is a metric 3-Hom–Lie algebra.

If there exists a metric B and a symplectic structure $\omega$ on the 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$, then $(L,[·,·,·],\alpha ,B,\omega )$ is called a metric symplectic 3-Hom–Lie algebra.

Let $(L,[·,·,·],\alpha ,B)$ be a metric 3-Hom–Lie algebra, we denote

$$\begin{array}{c}\hfill De{r}_B\left(L\right)=\{D\in Der\left(L\right)|B(Dx,y)+B(x,Dy)=0,\forall x,y\in L\}.\end{array}$$

Theorem 4.

Let $(L,[·,·,·],\alpha ,B)$ be a metric 3-Hom–Lie algebra. Then, there exists a symplectic structure on L if and only if there exists a skew-symmetric invertible derivation $D\in De{r}_B\left(L\right)$.

Proof. 

Suppose that $(L,[·,·,·],\alpha ,B)$ is a metric 3-Hom–Lie algebra, then for any $x,y\in L$, define $D:L\to L$ by

$$\begin{array}{c}\hfill B(Dx,y)=\omega \left(\alpha \right(x),y).\end{array}$$

(25)

It is clear that D is invertible. Next we will check that D is a skew-symmetric invertible derivation of $(L,[·,·,·],\alpha ,B)$. In fact, for any $x,y,z,w\in L$, we have

$$\begin{array}{ccc}& & B\left(\right[Dx,y,z],w)+B\left(\right[x,Dy,z],w)+B\left(\right[x,y,Dz],w)+B\left(D\right[x,y,z],w)\hfill \\ & =& -B\left(\right[y,z,w],Dx)+B\left(\right[x,z,w],Dy)-B\left(\right[x,y,w],Dz)+B\left(\right[x,y,z],Dw)\hfill \\ & =& \omega \left(\right[x,y,z],\alpha (w\left)\right)-\omega \left(\right[y,z,w],\alpha (x\left)\right)+\omega \left(\right[z,w,x],\alpha (y\left)\right)-\omega \left(\right[w,x,y],\alpha (z\left)\right)=0,\hfill \end{array}$$

that is, $D\in De{r}_B\left(L\right)$.

Conversely, assume that $D\in De{r}_B\left(L\right)$ is a skew-symmetric invertible derivation. Define $\omega$ by Equation (25), then there exists a symplectic structure on L satisfies Equation (23). □

Example 9.

Let $(L,[·,·,·],\alpha )$ be a 3-Hom–Lie algebra and

$$\begin{array}{c}\hfill F\left[t\right]=\{f\left(t\right)=\sum _{i=0}^m{a}_i{t}^i|a_i\in F,m\in N\}\end{array}$$

be the algebra of polynomials over F. We consider

$$\begin{array}{c}\hfill L_n=L\otimes (tF\left[t\right]/t^{n}F\left[t\right]),\end{array}$$

where $tF\left[t\right]/t^{n}F\left[t\right]$ is the quotient space of $tF\left[t\right]$ module $t^{n}F\left[t\right]$. Then, $L_n$ is a nilpotent 3-Hom–Lie algebra, with a linear map ${\alpha }^{\prime }:L_n\to L_n$ and the following multiplication:

$$\begin{array}{c}\hfill {\alpha }^{\prime }(x\otimes t^{\overline{p}})=\alpha \left(x\right)\otimes t^{\overline{p}},{[x\otimes t^{\overline{p}},y\otimes t^{\overline{q}},z\otimes t^{\overline{r}}]}^{\prime }=[x,y,z]\otimes t^{\overline{p+q+r}},\end{array}$$

for any $x,y,z\in L$ and $p,q,r\in N\backslash \left\{0\right\}.$ Define an endomorphism D of $L_n$ by

$$\begin{array}{c}\hfill D(x\otimes t^{\overline{p}})=p(x\otimes t^{\overline{p}}),\forall x\in L,p=1,...,n-1.\end{array}$$

Then D is an invertible derivation of the 3-Hom–Lie algebra $L_n$.

Let ${\tilde{L}}_n=L_n\oplus L_n^{*}$, where $L_n^{*}$ is the dual space of $L_n$. Then, $({\tilde{L}}_n,B)$ ia a metric 3-Hom–Lie algebra with the multiplication

$$\begin{array}{ccc}& & [x+f,y+g,z+h]=[x,y,z]+a{d}^{*}(y,z)f-a{d}^{*}(x,z)g+a{d}^{*}(x,y)h,\hfill \\ & & B(x+f,y+g)=f\left(y\right)+g\left(x\right),\hfill \end{array}$$

for any $x,y,z\in L_n$ and $f,g,h\in L_n^{*}$. And define linear maps $\widehat{D},\tilde{\alpha }:{\tilde{L}}_n\to {\tilde{L}}_n$ by

$$\begin{array}{c}\hfill \widehat{D}(x+f)=Dx+D^{*}f,\tilde{\alpha }(x+f)=\alpha \left(x\right)+f\circ \alpha ,\end{array}$$

where $D^{*}f=-fD$. Then, $\widehat{D}$ is invertible. Hence $({\tilde{L}}_n,\tilde{\alpha },B,\omega )$ is a metric symplectic 3-Hom–Lie algebra, where ω is defined as follows:

$$\begin{array}{c}\hfill \omega (\tilde{\alpha }(x+f),y+g)=B(\widehat{D}(x+f),y+g)=-f\left(Dy\right)+g\left(Dx\right).\end{array}$$

Proposition 10.

Let $(L,[·,·,·],\alpha ,\omega )$ be a symplectic 3-Hom–Lie algebra. Then, there exists a compatible 3-Hom–pre-Lie algebra structure $\{·,·,·\}$ on L given by

$$\begin{array}{c}\hfill \omega \left(\right\{x,y,z\},\alpha (w\left)\right)=-\omega \left(\alpha \right(z),[x,y,w\left]\right),\forall x,y,z,w\in L.\end{array}$$

(26)

Proof. 

For any $x,y\in L$, define the map $T:L^{*}\to L$ by $\langle T^{-1}x,y\rangle =\omega (x,y)$. By Equation (23), we obtain that T is an invertible $\mathcal{O}$-operator associated to the coadjoint representation $(L^{*},a{d}^{*},{\alpha }^{*})$, and there exists a compatible 3-Hom–pre-Lie algebra on L given by $\{x,y,z\}=T\left(a{d}_{x,y}^{*}{T}^{-1}z\right)$. For any $x,y,z,w\in L$, we have

$$\begin{array}{ccc}\hfill \omega \left(\right\{x,y,z\},\alpha (w\left)\right)& =& \omega (T\left(a{d}_{x,y}^{*}{T}^{-1}z\right),\alpha \left(w\right))=\langle a{d}_{x,y}^{*}{T}^{-1}z,\alpha \left(w\right)\rangle \hfill \\ & =& \langle T^{-1}\left(\alpha \left(z\right)\right),-[x,y,w]\rangle =-\omega (\alpha \left(z\right),[x,y,w]),\hfill \end{array}$$

as desired. The proof is finished. □

Let V be a vector space and $V^{*}$ its dual space. Then, there is a natural nondegenerate skew-symmetric bilinear form $\omega$ on $T^{*}V=V\oplus V^{*}$ given by:

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

(27)

Definition 15.

Let $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ be two admissible 3-Hom–Lie algebras. If there is a 3-Hom–Lie algebra structure $[·,·,·]$ on the direct sum vector space $T^{*}L=L\oplus L^{*}$ such that $(L\oplus L^{*},[·,·,·],\alpha \oplus {\alpha }^{*},\omega )$ is a symplectic 3-Hom–Lie algebra, where ω given by Equation (27), $(L,[·,·,·],\alpha )$ and $(L^{*},{[·,·,·]}^{*},{\alpha }^{*})$ are two 3-Hom–Lie subalgebras of $(L\oplus L^{*},[·,·,·],\alpha \oplus {\alpha }^{*})$. Then the symplectic 3-Hom–Lie algebra $(L\oplus L^{*},[·,·,·],\alpha \oplus {\alpha }^{*},\omega )$ is called a phase space of the 3-Hom–Lie algebra $(L,[·,·,·],\alpha )$.

Next, we will study the relation between 3-Hom–pre-Lie algebras and phase spaces of 3-Hom–Lie algebras.

Theorem 5.

A 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra.

Proof. 

⇐ Assume $(L,\{·,·,·\},\alpha )$ is a 3-Hom–pre-Lie algebra. By Proposition 5, the left multiplication $\mathcal{L}$ is a representation of the sub-adjacent 3-Lie algebra $L^C$ on L, ${\mathcal{L}}^{*}$ is a representation of the sub-adjacent 3-Lie algebra $L^c$ on $L^{*}$, then we have a 3-Hom–Lie algebra $(L^c\oplus L^{*},{[·,·,·]}_{L^{*}},\alpha \oplus {\alpha }^{*})$. For any $x_1,x_2,x_3,x_4\in L$ and $f_1,f_2,f_3,f_4\in L^{*}$, we have

$$\begin{array}{ccc}& & \omega ({[x_1+f_1,x_2+f_2,x_3+f_3]}_{L^{*}},\alpha \left(x_4\right)+{\alpha }^{*}\left(f_4\right))\hfill \\ & =& \omega ({[x_1,x_2,x_3]}_C+{\mathcal{L}}^{*}(x_1,x_3)f_3+{\mathcal{L}}^{*}(x_2,x_3)f_1+{\mathcal{L}}^{*}(x_3,x_1)f_2,\alpha \left(x_4\right)+{\alpha }^{*}\left(f_4\right))\hfill \\ & =& \langle {\mathcal{L}}^{*}(x_1,x_3)f_3+{\mathcal{L}}^{*}(x_2,x_3)f_1+{\mathcal{L}}^{*}(x_3,x_1)f_2,\alpha \left(x_4\right)\rangle -\langle {\alpha }^{*}\left(f_4\right),{[x_1,x_2,x_3]}_C\rangle \hfill \\ & =& -\langle {\alpha }^{*}\left(f_3\right),\{x_1,x_2,x_3\}\rangle -\langle {\alpha }^{*}\left(f_1\right),\{x_2,x_3,x_4\}\rangle -\langle {\alpha }^{*}\left(f_2\right),\{x_3,x_1,x_4\}\rangle \hfill \\ & & -\langle {\alpha }^{*}\left(f_4\right),\{x_1,x_2,x_3\}\rangle -\langle {\alpha }^{*}\left(f_4\right),\{x_2,x_3,x_1\}\rangle -\langle {\alpha }^{*}\left(f_4\right),\{x_3,x_1,x_2\}\rangle .\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}& & \omega ({[x_2+f_2,x_3+f_3,x_4+f_4]}_{L^{*}},\alpha \left(x_1\right)+{\alpha }^{*}\left(f_1\right))\hfill \\ & =& -\langle {\alpha }^{*}\left(f_4\right),\{x_2,x_3,x_1\}\rangle -\langle {\alpha }^{*}\left(f_2\right),\{x_3,x_4,x_1\}\rangle -\langle {\alpha }^{*}\left(f_3\right),\{x_4,x_2,x_1\}\rangle \hfill \\ & & -\langle {\alpha }^{*}\left(f_1\right),\{x_2,x_3,x_4\}\rangle -\langle {\alpha }^{*}\left(f_1\right),\{x_3,x_4,x_2\}\rangle -\langle {\alpha }^{*}\left(f_1\right),\{x_4,x_2,x_3\}\rangle ,\hfill \\ & & \omega ({[x_3+f_3,x_4+f_4,x_1+f_1,]}_{L^{*}},\alpha \left(x_2\right)+{\alpha }^{*}\left(f_2\right))\hfill \end{array}$$

$$\begin{array}{ccc}& =& -\langle {\alpha }^{*}\left(f_1\right),\{x_3,x_4,x_2\}\rangle -\langle {\alpha }^{*}\left(f_3\right),\{x_4,x_1,x_2\}\rangle -\langle {\alpha }^{*}\left(f_4\right),\{x_1,x_3,x_2\}\rangle \hfill \\ & & -\langle {\alpha }^{*}\left(f_2\right),\{x_3,x_4,x_1\}\rangle -\langle {\alpha }^{*}\left(f_2\right),\{x_4,x_1,x_3\}\rangle -\langle {\alpha }^{*}\left(f_2\right),\{x_1,x_3,x_4\}\rangle ,\hfill \\ & & \omega ({[x_4+f_4,x_1+f_1,x_2+f_3,]}_{L^{*}},\alpha \left(x_3\right)+{\alpha }^{*}\left(f_3\right))\hfill \\ & =& -\langle {\alpha }^{*}\left(f_2\right),\{x_4,x_1,x_3\}\rangle -\langle {\alpha }^{*}\left(f_4\right),\{x_1,x_2,x_3\}\rangle -\langle {\alpha }^{*}\left(f_1\right),\{x_2,x_4,x_3\}\rangle \hfill \\ & & -\langle {\alpha }^{*}\left(f_3\right),\{x_4,x_1,x_2\}\rangle -\langle {\alpha }^{*}\left(f_3\right),\{x_1,x_2,x_4\}\rangle -\langle {\alpha }^{*}\left(f_3\right),\{x_2,x_4,x_1\}\rangle .\hfill \end{array}$$

So $\omega$ is a symplectic structure on the semidirect product 3-Hom–Lie algebra $(L^c\oplus L^{*},$ ${[·,·,·]}_{L^{*}},\alpha \oplus {\alpha }^{*})$. Thus the symplectic 3-Hom–Lie algebra $(L^c\oplus L^{*},{[·,·,·]}_{L^{*}},\alpha \oplus {\alpha }^{*},\omega )$ is a phase space of the sub-adjacent 3-Hom–Lie algebra $(L^c,{[·,·,·]}_C,\alpha )$.

⇒ Clearly. □