Another approach to Hom-Lie bialgebras via Manin triples

TL;DR

This paper introduces a new framework for Hom-Lie bialgebras using Manin triples, enabling the study of coboundary structures and solutions to the Hom-Yang-Baxter equation without skew-symmetry constraints.

Contribution

It develops a novel approach to Hom-Lie bialgebras via Manin triples and explores their coboundary cases and solutions to the Hom-Yang-Baxter equation.

Findings

Defined Hom-Lie bialgebras with Manin triples and invariance conditions.

Constructed coboundary Hom-Lie bialgebras without skew-symmetry.

Derived solutions to the classical Hom-Yang-Baxter equation from $\\mathcal{O}$-operators and Hom-left-symmetric algebras.

Abstract

In this paper, we study Hom-Lie bialgebras by a new notion of the dual representation of a representation of a Hom-Lie algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we introduce 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. With this notion, coboundary Hom-Lie bialgebras can be studied without a skew-symmetric condition of $r\in\mathfrak{g}\otimes \mathfrak{g}$, naturally leading to the classical Hom-Yang-Baxter equation whose solutions are used to construct coboundary Hom-Lie bialgebras. In particular, they are used to obtain a canonical Hom-Lie bialgebra structure on the double space of a Hom-Lie bialgebra. We also derive solutions of the classical Hom-Yang-Baxter equation from $\mathcal{O}$-operators and Hom-left-symmetric algebras.

Equations

$$\mathfrak{B}(x+u^{\ast },y+v^{\ast }):=⟨x,v^{\ast }⟩+⟨u^{\ast },y⟩,\forall x,y\in \mathfrak{g},u^{\ast },v^{\ast }\in {\mathfrak{g}}^{\ast },$$

$$\mathfrak{B}([a,b{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }},c)=\mathfrak{B}(a,[b,c{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}),\forall a,b,c\in \mathfrak{g}\oplus {\mathfrak{g}}^{\ast }.$$

$$\Delta [x,y{]}_{\mathfrak{g}}={\mathrm{ad}}_{{\varphi }_{{}_{\mathfrak{g}}}(x)}\Delta (y)-{\mathrm{ad}}_{{\varphi }_{{}_{\mathfrak{g}}}(y)}\Delta (x),\forall x,y\in \mathfrak{g}.$$

$$[(\mathrm{Id}-{\varphi }_{{}_{\mathfrak{g}}}^2)(x),{\varphi }_{{}_{\mathfrak{g}}}(y){]}_{\mathfrak{g}}$$

$$[(\mathrm{Id}-{\varphi }_{{}_{\mathfrak{g}}}^2)(x),[{\varphi }_{{}_{\mathfrak{g}}}(y),z{]}_{\mathfrak{g}}{]}_{\mathfrak{g}}$$

$$⟨\Delta [x,y{]}_{\mathfrak{g}},{\varphi }_{{}_{\mathfrak{g}}}^{\ast }(\xi )\wedge \eta ⟩=⟨{\mathrm{ad}}_{{\varphi }_{{}_{\mathfrak{g}}}(x)}\Delta (y)-{\mathrm{ad}}_{{\varphi }_{{}_{\mathfrak{g}}}(y)}\Delta (x),{\varphi }_{{}_{\mathfrak{g}}}^{\ast }(\xi )\wedge \eta ⟩,\forall x,y\in \mathfrak{g},\xi ,\eta \in {\mathfrak{g}}^{\ast }.$$

$$\mathfrak{B}([a,b{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }},c)=\mathfrak{B}(a,[b,c{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}),\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(a),b)=\mathfrak{B}(a,{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(b)),\forall a,b,c\in \mathfrak{g}\oplus {\mathfrak{g}}^{\ast }.$$

$$\Delta [x,y{]}_{\mathfrak{g}}={\mathrm{ad}}_{{\varphi }_{{}_{\mathfrak{g}}}^{-1}(x)}\Delta (y)-{\mathrm{ad}}_{{\varphi }_{{}_{\mathfrak{g}}}^{-1}(y)}\Delta (x),\forall x,y\in \mathfrak{g}.$$

$$\mathfrak{B}([a,b{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }},{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(c))=\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(a),[b,c{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}),\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(a),{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(b))=\mathfrak{B}(a,b),$$

$$\mathfrak{B}([a,b{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }},c)=\mathfrak{B}(a,[{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(b),c{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}),\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(a),b)=\mathfrak{B}(a,{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(b)),$$

$$\mathfrak{B}([a,b{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }},{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(c))=\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(a),[b,c{]}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}),\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(a),b)=\mathfrak{B}(a,{\varphi }_{{}_{\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }}}(b)),$$

$$[{\varphi }_{{}_{\mathfrak{g}}}(x),[y,z{]}_{\mathfrak{g}}{]}_{\mathfrak{g}}+[{\varphi }_{{}_{\mathfrak{g}}}(y),[z,x{]}_{\mathfrak{g}}{]}_{\mathfrak{g}}+[{\varphi }_{{}_{\mathfrak{g}}}(z),[x,y{]}_{\mathfrak{g}}{]}_{\mathfrak{g}}=0.$$

$$f([x,y{]}_{\mathfrak{g}})$$

$$f{\varphi }_{{}_{\mathfrak{g}}}$$

$$\begin{array}{rl}[(x,u),(y,v){]}_{\mathfrak{g}{⋉}_{\rho }V}& :=([x,y{]}_{\mathfrak{g}},\rho (x)(v)-\rho (y)(u)),\\ ({\varphi }_{{}_{\mathfrak{g}}}\oplus \beta )(x,u)& :=({\varphi }_{{}_{\mathfrak{g}}}(x),\beta (u)).\end{array}$$

$$\phi \rho (x)={\rho }^{\prime }(x)\phi ,{\beta }^{\prime }\phi (x)=\phi \beta (x),\forall x\in \mathfrak{g}.$$

$$⟨{\varphi }^{\ast }(\xi ),v⟩=⟨\xi ,\varphi (v)⟩,\forall v\in V_1,\xi \in V_2^{\ast }.$$

$$⟨{\rho }^{\ast }(x)(\xi ),v⟩=-⟨\xi ,\rho (x)v⟩,\forall x\in \mathfrak{g},v\in V,\xi \in V^{\ast }.$$

$${\rho }^{\circ }:={\rho }^{\ast }{\varphi }_{{}_{\mathfrak{g}}}:\mathfrak{g}⟶\mathfrak{gl}(V^{\ast }).$$

$$⟨{\rho }^{\circ }(x)(\xi ),v⟩=-⟨\xi ,\rho ({\varphi }_{{}_{\mathfrak{g}}}(x))v⟩,\forall x\in \mathfrak{g},v\in V,\xi \in V^{\ast }.$$

$$({\mathrm{ad}}_x^{\ast }{{\varphi }_{{}_{\mathfrak{g}}}^{\ast }}^2)(a)={\mathrm{ad}}_x^{\ast }(a),\forall x\in \mathfrak{g},a\in {\mathfrak{g}}^{\ast }.$$

$$\begin{array}{r}⟨({\mathrm{ad}}_x^{\ast }{{\varphi }_{{}_{\mathfrak{g}}}^{\ast }}^2)(a),y⟩=-⟨a,{\varphi }_{{}_{\mathfrak{g}}}^2[x,y{]}_{\mathfrak{g}}⟩=-⟨a,[x,y{]}_{\mathfrak{g}}⟩=⟨{\mathrm{ad}}_x^{\ast }(a),y⟩.\end{array}$$

$$\mathfrak{B}([x,y{]}_{\mathfrak{g}},z)=\mathfrak{B}(x,[{\varphi }_{{}_{\mathfrak{g}}}(y),z{]}_{\mathfrak{g}}),\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}}}(x),y)=\mathfrak{B}(x,{\varphi }_{{}_{\mathfrak{g}}}(y)),\forall x,y,z\in \mathfrak{g}.$$

$$\mathfrak{B}([x,y{]}_{\mathfrak{g}},{\varphi }_{{}_{\mathfrak{g}}}(z))=\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}}}(x),[y,z{]}_{\mathfrak{g}}),\mathfrak{B}({\varphi }_{{}_{\mathfrak{g}}}(x),y)=\mathfrak{B}(x,{\varphi }_{{}_{\mathfrak{g}}}(y)).$$

$$\begin{array}{r}\mathfrak{B}(x,[y,z{]}_{\mathfrak{g}})=-\mathfrak{B}([z,y{]}_{\mathfrak{g}},x)=-\mathfrak{B}(z,[{\varphi }_{{}_{\mathfrak{g}}}(y),x{]}_{\mathfrak{g}})=\mathfrak{B}([x,{\varphi }_{{}_{\mathfrak{g}}}(y){]}_{\mathfrak{g}},z).\end{array}$$

$$\begin{array}{r}\mathfrak{B}([x,y{]}_{\mathfrak{g}},z)=\mathfrak{B}(x,[{\varphi }_{{}_{\mathfrak{g}}}(y),z{]}_{\mathfrak{g}})=\mathfrak{B}([x,{\varphi }_{{}_{\mathfrak{g}}}^2(y){]}_{\mathfrak{g}},z).\end{array}$$

$$\phi (x)(y):=⟨\phi (x),y⟩=B(x,y),\forall x,y\in \mathfrak{g}.$$

$$\phi (\mathrm{ad}(x)(y))(z)$$

$$\phi {\varphi }_{{}_{\mathfrak{g}}}(x)(y)$$

$$\psi \mathrm{ad}(x)={\mathrm{ad}}^{\circ }(x)\psi ,\psi {\varphi }_{{}_{\mathfrak{g}}}(x)={\varphi }_{{}_{\mathfrak{g}}}^{\ast }\psi (x),\forall x\in \mathfrak{g}.$$

$$B(x,y):=\psi (x)y=⟨\psi (x),y⟩,\forall x,y\in \mathfrak{g}.$$

$$\begin{array}{r}{\varphi }_{{}_d}:\mathfrak{g}\oplus {\mathfrak{g}}^{\prime }⟶\mathfrak{g}\oplus {\mathfrak{g}}^{\prime },{\varphi }_{{}_d}(x,x{}^{\prime }):=({\varphi }_{{}_{\mathfrak{g}}}(x),{\varphi }_{{}_{{\mathfrak{g}}^{\prime }}}(x{}^{\prime })),\end{array}$$

Full text

1. 1 Introduction
2. 2 Hom-Lie algebras, their representations and bilinear forms
3. 3 Matched pairs, Manin triples and Hom-Lie bialgebras
4. 4 Coboundary Hom-Lie bialgebras and the classical Hom-Yang-Baxter equation
5. 5 Operator forms of the classical Hom-Yang-Baxter equation

1. Introduction

The notion of Hom-Lie algebras was introduced in [8] in the context of deformations of the Witt algebra and the Virasoro algebra. In a Hom-Lie algebra $\mathfrak{g}$, the Jacobi identity defining a Lie algebra is twisted by a linear map $\phi_{{}_{\mathfrak{g}}}$, called the Hom-Jacobi identity, which recovers the Jacobi identity in the special case when $\phi_{{}_{\mathfrak{g}}}$ is the identity map. With this generalization of the Lie algebra, some $q$-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [8]. Due to their close relationship with discrete and deformed vector fields and differential calculus [8, 9, 10], Hom-Lie algebras have been studied in broad areas [1, 2, 3, 4, 5, 11, 12, 14, 15, 16, 17, 18, 20].

While the generalization of Hom-Lie algebra from Lie algebra allows wider potential applications, this generalization has often posed challenges in extending the well-established theory of Lie algebras.

This is the case of the bialgebra theory which is an essential aspect of Lie algebra due to its important applications. For example, the Lie bialgebra is the algebraic structure corresponding to a Poisson-Lie group and the classical structure of a quantized universal enveloping algebra [6, 7]. A Lie bialgebra consists of a Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}}$) where $[\cdot,\cdot]_{\mathfrak{g}}:\otimes^{2}\mathfrak{g}\to\mathfrak{g}$ is a Lie bracket, a Lie coalgebra $(\mathfrak{g},\Delta)$ where $\Delta:\mathfrak{g}\to\otimes^{2}\mathfrak{g}$ is a Lie comultiplication, and a suitable compatibility condition between the Lie bracket $[\cdot,\cdot]_{\mathfrak{g}}$ and the Lie comultiplication $\Delta$.

The great importance of the Lie bialgebra in the related fields relies on the fact that a Lie bialgebra can be equivalently described in terms of a Manin triple. More precisely, the compatibility condition of a Lie bialgebra is given by the condition that the Lie algebra $\mathfrak{g}$ and the Lie algebra $\mathfrak{g}^{*}$ from the Lie coalgebra $(\mathfrak{g},\Delta)$ are subalgebras of a third Lie algebra $\mathfrak{g}\oplus\mathfrak{g}^{*}$ such that, for the usual pairing $\langle\cdot,\cdot\rangle$ between $\mathfrak{g}$ and $\mathfrak{g}^{*}$, the bilinear form

[TABLE]

is invariant on $\mathfrak{g}\oplus\mathfrak{g}^{*}$ in the sense that

[TABLE]

Thus a suitable generalization of the theory of Lie bialgebras to the Hom context should include a meaningful generalization of Manin triples in the picture, which turns out to be a nontrivial requirement. A notion of Hom-Lie bialgebra was proposed in [20], where the compatibility condition is that the Hom-Lie cobracket $\Delta$ is a 1-cocycle on the Hom-Lie algebras $\mathfrak{g}$ with the coefficients in $\wedge^{2}\mathfrak{g}$:

[TABLE]

Due to the lack of a Manin triple interpretation of this definition, other notions of Hom-Lie bialgebras were subsequently introduced [5, 16], incorporating Hom variations of the nondegenerate symmetric bilinear form in Eq. (1.1).

Different ways of integrating the Hom-map $\phi_{{}_{\mathfrak{g}}}$ into the bilinear form in Eq. (1.1) give rise to variations in generalizing the invariant nondegenerate symmetric bilinear forms and the resulting Manin triples to the Hom case, yielding various notions of Hom-Lie bialgebras. Thus a Hom-Lie bialgebra is a triple $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\Delta)$ such that $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ defines a Hom-Lie algebra and $(\mathfrak{g},\Delta)$ defines a Hom-Lie coalgebra (that is, $\Delta^{*}$ defines a Hom-Lie algebra on $\mathfrak{g}^{*}$ when $\mathfrak{g}$ is finite-dimensional) which satisfy certain compatibility conditions depending on the appearance of $\phi_{{}_{\mathfrak{g}}}$. We compare the approaches in [16] and [5] in order to motivate the approach taken in this paper.

In [16], the authors introduced a notion of Hom-Lie bialgebras on Hom-Lie algebras which are admissible in the sense that the following equations hold:

[TABLE]

The compatibility condition is still given by Eq. (1.3), but is only required to hold on a subspace of $\wedge^{2}\mathfrak{g}^{*}$, that is,

[TABLE]

It is verified in [16] that this variation of Hom-Lie bialgebras correspond to the Manin triples of Hom-Lie algebras associated to the nondegenerate symmetric bilinear form satisfying

[TABLE]

In [5], the authors introduced another notion of Hom-Lie bialgebras, called purely Hom-Lie bialgebras, defined on Hom-Lie algebras which are regular, that is, such that $\phi_{{}_{\mathfrak{g}}}$ is invertible. The compatibility condition is changed to

[TABLE]

Such Hom-Lie bialgebras correspond to the Manin triples of Hom-Lie algebras associated to the nondegenerate symmetric bilinear form satisfying

[TABLE]

for any $a,b,c\in\mathfrak{g}\oplus\mathfrak{g}^{*}$.

As can be noted at this point, there should be a variety of Hom-Lie bialgebras and it is worthwhile to explore other possibilities in order to gain further insight on the interesting role played by the Hom-map $\phi_{{}_{\mathfrak{g}}}$ in Hom-Lie bialgebras. In this paper, we consider a new kind of Hom-Lie bialgebras which correspond to the Manin triples of Hom-Lie algebras associated to the nondegenerate symmetric bilinear form satisfying the condition

[TABLE]

for any $a,b,c\in\mathfrak{g}\oplus\mathfrak{g}^{*}$. Note that if $\phi_{{}_{\mathfrak{g}\oplus\mathfrak{g}^{*}}}$ is invertible, then Eq. (1.8) is equivalent to

[TABLE]

for any $a,b,c\in\mathfrak{g}\oplus\mathfrak{g}^{*}$, that is, a certain hybrid of Eqs. (1.6) and (1.7). It is also interesting to note that the compatible condition is exactly Eq. (1.3), on weakly involutive Hom-Lie algebras. Thus this new kind of Hom-Lie bialgebras incorporate features of all the previous three notions of Hom-Lie bialgebras.

Another complexity in defining Hom-Lie bialgebras is their dependence on a suitable notion of the dual representation, since the natural choice $(V^{*},\beta^{*},\rho^{*})$ for the dual representation of a representation $(V,\beta,\rho)$ of a Hom-Lie algebra does not work for all Hom-Lie algebras [2]. To work around this difficulty, the authors of [16] imposed a condition on Hom-Lie bialgebras called admissible such that $(V^{*},\beta^{*},\rho^{*})$ is still the dual representation of $(V,\beta,\rho)$. Taking a different approach, a notion of dual representation, under the condition that $\beta$ is invertible, was introduced in [5] in order to define their Hom-Lie bialgebras. In this paper, we introduce an alternative notion of the dual representation $(V^{*},\beta^{*},\rho^{\circ})$, where $\rho^{\circ}=\rho^{*}\,\phi_{{}_{\mathfrak{g}}}:\mathfrak{g}\longrightarrow\mathfrak{gl}(V^{*})$111The composition sign $\circ$ is suppressed throughout the paper to simplify notations., under the weakly involutive condition $\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x))=\rho(x)$. In particular, for the adjoint representation $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$, $\mathrm{ad}(\phi_{{}_{\mathfrak{g}}}^{2}(x))=\mathrm{ad}(x)$ for all $x\in\mathfrak{g}$ gives the notion of a weakly involutive Hom-Lie algebra.

One importance of Lie bialgebras lies with the essential role played by coboundary Lie bialgebras and quasitriangular Lie bialgebras which led to the introduction of the classical Yang-Baxter equation whose solutions produce such bialgebras. Another advantage of our approach of Hom-Lie bialgebras is that it works, without any restrictions, with the coboundary Hom-Lie bialgebras and quasitriangular Hom-Lie bialgebras and their connection with the classical Hom-Yang-Baxter equation, while conditions like $r\in\mathfrak{g}\otimes\mathfrak{g}$ being skew-symmetric needed to be imposed in [5, 16]. In particular, there is a canonical (quasitriangular) Hom-Lie bialgebra structure on the double space of a Hom-Lie bialgebra induced from a non-skew-symmetric solution of the classical Hom-Yang-Baxter equation.

The paper is organized as follows. In Section 2, we introduce a new dual representation $(V^{*},\beta^{*},\rho^{\circ})$ , leading to the basic notion of weakly involutive Hom-Lie algebra. This weakly involutive condition is shown to provide the right context for a nondegenerate symmetric invariant bilinear form on a Hom-Lie algebra (Proposition 2.15). These nondegenerate symmetric invariant bilinear forms are applied in Section 3 to study matched pairs and Manin triples of weakly involutive Hom-Lie algebras. With these preparations, we are naturally led to our notion of a Hom-Lie bialgebra together with its characterizations by matched pairs and Manin triples (Theorem 3.12). As a further evidence that we have a suitable notion of Hom-Lie bialgebras, in Section 4, we study coboundary Hom-Lie bialgebras and quasitriangular Hom-Lie bialgebras in terms of $r$-matrices (Theorem 4.4), showing that skew-symmetric solutions of the classical Hom-Yang-Baxter equation in a weakly involutive Hom-Lie algebra naturally give rise to Hom-Lie bialgebras. Finally, in Section 5, we study the operator forms of the classical Hom-Yang-Baxter equation and then construct solutions of the classical Hom-Yang-Baxter equation in semidirect products of weakly involutive Hom-Lie algebras using $\mathcal{O}$-operators (Corollary 5.10) and Hom-left-symmetric algebras (Corollary 5.11).

Notations. Throughout this paper, $\mathfrak{g}$ denotes a Hom-Lie algebra and $V$ denotes a vector space, both assumed to be finite-dimensional. $\mathfrak{g}^{*}$ and $V^{*}$ denote the dual of $\mathfrak{g}$ and $V$ respectively. Further $x,y,z,x_{i},y_{i}$ denote elements in $\mathfrak{g}$, $a,b,c$ denote elements in $\mathfrak{g}^{*}$, $u,v,w$ denote elements in $V$ and $\xi,\eta$ denote elements in $V^{*}$. $\mathrm{ad}$ and $\mathfrak{ad}$ are the adjoint representations of the Hom-Lie algebras $\mathfrak{g}$ and $\mathfrak{g}^{*}$ respectively.

2. Hom-Lie algebras, their representations and bilinear forms

In this section, we first recall some basic concepts and facts on Hom-Lie algebras and their representations [8, 15]. We then introduce a new notion of the dual representation of a representation of a Hom-Lie algebra. The related bilinear forms are also studied.

Definition 2.1** ([8]).**

A Hom-Lie algebra is a triple $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ consisting of a linear space $\mathfrak{g}$, a skew-symmetric bilinear map (bracket) $[\cdot,\cdot]_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ and a bracket preserving linear map $\phi_{{}_{\mathfrak{g}}}:\mathfrak{g}\longrightarrow\mathfrak{g}$ satisfying the Hom-Jacobi identity

[TABLE]

A Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ is called involutive if $\phi_{{}_{\mathfrak{g}}}$ satisfies $\phi_{{}_{\mathfrak{g}}}^{2}=\mathrm{Id}$.

Definition 2.2**.**

A homomorphism of Hom-Lie algebras $f:(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})\longrightarrow(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}},\phi_{{}_{\mathfrak{k}}})$ is a linear map $f:\mathfrak{g}\longrightarrow\mathfrak{k}$ such that

[TABLE]

The notion of a representation of a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ was introduced in [15].

Definition 2.3**.**

A representation of a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ is a triple $(V,\beta,\rho)$ consisting of a vector space $V$, an element $\beta\in\mathfrak{gl}(V)$ and a linear map $\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$, such that for any $x,y\in\mathfrak{g}$, the following equalities hold:

• (i)

$\rho(\phi_{{}_{\mathfrak{g}}}(x))\,\beta=\beta\,\rho(x)$;

• (ii)

$\rho([x,y]_{\mathfrak{g}})\,\beta=\rho(\phi_{{}_{\mathfrak{g}}}(x))\,\rho(y)-\rho(\phi_{{}_{\mathfrak{g}}}(y))\,\rho(x)$.

It is straightforward to see that $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$, where $\mathrm{ad}_{x}(y):=[x,y]$ for all $x,y\in\mathfrak{g}$, is a representation of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$, called the adjoint representation.

As in the Lie algebra case, given a representation $(V,\beta,\rho)$  of a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$, we obtain a new Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho}V:=(\mathfrak{g}\oplus V,[\cdot,\cdot]_{\mathfrak{g}\ltimes_{\rho}V},\phi_{{}_{\mathfrak{g}}}\oplus\beta)$ as the semidirect product:

[TABLE]

Definition 2.4**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra. Two representations $(V,\beta,\rho)$ and $(V^{\prime},\beta^{\prime},\rho^{\prime})$ are called equivalent if there exists a linear isomorphism $\varphi:V\rightarrow V^{\prime}$ such that

[TABLE]

Let $V_{1}$ and $V_{2}$ be two vector spaces. For a linear map $\phi:V_{1}\longrightarrow V_{2}$, we define the map $\phi^{*}:V_{2}^{*}\longrightarrow V_{1}^{*}$ by

[TABLE]

Given a representation $(V,\beta,\rho)$  of a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$, define $\rho^{*}:\mathfrak{g}\longrightarrow\mathfrak{gl}(V^{*})$ by

[TABLE]

As noted in [2], in general $(V^{*},\beta^{*},\rho^{*})$ is not a representation of $\mathfrak{g}$ on $V^{*}$ with respect to $\beta^{*}$.

We now introduce a Hom-variation of dual representation. Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and let $(V,\beta,\rho)$ be a representation of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$. Define

[TABLE]

In other words,

[TABLE]

In general, $(V^{*},\beta^{*},\rho^{\circ})$ is not a representation of $\mathfrak{g}$. However, by the definition of a representation of $\mathfrak{g}$, we have

Lemma 2.5**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a representation. Then $(V^{*},\beta^{*},\rho^{\circ})$ is a representation if and only if the following equations hold:

• (i)

$\beta\,\rho(x)=\beta\,\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x))$;

• (ii)

$\rho(\phi_{{}_{\mathfrak{g}}}^{2}[x,y]_{\mathfrak{g}})\,\beta=\rho(\phi_{{}_{\mathfrak{g}}}(x))\,\rho(\phi_{{}_{\mathfrak{g}}}^{2}(y))-\rho(\phi_{{}_{\mathfrak{g}}}(y))\,\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x)),\;\;\forall x,y\in\mathfrak{g}$.

In this case, $(V^{*},\beta^{*},\rho^{\circ})$ is called the Hom-dual representation of $(V,\beta,\rho)$.

As can be expected for a dual representation, $\rho^{\circ}$ should satisfy $(\rho^{\circ})^{\circ}=\rho$, that is, $\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x))=\rho(x)$ for all $x\in\mathfrak{g}$. Moreover, the following obvious result further highlights the relevance of the latter condition to representations.

Lemma 2.6**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a representation.

1. (a)

If $\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x))=\rho(x)$ for any $x\in\mathfrak{g}$, then $(V^{*},\beta^{*},\rho^{\circ})$ is a representation of $\mathfrak{g}$. 2. (b)

If $\beta$ is invertible and $(V^{*},\beta^{*},\rho^{\circ})$ is a representation of $\mathfrak{g}$, then we have $\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x))=\rho(x)$.

Motivated by Lemma 2.6, a representation $(V,\beta,\rho)$ is called weakly involutive if $\rho(\phi_{{}_{\mathfrak{g}}}^{2}(x))=\rho(x)$. By the definitions of $\rho^{\circ}$ and $\rho^{*}$ we obtain

Lemma 2.7**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a weakly involutive representation. Then the Hom-dual representation $(V^{*},\beta^{*},\rho^{\circ})$ is also a weakly involutive representation.

Focusing on the adjoint representation, we give

Definition 2.8**.**

A Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ is called weakly involutive if the adjoint representation $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$ is weakly involutive, more precisely, $[\phi_{{}_{\mathfrak{g}}}^{2}(x),y]_{\mathfrak{g}}=[x,y]_{\mathfrak{g}}$ for all $x,y\in\mathfrak{g}$.

Corollary 2.9**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra. Then we have

[TABLE]

In particular, we have $(\mathrm{ad}^{\circ}_{x}{\phi_{{}_{\mathfrak{g}}}^{*}}^{2})(a)=\mathrm{ad}^{\circ}_{x}(a)$.

Proof.

Since $\mathfrak{g}$ is weakly involutive, we have $[\phi_{{}_{\mathfrak{g}}}^{2}(x),y]_{\mathfrak{g}}=[x,y]_{\mathfrak{g}}$. Then we have

[TABLE]

Therefore the conclusion holds. ∎

It is straightforward to get the following conclusion.

Proposition 2.10**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a representation. Then the Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho}V$ is weakly involutive if and only if

• (i)

$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$* is weakly involutive;*

• (ii)

$(V,\beta,\rho)$* is weakly involutive;*

• (iii)

$\rho(x)\,\beta^{2}=\rho(x)$* for all $x\in\mathfrak{g}$.*

Corollary 2.11**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a weakly involutive representation. Then the Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$ is weakly involutive if and only if

• (i)

$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$* is weakly involutive;*

• (ii)

$\rho(\phi_{{}_{\mathfrak{g}}}(x))\,\beta^{2}=\rho(\phi_{{}_{\mathfrak{g}}}(x))$* for all $x\in\mathfrak{g}$.*

Proof.

Since $(V,\beta,\rho)$ is weakly involutive, $(V^{*},\beta^{*},\rho^{\circ})$ is weakly involutive by Lemma 2.7. Moreover $\rho^{\circ}(x)\,(\beta^{*})^{2}=\rho^{\circ}(x)$ for all $x\in\mathfrak{g}$ if and only if $\rho(\phi_{{}_{\mathfrak{g}}}(x))\,\beta^{2}=\rho(\phi_{{}_{\mathfrak{g}}}(x))$ for all $x\in\mathfrak{g}$. Hence the conclusion holds by Proposition 2.10. ∎

Corollary 2.12**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra, $(V,\beta,\rho)$ be a weakly involutive representation and $\mathfrak{g}\ltimes_{\rho}V$ be a weakly involutive Hom-Lie algebra. Then the Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$ is weakly involutive.

Definition 2.13**.**

A bilinear form $\mathfrak{B}(\cdot,\cdot)$ on a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ is called invariant if

[TABLE]

Remark 2.14**.**

If $\phi_{{}_{\mathfrak{g}}}$ is invertible, then the above invariant conditions are equivalent to

[TABLE]

The following proposition gives the close relationship between weakly involutive condition on a Hom-Lie algebra and the existence of a nondegenerate invariant bilinear form on the Hom-Lie algebra.

Proposition 2.15**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra with a nondegenerate symmetric invariant bilinear form $\mathfrak{B}(\cdot,\cdot)$. Then $\mathfrak{g}$ is a weakly involutive Hom-Lie algebra, and $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$ and $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$ are equivalent representations of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ . Conversely, if $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ is a weakly involutive Hom-Lie algebra, and $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$ and $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$ are equivalent representations of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ , then there exists a nondegenerate invariant bilinear form $\mathfrak{B}(\cdot,\cdot)$ on $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ .

Note that the converse statement does not include the symmetric condition.

Proof.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra with a nondegenerate symmetric invariant bilinear form $\mathfrak{B}(\cdot,\cdot)$. For $x,y,z\in\mathfrak{g}$, we have

[TABLE]

Then we get

[TABLE]

Since $\mathfrak{B}(\cdot,\cdot)$ is nondegenerate, we get $[x,y]_{\mathfrak{g}}=[x,\phi_{{}_{\mathfrak{g}}}^{2}(y)]_{\mathfrak{g}}$. Moreover, define a linear transformation $\varphi:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$ by

[TABLE]

Then $\varphi$ is a linear isomorphism and, for all $x,y,z\in\mathfrak{g}$, we have

[TABLE]

Hence $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$ and $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$ are equivalent representations of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ .

Conversely, let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra. If $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$ and $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$ are equivalent representations of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ , then there exists a linear isomorphism $\psi:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$ satisfying

[TABLE]

Define a bilinear form on $\mathfrak{g}$ by

[TABLE]

Then $B$ is nondegenerate and, by a similar argument, one can show that $B$ is invariant. ∎

3. Matched pairs, Manin triples and Hom-Lie bialgebras

We first recall the notion of matched pairs of Hom-Lie algebras. Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{\prime},[\cdot,\cdot]_{\mathfrak{g}^{\prime}},\phi_{{}_{\mathfrak{g}^{\prime}}})$ be two Hom-Lie algebras. Let $\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g}^{\prime})$ and $\rho^{\prime}:\mathfrak{g}^{\prime}\longrightarrow\mathfrak{gl}(\mathfrak{g})$ be two linear maps. On the direct sum $\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ of the underlying vector spaces, define

[TABLE]

and define a skew-symmetric bilinear map $[\cdot,\cdot]_{d}:\otimes^{2}(\mathfrak{g}\oplus\mathfrak{g}^{\prime})\longrightarrow\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ by

[TABLE]

Theorem 3.1** ([16]).**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{\prime},[\cdot,\cdot]_{\mathfrak{g}^{\prime}},\phi_{{}_{\mathfrak{g}^{\prime}}})$ be two Hom-Lie algebras. Then $(\mathfrak{g}\oplus\mathfrak{g}^{\prime},[\cdot,\cdot]_{d},\phi_{{}_{d}})$ is a Hom-Lie algebra if and only if $(\mathfrak{g}^{\prime},\phi_{{}_{\mathfrak{g}^{\prime}}},\rho)$ and $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\rho^{\prime})$ are representations of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{\prime},[\cdot,\cdot]_{\mathfrak{g}^{\prime}},\phi_{{}_{\mathfrak{g}^{\prime}}})$ respectively and

[TABLE]

[TABLE]

Definition 3.2** ([16]).**

A matched pair of Hom-Lie algebras is a quadruple $(\mathfrak{g},\mathfrak{g}^{\prime};\rho,\rho^{\prime})$ consisting of two Hom-Lie algebras $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{\prime},[\cdot,\cdot]_{\mathfrak{g}^{\prime}},\phi_{{}_{\mathfrak{g}^{\prime}}})$, together with representations $\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g}^{\prime})$ and $\rho^{\prime}:\mathfrak{g}^{\prime}\longrightarrow\mathfrak{gl}(\mathfrak{g})$ with respect to $\phi_{{}_{\mathfrak{g}^{\prime}}}$ and $\phi_{{}_{\mathfrak{g}}}$ respectively, such that the compatibility conditions (3.3) and (3.4) are satisfied.

It is straightforward to reach the following conclusion.

Proposition 3.3**.**

Let $(\mathfrak{g},\mathfrak{g}^{\prime};\rho,\rho^{\prime})$ be a matched pair of Hom-Lie algebras. Then the Hom-Lie algebra $(\mathfrak{g}\oplus\mathfrak{g}^{\prime},[\cdot,\cdot]_{d},\phi_{{}_{d}})$ is weakly involutive if and only if the following conditions hold:

• (i)

$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$* is weakly involutive and $(\mathfrak{g}^{\prime},\phi_{{}_{\mathfrak{g}^{\prime}}},\rho)$ is a weakly involutive representation of $\mathfrak{g}$;*

• (ii)

$(\mathfrak{g}^{\prime},[\cdot,\cdot]_{\mathfrak{g}^{\prime}},\phi_{{}_{\mathfrak{g}^{\prime}}})$* is weakly involutive and $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\rho^{\prime})$ is a weakly involutive representation of $\mathfrak{g}^{\prime}$;*

• (iii)

$\rho(x)\,\phi_{{}_{\mathfrak{g}^{\prime}}}^{2}=\rho(x)$* for all $x\in\mathfrak{g}$;*

• (iv)

$\rho^{\prime}(x^{\prime})\,\phi_{{}_{\mathfrak{g}}}^{2}=\rho^{\prime}(x^{\prime})$* for all $x^{\prime}\in\mathfrak{g}^{\prime}$.*

We now introduce the closely related notion of Manin triples of Hom-Lie algebras.

Definition 3.4**.**

A Manin triple of Hom-Lie algebras is a triple of Hom-Lie algebras $(\mathfrak{k};\mathfrak{g},\mathfrak{g}^{\prime})$ together with a nondegenerate symmetric invariant bilinear form $\mathfrak{B}(\cdot,\cdot)$ on $\mathfrak{k}$ such that $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are isotropic Hom-Lie subalgebras of $\mathfrak{k}$ and $\mathfrak{k}=\mathfrak{g}\oplus\mathfrak{g}^{\prime}$ as vector spaces.

By Proposition 2.15, we have

Corollary 3.5**.**

Let $(\mathfrak{k};\mathfrak{g},\mathfrak{g}^{\prime})$ be a Manin triple of Hom-Lie algebras. Then $\mathfrak{k}$ and hence $\mathfrak{g},\mathfrak{g}^{\prime}$ are weakly involutive Hom-Lie algebras.

Proposition 3.6**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}^{*}}})$ be two weakly involutive Hom-Lie algebras. Then $(\mathfrak{g}\oplus\mathfrak{g}^{*};\mathfrak{g},\mathfrak{g}^{*})$ is a Manin triple of Hom-Lie algebras associated to the nondegenerate symmetric invariant bilinear form on $\mathfrak{g}\oplus\mathfrak{g}^{*}$ defined by

[TABLE]

if and only if $\phi_{{}_{\mathfrak{g}^{*}}}=\phi_{{}_{\mathfrak{g}}}^{*}$ and $(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$ is a matched pair of Hom-Lie algebras. Here $\mathfrak{ad}^{\circ}$ is the Hom-dual representation of the adjoint representation $\mathfrak{ad}$ of the Hom-Lie algebra $\mathfrak{g}^{*}$.

Proof.

Suppose that $\phi_{{}_{\mathfrak{g}^{*}}}=\phi_{{}_{\mathfrak{g}}}^{*}$ and $(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$ is a matched pair of Hom-Lie algebras. Then $(\mathfrak{g}\oplus\mathfrak{g}^{*},[\cdot,\cdot]_{d},\phi_{{}_{\mathfrak{g}}}\oplus\phi_{{}_{\mathfrak{g}}}^{*})$ is a Hom-Lie algebra, where $[\cdot,\cdot]_{d}$ is given by

[TABLE]

Let $x,y\in\mathfrak{g}$ and $a,b\in\mathfrak{g}^{*}$. We just need to prove that the bilinear form defined by Eq. (3.5) is invariant. Setting $\phi_{{}_{d}}=\phi_{{}_{\mathfrak{g}}}\oplus\phi_{{}_{\mathfrak{g}}}^{*}$, we have

[TABLE]

Therefore the bilinear form defined by Eq. (3.5) is invariant.

Conversely, let $(\mathfrak{g}\oplus\mathfrak{g}^{*};\mathfrak{g},\mathfrak{g}^{*})$ be a Manin triple of Hom-Lie algebras associated to the invariant bilinear form $\mathfrak{B}$ given by Eq. (3.5). Then for any $x,y\in\mathfrak{g}$ and $a,b\in\mathfrak{g}^{*}$, due to the invariance of $\mathfrak{B}$, we have

[TABLE]

which implies $\phi_{{}_{\mathfrak{g}^{*}}}=\phi_{{}_{\mathfrak{g}}}^{*}$. Since

[TABLE]

we have

[TABLE]

So the Hom-Lie bracket on $\mathfrak{g}\oplus\mathfrak{g}^{*}$ is given by Eq. (3.6). Therefore, $(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$ is a matched pair of Hom-Lie algebras. ∎

For a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ (resp. $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$), let $\Delta_{\mathfrak{g}^{*}}:\mathfrak{g}^{*}\longrightarrow\otimes^{2}\mathfrak{g}^{*}$ (resp. $\Delta_{\mathfrak{g}}:\mathfrak{g}\longrightarrow\otimes^{2}\mathfrak{g}$) be the dual map of $[\cdot,\cdot]_{\mathfrak{g}}:\otimes^{2}\mathfrak{g}\longrightarrow\mathfrak{g}$ (resp. $[\cdot,\cdot]_{\mathfrak{g}^{*}}:\otimes^{2}\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$), i.e.,

[TABLE]

In particular, we set $\Delta:=\Delta_{\mathfrak{g}}$.

Theorem 3.7**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ be two weakly involutive Hom-Lie algebras. Then $(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$ is a matched pair of Hom-Lie algebras if and only if

[TABLE]

where $\Delta=:\Delta_{\mathfrak{g}}$ is given by Eq. (3.7).

Proof.

By Theorem 3.1, $(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$ is a matched pair of Hom-Lie algebras if and only if

[TABLE]

Then we get

[TABLE]

which is exactly Eq. (3.8). Similarly, we could deduce that Eq. (3.10) is equivalent to the equation

[TABLE]

Next we prove that Eq. (3.8) and Eq. (3.11) are equivalent. In fact, we have

[TABLE]

Therefore the two equations are equivalent. This gives what we need. ∎

Definition 3.8**.**

A pair of weakly involutive Hom-Lie algebras $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ with $\Delta=:\Delta_{\mathfrak{g}}$ given by Eq. (3.7) is called a Hom-Lie bialgebra if Eq. (3.8) holds. We denote it by $(\mathfrak{g},\mathfrak{g}^{*})$ or $(\mathfrak{g},\Delta)$.

Remark 3.9**.**

We note that this notion of a Hom-Lie bialgebra is different from either of the three notions of a Hom-Lie bialgebra given in [20, 16, 5]. Even though it has the same compatibility condition as in [20], it is just for weakly involutive Hom-Lie algebras, not all Hom-Lie algebras. Under this compatibility condition, there is a natural Hom-Lie algebra structure on $\mathfrak{g}\oplus\mathfrak{g}^{*}$ as follows, which does not exist in [20].

Corollary 3.10**.**

Let $(\mathfrak{g},\mathfrak{g}^{*})$ be a Hom-Lie bialgebra. Then the Hom-Lie algebra $(\mathfrak{g}\oplus\mathfrak{g}^{*},[\cdot,\cdot]_{d},\phi_{{}_{\mathfrak{g}}}\oplus\phi_{{}_{\mathfrak{g}}}^{*})$ is weakly involutive.

Proof.

Since $(\mathfrak{g},\mathfrak{g}^{*})$ is a Hom-Lie bialgebra, it follows that $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ are weakly involutive Hom-Lie algebras, and $(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$ is a matched pair of Hom-Lie algebras. Then $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathrm{ad})$ and $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathfrak{ad})$ are weakly involutive representations of $\mathfrak{g}$ and $\mathfrak{g}^{*}$ respectively. Hence by Lemma 2.7, $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$ and $(\mathfrak{g},\phi_{{}_{\mathfrak{g}}},\mathfrak{ad}^{\circ})$ are weakly involutive representations of $\mathfrak{g}$ and $\mathfrak{g}^{*}$ respectively. By Corollary 2.9, we have

[TABLE]

for any $x\in\mathfrak{g},a\in\mathfrak{g}^{*}$. Finally, Proposition 3.3 gives the conclusion. ∎

Definition 3.11**.**

A homomorphism of Hom-Lie bialgebras $f:(\mathfrak{g},\Delta_{\mathfrak{g}})\longrightarrow(\mathfrak{k},\Delta_{\mathfrak{k}})$ is a homomorphism of Hom-Lie algebras such that

[TABLE]

Combining Proposition 3.6, Theorem 3.7 and Definition 3.8, we arrive at the following conclusion which, when $\phi_{{}_{\mathfrak{g}}}$ is the identity, recovers the classical characterization of Lie bialgebras in terms of matched pairs and Manin triples.

Theorem 3.12**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ be two weakly involutive Hom-Lie algebras. Then the following conditions are equivalent.

• (i)

$(\mathfrak{g},\mathfrak{g}^{*})$* is a Hom-Lie bialgebra.*

• (ii)

$(\mathfrak{g},\mathfrak{g}^{*};\mathrm{ad}^{\circ},\mathfrak{ad}^{\circ})$* is a matched pair of Hom-Lie algebras.*

• (iii)

$(\mathfrak{g}\oplus\mathfrak{g}^{*};\mathfrak{g},\mathfrak{g}^{*})$* is a Manin triple of Hom-Lie algebras associated to the invariant bilinear form given by Eq. (3.5).*

4. Coboundary Hom-Lie bialgebras and the classical Hom-Yang-Baxter equation

In this section, we study coboundary Hom-Lie bialgebras and their relationship with the classical Hom-Yang-Baxter equation. For a weakly involutive Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $r\in\mathfrak{g}\otimes\mathfrak{g}$, define linear maps

[TABLE]

and

[TABLE]

In order for $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ to be a weakly involutive Hom-Lie algebra and for $(\mathfrak{g},\mathfrak{g}^{*})$ to be a Hom-Lie bialgebra, the following conditions should hold.

1. (a)

$\phi_{{}_{\mathfrak{g}}}^{*}$ is a Lie algebra homomorphism with respect to $[\cdot,\cdot]_{\mathfrak{g}^{*}}$, or equivalently, $\Delta(\phi_{{}_{\mathfrak{g}}}(x))=(\phi_{{}_{\mathfrak{g}}}\otimes\phi_{{}_{\mathfrak{g}}})\Delta(x)$; 2. (b)

$[{\phi_{{}_{\mathfrak{g}}}^{*}}^{2}(a),b]_{\mathfrak{g}^{*}}=[a,b]_{\mathfrak{g}^{*}}$, or equivalently, $(\phi_{{}_{\mathfrak{g}}}^{2}\otimes\mathrm{Id})\Delta(x)=\Delta(x)$; 3. (c)

$\Delta[x,y]_{\mathfrak{g}}-(\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}\Delta(y)-\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(y)}\Delta(x))=0$; 4. (d)

$(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ is a Hom-Lie algebra.

So we first investigate these conditions. For the first three conditions we have the following result, noting the common factor $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id}-\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$ on the right hand sides.

Lemma 4.1**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra and $r\in\mathfrak{g}\otimes\mathfrak{g}$. Define a linear map $\Delta:\mathfrak{g}\longrightarrow\mathfrak{g}\otimes\mathfrak{g}$ by Eq. (4.1). Then we have

1. (a)

$\Delta(\phi_{{}_{\mathfrak{g}}}(x))-(\phi_{{}_{\mathfrak{g}}}\otimes\phi_{{}_{\mathfrak{g}}})\Delta(x)=(\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}\,\phi_{{}_{\mathfrak{g}}}\otimes\phi_{{}_{\mathfrak{g}}}-\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}\,\phi_{{}_{\mathfrak{g}}})(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id}-\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r;$** 2. (b)

$(\phi_{{}_{\mathfrak{g}}}^{2}\otimes\mathrm{Id})\Delta(x)-\Delta(x)=(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{ad}_{x})(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id}+\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id}-\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r;$** 3. (c)

$\Delta[x,y]_{\mathfrak{g}}-(\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}\Delta(y)-\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(y)}\Delta(x))\\ =(\mathrm{ad}_{[x,y]_{\mathfrak{g}}}\,\phi_{{}_{\mathfrak{g}}}\otimes\phi_{{}_{\mathfrak{g}}}-\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{ad}_{[x,y]_{\mathfrak{g}}}\,\phi_{{}_{\mathfrak{g}}})(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id}-\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r.$**

Proof.

(a). For $x,y\in\mathfrak{g}$, we have

[TABLE]

and

[TABLE]

Thus we get

[TABLE]

(b). For $x,y\in\mathfrak{g}$, we have

[TABLE]

and

[TABLE]

Then we have

[TABLE]

(c). For $x,y\in\mathfrak{g}$, we have

[TABLE]

and

[TABLE]

This gives the desired equation. ∎

Therefore, to give the definition of coboundary Hom-Lie bialgebras, requiring $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$ is a natural choice. This leads us to the following definition.

Definition 4.2**.**

A coboundary Hom-Lie bialgebra is a Hom-Lie bialgebra $(\mathfrak{g},\Delta)$ such that the linear map $\Delta:\mathfrak{g}\rightarrow\mathfrak{g}\otimes\mathfrak{g}$ is given by Eq. (4.1), where $r\in\mathfrak{g}\otimes\mathfrak{g}$ satisfies

[TABLE]

For a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ and $r=\sum\limits_{i}x_{i}\otimes y_{i}\in\mathfrak{g}\otimes\mathfrak{g}$, define $[r,r]_{\mathfrak{g}}\in\otimes^{3}\mathfrak{g}$ by

[TABLE]

and set $\sigma(r):=\sum\limits_{i}y_{i}\otimes x_{i}$.

For any linear map $\Delta:\mathfrak{g}\longrightarrow\mathfrak{g}\otimes\mathfrak{g}$ and any $x\in\mathfrak{g}$, let

[TABLE]

where ‘$\sum\limits_{c.p.}$’ means that the sum is taken of the term after the summation sign and together with the two similar terms obtained by cyclic permutations of the factors in the tensor product $\otimes^{3}\mathfrak{g}$. Since

[TABLE]

it is clear that $[\cdot,\cdot]_{\mathfrak{g}^{*}}$ satisfies the Hom-Jacobi identity if and only if $\mathrm{Jac}_{{}_{\Delta}}$ is the zero map.

Lemma 4.3**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra. Define a linear map $\Delta:\mathfrak{g}\longrightarrow\mathfrak{g}\otimes\mathfrak{g}$ by Eq. (4.1) with some $r\in\mathfrak{g}\otimes\mathfrak{g}$ satisfying $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$ and $[x,r+\sigma(r)]_{\mathfrak{g}}=0$ for all $x\in\mathfrak{g}$. Then

[TABLE]

for all $x\in\mathfrak{g}$.

Proof.

Let $r=\sum\limits_{i}x_{i}\otimes y_{i}$. The following computations make repeated use of the Hom-Jacobi identity in $\mathfrak{g}$, and $[x,r+\sigma(r)]_{\mathfrak{g}}=0$ for all $x\in\mathfrak{g}$ which is equivalent to

[TABLE]

If we write out $\mathrm{Jac}_{{}_{\Delta}}(x)$ explicitly, using the expression $r=\sum\limits_{i}x_{i}\otimes y_{i}$ and with a summation over repeated indices understood, we have

[TABLE]

Since $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ is a weakly involutive Hom-Lie algebra and $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$, the first term in Eq. (4.6) becomes

[TABLE]

Doing the same to other terms in Eq. (4.6), we obtain

[TABLE]

Then we write out $\mathrm{Jac}_{{}_{\Delta}}(x)-\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}[r,r]_{\mathfrak{g}}$ explicitly. Note that $\mathrm{Jac}_{{}_{\Delta}}(x)$ is a sum of twelve terms and that $\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}[r,r]_{\mathfrak{g}}$ is a sum of nine terms, but two terms appear in both sums and hence are canceled. Thus $\mathrm{Jac}_{{}_{\Delta}}(x)-\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}[r,r]_{\mathfrak{g}}$ is a sum of seventeen terms. After rearranging the terms suitably, we obtain

[TABLE]

Interchanging the indices $i$ and $j$ in the first term and using the Hom-Jacobi identity in $\mathfrak{g}$, the first term becomes

[TABLE]

The sum of $-[\phi_{{}_{\mathfrak{g}}}(y_{j}),[x,x_{i}]_{\mathfrak{g}}]_{\mathfrak{g}}\otimes\phi_{{}_{\mathfrak{g}}}^{2}(y_{i})\otimes\phi_{{}_{\mathfrak{g}}}^{2}(x_{j})$ and the third and fourth terms is

[TABLE]

using the Eq. (4.5).

Similarly, the sum of $-\phi_{{}_{\mathfrak{g}}}[x_{i},y_{j}]\otimes\phi_{{}_{\mathfrak{g}}}[x,y_{i}]_{\mathfrak{g}}\otimes\phi_{{}_{\mathfrak{g}}}^{2}(x_{j})$ and the fifth term becomes

[TABLE]

and the sum of the sixth and seventh terms is

[TABLE]

Finally, the sum of $-[\phi_{{}_{\mathfrak{g}}}(y_{j}),[x,x_{i}]_{\mathfrak{g}}]_{\mathfrak{g}}\otimes\phi_{{}_{\mathfrak{g}}}^{2}(y_{i})\otimes\phi_{{}_{\mathfrak{g}}}^{2}(x_{j})$ and the second term in the sum of the expression of $\mathrm{Jac}_{{}_{\Delta}}(x)-\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}[r,r]_{\mathfrak{g}}$ becomes

[TABLE]

Inserting these results, we find that the expression of $\mathrm{Jac}_{{}_{\Delta}}(x)-\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}[r,r]_{\mathfrak{g}}$ can be written in the form $\sum\limits_{i}(\phi_{{}_{\mathfrak{g}}}^{2}(x_{i})\otimes u_{i}+\phi_{{}_{\mathfrak{g}}}^{2}(y_{i})\otimes v_{i})$ for certain $u_{i},v_{i}\in\mathfrak{g}$. In fact,

[TABLE]

On the right-hand side, the sum of the first four terms is zero by Eq. (4.5), and that of the next three terms becomes

[TABLE]

By the Hom-Jacobi identity in $\mathfrak{g}$, the sum of $[\phi_{{}_{\mathfrak{g}}}(x),[y_{i},y_{j}]_{\mathfrak{g}}]_{\mathfrak{g}}\otimes\phi_{{}_{\mathfrak{g}}}^{2}(x_{j})$ and the eighth term is

[TABLE]

Finally, the sum of $[[x,y_{i}]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}}(y_{j})]_{\mathfrak{g}}\otimes\phi_{{}_{\mathfrak{g}}}^{2}(x_{j})$ and the last three terms becomes

[TABLE]

if we replace $x$ in Eq. (4.5) by $[x,y_{i}]_{\mathfrak{g}}$. Hence, we get $u_{i}=0$. A similar, but shorter, argument proves that

[TABLE]

Hence the conclusion holds. ∎

Theorem 4.4**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra. Define a bilinear map $[\cdot,\cdot]_{\mathfrak{g}^{*}}:\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$ by Eq. (4.2), where $\Delta$ is defined by Eq. (4.1) with some $r\in\mathfrak{g}\otimes\mathfrak{g}$ satisfying $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$. Then $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ is a weakly involutive Hom-Lie algebra if and only if the following conditions are satisfied:

• (i)

$[x,r+\sigma(r)]_{\mathfrak{g}}=0$* for all $x\in\mathfrak{g}$,*

• (ii)

$\mathrm{ad}_{\phi_{{}_{\mathfrak{g}}}(x)}[r,r]_{\mathfrak{g}}=0$* for all $x\in\mathfrak{g}$.*

Under these conditions, $(\mathfrak{g},\mathfrak{g}^{*})$ is a coboundary Hom-Lie bialgebra.

Proof.

The bracket $[\cdot,\cdot]_{\mathfrak{g}^{*}}$ is determined by the cobracket $\Delta(x)=[x,r]_{\mathfrak{g}}$. Note that $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$. Applying Lemma 4.1, we find that $\phi_{{}_{\mathfrak{g}}}^{*}$ is an algebra homomorphism with respect to $[\cdot,\cdot]_{\mathfrak{g}^{*}}$ and $[{\phi_{{}_{\mathfrak{g}}}^{*}}^{2}(a),b]_{\mathfrak{g}^{*}}=[a,b]_{\mathfrak{g}^{*}}$. Hence $(\mathfrak{g}^{*},[\cdot,\cdot]_{\mathfrak{g}^{*}},\phi_{{}_{\mathfrak{g}}}^{*})$ is a weakly involutive Hom-Lie algebra if and only if $[\cdot,\cdot]_{\mathfrak{g}^{*}}$ is skew-symmetric and satisfies the Hom-Jacobi identity.

The proof that $r$ satisfies (i) if and only if $[\cdot,\cdot]_{\mathfrak{g}^{*}}$ is skew-symmetric is straightforward. The proof that (ii) is equivalent to the condition that $[\cdot,\cdot]_{\mathfrak{g}^{*}}$ satisfies the Hom-Jacobi identity follows from Lemma 4.3.

Since $\Delta(x)=[x,r]_{\mathfrak{g}}$ and $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$, by Lemma 4.1.c, the compatibility conditions for a Hom-Lie bialgebra in Definition 3.8 hold naturally. Therefore the conclusion follows. ∎

Remark 4.5**.**

An easy way to satisfy conditions (i) and $(\mathrm{i}\mathrm{i})$ in Theorem 4.4 is to assume that $r$ is skew-symmetric and

[TABLE]

respectively. Eq. (4.7) is the classical Hom-Yang-Baxter equation. A quasitriangular Hom-Lie bialgebra is a coboundary Hom-Lie bialgebra, in which $r$ is a solution of the classical Hom-Yang-Baxter equation. A triangular Hom-Lie bialgebra is a coboundary Hom-Lie bialgebra, in which $r$ is a skew-symmetric solution of the classical Hom-Yang-Baxter equation.

Proposition 4.6**.**

Let $(\mathfrak{g},\mathfrak{g}^{*})$ be a Hom-Lie bialgebra. Then there is a canonical Hom-Lie bialgebra structure on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^{*}$ of the underlying vector spaces of $\mathfrak{g}$ and $\mathfrak{g}^{*}$ such that $\phi_{{}_{\mathfrak{g}}}:\mathfrak{g}\hookrightarrow\mathfrak{g}\oplus\mathfrak{g}^{*}$ and $\phi_{{}_{\mathfrak{g}}}^{*}:\mathfrak{g}^{*}\hookrightarrow\mathfrak{g}\oplus\mathfrak{g}^{*}$ into the two summands are homomorphisms of Hom-Lie bialgebras. Here the Hom-Lie bialgebra structure on $\mathfrak{g}^{*}$ is $(\mathfrak{g}^{*},-\Delta_{\mathfrak{g}^{*}})$, where $\Delta_{\mathfrak{g}^{*}}$ is given by Eq. (3.7).

Proof.

Let $r\in\mathfrak{g}\otimes\mathfrak{g}^{*}\subset(\mathfrak{g}\oplus\mathfrak{g}^{*})\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$ correspond to the identity map $\mathrm{id}:\mathfrak{g}\longrightarrow\mathfrak{g}$. Let $\{e_{1},\ldots,e_{n}\}$ be a basis of $\mathfrak{g}$ and $\{f_{1},\ldots,f_{n}\}$ be its dual basis. Then $r=\sum\limits_{i}e_{i}\otimes f_{i}$. We denote $\mathfrak{g}\oplus\mathfrak{g}^{*}$ by $\mathcal{HD}(\mathfrak{g})$, and $\phi_{{}_{\mathfrak{g}}}\oplus\phi_{{}_{\mathfrak{g}}}^{*}$ by $\phi_{{}_{\mathcal{HD}}}$. By Corollary 3.10, there is a weakly involutive Hom-Lie algebra structure $(\mathcal{HD}(\mathfrak{g}),[\cdot,\cdot]_{{}_{\mathcal{HD}(\mathfrak{g})}},\phi_{{}_{\mathcal{HD}}})$ on $\mathcal{HD}(\mathfrak{g})$. Since

[TABLE]

[TABLE]

we have $\sum\limits_{i}\phi_{{}_{\mathfrak{g}}}(e_{i})\otimes f_{i}=\sum\limits_{i}e_{i}\otimes\phi_{{}_{\mathfrak{g}}}^{*}(f_{i})$. Note that $\phi_{{}_{\mathcal{HD}}}(x+a)=\phi_{{}_{\mathfrak{g}}}(x)+\phi_{{}_{\mathfrak{g}}}^{*}(a)$ for all $x\in\mathfrak{g}$, $a\in\mathfrak{g}^{*}$. We get $(\phi_{{}_{\mathcal{HD}}}\otimes\mathrm{id})r=(\mathrm{id}\otimes\phi_{{}_{\mathcal{HD}}})r$. Set

[TABLE]

Since

[TABLE]

we get $[r,r]_{{}_{\mathcal{HD}(\mathfrak{g})}}=0$. Similarly, we prove that $[u,r+\sigma(r)]_{{}_{\mathcal{HD}(\mathfrak{g})}}=0$ for all $u\in\mathcal{HD}(\mathfrak{g})$. Hence $\mathcal{HD}(\mathfrak{g})$ is a quasitriangular Hom-Lie bialgebra by Theorem 4.4.

For $e_{i}\in\mathfrak{g}$, we have

[TABLE]

So $\Delta_{\mathcal{HD}}(\phi_{{}_{\mathfrak{g}}}(e_{i}))=(\phi_{{}_{\mathfrak{g}}}\otimes\phi_{{}_{\mathfrak{g}}})\Delta(e_{i})$. Therefore $\phi_{{}_{\mathfrak{g}}}:\mathfrak{g}\hookrightarrow\mathfrak{g}\oplus\mathfrak{g}^{*}$ is a homomorphism of Hom-Lie bialgebras. Similarly, $\phi_{{}_{\mathfrak{g}}}^{*}:\mathfrak{g}^{*}\hookrightarrow\mathfrak{g}\oplus\mathfrak{g}^{*}$ is also a homomorphism of Hom-Lie bialgebras since $\Delta_{\mathcal{HD}}(\phi_{{}_{\mathfrak{g}}}^{*}(f_{i}))=(\phi_{{}_{\mathfrak{g}}}^{*}\otimes\phi_{{}_{\mathfrak{g}}}^{*})(-\Delta_{\mathfrak{g}^{*}})(f_{i})$. ∎

Remark 4.7**.**

With the above Hom-Lie bialgebra structure, $\mathfrak{g}\oplus\mathfrak{g}^{*}$ is called the Hom-double of $\mathfrak{g}$, and is denoted by $\mathcal{HD}(\mathfrak{g})$.

For any $r\in\mathfrak{g}\otimes\mathfrak{g}$, the induced linear map $r^{\sharp}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}$ is defined by

[TABLE]

Then it is easy to see that $(\phi_{{}_{\mathfrak{g}}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{\mathfrak{g}}})r$ is equivalent to

[TABLE]

Setting $r_{21}=\sigma(r)$, we have

Proposition 4.8**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra and $r\in\mathfrak{g}\otimes\mathfrak{g}$ such that $\phi_{{}_{\mathfrak{g}}}\,r^{\sharp}=r^{\sharp}\,\phi_{{}_{\mathfrak{g}}}^{*}$. Define linear maps $\Delta:\mathfrak{g}\longrightarrow\mathfrak{g}\otimes\mathfrak{g}$ by Eq. (4.1) and $[\cdot,\cdot]_{\mathfrak{g}^{*}}:\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$ by Eq. (4.2). Then we have

[TABLE]

Proof.

For convenience, we assume that $r=x\otimes y$ is a pure tensor. Then we have

[TABLE]

Hence the conclusion holds. ∎

Lemma 4.9**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra. If $r\in\mathfrak{g}\otimes\mathfrak{g}$ satisfies $\phi_{{}_{\mathfrak{g}}}\,r^{\sharp}=r^{\sharp}\,\phi_{{}_{\mathfrak{g}}}^{*}$ and $[\cdot,\cdot]_{\mathfrak{g}^{*}}:\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$ is given by Eq. (4.2), where $\Delta$ is defined by Eq. (4.1), then we have

[TABLE]

Proof.

For any $a,b,c\in\mathfrak{g}$, we have

[TABLE]

It is straightforward to deduce that

[TABLE]

Therefore the conclusion holds. ∎

Corollary 4.10**.**

Let $(\mathfrak{g},\Delta)$ be a quasitriangular Hom-Lie bialgebra, where $\Delta$ is given by Eq. (4.1) for a solution $r$ of the classical Hom-Yang-Baxter equation. Then $r^{\sharp}\,\phi_{{}_{\mathfrak{g}}}^{*}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}$ is a homomorphism of Hom-Lie algebras.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra and $r\in\mathfrak{g}\otimes\mathfrak{g}$ be invertible (that is, $r^{\sharp}$ is invertible). Define a bilinear form $B\in(\mathfrak{g}\otimes\mathfrak{g})^{*}$ by

[TABLE]

Then it is easy to see that $B$ is skew-symmetric if and only if $r$ is skew-symmetric.

Proposition 4.11**.**

With the above notations, suppose that $r$ is skew-symmetric. Then $\phi_{{}_{\mathfrak{g}}}\,r^{\sharp}=r^{\sharp}\,\phi_{{}_{\mathfrak{g}}}^{*}$ and $r$ is a solution of the classical Hom-Yang-Baxter equation (4.7) if and only if

[TABLE]

Proof.

For any $x,y,z\in\mathfrak{g}$, set $x=r^{\sharp}(a)$, $y=r^{\sharp}(b)$, $z=r^{\sharp}(c)$. Since $r^{\sharp}\,\phi_{{}_{\mathfrak{g}}}^{*}=\phi_{{}_{\mathfrak{g}}}\,r^{\sharp}$, we have

[TABLE]

By Lemma 4.9, if $r$ satisfies the classical Hom-Yang-Baxter equation, then we have

[TABLE]

The converse can be proved similarly. ∎

5. Operator forms of the classical Hom-Yang-Baxter equation

In this section, we give a further interpretation of the classical Hom-Yang-Baxter equation. We discuss the relationship between an $\mathcal{O}$-operator associated to an arbitrary weakly involutive representation and the classical Hom-Yang-Baxter equation, which leads to a construction of solutions of the classical Hom-Yang-Baxter equation by means of $\mathcal{O}$-operators and Hom-left-symmetric algebras.

Definition 5.1** ([16]).**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a representation of $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$. A linear map $T:V\longrightarrow\mathfrak{g}$ is called an $\mathcal{O}$-operator associated to $(V,\beta,\rho)$ if $T$ satisfies

• (i)

$T\,\beta=\phi_{{}_{\mathfrak{g}}}\,T$,

• (ii)

$[T(u),T(v)]_{\mathfrak{g}}=T(\rho(T(u))v-\rho(T(v))u)$, $\forall\ u,v\in V$.

Example 5.2**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra. Suppose that $r\in\wedge^{2}\mathfrak{g}$ satisfies Eq. (4.3). By Lemma 4.9, $r$ satisfies the classical Hom-Yang-Baxter equation (4.7) if $r^{\sharp}$ is an $\mathcal{O}$-operator associated to the representation $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$. Conversely, if $r$ satisfies the classical Hom-Yang-Baxter equation (4.7) and in addition, $\phi_{{}_{\mathfrak{g}}}$ is invertible, then $r^{\sharp}$ is an $\mathcal{O}$-operator associated to the representation $(\mathfrak{g}^{*},\phi_{{}_{\mathfrak{g}}}^{*},\mathrm{ad}^{\circ})$.

There is a class of Hom-Lie algebras coming from the following structure:

Definition 5.3** ([13, 19]).**

A Hom-left-symmetric algebra is a triple $(V,\cdot,\psi)$ consisting of a linear space $V$, a bilinear map $\cdot:V\otimes V\longrightarrow V$ and an algebra homomorphism $\psi:V\longrightarrow V$ satisfying

[TABLE]

Indeed,

Proposition 5.4** ([16]).**

Let $(V,\cdot,\psi)$ be a Hom-left-symmetric algebra. Then

• (i)

$(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$* is a Hom-Lie algebra, where $\mathfrak{g}(V)=V$ as a vector space, and the bracket $[\cdot,\cdot]_{V}$ is given by*

[TABLE]

We call $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ the commutator Hom-Lie algebra.

• (ii)

Let $L:V\longrightarrow\mathfrak{gl}(V)$ be the linear map with $u\mapsto L_{u}$, where $L_{u}(v)=u\cdot v$ for any $u,v\in V$. Then $(V,\psi,L)$ is a representation of the Hom-Lie algebra $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ on $V$.

Example 5.5**.**

Let $(V,\cdot,\psi)$ be a Hom-left-symmetric algebra. Then $\mathrm{Id}$ is an $\mathcal{O}$-operator of the commutator Hom-Lie algebra $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ associated to the representation $(V,\psi,L)$.

Corollary 5.6**.**

Let $(V,\cdot,\psi)$ be a Hom-left-symmetric algebra. Then $(V,\psi,L)$ is a weakly involutive representation of the commutator Hom-Lie algebra $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ if and only if

[TABLE]

Under this condition, $\psi^{2}$ is an $\mathcal{O}$-operator of $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ associated to $(V,\psi,L)$.

Proof.

By definition, $(V,\psi,L)$ is a weakly involutive representation of $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ if and only if $L_{u}=L_{\psi^{2}(u)}$ for all $u\in V$, i.e. $u\cdot v=\psi^{2}(u)\cdot v$ for all $u,v\in V$. Under this condition, we have

[TABLE]

and $\psi^{2}\,\psi=\psi\,\psi^{2}$. Hence the conclusion holds. ∎

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a weakly involutive representation. Then we have the semidirect product Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$, and any linear map $T:V\longrightarrow\mathfrak{g}$ can be viewed as an element $\overline{T}\in(\mathfrak{g}\oplus V^{*})\otimes(\mathfrak{g}\oplus V^{*})$ via

[TABLE]

Then $r=\overline{T}-\sigma(\overline{T})$ is skew-symmetric.

Lemma 5.7**.**

With the above notations, let $T:V\longrightarrow\mathfrak{g}$ be a linear map satisfying $T\,\beta=\phi_{{}_{\mathfrak{g}}}\,T$. Then $r=\overline{T}-\sigma(\overline{T})$ satisfying $(\phi_{{}_{d}}\otimes\mathrm{Id})r=(\mathrm{Id}\otimes\phi_{{}_{d}})r$, where $\phi_{{}_{d}}=\phi_{{}_{\mathfrak{g}}}\oplus\beta^{*}$.

Proof.

Let $\{v_{1},\ldots,v_{n}\}$ be a basis of $V$ and $\{v^{1},\ldots,v^{n}\}$ be its dual basis. Use the Einstein summation convention, $r$ can be expressed by $r=v^{i}\otimes T(v_{i})-T(v_{i})\otimes v^{i}$. Then we have

[TABLE]

Since

[TABLE]

we get $\beta^{*}(v^{i})\otimes T(v_{i})=v^{i}\otimes T\,\beta(v_{i})$. Similarly, we have $T(v_{i})\otimes\beta^{*}(v^{i})=T\,\beta(v_{i})\otimes v^{i}$. Hence the conclusion holds. ∎

Theorem 5.8**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a Hom-Lie algebra and $(V,\beta,\rho)$ be a weakly involutive representation. Let $T:V\longrightarrow\mathfrak{g}$ be a linear map satisfying $T\,\beta=\phi_{{}_{\mathfrak{g}}}\,T$. Then $r=\overline{T}-\sigma(\overline{T})$ is a solution of the classical Hom-Yang-Baxter equation in the Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$ if $T$ is an $\mathcal{O}$-operator associated to $(V,\beta,\rho)$.

Proof.

Let $\{v_{1},\ldots,v_{n}\}$ be a basis of $V$ and $\{v^{1},\ldots,v^{n}\}$ be its dual basis. Use the Einstein summation convention, $\overline{T}$ can be expressed by $\overline{T}=v^{i}\otimes T(v_{i})$. Then we have

[TABLE]

Set

[TABLE]

Note that

[TABLE]

and

[TABLE]

Then we get

[TABLE]

Therefore $r=\overline{T}-\sigma(\overline{T})$ is a solution of the classical Hom-Yang-Baxter equation in the Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$ if $T$ is an $\mathcal{O}$-operator associated to $(V,\beta,\rho)$. ∎

Remark 5.9**.**

Suppose that in addition, $\phi_{{}_{\mathfrak{g}}}$ in Theorem 5.8 is invertible. Then $r=\overline{T}-\sigma(\overline{T})$ is a solution of the classical Hom-Yang-Baxter equation in the Hom-Lie algebra $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$ if and only if $T$ is an $\mathcal{O}$-operator associated to $(V,\beta,\rho)$.

Combining Corollary 2.11, Theorem 4.4, Lemma 5.7 and Theorem 5.8, we get the following conclusion.

Corollary 5.10**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{{}_{\mathfrak{g}}})$ be a weakly involutive Hom-Lie algebra and $(V,\beta,\rho)$ be a weakly involutive representation satisfying $\rho(\phi_{\mathfrak{g}}(x))\,\beta^{2}=\rho(\phi_{\mathfrak{g}}(x))$ for all $x\in\mathfrak{g}$. Let $T:V\longrightarrow\mathfrak{g}$ be an $\mathcal{O}$-operator associated to $(V,\beta,\rho)$. Then there is a coboundary Hom-Lie bialgebra structure on $\mathfrak{g}\ltimes_{\rho^{\circ}}V^{*}$ induced by $r=\overline{T}-\sigma(\overline{T})$.

Corollary 5.11**.**

Let $(V,\cdot,\psi)$ be a Hom-left-symmetric algebra. Suppose that

[TABLE]

Let $\{v_{1},\ldots,v_{n}\}$ be a basis of $V$ and $\{v^{1},\ldots,v^{n}\}$ be its dual basis. Then we have the following conclusions.

1. (a)

$r_{1}=v^{i}\wedge v_{i}$* and $r_{2}=v^{i}\wedge\psi^{2}(v_{i})$ are solutions of the classical Hom-Yang-Baxter equation in the Hom-Lie algebra $\mathfrak{g}(V)\ltimes_{L^{\circ}}V^{*}$, where $L^{\circ}:=L^{*}\,\phi_{\mathfrak{g}}$ with the notation in Eq. (2.7).* 2. (b)

Suppose in addition that the commutator Hom-Lie algebra $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ is weakly involutive. Then there are coboundary Hom-Lie bialgebra structures on $\mathfrak{g}(V)\ltimes_{L^{\circ}}V^{*}$ induced by $r_{1}=v^{i}\wedge v_{i}$ and $r_{2}=v^{i}\wedge\psi^{2}(v_{i})$ respectively. Moreover, these two induced coboundary Hom-Lie bialgebra structures are the same.

Proof.

(a) By Corollary 5.6, $(V,\psi,L)$ is a weakly involutive representation of the Hom-Lie algebra $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$. Moreover, both the identity map $\mathrm{Id}$ and $\psi^{2}$ are $\mathcal{O}$-operators of $\mathfrak{g}(V)$ associated to $(V,\psi,L)$. Therefore, the conclusion follows from Theorem 5.8.

(b) In fact, $(\mathfrak{g}(V),[\cdot,\cdot]_{V},\psi)$ is weakly involutive if and only if

[TABLE]

Hence we get $v\cdot\psi^{2}(u)=v\cdot u$ for all $u,v\in V$, i.e., $L_{v}\,\psi^{2}=L_{v}$ for all $v\in V$. Note that the identity map $\mathrm{Id}$ and $\psi^{2}$ are $\mathcal{O}$-operators of $\mathfrak{g}(V)$ associated to $(V,\psi,L)$. By Corollary 5.10, there are coboundary Hom-Lie bialgebra structures on $\mathfrak{g}(V)\ltimes_{L^{\circ}}V^{*}$ induced by $r_{1}$ and $r_{2}$ respectively. Define linear maps $\Delta_{k}$ $(k=1,2)$ by

[TABLE]

[TABLE]

For any $u,v\in V$ and $\xi,\eta\in V^{*}$, we have

[TABLE]

Since

[TABLE]

we get $L^{\circ}_{u}\,(\psi^{*})^{2}=L^{\circ}_{u}$. Then we have

[TABLE]

Hence we get $L^{\circ}_{u}(v^{i})\otimes\psi^{3}(v_{i})=L^{\circ}_{u}(v^{i})\otimes\psi(v_{i})$. Similarly, we have $\psi^{3}(v_{i})\otimes L^{\circ}_{u}(v^{i})=\psi(v_{i})\otimes L^{\circ}_{u}(v^{i})$. Thus we get $\Delta_{1}=\Delta_{2}$. Hence the conclusion holds. ∎

Acknowledgements The authors acknowledge supports from the National Natural Science Foundation of China (Grant Nos. 11425104 and 11771190) and the Fundamental Research Funds for the Central Universities.