(Almost) Ricci Solitons in Lorentzian–Sasakian Hom-Lie Groups

Abstract

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional $\mathfrak{sl}(2,\mathbb{R})$ and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci curvature tensor is v-parallel are given. Ricci solitons for Lorentzian–Sasakian Hom-Lie algebras are also studied. It is shown that a Ricci soliton vector field $\zeta$ is conformal whenever the Lorentzian–Sasakian Hom-Lie algebra is Ricci semisymmetric. To illustrate the use of the theory, a two-parameter family of three-dimensional Lorentzian–Sasakian Hom-Lie algebras which are not Lie algebras is given and their Ricci solitons are computed.

1. Introduction

Lorentzian geometry was born as a geometric theory in which general relativity can be expressed mathematically. A Lorentzian manifold is a subclass of the pseudo-Riemannian manifolds in which the signature of the metric is $(1,n-1)$. Such metrics are called Lorentzian metrics and play an important role in mathematical physics, especially in the development of the theory of relativity and cosmology.

Lorentzian geometry has been very extensively studied and constitutes a very active area of research in differential geometry and mathematical physics. It is noteworthy that many mathematical branches are involved in this field such as functional analysis, geometric analysis, Lie groups and Lie algebras. For some recent results on Lorentzian geometry, we may refer to [1,2,3,4] and references therein.

Contact geometry is an essential tool for many theoretical physicists, particularly in the study of mechanics, thermodynamics and electrodynamics, gauge fields and gravity. The relevance of contact pseudo-Riemannian structures for physics was pointed out in [5,6]. Odd-dimensional almost-contact manifolds were introduced by Gray in 1959 (see [7]). A contact manifold M satisfies ${\mathcal{F}}^2=-Id+u\otimes v$, where v is a nowhere-vanishing vector field, u is a 1-form satisfying $u\left(v\right)=1$ (called a contact form), and $\mathcal{F}$ is a tensor of type $(1,1)$ on M. If M is also equipped with a Lorentzian metric g such that $g(\mathcal{F}{z}_1,\mathcal{F}{z}_2)=g(z_1,z_2)+u\left(z_1\right)u\left(z_2\right)$ for all vector fields $z_1$ and $z_2$ on M, $(\mathcal{F},u,v,g)$ is called a Lorentzian almost-contact structure on M. Moreover, M is called a Lorentzian contact metric manifold if $du=2\Phi$, where $\Phi (z_1,z_2)=g(z_1,\mathcal{F}{z}_2)$. A Lorentzian almost-contact structure is called normal if the Nijenhuis tensor $N_{\mathcal{F}}$ associated to the tensor $\mathcal{F}$, is given by $N_{\mathcal{F}}=-du\otimes v$. A Lorentzian normal contact manifold M is called a Lorentzian–Sasakian manifold (see [8,9]).

In the above statements, if M is a Lie group H, then the metric g, $(1,1)$ tensor $\mathcal{F}$, the vector field v and 1-form u are left-invariant, which is given by their restrictions to the Lie algebra $\mathfrak{h}$ of H. In this situation, $(\mathfrak{h},{\mathcal{F}}_e,u_e,v_e,g_e)$ is called a Lorentzian almost-contact Lie algebra.

Hom-Lie algebras originated in the study of Virasoro and Witt algebras in [10], which are a generalization of Lie algebras. It is known that some q deformations of the Witt and the Virasoro algebras carry the structure of a Hom-Lie algebra [10,11]. These algebras play a chief role in research fields (for instance, see [12,13,14,15,16,17]).

Hom-groups were recently introduced in [18]. Shortly after, Hom-Lie groups were given in [19]. A Hom-Lie group is a Hom-group $(H,\diamond ,e_{\psi },\psi )$ such that H is a smooth manifold, the Hom-group operations are smooth maps, and the underlying structure map $\psi :H\to H$ is a diffeomorphism. Recently, many scholars have been very interested in the geometric and algebraic problems in Hom-Lie groups, Hom-Lie algebras and dependent spaces (see [20,21,22]). For instance, in [22], the authors showed that any Sasakian Hom-Lie algebra is a K-contact Hom-Lie algebra.

A Ricci soliton is a Riemannian manifold $(M,g)$ admitting a smooth vector field V such that

$$\begin{array}{c}\hfill {\pounds }_{V}g+2Ric+2\lambda g=0,\end{array}$$

where $Ric$ is the Ricci tensor, ${\pounds }_V$ denotes the Lie derivative operator in the direction of V and $\lambda$ is a constant. The Ricci soliton is said to be shrinking, steady or expanding depending on whether $\lambda$ is negative, zero or positive, respectively. In [23], Sharma studied Ricci solitons in K-contact manifolds and showed that a complete K-contact gradient soliton is compact Einstein and Sasakian. Recently, Ashoka and Bagewadi studied Ricci solitons in $\alpha$-Sasakian manifolds [24].

In this paper, we study Lorentzian almost-contact, K-contact and Lorentzian–Sasakian structures on Hom-Lie groups by using the corresponding Hom-Lie algebras. We also study the (almost) Ricci solitons in Lorentzian–Sasakian Hom-Lie algebras.

Notice that in this paper, we work over the field $\mathbb{R}$.

2. Lorentzian Almost-Contact Hom-Lie Algebras

Consider a linear space $\mathfrak{h}$ equipped with a skew-symmetric bilinear map (bracket) ${[·,·]}_{\mathfrak{h}}:\mathfrak{h}\times \mathfrak{h}\to \mathfrak{h}$ and an algebra morphism ${\varrho }_{\mathfrak{h}}:\mathfrak{h}\to \mathfrak{h}$. The triple $(\mathfrak{h},{[·,·]}_{\mathfrak{h}},{\varrho }_{\mathfrak{h}})$ is called a Hom-Lie algebra if

$${[{\varrho }_{\mathfrak{h}}\left(u_1\right),{[u_2,u_3]}_{\mathfrak{h}}]}_{\mathfrak{h}}+{[{\varrho }_{\mathfrak{h}}\left(u_2\right),{[u_3,u_1]}_{\mathfrak{h}}]}_{\mathfrak{h}}+{[{\varrho }_{\mathfrak{h}}\left(u_3\right),{[u_1,u_2]}_{\mathfrak{h}}]}_{\mathfrak{h}}=0,$$

for all $u_1,u_2,u_3\in \mathfrak{h}$. Moreover, the Hom-Lie algebra is said to be regular (involutive) if ${\varrho }_{\mathfrak{h}}$ is an invertible map (${\varrho }_{\mathfrak{h}}^2=I{d}_{\mathfrak{h}}$). In the following, we always assume that all Hom-Lie algebras are regular.

Let H be a differential manifold. We consider a smooth map $\psi :H\to H$ and its pullback map ${\psi }^{\ast }:C^{\infty }\left(H\right)\to C^{\infty }\left(H\right)$, which is a morphism of the function ring $C^{\infty }\left(H\right)$. We denote by $\Gamma \left(E\right)$ the $C^{\infty }\left(H\right)$ module of the sections of a vector bundle map $E\to H$. Moreover, if an algebra morphism ${\varrho }_E:\Gamma \left(E\right)\to \Gamma \left(E\right)$ satisfies ${\varrho }_E\left(f{z}_1\right)={\psi }^{\ast }\left(f\right){\varrho }_A\left(z_1\right),$ for any $z_1\in \Gamma \left(E\right)$ and $f\in C^{\infty }\left(H\right)$, the triple $(E\to H,\psi ,{\varrho }_E)$ is called a Hom-bundle [21]. As an example, the triple $({\psi }^{!}TH,\psi ,A{d}_{{\psi }^{\ast }})$ forms a Hom-bundle where ${\psi }^{!}TH$ is the pullback bundle of the tangent bundle $TH$ along the diffeomorphism $\psi :H\to H$ and $A{d}_{{\psi }^{\ast }}:\Gamma \left({\psi }^{!}TH\right)\to \Gamma \left({\psi }^{!}TH\right)$ is determined by $A{d}_{{\psi }^{\ast }}\left(z_1\right)={\psi }^{\ast }\circ z_1\circ {\left({\psi }^{\ast }\right)}^{-1},$ for all $z_1\in \Gamma \left({\psi }^{!}TH\right)$. One sees immediately that $(\Gamma \left({\psi }^{!}TH\right),{[·,·]}_{\psi },A{d}_{{\psi }^{\ast }})$ is a Hom-Lie algebra such that

$$\begin{array}{cc}\hfill {[z_1,z_2]}_{\psi }& ={\psi }^{\ast }\circ z_1\circ {\left({\psi }^{-1}\right)}^{\ast }\circ z_2\circ {\left({\psi }^{-1}\right)}^{\ast }-{\psi }^{\ast }\circ z_2\circ {\left({\psi }^{-1}\right)}^{\ast }\circ z_1\circ {\left({\psi }^{-1}\right)}^{\ast },\hfill \end{array}$$

for all $z_1,z_2\in \Gamma \left({\psi }^{!}TH\right)$. A Hom-group $(H,\diamond ,e_{\psi },\psi )$ is called a Hom-Lie group if H is also a smooth manifold such that the map $\psi :H\to H$ is a diffeomorphism and the product and inversion operations are smooth maps [19]. Let denote by ${\mathfrak{h}}^{!}$ the fibre of $e_{\psi }$ in the pullback bundle ${\psi }^{!}TH$. Thus, ${\psi }^{!}{T}_{e_{\psi }}H={\mathfrak{h}}^{!}$ and also ${\mathfrak{h}}^{!}$ is in one-to-one correspondence with ${\Gamma }_L\left({\psi }^{!}TH\right)=\{z_1\in \Gamma \left({\psi }^{!}TH\right)|{z_1}_{\mathfrak{g}}={(l_{\mathfrak{g}}\circ {\psi }^{-1})}_{{\ast }_{e_{\psi }}}({z_1}_{e_{\psi }}),\forall \mathfrak{g}\in H\}$ (see [19]). In addition, considering a bracket ${[·,·]}_{{\mathfrak{h}}^{!}}$ and the isomorphisms ${\varrho }_{{\mathfrak{h}}^{!}}:{\mathfrak{h}}^{!}\to {\mathfrak{h}}^{!}$ by ${[z_1\left(e_{\psi }\right),z_2\left(e_{\psi }\right)]}_{{\mathfrak{h}}^{!}}={[z_1,z_2]}_{\psi }\left(e_{\psi }\right)$ and ${\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\left(e_{\psi }\right)\right)=\left(A{d}_{{\psi }^{\ast }}\left(z_1\right)\right)\left(e_{\psi }\right),$ respectively, for all $z_1,z_2\in {\Gamma }_L\left({\psi }^{!}TH\right)$, the triple $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ forms a Hom-Lie algebra which is isomorphic to the Hom-Lie algebra $({\Gamma }_L\left({\psi }^{!}TH\right),{[·,·]}_{\psi },A{d}_{{\psi }^{\ast }})$.

On a $(2n+1)$-dimension Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$, an almost-contact structure satisfies the following conditions [22]:

$$\begin{array}{c}\hfill {({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2=-I{d}_{{\mathfrak{h}}^{!}}+u\otimes v,u\left(v\right)=1,{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}=\mathcal{F}\circ {\varrho }_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}^2\left(v\right)=v,\end{array}$$

(1)

where $\mathcal{F}\in {\mathcal{T}}_1^1\left({\mathfrak{h}}^{!}\right)$, $v\in {\mathfrak{h}}^{!}$ and $u\in {\mathfrak{h}}^{!\ast }$. It follows that

$$\begin{array}{c}\hfill ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})v=0,u\circ {\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}=0,rank({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})=2n.\end{array}$$

(2)

Considering a finite-dimensional Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$, a pseudo-Riemannian metric on ${\mathfrak{h}}^{!}$ is a bilinear symmetric nondegenerate form $\langle ·,·\rangle$ which satisfies

$$\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle =\langle z_1,z_2\rangle ,\forall z_1,z_2\in {\mathfrak{h}}^{!}.$$

(3)

Definition 1.

A pseudo-Riemannian metric Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\langle ·,·\rangle )$ is said to be Lorentzian if the signature $\langle ·,·\rangle$ is $(-,+,+,\cdots ,+)$, i.e., a matrix representation of $\langle ·,·\rangle$ has one negative eigenvalue and all other eigenvalues are positive. A nonzero tensor $z_1\in {\mathfrak{h}}^{!}$ is called space-like, time-like and null if it satisfies $\langle z_1,z_1\rangle >0$, $\langle z_1,z_1\rangle <0$ and $\langle z_1,z_1\rangle =0$, respectively.

Definition 2.

On a Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$, a Lorentzian metric $\langle ·,·\rangle$ is said to be compatible with the almost-contact structure $(\mathcal{F},v,u)$ if

$$\begin{array}{cc}\hfill \langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =& \langle z_1,z_2\rangle +u\left(z_1\right)u\left(z_2\right),\forall z_1,z_2\in {\mathfrak{h}}^{!}.\hfill \end{array}$$

(4)

In this case, the quadruple $(\mathcal{F},v,u,\langle ·,·\rangle )$ is called a Lorentzian almost-contact structure and ${\mathfrak{h}}^{!}$ is said to be a Lorentzian almost-contact Hom-Lie algebra.

For a Lorentzian almost-contact Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$, (4) implies $u\left(z_1\right)=-\langle z_1,v\rangle$. So, v is a time-like tensor. We consider a local basis $\{e_1,\cdots ,e_{2n},v\}$ for ${\mathfrak{h}}^{!}$, such that

$$\langle e_i,e_j\rangle ={\psi }_{ij},\langle v,v\rangle =-1,$$

i.e., $\{e_1,\cdots ,e_{2n}\}$ are space-like tensors. Let $e_1\in {\mathfrak{h}}^{!}$ be orthogonal to v and $|e_1{|}^2=1$. Then, $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_1$ is orthogonal to $e_1$ and v, and $|({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_1{|}^2=1$. By choosing $e_2$ orthogonal to v, $e_1$ and $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_1$, then $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_2$ is orthogonal to v, $e_1$, $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(e_1\right)$ and $e_2$ such that $|({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_2{|}^2=1$. Proceeding in this way, we obtain an orthonormal basis ${\{e_i,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,v\}}_{i=1}^n$, i.e., an $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})$ basis.

Example 1.

We consider the Heisenberg Hom-Lie algebra $(\mathcal{H},{[·,·]}_{\mathcal{H}},{\varrho }_{\mathcal{H}})$ spanned by

$$e_1=\sqrt{2}\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),e_2=\sqrt{2}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),e_3=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),$$

where the bracket ${[·,·]}_{\mathcal{H}}$ on $\mathcal{H}$ is determined by

$$\begin{array}{c}\hfill {[e_1,e_2]}_{\mathcal{H}}=2e_3,{[e_2,e_3]}_{\mathcal{H}}=0,{[e_1,e_3]}_{\mathcal{H}}=0.\end{array}$$

Describing the linear map ${\varrho }_{\mathcal{H}}$ ${\varrho }_{\mathcal{H}}\left(e_1\right)=e_2,{\varrho }_{\mathcal{H}}\left(e_2\right)=-e_1$ and ${\varrho }_{\mathcal{H}}\left(e_3\right)=e_3,$ we set $v=e_3$ and $u=e^3$. Defining the map $\mathcal{F}$ on $\mathcal{H}$ as $\mathcal{F}\left(e_1\right)=e_1,\mathcal{F}\left(e_2\right)=e_2$ and $\mathcal{F}\left(e_3\right)=0,$ it follows that

$$\begin{array}{cc}& ({\varrho }_{\mathcal{H}}\circ \mathcal{F})e_1=e_2=(\mathcal{F}\circ {\varrho }_{\mathcal{H}})e_1,({\varrho }_{\mathcal{H}}\circ \mathcal{F})e_2=-e_1=(\mathcal{F}\circ {\varrho }_{\mathcal{H}})e_2,\hfill \\ & ({\varrho }_{\mathcal{H}}\circ \mathcal{F})e_3=0=(\mathcal{F}\circ {\varrho }_{\mathcal{H}})e_3,\hfill \end{array}$$

and

$$\begin{array}{cc}& {({\varrho }_{\mathcal{H}}\circ \mathcal{F})}^2{e}_1=-e_1=-e_1+u\left(e_1\right)v,{({\varrho }_{\mathcal{H}}\circ \mathcal{F})}^2{e}_2=-e_2=-e_2+u\left(e_2\right)v,\hfill \\ & {({\varrho }_{\mathcal{H}}\circ \mathcal{F})}^{2}v=0=-v+u\left(v\right)v.\hfill \end{array}$$

Hence, $(\mathcal{H},{[·,·]}_{\mathcal{H}},{\varrho }_{\mathcal{H}},\mathcal{F},v,u)$ is an almost-contact Hom-Lie algebra. By describing a bilinear symmetric nondegenerate form $\langle ·,·\rangle$ as $\langle e_1,e_1\rangle =\langle e_2,e_2\rangle =-\langle e_3,e_3\rangle =1,$ then for all $i,j=1,2,3$, we obtain

$$\begin{array}{c}\hfill \langle {\varrho }_{\mathcal{H}}\left(e_i\right),{\varrho }_{\mathcal{H}}\left(e_j\right)\rangle =0=\langle e_i,e_j\rangle ,\langle ({\varrho }_{\mathcal{H}}\circ \mathcal{F})\left(e_i\right),({\varrho }_{\mathcal{H}}\circ \mathcal{F})\left(e_j\right)\rangle =0=\langle e_i,e_j\rangle +u\left(e_i\right)u\left(e_j\right),\end{array}$$

except

$$\begin{array}{cc}\hfill & \langle {\varrho }_{\mathcal{H}}\left(e_1\right),{\varrho }_{\mathcal{H}}\left(e_1\right)\rangle =1=\langle e_1,e_1\rangle ,\langle {\varrho }_{\mathcal{H}}\left(e_2\right),{\varrho }_{\mathcal{H}}\left(e_2\right)\rangle =1=\langle e_2,e_2\rangle ,\hfill \\ & \langle {\varrho }_{\mathcal{H}}\left(e_3\right),{\varrho }_{\mathcal{H}}\left(e_3\right)\rangle =-1=\langle e_3,e_3\rangle ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \langle ({\varrho }_{\mathcal{H}}\circ \mathcal{F})\left(e_1\right),({\varrho }_{\mathcal{H}}\circ \mathcal{F})\left(e_1\right)\rangle =1=\langle e_1,e_1\rangle +u\left(e_1\right)u\left(e_1\right),\hfill \\ \hfill & \langle ({\varrho }_{\mathcal{H}}\circ \mathcal{F})\left(e_2\right),({\varrho }_{\mathcal{H}}\circ \mathcal{F})\left(e_2\right)\rangle =1=\langle e_2,e_2\rangle +u\left(e_2\right)u\left(e_2\right).\hfill \end{array}$$

Therefore, $(\mathcal{H},{[·,·]}_{\mathcal{H}},{\varrho }_{\mathcal{H}},\mathcal{F},v,u,\langle ·,·\rangle )$ is a Lorentzian almost-contact Hom-Lie algebra.

Example 2.

Consider that the Hom-Lie algebra $(\mathfrak{h}=\mathfrak{sl}(2,\mathbb{R}),{[·,·]}_{\mathfrak{h}},{\varrho }_{\mathfrak{h}})$ consists of traceless $2\times 2$ matrices with entries in $\mathbb{R}$ with an orthonormal basis

$$e_1=2\sqrt{2}\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),e_2=2\sqrt{2}\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),e_3=4\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$$

such that the bracket ${[·,·]}_{\mathfrak{h}}$ and the linear map ${\varrho }_{\mathfrak{h}}$ on $\mathfrak{h}$ are defined by

$$\begin{array}{c}\hfill {[e_1,e_2]}_{\mathfrak{h}}=2e_3,{[e_2,e_3]}_{\mathfrak{h}}=8e_2,{[e_1,e_3]}_{\mathfrak{h}}=-8e_1,\end{array}$$

and

$${\varrho }_{\mathfrak{h}}\left(e_1\right)=-e_1,{\varrho }_{\mathfrak{h}}\left(e_2\right)=-e_2,{\varrho }_{\mathfrak{h}}\left(e_3\right)=e_3.$$

We set $v=e_3$ and $u=e^3$. We define $\mathcal{F}\left(e_1\right)=-e_2,\mathcal{F}\left(e_2\right)=e_1,\mathcal{F}\left(e_3\right)=0$, $\langle e_1,e_1\rangle =\langle e_2,e_2\rangle =1$ and $\langle e_3,e_3\rangle =-1.$ Therefore, $\mathfrak{h}=\mathfrak{sl}(2,\mathbb{R})$ forms a Lorentzian almost-contact Hom-Lie algebra.

In a Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$, the Nijenhuis torsion of an algebra morphism $F:{\mathfrak{h}}^{!}\to {\mathfrak{h}}^{!}$ for any $z_1,z_2\in {\mathfrak{h}}^{!}$ is determined by

$$\begin{array}{cc}\hfill N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ F}(z_1,z_2)=& {({\varrho }_{{\mathfrak{h}}^{!}}\circ F)}^2{[z_1,z_2]}_{{\mathfrak{h}}^{!}}+{[({\varrho }_{{\mathfrak{h}}^{!}}\circ F)z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ F)z_2]}_{{\mathfrak{h}}^{!}}\hfill \\ & -{\varrho }_{{\mathfrak{h}}^{!}}\circ F{[({\varrho }_{{\mathfrak{h}}^{!}}\circ F)z_1,z_2]}_{{\mathfrak{h}}^{!}}-{\varrho }_{{\mathfrak{h}}^{!}}\circ F{[z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ F)z_2]}_{{\mathfrak{h}}^{!}}.\hfill \end{array}$$

(5)

Considering the Hom-Lie algebra ${\mathfrak{h}}^{!}{⋉}_0\mathbb{R}=({\mathfrak{h}}^{!}\oplus \mathbb{R},[·,·],\gamma ={\varrho }_{{\mathfrak{h}}^{!}}\oplus {\beta }_{\mathbb{R}})$, where ${\beta }_{\mathbb{R}}^2=I{d}_{\mathbb{R}}$, we describe the isomorphism $J:{\mathfrak{h}}^{!}\oplus \mathbb{R}\to {\mathfrak{h}}^{!}\oplus \mathbb{R}$ as $J(z_1,k)=(\mathcal{F}\left(z_1\right)-k{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),u\left(z_1\right)),$ where $z_1\in {\mathfrak{h}}^{!}$ and $k\in \mathbb{R}$. The almost-contact structure $(\mathcal{F},v,u)$ is said to be normal if and only if the almost-complex structure J is integrable, i.e., $N_{\gamma \circ J}=0$. Hence, we obtain

$$\begin{array}{cc}\hfill & N_{\gamma \circ J}((z_1,0),(z_2,0))=(N^{\left(1\right)}(z_1,z_2),N^{\left(2\right)}(z_1,z_2){\beta }_{\mathbb{R}}\left(1\right)),\hfill \\ \hfill & N_{\gamma \circ J}((z_1,0),(0,1))=(N^{\left(3\right)}\left(z_1\right),N^{\left(4\right)}\left(z_1\right){\beta }_{\mathbb{R}}\left(1\right)),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill N^{\left(1\right)}(z_1,z_2)& :=N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(z_1,z_2)+du(z_1,z_2)v,N^{\left(3\right)}\left(z_1\right):=\left({\pounds }_v({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_1,\hfill \\ \hfill N^{\left(2\right)}(z_1,z_2)& :=({\pounds }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}u)z_2-({\pounds }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2}u)z_1,N^{\left(4\right)}\left(z_1\right):=\left({\pounds }_{v}u\right)z_1,\hfill \end{array}$$

(6)

and the Lie derivative operator £ is determined by ${\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}}=i\circ d+d\circ i_{{\varrho }_{{\mathfrak{h}}^{!}}}$ [22]. Moreover, the vanishing of $N^{\left(1\right)}=0$ yields $N^{\left(2\right)}=N^{\left(3\right)}=N^{\left(4\right)}=0$. So, $N^{\left(1\right)}=0$ is a necessary and sufficient condition for the integrability of J.

Definition 3.

A Lorentzian almost-contact structure $(\mathcal{F},v,u,\langle ·,·\rangle )$ on a Hom-Lie algebra

$({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ is called a Lorentzian contact structure if $2\Phi =du$, where Φ is a skew-symmetric 2-form given by

$$\begin{array}{c}\hfill \Phi (z_1,z_2)=\langle z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle ,\forall z_1,z_2\in {\mathfrak{h}}^{!}.\end{array}$$

(7)

An immediate corollary of the above is that

$$\begin{array}{c}\hfill du(z_1,v)=0=du(z_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)),du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2)=du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_1),\end{array}$$

(8)

and

$${\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}\Phi =0={\pounds }_v\Phi .$$

Lemma 1.

Let $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ be a Hom-Lie algebra equipped with a Lorentzian almost-contact structure $(\mathcal{F},v,u,\langle ·,·\rangle )$. Then,

$$\begin{array}{cc}\hfill 2\langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle & =d\Phi (z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3)-d\Phi (z_1,z_2,z_3)\hfill \\ \hfill & +\langle N^{\left(1\right)}(z_2,z_3),{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle -N^{\left(2\right)}(z_2,z_3)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_1\right)\hfill \\ & -du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_1)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_3\right)+du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_2\right),\hfill \end{array}$$

(9)

for all $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$, where ∇ is the Hom-Levi-Civita connection.

Proof. 

Since

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2={\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\nabla }_{z_1}{z}_2,\end{array}$$

(10)

we can write

$$\begin{array}{c}\hfill \langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =\langle {\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle +\langle {\nabla }_{z_1}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle ,\end{array}$$

for all $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$. By Koszul’s formula, we have [17]

$$2\langle {\nabla }_{z_1}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =\langle {[z_1,z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle +\langle {[z_3,z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\rangle +\langle {[z_3,z_1]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle .$$

(11)

Thus, the above two equations imply

$$\begin{array}{cc}\hfill & 2\langle {\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle +2\langle {\nabla }_{z_1}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle =\langle {[z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle \hfill \\ \hfill & +\langle {[z_3,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\rangle +\langle {[z_3,z_1]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle ,+\langle {[z_1,z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle \hfill \\ \hfill & +\langle {[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\rangle +\langle {[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle .\hfill \end{array}$$

On the other hand, we have [20]

$$\begin{array}{cc}\hfill d\omega (z_1,z_2,z_3)=& -\omega ({[z_1,z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))+\omega ({[z_1,z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right))-\omega ({[z_2,z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right));\hfill \end{array}$$

thus, using (7), it follows that

$$\begin{array}{cc}\hfill d\Phi (z_1,z_2,z_3)=& -\langle {[z_1,z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle -\langle {[z_3,z_1]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle \hfill \\ \hfill & -\langle {[z_2,z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & d\Phi (z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3)=\langle {[z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle \hfill \\ \hfill & +u\left({[z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2]}_{{\mathfrak{h}}^{!}}\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_3\right)+\langle {[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}{z}_2\rangle \hfill \\ \hfill & +u\left({[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1]}_{{\mathfrak{h}}^{!}}\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_2\right)-\langle {[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle .\hfill \end{array}$$

Using (6), we also obtain

$$\begin{array}{cc}\hfill & \langle N^{\left(1\right)}(z_2,z_3),{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle =\langle {[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3]}_{{\mathfrak{h}}^{!}}-{[z_2,z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle \hfill \\ \hfill & -\langle {[z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}{z}_1)\rangle -\{u\left({[z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3]}_{{\mathfrak{h}}^{!}}\right)\hfill \\ \hfill & +u\left({[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_3]}_{{\mathfrak{h}}^{!}}\right)\}u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_1\right)-\langle {[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_3]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}}{z}_1)\rangle ,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill N^{\left(2\right)}(z_2,z_3)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_1\right)=u\left({[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_2]}_{{\mathfrak{h}}^{!}}\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_1\right)-u\left({[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_3]}_{{\mathfrak{h}}^{!}}\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_1\right).\end{array}$$

Moreover, we have

$$\begin{array}{cc}\hfill du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_1)& u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_3\right)-du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_2\right)\hfill \\ \hfill =& u\left({[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1]}_{{\mathfrak{h}}^{!}}\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_2\right)-u\left({[({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_1]}_{{\mathfrak{h}}^{!}}\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_3\right).\hfill \end{array}$$

From the above equations, we have the assertion. □

The Lie derivative of a pseudo-Riemannian metric $\langle ·,·\rangle$ is described by

$$\begin{array}{cc}\hfill \left({\pounds }_{z_1}\langle ·,·\rangle \right)(z_2,z_3)& =\langle {\nabla }_{z_2}{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle +\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\nabla }_{z_3}{z}_1\rangle ,\forall z_1,z_2,z_3\in {\mathfrak{h}}^{!}.\hfill \end{array}$$

(12)

Definition 4.

Let $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\langle ·,·\rangle )$ be a pseudo-Riemannian metric Hom-Lie algebra. A tensor $\zeta \in {\mathfrak{h}}^{!}$ is called conformal if there is a real scalar ρ such that

$${\pounds }_{\zeta }\langle ·,·\rangle =2\rho \langle ·,{\varrho }_{{\mathfrak{h}}^{!}}^2(·)\rangle .$$

Also, ζ is said to be Killing if ρ is zero.

According to (12) and the above definition, $\zeta \in {\mathfrak{h}}^{!}$ is Killing if and only if

$$\begin{array}{c}\hfill \langle {\nabla }_{z_1}\zeta ,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle +\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\nabla }_{z_2}\zeta \rangle =0,\forall z_1,z_2\in {\mathfrak{h}}^{!}.\end{array}$$

(13)

Definition 5.

A Lorentzian contact structure $(\mathcal{F},v,u,\langle ·,·\rangle )$ on a Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ is said to be K-contact if the Reeb tensor v is Killing.

On a Lorentzian contact Hom-Lie algebra, we define a tensor $h\in {\mathcal{T}}_1^1\left({\mathfrak{h}}^{!}\right)$ by $h={\displaystyle \frac{1}{2}}{\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}).$

Corollary 1.

A Lorentzian contact Hom-Lie algebra is K-contact if and only if $h=0$.

Proof. 

For any $z_1,z_2\in {\mathfrak{h}}^{!}$, a simple computation yields

$$0=({\pounds }_v\Phi )(z_1,z_2)=-\left({\pounds }_v\langle ·,·\rangle \right)(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2)-\langle \left({\pounds }_v({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle ,$$

which gives

$$({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}\langle ·,·\rangle )(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2)=-\langle ({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}))z_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle .$$

Hence, $({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}\langle ·,·\rangle )(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2)=-2\langle h{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle$. Therefore, $h=0$ if and only if ${\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}\langle ·,·\rangle =0$, which completes the proof. □

Considering the definition of h, the first property to note is immediate, namely $hv=0$. If ${\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)=v$, then

$$h\circ {\varrho }_{{\mathfrak{h}}^{!}}={\varrho }_{{\mathfrak{h}}^{!}}\circ h.$$

We now exhibit a number of other important properties of h.

Proposition 1.

For a Lorentzian contact Hom-Lie algebra for any $z_1,z_2\in {\mathfrak{h}}^{!}$, we have

$$\begin{array}{cc}\hfill & {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}=0,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \langle h{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle =\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),h{z}_2\rangle ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})h=-h({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)={\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})h\left(z_1\right).\hfill \end{array}$$

(17)

Proof. 

By replacing $z_1$ by ${\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)$ in (9) and using (8), we conclude (14). We have

$$\begin{array}{cc}\hfill \langle h{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle & ={\displaystyle \frac{1}{2}}\langle ({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}))z_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle \hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\langle {\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle .\hfill \end{array}$$

On the other hand, since

$$\begin{array}{cc}\hfill {[z_1,z_2]}_{{\mathfrak{h}}^{!}}=& {\nabla }_{z_1}{z}_2-{\nabla }_{z_2}{z}_1,\hfill \end{array}$$

(18)

and using (14), it follows that

$$\langle h{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle =-{\displaystyle \frac{1}{2}}\langle {\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle -{\displaystyle \frac{1}{2}}\langle {\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle .$$

Applying Equation (11) in the last equation, we obtain

$$\langle h{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle ={\displaystyle \frac{1}{2}}\langle h{z}_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle +{\displaystyle \frac{1}{2}}\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),h{z}_2\rangle ,$$

which gives us (15). The definition of h implies

$$({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})h{z}_1={\displaystyle \frac{1}{2}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_1={\displaystyle \frac{1}{2}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})[{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1]+{\displaystyle \frac{1}{2}}[{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),z_1].$$

Similarly, we have $h({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1=-{\displaystyle \frac{1}{2}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})[{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1]-{\displaystyle \frac{1}{2}}[{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),z_1].$ Thus, (16) holds. Putting $z_2={\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)$ in (9), one can see

$$\begin{array}{cc}\hfill 2\langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =& -\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_3,{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle \hfill \\ \hfill & +2\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle .\hfill \end{array}$$

Using (4) and (15) in the last equation, we obtain

$$\begin{array}{cc}\hfill 2\langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =& -\langle \left({\pounds }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_1,{\varrho }_{{\mathfrak{h}}^{!}}{z}_3\rangle +2\langle z_1,z_3\rangle +2u\left(z_1\right)u\left(z_3\right),\hfill \end{array}$$

which gives

$$-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)=-h\left(z_1\right)+{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)-u\left(z_1\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right).$$

Applying $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})$ and noting that $u\left({\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)\right)=0$, from the above equation, we deduce (17). □

3. Lorentzian–Sasakian Hom-Lie Algebras

Definition 6.

A Lorentzian–Sasakian Hom-Lie algebra is a Lorentzian contact Hom-Lie algebra which admits a normal structure.

Example 3.

On a three-dimensional linear space $\mathfrak{h}$, we define ${[e_1,e_2]}_{\mathfrak{h}}=2e_3,{[e_2,e_3]}_{\mathfrak{h}}=-4e_2$ and ${[e_1,e_3]}_{\mathfrak{h}}=-4e_1,$ and set ${\varrho }_{\mathfrak{h}}\left(e_1\right)=e_2,{\varrho }_{\mathfrak{h}}\left(e_2\right)=-e_1$ and ${\varrho }_{\mathfrak{h}}\left(e_3\right)=e_3.$ We set $v=e_3$ and $u=e^3$. We define $\mathcal{F}\left(e_1\right)=e_1,\mathcal{F}\left(e_2\right)=e_2,\mathcal{F}\left(e_3\right)=0,\langle e_1,e_1\rangle =\langle e_2,e_2\rangle =1$ and $\langle e_3,e_3\rangle =-1.$ Therefore, $\mathfrak{h}$ forms a Lorentzian almost-contact Hom-Lie algebra. We also obtain that $d{e}^3(e_i,e_j)=0,i,j=1,2,3$, except $d{e}^3(e_1,e_2)=-2=2\Phi (e_1,e_2)$. Moreover, it results that $N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(e_i,e_j)+du(e_i,e_j)v=0,$ for all $i,j=1,2,3$. So, $(\mathfrak{h},{[·,·]}_{\mathfrak{h}},{\varrho }_{\mathfrak{h}},\mathcal{F},v,u,\langle ·,·\rangle )$ forms a Lorentzian–Sasakian Hom-Lie algebra.

Example 4.

Similar to the above example, we can see that the Lorentzian almost-contact Heisenberg Hom-Lie algebra $(\mathcal{H},{[·,·]}_{\mathcal{H}},{\varrho }_{\mathcal{H}},\mathcal{F},v,u,\langle ·,·\rangle )$ in Example 1 is Sasakian.

Example 5.

Considering Example 3, we obtain that $d{e}^3(e_i,e_j)=0,i,j=1,2,3$, except $d{e}^3(e_1,e_2)=-2=2\Phi (e_1,e_2)$. Moreover, it results that $N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(e_i,e_j)+du(e_i,e_j)v\ne 0,$ for all $i,j=1,2,3$. So, $(\mathfrak{h}=\mathfrak{sl}(2,\mathbb{R}),{[·,·]}_{\mathfrak{h}},{\varrho }_{\mathfrak{h}},\mathcal{F},v,u,\langle ·,·\rangle )$ is not a Lorentzian–Sasakian Hom-Lie algebra.

Theorem 1.

A Lorentzian almost-contact structure $(\mathcal{F},v,u,\langle ·,·\rangle )$ on a Hom-Lie algebra

$({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ is Lorentzian–Sasakian if and only if

$$\begin{array}{cc}\hfill \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2=& \langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle v\hfill \\ & +u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)-u\left(z_1\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),\hfill \end{array}$$

(19)

for all $z_1,z_2\in {\mathfrak{h}}^{!}$. In particular, if ${\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)=v$, then

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2=\langle z_1,z_2\rangle v+u\left(z_2\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right).\end{array}$$

(20)

Proof. 

First suppose that $(\mathcal{F},v,u,\langle ·,·\rangle )$ is a Lorentzian–Sasakian structure on ${\mathfrak{h}}^{!}$. Hence, Lemma 1 implies

$$2\langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =-du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_1)u\left({\varrho }_{{\mathfrak{h}}^{!}}{z}_3\right)+du(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1)u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right).$$

Equation (7) and the above equation imply

$$\begin{array}{cc}\hfill \langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =& -\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\right)\hfill \\ & +\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right),\hfill \end{array}$$

which gives

$$\begin{array}{cc}\hfill \langle \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =& \langle \langle z_2,z_1\rangle v+u\left(z_1\right)u\left(z_2\right)v+{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right)\hfill \\ & -{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)u\left(z_1\right)u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle .\hfill \end{array}$$

Thus, (19) follows. Conversely, if we set $z_2={\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)$ in (19), then

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)={\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)-u\left(z_1\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right).\end{array}$$

From (10) and the last equation, we conclude

$$\begin{array}{c}\hfill -({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)={\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)-u\left(z_1\right){\varrho }_{{\mathfrak{h}}^{!}}\left(v\right).\end{array}$$

Applying ${\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}$ to both sides of the last equation, we have

$$\begin{array}{c}\hfill {\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)=({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\right).\end{array}$$

Using the above equation, we also obtain

$$\begin{array}{cc}\hfill du(z_1,z_2)=& -u\left({[z_1,z_2]}_{{\mathfrak{h}}^{!}}\right)\hfill \\ \hfill =& -(\langle {\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle +\langle {\nabla }_{z_2}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\rangle )=2\langle z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle .\hfill \end{array}$$

(21)

So, $(\mathcal{F},v,u,\langle ·,·\rangle )$ is a Lorentzian contact metric structure on the Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$. Moreover, (5) and (18) imply

$$\begin{array}{cc}\hfill & N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(z_1,z_2)={({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2({\nabla }_{z_1}{z}_2-{\nabla }_{z_2}{z}_1)+{\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2-{\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\hfill \\ \hfill & -{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}({\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}{z}_2-{\nabla }_{z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)-{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2-{\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2}{z}_1).\hfill \end{array}$$

(22)

On the other hand, from (10), it follows that

$$\begin{array}{cc}\hfill & \left({\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2={\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}{z}_2\right),\hfill \\ \hfill & ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(\left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2\right)=({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\right)-{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2\left({\nabla }_{z_1}{z}_2\right).\hfill \end{array}$$

Applying the last two equations in (22), we have

$$\begin{array}{cc}\hfill N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(z_1,z_2)=& \left({\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2-\left({\nabla }_{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_1\hfill \\ & +({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(\left({\nabla }_{z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_1\right)-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(\left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2\right).\hfill \end{array}$$

Substituting (19) into the above equation, it follows that

$$\begin{array}{cc}\hfill N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(z_1,z_2)=& \langle {({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2{z}_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle v+u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right){\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\hfill \\ & -\langle {({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2{z}_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1\rangle v-u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\right){\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\hfill \\ & +u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\right)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)-u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\right)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right).\hfill \end{array}$$

(1), (21) and the last equation yield

$$\begin{array}{cc}\hfill N_{{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}}(z_1,z_2)=& -2\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_1\rangle v=-du(z_1,z_2)v.\hfill \end{array}$$

Thus, the normality condition holds and the proof completes. □

Corollary 2.

On any Lorentzian–Sasakian Hom-Lie algebra, we have

$$\begin{array}{cc}\hfill & {\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)={\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,\forall z_1\in {\mathfrak{h}}^{!}.\hfill \end{array}$$

Corollary 3.

A Lorentzian–Sasakian Hom-Lie algebra is K-contact.

Example 6.

We consider the Lorentzian almost-contact Hom-Lie algebra $\mathfrak{h}$ in Example 3. From Koszul’s formula given by (11), we obtain

$$\begin{array}{cc}\hfill & {\nabla }_{e_1}{e}_1=0,{\nabla }_{e_1}{e}_2=e_3,{\nabla }_{e_1}{e}_3=-e_1,{\nabla }_{e_2}{e}_1=-e_3,{\nabla }_{e_2}{e}_2=0,\hfill \\ \hfill & {\nabla }_{e_2}{e}_3=-e_2,{\nabla }_{e_3}{e}_1=3e_1,{\nabla }_{e_3}{e}_2=3e_2,{\nabla }_{e_3}{e}_3=0.\hfill \end{array}$$

Since ${\varrho }_{\mathfrak{h}}\left(e_3\right)=e_3$, we obtain $\left({\nabla }_{e_i}({\varrho }_{\mathfrak{h}}\circ \mathcal{F})\right)e_j=0=\langle e_i,e_j\rangle v+u\left(e_j\right){\varrho }_{\mathfrak{h}}\left(e_i\right),i,j=1,2,3$ except

$$\begin{array}{cc}\hfill & \left({\nabla }_{e_1}({\varrho }_{\mathfrak{h}}\circ \mathcal{F})\right)e_1=e_3=\langle e_1,e_1\rangle v+u\left(e_1\right){\varrho }_{\mathfrak{h}}\left(e_1\right),\hfill \\ & \left({\nabla }_{e_1}({\varrho }_{\mathfrak{h}}\circ \mathcal{F})\right)e_3=e_2=\langle e_1,e_3\rangle v+u\left(e_3\right){\varrho }_{\mathfrak{h}}\left(e_1\right),\hfill \\ \hfill & \left({\nabla }_{e_2}({\varrho }_{\mathfrak{h}}\circ \mathcal{F})\right)e_2=e_3=\langle e_2,e_2\rangle v+u\left(e_2\right){\varrho }_{\mathfrak{h}}\left(e_2\right),\hfill \\ & \left({\nabla }_{e_2}({\varrho }_{\mathfrak{h}}\circ \mathcal{F})\right)e_3=-e_1=\langle e_2,e_3\rangle v+u\left(e_3\right){\varrho }_{\mathfrak{h}}\left(e_2\right).\hfill \end{array}$$

Hence, (20) holds. Therefore, $\mathfrak{h}$ is a Lorentzian–Sasakian Hom-Lie algebra and consequently has a K-contact structure.

In the following, we always consider ${\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)=v$. Thus, (3) implies

$$\begin{array}{c}\hfill u\left({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\right)=-\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),v\rangle =-\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(v\right)\rangle =-\langle z_1,v\rangle =u\left(z_1\right),\forall z_1\in {\mathfrak{h}}^{!}.\end{array}$$

(23)

4. Curvature Tensor of Lorentzian Contact Hom-Lie Algebras

Proposition 2.

In a Lorentzian contact Hom-Lie algebra for any $z_1\in {\mathfrak{h}}^{!}$, the following formulas hold

$$\begin{array}{cc}\hfill & \left(i\right)\left({\nabla }_{v}h\right)z_1={\varrho }_{{\mathfrak{h}}^{!}}^2({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1-h^2({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1+({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(v,z_1)v,\hfill \\ \hfill & \left(ii\right){\displaystyle \frac{1}{2}}\{R(v,z_1)v-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)v\}={\varrho }_{{\mathfrak{h}}^{!}}^2{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2{z}_1+h^2\left(z_1\right).\hfill \end{array}$$

Proof. 

The curvature tensor $R\in {\mathcal{T}}_3^1\left({\mathfrak{h}}^{!}\right)$ of the Hom-Levi-Civita connection ∇ is defined by [20]

$$\begin{array}{cc}\hfill R(z_1,z_2)z_3=& {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}{z}_3-{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}{\nabla }_{z_1}{z}_3-{\nabla }_{{[z_1,z_2]}_{{\mathfrak{h}}^{!}}}{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right),\hfill \end{array}$$

(24)

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$. Setting (17) in (24), we obtain

$$\begin{array}{cc}\hfill R(v,z_1)v=& ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\{{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}v-\left({\nabla }_{v}h\right)z_1-h\left({\nabla }_{z_1}v\right)\}.\hfill \end{array}$$

Applying $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})$, it follows that

$$({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(v,z_1)v=-{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}v+\left({\nabla }_{v}h\right)z_1+h\left({\nabla }_{z_1}v\right)+u\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}v-\left({\nabla }_{v}h\right)z_1-h\left({\nabla }_{z_1}v\right)\right)v.$$

On the other hand, we obtain $u\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}v-\left({\nabla }_{v}h\right)z_1-h\left({\nabla }_{z_1}v\right)\right)=0$. Thus, (17) and the above equation yield

$$\begin{array}{cc}\hfill ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(v,z_1)v=& -{\varrho }_{{\mathfrak{h}}^{!}}^2({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1+\left({\nabla }_{v}h\right)z_1+h^2({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,\hfill \end{array}$$

which gives (i). Now, replacing $z_1$ by $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1$ in (i), we obtain

$$\begin{array}{cc}\hfill ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)v=& -{\varrho }_{{\mathfrak{h}}^{!}}^2{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2{z}_1+\left({\nabla }_{v}h\right)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1-h^2{z}_1.\hfill \end{array}$$

As $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\nabla }_{v}h\right)z_1=-\left({\nabla }_{v}h\right)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1$, the last equation together with (i) implies (ii). □

The Ricci curvature tensor $Ric\in {\mathcal{T}}_2^0\left({\mathfrak{h}}^{!}\right)$ is described by

$$\begin{array}{cc}\hfill Ric(z_1,z_2):=& tr\{z_3\to R(z_3,z_1)z_2\},\hfill \end{array}$$

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$. Moreover, $Ric$ is a symmetric tensor if the Hom-Lie algebra is involutive [25].

Corollary 4.

On a Lorentzian contact Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$, the Ricci curvature tensor in the direction v is given by

$$Ric(v,v)=tr({\varrho }_{{\mathfrak{h}}^{!}}^2-h^2)+1.$$

Proof. 

Suppose that ${\{e_i,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,v\}}_{i=1}^n$ is an $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})$ basis on ${\mathfrak{h}}^{!}$. We have

$$Ric(v,v)=\sum _{i=1}^n\langle R(e_i,v)v,e_i\rangle +\sum _{i=1}^n\langle R(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,v)v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i\rangle .$$

On the other hand, since [20]

$$\begin{array}{c}\hfill \langle R(z_1,z_2)z_3,w\rangle =-\langle R(z_2,z_1)z_3,w\rangle ,\end{array}$$

(25)

by using (4), (25) and the part (ii) of Proposition 2, it follows that

$$\begin{array}{cc}\hfill Ric(v,v)& =-\sum _{i=1}^n\langle R(v,e_i)v,e_i\rangle +\sum _{i=1}^n\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i)v,e_i\rangle \hfill \\ \hfill & =-2\sum _{i=1}^n\langle {\varrho }_{{\mathfrak{h}}^{!}}^2{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2{e}_i+h^2\left(e_i\right),e_i\rangle .\hfill \end{array}$$

Since $tr{\varrho }_{{\mathfrak{h}}^{!}}^2=2\langle {\varrho }_{{\mathfrak{h}}^{!}}^2{e}_i,e_i\rangle -1$ and $tr{h}^2=2\langle h^2{e}_i,e_i\rangle$, the last equation implies the assertion. □

The following theorem can be obtained from the above corollary.

Theorem 2.

A Lorentzian contact Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$ is K-contact if and only if

$$Ric(v,v)=tr{\varrho }_{{\mathfrak{h}}^{!}}^2+1.$$

Proposition 3.

Let $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\langle ·,·\rangle )$ be a pseudo-Riemannian metric Hom-Lie algebra. If a tensor $v\in {\mathfrak{h}}^{!}$ is Killing, then the curvature tensor R for any $z_1,z_2\in {\mathfrak{h}}^{!}$ satisfies

$$\begin{array}{c}\hfill R(z_1,v)z_2={\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}v-{\nabla }_{{\nabla }_{z_1}{z}_2}v.\end{array}$$

Proof. 

According to the first Bianchi identity, we have

$$\langle R(z_1,v)z_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =-\langle R(v,z_2)z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle -\langle R(z_2,z_1)v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle .$$

Using (24) and the following equation [20]

$$\begin{array}{c}\hfill \langle R(z_1,z_2)z_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(w\right)\rangle =\langle R(z_3,w)z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle ,\forall z_1,z_2,z_3,w\in {\mathfrak{h}}^{!},\end{array}$$

(26)

we obtain

$$\begin{array}{cc}\hfill \langle R(z_1,v)z_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =& \langle -{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_3}v+{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)}{\nabla }_{z_1}v+{\nabla }_{{[z_1,z_3]}_{{\mathfrak{h}}^{!}}}v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle \hfill \\ \hfill & +\langle -{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}{\nabla }_{z_1}v+{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}v+{\nabla }_{{[z_2,z_1]}_{{\mathfrak{h}}^{!}}}v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle .\hfill \end{array}$$

(27)

On the other hand, since v is a Killing tensor, (13) implies

$$\begin{array}{c}\hfill \langle {\nabla }_{z_3}v,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle =-\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right),{\nabla }_{z_2}v\rangle .\end{array}$$

(28)

By affecting ${\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}$ on the parties of the above equation, it follows that

$$\begin{array}{cc}\hfill \langle {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_3}v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle +\langle {\varrho }_{{\mathfrak{h}}^{!}}\left({\nabla }_{z_3}v\right),{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle =& -\langle {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right),{\varrho }_{{\mathfrak{h}}^{!}}\left({\nabla }_{z_2}v\right)\rangle \hfill \\ \hfill & -\langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right),{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}v\rangle .\hfill \end{array}$$

Applying the last two equations to (27), we obtain

$$\begin{array}{c}\hfill \langle R(z_1,v)z_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =\langle R(z_2,z_3)v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle +2\langle {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}v-{\nabla }_{{\nabla }_{z_1}{z}_2}v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle .\end{array}$$

From (26) and the above equation, the result is obtained. □

The sectional curvature spanned by $z_1,z_2\in {\mathfrak{h}}^{!}$ is as follows [26]:

$$\mathcal{K}(z_1,z_2)={\displaystyle \frac{\langle R(z_1,z_2)z_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle }{|z_1{|}^2{\left|z_2\right|}^2-{\langle z_1,z_2\rangle }^2}},$$

where $|z_1{|}^2=\langle z_1,z_1\rangle$.

Theorem 3.

A pseudo-Riemannian metric Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\langle ·,·\rangle )$ of dimension $2n+1$ is a Lorentzian contact Hom-Lie algebra with K-contact structure if and only if it admits a Killing tensor v such that $\langle v,v\rangle =-1$ and $R(z_1,v)v={\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)$ for any $z_1$ orthogonal to v. In addition, in this case, $\mathcal{K}(z_1,v)=-1$.

Proof. 

First, we assume that $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$ is a Lorentzian Hom-Lie algebra with K-contact structure. Since v is Killing, that is $h=0$, the part (i) of Proposition 2 implies $R(z_1,v)v={\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)$. So, we have

$$\mathcal{K}(z_1,v)={\displaystyle \frac{\langle R(z_1,v)v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle }{|z_1{|}^2{\left|v\right|}^2}}=-{\displaystyle \frac{\langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle }{|z_1{|}^2}}=-1.$$

Conversely, as v is a Killing tensor with $\langle v,v\rangle =-1$, we define $u\left(z_1\right):=-\langle z_1,v\rangle$ and ${\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(z_1\right):={\nabla }_{z_1}v$. Hence, $u\left(v\right)=1$, and also, from (28), it follows that ${\nabla }_{v}v=0$. Thus, $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(v\right)={\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(v\right)={\nabla }_{v}v=0$. Also, for $z_1$ orthogonal to v, Proposition 3 implies

$${\varrho }_{{\mathfrak{h}}^{!}}^2{({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2\left(z_1\right)={\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\nabla }_{z_1}v\right)={\nabla }_{{\nabla }_{z_1}v}v=-R(z_1,v)v=-{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right).$$

Because ${\mathfrak{h}}^{!}$ is regular, the above equation leads to ${({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2\left(z_1\right)=-z_1$. So, ${({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})}^2=-I_{{\mathfrak{h}}^{!}}+u\otimes v$. Furthermore, we obtain

$$\begin{array}{cc}\hfill du(z_1,z_2)=& -u\left([z_1,z_2]\right)=\langle [z_1,z_2],v\rangle \hfill \\ \hfill =& \langle {\nabla }_{z_1}{z}_2-{\nabla }_{z_2}{z}_1,v\rangle =-\langle {\nabla }_{z_1}v,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle +\langle {\nabla }_{z_2}v,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\rangle \hfill \\ \hfill =& 2\langle z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left(z_2\right)\rangle .\hfill \end{array}$$

Thus, $du=2\mathcal{F}$ and $(\mathcal{F},v,u,\langle ·,·\rangle )$ is a Lorentzian contact structure on ${\mathfrak{h}}^{!}$. Since v is Killing, the Lorentzian contact structure is K-contact. □

Proposition 4.

For a Lorentzian–Sasakian Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$, for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$, the following hold:

$$\begin{array}{cc}\hfill & R(z_1,z_2)v=u\left(z_2\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)-u\left(z_1\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & R(v,z_1)z_2=-\langle z_1,z_2\rangle v-u\left(z_2\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & R(v,z_1)v=u\left(z_1\right)v-{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & u\left(R(z_1,z_2)z_3\right)=u\left(z_2\right)\langle z_1,z_3\rangle -u\left(z_1\right)\langle z_2,z_3\rangle ,\hfill \end{array}$$

(32)

Proof. 

According to (24), we can write

$$\begin{array}{c}\hfill R(z_1,z_2)v={\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}v-{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}{\nabla }_{z_1}v-{\nabla }_{{[z_1,z_2]}_{{\mathfrak{h}}^{!}}}v.\end{array}$$

From Corollary 2 and the above equation, we obtain

$$\begin{array}{c}\hfill R(z_1,z_2)v=\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)-\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right).\end{array}$$

Setting (20) in the last equation, (29) follows. Similarly, proof of other cases results. □

Theorem 4.

A Lorentzian contact Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$ with K-contact structure is a Lorentzian–Sasakian Hom Lie algebra if and only if

$$\begin{array}{c}\hfill R(z_1,z_2)v=u\left(z_2\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)-u\left(z_1\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right),\forall z_1,z_2\in {\mathfrak{h}}^{!}.\end{array}$$

(33)

Proof. 

Assuming (33), it suffices to show that (20) holds. Using Proposition 3, we find

$$\begin{array}{cc}\hfill \left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)=& {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\hfill \\ \hfill =& {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}v-{\nabla }_{{\nabla }_{z_1}{z}_2}v\hfill \\ \hfill =& -R(v,z_1)z_2.\hfill \end{array}$$

Thus, the above equation and (26) yield

$$\begin{array}{cc}\hfill & \langle \left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =-\langle R(v,z_1)z_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =-\langle R(z_2,z_3)v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle \hfill \\ \hfill & =-\langle u\left(z_3\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)-u\left(z_2\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle =\langle z_3,v\rangle \langle z_1,z_2\rangle +u\left(z_2\right)\langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle ,\hfill \end{array}$$

which gives

$$\langle \left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =\langle \langle z_1,z_2\rangle v+u\left(z_2\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle .$$

On the other hand, $\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\rangle =\left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle$, so from this and the last equation, we obtain the assertion. □

Theorem 5.

Let $(\mathcal{F},v,u,\langle ·,·\rangle )$ be a Lorentzian–Sasakian structure on a Hom-Lie algebra

$({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$. Then,

$$\begin{array}{cc}\hfill \left(i\right)R(z_1,z_2)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3& -({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(z_1,z_2)z_3\hfill \\ \hfill =& \langle z_2,z_3\rangle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)-\langle z_1,z_3\rangle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\hfill \\ \hfill -& \langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_3\rangle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)+\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,z_3\rangle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left(ii\right)& R(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2)z_3=R(z_1,z_2)z_3+\langle z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\hfill \\ \hfill & -\langle z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\rangle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)+\langle z_2,z_3\rangle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)-\langle z_1,z_3\rangle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right),\hfill \end{array}$$

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$.

Proof. 

Replacing $z_3$ by $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3$ in (24), we have

$$\begin{array}{cc}\hfill R(z_1,z_2)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3=& {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3-{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}{\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3\hfill \\ \hfill -& {\nabla }_{[z_1,z_2}{]}_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right),\hfill \end{array}$$

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$. On the other hand, (10) gives

$$\begin{array}{cc}\hfill {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3=& {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}\left({\nabla }_{z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_3+\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\nabla }_{z_2}{z}_3\hfill \\ \hfill & +({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\nabla }_{z_2}{z}_3\right).\hfill \end{array}$$

From the above two equations and (24), it follows that

$$\begin{array}{cc}\hfill R(z_1,z_2)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3-& ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(z_1,z_2)z_3={\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}\left({\nabla }_{z_2}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_3\hfill \\ \hfill +& \left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\nabla }_{z_2}{z}_3-{\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}\left({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right)z_3\hfill \\ \hfill -& \left({\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\nabla }_{z_1}{z}_3-\left({\nabla }_{{[z_1,z_2]}_{{\mathfrak{h}}^{!}}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\right){\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right).\hfill \end{array}$$

Using (10) and (20) in the last equation, we find

$$\begin{array}{cc}\hfill & R(z_1,z_2)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3-({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(z_1,z_2)z_3=\langle z_2,z_3\rangle {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}v+u\left(z_3\right){\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)}{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\hfill \\ \hfill & +\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\nabla }_{z_2}{z}_3\rangle v+u\left({\nabla }_{z_2}{z}_3\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)-\langle z_1,z_3\rangle {\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}v-u\left(z_3\right){\nabla }_{{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)}{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)\hfill \\ \hfill & -\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\nabla }_{z_1}{z}_3\rangle v-u\left({\nabla }_{z_1}{z}_3\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)-\langle [z_1,z_2],{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle v-u\left(z_3\right){\varrho }_{{\mathfrak{h}}^{!}}[z_1,z_2].\hfill \end{array}$$

Applying (20) again, the above equation leads to (i). To prove (ii), considering (i), we can write

$$\begin{array}{cc}\hfill & \langle R(z_3,z_4)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle =\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})R(z_3,z_4)z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle \hfill \\ \hfill & +\langle z_4,z_1\rangle \langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right)\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle -\langle z_3,z_1\rangle \langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})\left({\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right)\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle \hfill \\ \hfill & -\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1\rangle \langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle +\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_4,z_1\rangle \langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_3\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle .\hfill \end{array}$$

Equations (4), (32) and the above equation imply

$$\begin{array}{cc}\hfill & \langle R(z_3,z_4)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle =\langle R(z_3,z_4)z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle +\langle z_1,z_3\rangle u\left(z_4\right)u\left(z_2\right)\hfill \\ \hfill & -\langle z_4,z_1\rangle u\left(z_3\right)u\left(z_2\right)+\langle z_4,z_1\rangle (\langle z_3,z_2\rangle +u\left(z_3\right)u\left(z_2\right))-\langle z_3,z_1\rangle (\langle z_4,z_2\rangle +u\left(z_4\right)u\left(z_2\right))\hfill \\ \hfill & -\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1\rangle \langle z_4,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle +\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_4,z_1\rangle \langle z_3,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle .\hfill \end{array}$$

Setting (3) in the last equation, we infer

$$\begin{array}{cc}\hfill & \langle R(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2)z_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right)\rangle =\langle R(z_1,z_2)z_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right)\rangle +\langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle \langle z_3,z_2\rangle \hfill \\ \hfill & -\langle z_3,z_1\rangle \langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right),{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle -\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,z_1\rangle \langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle \hfill \\ \hfill & -\langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right)\rangle \langle z_3,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle ,\hfill \end{array}$$

which gives us the assertion. □

The following corollary follows from the above theorem.

Corollary 5.

In a Lorentzian–Sasakian Hom-Lie algebra, we have

$$\begin{array}{c}\hfill \langle R(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right)\rangle =\langle R(z_1,z_2)z_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right)\rangle ,\end{array}$$

(34)

where $z_1,z_2,z_3,z_4\in {\mathfrak{h}}^{!}$ are orthogonal to v.

Proposition 5.

The Ricci curvature tensor of a Lorentzian–Sasakian Hom-Lie algebra satisfies the following:

$$\begin{array}{cc}\hfill Ric(z_1,v)=& u\left(z_1\right)(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)=Ric(v,z_1),\forall z_1\in {\mathfrak{h}}^{!}.\hfill \end{array}$$

Proof. 

Choose ${\{e_i,e_{i+n}=({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,v\}}_{i=1}^n$ as an $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})$-basis of ${\mathfrak{h}}^{!}$. By the definition of the Ricci curvature tensor and using (29), we obtain

$$Ric(z_1,v)=\sum _{i=1}^{2n+1}\langle R(e_i,z_1)v,e_i\rangle =\sum _{i=1}^{2n+1}\langle u\left(z_1\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_i\right)-u\left(e_i\right){\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_1\right),e_i\rangle =u\left(z_1\right)(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2).$$

From (25) and (30), it follows also that

$$\begin{array}{cc}\hfill Ric(v,z_1)=& \sum _{i=1}^{2n+1}\langle R(e_i,v)z_1,e_i\rangle =-\sum _{i=1}^{2n+1}\langle R(v,e_i)z_1,e_i\rangle =\sum _{i=1}^{2n+1}\langle z_1,e_i\rangle \langle v,e_i\rangle \hfill \\ \hfill +& u\left(z_1\right)\sum _{i=1}^{2n+1}\langle {\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_i\right),e_i\rangle .\hfill \end{array}$$

Thus, $Ric(v,z_1)=u\left(z_1\right)(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)$, which completes the proof. □

Corollary 6.

In a Lorentzian–Sasakian Hom-Lie algebra, the following relations hold:

$$\begin{array}{cc}\hfill & Qv=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)v,Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})=({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})Q,\hfill \end{array}$$

(35)

where Q is the Ricci operator determined by $\langle Q·,·\rangle :=Ric(·,·)$.

Proof. 

From Proposition 5, we infer that $\langle Qv,v\rangle =Ric(v,v)=1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2$. Since $\langle v,v\rangle =-1$, $Qv=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)v$. Now, suppose that $z_1$ and $z_2$ are orthogonal to v. (4) implies $\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})Q\left(z_1\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =\langle Q\left(z_1\right),z_2\rangle$. So, to show $Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})=({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})Q$, it suffices to prove $\langle Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =\langle Q\left(z_1\right),z_2\rangle$. Since

$$\begin{array}{cc}\hfill & \langle Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2);\hfill \end{array}$$

thus,

$$\begin{array}{cc}\hfill \langle Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =& \sum _{i=1}^{2n}\langle R(e_i,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,e_i\rangle \hfill \\ \hfill +& \langle R(v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,v\rangle .\hfill \end{array}$$

On the other hand, (4) and (30) imply

$$\langle R(v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,v\rangle =-\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle \langle v,v\rangle =\langle z_1,z_2\rangle .$$

From the above two equations, we have

$$\langle Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =\sum _{i=1}^{2n}\langle R(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i\rangle +\langle z_1,z_2\rangle .$$

Similarly, it follows that

$$\begin{array}{c}\hfill \langle Q\left(z_1\right),z_2\rangle =\sum _{i=1}^{2n}\langle R(e_i,z_1)z_2,e_i\rangle +\langle z_1,z_2\rangle .\end{array}$$

Replacing $z_1$ and ${\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_4\right)$ by $e_i$ in (34) and using the last two equations, we find

$$Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})=({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})Q.$$

□

Lemma 2.

On a Lorentzian–Sasakian Hom-Lie algebra, we have

$$\begin{array}{cc}\hfill & \left(i\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2)=Ric(z_1,z_2)-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)u\left(z_1\right)u\left(z_2\right),\hfill \\ \hfill & \left(ii\right)\left({\nabla }_{z_1}Ric\right)(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3)=\left({\nabla }_{z_1}Ric\right)(z_2,z_3)+(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\{\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_3\rangle u\left(z_2\right)\hfill \\ \hfill & +\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2\rangle u\left(z_3\right)\}-u\left(z_2\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))\hfill \\ \hfill & -u\left(z_3\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)),\hfill \end{array}$$

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$.

Proof. 

According to (4) and (35), it follows that

$$\begin{array}{cc}\hfill Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2)& =\langle Q({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle =\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})Q{z}_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle \hfill \\ \hfill & =\langle Q\left(z_1\right),z_2\rangle +u\left(Q{z}_1\right)u\left(z_2\right)=Ric(z_1,z_2)-\langle Q{z}_1,v\rangle u\left(z_2\right).\hfill \end{array}$$

Using Proposition 5 in the above equation, we conclude (i). We have

$$\begin{array}{cc}\hfill \left({\nabla }_{z_1}Ric\right)(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3)=& -Ric({\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,{\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3)\hfill \\ \hfill & -Ric({\varrho }_{{\mathfrak{h}}^{!}}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,{\nabla }_{z_1}({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3).\hfill \end{array}$$

Equations (10), (20) and the above equation imply

$$\begin{array}{cc}\hfill \left({\nabla }_{z_1}Ric\right)& (({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3)=-\langle z_1,z_2\rangle Ric(v,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}{z}_3)\hfill \\ & -u\left(z_2\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}{z}_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}{z}_3)-\langle z_1,z_3\rangle Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}{z}_2,v)\hfill \\ & -u\left(z_3\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}{z}_1)\hfill \\ \hfill & -Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\nabla }_{z_1}{z}_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}{z}_3)-Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}{z}_2,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\nabla }_{z_1}{z}_3).\hfill \end{array}$$

Applying (i) in the above equation, (ii) follows. □

Definition 7.

The Ricci tensor of a Lorentzian contact Hom-Lie algebra is said to be u-parallel if

$$(\nabla Ric)({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F},{\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})=0.$$

Corollary 7.

The Ricci tensor of a Lorentzian–Sasakian Hom-Lie algebra is u-parallel if and only if

$$\begin{array}{cc}\hfill \left({\nabla }_{z_1}Ric\right)(z_2,z_3)=& -(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)(\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_3\rangle u\left(z_2\right)+\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2\rangle u\left(z_3\right))\hfill \\ \hfill +& u\left(z_2\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))+u\left(z_3\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right)),\hfill \end{array}$$

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$.

Example 7.

Consider the Lorentzian–Sasakian Hom-Lie algebra $(\mathcal{H},{[·,·]}_{\mathcal{H}},{\varrho }_{\mathcal{H}},\mathcal{F},v,u,\langle ·,·\rangle )$ in Example 4. Using (24), one obtains that $R(e_i,e_j)e_k=0,i,j,k=1,2,3$, except

$$\begin{array}{cc}\hfill & R(e_1,e_2)e_2=-3e_1=-R(e_2,e_1)e_2,R(e_1,e_3)e_3=-e_1=-R(e_3,e_1)e_3,\hfill \\ \hfill & R(e_1,e_2)e_1=3e_2=-R(e_2,e_1)e_1,R(e_2,v)v=-e_2=-R(v,e_2)v,\hfill \\ \hfill & R(e_1,v)e_1=v=-R(v,e_1)e_1,R(e_2,v)e_2=v=-R(v,e_2)e_2.\hfill \end{array}$$

From the above equations, we obtain $Ric(e_i,e_j)=0,i,j=1,2,3$, but

$$Ric(e_1,e_1)=Ric(e_2,e_2)=Ric(v,v)=-2.$$

It is easy to check that $\left({\nabla }_{e_i}Ric\right)(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_j,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_k)=0,i,j,k=1,2,3$. We also obtain $tr{\varrho }_{\mathcal{H}}^2=-3,$ and

$$\begin{array}{cc}\hfill \left({\nabla }_{e_i}Ric\right)(e_j,e_k)=& 0=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)(\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,e_k\rangle u\left(e_j\right)+\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_i,e_j\rangle u\left(e_k\right))\hfill \\ \hfill & +u\left(e_j\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(e_i\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(e_k\right))+u\left(e_k\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(e_j\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(e_i\right)),\hfill \end{array}$$

where $i,j,k=1,2,3$ and $\left({\nabla }_{e_i}Ric\right)(e_j,e_k)=\left({\nabla }_{e_i}Ric\right)(e_k,e_j)$ except

$$\begin{array}{cc}\hfill \left({\nabla }_{e_1}Ric\right)(e_2,e_3)=& 4=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)(\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_1,e_3\rangle u\left(e_2\right)+\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_1,e_2\rangle u\left(e_3\right))\hfill \\ \hfill +& u\left(e_2\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(e_1\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(e_3\right))+u\left(e_3\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(e_2\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(e_1\right)),\hfill \\ \hfill \left({\nabla }_{e_2}Ric\right)(e_1,e_3)=& -4=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)(\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_2,e_3\rangle u\left(e_1\right)+\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})e_2,e_1\rangle u\left(e_3\right))\hfill \\ \hfill +& u\left(e_1\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(e_2\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(e_3\right))+u\left(e_3\right)Ric(({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(e_1\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(e_2\right)).\hfill \end{array}$$

Therefore, Corollary 7 holds, and hence, the Lorentzian–Sasakian Hom-Lie algebra is u-parallel.

5. (Almost) Ricci Solitons in Lorentzian–Sasakian Hom-Lie Algebras

Definition 8.

Let $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\langle ·,·\rangle )$ be a pseudo-Riemannian Hom-Lie algebra. A triple $(\langle ·,·\rangle ,\zeta ,\lambda )$ consisting of a pseudo-Riemannian metric $\langle ·,·\rangle$, $\zeta \in {\mathfrak{h}}^{!}$ and a real scalar λ is called

(i) 

a Ricci soliton if

$$\begin{array}{c}\hfill {\pounds }_{\zeta }\langle ·,·\rangle +2Ric(·,·)+2\lambda \langle ·,{\varrho }_{{\mathfrak{h}}^{!}}^2(·)\rangle =0;\end{array}$$

(36)

(ii) 

an almost Ricci soliton if

$$\begin{array}{c}\hfill {\pounds }_{\zeta }\langle ·,·\rangle +2Ric(·,{\varrho }_{{\mathfrak{h}}^{!}}(·))+2\lambda \langle ·,{\varrho }_{{\mathfrak{h}}^{!}}^2(·)\rangle =0.\end{array}$$

(37)

The (almost) Ricci soliton $(\langle ·,·\rangle ,\zeta ,\lambda )$ on ${\mathfrak{h}}^{!}$ is said to be shrinking, steady and expanding if $\lambda <0$, $\lambda =0$ and $\lambda >0$, respectively.

A pseudo-Riemannian Hom-Lie algebra $({\mathfrak{h}}^{!},[·,·],{\varrho }_{{\mathfrak{h}}^{!}},\langle ·,·\rangle )$ is called Einstein if

$$\begin{array}{c}\hfill Ric(z_1,z_2)=a\langle z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle ,\end{array}$$

(38)

where a is real scaler, for all $z_1,z_2\in {\mathfrak{h}}^{!}$.

Theorem 6.

If a Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ with a Lorentzian–Sasakian structure $(\mathcal{F},v,u,\langle ·,·\rangle )$ is Einstein, then

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}Ric\right)(z_2,z_3)=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),\left({\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}^2\right)z_3\rangle ,\end{array}$$

(39)

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$,

Proof. 

We have

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}Ric\right)(z_2,z_3)=-Ric({\nabla }_{z_1}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))-Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\nabla }_{z_1}{z}_3),\end{array}$$

for any $z_1,z_2,z_3\in {\mathfrak{h}}^{!}$. Since ${\mathfrak{h}}^{!}$ is Einstein, (38) and the above equation imply

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}Ric\right)(z_2,z_3)=-a\langle {\nabla }_{z_1}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}^3\left(z_3\right)\rangle -a\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}^2{\nabla }_{z_1}{z}_3\rangle .\end{array}$$

On the other hand, using the equation [17]

$$\begin{array}{cc}\hfill \langle {\nabla }_{z_1}{z}_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right)\rangle =& -\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\nabla }_{z_1}{z}_3\rangle ,\hfill \end{array}$$

(40)

it follows that

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}Ric\right)(z_2,z_3)=a\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}^2{z}_3\rangle -a\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}^2{\nabla }_{z_1}{z}_3\rangle =a\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right),\left({\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}^2\right)z_3\rangle .\end{array}$$

Proposition 5 and (38) yield $a=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)$. Thus, (39) follows. □

Theorem 7.

Let $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}})$ be a Hom-Lie algebra with a Lorentzian–Sasakian structure $(\mathcal{F},v,u,\langle ·,·\rangle )$. If (39) holds, then

$$\begin{array}{c}\hfill Ric(z_1,z_2)=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle z_1,z_2\rangle ,\forall z_1,z_2\in {\mathfrak{h}}^{!}.\end{array}$$

(41)

Proof. 

Suppose that (39) holds. Replacing $z_2$ by v in part (ii) of Lemma 2, we obtain

$$\begin{array}{c}\hfill \left({\nabla }_{z_1}Ric\right)(v,z_3)+(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_3\rangle -Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))=0.\end{array}$$

On the other hand, we see that $\langle v,\left({\nabla }_{z_1}{\varrho }_{{\mathfrak{h}}^{!}}^2\right)z_3\rangle =0$. So, it follows from (39) that $\left({\nabla }_{z_1}Ric\right)(v,z_3)=0$. Thus, the above equation yields

$$Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F}){\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))=(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_3\rangle .$$

Replacing $z_3$ by $({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_3$ in the above equation, we obtain

$$-Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))+u\left(z_3\right)Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),v)=(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\{\langle z_1,z_3\rangle +u\left(z_1\right)u\left(z_3\right)\}.$$

The last equation and Proposition 5 imply

$$Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right))=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle z_1,z_3\rangle .$$

From (3) and the above equation, (41) follows. □

Proposition 6.

Let $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$ be a Lorentzian–Sasakian Hom-Lie algebra. If the metric $\langle ·,·\rangle$ is a Ricci soliton with $\zeta =v$, then ${\mathfrak{h}}^{!}$ is Einstein.

Proof. 

From (12), we have

$$\begin{array}{c}\hfill \left({\pounds }_v\langle ·,·\rangle \right)(z_1,z_2)=\langle {\nabla }_{z_1}v,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_2\right)\rangle +\langle {\varrho }_{{\mathfrak{h}}^{!}}\left(z_1\right),{\nabla }_{z_2}v\rangle .\end{array}$$

According to Corollary 2 and the above equation, it follows that

$$\begin{array}{cc}\hfill \left({\pounds }_v\langle ·,·\rangle \right)(z_1,z_2)& =\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2\rangle +\langle z_1,({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_2\rangle \hfill \\ & =\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2\rangle -\langle ({\varrho }_{{\mathfrak{h}}^{!}}\circ \mathcal{F})z_1,z_2\rangle =0.\hfill \end{array}$$

Hence, from (36) and (38), we conclude the assertion. □

Theorem 8.

A Lorentzian–Sasakian Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$ equipped with a Ricci soliton structure $(\langle ·,·\rangle ,\zeta ,\lambda )$ is Einstein if ζ is conformal.

Proof. 

Assume that $\zeta$ is conformal, for any $z_1,z_2\in {\mathfrak{h}}^{!}$, (36) gives

$$\begin{array}{c}\hfill 2\rho \langle z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle +2Ric(z_1,z_2)+2\lambda \langle z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle =0.\end{array}$$

Thus, $Ric(z_1,z_2)=-(\rho +\lambda )\langle z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle$, which completes the proof. □

Theorem 9.

Let $(\langle ·,·\rangle ,\zeta ,\lambda )$ be an almost Ricci soliton in a Lorentzian–Sasakian Hom-Lie algebra $({\mathfrak{h}}^{!},{[·,·]}_{{\mathfrak{h}}^{!}},{\varrho }_{{\mathfrak{h}}^{!}},\mathcal{F},v,u,\langle ·,·\rangle )$. If ${\mathfrak{h}}^{!}$ is Ricci-semisymmetric, i.e., $R(z_1,z_2)·Ric=0$ for any $z_1,z_2\in {\mathfrak{h}}^{!}$, then ζ is conformal.

Proof. 

Assuming ${\mathfrak{h}}^{!}$ is Ricci-semisymmetric, we have

$$Ric(R(z_1,z_2)z_3,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_4\right))+Ric({\varrho }_{{\mathfrak{h}}^{!}}\left(z_3\right),R(z_1,z_2)z_4)=0,$$

for any $z_1,z_2,z_3,z_4\in {\mathfrak{h}}^{!}$. Replacing $z_1$ and $z_3$ by v in the above equation, it follows that

$$Ric(R(v,z_2)v,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_4\right))+Ric(v,R(v,z_2)z_4)=0.$$

(30) and the last equation imply

$$-u\left(z_2\right)Ric(v,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_4\right))+Ric({\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_4\right))+\langle z_2,z_4\rangle Ric(v,v)+u\left(z_4\right)Ric(v,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right))=0.$$

Applying Proposition 5 in the above equation, we obtain

$$Ric({\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right),{\varrho }_{{\mathfrak{h}}^{!}}\left(z_4\right))=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle z_2,z_4\rangle ,$$

which gives

$$\begin{array}{c}\hfill Ric(z_2,z_4)=-(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)\langle z_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(z_4\right)\rangle .\end{array}$$

(42)

Substituting (42) in (37), we infer

$$\left({\pounds }_{\zeta }\langle ·,·\rangle \right)(z_1,z_2)=2\rho \langle z_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(z_2\right)\rangle ,$$

where $\rho =(1+tr{\varrho }_{{\mathfrak{h}}^{!}}^2)-\lambda$, i.e., $\zeta$ is conformal. □

Example 8.

On the Lorentzian–Sasakian Heisenberg Hom-Lie algebra $(\mathcal{H},{[·,·]}_{\mathcal{H}},{\varrho }_{\mathcal{H}},\mathcal{F},v,u,\langle ·,·\rangle )$ in Example 7, the triplet $(\langle ·,·\rangle ,e_3,\lambda =-2)$ defines a Ricci soliton. Indeed, from (36), an easy computation shows that ${\pounds }_{e_3}\langle e_i,e_j\rangle +2Ric(e_i,e_j)+2\lambda \langle e_i,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_j\right)\rangle =0,i,j=1,2,3$, except

$$\begin{array}{cc}\hfill & {\pounds }_{e_3}\langle e_1,e_1\rangle +2Ric(e_1,e_1)+2\lambda \langle e_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_1\right)\rangle =-4-2\lambda ,\hfill \\ \hfill & {\pounds }_{e_3}\langle e_2,e_2\rangle +2Ric(e_2,e_2)+2\lambda \langle e_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_2\right)\rangle =-4-2\lambda ,\hfill \\ \hfill & {\pounds }_{e_3}\langle e_3,e_3\rangle +2Ric(e_3,e_3)+2\lambda \langle e_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_3\right)\rangle =-4-2\lambda .\hfill \end{array}$$

Example 9.

Consider a three-dimensional Hom-Lie algebra $(\mathfrak{h},{[·,·]}_{\mathfrak{h}},{\varrho }_{\mathfrak{h}})$ with an arbitrary basis $\{e_1,e_2,e_3\}$ where ${\varrho }_{\mathfrak{h}}\left(e_1\right)=e_2,{\varrho }_{\mathfrak{h}}\left(e_2\right)=-e_1,{\varrho }_{\mathfrak{h}}\left(e_3\right)=e_3,$ and ${[e_1,e_2]}_{\mathfrak{h}}=A{e}_3,{[e_1,e_3]}_{\mathfrak{h}}=B{e}_1,{[e_2,e_3]}_{\mathfrak{h}}=B{e}_2.$ Defining $v=e_3,u=e^3$, $\mathcal{F}\left(e_1\right)=e_1,\mathcal{F}\left(e_2\right)=e_2$ and $\mathcal{F}\left(v\right)=0$, it follows that

$$\begin{array}{c}\hfill ({\varrho }_{\mathfrak{h}}\circ \mathcal{F})e_1=e_2=(\mathcal{F}\circ {\varrho }_{\mathfrak{h}})e_1,({\varrho }_{\mathfrak{h}}\circ \mathcal{F})e_2=-e_1=(\mathcal{F}\circ {\varrho }_{\mathfrak{h}})e_2,({\varrho }_{\mathfrak{h}}\circ \mathcal{F})e_3=0=(\mathcal{F}\circ {\varrho }_{\mathfrak{h}})e_3.\end{array}$$

We also see that ${({\varrho }_{\mathfrak{h}}\circ \mathcal{F})}^2{e}_i=-e_i,i=1,2$ and ${({\varrho }_{\mathfrak{h}}\circ \mathcal{F})}^2{e}_3=0$. Thus, $\mathfrak{h}$ is an almost-contact Hom-Lie algebra. Considering a Lorentzian metric on $\mathfrak{h}$, as $\langle e_1,e_1\rangle =\langle e_2,e_2\rangle ={\displaystyle \frac{A}{2}}$ and $\langle e_3,e_3\rangle =-1$, it is easy to check that $(\mathcal{F},e_3,u,\langle ·,·\rangle )$ forms a Lorentzian–Sasakian structure. The non-vanishing components of the curvature tensor are computed as follows:

$$\begin{array}{cc}\hfill & R(e_1,e_2)e_2={\displaystyle \frac{A}{4}}(-{\displaystyle \frac{3}{2}}+B)e_1=-R(e_2,e_1)e_2,R(e_1,e_3)e_3=-{\displaystyle \frac{1}{4}}{e}_1=-R(e_3,e_1)e_3,\hfill \\ \hfill & R(e_1,e_2)e_1={\displaystyle \frac{A}{4}}({\displaystyle \frac{3}{2}}-B)e_2=-R(e_2,e_1)e_1,R(e_2,e_3)e_3=-{\displaystyle \frac{1}{4}}{e}_2=-R(e_3,e_2)e_3,\hfill \\ \hfill & R(e_1,e_3)e_1=-R(e_3,e_1)e_1=R(e_2,e_3)e_2=-R(e_3,e_2)e_2={\displaystyle \frac{A}{8}}{e}_3.\hfill \end{array}$$

From the above expression of the curvature tensor, we can also obtain the Ricci tensor:

$$Ric(e_1,e_1)=Ric(e_2,e_2)={\displaystyle \frac{A}{8}}(-{\displaystyle \frac{3}{2}}A+AB+1),Ric(e_3,e_3)=-{\displaystyle \frac{1}{4}}A.$$

(36) implies ${\pounds }_{e_k}\langle e_i,e_j\rangle +2Ric(e_i,e_j)+2\lambda \langle e_i,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_j\right)\rangle =0,,k,i,j=1,2,3$, except

$$\begin{array}{cc}\hfill & {\pounds }_{e_k}\langle e_i,e_i\rangle +2Ric(e_i,e_i)+2\lambda \langle e_i,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_i\right)\rangle =A(-{\displaystyle \frac{3}{8}}A+{\displaystyle \frac{1}{4}}AB+{\displaystyle \frac{1}{4}}-\lambda ),i=1,2,k=1,2,3,\hfill \\ \hfill & {\pounds }_{e_k}\langle e_i,e_3\rangle +2Ric(e_i,e_3)+2\lambda \langle e_i,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_3\right)\rangle ={\pounds }_{e_k}\langle e_3,e_i\rangle +2Ric(e_3,e_i)+2\lambda \langle e_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_i\right)\rangle \hfill \\ \hfill & ={\displaystyle \frac{A}{2}}(1-{\displaystyle \frac{B}{2}}),i,k=1,2,i\ne k,\hfill \\ \hfill & {\pounds }_{e_k}\langle e_3,e_3\rangle +2Ric(e_3,e_3)+2\lambda \langle e_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_3\right)\rangle =-{\displaystyle \frac{A}{2}}-2\lambda ,k=1,2,3.\hfill \end{array}$$

According to the above equations, $\mathfrak{h}$ admits

(i) 

an expanding Ricci soliton $(\langle ·,·\rangle ,e_1,\lambda ={\displaystyle \frac{1}{6}})$ if $A=-{\displaystyle \frac{2}{3}}$ and $B=2$.

(ii) 

an expanding Ricci soliton $(\langle ·,·\rangle ,e_2,\lambda ={\displaystyle \frac{1}{6}})$ if $A=-{\displaystyle \frac{2}{3}}$ and $B=2$.

(iii) 

a steady Ricci soliton $(\langle ·,·\rangle ,e_1,\lambda =0)$ if $A=0$.

(iv) 

a steady Ricci soliton $(\langle ·,·\rangle ,e_2,\lambda =0)$ if $A=0$.

(v) 

a shrinking Ricci soliton $(\langle ·,·\rangle ,e_3,\lambda =-{\displaystyle \frac{1}{2}})$ if $A=2$ and $B=0$.

From (37), it also follows that

$${\pounds }_{e_3}\langle e_i,e_j\rangle +2Ric(e_i,{\varrho }_{{\mathfrak{h}}^{!}}\left(e_j\right))+2\lambda \langle e_i,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_j\right)\rangle =0,i,j=1,2,3,$$

unless

$$\begin{array}{cc}\hfill & {\pounds }_{e_3}\langle e_i,e_i\rangle +2Ric(e_i,{\varrho }_{{\mathfrak{h}}^{!}}\left(e_i\right))+2\lambda \langle e_i,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_i\right)\rangle =\lambda A,i=1,2,\hfill \\ \hfill {\pounds }_{e_3}\langle e_1,e_2\rangle & +2Ric(e_1,{\varrho }_{{\mathfrak{h}}^{!}}\left(e_2\right))+2\lambda \langle e_1,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_2\right)\rangle \hfill \\ & =-{\pounds }_{e_3}\langle e_2,e_1\rangle -2Ric(e_2,{\varrho }_{{\mathfrak{h}}^{!}}\left(e_1\right))-2\lambda \langle e_2,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_1\right)\rangle \hfill \\ \hfill & ={\displaystyle \frac{A}{4}}({\displaystyle \frac{3}{2}}A-AB-1),\hfill \\ \hfill & {\pounds }_{e_3}\langle e_3,e_3\rangle +2Ric(e_3,{\varrho }_{{\mathfrak{h}}^{!}}\left(e_3\right))+2\lambda \langle e_3,{\varrho }_{{\mathfrak{h}}^{!}}^2\left(e_3\right)\rangle =-{\displaystyle \frac{A}{2}}-2\lambda .\hfill \end{array}$$

Therefore, $\mathfrak{h}$ has a steady almost Ricci soliton $(\langle ·,·\rangle ,e_3,\lambda =0)$ if $A=0$.