Classification of Algebraic Schouten Solitons on Lorentzian Lie Groups Associated with the Perturbed Canonical Connection and the Perturbed Kobayashi–Nomizu Connection

Abstract

In this paper, we investigate the algebraic conditions of algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. Furthermore, we provide the complete classification for these algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the algebraic Schouten solitons. The main results indicate that $G_4$ does not possess algebraic Schouten solitons related to the perturbed Kobayashi–Nomizu connection, $G_1,G_2,G_3,G_6$, and $G_7$ possess algebraic Schouten solitons, and the result for $G_5$ is trivial.

1. Introduction

Einstein metrics are cornerstone concepts in differential geometry and mathematical physics, and they have been extensively studied within the Lorentzian manifolds. In [1], Hamilton introduced the Ricci flow as a natural generalization of the Einstein metric. Recall that a Riemannian manifold $(M,g)$ is called a Ricci soliton if there exists a smooth vector field V and a real constant $\lambda$ satisfying

$$Ric+{\displaystyle \frac{1}{2}}{\mathcal{L}}_{V}g=\lambda g,$$

(1)

where $Ric$ represents the Ricci tensor, and ${\mathcal{L}}_{V}g$ denotes the Lie derivative of the metric g along the vector field V. Additionally, Perelman [2] applied the Ricci flow to prove the long-standing and well-known Poincaré conjecture. Some applications of the Ricci flow and Ricci solitons in Physics can be found in [3,4]. Subsequently, in [5], Lauret introduced the definition of an algebraic Ricci soliton in the Riemannian setting and proved that algebraic Ricci solitons on homogeneous Riemannian manifolds are Ricci solitons. In [6], Onda and Batat extended the concept to pseudo-Riemannian Lie groups and classified all algebraic Ricci solitons on three-dimensional Lorentzian Lie groups. They proved that, unlike in the Riemannian case, Lorentzian Ricci solitons are not necessarily algebraic Ricci solitons. Inspired by Lauret’s research, Wears formulated the concept of algebraic T-solitons and established a relationship between T-solitons and algebraic T-solitons. Using Milnor frames, Wears provided a complete classification of algebraic T-solitons on three-dimensional unimodular Lie groups in [7]. In [8], Azami defined the affine generalized Ricci soliton on three-dimensional Lie groups, which could be considered as a type of Schouten soliton.

In [9], Etayo and Santamaria explored various linear connections that can be defined on four geometric types of $(J^2=\pm 1)$-metric manifolds. Motivated by this research, mathematicians started to study algebraic Ricci solitons associated with different affine connections. For instance, in [10], Wang provided a thorough classification of affine Ricci solitons related to canonical connections, perturbed canonical connections, Kobayashi–Nomizu connections, and perturbed Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups that possess a specific product structure. In [11], Liu presented the classification results of algebraic Schouten solitons on three-dimensional Lie groups in relation to certain affine connections. Additionally, Yang and Miao studied the algebraic Schouten soliton on three-dimensional Lie groups associated with the Yano connection. For additional findings related to algebraic Ricci solitons, see [8,12,13,14,15]. Ricci solitons are special solutions to the heat flow that exhibit self-similarity, and they have a close relationship with the symmetric properties of pseudo-Riemannian manifolds. Additionally, algebraic Schouten solitons, which are generalizations of Schouten solitons, also have a natural link to symmetry. Thus, we investigate the algebraic conditions of algebraic Schouten solitons on three-dimensional Lie groups associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. Subsequently, we classify algebraic Schouten solitons related to these connections on three-dimensional Lorentzian Lie groups.

The paper is organized as follows: In Section 2, we introduce some basic concepts related to the Lorentzian Lie groups, particularly concerning the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. In Section 3 and Section 4, we give the classification of algebraic Schouten solitons associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection on three-dimensional Lorentzian Lie groups that possess a product structure. In Section 5, we summarize the findings and outline potential directions for future research.

2. Preliminaries

In [16], Milnor reviewed both historical and recent findings concerning left-invariant Riemannian metrics on Lie groups, with a particular emphasis on three-dimensional unimodular Lie groups, which he comprehensively classified. Additionally, Rahmani undertook the classification of three-dimensional unimodular Lie groups equipped with a left-invariant Lorentzian metric in [17]. Furthermore, the non-unimodular cases were solved in [18,19]. In this paper, we will use ${\left\{G_i\right\}}_{i=1}^7$ to represent the connected three-dimensional Lie groups endowed with seven left-invariant Lorentzian metrics, and ${\left\{{\mathfrak{g}}_{\mathfrak{i}}\right\}}_{i=1}^7$ to denote their corresponding Lie algebras (refer to [6]). Let J be defined as a product structure on $G_i$ and is given by

$$J{\tilde{e}}_1={\tilde{e}}_1,J{\tilde{e}}_2={\tilde{e}}_2,J{\tilde{e}}_3={\tilde{e}}_3,$$

(2)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ constitutes a pseudo-orthonormal basis for ${\mathfrak{g}}_i$, with ${\tilde{e}}_3$ being timelike. Then, $J^2=id$ and $g(J{\tilde{e}}_i,J{\tilde{e}}_i)=g({\tilde{e}}_i,{\tilde{e}}_i)$. The canonical connection ${\nabla }^0$ and the Kobayashi–Nomizu connection ${\nabla }^1$, as described in [9], are defined as follows:

$$\begin{array}{cc}\hfill & {\nabla }_W^{0}V={\nabla }_{W}V-{\displaystyle \frac{1}{2}}\left({\nabla }_{W}J\right)JV,\hfill \\ \hfill & {\nabla }_W^{1}V={\nabla }_W^{0}V-{\displaystyle \frac{1}{4}}[\left({\nabla }_{V}J\right)JW-\left({\nabla }_{JV}J\right)W].\hfill \end{array}$$

(3)

where $W,V\in \mathcal{X}\left(G_i\right)$. On each $G_i$, where i ranges from 1 to 7, the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection are defined as follows:

$$\begin{array}{cc}\hfill & {\overline{\nabla }}_W^{0}V={\nabla }_W^{0}V+{\rho }_0{\tilde{e}}_3^{♭}\left(W\right){\tilde{e}}_3^{♭}\left(V\right)e_3,\hfill \\ \hfill & {\overline{\nabla }}_W^{1}V={\nabla }_W^{1}V+{\rho }_0{\tilde{e}}_3^{♭}\left(W\right){\tilde{e}}_3^{♭}\left(V\right)e_3,\hfill \end{array}$$

(4)

where ${\rho }_0$ is a constant. Subsequently, for $k,l\in \{1,2\}$, we have

$$\begin{array}{c}\hfill {\overline{\nabla }}_{{\tilde{e}}_i}^0{\tilde{e}}_j={\nabla }_{{\tilde{e}}_k}^0{\tilde{e}}_l,{\overline{\nabla }}_{{\tilde{e}}_3}^0{\tilde{e}}_3={\rho }_0{\tilde{e}}_3,{\overline{\nabla }}_{{\tilde{e}}_i}^1{\tilde{e}}_j={\nabla }_{{\tilde{e}}_k}^1{\tilde{e}}_l,{\overline{\nabla }}_{{\tilde{e}}_3}^1{\tilde{e}}_3={\rho }_0{\tilde{e}}_3.\end{array}$$

(5)

Now, we proceed to define the Ricci curvature as follows:

$$R^i(W,V)T=[{\overline{\nabla }}_W^i,{\overline{\nabla }}_V^i]T-{\overline{\nabla }}_{[W,V]}^{i}T,i\in \{0,1\}.$$

(6)

The Ricci tensor ${\rho }^i$ of $(G_i,g,J)$ is given by the following:

$${\rho }^i(W,V)=-g(R^i(W,{\tilde{e}}_1)V,{\tilde{e}}_1)-g(R^i(W,{\tilde{e}}_2)V,{\tilde{e}}_2)+g(R^i(W,{\tilde{e}}_3)V,{\tilde{e}}_3),i\in \{0,1\},$$

(7)

where $e_1,e_2$, and $e_3$ form a pseudo-orthonormal basis for ${\mathfrak{g}}_i$, with $e_3$ being timelike. And we define

$${\tilde{\rho }}^i(W,V)={\displaystyle \frac{{\rho }^i(W,V)+{\rho }^i(V,W)}{2}},i\in \{0,1\}.$$

(8)

Subsequently, the Ricci operator ${\widetilde{Ric}}^i$ is defined as follows:

$$g({\widetilde{Ric}}^i\left(W\right),V)={\tilde{\rho }}^i(W,V),i\in \{0,1\}.$$

(9)

Furthermore, the Schouten tensor $S^i$ can be expressed as follows:

$$S^i({\tilde{e}}_i,{\tilde{e}}_j)={\tilde{\rho }}^i(e_i,e_j)-s^i{\lambda }_{0}g(e_i,e_j),i\in \{0,1\},$$

(10)

where $s^i$ denotes the scalar curvature for different connection, and ${\lambda }_0$ is a real number. According to [20], the expression for the scalar curvature $s^i$ is given by the following:

$$s^i={\tilde{\rho }}^i({\tilde{e}}_1,{\tilde{e}}_1)+{\tilde{\rho }}^i({\tilde{e}}_2,{\tilde{e}}_2)-{\tilde{\rho }}^i({\tilde{e}}_3,{\tilde{e}}_3),i\in \{0,1\}.$$

(11)

Definition 1. 

The triple $(G_i,g,J)$ is called an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if it satisfies the equation

$${\widetilde{Ric}}^0=(s^0{\lambda }_0+c)Id+D^0,$$

(12)

where c and ${\lambda }_0$ are constants, and $D^0$ is a derivation on ${\mathfrak{g}}_{\mathfrak{i}}$; i.e., it satisfies the identity

$$D^0[W,V]=[D^{0}W,V]+[W,D^{0}V]forallW,V\in {\mathfrak{g}}_{\mathfrak{i}}.$$

(13)

The triple $(G_i,g,J)$ is called the algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ if it satisfies the equation

$${\widetilde{Ric}}^1=(s^1{\lambda }_0+c)Id+D^1,$$

(14)

where c and ${\lambda }_0$ are constants, and $D^1$ is a derivation on ${\mathfrak{g}}_{\mathfrak{i}}$; i.e., it satisfies the identity

$$D^1[W,V]=[D^{1}W,V]+[W,D^{1}V]forallW,V\in {\mathfrak{g}}_{\mathfrak{i}}.$$

(15)

3. Algebraic Schouten Solitons on Lorentzian Lie Groups Associated with the Perturbed Canonical Connections

In this section, we present the geometric properties of $G_i$, where i ranges from 1 to 7, and derive the algebraic conditions for algebraic Schouten solitons that are associated with the perturbed canonical connection on Lie groups $G_i$. Throughout the paper, we use the expression for the Lie algebras ${\mathfrak{g}}_{\mathfrak{i}}$ in [19,21], where ${\tilde{e}}_1$, ${\tilde{e}}_2$, and ${\tilde{e}}_3$ form a pseudo-orthonormal basis, with ${\tilde{e}}_3$ being timelike.

3.1. Algebraic Schouten Soliton of $G_1$

By [6], the non-vanishing brackets on the Lie algebra ${\mathfrak{g}}_{\mathfrak{1}}$ is as follows:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=\alpha {\tilde{e}}_1-\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\beta {\tilde{e}}_1+\alpha {\tilde{e}}_2+\alpha {\tilde{e}}_3,\alpha \ne 0$$

(16)

Then, we can formulate the following theorem concerning the algebraic condition for an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_1,g,J)$.

Theorem 1. 

The Lie group $(G_1,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if $\alpha \ne 0$, $\beta ={\rho }_0=0$, and $c={\displaystyle \frac{1}{2}}{\alpha }^2-2{\alpha }^2{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^0$ as follows:

$${\widetilde{Ric}}^0\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-({\alpha }^2+{\displaystyle \frac{1}{2}}{\beta }^2)& 0& -{\displaystyle \frac{1}{4}}\alpha \beta \\ \alpha \beta & -({\alpha }^2+{\displaystyle \frac{1}{2}}{\beta }^2)& -{\displaystyle \frac{1}{2}}({\alpha }^2+\alpha {\rho }_0)\\ {\displaystyle \frac{1}{4}}\alpha \beta & {\displaystyle \frac{1}{2}}{\alpha }^2& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(17)

Therefore, the scalar curvature can be derived as $s^0=-(2{\alpha }^2+{\beta }^2)$. Additionally, we can write $D^0$ as follows:

$$\left\{\begin{array}{c}{D}^0{\tilde{e}}_1=-({\alpha }^2+{\displaystyle \frac{{\beta }^2}{2}}-(2{\alpha }^2+{\beta }^2){\lambda }_0+c){\tilde{e}}_1-{\displaystyle \frac{\alpha \beta }{4}}{\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_2=-({\alpha }^2+{\displaystyle \frac{{\beta }^2}{2}}-(2{\alpha }^2+{\beta }^2){\lambda }_0+c){\tilde{e}}_2-{\displaystyle \frac{{\alpha }^2+\alpha {\rho }_0}{2}}{\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_3={\displaystyle \frac{\alpha \beta }{4}}{\tilde{e}}_1+{\displaystyle \frac{{\alpha }^2}{2}}{\tilde{e}}_2+((2{\alpha }^2+{\beta }^2){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(18)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\alpha ({\alpha }^2+{\displaystyle \frac{{\beta }^2}{2}}-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{\alpha {\beta }^2}{2}}-{\displaystyle \frac{{\alpha }^3+{\alpha }^2{\rho }_0}{2}}=0,\hfill \\ 5{\alpha }^2\beta +2\alpha \beta {\rho }_0=0,\hfill \\ \beta (2{\alpha }^2+{\beta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{{\alpha }^2\beta }{4}}=0,\hfill \\ \alpha ((2{\alpha }^2+{\beta }^2){\lambda }_0-c)-{\displaystyle \frac{{\alpha }^3}{2}}=0,\hfill \\ \beta ((2{\alpha }^2+{\beta }^2){\lambda }_0-c)=0,\hfill \\ \beta ((2{\alpha }^2+{\beta }^2){\lambda }_0-c)-{\displaystyle \frac{{\alpha }^2\beta }{2}}=0.\hfill \end{array}\right.$$

(19)

Since $\alpha \ne 0$, the last two equations of system (19) yield $\beta =0$. Then, the system (19) reduces to

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}{\alpha }^2-2{\alpha }^2{\lambda }_0+c-{\displaystyle \frac{\alpha {\rho }_0}{2}}=0,\hfill \\ -{\displaystyle \frac{{\alpha }^2}{2}}+2{\alpha }^2{\lambda }_0-c=0.\hfill \end{array}\right.$$

(20)

From this, we obtain ${\rho }_0=0$, $c={\displaystyle \frac{1}{2}}{\alpha }^2-2{\alpha }^2{\lambda }_0$. □

3.2. Algebraic Schouten Soliton of $G_2$

By [6], the non-vanishing brackets on the Lie algebra ${\mathfrak{g}}_{\mathfrak{2}}$ is as follows:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=\gamma {\tilde{e}}_2-\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=-\beta {\tilde{e}}_2-\gamma {\tilde{e}}_3,[{\tilde{e}}_2,{\tilde{e}}_3]=\alpha {\tilde{e}}_1,$$

(21)

where $\gamma \ne 0$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_2,g,J)$.

Theorem 2. 

The Lie group $(G_2,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if $\alpha =\beta =0$, $\gamma \ne 0$, and $c=-{\gamma }^2+2{\gamma }^2{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^0$ as follows:

$${\widetilde{Ric}}^0\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-({\gamma }^2+{\displaystyle \frac{1}{2}}\alpha \beta )& 0& -{\displaystyle \frac{1}{2}}\gamma {\rho }_0\\ 0& -({\gamma }^2+{\displaystyle \frac{1}{2}}\alpha \beta )& -{\displaystyle \frac{1}{2}}\beta \gamma +{\displaystyle \frac{1}{4}}\alpha \gamma \\ 0& {\displaystyle \frac{1}{2}}\beta \gamma -{\displaystyle \frac{1}{4}}\alpha \gamma & 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(22)

Therefore, the scalar curvature can be derived as $s^0=-(2{\gamma }^2+\alpha \beta )$. Additionally, we can write $D^0$ as follows:

$$\left\{\begin{array}{c}{D}^0{\tilde{e}}_1=-({\gamma }^2+{\displaystyle \frac{\alpha \beta }{2}}-(2{\gamma }^2+\alpha \beta ){\lambda }_0+c){\tilde{e}}_1+{\displaystyle \frac{\gamma {\rho }_0}{2}}{\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_2=-({\gamma }^2+{\displaystyle \frac{\alpha \beta }{2}}-(2{\gamma }^2+\alpha \beta ){\lambda }_0+c){\tilde{e}}_2-({\displaystyle \frac{\beta \gamma }{2}}-{\displaystyle \frac{\alpha \gamma }{4}}){\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_3=({\displaystyle \frac{\beta \gamma }{2}}-{\displaystyle \frac{\alpha \gamma }{4}}){\tilde{e}}_2+((2{\gamma }^2+\alpha \beta ){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(23)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\gamma ({\gamma }^2+{\displaystyle \frac{\alpha \beta }{2}}-(2{\gamma }^2+\alpha \beta ){\lambda }_0+c)-\beta (\beta \gamma -{\displaystyle \frac{\alpha \gamma }{2}})=0,\hfill \\ \alpha \gamma {\rho }_0=0,\hfill \\ \beta (3{\gamma }^2+{\displaystyle \frac{\alpha \beta }{2}}-(2{\gamma }^2+\alpha \beta ){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\alpha {\gamma }^2=0,\hfill \\ \beta ({\gamma }^2-(2{\gamma }^2+\alpha \beta ){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\alpha {\gamma }^2=0,\hfill \\ \alpha ((2{\gamma }^2+\alpha \beta ){\lambda }_0-c)=0.\hfill \end{array}\right.$$

(24)

Since $\gamma \ne 0$, the second equation of (24) implies either $\alpha =0$ or ${\rho }_0=0$. If we set $\alpha =0$, then system (24) becomes

$$\left\{\begin{array}{c}\gamma ({\gamma }^2-{\beta }^2-2{\gamma }^2{\lambda }_0+c)=0,\hfill \\ \beta (3{\gamma }^2-2{\gamma }^2{\lambda }_0+c)=0,\hfill \\ \beta ({\gamma }^2-2{\gamma }^2{\lambda }_0+c)=0.\hfill \end{array}\right.$$

(25)

The last two equations above indicates that $\beta =0$ and $c=-{\gamma }^2+2{\gamma }^2{\lambda }_0$. Now, let ${\rho }_0=0$. The last two equations in (24) give $\beta \gamma -{\displaystyle \frac{1}{2}}\alpha \gamma =0$. Substituting this into the first two equations in (24) results in ${\gamma }^2=0$, which is a contradiction. □

3.3. Algebraic Schouten Soliton of $G_3$

By [6], the non-vanishing brackets on the Lie algebra ${\mathfrak{g}}_{\mathfrak{3}}$ is as follows:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=-\gamma {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=-\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\alpha {\tilde{e}}_1.$$

(26)

Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_3,g,J)$.

Theorem 3. 

The Lie group $(G_3,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if one of the following holds:

1. 

$\alpha =\beta =\gamma =0$, for all c;

2. 

$\alpha \ne 0$, $\beta =\gamma =0$, $c=0$;

3. 

$\alpha =\gamma =0$, $\beta \ne 0$, $c=0$;

4. 

$\alpha \ne 0$, $\beta \ne 0$, $\gamma =0$, $c=0$;

5. 

$\alpha =\beta =0$, $\gamma \ne 0$, $c=-{\gamma }^2+{\gamma }^2{\lambda }_0$;

6. 

$\alpha =\gamma \ne 0$, $\beta =0$, $c=0$;

7. 

$\alpha =0$, $\beta \ne 0$, $\gamma \ne 0$, and $\beta =\gamma$, $c=0$;

8. 

$\alpha \ne 0$, $\beta \ne 0$, $\gamma \ne 0$, and $\alpha +\beta =\gamma$, $c=0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^0$ as follows:

$${\widetilde{Ric}}^0\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-{\displaystyle \frac{1}{2}}\gamma (\alpha +\beta -\gamma )& 0& 0\\ 0& -{\displaystyle \frac{1}{2}}\gamma (\alpha +\beta -\gamma )& 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(27)

Therefore, the scalar curvature can be derived as $s^0=-\gamma (\alpha +\beta -\gamma )$. Additionally, we can write $D^0$ as follows:

$$\left\{\begin{array}{c}{D}^0{\tilde{e}}_1=-(\gamma (\alpha +\beta -\gamma )({\displaystyle \frac{1}{2}}-{\lambda }_0)+c){\tilde{e}}_1,\hfill \\ D^0{\tilde{e}}_2=-(\gamma (\alpha +\beta -\gamma )({\displaystyle \frac{1}{2}}-{\lambda }_0)+c){\tilde{e}}_2,\hfill \\ D^0{\tilde{e}}_3=(\gamma (\alpha +\beta -\gamma ){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(28)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}{\gamma }^2(\alpha +\beta -\gamma )(1-{\lambda }_0)+\gamma c=0,\hfill \\ \beta \gamma (\alpha +\beta -\gamma ){\lambda }_0-\beta c=0,\hfill \\ \alpha \gamma (\alpha +\beta -\gamma ){\lambda }_0-\alpha c=0.\hfill \end{array}\right.$$

(29)

The first equation above implies that either $\gamma =0$ or $\gamma \ne 0$. First, if we assume $\gamma =0$, we encounter cases (1)–(4). Next, let $\gamma \ne 0$; then, $c=\gamma (\alpha +\beta -\gamma )({\lambda }_0-1)$. Meanwhile, if $\alpha =0$, we have cases (5)–(7). On the other hand, if $\alpha \ne 0$, we are led to case (8), where system (29) holds. □

3.4. Algebraic Schouten Soliton of $G_4$

By [6], the non-vanishing brackets on the Lie algebra ${\mathfrak{g}}_{\mathfrak{4}}$ is as follows:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=-{\tilde{e}}_2+(2\eta -\beta ){\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]={\tilde{e}}_3-\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\alpha {\tilde{e}}_1,\eta =1or-1,$$

(30)

Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_4,g,J)$.

Theorem 4. 

The Lie group $(G_4,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if $\alpha =0$, ${\rho }_0\ne 0$, $c=0$, and β satisfies $\beta =\eta$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^0$ as follows:

$${\widetilde{Ric}}^0\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}(2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )-1& 0& -{\displaystyle \frac{{\rho }_0}{2}}\\ 0& (2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )-1& -({\displaystyle \frac{\alpha }{4}}+{\displaystyle \frac{\eta }{2}}-{\displaystyle \frac{\beta }{2}})\\ 0& ({\displaystyle \frac{\alpha }{4}}+{\displaystyle \frac{\eta }{2}}-{\displaystyle \frac{\beta }{2}})& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(31)

Therefore, the scalar curvature can be derived as $s^0=2(2\eta -\beta )({\displaystyle \frac{1}{2}}\alpha +\eta )-2$. Additionally, we can write $D^0$ as follows:

$$\left\{\begin{array}{c}{D}^0{\tilde{e}}_1=(({\displaystyle \frac{1}{2}}\alpha +\eta )(2\eta -\beta )(1-2{\lambda }_0)-1+2{\lambda }_0-c){\tilde{e}}_1-{\displaystyle \frac{{\rho }_0}{2}}{\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_2=(({\displaystyle \frac{1}{2}}\alpha +\eta )(2\eta -\beta )(1-2{\lambda }_0)-1+2{\lambda }_0-c){\tilde{e}}_2-({\displaystyle \frac{\alpha }{4}}+{\displaystyle \frac{\eta }{2}}-{\displaystyle \frac{\beta }{2}}){\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_3=({\displaystyle \frac{\alpha }{4}}+{\displaystyle \frac{\eta }{2}}-{\displaystyle \frac{\beta }{2}}){\tilde{e}}_2-((2(2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )-2){\lambda }_0+c){\tilde{e}}_3.\hfill \end{array}\right.$$

(32)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}{\displaystyle \frac{\alpha {\rho }_0}{2}}=0,\hfill \\ (2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )(1-2{\lambda }_0)-1+2{\lambda }_0-c+(\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta -\beta )=0,\hfill \\ (2\eta -\beta )(2(2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )(1-{\lambda }_0)-2+2{\lambda }_0+c-({\displaystyle \frac{\alpha }{2}}+\eta -\beta )=0,\hfill \\ \beta ((2(2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )-2){\lambda }_0+c)-({\displaystyle \frac{\alpha }{2}}+\eta -\beta )=0,\hfill \\ \alpha ((2(2\eta -\beta )({\displaystyle \frac{\alpha }{2}}+\eta )-2){\lambda }_0+c)=0.\hfill \end{array}\right.$$

(33)

The first equation above implies that either $\alpha =0$ or ${\rho }_0=0$. First, we consider that $\alpha =0$, and we obtain

$$\left\{\begin{array}{c}\eta (2\eta -\beta )(1-2{\lambda }_0)-1+2{\lambda }_0=c-(\eta -\beta )(\eta -\beta ),\hfill \\ (2\eta -\beta )(2\eta (2\eta -\beta )(1-{\lambda }_0)-2+2{\lambda }_0-c)=(\eta -\beta ),\hfill \\ \beta ((2\eta (2\eta -\beta )-2){\lambda }_0+c)=(\eta -\beta ).\hfill \end{array}\right.$$

(34)

Then, we obtain $\alpha =0$, $\beta =\eta$, ${\rho }_0\ne 0$, and $c=0$.

Next, we consider the second case where ${\rho }_0=0$ and $\alpha \ne 0$. Then, $c=-(2(2\eta -\beta )({\displaystyle \frac{1}{2}}\alpha +\eta )-2){\lambda }_0$. Meanwhile, we have

$$\left\{\begin{array}{c}(2\eta -\beta )(\alpha +2\eta )=2,\hfill \\ (2\eta -\beta )\left(\right(2\eta -\beta \left)\right(\alpha +2\eta )-2)=0.\hfill \end{array}\right.$$

(35)

This is a contradiction. □

3.5. Algebraic Schouten Soliton of $G_5$

By [6], the non-vanishing brackets on the Lie algebra ${\mathfrak{g}}_{\mathfrak{5}}$ is as follows:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=0,[{\tilde{e}}_1,{\tilde{e}}_3]=\alpha {\tilde{e}}_1+\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\gamma {\tilde{e}}_1+\delta {\tilde{e}}_2,$$

(36)

where $\alpha +\delta \ne 0$ and $\alpha \gamma -\beta \delta =0$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_5,g,J)$.

Theorem 5. 

The Lie group $(G_5,g,J)$ possesses a trivial algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if $c=0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive that ${\widetilde{Ric}}^0$ is a zero matrix. Therefore, the scalar curvature can be obtained as $s^0=0$. We can express $D^0$ as

$$\left\{\begin{array}{c}{D}^0{\tilde{e}}_1=-c{\tilde{e}}_1,\hfill \\ D^0{\tilde{e}}_2=-c{\tilde{e}}_2,\hfill \\ D^0{\tilde{e}}_3=-c{\tilde{e}}_3.\hfill \end{array}\right.$$

(37)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if $c=0$. □

3.6. Algebraic Schouten Soliton of $G_6$

By [6], the non-vanishing brackets on the Lie algebra ${\mathfrak{g}}_{\mathfrak{6}}$ is as follows:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=\alpha {\tilde{e}}_2+\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=\gamma {\tilde{e}}_2+\delta {\tilde{e}}_3,\alpha +\delta \ne 0,\alpha \gamma -\beta \delta =0.$$

(38)

Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_6,g,J)$.

Theorem 6. 

The Lie group $(G_6,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if one of the following holds:

1. 

$\alpha =\beta =\gamma =0$, $\delta \ne 0$, $c=-s{\lambda }_0$;

2. 

$\alpha \ne 0$, $\beta \ne 0$, $\gamma =\delta =0$, and ${\beta }^2=2{\alpha }^2$, $c=-s{\lambda }_0$;

3. 

$\alpha \ne 0$, $\beta =\gamma =\delta =0$, $c=-{\alpha }^2-s{\lambda }_0$;

4. 

$\alpha \ne 0$, $\beta =\gamma =0$, $\delta \ne 0$, and $\alpha =\delta$, $c=-{\alpha }^2-s{\lambda }_0$;

5. 

$\alpha \ne 0$, $\beta =\gamma =0$, $\delta \ne 0$, and $\alpha \ne \delta$, $c=-{\alpha }^2-s{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^0$ as follows:

$${\widetilde{Ric}}^0\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}{\displaystyle \frac{1}{2}}{\beta }^2-{\displaystyle \frac{1}{2}}\beta \gamma -{\alpha }^2& 0& -{\displaystyle \frac{1}{2}}\delta {\rho }_0\\ 0& {\displaystyle \frac{1}{2}}{\beta }^2-{\displaystyle \frac{1}{2}}\beta \gamma -{\alpha }^2& -{\displaystyle \frac{1}{2}}(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma )\\ 0& {\displaystyle \frac{1}{2}}(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma )& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(39)

Therefore, the scalar curvature can be derived as $s^0=\beta (\beta -\gamma )-2{\alpha }^2$. Additionally, we can write $D^0$ as follows:

$$\left\{\begin{array}{c}{D}^0{\tilde{e}}_1=(\beta ({\displaystyle \frac{1}{2}}-{\lambda }_0)(\beta -\gamma )-{\alpha }^2+2{\alpha }^2{\lambda }_0-c){\tilde{e}}_1-{\displaystyle \frac{\delta {\rho }_0}{2}}{\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_2=(\beta ({\displaystyle \frac{1}{2}}-{\lambda }_0)(\beta -\gamma )-{\alpha }^2+2{\alpha }^2{\lambda }_0-c){\tilde{e}}_2-{\displaystyle \frac{1}{2}}(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma ){\tilde{e}}_3,\hfill \\ D^0{\tilde{e}}_3={\displaystyle \frac{1}{2}}(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma ){\tilde{e}}_2-((\beta (\beta -\gamma )-2{\alpha }^2){\lambda }_0+c){\tilde{e}}_3.\hfill \end{array}\right.$$

(40)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}2\alpha (\beta (\beta -\gamma )({\displaystyle \frac{1}{2}}-{\lambda }_0)-{\alpha }^2+2{\alpha }^2{\lambda }_0-c)+(\beta +\gamma )(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma )=0,\hfill \\ 2\beta (\beta (\beta -\gamma )(1-{\lambda }_0)-2{\alpha }^2+2{\alpha }^2{\lambda }_0+c+(\delta -\alpha )(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma )=0,\hfill \\ 2\gamma ((\beta (\beta -\gamma )-2{\alpha }^2){\lambda }_0+c)+(\delta -\alpha )(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta (\beta -\gamma ))=0,\hfill \\ 2\delta ({\displaystyle \frac{1}{2}}\beta (\beta -\gamma )-{\alpha }^2-(\beta (\beta -\gamma )-2{\alpha }^2){\lambda }_0-c)+(\beta +\gamma )(-\alpha \gamma +{\displaystyle \frac{1}{2}}\delta \beta -{\displaystyle \frac{1}{2}}\delta \gamma )=0.\hfill \end{array}\right.$$

(41)

The first equation above implies that either $\alpha =0$ or $\alpha \ne 0$. First, assume that $\alpha =0$. By the condition in (38), we can deduce that $\delta \ne 0$ and $\beta =\gamma =0$. Therefore, case (1) is true.

Now, consider $\alpha \ne 0$. Let $\delta =0$. Since $\alpha \gamma -\beta \delta =0$, we can derive that $\gamma =0$. Then, system (41) reduces to

$$\left\{\begin{array}{c}\alpha (({\beta }^2-2{\alpha }^2)({\displaystyle \frac{1}{2}}-{\lambda }_0)-c)=0,\hfill \\ \beta (({\beta }^2-2{\alpha }^2)({\displaystyle \frac{1}{2}}-{\lambda }_0)-c)=0.\hfill \end{array}\right.$$

(42)

If $\beta \ne 0$, then it follows that ${\beta }^2=2{\alpha }^2$ and $c=0$. Conversely, if $\beta =0$, then we have $c=-{\alpha }^2+2{\alpha }^2{\lambda }_0$. Therefore, cases (2) and (3) hold.

Next, let $\delta \ne 0$. From the first and the fourth equations of system (41), we obtain the equation $(\alpha -\delta )({\displaystyle \frac{1}{2}}\beta (\beta -\gamma )-{\alpha }^2-s{\lambda }_0-c)=0$, which yields either $\alpha =\delta$ or $\alpha \ne \delta$. Let $\alpha =\delta$; then, we can derive $\beta =\gamma$. In this case, system (41) becomes

$$\left\{\begin{array}{c}\alpha (-{\alpha }^2+2{\alpha }^2{\lambda }_0-c)-{\displaystyle \frac{1}{2}}\alpha \gamma (\beta +\gamma )+{\displaystyle \frac{1}{4}}\delta ({\beta }^2-{\gamma }^2)=0,\hfill \\ \beta (2{\alpha }^2({\lambda }_0-1)-c)=0,\hfill \\ \gamma (2{\alpha }^2{\lambda }_0-c)=0.\hfill \end{array}\right.$$

(43)

If $\beta =\gamma =0$, then we have $c=-{\alpha }^2+2{\alpha }^2{\lambda }_0$, and case (4) is true. On the other hand, if $\beta \ne 0$, the last two equations of system (43) yield $2{\alpha }^2=0$, which is a contradiction.

Now, consider $\alpha \ne \delta$. From the first and the fourth equations of system (41), we obtain ${\displaystyle \frac{1}{2}}\beta (\beta -\gamma )-{\alpha }^2-(\beta (\beta -\gamma )-2{\alpha }^2){\lambda }_0-c=0$. Then, system (41) becomes

$$\left\{\begin{array}{c}\beta \delta -\delta \gamma -2\alpha \gamma =0,\hfill \\ \beta (\beta (1-{\lambda }_0)(\beta -\gamma )-2{\alpha }^2+2{\alpha }^2{\lambda }_0-c)=0,\hfill \\ \gamma ((\beta (\beta -\gamma )-2{\alpha }^2){\lambda }_0+c)=0.\hfill \end{array}\right.$$

(44)

From the second equation above, we can derive that $\beta +\gamma =0$. Since $\alpha \gamma =\beta \delta$, we deduce that $\beta =\gamma =0$. Therefore, case (5) holds. □

3.7. Algebraic Schouten Soliton of $G_7$

By [6], the non-vanishing bracket on the Lie algebra ${\mathfrak{g}}_{\mathfrak{7}}$ is as follows:

$$\begin{array}{cc}\hfill & [{\tilde{e}}_1,{\tilde{e}}_2]=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2-\beta {\tilde{e}}_3,\hfill \\ \hfill & [{\tilde{e}}_1,{\tilde{e}}_3]=\alpha {\tilde{e}}_1+\beta {\tilde{e}}_2+\beta {\tilde{e}}_3,\hfill \\ \hfill & [{\tilde{e}}_2,{\tilde{e}}_3]=\gamma {\tilde{e}}_1+\delta {\tilde{e}}_2+\delta {\tilde{e}}_3,\hfill \end{array}$$

(45)

where $\alpha +\delta \ne 0$ and $\alpha \gamma =0$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed canonical connections on $(G_7,g,J)$.

Theorem 7. 

The Lie group $(G_7,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^0$ if one of the following holds:

1. 

$\alpha \ne 0$, $\beta =\gamma =0$, $\delta =0$, ${\rho }_0\ne 0$, $c=-{\displaystyle \frac{1}{2}}{\alpha }^2+2{\alpha }^2{\lambda }_0$;

2. 

$\alpha \ne 0$, $\beta =\gamma =0$, $\delta \ne 0$, ${\rho }_0=0$, $c=-{\displaystyle \frac{1}{2}}{\alpha }^2+2{\alpha }^2{\lambda }_0$;

3. 

$\alpha =\gamma =0$, $\delta \ne 0$, $\rho =0$, and $c=0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^0$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^0$ as follows:

$${\widetilde{Ric}}^0\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right)=\left(\begin{array}{ccc}-({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma )& 0& {\displaystyle \frac{1}{2}}(\alpha \gamma +{\displaystyle \frac{1}{2}}\gamma \delta -\beta {\rho }_0)\\ 0& -({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma )& -{\displaystyle \frac{1}{2}}({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma +\delta {\rho }_0)\\ -{\displaystyle \frac{1}{2}}(\alpha \gamma +{\displaystyle \frac{1}{2}}\gamma \delta )& {\displaystyle \frac{1}{2}}({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma )& 0\end{array}\right)\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right).$$

(46)

Therefore, the scalar curvature can be derived as $s^0=-2{\alpha }^2-\beta \gamma$. Additionally, we can write $D^0$ as follows:

$$\left\{\begin{array}{c}{D}^0{e}_1=-({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)e_1+{\displaystyle \frac{1}{2}}(\alpha \gamma +{\displaystyle \frac{1}{2}}\gamma \delta -\beta {\rho }_0)e_3,\hfill \\ D^0{e}_2=-({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)e_2-{\displaystyle \frac{1}{2}}({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma +\delta {\rho }_0)e_3,\hfill \\ D^0{e}_3=-{\displaystyle \frac{1}{2}}(\alpha \gamma +{\displaystyle \frac{1}{2}}\gamma \delta )e_1+{\displaystyle \frac{1}{2}}({\alpha }^2+{\displaystyle \frac{1}{2}}\beta \gamma )e_2+((2{\alpha }^2-\beta \gamma ){\lambda }_0-c)e_3.\hfill \end{array}\right.$$

(47)

Consequently, according to (13), the existence of an algebraic Schouten soliton in relation to the perturbed canonical connection ${\overline{\nabla }}^0$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\alpha ({\displaystyle \frac{1}{2}}{\alpha }^2+{\displaystyle \frac{\beta \gamma }{4}}-{\displaystyle \frac{1}{2}}\delta {\rho }_0-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\gamma (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}}-\beta {\rho }_0)-{\displaystyle \frac{1}{2}}\beta (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}})=0,\hfill \\ \beta ({\alpha }^2+{\displaystyle \frac{\beta \gamma }{2}}-{\displaystyle \frac{1}{2}}\delta {\rho }_0-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\delta (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}}-\beta {\rho }_0)=0,\hfill \\ \beta ({\alpha }^2+{\displaystyle \frac{\beta \gamma }{2}}-\delta {\rho }_0-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)+{\displaystyle \frac{1}{2}}(\alpha -\delta )(\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}}-\beta {\rho }_0)=0,\hfill \\ \alpha ({\displaystyle \frac{1}{2}}{\alpha }^2+{\displaystyle \frac{\beta \gamma }{4}}-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}})=0,\hfill \\ \beta ({\alpha }^2+{\displaystyle \frac{\beta \gamma }{2}}-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)=0,\hfill \\ \beta ({\displaystyle \frac{1}{2}}{\alpha }^2+{\displaystyle \frac{\beta \gamma }{4}}-\delta {\rho }_0-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)+{\displaystyle \frac{1}{2}}\alpha (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}}-\beta {\rho }_0)=0,\hfill \\ \gamma ((2{\alpha }^2+\beta \gamma ){\lambda }_0-c)-{\displaystyle \frac{1}{2}}(\alpha -\delta )(\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}})=0,\hfill \\ \delta ({\displaystyle \frac{1}{2}}{\alpha }^2+{\displaystyle \frac{\beta \gamma }{4}}-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)+{\displaystyle \frac{1}{2}}\beta (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}})=0,\hfill \\ \delta ({\displaystyle \frac{1}{2}}{\alpha }^2+{\displaystyle \frac{\beta \gamma }{4}}-\delta {\rho }_0-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)+{\displaystyle \frac{1}{2}}\beta (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}})+{\displaystyle \frac{1}{2}}\gamma (\alpha \gamma +{\displaystyle \frac{\gamma \delta }{2}}-\beta {\rho }_0)=0.\hfill \end{array}\right.$$

(48)

Since $\alpha \gamma =0$ and $\alpha +\delta \ne 0$, we first consider the case where $\alpha =0$ and $\gamma \ne 0$. From the fourth equation above, we can derive that $\beta =0$. Substituting it into the first equation above, we have ${\displaystyle \frac{1}{4}}{\gamma }^2\delta =0$; this is a contradiction to our initial assumption. Next, we consider the second case where $\alpha \ne 0$ and $\gamma =0$. Under this assumption, system (48) becomes

$$\left\{\begin{array}{c}\alpha ({\alpha }^2-2{\alpha }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\alpha ({\alpha }^2+\delta {\rho }_0)=0,\hfill \\ \beta ({\alpha }^2-2{\alpha }^2{\lambda }_0+c)=0,\hfill \\ \beta ({\alpha }^2-\delta {\rho }_0-2{\alpha }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}(\alpha -\delta )\beta {\rho }_0=0,\hfill \\ {\displaystyle \frac{1}{2}}{\alpha }^2-2{\alpha }^2{\lambda }_0+c=0,\hfill \\ \beta ({\alpha }^2-2{\alpha }^2{\lambda }_0+c)=0,\hfill \\ \beta ({\alpha }^2-{\displaystyle \frac{1}{2}}\delta {\rho }_0-2{\alpha }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\alpha \beta {\rho }_0=0,\hfill \\ \delta ({\displaystyle \frac{1}{2}}{\alpha }^2-2{\alpha }^2{\lambda }_0+c)=0,\hfill \\ \delta ({\displaystyle \frac{1}{2}}{\alpha }^2-{\displaystyle \frac{1}{2}}\delta {\rho }_0-2{\alpha }^2{\lambda }_0+c)=0.\hfill \end{array}\right.$$

(49)

From the fourth and fifth equations above, we obtain $\beta =0$. Then, the third equation above gives $\delta {\rho }_0=0$, which implies either $\delta =0$ or ${\rho }_0=0$. First, if we assume $\delta =0$, then the first equation above yields $c=-{\displaystyle \frac{1}{2}}{\alpha }^2+2{\alpha }^2{\lambda }_0$. On the other hand, if $\rho =0$, then we have $c=-{\displaystyle \frac{1}{2}}{\alpha }^2+2{\alpha }^2{\lambda }_0$. Therefore, cases (1)–(2) hold.

Finally, we consider the third case where $\alpha =\gamma =0$; then, it follows that $\delta \ne 0$. In this case, system (48) becomes

$$\left\{\begin{array}{c}\beta c=0,\hfill \\ \beta c-{\displaystyle \frac{1}{2}}\beta \delta \rho =0,\hfill \\ \delta c=0,\hfill \\ \delta c-{\displaystyle \frac{1}{2}}{\delta }^2\rho =0.\hfill \end{array}\right.$$

(50)

Then, we have ${\rho }_0=0$ and $c=0$. Therefore, we conclude that case (3) holds. □

4. Algebraic Schouten Solitons on Lorentzian Lie Groups Associated with the Perturbed Kobayashi–Nomizu Connections

In this section, we present the geometry properties of $G_i$, where i ranges from 1 to 7, and derive the algebraic conditions for algebraic Schouten solitons that are associated with the perturbed Kobayashi–Nomizu connection on Lie groups $G_i$.

4.1. Algebraic Schouten Soliton of $G_1$

Similarly to Section 3.1, we have the expression of the Lie algebra of $G_1$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_1,g,J)$.

Theorem 8. 

The Lie group $(G_1,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^1$ if $\alpha \ne 0$, $\beta ={\rho }_0=0$, and $c=-{\displaystyle \frac{1}{2}}{\alpha }^2+2{\alpha }^2{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^1$ as follows:

$${\widetilde{Ric}}^1\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-({\alpha }^2+{\beta }^2)& \alpha \beta & {\displaystyle \frac{1}{2}}\alpha \beta \\ \alpha \beta & -({\alpha }^2+{\beta }^2)& -{\displaystyle \frac{1}{2}}({\alpha }^2+\alpha {\rho }_0)\\ -{\displaystyle \frac{1}{2}}\alpha \beta & {\displaystyle \frac{1}{2}}{\alpha }^2& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(51)

Therefore, the scalar curvature can be derived as $s^1=-2({\alpha }^2+{\beta }^2)$. Additionally, we can write $D^1$ as follows:

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-(({\alpha }^2+{\beta }^2)(1-2{\lambda }_0)+c){\tilde{e}}_1+\alpha \beta {\tilde{e}}_2+{\displaystyle \frac{\alpha \beta }{2}}{\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_2=\alpha \beta {\tilde{e}}_1-(({\alpha }^2+{\beta }^2)(1-2{\lambda }_0)+c){\tilde{e}}_2-{\displaystyle \frac{{\alpha }^2+\alpha {\rho }_0}{2}}{\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_3=-{\displaystyle \frac{1}{2}}\alpha \beta {\tilde{e}}_1+{\displaystyle \frac{1}{2}}{\alpha }^2{\tilde{e}}_2+(2({\alpha }^2+{\beta }^2){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(52)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\alpha ({\alpha }^2+{\beta }^2-2({\alpha }^2+{\beta }^2){\lambda }_0+c)={\displaystyle \frac{{\alpha }^3+{\alpha }^2{\rho }_0}{2}}-\alpha {\beta }^2,\hfill \\ {\alpha }^2\beta -\alpha \beta {\rho }_0=0,\hfill \\ \beta (2{\alpha }^2+2{\beta }^2-2({\alpha }^2+{\beta }^2){\lambda }_0+c)={\alpha }^2\beta ,\hfill \\ \alpha (2({\alpha }^2+{\beta }^2){\lambda }_0-c)=2\alpha {\beta }^2+{\displaystyle \frac{{\alpha }^3}{2}},\hfill \\ \beta (2({\alpha }^2+{\beta }^2){\lambda }_0-c)=2{\alpha }^2\beta ,\hfill \\ \beta (2({\alpha }^2+{\beta }^2){\lambda }_0-c)={\alpha }^2\beta .\hfill \end{array}\right.$$

(53)

Since $\alpha \ne 0$, the last two equations of (53) imply that $\beta =0$. Then, the system (53) becomes

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}{\alpha }^2-2{\alpha }^2{\lambda }_0+c={\displaystyle \frac{\alpha {\rho }_0}{2}},\hfill \\ {\displaystyle \frac{{\alpha }^2}{2}}-2{\alpha }^2{\lambda }_0+c=0.\hfill \end{array}\right.$$

(54)

Therefore, we have ${\rho }_0=0$, and $c=-{\displaystyle \frac{1}{2}}{\alpha }^2+2{\alpha }^2{\lambda }_0$. □

4.2. Algebraic Schouten Soliton of $G_2$

Similarly to Section 3.2, we have the expression of the Lie algebra of $G_2$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_2,g,J)$.

Theorem 9. 

The Lie group $(G_2,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^1$ if $\alpha =\beta =0$, $\gamma \ne 0$, and $c=-{\gamma }^2+2{\gamma }^2{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^1$ as follows:

$${\widetilde{Ric}}^1\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-({\gamma }^2+{\beta }^2)& 0& {\displaystyle \frac{1}{2}}\gamma {\rho }_0\\ 0& -({\gamma }^2+\alpha \beta )& {\displaystyle \frac{1}{2}}\alpha \gamma \\ 0& -{\displaystyle \frac{1}{2}}\alpha \gamma & 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(55)

Therefore, the scalar curvature can be derived as $s^1=-(2{\gamma }^2+{\beta }^2+\alpha \beta )$. Additionally, we can write $D^1$ as follows:

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-({\gamma }^2+{\beta }^2-(2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0+c){\tilde{e}}_1+{\displaystyle \frac{1}{2}}\gamma {\rho }_0{\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_2=-({\gamma }^2+\alpha \beta -(2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0+c){\tilde{e}}_2+{\displaystyle \frac{1}{2}}\alpha \gamma {\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_3=-{\displaystyle \frac{1}{2}}\alpha \gamma {\tilde{e}}_2+((2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(56)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\gamma ({\gamma }^2+{\beta }^2-(2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0+c)=-\alpha \beta \gamma ,\hfill \\ \beta (2{\gamma }^2+{\beta }^2+\alpha \beta -(2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0+c)=\alpha {\gamma }^2,\hfill \\ \alpha \gamma {\rho }_0=0,\hfill \\ \beta ((2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0-c)+\alpha {\gamma }^2=0.\hfill \end{array}\right.$$

(57)

Since $\gamma \ne 0$, the third equation in (57) yields either $\alpha =0$ or ${\rho }_0=0$. Let us first consider $\alpha =0$. In this case, system (57) becomes

$$\left\{\begin{array}{c}\gamma ({\gamma }^2+{\beta }^2-(2{\gamma }^2+{\beta }^2){\lambda }_0+c)=0,\hfill \\ \beta (2{\gamma }^2+{\beta }^2-(2{\gamma }^2+{\beta }^2){\lambda }_0+c)=0,\hfill \\ \beta (-(2{\gamma }^2+{\beta }^2){\lambda }_0+c)=0.\hfill \end{array}\right.$$

(58)

Assume $\beta \ne 0$. From the first and last equations above, we obtain ${\beta }^2+{\gamma }^2=0$, which is a contradiction. Therefore, $\beta =0$ and $c=-{\gamma }^2+2{\gamma }^2{\lambda }_0$.

Next, let us consider ${\rho }_0=0$ and $\alpha \ne 0$. If $\beta =0$, from the last equation in (57), we obtain $\alpha {\gamma }^2=0$, which is a contradiction. Therefore, we have $\beta \ne 0$, and from the second and the last equations in (57), we deduce $\beta (2{\gamma }^2+{\beta }^2+\alpha \beta )=0$. Then, system (57) becomes

$$\left\{\begin{array}{c}\gamma ({\gamma }^2+{\beta }^2-(2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0+c)+\alpha \beta \gamma =0,\hfill \\ \beta (2{\gamma }^2+{\beta }^2+\alpha \beta ){\lambda }_0-\beta c+\alpha {\gamma }^2=0,\hfill \\ 2{\gamma }^2+{\beta }^2+\alpha \beta =0.\hfill \end{array}\right.$$

(59)

From the equations above, we can derive $\alpha =-\beta$. Substituting into the last equation, we obtain ${\gamma }^2=0$, which is a contradiction. □

4.3. Algebraic Schouten Soliton of $G_3$

Similarly to Section 3.3, we have the expression of the Lie algebra of $G_3$. Then, we formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_3,g,J)$.

Theorem 10. 

The Lie group $(G_3,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ if one of the following holds:

1. 

$\alpha =\beta =\gamma =0$, for all c;

2. 

$\alpha \ne 0$, $\beta =\gamma =0$, $c=0$;

3. 

$\alpha =\gamma =0$, $\beta \ne 0$, $c=0$;

4. 

$\alpha \ne 0$, $\beta \ne 0$, $\gamma =0$, $c=0$;

5. 

$\alpha =\beta =0$, $\gamma \ne 0$, $c=0$;

6. 

$\alpha \ne 0$, $\gamma \ne 0$, $\beta =0$, and $\alpha =\gamma$, $c=-{\alpha }^2+{\alpha }^2{\lambda }_0$;

7. 

$\alpha =0$, $\beta \ne 0$, $\gamma \ne 0$, $c=-{\beta }^2+{\beta }^2{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^1$ as follows:

$${\widetilde{Ric}}^1\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-\beta \gamma & 0& 0\\ 0& -\alpha \gamma & 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(60)

Therefore, the scalar curvature can be derived as $s^1=-(\alpha \gamma +\beta \gamma )$. Additionally, we can write $D^1$ as follows:

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-(\beta \gamma -(\alpha \gamma +\beta \gamma ){\lambda }_0+c){\tilde{e}}_1,\hfill \\ D^1{\tilde{e}}_2=-(\alpha \gamma -(\alpha \gamma +\beta \gamma ){\lambda }_0+c){\tilde{e}}_2,\hfill \\ D^1{\tilde{e}}_3=((\alpha \gamma +\beta \gamma ){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(61)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if it satisfies the following system of equations:

$$\left\{\begin{array}{c}\gamma (\alpha \gamma +\beta \gamma -(\alpha \gamma +\beta \gamma ){\lambda }_0+c)=0,\hfill \\ \beta (\beta \gamma -\alpha \gamma -(\alpha \gamma +\beta \gamma ){\lambda }_0+c)=0,\hfill \\ \alpha (\alpha \gamma -\beta \gamma -(\alpha \gamma +\beta \gamma ){\lambda }_0+c)=0.\hfill \end{array}\right.$$

(62)

The first equation above implies that either $\gamma =0$ or $\gamma \ne 0$. Assume first $\gamma =0$. We encounter cases (1)–(4). Next, let $\gamma \ne 0$; then, $c=-\alpha \gamma -\beta \gamma +(\alpha \gamma +\beta \gamma ){\lambda }_0$. Meanwhile, if $\beta =0$, we have cases (5) and (6); if $\beta \ne 0$, then for case (7), system (62) holds. □

4.4. Algebraic Schouten Soliton of $G_4$

Similarly to Section 3.4, we have the expression of the Lie algebra of $G_4$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_4,g,J)$.

Theorem 11. 

The Lie group $(G_4,g,J)$ is not the algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^1$ as follows:

$${\widetilde{Ric}}^1\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-(1+(\beta -2\eta \left)\beta \right)& 0& -{\displaystyle \frac{1}{2}}{\rho }_0\\ 0& -(1+(\beta -2\eta \left)\alpha \right)& -{\displaystyle \frac{1}{2}}\alpha \\ 0& {\displaystyle \frac{1}{2}}\alpha & 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(63)

Therefore, the scalar curvature can be derived as $s^1=-(2+(\beta -2\eta )(\beta +\alpha ))$. Additionally, we can write $D^1$ as follows:

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-(1+(\beta -2\eta )\beta -(2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0+c){\tilde{e}}_1-{\displaystyle \frac{1}{2}}{\rho }_0{\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_2=-(1+(\beta -2\eta )\beta -(2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0+c){\tilde{e}}_2-{\displaystyle \frac{1}{2}}\alpha {\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_3={\displaystyle \frac{1}{2}}\alpha {\tilde{e}}_2+((2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(64)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if it satisfies the following system of equations:

$$\left\{\begin{array}{c}\alpha {\rho }_0=0,\hfill \\ 1+(\beta -2\eta )\beta -(2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0+c=\alpha (\eta -\beta ),\hfill \\ (2\eta -\beta )(2+(\beta -2\eta )(\alpha +\beta )-(2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0+c)=-\alpha ,\hfill \\ \beta ((\beta -2\eta )(\beta -\alpha )-(2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0+c)=\alpha ,\hfill \\ \alpha ((\beta -2\eta )(\alpha -\beta )-(2+(\beta -2\eta )(\beta +\alpha )){\lambda }_0+c)=0.\hfill \end{array}\right.$$

(65)

Assume first that $\alpha =0$. Then, we have the following equations:

$$\left\{\begin{array}{c}1+\beta (\beta -2\eta )-(2\beta +\beta (\beta -2\eta )){\lambda }_0+c=0,\hfill \\ {\beta }^2(\beta -2\eta )-{\beta }^2(2+(\beta -2\eta )){\lambda }_0+\beta c=0.\hfill \end{array}\right.$$

(66)

This is a contradiction.

If ${\rho }_0=0$ and $\alpha \ne 0$, then we have the following equations:

$$\left\{\begin{array}{c}1+(\beta -2\eta )(2\beta -\alpha )=\alpha (\eta -\beta ),\hfill \\ (2\eta -\beta )(2+2\beta (\beta -2\eta \left)\right(\alpha +\beta \left)\right)=-\alpha ,\hfill \\ 2\beta (\beta -2\eta )(\beta -\alpha )=\alpha .\hfill \end{array}\right.$$

(67)

The above system of equations has no solution, thus proving the theorem. □

4.5. Algebraic Schouten Soliton of $G_5$

Similarly to Section 3.5, we have the expression of the Lie algebra of $G_5$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_5,g,J)$.

Theorem 12. 

The Lie group $(G_5,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed canonical connection ${\overline{\nabla }}^1$ if $c=0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive that ${\widetilde{Ric}}^1=\mathbf{0}$. Therefore, the scalar curvature can be obtained as $s^0=0$. We can express $D^0$ as

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-c{\tilde{e}}_1,\hfill \\ D^1{\tilde{e}}_2=-c{\tilde{e}}_2,\hfill \\ D^1{\tilde{e}}_3=-c{\tilde{e}}_3.\hfill \end{array}\right.$$

(68)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if $c=0$. □

4.6. Algebraic Schouten Soliton of $G_6$

Similarly to Section 3.6, we have the expression of the Lie algebra of $G_6$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_6,g,J)$.

Theorem 13. 

The Lie group $(G_6,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ if one of the following holds:

1. 

$\alpha =\beta =\gamma =0$, $\delta \ne 0$, $c=-s{\lambda }_0$;

2. 

$\alpha =\beta =0$, $\gamma \ne 0$, $\delta \ne 0$, $c=-s{\lambda }_0$;

3. 

$\alpha \ne 0$, $\beta =\gamma =\delta =0$, $c=-{\alpha }^2-s{\lambda }_0$;

4. 

$\alpha \ne 0$, $\beta =\gamma =0$, $\delta \ne 0$, $c=-{\alpha }^2-s{\lambda }_0$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^1$ as follows:

$${\widetilde{Ric}}^1\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-({\alpha }^2+\beta \gamma )& 0& -{\displaystyle \frac{1}{2}}\delta {\rho }_0\\ 0& -{\alpha }^2& 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(69)

Therefore, the scalar curvature can be derived as $s^1=-(2{\alpha }^2+\beta \gamma )$. Additionally, we can write $D^1$ as follows:

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-({\alpha }^2+\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c){\tilde{e}}_1-{\displaystyle \frac{\delta {\rho }_0}{2}}{\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_2=-({\alpha }^2-(2{\alpha }^2+\beta \gamma ){\lambda }_0+c){\tilde{e}}_2,\hfill \\ D^1{\tilde{e}}_3=((2{\alpha }^2+\beta \gamma ){\lambda }_0-c){\tilde{e}}_3.\hfill \end{array}\right.$$

(70)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\alpha ({\alpha }^2+\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)=0,\hfill \\ \beta (2{\alpha }^2+\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)=0,\hfill \\ \gamma (\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)=0,\hfill \\ \delta ({\alpha }^2+\beta \gamma -(2{\alpha }^2+\beta \gamma ){\lambda }_0+c)=0.\hfill \end{array}\right.$$

(71)

The first equation above implies that either $\alpha =0$ or $\alpha \ne 0$. Assume that $\alpha =0$. Using the conditions for parameters in (38), we deduce that $\beta =0$ and $\delta \ne 0$. Then, the system (71) reduces to

$$\left\{\begin{array}{c}\gamma c=0,\hfill \\ \delta c=0.\hfill \end{array}\right.$$

(72)

If $\gamma =0$, then $c=0$. On the other hand, if $\gamma \ne 0$, then $c=0$. Thus, for cases (1) and (2), system (71) holds.

Now, let $\alpha \ne 0$. Consider the subcase where $\delta =0$; we derived that $\gamma =0$. Then, system (71) reduces to

$$\left\{\begin{array}{c}{\alpha }^3-2{\alpha }^3{\lambda }_0+\alpha c=0,\hfill \\ 2{\alpha }^2\beta -2{\alpha }^2\beta {\lambda }_0+\beta c=0,\hfill \end{array}\right.$$

(73)

which leads to $\beta =0$ and $c=-{\alpha }^2+2{\alpha }^2{\lambda }_0$. If $\delta \ne 0$, then the second and third equations of system (71) give ${\alpha }^2\beta =\beta {\gamma }^2=0$, which provides $\beta =\gamma =0$. Therefore, case (4) holds. □

4.7. Algebraic Schouten Soliton of $G_7$

Similarly to Section 3.7, we have the expression of the Lie algebra of $G_7$. Then, we can formulate the following theorem concerning the algebraic condition of an algebraic Schouten soliton associated with the perturbed Kobayashi–Nomizu connections on $(G_7,g,J)$.

Theorem 14. 

The Lie group $(G_7,g,J)$ is considered as an algebraic Schouten soliton with respect to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ if one of the following holds:

1. 

$\alpha \ne 0$, $\beta =0$, $\gamma =0$, $\alpha =\delta$ and ${\rho }_0=-4\alpha$, $c=-{\displaystyle \frac{3}{2}}{\delta }^2+2{\alpha }^2{\lambda }_0$,

2. 

$\alpha =\beta =\gamma =0$, $\delta \ne 0$, and ${\rho }_0=-4\delta$, $c=-{\delta }^2$.

Proof. 

From [15], we obtain the expression for ${\tilde{\rho }}^1$. By utilizing Equation (9), we further derive the expression for ${\widetilde{Ric}}^1$ as follows:

$${\widetilde{Ric}}^1\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right)=\left(\begin{array}{ccc}-{\alpha }^2& {\displaystyle \frac{1}{2}}\beta \delta -{\displaystyle \frac{1}{2}}\alpha \beta & -(\alpha \beta +\beta \delta +{\displaystyle \frac{\beta {\rho }_0}{2}})\\ {\displaystyle \frac{1}{2}}\beta \delta -{\displaystyle \frac{1}{2}}\alpha \beta & -({\alpha }^2+{\beta }^2+\beta \gamma )& -{\displaystyle \frac{1}{2}}(\beta \gamma +\alpha \delta +2{\delta }^2+\delta {\rho }_0)\\ \alpha \beta +\beta \delta & {\displaystyle \frac{1}{2}}(\beta \gamma +\alpha \delta +2{\delta }^2)& 0\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_1\\ {\tilde{e}}_2\\ {\tilde{e}}_3\end{array}\right).$$

(74)

Therefore, the scalar curvature can be derived as $s^1=-(2{\alpha }^2+{\beta }^2+\beta \gamma )$. Additionally, we can write $D^1$ as follows:

$$\left\{\begin{array}{c}{D}^1{\tilde{e}}_1=-({\alpha }^2+s^1{\lambda }_0+c){\tilde{e}}_1+{\displaystyle \frac{1}{2}}\beta (\delta -\alpha ){\tilde{e}}_2-\beta (\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}}){\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_2={\displaystyle \frac{1}{2}}\beta (\delta -\alpha ){\tilde{e}}_1-({\alpha }^2+{\beta }^2+\beta \gamma +s^1{\lambda }_0+c){\tilde{e}}_2-{\displaystyle \frac{1}{2}}(\beta \gamma +\alpha \delta +2{\delta }^2+\delta {\rho }_0){\tilde{e}}_3,\hfill \\ D^1{\tilde{e}}_3=\beta (\alpha +\delta ){\tilde{e}}_1+{\displaystyle \frac{1}{2}}(\beta \gamma +\alpha \delta +2{\delta }^2){\tilde{e}}_2-(s^1{\lambda }_0+c){\tilde{e}}_3.\hfill \end{array}\right.$$

(75)

Consequently, according to (15), the existence of an algebraic Schouten soliton in relation to the perturbed Kobayashi–Nomizu connection ${\overline{\nabla }}^1$ can be established if it satisfies the following:

$$\left\{\begin{array}{c}\alpha ({\alpha }^2+{\beta }^2+{\displaystyle \frac{1}{2}}\beta \gamma -{\displaystyle \frac{1}{2}}\alpha \delta -{\delta }^2-{\displaystyle \frac{1}{2}}\delta {\rho }_0+s^1{\lambda }_0+c)+\beta (\gamma (\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}})+{\displaystyle \frac{1}{2}}\beta (\delta -\alpha ))=0,\hfill \\ \beta ({\alpha }^2+s^1{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta \delta {\rho }_0+{\displaystyle \frac{1}{2}}\alpha \beta (\delta -\alpha )+\beta \delta (\alpha +\delta )=0,\hfill \\ \beta (2{\alpha }^2+{\beta }^2+s^1{\lambda }_0+c)-\beta (\alpha \delta +2{\delta }^2+\delta {\rho }_0)-\delta (\alpha -\beta )(\alpha +\delta +{\displaystyle \frac{\rho }{2}})=0,\hfill \\ \alpha ({\displaystyle \frac{1}{2}}\beta \gamma +{\displaystyle \frac{1}{2}}\alpha \delta +{\delta }^2+s^1{\lambda }_0+c)+\beta ({\displaystyle \frac{1}{2}}(\beta -\gamma )(\delta -\alpha )+\beta (\alpha +\delta ))=0,\hfill \\ \beta (\alpha \delta +2{\delta }^2-{\beta }^2+s^1{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta {(\alpha -\delta )}^2=0,\hfill \\ \beta ({\alpha }^2+s^1{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta \delta {\rho }_0-\alpha \beta (\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}})-{\displaystyle \frac{1}{2}}\beta \delta (\delta -\alpha )=0,\hfill \\ \gamma ({\beta }^2+\beta \gamma +s^1{\lambda }_0+c)+{\displaystyle \frac{1}{2}}\beta (\delta -\alpha )(3\delta +\alpha )=0,\hfill \\ \delta ({\displaystyle \frac{1}{2}}\beta \gamma +{\displaystyle \frac{1}{2}}\alpha \delta +{\delta }^2+s^1{\lambda }_0+c)+{\displaystyle \frac{1}{2}}\beta (\gamma -\beta )(\delta -\alpha )-{\beta }^2(\alpha +\delta )=0,\hfill \\ \delta ({\alpha }^2+{\beta }^2-{\displaystyle \frac{1}{2}}\alpha \delta -{\delta }^2-{\displaystyle \frac{1}{2}}\delta {\rho }_0+{\displaystyle \frac{1}{2}}\beta \gamma +s^1{\lambda }_0+c)-\beta \gamma (\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}})={\displaystyle \frac{1}{2}}{\beta }^2(3\delta +\alpha ).\hfill \end{array}\right.$$

(76)

Since $\alpha +\delta \ne 0$ and $\alpha \gamma =0$, we first assume that $\alpha =0$ and $\gamma \ne 0$. The second and fourth equations above yield $\beta (3\delta +{\rho }_0)=0$. If $\beta =0$, then the eighth equation above results in ${\delta }^3=0$, which is a contradiction. On the other hand, if $3\delta +{\rho }_0=0$ and $\beta \ne 0$, substituting them into the first equation above gives us $\beta =4\gamma$, but the fourth equation then yields $3\beta =2\gamma$, which is also a contradiction. Then, we consider the second case where $\alpha \ne 0$ and $\gamma =0$. In this case, (76) becomes

$$\left\{\begin{array}{c}\alpha ({\alpha }^2+{\beta }^2-{\displaystyle \frac{1}{2}}\alpha \delta -{\delta }^2-{\displaystyle \frac{1}{2}}\delta {\rho }_0-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)+{\displaystyle \frac{1}{2}}{\beta }^2(\delta -\alpha )=0,\hfill \\ \beta ({\alpha }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta \delta {\rho }_0+{\displaystyle \frac{1}{2}}\alpha \beta (\delta -\alpha )+\beta \delta (\alpha +\delta )=0,\hfill \\ \beta (2{\alpha }^2+{\beta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-\beta (\alpha \delta +2{\delta }^2+\delta {\rho }_0)+\delta (\beta -\alpha )(\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}})=0,\hfill \\ \alpha ({\displaystyle \frac{1}{2}}\alpha \delta +{\delta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)+{\displaystyle \frac{1}{2}}{\beta }^2(3\delta +\alpha )=0,\hfill \\ \beta (\alpha \delta +2{\delta }^2-{\beta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta {(\alpha -\delta )}^2=0,\hfill \\ \beta ({\alpha }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta \delta {\rho }_0-\alpha \beta (\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}})-{\displaystyle \frac{1}{2}}\beta \delta (\delta -\alpha )=0,\hfill \\ {\displaystyle \frac{1}{2}}\beta (\delta -\alpha )(3\delta +\alpha )=0,\hfill \\ \delta ({\displaystyle \frac{1}{2}}\alpha \delta +{\delta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{1}{2}}{\beta }^2(3\delta +\alpha )=0,\hfill \\ \delta ({\alpha }^2+{\beta }^2-{\displaystyle \frac{1}{2}}\alpha \delta -{\delta }^2-{\displaystyle \frac{1}{2}}\delta {\rho }_0-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)={\displaystyle \frac{1}{2}}{\beta }^2(3\delta +\alpha ).\hfill \end{array}\right.$$

(77)

The seventh equation of system (77) gives us three subcases of $\beta =0$, $\alpha =\delta$, or $\alpha =-3\delta$. Assume first that $\beta =0$. In this case, system (77) becomes

$$\left\{\begin{array}{c}\alpha ({\alpha }^2-2{\alpha }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\alpha (\alpha \delta +2{\delta }^2+\delta {\rho }_0)=0,\hfill \\ \alpha \delta (\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}})=0,\hfill \\ \alpha (-2{\alpha }^2{\lambda }_0+c)+{\displaystyle \frac{1}{2}}\alpha (\alpha \delta +2{\delta }^2)=0,\hfill \\ \delta (-2{\alpha }^2{\lambda }_0+c)+{\displaystyle \frac{1}{2}}\delta (\alpha \delta +2{\delta }^2)=0,\hfill \\ \delta ({\alpha }^2-2{\alpha }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\delta (\alpha \delta +2{\delta }^2+\delta {\rho }_0)=0.\hfill \end{array}\right.$$

(78)

Through some direct calculations, we derive $\alpha =\delta$ and ${\rho }_0=-4\alpha$. Therefore, case (1) of the theorem holds. Now, assuming $\alpha =\delta$, substitute this into the second and sixth equations of (77), and we find that the equation yields $\alpha +\delta +{\displaystyle \frac{{\rho }_0}{2}}=0$. In this case, the first and the third equations in (77) can be simplified as

$$\left\{\begin{array}{c}{\alpha }^2+{\beta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c+{\displaystyle \frac{1}{2}}\alpha \delta =0,\hfill \\ 2{\alpha }^2+{\beta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c+\alpha \delta =0.\hfill \end{array}\right.$$

(79)

The above equation yields $\alpha =-{\displaystyle \frac{\delta }{2}}$, which is a contradiction. Finally, we consider the case where $\alpha =-3\delta$. Substituting this into the second and sixth equations of (77), we find that $\rho =0$. Now we examine the last two equations in (77), which give us

$$\left\{\begin{array}{c}\delta (-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)-{\displaystyle \frac{1}{2}}{\delta }^3=0,\hfill \\ \delta ({\alpha }^2+{\beta }^2-(2{\alpha }^2+{\beta }^2){\lambda }_0+c)+{\displaystyle \frac{1}{2}}{\delta }^3=0.\hfill \end{array}\right.$$

(80)

This is a contradiction.

Now, we assume $\alpha =\gamma =0$. Since $\alpha +\delta \ne 0$, we have $\delta \ne 0$. In this case, we find that system (77) becomes

$$\left\{\begin{array}{c}{\beta }^2\delta =0,\hfill \\ \beta \delta (\delta +{\displaystyle \frac{{\rho }_0}{2}})=0,\hfill \\ \beta (-{\beta }^2-{\beta }^2{\lambda }_0+c)+{\displaystyle \frac{3}{2}}\beta {\delta }^2=0,\hfill \\ \beta (-{\beta }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\beta \delta {\rho }_0=0,\hfill \\ \delta (-{\beta }^2\lambda +c)-{\displaystyle \frac{3}{2}}{\beta }^2\delta +{\delta }^3=0,\hfill \\ \delta ({\beta }^2-{\beta }^2{\lambda }_0+c)-{\displaystyle \frac{1}{2}}\delta (2{\delta }^2+\delta {\rho }_0)=0.\hfill \end{array}\right.$$

(81)

By solving the above equations, we find that $\beta =0$ and ${\rho }_0=-4\delta$. Therefore, case (2) holds. □

5. Conclusions

In this study, we investigate the necessary conditions for $(G_i,g,J)$, where i ranges from 1 to 7, to constitute algebraic Schouten solitons associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. Our results indicate that $G_4$ does not possess algebraic Schouten solitons related to the perturbed Kobayashi–Nomizu connection, and $G_1,G_2,G_3,G_6$, and $G_7$ exhibit algebraic Schouten solitons in relation to the above connections, while the result for $G_5$ is found to be trivial. In the future, we plan to focus on metric Lie groups; in particular, we intend to address the classification problem of algebraic Ricci solitons on metric Lie groups under certain tensor conditions [22,23].