Nijenhuis Operators on 2D Pre-Lie Algebras and 3D Associative Algebras

Abstract

In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions to the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.

1. Introduction

This paper aims to describe Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras, and to explore their applications in the solutions of the classical Yang-Baxter equation.

1.1. Nijenhuis Algebras and Nijenhuis Torsion

A Nijenhuis algebra is a triple $(A,·,N)$ including an algebra $(A,·)$ and a Nijenhuis operator N on A, i.e., a linear map $N:A\to A$ satisfies

$$N\left(x\right)·N\left(y\right)=N\left(N\right(x)·y+x·N(y)-N(x·y\left)\right),\forall x,y\in A.$$

(1)

Nijenhuis torsion was first introduced by Nijenhuis in 1951 [1]. Let M be a smooth n-dimensional manifold, N a smooth operator field on it, and v, u smooth vector fields. Then the Nijenhuis torsion ${\left({\mathcal{T}}_N\right)}_{jk}^i$ of N is a tensor of type (1, 2) given by the formula

$${\mathcal{T}}_N(v,u)=[Nv,Nu]+N^2[v,u]-N[Nv,u]-N[v,Nu],$$

or in local coordinates $\mathbf{x}=(x^1,\dots ,x^n)$,

$${\left({\mathcal{T}}_N\right)}_{jk}^i=N_j^l{\displaystyle \frac{\partial N_k^i}{\partial x^l}}-N_k^l{\displaystyle \frac{\partial N_j^i}{\partial x^l}}-N_l^i{\displaystyle \frac{\partial N_k^l}{\partial x^j}}+N_l^i{\displaystyle \frac{\partial N_j^l}{\partial x^k}}.$$

An operator N is called a Nijenhuis operator if ${\mathcal{T}}_N=0$. The vanishing of the Nijenhuis torsion is a necessary (but not sufficient) condition for the integrability of an operator field [2]. Moreover, Nijenhuis operators play a central role in various fields such as complex differential geometry and complex Lie algebra theory [3], deformations of associative (resp. Lie) algebras [4,5], and pairs of Dirac structures [6,7]. Many interesting generalizations have been made, including applications to pre-Lie algebras [8], n-Lie algebras [9], Hom-Lie algebras [10], and Courant algebroids with Nijenhuis structures [11]. The theory of Nijenhuis operators has recently been enriched by studies extending them to broader frameworks, including Banach homogeneous spaces [12], Leibniz algebras [13], associative D-bialgebras [14], and 3-Hom-Lie algebras [15], among others.

1.2. Nijenhuis Operators on Pre-Lie Algebras

Pre-Lie algebras, also known as left-symmetric algebras, were first introduced by Cayley in the context of rooted tree algebras [16]. They have been used in the study of homogeneous convex cones [17], affine manifolds, and affine structures on Lie groups [18,19]. In [20], it was shown that an operator depending linearly on local coordinates is a Nijenhuis operator if and only if its coefficients are structure constants of a pre-Lie algebra. In [8], Wang, Sheng, Bai and Liu defined Nijenhuis operators on pre-Lie algebras, which generate a trivial deformation of pre-Lie algebras. The local behavior of these operators at singular points is important in the study of complex integrable systems [21]. So far, only a few Nijenhuis operators on (simple) pre-Lie algebras are known explicitly, and a full classification of simple pre-Lie algebras has not yet been completed [22,23,24,25]. Constructing examples of Nijenhuis operators on low-dimensional pre-Lie algebras can help better understand Nijenhuis operators and related topics.

A Rota-Baxter operator of weight zero on the sub-adjacent Lie algebra can be obtained from a Nijenhuis operator on a pre-Lie algebra. Let A be a pre-Lie algebra (product of x and y is denoted by $xy$) over a field $\mathbb{C}$. A linear map $R:A\to A$ is called a Rota-Baxter operator of weight $\lambda \in \mathbb{C}$ on A if

$$R\left(x\right)R\left(y\right)=R\left(R\right(x)y+xR(y)+\lambda xy),\forall x,y\in A.$$

(2)

If a Nijenhuis operator N on a pre-Lie algebra satisfies $N^2=0$, then N is a Rota-Baxter operator of weight zero on A (see Proposition 6). Let $\mathfrak{g}$ be a Lie algebra. A linear map $R:\mathfrak{g}\to \mathfrak{g}$ is called an operator form of the classical Yang-Baxter equation (CYBE) if,

$$\left[R\right(x),R(y\left)\right]=R\left(\right[R\left(x\right),y]+[x,R\left(y\right)\left]\right),\forall x,y\in \mathfrak{g},$$

(3)

which is also called a Rota-Baxter operator on $\mathfrak{g}$. The CYBE originated from the study of inverse scattering theory, which is recognized as the “semi-classical limit” of the quantum Yang-Baxter equation [26,27,28]. If R is a Rota-Baxter operator of weight zero on a pre-Lie algebra A, then R is a Rota-Baxter operator of weight zero on the sub-adjacent Lie algebra $\mathfrak{g}\left(A\right)$ (see Proposition 7).

Based on the above relationship between Nijenhuis operators and Rota-Baxter operators, solutions of the CYBE on semidirect product sub-adjacent Lie algebras can be derived. Recall that $r={\displaystyle \sum _i}{a}_i\otimes b_i\in \mathfrak{g}\otimes \mathfrak{g}$ is a solution of the CYBE in $\mathfrak{g}$ if

$$[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0\mathrm{in}U\left(\mathfrak{g}\right),$$

where $U\left(\mathfrak{g}\right)$ is the universal enveloping algebra of $\mathfrak{g}$, and

$$r_{12}=\sum _i{a}_i\otimes b_i\otimes 1,r_{13}=\sum _i{a}_i\otimes 1\otimes b_i,r_{23}=\sum _{i}1\otimes a_i\otimes b_i.$$

Semonov-Tian-Shansky [29] showed that if a nondegenerate symmetric invariant bilinear form exists on a Lie algebra $\mathfrak{g}$, then any skew-symmetric solution r of the CYBE satisfies the Rota-Baxter equation. Bai [30] proved that a linear map $R:\mathfrak{g}\to \mathfrak{g}$ is a Rota-Baxter operator if and only if $r=R-R^{21}$ is a skew-symmetric solution of the CYBE in $\mathfrak{g}{⋉}_{{\mathrm{ad}}^{*}}{\mathfrak{g}}^{*}$. In [31], all Rota-Baxter operators (of weight zero) on $\mathfrak{sl}(2,\mathbb{C})$ were determined under the Cartan-Weyl basis, and corresponding solutions of the classical Yang-Baxter equation were found in the 6-dimensional Lie algebra $\mathfrak{sl}(2,\mathbb{C}){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{sl}{(2,\mathbb{C})}^{*}$. As an application, this paper also presents the solution of CYBE derived from Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras.

Classifying algebras of small dimensions is a natural first step toward understanding higher-dimensional cases. In fact, 2-dimensional and 3-dimensional pre-Lie algebras have been completely classified, and some special cases in 4 dimensions have been classified (for example, 4-dimensional nilpotent pre-Lie algebras [32]). Therefore, “2-dimensional” and “3-dimensional” are the natural minimal dimensions for a nontrivial classification of this kind. There are 5 classes of 2-dimensional commutative pre-Lie algebras and 6 classes of 2-dimensional non-commutative pre-Lie algebras [23]. The classification of 3-dimensional pre-Lie algebras falls into the following three categories: 3-dimensional simple pre-Lie algebras, pre-Lie algebras on the 3-dimensional Heisenberg Lie algebra, and 3-dimensional associative algebras [22]. However, their Nijenhuis operators remain unexplored.

The novelty of this work lies in describing all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. In particular, we emphasize that this study provides a systematic derivation of induced solutions to the classical Yang-Baxter equation. The Nijenhuis operators presented are parametrically complete and list all possible Nijenhuis operators satisfying the defining equation. However, we do not perform a full orbit analysis under the automorphism group $\mathrm{Aut}\left(A\right)$. This paper aims to explicitly list Nijenhuis operators, enabling future conjugacy classification.

1.3. Outline of the Paper

This paper is organized as follows. Section 2 reviews basic concepts and known results on pre-Lie algebras and Nijenhuis operators. Section 3 presents all Nijenhuis operators on 2-dimensional complex pre-Lie algebras (Theorems 1 and 2). Section 4 lists all Nijenhuis operators on 3-dimensional complex associative algebras (Theorems 3 and 4). In Section 5, we first outline the general construction pipeline that transforms Nijenhuis operators on pre-Lie algebras into Rota-Baxter operators on their sub-adjacent Lie algebras, and subsequently into solutions of the classical Yang-Baxter equation. We then provide a fully worked, step-by-step example for the 2-dimensional pre-Lie algebra $B_1$ to illustrate this procedure concretely. Section 5.3 lists the sub-adjacent Lie algebras that arise from all classified 2-dimensional pre-Lie and 3-dimensional associative algebras. Section 5.4 presents the Rota-Baxter operators induced by the Nijenhuis operators on these Lie algebras. Finally, Section 5.5 gives the explicit solutions of the CYBE on the corresponding Lie algebras $\mathfrak{g}\left(A\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(A\right)}^{*}$.

All pre-Lie and associative algebras discussed in this paper are finite-dimensional over $\mathbb{C}$, and juxtaposition $kx$ represents the product of the scalar k and the element x in the vector space, while the dotted notation $x·y$ indicates the product in a pre-Lie algebra. Denote the sub-adjacent Lie algebra $\mathfrak{g}\left(B_1\right)$, $\mathfrak{g}\left(B_2\right)$, $\mathfrak{g}\left(B_3\right)$, $\mathfrak{g}\left(B_4\right)$, and $\mathfrak{g}\left(B_5\right)$ by ${\mathfrak{g}}_1$. Denote $\mathfrak{g}\left(D_1\right)$, $\mathfrak{g}\left(D_2\right)$ and $\mathfrak{g}\left(D_3\right)$ by ${\mathfrak{g}}_2$. Denote $\mathfrak{g}\left(D_4\right)$, $\mathfrak{g}\left(D_6\right)$, $\mathfrak{g}\left(D_8\right)$, and $\mathfrak{g}\left(D_9\right)$ by ${\mathfrak{g}}_3$. Denote $\mathfrak{g}\left(D_5\right)$, $\mathfrak{g}\left(D_7\right)$ by ${\mathfrak{g}}_4$.

2. Preliminaries

In this section, we provide some preliminaries and classification results of 2-dimensional pre-Lie algebras and 3-dimensional associative algebras in [22,23,33].

Definition 1 ([16]). Let A be a vector space over $\mathbb{C}$ with a bilinear product $·:A\times A\to A$. Then $(A,·)$ is called a pre-Lie algebra if for any $x,y,z\in A$,

$$(x·y)·z-x·(y·z)=(y·x)·z-y·(x·z).$$

(4)

For a pre-Lie algebra $(A,·)$, the commutator

$$[x,y]=x·y-y·x,$$

(5)

defines a Lie algebra $\mathfrak{g}\left(A\right)$, which is called the sub-adjacent Lie algebra of $(A,·)$.

Proposition 1 ([23,33]). Let $\left(A,·\right)$ be a 2-dimensional commutative pre-Lie algebra and $\left\{e_1,e_2\right\}$ be a basis of A. Then $(A,·)$ isomorphic to one of the following algebras:

$\left(A_1\right)$$e_1·e_1=e_1,e_2·e_2=e_2$;

$\left(A_2\right)$$e_1·e_1=e_1,e_1.e_2=e_2·e_1=e_2$;

$\left(A_3\right)$$e_1·e_1=e_1$;

$\left(A_4\right)$$e_i·e_j=0,i,j=1,2$;

$\left(A_5\right)$$e_1·e_1=e_2$.

Proposition 2 ([23,33]). Let $\left(B,·\right)$ be a 2-dimensional non-commutative pre-Lie algebra and $\left\{e_1,e_2\right\}$ be a basis of B. Then $(B,·)$ isomorphic to one of the following algebras:

$\left(B_1\right)$$e_2·e_1=-e_1,e_2·e_2=e_1-e_2$;

$\left(B_2\right)$$e_2·e_1=-e_1,e_2·e_2=-e_2$;

$\left(B_3\right)$$e_2·e_1=-e_1,e_2·e_2=k{e}_2,k\ne -1,k\in \mathbb{C}$;

$\left(B_4\right)$$e_1·e_2=e_1,e_2·e_2=e_2$;

$\left(B_5\right)$$e_1·e_2=l{e}_1,e_2·e_1=(l-1)e_1,e_2·e_2=e_1+l{e}_2,l\ne 0,l\in \mathbb{C}$;

$\left(B_6\right)$$e_1·e_1=2e_1,e_1·e_2=e_2,e_2·e_2=e_1$.

Proposition 3 ([22]). Let $\left(C,·\right)$ be a 3-dimensional commutative associative algebra and $\left\{e_1,e_2,e_3\right\}$ be a basis of C. Then $(C,·)$ isomorphic to one of the following algebras:

$\left(C_1\right)$$e_3·e_3=e_1$;

$\left(C_2\right)$$e_i·e_j=0,i,j=1,2,3$;

$\left(C_3\right)$$e_2·e_2=e_1,e_3·e_3=e_1$;

$\left(C_4\right)$$e_2·e_3=e_3·e_2=e_1,e_3·e_3=e_2$;

$\left(C_5\right)$$e_1·e_1=e_1,e_2·e_2=e_2,e_3·e_3=e_3$;

$\left(C_6\right)$$e_2·e_2=e_2,e_3·e_3=e_3$;

$\left(C_7\right)$$e_1·e_3=e_3·e_1=e_1,e_2·e_2=e_2,e_3·e_3=e_3$;

$\left(C_8\right)$$e_3·e_3=e_3$;

$\left(C_9\right)$$e_1·e_3=e_3·e_1=e_1,e_3·e_3=e_3$;

$\left(C_{10}\right)$$e_1·e_3=e_3·e_1=e_1,e_2·e_3=e_3·e_2=e_2,e_3·e_3=e_3$;

$\left(C_{11}\right)$$e_1·e_1=e_2,e_3·e_3=e_3$;

$\left(C_{12}\right)$$e_1·e_1=e_2,e_1·e_3=e_3·e_1=e_1,e_2·e_3=e_3·e_2=e_2,e_3·e_3=e_3$.

Proposition 4 ([22]). Let $\left(D,·\right)$ be a 3-dimensional non-commutative associative algebra and $\left\{e_1,e_2,e_3\right\}$ be a basis of D. Then $(D,·)$ isomorphic to one of the following algebras:

$\left(D_1\right)$$e_1·e_2={\displaystyle \frac{1}{2}}{e}_3,e_2·e_1=-{\displaystyle \frac{1}{2}}{e}_3$;

$\left(D_2\right)$$e_2·e_1=-e_3$;

$\left(D_3\right)$$e_1·e_1=e_3,e_1·e_2=e_3,e_2·e_2=\lambda e_3,\lambda \ne 0,\lambda \in \mathbb{C}$;

$\left(D_4\right)$$e_3·e_2=e_2,e_3·e_3=e_3$;

$\left(D_5\right)$$e_2·e_3=e_2,e_3·e_3=e_3$;

$\left(D_6\right)$$e_1·e_1=e_1,e_3·e_2=e_2,e_3·e_3=e_3$;

$\left(D_7\right)$$e_1·e_1=e_1,e_2·e_3=e_2,e_3·e_3=e_3$;

$\left(D_8\right)$$e_1·e_3=e_1,e_3·e_1=e_1,e_3·e_2=e_2,e_3·e_3=e_3$;

$\left(D_9\right)$$e_1·e_1=e_1,e_1·e_2=e_2·e_1=e_2,e_1·e_3=e_3·e_1=e_3,e_3·e_2=e_2,e_3·e_3=e_3$;

$\left(D_{10}\right)$$e_3·e_1=e_1,e_3·e_2=e_2,e_3·e_3=e_3$;

$\left(D_{11}\right)$$e_1·e_3=e_1,e_2·e_3=e_2,e_3·e_3=e_3$;

$\left(D_{12}\right)$$e_3·e_1=e_1,e_2·e_3=e_2,e_3·e_3=e_3$.

Definition 2 ([8]). Let $(A,·)$ be a pre-Lie algebra. A linear map $N:A\to A$ is called a Nijenhuis operator on $(A,·)$ if

$$N\left(x\right)·N\left(y\right)=N\left(N\right(x)·y+x·N(y)-N(x·y\left)\right),\forall x,y\in A.$$

(6)

Proposition 5.

Let $(A,·)$ be an n-dimensional pre-Lie algebra and $\left\{e_1,\cdots ,e_n\right\}$ be a basis of A. For all positive integers $1\le i,j,t\le n$ and structural constants $C_{ij}^t\in \mathbb{C}$, set

$$e_i·e_j=\sum _{t=1}^n{C}_{ij}^t{e}_t.$$

(7)

If $N:A\to A$ is a linear map given by

$$N\left(e_i\right)=\sum _{j=1}^n{n}_{ij}{e}_j,n_{ij}\in \mathbb{C},$$

(8)

then N is a Nijenhuis operator on $(A,·)$ if and only if $n_{ij}$ satisfies the following equations:

$$\sum _{k,l,m,t=1}^n\left(C_{kl}^m{n}_{ik}{n}_{jl}+C_{ij}^t{n}_{tl}{n}_{lm}-C_{kj}^l{n}_{ik}{n}_{lm}-C_{il}^k{n}_{jl}{n}_{km}\right)=0.$$

(9)

Proof. 

For all $e_i,e_j\in \left\{e_1,\cdots ,e_n\right\}$, $1\le i,j\le n$, set

$$N\left(e_i\right)=\sum _{k=1}^n{n}_{ik}{e}_k,N\left(e_j\right)=\sum _{l=1}^n{n}_{jl}{e}_l,e_i·e_j=\sum _{t=1}^n{C}_{ij}^t{e}_t,n_{ik},n_{jl},C_{ij}^t\in \mathbb{C}.$$

Then we have

$$N\left(e_i\right)·N\left(e_j\right)=\sum _{k,l=1}^n{n}_{ik}{n}_{jl}{e}_k·e_l=\sum _{k,l,m=1}^n{C}_{kl}^m{n}_{ik}{n}_{jl}{e}_m,$$

and

$$\begin{array}{ccc}& & N(N\left(e_i\right)·e_j+e_i·N\left(e_j\right)-N(e_i·e_j))\hfill \\ & =& N(\sum _{k=1}^n{n}_{ik}{e}_k·e_j+e_i·\sum _{l=1}^n{n}_{jl}{e}_l-N\left(\sum _{t=1}^n{C}_{ij}^t{e}_t\right))\hfill \\ & =& N(\sum _{k,l=1}^n{C}_{kj}^l{n}_{ik}{e}_l+\sum _{l,k=1}^n{C}_{il}^k{n}_{jl}{e}_k-\sum _{t,l=1}^n{C}_{ij}^t{n}_{tl}{e}_l)\hfill \\ & =& \sum _{k,l,m,t=1}^n(C_{kj}^l{n}_{ik}{n}_{lm}{e}_m+C_{il}^k{n}_{jl}{n}_{km}{e}_m-C_{ij}^t{n}_{tl}{n}_{lm}{e}_m).\hfill \end{array}$$

If N is a Nijenhuis operator, then

$$\begin{array}{ccc}& & N\left(e_i\right)·N\left(e_j\right)-N(N\left(e_i\right)·e_j+e_i·N\left(e_j\right)-N(e_i·e_j))\hfill \\ & =& \sum _{k,l,m=1}^n{C}_{kl}^m{n}_{ik}{n}_{jl}{e}_m-\sum _{k,l,m,t=1}^n(C_{kj}^l{n}_{ik}{n}_{lm}{e}_m+C_{il}^k{n}_{jl}{n}_{km}{e}_m-C_{ij}^t{n}_{tl}{n}_{lm}{e}_m)\hfill \\ & =& \sum _{k,l,m,t=1}^n(C_{kl}^m{n}_{ik}{n}_{jl}{e}_m+C_{ij}^t{n}_{tl}{n}_{lm}{e}_m-C_{kj}^l{n}_{ik}{n}_{lm}{e}_m-C_{il}^k{n}_{jl}{n}_{km}{e}_m)\hfill \\ & =& 0.\hfill \end{array}$$

Therefore, N is a Nijenhuis operator on $(A,·)$ if and only if (9) holds. □

3. Nijenhuis Operators on 2-Dimensional Pre-Lie Algebras

Based on the classification results of 2-dimensional commutative and non-commutative complex pre-Lie algebras in Propositions 1 and 2, this section determines all Nijenhuis operators on $(A_i,·)$ and $(B_i,·)$, respectively.

3.1. Nijenhuis Operators on 2-Dimensional Commutative Pre-Lie Algebras

In this subsection, we determine all Nijenhuis operators on $(A_i,·)$.

Reading guide for Theorem 1: we use $N_{A_i}^j$ to denote the j-th Nijenhuis operator on the pre-Lie algebra $(A_i,·)$, where the algebras $(A_i,·)$ are taken from the complete classification in [33]. The parameters $n_{ij}$, where $i,j=1,2$, of the matrix representing N are arbitrary complex numbers unless otherwise specified.

Theorem 1.

Let N: $A_i\to A_i,1\le i\le 5$, be a linear map on the pre-Lie algebra $(A_i,·)$ defined by (8). Then the Nijenhuis operators on 2-dimensional commutative pre-Lie algebras are as follows:

$\left(1\right)$ The Nijenhuis operators on the pre-Lie algebra $(A_1,·)$ are:

• $N_{A_1}^1\left(e_1\right)=n_{11}{e}_1,N_{A_1}^1\left(e_2\right)=n_{21}{e}_1+(n_{11}+n_{21})e_2,n_{21}\ne 0.$
• $N_{A_1}^2\left(e_1\right)=n_{11}{e}_1,N_{A_1}^2\left(e_2\right)=n_{22}{e}_2.$
• $N_{A_1}^3\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{A_1}^3\left(e_2\right)=(n_{11}-n_{12})e_2,n_{12}\ne 0.$

$\left(2\right)$ The Nijenhuis operators on the pre-Lie algebra $(A_2,·)$ are:

• $N_{A_2}^1\left(e_1\right)=n_{11}{e}_1,N_{A_2}^1\left(e_2\right)=n_{22}{e}_2.$
• $N_{A_2}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{A_2}^2\left(e_2\right)=n_{11}{e}_2,n_{12}\ne 0.$

$\left(3\right)$ The Nijenhuis operators on the pre-Lie algebra $(A_3,·)$ are:

• $N_{A_3}^1\left(e_1\right)=n_{11}{e}_1,N_{A_3}^1\left(e_2\right)=n_{22}{e}_2.$
• $N_{A_3}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{A_3}^2\left(e_2\right)=n_{11}{e}_2,n_{12}\ne 0.$

$\left(4\right)$ The Nijenhuis operators on the pre-Lie algebra $(A_4,·)$ are:

• $N_{A_4}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{A_4}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2.$

$\left(5\right)$ The Nijenhuis operators on the pre-Lie algebra $(A_5,·)$ are:

• $N_{A_5}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{A_5}^1\left(e_2\right)=n_{11}{e}_2.$

Proof. 

Let $\left\{e_1,e_2\right\}$ be a basis of 2-dimensional commutative pre-Lie algebras $(A_i,·)$. Set

$$N\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2.$$

Since N is a Nijenhuis operator, we have

$$\begin{array}{ccc}\hfill N\left(e_1\right)·N\left(e_1\right)& =& N(N\left(e_1\right)·e_1+e_1·N\left(e_1\right)-N(e_1·e_1)),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill N\left(e_1\right)·N\left(e_2\right)& =& N(N\left(e_1\right)·e_2+e_1·N\left(e_2\right)-N(e_1·e_2)),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill N\left(e_2\right)·N\left(e_1\right)& =& N(N\left(e_2\right)·e_1+e_2·N\left(e_1\right)-N(e_2·e_1)),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill N\left(e_2\right)·N\left(e_2\right)& =& N(N\left(e_2\right)·e_2+e_2·N\left(e_2\right)-N(e_2·e_2)).\hfill \end{array}$$

(13)

Calculating (10) for the pre-Lie algebra $(A_1,·)$, we have:

$$N\left(e_1\right)·N\left(e_1\right)={n_{11}}^2{e}_1+{n_{12}}^2{e}_2,$$

and

$$N(N\left(e_1\right)·e_1+e_1·N\left(e_1\right)-N(e_1·e_1))=(n_{11}^2-n_{12}{n}_{21})e_1+(n_{11}{n}_{12}-n_{12}{n}_{22})e_2.$$

Then

$$n_{11}^2=n_{11}^2-n_{12}{n}_{21},{n_{12}}^2=n_{11}{n}_{12}-n_{12}{n}_{22}.$$

(14)

Similarly, by (11)–(13), we have:

$$\begin{array}{c}\hfill n_{11}{n}_{21}=n_{11}{n}_{21}+n_{12}{n}_{21},n_{12}{n}_{22}=n_{12}{n}_{21}+n_{12}{n}_{22}.\end{array}$$

(15)

$$\begin{array}{c}\hfill {n_{21}}^2=n_{21}{n}_{22}-n_{11}{n}_{21},{n_{22}}^2={n_{22}}^2-n_{12}{n}_{21}.\end{array}$$

(16)

Simplifying (14)–(16), we obtain the following three equations:

$$\left\{\begin{array}{c}{n}_{12}{n}_{21}=0,\hfill \\ {n_{12}}^2=n_{11}{n}_{12}-n_{12}{n}_{22},\hfill \\ {n_{21}}^2=n_{21}{n}_{22}-n_{11}{n}_{21}.\hfill \end{array}\right.$$

(17)

To solve the quadratic equations, we distinguish the two cases depending on whether or not $n_{12}=0$.

$\mathbf{Case}\mathbf{1}$: If $n_{12}=0$, then (17) implies $n_{21}(n_{21}-n_{22}+n_{11})=0$. There are two subcases: $n_{21}=0$ and $n_{21}\ne 0$. If $n_{21}\ne 0$, we have $n_{11}+n_{21}=n_{22}$. Then we get $N_{A_1}^1$. If $n_{21}=0$, then we get $N_{A_1}^2$.

$\mathbf{Case}\mathbf{2}$: If $n_{12}\ne 0$, we have $n_{21}=0,n_{22}=n_{11}-n_{12}$. Then we get $N_{A_1}^3$. Therefore, we obtain all Nijenhuis operators on the pre-Lie algebra $(A_1,·)$.

Similarly, we can obtain all Nijenhuis operators on other 2-dimensional commutative pre-Lie algebras. This completes the proof. □

3.2. Nijenhuis Operators on 2-Dimensional Non-Commutative Pre-Lie Algebras

In this subsection, we determine all Nijenhuis operators on $(B_i,·)$.

Reading guide for Theorem 2: we use $N_{B_i}^j$ to denote the j-th Nijenhuis operator on the pre-Lie algebra $(B_i,·)$, where the algebras $(B_i,·)$ are taken from the complete classification in [33]. The parameters $n_{ij}$, where $i,j=1,2$, of the matrix representing N are arbitrary complex numbers unless otherwise specified. The following results are obtained through direct computation.

Theorem 2.

Let $N:B_i\to B_i$, $1\le i\le 6$, be the linear map on the pre-Lie algebras $(B_i,·)$ defined by (8). Then the Nijenhuis operators on 2-dimensional non-commutative pre-Lie algebras are as follows:

$\left(1\right)$ The Nijenhuis operators on the pre-Lie algebra $(B_1,·)$ are:

• $N_{B_1}^1\left(e_1\right)=n_{11}{e}_1,N_{B_1}^1\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2.$

$\left(2\right)$ The Nijenhuis operators on the pre-Lie algebra $(B_2,·)$ are:

• $N_{B_2}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{B_2}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2.$

$\left(3\right)$ The Nijenhuis operators on the pre-Lie algebra $(B_3,·)$ are:

• $N_{B_3}^1\left(e_1\right)=n_{11}{e}_1,N_{B_3}^1\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,n_{21}\ne 0.$
• $N_{B_3}^2\left(e_1\right)=n_{11}{e}_1,N_{B_3}^2\left(e_2\right)=n_{22}{e}_2.$

$\left(4\right)$ The Nijenhuis operators on the pre-Lie algebra $(B_4,·)$ are:

• $N_{B_4}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{B_4}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2.$

$\left(5\right)$ The Nijenhuis operators on the pre-Lie algebra $(B_5,·)$ are:

• $N_{B_5}^1\left(e_1\right)=n_{11}{e}_1,N_{B_5}^1\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2.$

• $N_{B_5}^2\left(e_1\right)=n_{11}{e}_1,N_{B_5}^2\left(e_2\right)={\displaystyle \frac{n_{11}-n_{22}}{l-1}}{e}_1+n_{22}{e}_2,n_{11}\ne n_{22},l\ne 1.$

$\left(6\right)$ The Nijenhuis operators on the pre-Lie algebra $(B_6,·)$ are:

• $N_{B_6}^1\left(e_1\right)=n_{11}{e}_1,N_{B_6}^1\left(e_2\right)=n_{21}{e}_1+(n_{11}\pm n_{21}\sqrt{-1})e_2.$

• $N_{B_6}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{B_6}^2\left(e_2\right)=-n_{12}{e}_1+n_{11}{e}_2,n_{12}\ne 0.$

Remark 1.

Theorems 1 and 2 encompass all possible Nijenhuis operators on 2-dimensional pre-Lie algebras, and the above results are summarized in

Table A1. Nijenhuis operators on 2-dimensional pre-Lie algebras.

Pre-Lie	Dim.	Multiplication Table	Commutative?	Nijenhuis Operators	Parameters
$A_1$	2	$\left\{\begin{array}{c}{e}_1{e}_1=e_1\hfill \\ e_2{e}_2=e_2\hfill \end{array}\right.$	Yes	$N_{A_1}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{11}+n_{21}\end{array}\right)$	$n_{21}\ne 0$
				$N_{A_1}^2=\left(\begin{array}{cc}{n}_{11}& 0\\ 0& n_{22}\end{array}\right)$	/
				$N_{A_1}^3=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ 0& n_{11}-n_{12}\end{array}\right)$	$n_{12}\ne 0$
$A_2$	2	$\left\{\begin{array}{c}{e}_1{e}_1=e_1\hfill \\ e_1{e}_2=e_2{e}_1=e_2\hfill \end{array}\right.$	Yes	$N_{A_2}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ 0& n_{22}\end{array}\right)$	/
				$N_{A_2}^2=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ 0& n_{11}\end{array}\right)$	$n_{12}\ne 0$
$A_3$	2	$e_1{e}_1=e_1$	Yes	$N_{A_3}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ 0& n_{22}\end{array}\right)$	/
				$N_{A_3}^2=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ 0& n_{11}\end{array}\right)$	$n_{12}\ne 0$
$A_4$	2	$e_i{e}_j=0,i,j=1,2$	Yes	$N_{A_4}^1=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ n_{21}& n_{22}\end{array}\right)$	/
$A_5$	2	$e_1{e}_1=e_2$	No	$N_{A_5}^1=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ 0& n_{11}\end{array}\right)$	/
$B_1$	2	$\left\{\begin{array}{c}{e}_2{e}_1=-e_1\hfill \\ e_2{e}_2=e_1-e_2\hfill \end{array}\right.$	No	$N_{B_1}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{11}\end{array}\right)$	/
$B_2$	2	$\left\{\begin{array}{c}{e}_2{e}_1=-e_1\hfill \\ e_2{e}_2=-e_2\hfill \end{array}\right.$	No	$N_{B_2}^1=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ n_{21}& n_{22}\end{array}\right)$	/
$B_3$	2	$\left\{\begin{array}{c}{e}_2{e}_1=-e_1\hfill \\ e_2{e}_2=k{e}_2,k\ne -1\hfill \end{array}\right.$	No	$N_{B_3}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{B_3}^2=\left(\begin{array}{cc}{n}_{11}& 0\\ 0& n_{22}\end{array}\right)$	/
$B_4$	2	$\left\{\begin{array}{c}{e}_1{e}_2=e_1\hfill \\ e_2{e}_2=e_2\hfill \end{array}\right.$	No	$N_{B_4}^1=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ n_{21}& n_{22}\end{array}\right)$	/
$B_5$	2	$\left\{\begin{array}{c}{e}_1{e}_2=l{e}_1\hfill \\ e_2{e}_1=(l-1)e_1\hfill \\ e_2{e}_2=e_1+l{e}_2,l\ne 0\hfill \end{array}\right.$	No	$l=1$: $N_{B_5}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{11}\end{array}\right)$	/
				$l\ne 1$: $N_{B_5}^2=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{22}\end{array}\right)$	$n_{11}=n_{22}$
				$\phantom{l\ne 1:}$ $N_{B_5}^3=\left(\begin{array}{cc}{n}_{11}& 0\\ {\displaystyle \frac{n_{11}-n_{22}}{l-1}}& n_{22}\end{array}\right)$	$n_{11}\ne n_{22}$
$B_6$	2	$\left\{\begin{array}{c}{e}_1{e}_1=2e_1\hfill \\ e_1{e}_2=e_2\hfill \\ e_2{e}_2=e_1\hfill \end{array}\right.$	No	$N_{B_6}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{11}\pm n_{21}\sqrt{-1}\end{array}\right)$	/
				$N_{B_6}^2=\left(\begin{array}{cc}{n}_{11}& n_{12}\\ -n_{12}& n_{11}\end{array}\right)$	$n_{12}\ne 0$

of the Appendix A.

4. Nijenhuis Operators on 3-Dimensional Associative Algebras

Associative algebras are a special type of pre-Lie algebras. In this section, we use the classification results of commutative and non-commutative associative algebras from Proposition 3 and 4 to find all Nijenhuis operators on $(C_i,·)$ and $(D_i,·)$, respectively.

4.1. Nijenhuis Operators on 3-Dimensional Commutative Associative Algebras

In this subsection, we determine all Nijenhuis operators on $(C_i,·)$.

Reading guide for Theorem 3: we use $N_{C_i}^j$ to denote the j-th Nijenhuis operator on the commutative associative algebra $(C_i,·)$, where the algebra $(C_i,·)$ are taken from the complete classification in [22]. The parameters $n_{ij}$, where $i,j=1,2,3$, of the matrix representing N are arbitrary complex numbers unless otherwise specified.

Theorem 3.

Let $N:C_i\to C_i$, $1\le i\le 12$, be a linear map on commutative associative algebras $(C_i,·)$ defined by (8). Then the Nijenhuis operators on 3-dimensional commutative associative algebras are as follows:

$\left(1\right)$ The Nijenhuis operators on the associative algebra $(C_1,·)$ are:

• $N_{C_1}^1\left(e_1\right)=n_{11}{e}_1,N_{C_1}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_1}^1\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3.$
• $N_{C_1}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_1}^2\left(e_2\right)=-{\displaystyle \frac{{(n_{11}-n_{33})}^2}{n_{12}}}{e}_1+(2n_{33}-n_{11})e_2,N_{C_1}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3,n_{12}\ne 0.$

$\left(2\right)$ The Nijenhuis operators on the associative algebra $(C_2,·)$ are:

• $N_{C_2}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{C_2}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{C_2}^1\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3.$

$\left(3\right)$ The Nijenhuis operators on the associative algebra $(C_3,·)$ are:

• $N_{C_3}^1\left(e_1\right)=n_{11}{e}_1,N_{C_3}^1\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,N_{C_3}^1\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3.$
• $N_{C_3}^2\left(e_1\right)=n_{11}{e}_1,N_{C_3}^2\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,N_{C_3}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}\pm n_{32}\sqrt{-1})e_3,n_{32}\ne 0.$
• $N_{C_3}^3\left(e_1\right)=n_{11}{e}_1,N_{C_3}^3\left(e_2\right)=n_{21}{e}_1+(n_{11}\pm n_{23}\sqrt{-1})e_2+n_{23}{e}_3,N_{C_3}^3\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{23}\ne 0.$
• $N_{C_3}^4\left(e_1\right)=n_{11}{e}_1,N_{C_3}^4\left(e_2\right)=n_{21}{e}_1+(n_{11}-n_{23}\sqrt{-1})e_2+n_{23}{e}_3,N_{C_3}^4\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}+n_{32}\sqrt{-1})e_3,n_{23},n_{32}\ne 0.$
• $N_{C_3}^5\left(e_1\right)=n_{11}{e}_1,N_{C_3}^5\left(e_2\right)=n_{21}{e}_1+(n_{11}+n_{23}\sqrt{-1})e_2+n_{23}{e}_3,N_{C_3}^5\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}-n_{32}\sqrt{-1})e_3,n_{23},n_{32}\ne 0.$

$\left(4\right)$ The Nijenhuis operators on the associative algebra $(C_4,·)$ are:

• $N_{C_4}^1\left(e_1\right)=n_{11}{e}_1,N_{C_4}^1\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,N_{C_4}^1\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3.$

$\left(5\right)$ The Nijenhuis operators on the associative algebra $(C_5,·)$ are:

• $N_{C_5}^1\left(e_1\right)=n_{11}{e}_1,N_{C_5}^1\left(e_2\right)=n_{22}{e}_2,N_{C_5}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_5}^2\left(e_1\right)=n_{11}{e}_1,N_{C_5}^2\left(e_2\right)=n_{22}{e}_2,N_{C_5}^2\left(e_3\right)=n_{31}{e}_1+(n_{11}+n_{31})e_3,n_{31}\ne 0.$
• $N_{C_5}^3\left(e_1\right)=n_{11}{e}_1,N_{C_5}^3\left(e_2\right)=n_{22}{e}_2,N_{C_5}^3\left(e_3\right)=n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{32}\ne 0.$
• $N_{C_5}^4\left(e_1\right)=n_{11}{e}_1,N_{C_5}^4\left(e_2\right)=n_{21}{e}_1+(n_{11}+n_{21})e_2,N_{C_5}^4\left(e_3\right)=n_{33}{e}_3,n_{21}\ne 0.$
• $N_{C_5}^5\left(e_1\right)=n_{11}{e}_1,N_{C_5}^5\left(e_2\right)=(n_{23}+n_{33})e_2+n_{23}{e}_3,N_{C_5}^5\left(e_3\right)=n_{33}{e}_3,n_{23}\ne 0.$
• $N_{C_5}^6\left(e_1\right)=n_{11}{e}_1,N_{C_5}^6\left(e_2\right)=n_{21}{e}_1+(n_{11}+n_{21})e_2,N_{C_5}^6\left(e_3\right)=n_{32}{e}_1+n_{32}{e}_2+(n_{11}+n_{21}+n_{32})e_3,n_{21},n_{32}\ne 0.$
• $N_{C_5}^7\left(e_1\right)=(n_{22}+n_{32}-n_{31})e_1,N_{C_5}^7\left(e_2\right)=n_{22}{e}_2,N_{C_5}^7\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{31},n_{32}\ne 0.$
• $N_{C_5}^8\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{C_5}^8\left(e_2\right)=n_{22}{e}_2,N_{C_5}^8\left(e_3\right)=(n_{11}+n_{13})e_3,n_{13}\ne 0.$
• $N_{C_5}^9\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_5}^9\left(e_2\right)=(n_{11}-n_{12})e_2,N_{C_5}^9\left(e_3\right)=n_{33}{e}_3,n_{12}\ne 0.$
• $N_{C_5}^{10}\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{C_5}^{10}\left(e_2\right)=n_{23}{e}_1+(n_{11}+n_{23})e_2+n_{23}{e}_3,N_{C_5}^{10}\left(e_3\right)=(n_{11}-n_{13})e_3,n_{13},n_{23}\ne 0.$
• $N_{C_5}^{11}\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_5}^{11}\left(e_2\right)=(n_{11}-n_{12})e_2,N_{C_5}^{11}\left(e_3\right)=n_{31}{e}_1+n_{31}{e}_2+(n_{31}+n_{11})e_3,n_{12},n_{31}\ne 0.$
• $N_{C_5}^{12}\left(e_1\right)=(n_{23}+n_{33}-n_{21})e_1,N_{C_5}^{12}\left(e_2\right)=n_{21}{e}_1+(n_{23}+n_{33})e_2+n_{23}{e}_3,N_{C_5}^{12}\left(e_3\right)=n_{33}{e}_3,n_{21},n_{23}\ne 0.$
• $N_{C_5}^{13}\left(e_1\right)=(n_{22}-n_{23}-n_{31})e_1,N_{C_5}^{13}\left(e_2\right)=n_{23}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{C_5}^{13}\left(e_3\right)=n_{31}{e}_1+(n_{22}-n_{23})e_3,n_{31},n_{23}\ne 0.$
• $N_{C_5}^{14}\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{C_5}^{14}\left(e_2\right)=(n_{11}-n_{12})e_2,N_{C_5}^{14}\left(e_3\right)=(n_{11}-n_{13})e_3,n_{12},n_{13}\ne 0.$
• $N_{C_5}^{15}\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{12}{e}_3,N_{C_5}^{15}\left(e_2\right)=(n_{11}-n_{12}-n_{32})e_2,N_{C_5}^{15}\left(e_3\right)=n_{32}{e}_2+(n_{11}-n_{12})e_3,n_{12},n_{13},n_{32}\ne 0.$
• $N_{C_5}^{16}\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{12}{e}_3,N_{C_5}^{16}\left(e_2\right)=(n_{11}-n_{12})e_2+n_{23}{e}_3,N_{C_5}^{16}\left(e_3\right)=(n_{11}-n_{12}-n_{23})e_3,n_{12},n_{13},n_{23}\ne 0.$

$\left(6\right)$ The Nijenhuis operators on the associative algebra $(C_6,·)$ are:

• $N_{C_6}^1\left(e_1\right)=n_{11}{e}_1,N_{C_6}^1\left(e_2\right)=n_{22}{e}_2,N_{C_6}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_6}^2\left(e_1\right)=n_{11}{e}_1,N_{C_6}^2\left(e_2\right)=n_{22}{e}_2,N_{C_6}^2\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{31}\ne 0.$
• $N_{C_6}^3\left(e_1\right)=n_{11}{e}_1,N_{C_6}^3\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,N_{C_6}^3\left(e_3\right)=n_{33}{e}_3,n_{21}\ne 0.$
• $N_{C_6}^4\left(e_1\right)=n_{11}{e}_1,N_{C_6}^4\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,N_{C_6}^4\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{21},n_{31}\ne 0.$
• $N_{C_6}^5\left(e_1\right)=n_{11}{e}_1,N_{C_6}^5\left(e_2\right)=n_{22}{e}_2,N_{C_6}^5\left(e_3\right)=n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{32}\ne 0.$
• $N_{C_6}^6\left(e_1\right)=n_{11}{e}_1,N_{C_6}^6\left(e_2\right)=(n_{23}+n_{33})e_2+n_{23}{e}_3,N_{C_6}^6\left(e_3\right)=n_{33}{e}_3,n_{23}\ne 0.$
• $N_{C_6}^7\left(e_1\right)=(n_{22}+n_{32})e_1,N_{C_6}^7\left(e_2\right)=n_{22}{e}_2,N_{C_6}^7\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{31},n_{32}\ne 0.$
• $N_{C_6}^8\left(e_1\right)=(n_{23}+n_{33})e_1,N_{C_6}^8\left(e_2\right)=n_{21}{e}_1+(n_{23}+n_{33})e_2+n_{23}{e}_3,N_{C_6}^8\left(e_3\right)=n_{33}{e}_3,n_{21},n_{23}\ne 0.$

$\left(7\right)$ The Nijenhuis operators on the associative algebra $(C_7,·)$ are:

• $N_{C_7}^1\left(e_1\right)=n_{11}{e}_1,N_{C_7}^1\left(e_2\right)=n_{22}{e}_2,N_{C_7}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_7}^2\left(e_1\right)=n_{11}{e}_1,N_{C_7}^2\left(e_2\right)=n_{22}{e}_2,N_{C_7}^2\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{31}\ne 0.$
• $N_{C_7}^3\left(e_1\right)=n_{11}{e}_1,N_{C_7}^3\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,N_{C_7}^3\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{21}\ne 0.$
• $N_{C_7}^4\left(e_1\right)=(n_{22}+n_{32})e_1,N_{C_7}^4\left(e_2\right)=n_{22}{e}_2,N_{C_7}^4\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{31},n_{32}\ne 0.$
• $N_{C_7}^5\left(e_1\right)=n_{11}{e}_1,N_{C_7}^5\left(e_2\right)=n_{22}{e}_2,N_{C_7}^5\left(e_3\right)=n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{32}\ne 0.$
• $N_{C_7}^6\left(e_1\right)=n_{11}{e}_1,N_{C_7}^6\left(e_2\right)=(n_{23}+n_{33})e_2+n_{23}{e}_3,N_{C_7}^6\left(e_3\right)=n_{33}{e}_3,n_{23}\ne 0.$
• $N_{C_7}^7\left(e_1\right)=n_{22}{e}_1,N_{C_7}^7\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_7}^7\left(e_3\right)=-n_{21}{e}_1+n_{32}{e}_2+(n_{22}+n_{32})e_3,n_{21},n_{32}\ne 0.$
• $N_{C_7}^8\left(e_1\right)=n_{11}{e}_1,N_{C_7}^8\left(e_2\right)=n_{21}{e}_1+(n_{23}+n_{11})e_2+n_{23}{e}_3,N_{C_7}^8\left(e_3\right)=-n_{21}{e}_1+n_{11}{e}_3,n_{21},n_{23}\ne 0.$

$\left(8\right)$ The Nijenhuis operators on the associative algebra $(C_8,·)$ are:

• $N_{C_8}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_8}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_8}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_8}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_8}^2\left(e_2\right)=n_{22}{e}_2,N_{C_8}^2\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{32}\ne 0.$
• $N_{C_8}^3\left(e_1\right)=(n_{33}-{\displaystyle \frac{n_{21}{n}_{32}}{n_{31}}})e_1+{\displaystyle \frac{n_{32}{n}_{33}-n_{22}{n}_{32}}{n_{31}}}{e}_2,N_{C_8}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_8}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3,n_{31}\ne 0.$

$\left(9\right)$ The Nijenhuis operators on the associative algebra $(C_9,·)$ are:

• $N_{C_9}^1\left(e_1\right)=n_{11}{e}_1,N_{C_9}^1\left(e_2\right)=n_{22}{e}_2,N_{C_9}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_9}^2\left(e_1\right)=n_{11}{e}_1,N_{C_9}^2\left(e_2\right)=n_{22}{e}_2,N_{C_9}^2\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{32}\ne 0.$
• $N_{C_9}^3\left(e_1\right)=n_{11}{e}_1,N_{C_9}^3\left(e_2\right)=n_{22}{e}_2,N_{C_9}^3\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{31}\ne 0.$
• $N_{C_9}^4\left(e_1\right)=n_{11}{e}_1,N_{C_9}^4\left(e_2\right)=n_{11}{e}_2,N_{C_9}^4\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{31},n_{32}\ne 0.$
• $N_{C_9}^5\left(e_1\right)=n_{11}{e}_1,N_{C_9}^5\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_9}^5\left(e_3\right)=n_{31}{e}_1+n_{11}{e}_3,n_{21}\ne 0.$
• $N_{C_9}^6\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_9}^6\left(e_2\right)=n_{22}{e}_2,N_{C_9}^6\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{12}\ne 0.$

$\left(10\right)$ The Nijenhuis operators on the associative algebra $(C_{10},·)$ are:

• $N_{C_{10}}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_{10}}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_{10}}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_{10}}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_{10}}^2\left(e_2\right)=n_{22}{e}_2,N_{C_{10}}^2\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{32}\ne 0.$
• $N_{C_{10}}^3\left(e_1\right)=(n_{33}-{\displaystyle \frac{n_{21}{n}_{32}}{n_{31}}})e_1+{\displaystyle \frac{n_{32}{n}_{33}-n_{22}{n}_{32}}{n_{31}}}{e}_2,N_{C_{10}}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{C_{10}}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3,n_{31}\ne 0.$

$\left(11\right)$ The Nijenhuis operators on the associative algebra $(C_{11},·)$ are:

• $N_{C_{11}}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_{11}}^1\left(e_2\right)=n_{11}{e}_2,N_{C_{11}}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_{11}}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_{11}}^2\left(e_2\right)=n_{11}{e}_2,N_{C_{11}}^2\left(e_3\right)=n_{32}{e}_2+n_{11}{e}_3,n_{32}\ne 0.$
• $N_{C_{11}}^3\left(e_1\right)=n_{11}{e}_1-n_{31}{e}_2,N_{C_{11}}^3\left(e_2\right)=n_{11}{e}_2,N_{C_{11}}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{31}\ne 0.$

$\left(12\right)$ The Nijenhuis operators on the associative algebra $(C_{12},·)$ are:

• $N_{C_{12}}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_{12}}^1\left(e_2\right)=n_{11}{e}_2,N_{C_{12}}^1\left(e_3\right)=n_{33}{e}_3.$
• $N_{C_{12}}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{C_{12}}^2\left(e_2\right)=n_{11}{e}_2,N_{C_{12}}^2\left(e_3\right)=n_{32}{e}_2+n_{11}{e}_3,n_{32}\ne 0.$
• $N_{C_{12}}^3\left(e_1\right)=n_{11}{e}_1+n_{31}{e}_2,N_{C_{12}}^3\left(e_2\right)=n_{11}{e}_2,N_{C_{12}}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{31}\ne 0.$

Proof. 

Let $\left\{e_1,e_2,e_3\right\}$ be a basis of 3-dimensional associative algebras $(C_i,·)$, and let N be a linear map on $C_i$. For all $n_{ij}\in \mathbb{C},1⩽i,j⩽3$, set

$$N\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3.$$

The non-zero structural constants of $(C_1,·)$ is

$$C_{33}^1=1.$$

By Proposition 5, N is a Nijenhuis operator on $C_1$ if and only if $n_{ij}$ satisfies the following equations:

$$\left\{\begin{array}{c}{n}_{13}=0,\hfill \\ n_{23}=0,\hfill \\ {(n_{11}-n_{33})}^2=-n_{12}{n}_{21},\hfill \\ n_{12}(2n_{33}-n_{11}-n_{22})=0.\hfill \end{array}\right.$$

(18)

To solve (18), we distinguish the two cases depending on whether $n_{12}$ is equal to zero or not.

$\mathbf{Case}\mathbf{1}$: If $n_{12}=0$, then $n_{11}=n_{33}$, and we obtain $N_{C_1}^1$.

$\mathbf{Case}\mathbf{2}$: If $n_{12}\ne 0$, then we have $n_{21}=-{\displaystyle \frac{{(n_{11}-n_{33})}^2}{n_{12}}},n_{22}=2n_{33}-n_{11}$. Thus we get $N_{C_1}^2$.

Similarly, we can obtain all Nijenhuis operators on other 3-dimensional commutative associative algebras. This completes the proof. □

4.2. Nijenhuis Operators on 3-Dimensional Non-Commutative Associative Algebras

In this subsection, we determine all Nijenhuis operators on $(D_i,·)$.

Reading guide for Theorem 4: we use $N_{D_i}^j$ to denote the j-th Nijenhuis operator on the non-commutative associative algebra $(D_i,·)$, where the algebras $(D_i,·)$ are taken from the complete classification in [22]. The parameters $n_{ij}$, where $i,j=1,2,3$, of the matrix representing N are arbitrary complex numbers unless otherwise specified.

Theorem 4.

Let $N:D_i\to D_i$, $1\le i\le 12$, be a linear map on non-commutative associative algebras $(D_i,·)$ defined by (8). The Nijenhuis operators on 3-dimensional non-commutative associative algebras are then given as follows:

$\left(1\right)$ The Nijenhuis operators on the associative algebra $(D_1,·)$ are:

• $N_{D_1}^1\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{D_1}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_1}^1\left(e_3\right)=n_{11}{e}_3.$
• $N_{D_1}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_1}^2\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_1}^2\left(e_3\right)=n_{11}{e}_3,n_{12}\ne 0.$
• $N_{D_1}^3\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_1}^3\left(e_2\right)=n_{21}{e}_1+(n_{33}-{\displaystyle \frac{n_{12}{n}_{21}}{n_{33}-n_{11}}})e_2+n_{23}{e}_3,N_{D_1}^3\left(e_3\right)=n_{33}{e}_3,n_{33}\ne n_{11}.$

$\left(2\right)$ The Nijenhuis operators on the associative algebra $(D_2,·)$ are:

• $N_{D_2}^1\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{D_2}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_2}^1\left(e_3\right)=n_{22}{e}_3,n_{21}\ne 0.$
• $N_{D_2}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_2}^2\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_2}^2\left(e_3\right)=n_{11}{e}_3,n_{12}\ne 0.$
• $N_{D_2}^3\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{D_2}^3\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_2}^3\left(e_3\right)=n_{22}{e}_3.$
• $N_{D_2}^4\left(e_1\right)=n_{33}{e}_1+n_{13}{e}_3,N_{D_2}^4\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_2}^4\left(e_3\right)=n_{33}{e}_3,n_{33}\ne n_{22}.$

$\left(3\right)$ The Nijenhuis operators on the associative algebra $(D_3,·)$ are:

• $N_{D_3}^1\left(e_1\right)=(n_{33}+{\displaystyle \frac{(-1+\sqrt{1-4\lambda })n_{12}}{2}})e_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_3}^1\left(e_2\right)={\displaystyle \frac{(-1+\sqrt{1-4\lambda })(n_{22}-n_{33})}{2}}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_3}^1\left(e_3\right)=n_{33}{e}_3,n_{12},\lambda \ne 0.$
• $N_{D_3}^2\left(e_1\right)=(n_{33}+{\displaystyle \frac{(-1-\sqrt{1-4\lambda })n_{12}}{2}})e_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_3}^2\left(e_2\right)={\displaystyle \frac{(-1-\sqrt{1-4\lambda })(n_{22}-n_{33})}{2}}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_3}^2\left(e_3\right)=n_{33}{e}_3,n_{12},\lambda \ne 0.$
• $N_{D_3}^3\left(e_1\right)=n_{33}{e}_1+n_{13}{e}_3,N_{D_3}^3\left(e_2\right)={\displaystyle \frac{(-1\pm \sqrt{1-4\lambda })(n_{22}-n_{33})}{2}}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_3}^3\left(e_3\right)=n_{33}{e}_3,\lambda \ne 0.$

$\left(4\right)$ The Nijenhuis operators on the associative algebra $(D_4,·)$ are:

• $N_{D_4}^1\left(e_1\right)=n_{11}{e}_1,N_{D_4}^1\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_4}^1\left(e_3\right)=n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_4}^2\left(e_1\right)=n_{11}{e}_1,N_{D_4}^2\left(e_2\right)=n_{22}{e}_2,N_{D_4}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{31}\ne 0.$
• $N_{D_4}^3\left(e_1\right)=n_{11}{e}_1,N_{D_4}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{D_4}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{21}\ne 0.$
• $N_{D_4}^4\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_4}^4\left(e_2\right)=n_{22}{e}_2,N_{D_4}^4\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{12}\ne 0.$

$\left(5\right)$ The Nijenhuis operators on the associative algebra $(D_5,·)$ are:

• $N_{D_5}^1\left(e_1\right)=n_{11}{e}_1,N_{D_5}^1\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_5}^1\left(e_3\right)=n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_5}^2\left(e_1\right)=n_{11}{e}_1,N_{D_5}^2\left(e_2\right)=n_{22}{e}_2,N_{D_5}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{31}\ne 0.$
• $N_{D_5}^3\left(e_1\right)=n_{11}{e}_1,N_{D_5}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{D_5}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{21}\ne 0.$
• $N_{D_5}^4\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_5}^4\left(e_2\right)=n_{22}{e}_2,N_{D_5}^4\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{12}\ne 0.$

$\left(6\right)$ The Nijenhuis operators on the associative algebra $(D_6,·)$ are:

• $N_{D_6}^1\left(e_1\right)=n_{11}{e}_1,N_{D_6}^1\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_6}^1\left(e_3\right)=n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_6}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_6}^2\left(e_2\right)=n_{11}{e}_2,N_{D_6}^2\left(e_3\right)=n_{32}{e}_2+n_{11}{e}_3,n_{12}\ne 0.$
• $N_{D_6}^3\left(e_1\right)=n_{11}{e}_1,N_{D_6}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{21}{e}_3,N_{D_6}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}+n_{31})e_3,n_{21}\ne 0.$
• $N_{D_6}^4\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{D_6}^4\left(e_2\right)=n_{22}{e}_2,N_{D_6}^4\left(e_3\right)=(n_{11}-n_{13})e_3,n_{13}\ne 0.$
• $N_{D_6}^5\left(e_1\right)=n_{11}{e}_1,N_{D_6}^5\left(e_2\right)=n_{22}{e}_2,N_{D_6}^5\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}+n_{31})e_3,n_{31}\ne 0.$
• $N_{D_6}^6\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_2,N_{D_6}^6\left(e_2\right)=n_{22}{e}_2,N_{D_6}^6\left(e_3\right)=\left({\displaystyle \frac{n_{11}{n}_{12}-n_{12}{n}_{13}-n_{12}{n}_{22}}{n_{13}}}\right)e_2+(n_{11}-n_{13})e_3,n_{12},n_{13}\ne 0.$

$\left(7\right)$ The Nijenhuis operators on the associative algebra $(D_7,·)$ are:

• $N_{D_7}^1\left(e_1\right)=n_{11}{e}_1,N_{D_7}^1\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_7}^1\left(e_3\right)=n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_7}^2\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_7}^2\left(e_2\right)=n_{11}{e}_2,N_{D_7}^2\left(e_3\right)=n_{32}{e}_2+n_{11}{e}_3,n_{12}\ne 0.$
• $N_{D_7}^3\left(e_1\right)=n_{11}{e}_1,N_{D_7}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{21}{e}_3,N_{D_7}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}+n_{31})e_3,n_{21}\ne 0.$
• $N_{D_7}^4\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{D_7}^4\left(e_2\right)=n_{22}{e}_2,N_{D_7}^4\left(e_3\right)=(n_{11}-n_{13})e_3,n_{13}\ne 0.$
• $N_{D_7}^5\left(e_1\right)=n_{11}{e}_1,N_{D_7}^5\left(e_2\right)=n_{22}{e}_2,N_{D_7}^5\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+(n_{11}+n_{31})e_3,n_{31}\ne 0.$
• $N_{D_7}^6\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_2,N_{D_7}^6\left(e_2\right)=n_{22}{e}_2,N_{D_7}^6\left(e_3\right)=\left({\displaystyle \frac{n_{11}{n}_{12}-n_{12}{n}_{13}-n_{12}{n}_{22}}{n_{13}}}\right)e_2+(n_{11}-n_{13})e_3,n_{12},n_{13}\ne 0.$

$\left(8\right)$ The Nijenhuis operators on the associative algebra $(D_8,·)$ are:

• $N_{D_8}^1\left(e_1\right)=n_{11}{e}_1,N_{D_8}^1\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_8}^1\left(e_3\right)=n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_8}^2\left(e_1\right)=n_{11}{e}_1,N_{D_8}^2\left(e_2\right)=n_{22}{e}_2,N_{D_8}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{31}\ne 0.$
• $N_{D_8}^3\left(e_1\right)=n_{11}{e}_1,N_{D_8}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{D_8}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{21}\ne 0.$
• $N_{D_8}^4\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_8}^4\left(e_2\right)=n_{22}{e}_2,N_{D_8}^4\left(e_3\right)=n_{32}{e}_2+n_{22}{e}_3,n_{12}\ne 0.$

$\left(9\right)$ The Nijenhuis operators on the associative algebra $(D_9,·)$ are:

• $N_{D_9}^1\left(e_1\right)=n_{11}{e}_1,N_{D_9}^1\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_9}^1\left(e_3\right)=n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_9}^2\left(e_1\right)=(n_{31}+n_{33})e_1,N_{D_9}^2\left(e_2\right)=n_{22}{e}_2+n_{23}{e}_3,N_{D_9}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3,n_{31}\ne 0.$
• $N_{D_9}^3\left(e_1\right)=(n_{31}+n_{33})e_1,N_{D_9}^3\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2-n_{21}{e}_3,N_{D_9}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3,n_{21}\ne 0.$
• $N_{D_9}^4\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_9}^4\left(e_2\right)=n_{11}{e}_2,N_{D_9}^4\left(e_3\right)=n_{32}{e}_2+n_{11}{e}_3,n_{12}\ne 0.$
• $N_{D_9}^5\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_2,N_{D_9}^5\left(e_2\right)=n_{22}{e}_2,N_{D_9}^5\left(e_3\right)=\left({\displaystyle \frac{n_{11}{n}_{12}+n_{12}{n}_{13}-n_{12}{n}_{22}}{n_{13}}}\right)e_2+(n_{11}+n_{13})e_3,n_{13}\ne 0.$

$\left(10\right)$ The Nijenhuis operators on the associative algebra $(D_{10},·)$ are:

• $N_{D_{10}}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_{10}}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_{10}}^1\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3.$

$\left(11\right)$ The Nijenhuis operators on the associative algebra $(D_{11},·)$ are:

• $N_{D_{11}}^1\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_{11}}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_{11}}^1\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3.$

$\left(12\right)$ The Nijenhuis operators on the associative algebra $(D_{12},·)$ are:

• $N_{D_{12}}^1\left(e_1\right)=n_{11}{e}_1,N_{D_{12}}^1\left(e_2\right)=n_{22}{e}_2,N_{D_{12}}^1\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{33}{e}_3.$
• $N_{D_{12}}^2\left(e_1\right)=n_{11}{e}_1,N_{D_{12}}^2\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2,N_{D_{12}}^2\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{11}{e}_3,n_{21}\ne 0.$
• $N_{D_{12}}^3\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N_{D_{12}}^3\left(e_2\right)=n_{22}{e}_2,N_{D_{12}}^3\left(e_3\right)=n_{31}{e}_1+n_{32}{e}_2+n_{22}{e}_3,n_{12}\ne 0.$
• $N_{D_{12}}^4\left(e_1\right)=n_{11}{e}_1,N_{D_{12}}^4\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_{12}}^4\left(e_3\right)={\displaystyle \frac{n_{21}{n}_{33}-n_{11}{n}_{21}}{n_{23}}}{e}_1+n_{32}{e}_2+n_{33}{e}_3,n_{23}\ne 0.$
• $N_{D_{12}}^5\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2+n_{13}{e}_3,N_{D_{12}}^5\left(e_2\right)=n_{22}{e}_2,N_{D_{12}}^5\left(e_3\right)=n_{31}{e}_1+{\displaystyle \frac{n_{12}{n}_{33}-n_{12}{n}_{22}}{n_{13}}}{e}_2+n_{33}{e}_3,n_{13}\ne 0.$

Proof. 

The proof follows a similar approach to Proposition 3. □

Remark 2.

Theorems 3 and 4 encompass all possible Nijenhuis operators on 3-dimensional associative algebras, and the above results are summarized in

Table A2. Nijenhuis operators on 3-dimensional associative algebras.

Ass	Dim.	Multiplication Table	Commutative?	Nijenhuis Operators (${\mathit{C}}_1$– ${\mathit{C}}_5$)	Parameters
$C_1$	3	$e_3{e}_3=e_1$	Yes	$N_{C_1}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	/
				$N_{C_1}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ -{\displaystyle \frac{{(n_{11}-n_{33})}^2}{n_{12}}}& 2n_{33}-n_{11}& 0\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{12}\ne 0$
$C_2$	3	$e_i{e}_j=0,i,j=1,2,3$	Yes	$N_{C_2}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	/
$C_3$	3	$\left\{\begin{array}{c}{e}_2{e}_2=e_1\hfill \\ e_3{e}_3=e_1\hfill \end{array}\right.$	Yes	$N_{C_3}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	/
				$N_{C_3}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}& 0\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{33}=n_{11}\pm n_{32}\sqrt{-1},$ $n_{32}\ne 0$
				$N_{C_3}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& n_{23}\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{22}=n_{11}\pm n_{23}\sqrt{-1},$ $n_{23}\ne 0$
				$N_{C_3}^4=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{23}\ne 0$,$n_{32}\ne 0$ $n_{22}=n_{11}-n_{23}\sqrt{-1}$, $n_{33}=n_{11}+n_{32}\sqrt{-1}$
				$N_{C_3}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{23}\ne 0$,$n_{32}\ne 0$ $n_{22}=n_{11}+n_{23}\sqrt{-1}$, $n_{33}=n_{11}-n_{32}\sqrt{-1}$
$C_4$	3	$\left\{\begin{array}{c}{e}_2{e}_3=e_3{e}_2=e_1\hfill \\ e_3{e}_3=e_2\hfill \end{array}\right.$	Yes	$N_{C_4}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	/
$C_5$	3	$\left\{\begin{array}{c}{e}_1{e}_1=e_1\hfill \\ e_2{e}_2=e_2\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_5}^1=\mathrm{diag}(n_{11},n_{22},n_{33})$	/
				$N_{C_5}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& 0& n_{11}+n_{31}\end{array}\right)$	$n_{31}\ne 0$
				$N_{C_5}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_5}^4=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}+n_{21}& 0\\ 0& 0& n_{33}\end{array}\right)$	$n_{21}\ne 0$
				$N_{C_5}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{23}+n_{33}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{23}\ne 0$
				$N_{C_5}^6=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}+n_{21}& 0\\ n_{32}& n_{32}& n_{11}+n_{21}+n_{32}\end{array}\right)$	$n_{21},n_{32}\ne 0$
				$N_{C_5}^7=\left(\begin{array}{ccc}{n}_{22}+n_{32}-n_{31}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{31},n_{32}\ne 0$
				$N_{C_5}^8=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ 0& n_{22}& 0\\ 0& 0& n_{11}+n_{13}\end{array}\right)$	$n_{13}\ne 0$
				$N_{C_5}^9=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}-n_{12}& 0\\ 0& 0& n_{33}\end{array}\right)$	$n_{12}\ne 0$
				$N_{C_5}^{10}=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ n_{23}& n_{11}+n_{23}& n_{23}\\ 0& 0& n_{11}-n_{13}\end{array}\right)$	$n_{13},n_{23}\ne 0$
				$N_{C_5}^{11}=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}-n_{12}& 0\\ n_{31}& n_{31}& n_{31}+n_{11}\end{array}\right)$	$n_{12},n_{31}\ne 0$
				$N_{C_5}^{12}=\left(\begin{array}{ccc}{n}_{23}+n_{33}-n_{21}& 0& 0\\ n_{21}& n_{23}+n_{33}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{21},n_{23}\ne 0$
				$N_{C_5}^{13}=\left(\begin{array}{ccc}{n}_{22}-n_{23}-n_{31}& 0& 0\\ n_{23}& n_{22}& n_{23}\\ n_{31}& 0& n_{22}-n_{23}\end{array}\right)$	$n_{31},n_{23}\ne 0$
				$N_{C_5}^{14}=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{11}-n_{12}& 0\\ 0& 0& n_{11}-n_{13}\end{array}\right)$	$n_{12},n_{13}\ne 0$
				$N_{C_5}^{15}=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{12}\\ 0& n_{11}-n_{12}-n_{32}& 0\\ 0& n_{32}& n_{11}-n_{12}\end{array}\right)$	$n_{12},n_{32}\ne 0$
				$N_{C_5}^{16}=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{12}\\ 0& n_{11}-n_{12}& n_{23}\\ 0& 0& n_{11}-n_{12}-n_{23}\end{array}\right)$	$n_{12},n_{23}\ne 0$
$C_6$	3	$\left\{\begin{array}{c}{e}_2{e}_2=e_2\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_6}^1=\mathrm{diag}(n_{11},n_{22},n_{33})$	/
				$N_{C_6}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{31}\ne 0$
				$N_{C_6}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}& 0\\ 0& 0& n_{33}\end{array}\right)$	$n_{21}\ne 0$
				$N_{C_6}^4=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{21},n_{31}\ne 0$
				$N_{C_6}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_6}^6=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{23}+n_{33}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{23}\ne 0$
				$N_{C_6}^7=\left(\begin{array}{ccc}{n}_{22}+n_{32}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{31},n_{32}\ne 0$
				$N_{C_6}^8=\left(\begin{array}{ccc}{n}_{23}+n_{33}& 0& 0\\ n_{21}& n_{23}+n_{33}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{21},n_{23}\ne 0$
$C_7$	3	$\left\{\begin{array}{c}{e}_1{e}_3=e_3{e}_1=e_1\hfill \\ e_2{e}_2=e_2\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_7}^1=\mathrm{diag}(n_{11},n_{22},n_{33})$	/
				$N_{C_7}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{31}\ne 0$
				$N_{C_7}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{11}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{C_7}^4=\left(\begin{array}{ccc}{n}_{22}+n_{32}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{31},n_{32}\ne 0$
				$N_{C_7}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_7}^6=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{23}+n_{33}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{23}\ne 0$
				$N_{C_7}^7=\left(\begin{array}{ccc}{n}_{22}& 0& 0\\ n_{21}& n_{22}& 0\\ -n_{21}& n_{32}& n_{22}+n_{32}\end{array}\right)$	$n_{21},n_{32}\ne 0$
				$N_{C_7}^8=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{23}+n_{11}& n_{23}\\ -n_{21}& 0& n_{11}\end{array}\right)$	$n_{21},n_{23}\ne 0$
$C_8$	3	$e_3{e}_3=e_3$	Yes	$N_{C_8}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ n_{21}& n_{22}& 0\\ 0& 0& n_{33}\end{array}\right)$	/
				$N_{C_8}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_8}^3=\left(\begin{array}{ccc}{n}_{33}-{\displaystyle \frac{n_{21}{n}_{32}}{n_{31}}}& {\displaystyle \frac{n_{32}{n}_{33}-n_{22}{n}_{32}}{n_{31}}}& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{31}\ne 0$
$C_9$	3	$\left\{\begin{array}{c}{e}_1{e}_3=e_3{e}_1=e_1\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_9}^1=\mathrm{diag}(n_{11},n_{22},n_{33})$	/
				$N_{C_9}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_9}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{31}\ne 0$
				$N_{C_9}^4=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{11}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{31},n_{32}\ne 0$
				$N_{C_9}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& 0\\ n_{31}& 0& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{C_9}^6=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{12}\ne 0$
$C_{10}$	3	$\left\{\begin{array}{c}{e}_1{e}_3=e_3{e}_1=e_1\hfill \\ e_2{e}_3=e_3{e}_2=e_2\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_{10}}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ n_{21}& n_{22}& 0\\ 0& 0& n_{33}\end{array}\right)$	/
				$N_{C_{10}}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_{10}}^3=\left(\begin{array}{ccc}{n}_{33}-{\displaystyle \frac{n_{21}{n}_{32}}{n_{31}}}& {\displaystyle \frac{n_{32}{n}_{33}-n_{22}{n}_{32}}{n_{31}}}& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{31}\ne 0$
$C_{11}$	3	$\left\{\begin{array}{c}{e}_1{e}_1=e_2\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_{11}}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& 0& n_{33}\end{array}\right)$	/
				$N_{C_{11}}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& n_{32}& n_{11}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_{11}}^3=\left(\begin{array}{ccc}{n}_{11}& -n_{31}& 0\\ 0& n_{11}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{31}\ne 0$
$C_{12}$	3	$\left\{\begin{array}{c}{e}_1{e}_1=e_2\hfill \\ e_1{e}_3=e_3{e}_1=e_1\hfill \\ e_2{e}_3=e_3{e}_2=e_2\hfill \\ e_3{e}_3=e_3\hfill \end{array}\right.$	Yes	$N_{C_{12}}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& 0& n_{33}\end{array}\right)$	/
				$N_{C_{12}}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& n_{32}& n_{11}\end{array}\right)$	$n_{32}\ne 0$
				$N_{C_{12}}^3=\left(\begin{array}{ccc}{n}_{11}& n_{31}& 0\\ 0& n_{11}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{31}\ne 0$
$D_1$	3	$\left\{\begin{array}{c}{e}_1·e_2={\displaystyle \frac{1}{2}}{e}_3\hfill \\ e_2·e_1=-{\displaystyle \frac{1}{2}}{e}_3\hfill \end{array}\right.$	No	$N_{D_1}^1=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ n_{21}& n_{22}& n_{23}\\ 0& 0& n_{11}\end{array}\right)$	/
				$N_{D_1}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{22}& n_{23}\\ 0& 0& n_{11}\end{array}\right)$	$n_{12}\ne 0$
				$N_{D_1}^3=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{33}-{\displaystyle \frac{n_{12}{n}_{21}}{n_{33}-n_{11}}}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{33}\ne n_{11}$
$D_2$	3	$e_2·e_1=-e_3$	No	$N_{D_2}^1=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ n_{21}& n_{22}& n_{23}\\ 0& 0& n_{22}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_2}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{22}& n_{23}\\ 0& 0& n_{11}\end{array}\right)$	$n_{12}\ne 0$
				$N_{D_2}^3=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ 0& n_{22}& n_{23}\\ 0& 0& n_{22}\end{array}\right)$	/
				$N_{D_2}^4=\left(\begin{array}{ccc}{n}_{33}& 0& n_{13}\\ 0& n_{22}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{33}\ne n_{22}$
$D_3$	3	$\left\{\begin{array}{c}{e}_1·e_1=e_3\hfill \\ e_1·e_2=e_3\hfill \\ e_2·e_2=\lambda e_3,\lambda \ne 0\hfill \end{array}\right.$	No	$N_{D_3}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{12}\ne 0$, $n_{11}=n_{33}+{\displaystyle \frac{(-1+\sqrt{1-4\lambda })n_{12}}{2}}$, $n_{21}={\displaystyle \frac{(-1+\sqrt{1-4\lambda })(n_{22}-n_{33})}{2}}$
				$N_{D_3}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{12}\ne 0$, $n_{11}=n_{33}+{\displaystyle \frac{(-1-\sqrt{1-4\lambda })n_{12}}{2}}$, $n_{21}={\displaystyle \frac{(-1-\sqrt{1-4\lambda })(n_{22}-n_{33})}{2}}$
				$N_{D_3}^3=\left(\begin{array}{ccc}{n}_{33}& 0& n_{13}\\ n_{21}& n_{22}& n_{23}\\ 0& 0& n_{33}\end{array}\right)$	$n_{21}={\displaystyle \frac{(-1\pm \sqrt{1-4\lambda })(n_{22}-n_{33})}{2}}$
$D_4$	3	$\left\{\begin{array}{c}{e}_3·e_2=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_4}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& n_{23}\\ 0& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_4}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{31}\ne 0$
				$N_{D_4}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_4}^4=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{12}\ne 0$
$D_5$	3	$\left\{\begin{array}{c}{e}_2·e_3=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_5}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& n_{23}\\ 0& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_5}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{31}\ne 0$
				$N_{D_5}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_5}^4=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{12}\ne 0$
$D_6$	3	$\left\{\begin{array}{c}{e}_1·e_1=e_1\hfill \\ e_3·e_2=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_6}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& n_{23}\\ 0& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_6}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& n_{32}& n_{11}\end{array}\right)$	$n_{12}\ne 0$
				$N_{D_6}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& n_{21}\\ n_{31}& n_{32}& n_{11}+n_{31}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_6}^4=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ 0& n_{22}& 0\\ 0& 0& n_{11}-n_{13}\end{array}\right)$	$n_{13}\ne 0$
				$N_{D_6}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{11}+n_{31}\end{array}\right)$	$n_{31}\ne 0$
				$N_{D_6}^6=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{22}& 0\\ 0& n_{32}& n_{11}-n_{13}\end{array}\right)$	$n_{12}\ne 0,n_{13}\ne 0,$ $n_{32}={\displaystyle \frac{n_{11}{n}_{12}-n_{12}{n}_{13}-n_{12}{n}_{22}}{n_{13}}}$
$D_7$	3	$\left\{\begin{array}{c}{e}_1·e_1=e_1\hfill \\ e_2·e_3=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_7}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& n_{23}\\ 0& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_7}^2=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& n_{32}& n_{11}\end{array}\right)$	$n_{12}\ne 0$
				$N_{D_7}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& n_{21}\\ n_{31}& n_{32}& n_{11}+n_{31}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_7}^4=\left(\begin{array}{ccc}{n}_{11}& 0& n_{13}\\ 0& n_{22}& 0\\ 0& 0& n_{11}-n_{13}\end{array}\right)$	$n_{13}\ne 0$
				$N_{D_7}^5=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{11}+n_{31}\end{array}\right)$	$n_{31}\ne 0$
				$N_{D_7}^6=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{22}& 0\\ 0& {\displaystyle \frac{n_{11}{n}_{12}-n_{12}{n}_{13}-n_{12}{n}_{22}}{n_{13}}}& n_{11}-n_{13}\end{array}\right)$	$n_{12}\ne 0,n_{13}\ne 0$
$D_8$	3	$\left\{\begin{array}{c}{e}_1·e_3=e_1\hfill \\ e_3·e_1=e_1\hfill \\ e_3·e_2=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_8}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& n_{23}\\ 0& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_8}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{31}\ne 0$
				$N_{D_8}^3=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_8}^4=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ 0& n_{32}& n_{22}\end{array}\right)$	$n_{12}\ne 0$
$D_9$	3	$\left\{\begin{array}{c}{e}_1·e_1=e_1\hfill \\ e_1·e_2=e_2·e_1=e_2\hfill \\ e_1·e_3=e_3·e_1=e_3\hfill \\ e_3·e_2=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_9}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& n_{23}\\ 0& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_9}^2=\left(\begin{array}{ccc}{n}_{31}+n_{33}& 0& 0\\ 0& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{31}\ne 0$
				$N_{D_9}^3=\left(\begin{array}{ccc}{n}_{31}+n_{33}& 0& 0\\ n_{21}& n_{22}& -n_{21}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_9}^4=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{11}& 0\\ 0& n_{32}& n_{11}\end{array}\right)$	$n_{12}\ne 0$
				$N_{D_9}^5=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{22}& 0\\ 0& {\displaystyle \frac{n_{11}{n}_{12}+n_{12}{n}_{13}-n_{12}{n}_{22}}{n_{13}}}& n_{11}+n_{13}\end{array}\right)$	$n_{13}\ne 0$
$D_{10}$	3	$\left\{\begin{array}{c}{e}_3·e_1=e_1\hfill \\ e_3·e_2=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_{10}}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	/
$D_{11}$	3	$\left\{\begin{array}{c}{e}_1·e_3=e_1\hfill \\ e_2·e_3=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_{11}}^1=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	/
$D_{12}$	3	$\left\{\begin{array}{c}{e}_3·e_1=e_1\hfill \\ e_2·e_3=e_2\hfill \\ e_3·e_3=e_3\hfill \end{array}\right.$	No	$N_{D_{12}}^1=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{33}\end{array}\right)$	/
				$N_{D_{12}}^2=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& 0\\ n_{31}& n_{32}& n_{11}\end{array}\right)$	$n_{21}\ne 0$
				$N_{D_{12}}^3=\left(\begin{array}{ccc}{n}_{11}& n_{12}& 0\\ 0& n_{22}& 0\\ n_{31}& n_{32}& n_{22}\end{array}\right)$	$n_{12}\ne 0$
				$N_{D_{12}}^4=\left(\begin{array}{ccc}{n}_{11}& 0& 0\\ n_{21}& n_{22}& n_{23}\\ {\displaystyle \frac{n_{21}{n}_{33}-n_{11}{n}_{21}}{n_{23}}}& n_{32}& n_{33}\end{array}\right)$	$n_{23}\ne 0$
				$N_{D_{12}}^5=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ 0& n_{22}& 0\\ n_{31}& {\displaystyle \frac{n_{12}{n}_{33}-n_{12}{n}_{22}}{n_{13}}}& n_{33}\end{array}\right)$	$n_{13}\ne 0$

of the Appendix A.

Remark 3.

A full orbit analysis under $Aut\left(A\right)$ for each algebra is beyond the scope of this paper. As an illustration, consider $A_5$ in Proposition 1. Its Nijenhuis operators are

$$N\left(e_1\right)=n_{11}{e}_1+n_{12}{e}_2,N\left(e_2\right)=n_{11}{e}_2,n_{11},n_{12}\in \mathbb{C}.$$

The automorphism group $Aut\left(A_5\right)=\{\phi \mid \phi \left(e_1\right)=a{e}_1+b{e}_2,\phi \left(e_2\right)=a^2{e}_2,a\ne 0,b\in \mathbb{C}\}$ acts by conjugation:

$$\phi \circ N\circ {\phi }^{-1}:N\mapsto \left(n_{11}\mapsto n_{11},n_{12}\mapsto a^{-1}{n}_{12}\right).$$

The parameter b does not affect the action, so only a rescales $n_{12}$. This yields two non-isomorphic types: $(n_{12}=0)$: $N\left(e_1\right)=n_{11}{e}_1,N\left(e_2\right)=n_{11}{e}_2$; $(n_{12}\ne 0)$: by setting $a=n_{12}$, we obtain the canonical form $N\left(e_1\right)=n_{11}{e}_1+e_2,N\left(e_2\right)=n_{11}{e}_2$.

These types are not isomorphic because the $Aut\left(A_5\right)$-action preserves the condition $n_{12}=0$. Similar reductions can be applied to other families. We present the operators in parametric form, where the parameters in a family may represent fewer distinct isomorphism classes than initially apparent.

5. Applications of Nijenhuis Operators to the CYBE

In previous sections, we obtained Nijenhuis operators N on 2-dimensional pre-Lie algebras and 3-dimensional commutative and non-commutative associative algebras $(A,·)$. Using these results, we construct Rota-Baxter operators R of weight zero on the sub-adjacent Lie algebras $\mathfrak{g}\left(A\right)$ in this section. Consequently, these operators induce solutions of the CYBE on the Lie algebras $\mathfrak{g}\left(A\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(A\right)}^{*}$.

5.1. From Nijenhuis Operators to Rota-Baxter Operators and CYBE Solutions

Briefly recall the construction:

• For a pre-Lie algebra $(A,·)$, its sub-adjacent Lie algebra $\mathfrak{g}\left(A\right)=(A,[·,·\left]\right)$ is defined by $[x,y]=x·y-y·x$.
• A Nijenhuis operator N on A induces a Rota-Baxter operator R of weight zero on $\mathfrak{g}\left(A\right)$ via Propositions 6 and 7.
• A Rota-Baxter operator R on $\mathfrak{g}\left(A\right)$ yields a solution r of the CYBE on the double $\mathfrak{g}\left(A\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(A\right)}^{*}$ via the standard construction (Theorem 5).

This provides a systematic pipeline:

$$\mathrm{Nijenhuis}\mathrm{on}A\stackrel{\mathrm{Prop}. 6, 7}{\to }\mathrm{Rota}\text{-}\mathrm{Baxter}\mathrm{on}\mathfrak{g}\left(A\right)\stackrel{\mathrm{Thm}. 5}{\to }\mathrm{CYBE}\mathrm{solution}\mathrm{on}\mathfrak{g}\left(A\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(A\right)}^{*}$$

Proposition 6.

Let $\left(A,·\right)$ be a pre-Lie algebra and $N:A\to A$ be a linear map. If $N^2=0$, then N is a Nijenhuis operator if and only if N is a Rota-Baxter operator of weight zero on $(A,·)$.

Proof. 

Let $N:\mathfrak{g}\to \mathfrak{g}$ be a linear map such that $N^2=0$. If N is a Nijenhuis operator, then

$$\begin{array}{ccc}\hfill N\left(x\right)·N\left(y\right)& =& N\left(N\right(x)·y+x·N(y)-N(x·y\left)\right)\hfill \\ & =& N(N\left(x\right)·y+x·N\left(y\right))-N^2(x·y)\hfill \\ & =& N\left(N\right(x)·y+x·N(y\left)\right),\forall x,y\in A.\hfill \end{array}$$

Therefore N is a Rota-Baxter operator of weight zero on $(A,·)$. Conversely, if N is a Rota-Baxter operator of weight zero on $(A,·)$ such that $N^2=0$, then N is a Nijenhuis operator. □

Proposition 7.

If R is a Rota-Baxter operator of weight zero on a pre-Lie algebra $(A,·)$, then R is a Rota-Baxter operator of weight zero on its sub-adjacent Lie algebra $\mathfrak{g}\left(A\right)$.

Proof. 

Let R be a Rota-Baxter operator of weight zero on a pre-Lie algebra $(A,·)$. Then for all $x,y\in A$, we have

$$\begin{array}{c}\hfill R\left(x\right)·R\left(y\right)=R\left(R\right(x)·y+x·R(y\left)\right),\\ \hfill R\left(y\right)·R\left(x\right)=R\left(R\right(y)·x+y·R(x\left)\right).\end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \left[R\right(x),R(y\left)\right]& =& R\left(x\right)·R\left(y\right)-R\left(y\right)·R\left(x\right)\hfill \\ & =& R\left(R\right(x)·y+x·R(y\left)\right)-R\left(R\right(y)·x+y·R(x\left)\right)\hfill \\ & =& R\left(R\right(x)·y-y·R(x)+x·R(y)-R(y)·x)\hfill \\ & =& R\left(\right[R\left(x\right),y]+[x,R\left(y\right)\left]\right),\forall x,y\in A.\hfill \end{array}$$

Hence R is a Rota-Baxter operator of weight zero on its sub-adjacent Lie algebra $\mathfrak{g}\left(A\right).$ □

Let $\rho :\mathfrak{g}\to \mathfrak{gl}\left(V\right)$ be a representation of a Lie algebra $\mathfrak{g}$. On the vector space $\mathfrak{g}\oplus V$, we can define a Lie algebra structure (denoted by $\mathfrak{g}{⋉}_{\rho }V$) given by

$$[x_1+v_1,x_2+v_2]=[x_1,x_2]+\rho \left(x_1\right)v_2-\rho \left(x_2\right)v_1,\forall x_1,x_2\in \mathfrak{g},v_1,v_2\in V.$$

(19)

Define a linear map ${\rho }^{*}:\mathfrak{g}\to \mathfrak{gl}\left({\mathfrak{g}}^{*}\right)$ by

$$\langle {\rho }_x^{*}\left(\xi \right),y\rangle =-\langle \xi ,{\rho }_x\left(y\right)\rangle ,\forall x,y\in \mathfrak{g},\xi \in {\mathfrak{g}}^{*}.$$

(20)

Then ${\rho }^{*}$ is a representation of a Lie algebra $\mathfrak{g}$. It is the dual representation of $\rho$.

Proposition 8 ([30]). If $R:V\to \mathfrak{g}$ is a linear map, then R can be viewed as an element in $\mathfrak{g}\otimes V^{*}\subset \left(\mathfrak{g}{⋉}_{{\rho }^{*}}{V}^{*}\right)\otimes \left(\mathfrak{g}{⋉}_{{\rho }^{*}}{V}^{*}\right)$.

Proof. 

Let $\{e_1,\cdots ,e_n\}$ be a basis of $\mathfrak{g}$. Let $\{v_1,\cdots ,v_m\}$ be a basis of V and $\{v_1^{*},\cdots ,v_m^{*}\}$ be its dual basis, that is $v_i^{*}\left(v_j\right)={\delta }_{ij}$. Set $R\left(v_i\right)={\displaystyle \sum _{j=1}^n}{a}_{ij}{e}_j,$$i=1,2,\cdots ,n$. Since as vector spaces, $\mathrm{Hom}(V,\mathfrak{g})\cong \mathfrak{g}\otimes V^{*}$, we have

$$R=\sum _{i=1}^{m}R\left(v_i\right)\otimes v_i^{*}=\sum _{i=1}^m\sum _{j=1}^n{a}_{ij}{e}_j\otimes v_i^{*}\in \mathfrak{g}\otimes V^{*}\subset \left(\mathfrak{g}{⋉}_{{\rho }^{*}}{V}^{*}\right)\otimes \left(\mathfrak{g}{⋉}_{{\rho }^{*}}{V}^{*}\right).$$

(21)

Hence R can be viewed as an element in $\mathfrak{g}\otimes V^{*}\subset \left(\mathfrak{g}{⋉}_{{\rho }^{*}}{V}^{*}\right)\otimes \left(\mathfrak{g}{⋉}_{{\rho }^{*}}{V}^{*}\right)$. □

For any tensor element $r={\sum }_i{a}_i\otimes b_i\in V\otimes V$, denote $r^{21}={\sum }_i{b}_i\otimes a_i$.

Theorem 5 ([30]). Let $\mathfrak{g}$ be a Lie algebra. A linear map $R:\mathfrak{g}\to \mathfrak{g}$ is a Rota-Baxter operator if and only if $r=R-R^{21}$ is a skew-symmetric solution of CYBE in $\mathfrak{g}{⋉}_{{\mathrm{ad}}^{*}}{\mathfrak{g}}^{*}$.

5.2. A Complete Worked Example: From a 2D Pre-Lie Algebra $(B_1,·)$ to an Explicit r-Matrix

We now illustrate the pipeline from Nijenhuis operators to CYBE solutions through a concrete, end-to-end computation.

Step 1: Choice of a pre-Lie algebra and its Nijenhuis operator. Consider the 2-dimensional pre-Lie algebra $(B_1,·)$ with basis $\{e_1,e_2\}$ (appearing as algebra $A_{-1}$ in [33]) and non-zero products

$$e_2·e_1=-e_1,e_2·e_2=e_1-e_2.$$

(22)

From Theorem 2, the Nijenhuis operators on $(B_1,·)$ are given by the family $N_{B_1}^1$ with two free complex parameters $n_{11},n_{21}\in \mathbb{C}$:

$$N_{B_1}^1\left(e_1\right)=n_{11}{e}_1,N_{B_1}^1\left(e_2\right)=n_{21}{e}_1+n_{11}{e}_2,$$

(23)

or in matrix form (relative to $\{e_1,e_2\}$)

$$N_{B_1}^1=\left(\begin{array}{cc}{n}_{11}& 0\\ n_{21}& n_{11}\end{array}\right).$$

Step 2: Sub-adjacent Lie algebra. The sub-adjacent Lie algebra $\mathfrak{g}\left(B_1\right)=(B_1,[·,·])$ is obtained via $[x,y]=x·y-y·x$. Using (22),

$$\begin{array}{cc}\hfill [e_1,e_2]& =e_1·e_2-e_2·e_1=0-(-e_1)=e_1,\hfill \\ \hfill {⠀}_2,e_1]& =e_2·e_1-e_1·e_2=-e_1-0=-e_1,\hfill \end{array}$$

with all other brackets zero. Hence, the only non-trivial brackets are

$$[e_1,e_2]=e_1,[e_2,e_1]=-e_1.$$

Step 3: From Nijenhuis to Rota-Baxter operator. First compute the square of $N_{B_1}^1$:

$${\left(N_{B_1}^1\right)}^2\left(e_1\right)=n_{11}^2{e}_1,{\left(N_{B_1}^1\right)}^2\left(e_2\right)=2n_{11}{n}_{21}{e}_1+n_{11}^2{e}_2.$$

According to Proposition 6 and 7, a Nijenhuis operator N induces a weight-zero Rota-Baxter operator on $\mathfrak{g}\left(B_1\right)$ precisely when $N^2=0$. This forces

$$n_{11}=0,$$

while $n_{21}$ remains free. Consequently, the admissible Nijenhuis operators are

$$N_{B_1}^1\left(e_1\right)=0,N_{B_1}^1\left(e_2\right)=n_{21}{e}_1,$$

and the induced Rota-Baxter operator $R_{\mathfrak{g}\left(B_1\right)}^1$ on $\mathfrak{g}\left(B_1\right)$ has matrix

$$R_{\mathfrak{g}\left(B_1\right)}^1=\left(\begin{array}{cc}0& 0\\ n_{21}& 0\end{array}\right),\mathrm{i}.\mathrm{e}.,R\left(e_1\right)=0,R\left(e_2\right)=n_{21}{e}_1.$$

(24)

Step 4: Constructing the CYBE solution. Let $\{e_1^{*},e_2^{*}\}$ be the dual basis in $\mathfrak{g}{\left(B_1\right)}^{*}$. The remaining brackets in $\mathfrak{g}\left(B_1\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(B_1\right)}^{*}$ are determined by the coadjoint representation ${\mathrm{ad}}^{*}:\mathfrak{g}\left(B_1\right)\to \mathfrak{gl}\left(\mathfrak{g}{\left(B_1\right)}^{*}\right)$, defined by

$$\langle {\mathrm{ad}}_x^{*}\left(\xi \right),y\rangle =-\langle \xi ,[x,y]\rangle ,x,y\in \mathfrak{g}\left(B_1\right),\xi \in \mathfrak{g}{\left(B_1\right)}^{*}.$$

A direct computation gives

$${\mathrm{ad}}_{e_1}^{*}\left(e_1^{*}\right)=-e_2^{*},{\mathrm{ad}}_{e_1}^{*}\left(e_2^{*}\right)=0,{\mathrm{ad}}_{e_2}^{*}\left(e_1^{*}\right)=e_1^{*},{\mathrm{ad}}_{e_2}^{*}\left(e_2^{*}\right)=0,$$

which yields the brackets

$$[e_1,e_2]=e_1,[e_1,e_1^{*}]=-e_2^{*},[e_2,e_1^{*}]=e_1^{*}.$$

(25)

Following the standard construction (Theorem 5), a solution $r\in \mathfrak{g}\left(B_1\right)\otimes \mathfrak{g}{\left(B_1\right)}^{*}$ of the CYBE on the Lie algebra $\mathfrak{g}\left(B_1\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(B_1\right)}^{*}$ is obtained via

$$r=\sum _{i=1}^2\left(R\left(e_i\right)\otimes e_i^{*}-e_i^{*}\otimes R\left(e_i\right)\right).$$

(26)

Substituting the explicit action (24):

$$\begin{array}{cc}\hfill r& =\left(R\left(e_1\right)\otimes e_1^{*}-e_1^{*}\otimes R\left(e_1\right)\right)+\left(R\left(e_2\right)\otimes e_2^{*}-e_2^{*}\otimes R\left(e_2\right)\right)\hfill \\ \hfill & =\left(0\otimes e_1^{*}-e_1^{*}\otimes 0\right)+\left(n_{21}{e}_1\otimes e_2^{*}-e_2^{*}\otimes n_{21}{e}_1\right)\hfill \\ \hfill & =n_{21}\left(e_1\otimes e_2^{*}-e_2^{*}\otimes e_1\right).\hfill \end{array}$$

Step 5: Result. We thus obtain the one-parameter family of skew-symmetric r-matrices

$$r=n_{21}{e}_1\wedge e_2^{*},\mathrm{where}{e}_1\wedge e_2^{*}:=e_1\otimes e_2^{*}-e_2^{*}\otimes e_1,$$

(27)

which is a solution of the classical Yang–Baxter equation on $\mathfrak{g}\left(B_1\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(B_1\right)}^{*}$.

Remark 4.

The parameter $n_{21}$ is free; $n_{21}=0$ gives the trivial solution $r=0$, while any $n_{21}\ne 0$ yields a non-trivial solution. The whole chain is

$$\mathrm{pre}\text{-}\mathrm{Lie}{B}_1⟶\mathrm{Nijenhuis}{N}_{B_1}^1⟶\mathrm{Rota}\text{-}\mathrm{Baxter}{R}_{\mathfrak{g}\left(B_1\right)}^1⟶\mathrm{CYBE}\mathrm{solution}r$$

5.3. Sub-Adjacent Lie Algebras of Pre-Lie Algebras and Associative Algebras

Having illustrated the construction for a specific pre-Lie algebra $B_1$, we now list the sub-adjacent Lie algebras arising from all 2-dimensional pre-Lie and 3-dimensional associative algebras. These are computed directly following Step 2. The resulting Lie algebras will serve as the domains for the Rota-Baxter operators (and hence for the CYBE solutions) obtained via the Nijenhuis-operator pipeline.

Proposition 9.

Let $\mathfrak{g}\left(B\right)$ be the sub-adjacent Lie algebra of a 2-dimensional pre-Lie algebra $(B,·)$. Then $\mathfrak{g}\left(B\right)$ has a basis $\{e_1,e_2\}$ for which the non-zero product is one of the following:

$$\begin{array}{ccc}& & \mathfrak{g}\left(B_1\right):[e_1,e_2]=e_1;\mathfrak{g}\left(B_2\right):[e_1,e_2]=e_1;\hfill \\ & & \mathfrak{g}\left(B_3\right):[e_1,e_2]=e_1;\mathfrak{g}\left(B_4\right):[e_1,e_2]=e_1;\hfill \\ & & \mathfrak{g}\left(B_5\right):[e_1,e_2]=e_1;\mathfrak{g}\left(B_6\right):[e_1,e_2]=e_2.\hfill \end{array}$$

By Proposition 9, $\mathfrak{g}\left(B_1\right)$ equals $\mathfrak{g}\left(B_2\right)$, $\mathfrak{g}\left(B_3\right)$, $\mathfrak{g}\left(B_4\right)$, and $\mathfrak{g}\left(B_5\right)$; denote this common sub-adjacent Lie algebra by ${\mathfrak{g}}_1$. Keep the notation $\mathfrak{g}\left(B_6\right)$.

Proposition 10.

Let $\mathfrak{g}\left(D\right)$ be a non-trivial sub-adjacent Lie algebra of 3-dimensional associative algebras $(D,·)$. Then $\mathfrak{g}\left(D\right)$ has a basis $\left\{e_1,e_2,e_3\right\}$ in which the non-zero products are described by one of the following:

$$\begin{array}{ccc}& & \mathfrak{g}\left(D_1\right):[e_1,e_2]=e_3;\mathfrak{g}\left(D_2\right):[e_1,e_2]=e_3;\mathfrak{g}\left(D_3\right):[e_1,e_2]=e_3;\hfill \\ & & \mathfrak{g}\left(D_4\right):[e_2,e_3]=-e_2;\mathfrak{g}\left(D_5\right):[e_2,e_3]=e_2;\mathfrak{g}\left(D_6\right):[e_2,e_3]=-e_2;\hfill \\ & & \mathfrak{g}\left(D_7\right):[e_2,e_3]=e_2;\mathfrak{g}\left(D_8\right):[e_2,e_3]=-e_2;\mathfrak{g}\left(D_9\right):[e_2,e_3]=-e_2;\hfill \\ & & \mathfrak{g}\left(D_{10}\right):[e_1,e_3]=-e_1,[e_2,e_3]=-e_2;\mathfrak{g}\left(D_{11}\right):[e_1,e_3]=e_1,[e_2,e_3]=e_2;\hfill \\ & & \mathfrak{g}\left(D_{12}\right):[e_1,e_3]=-e_1,[e_2,e_3]=e_2.\hfill \end{array}$$

By Proposition 10, $\mathfrak{g}\left(D_1\right)=\mathfrak{g}\left(D_2\right)=\mathfrak{g}\left(D_3\right)$, denoted ${\mathfrak{g}}_2$. $\mathfrak{g}\left(D_4\right)=\mathfrak{g}\left(D_6\right)=\mathfrak{g}\left(D_8\right)=\mathfrak{g}\left(D_9\right)$, denoted ${\mathfrak{g}}_3$. $\mathfrak{g}\left(D_5\right)=\mathfrak{g}\left(D_7\right)$, denoted ${\mathfrak{g}}_4$. Keep the notations $\mathfrak{g}\left(D_{10}\right)$, $\mathfrak{g}\left(D_{11}\right)$ and $\mathfrak{g}\left(D_{12}\right)$.

Remark 5.

The sub-adjacent Lie algebras of the pre-Lie algebras $(A,·)$ and $(C,·)$ are trivial. Therefore, all linear maps on their sub-adjacent Lie algebras are Rota-Baxter operators of weight zero.

5.4. Rota-Baxter Operators on Sub-Adjacent Lie Algebras

We use the $2\times 2$ matrix $R_{\mathfrak{g}\left(A\right)}^j$ to represent the j-th Rota-Baxter operator of weight zero on the sub-adjacent Lie algebra $\mathfrak{g}\left(A\right)$. The following results are obtained using a method similar to that in Step 3.

Proposition 11.

The Rota-Baxter operators of weight zero on ${\mathfrak{g}}_1$, derived from the Nijenhuis operator $N_B$, are as follows:

Nijenhuis operators on B	Rota-Baxter operators of weight zero on ${\mathfrak{g}}_1$
$N_{B_1}^1,N_{B_5}^1$	$R_{{\mathfrak{g}}_1}^1=\left(\begin{array}{cc}0& 0\\ n_{21}& 0\end{array}\right)$
$N_{B_3}^2$	$R_{{\mathfrak{g}}_1}^2=0$
$N_{B_2}^1,N_{B_4}^1$	$R_{{\mathfrak{g}}_1}^3=\left\{N\right|N^2=0\}$
$N_{B_3}^1$	$R_{{\mathfrak{g}}_1}^4=\left(\begin{array}{cc}0& 0\\ n_{21}& 0\end{array}\right),n_{21}\ne 0$

Remark 6.

The Rota-Baxter operators of weight zero on the sub-adjacent Lie algebra $\mathfrak{g}\left(B_6\right)$ derived by Nijenhuis operators $N_B$ is 0.

Proposition 12.

The matrices of Rota-Baxter operators of weight zero on ${\mathfrak{g}}_2$, induced by Nijenhuis operators N, are as follows:

Nijenhuis operators on D	Rota-Baxter operators of weight zero on ${\mathfrak{g}}_2$
$N_{D_1}^1$	$R_{{\mathfrak{g}}_2}^1=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& 0& n_{23}\\ 0& 0& 0\end{array}\right)$
	$R_{{\mathfrak{g}}_2}^2=\left(\begin{array}{ccc}0& 0& n_{13}\\ 0& 0& n_{23}\\ 0& 0& 0\end{array}\right),n_{13}\ne 0$
$N_{D_1}^2,N_{D_2}^2$	$R_{{\mathfrak{g}}_2}^3=\left(\begin{array}{ccc}0& n_{12}& n_{13}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),n_{12}\ne 0$
$N_{D_1}^3$	$R_{{\mathfrak{g}}_2}^4=\left(\begin{array}{ccc}{n}_{11}& n_{12}& -{\displaystyle \frac{n_{12}{n}_{23}}{n_{11}}}\\ -{\displaystyle \frac{{n_{11}}^2}{n_{12}}}& -n_{11}& n_{23}\\ 0& 0& 0\end{array}\right),n_{11},n_{12}\ne 0$
$N_{D_2}^1$	$R_{{\mathfrak{g}}_2}^5=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& 0& n_{23}\\ 0& 0& 0\end{array}\right),n_{21}\ne 0$
$N_{D_2}^3,N_{D_3}^3$	$R_{{\mathfrak{g}}_2}^6=\left(\begin{array}{ccc}0& 0& n_{13}\\ 0& 0& n_{23}\\ 0& 0& 0\end{array}\right)$
$N_{D_3}^1$	$R_{{\mathfrak{g}}_2}^7=\left(\begin{array}{ccc}{k}_1{n}_{12}& n_{12}& n_{13}\\ -{k_1}^2{n}_{12}& -k_1{n}_{12}& -k_1{n}_{13}\\ 0& 0& 0\end{array}\right),\begin{array}{c}{k}_1={\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}},\\ n_{12},\lambda \ne 0\end{array}$
$N_{D_3}^2$	$R_{{\mathfrak{g}}_2}^8=\left(\begin{array}{ccc}{k}_2{n}_{12}& n_{12}& n_{13}\\ -{k_2}^2{n}_{12}& -k_2{n}_{12}& -k_2{n}_{13}\\ 0& 0& 0\end{array}\right),\begin{array}{c}{k}_2={\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}},\\ n_{12},\lambda \ne 0\end{array}$

Proof. 

The Nijenhuis operators $N_{D_1}^1$ on the pre-Lie algebra $(D_1,·)$ are

$$N_{D_1}^1\left(e_1\right)=n_{11}{e}_1+n_{13}{e}_3,N_{D_1}^1\left(e_2\right)=n_{21}{e}_1+n_{22}{e}_2+n_{23}{e}_3,N_{D_1}^1\left(e_3\right)=n_{11}{e}_3.$$

Then we have

$$\begin{array}{ccc}\hfill {\left(N_{D_1}^1\right)}^2\left(e_1\right)& =& {n_{11}}^2{e}_1+n_{11}{n}_{13}{e}_2,\hfill \\ \hfill {\left(N_{D_1}^1\right)}^2\left(e_2\right)& =& n_{21}(n_{11}+n_{22})e_1+{n_{22}}^2{e}_2+(n_{13}{n}_{21}+n_{22}{n}_{23}+n_{11}{n}_{23})e_3,\hfill \\ \hfill {\left(N_{D_1}^1\right)}^2\left(e_3\right)& =& {n_{11}}^2{e}_3.\hfill \end{array}$$

If ${\left(N_{D_1}^1\right)}^2=0$, then $n_{11}=0$, $n_{22}=0$, $n_{13}{n}_{21}=0$.

Case 1: If $n_{13}=0$, by Propositions 6 and 7, we obtain the Rota-Baxter operator $R_{{\mathfrak{g}}_2}^1$.

Case 2: If $n_{13}\ne 0$, then $n_{21}=0$, yielding the Rota-Baxter operator $R_{{\mathfrak{g}}_2}^2$. All other Rota-Baxter operators are obtained similarly. This completes the proof. □

The following Propositions 13–15 are direct applications of Theorem 4 and follow the same approach as in the proof of Proposition 12.

Proposition 13.

The Rota-Baxter operators of weight zero on ${\mathfrak{g}}_3$, induced by Nijenhuis operators N, are as follows:

Nijenhuis operators on D	Rota-Baxter operators of weight zero on ${\mathfrak{g}}_3$
$N_{D_4}^1,N_{D_6}^1,N_{D_8}^1,N_{D_9}^1$	$R_{{\mathfrak{g}}_3}^1=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& n_{32}& 0\end{array}\right)$
	$R_{{\mathfrak{g}}_3}^2=\left(\begin{array}{ccc}0& 0& 0\\ 0& n_{22}& n_{23}\\ 0& -{\displaystyle \frac{{n_{22}}^2}{n_{23}}}& -n_{22}\end{array}\right),n_{23}\ne 0$
$N_{D_4}^2,N_{D_8}^2$	$R_{{\mathfrak{g}}_3}^3=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ n_{31}& n_{32}& 0\end{array}\right),n_{31}\ne 0$
$N_{D_4}^3,N_{D_8}^3$	$R_{{\mathfrak{g}}_3}^4=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& 0& 0\\ n_{31}& 0& 0\end{array}\right),n_{21}\ne 0$
$N_{D_4}^4,N_{D_6}^2,N_{D_8}^4,N_{D_9}^4$	$R_{{\mathfrak{g}}_3}^5=\left(\begin{array}{ccc}0& n_{12}& 0\\ 0& 0& 0\\ 0& n_{32}& 0\end{array}\right),n_{12}\ne 0$
$N_{D_6}^3$	$R_{{\mathfrak{g}}_3}^6=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& -n_{31}& n_{21}\\ n_{31}& -{\displaystyle \frac{{n_{31}}^2}{n_{21}}}& n_{31}\end{array}\right),n_{21}\ne 0$
$N_{D_9}^3$	$R_{{\mathfrak{g}}_3}^7=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& -n_{33}& -n_{21}\\ -n_{33}& {\displaystyle \frac{{n_{33}}^2}{n_{21}}}& n_{33}\end{array}\right),n_{21}\ne 0$

Proposition 14.

The Rota-Baxter operators of weight zero on ${\mathfrak{g}}_4$, derived from Nijenhuis operators N, are as follows:

Nijenhuis operators on D	Rota-Baxter operators of weight zero on ${\mathfrak{g}}_4$
$N_{D_5}^1,N_{D_7}^1$	$R_{{\mathfrak{g}}_4}^1=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& n_{32}& 0\end{array}\right)$
	$R_{{\mathfrak{g}}_4}^2=\left(\begin{array}{ccc}0& 0& 0\\ 0& n_{22}& n_{23}\\ 0& -{\displaystyle \frac{{n_{22}}^2}{n_{23}}}& -n_{22}\end{array}\right),n_{23}\ne 0$
$N_{D_5}^2$	$R_{{\mathfrak{g}}_4}^3=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ n_{31}& n_{32}& 0\end{array}\right),n_{31}\ne 0$
$N_{D_5}^3$	$R_{{\mathfrak{g}}_4}^4=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& 0& 0\\ n_{31}& 0& 0\end{array}\right),n_{21}\ne 0$
$N_{D_5}^4,N_{D_7}^2$	$R_{{\mathfrak{g}}_4}^5=\left(\begin{array}{ccc}0& n_{12}& 0\\ 0& 0& 0\\ 0& n_{32}& 0\end{array}\right),n_{12}\ne 0$
$N_{D_7}^3$	$R_{{\mathfrak{g}}_4}^4=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& -n_{31}& n_{21}\\ n_{31}& -{\displaystyle \frac{{n_{31}}^2}{n_{21}}}& n_{31}\end{array}\right),n_{21}\ne 0$

Remark 7.

All linear maps on $(D_{10},·)$ and $(D_{11},·)$, whose squares are 0, are Rota-Baxter operators of weight zero on the sub-adjacent Lie algebras $\mathfrak{g}\left(D_{10}\right)$ and $\mathfrak{g}\left(D_{11}\right)$.

Proposition 15.

The Rota-Baxter operators of weight zero on $\mathfrak{g}\left(D_{12}\right)$ induced by Nijenhuis operators N are as follows:

Nijenhuis operators on D	Rota-Baxter operators of weight zero on $\mathfrak{g}\left(D_{12}\right)$
$N_{D_{12}}^1$	$R_{{\mathfrak{g}}_{\left(D_{12}\right)}}^1=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ n_{31}& n_{32}& 0\end{array}\right)$
$N_{D_{12}}^2$	$R_{{\mathfrak{g}}_{\left(D_{12}\right)}}^2=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& 0& 0\\ n_{31}& 0& 0\end{array}\right),n_{21}\ne 0$
$N_{D_{12}}^3$	$R_{{\mathfrak{g}}_{\left(D_{12}\right)}}^3=\left(\begin{array}{ccc}0& n_{12}& 0\\ 0& 0& 0\\ 0& n_{32}& 0\end{array}\right),n_{12}\ne 0$
$N_{D_{12}}^4$	$R_{{\mathfrak{g}}_{\left(D_{12}\right)}}^4=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& -n_{33}& n_{23}\\ {\displaystyle \frac{n_{21}{n}_{33}}{n_{23}}}& -{\displaystyle \frac{{n_{33}}^2}{n_{23}}}& n_{33}\end{array}\right),n_{23}\ne 0$
$N_{D_{12}}^5$	$R_{{\mathfrak{g}}_{\left(D_{12}\right)}}^5=\left(\begin{array}{ccc}-n_{33}& n_{12}& n_{13}\\ 0& 0& 0\\ -{\displaystyle \frac{{n_{33}}^2}{n_{13}}}& {\displaystyle \frac{n_{12}{n}_{33}}{n_{13}}}& n_{33}\end{array}\right),n_{13}\ne 0$

Remark 8.

There are the same Nijenhuis operators on different pre-Lie algebras. For example, $\left(1\right)N_{D_1}^2$ is equal to $N_{D_2}^2$. $\left(2\right)N_{D_4}^1$ is equal to $N_{D_5}^1$, $N_{D_6}^1$, $N_{D_7}^1$, $N_{D_8}^1$, $N_{D_9}^1$. $\left(3\right)N_{D_4}^2$ is equal to $N_{D_5}^2$, $N_{D_8}^2$. $\left(4\right)N_{D_4}^3$ is equal to $N_{D_5}^3$, $N_{D_8}^3$. $\left(5\right)N_{D_4}^4$ is equal to $N_{D_5}^4$, $N_{D_8}^4$. $\left(6\right)N_{D_6}^2$ is equal to $N_{D_7}^2$, $N_{D_9}^4$. $\left(7\right)N_{D_6}^3$ is equal to $N_{D_7}^3$. The sub-adjacent Lie algebras $\mathfrak{g}\left(D_4\right)$ and $\mathfrak{g}\left(D_{12}\right)$ are different, but the Rota-Baxter operators derived by $N_{D_4}^4$ and $N_{D_{12}}^3$ are the same. Therefore, the Rota-Baxter operators of weight zero on the different sub-adjacent Lie algebras derived by Nijenhuis operators may be identical.

5.5. Solutions of CYBE on Lie Algebras $g\left(A\right){⋉}_{{\mathrm{ad}}^{*}}g{\left(A\right)}^{*}$

Proposition 16.

Let $\{e_1,e_2,e_3\}$ be a basis of ${\mathfrak{g}}_2$ and $\{e_1^{*},e_2^{*},e_3^{*}\}$ be its dual basis. Then the Lie algebra ${\mathfrak{g}}_2{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_2}^{*}$ has a basis $\{e_1,e_2,e_3,e_1^{*},e_2^{*},e_3^{*}\}$, with the following non-zero Lie bracket:

$$[e_1,e_2]=e_3,[e_1,e_3^{*}]=-e_2^{*},[e_2,e_3^{*}]=e_1^{*}.$$

The classical Yang–Baxter equation on this Lie algebra has solutions as follows:

• $r_1=(n_{21}{e}_1+n_{23}{e}_2)\otimes e_2^{*}-e_2^{*}\otimes (n_{21}{e}_1+n_{23}{e}_2).$
• $r_2=(n_{11}{e}_1+n_{12}{e}_2-{\displaystyle \frac{n_{12}{n}_{23}}{n_{11}}}{e}_3)\otimes e_1^{*}+(-{\displaystyle \frac{{n_{11}}^2}{n_{12}}}{e}_1-n_{11}{e}_2+n_{23}{e}_3)\otimes e_2^{*}-e_1^{*}\otimes (n_{11}{e}_1+n_{12}{e}_2-{\displaystyle \frac{n_{12}{n}_{23}}{n_{11}}}{e}_3)-e_2^{*}\otimes (-{\displaystyle \frac{{n_{11}}^2}{n_{12}}}{e}_1-n_{11}{e}_2+n_{23}{e}_3)n_{11},n_{12}\ne 0.$
• $r_3=(n_{12}{e}_2+n_{13}{e}_3)\otimes e_1^{*}-e_1^{*}\otimes (n_{12}{e}_2+n_{13}{e}_3),n_{12}\ne 0.$
• $r_4=n_{13}{e}_3\otimes e_1^{*}+n_{23}{e}_3\otimes e_2^{*}-e_1^{*}\otimes n_{13}{e}_3-e_2^{*}\otimes n_{23}{e}_3.$
• $r_5=({\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3)\otimes e_1^{*}-{\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}}({\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3)\otimes e_2^{*}-e_1^{*}\otimes ({\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3)+e_2^{*}\otimes {\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}}({\displaystyle \frac{(-1+\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3),n_{12},\lambda \ne 0.$
• $r_6=({\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3)\otimes e_1^{*}-{\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}}({\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3)\otimes e_2^{*}-e_1^{*}\otimes ({\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3)+e_2^{*}\otimes {\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}}({\displaystyle \frac{(-1-\sqrt{1-4\lambda })}{2}}{n}_{12}{e}_1+n_{12}{e}_2+n_{13}{e}_3),n_{12},\lambda \ne 0.$

Proof. 

Let $\rho ={\mathrm{ad}}^{*}$. By (20), the dual representation ${\mathrm{ad}}^{*}:{\mathfrak{g}}_2\to \mathfrak{gl}\left({{\mathfrak{g}}_2}^{*}\right)$ is defined by

$$\langle {\mathrm{ad}}_x^{*}\left(\xi \right),y\rangle =-\langle \xi ,{\mathrm{ad}}_x\left(y\right)\rangle ,\forall x,y\in {\mathfrak{g}}_2,\xi \in {{\mathfrak{g}}_2}^{*}.$$

(28)

By (19) we have

$$[e_1,e_1]=0,[e_1,e_2]=-[e_2,e_1]=e_3,[e_1,e_3]=-[e_3,e_1]=0,$$

$$[e_2,e_2]=0,[e_2,e_3]=-[e_3,e_2]=0,[e_3,e_3]=0,$$

$$[e_1^{*},e_1^{*}]=0,[e_1^{*},e_2^{*}]=-[e_2^{*},e_1^{*}]=0,[e_1^{*},e_3^{*}]=-[e_3^{*},e_1^{*}]=0,$$

$$[e_2^{*},e_2^{*}]=0,[e_2^{*},e_3^{*}]=-[e_3^{*},e_2^{*}]=0,[e_3^{*},e_3^{*}]=0,$$

$$[e_1,e_1^{*}]=-[e_1^{*},e_1]={\mathrm{ad}}_{e_1}^{*}{e}_1^{*},[e_1,e_2^{*}]=-[e_2^{*},e_1]={\mathrm{ad}}_{e_1}^{*}{e}_2^{*},[e_1,e_3^{*}]=-[e_3^{*},e_1]={\mathrm{ad}}_{e_1}^{*}{e}_3^{*},$$

$$[e_2,e_1^{*}]=-[e_1^{*},e_2]={\mathrm{ad}}_{e_2}^{*}{e}_1^{*},[e_2,e_2^{*}]=-[e_2^{*},e_2]={\mathrm{ad}}_{e_2}^{*}{e}_2^{*},[e_2,e_3^{*}]=-[e_3^{*},e_2]={\mathrm{ad}}_{e_2}^{*}{e}_3^{*},$$

$$[e_3,e_1^{*}]=-[e_1^{*},e_3]={\mathrm{ad}}_{e_3}^{*}{e}_1^{*},[e_3,e_2^{*}]=-[e_2^{*},e_3]={\mathrm{ad}}_{e_3}^{*}{e}_2^{*}.[e_3,e_3^{*}]=-[e_3^{*},e_3]={\mathrm{ad}}_{e_3}^{*}{e}_3^{*}.$$

Set ${\mathrm{ad}}_{e_1}^{*}\left(e_1^{*}\right)=k_{11}{e}_1^{*}+k_{12}{e}_2^{*}+k_{13}{e}_3^{*},k_{11},k_{12},k_{13}\in \mathbb{C}.$ Then

$$\langle {\mathrm{ad}}_{e_1}^{*}\left(e_1^{*}\right),e_1\rangle =-\langle e_1^{*},{\mathrm{ad}}_{e_1}\left(e_1\right)\rangle =-\langle e_1^{*},0\rangle =0,$$

$$(k_{11}{e}_1^{*}+k_{12}{e}_2^{*}+k_{13}{e}_3^{*})\left(e_1\right)=k_{11}.$$

This implies that $k_{11}=0$. From

$$\langle {\mathrm{ad}}_{e_1}^{*}\left(e_1^{*}\right),e_2\rangle =-\langle e_1^{*},{\mathrm{ad}}_{e_1}\left(e_2\right)\rangle =-\langle e_1^{*},e_3\rangle =0,$$

$$(k_{11}{e}_1^{*}+k_{12}{e}_2^{*}+k_{13}{e}_3^{*})\left(e_2\right)=k_{12},$$

we get $k_{12}=0$. From

$$\langle {\mathrm{ad}}_{e_1}^{*}\left(e_1^{*}\right),e_3\rangle =-\langle e_1^{*},{\mathrm{ad}}_{e_1}\left(e_3\right)\rangle =-\langle e_1^{*},0\rangle =0,$$

$$(k_{11}{e}_1^{*}+k_{12}{e}_2^{*}+k_{13}{e}_3^{*})\left(e_3\right)=k_{13},$$

we get $k_{13}=0$. Thus, ${\mathrm{ad}}_{e_1}^{*}\left(e_1^{*}\right)=0$. Similarly, we have ${\mathrm{ad}}_{e_1}^{*}\left(e_2^{*}\right)=0$, ${\mathrm{ad}}_{e_1}^{*}\left(e_3^{*}\right)=-e_2^{*}$, ${\mathrm{ad}}_{e_2}^{*}\left(e_1^{*}\right)=0$, ${\mathrm{ad}}_{e_2}^{*}\left(e_2^{*}\right)=0$, ${\mathrm{ad}}_{e_2}^{*}\left(e_3^{*}\right)=e_1^{*}$, ${\mathrm{ad}}_{e_3}^{*}\left(e_1^{*}\right)=0$, ${\mathrm{ad}}_{e_3}^{*}\left(e_2^{*}\right)=0$, and ${\mathrm{ad}}_{e_3}^{*}\left(e_3^{*}\right)=0$. Therefore, the non-zero Lie brackets of ${\mathfrak{g}}_2{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_2}^{*}$ are

$$[e_1,e_2]=e_3,[e_1,e_3^{*}]=-e_2^{*},[e_2,e_3^{*}]=e_1^{*}.$$

The Rota-Baxter operators $R_{{\mathfrak{g}}_2}^1$ of weight zero on the sub-adjacent Lie algebra ${\mathfrak{g}}_2$ are

$$R_{{\mathfrak{g}}_2}^1=\left(\begin{array}{ccc}0& 0& 0\\ n_{21}& 0& n_{23}\\ 0& 0& 0\end{array}\right).$$

By Proposition 8, we have $R_{{\mathfrak{g}}_2}^1=(n_{21}{e}_1+n_{23}{e}_3)\otimes e_2^{*}$ and ${\left(R_{{\mathfrak{g}}_1}^1\right)}^{21}=e_2^{*}\otimes (n_{21}{e}_1+n_{23}{e}_3)$. By Lemma 5, the induced solutions of CYBE on the Lie algebra ${\mathfrak{g}}_2{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_2}^{*}$ are

$$r_1=R_{{\mathfrak{g}}_2}^1-{\left(R_{{\mathfrak{g}}_2}^1\right)}^{21}=(n_{21}{e}_1+n_{23}{e}_3)\otimes e_2^{*}-e_2^{*}\otimes (n_{21}{e}_1+n_{23}{e}_3).$$

The other solutions of CYBE on the Lie algebra ${\mathfrak{g}}_2{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_2}^{*}$ are obtained for the same step. This completes the proof. □

Using a similar proof as in Proposition 16, the solutions of CYBE on the 6-dimensional Lie algebras ${\mathfrak{g}}_3{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_3}^{*}$, ${\mathfrak{g}}_4{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_4}^{*}$, and $\mathfrak{g}\left(D_{12}\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(D_{12}\right)}^{*}$ can be obtained by applying Rota-Baxter operators of weight zero on the sub-adjacent Lie algebras from Propositions 13–15, as shown in the following Propositions 17–19.

Proposition 17.

Let $\{e_1,e_2,e_3\}$ be a basis of ${\mathfrak{g}}_3$ and $\{e_1^{*},e_2^{*},e_3^{*}\}$ be its dual basis. Then the Lie algebra ${\mathfrak{g}}_3{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_3}^{*}$ has a basis $\{e_1,e_2,e_3,e_1^{*},e_2^{*},e_3^{*}\}$, with the following non-zero Lie brackets:

$$[e_2,e_3]=-e_2,[e_2,e_2^{*}]=e_3^{*},[e_3,e_2^{*}]=-e_2^{*}.$$

The classical Yang-Baxter equation on this Lie algebra has solutions as follows:

• $r_1=n_{32}{e}_2\otimes e_3^{*}-e_3^{*}\otimes n_{32}{e}_2.$
• $r_2=(n_{22}{e}_2+n_{23}{e}_3)\otimes e_2^{*}-({\displaystyle \frac{{n_{22}}^2}{n_{23}}}{e}_2+n_{22}{e}_3)\otimes e_3^{*}-e_2^{*}\otimes (n_{22}{e}_2+n_{23}{e}_3)+e_3^{*}\otimes ({\displaystyle \frac{{n_{22}}^2}{n_{23}}}{e}_2+n_{22}{e}_3),n_{23}\ne 0.$
• $r_3=(n_{31}{e}_1+n_{32}{e}_2)\otimes e_3^{*}-e_3^{*}\otimes (n_{31}{e}_1+n_{32}{e}_2),n_{31}\ne 0.$
• $r_4=n_{21}{e}_1\otimes e_2^{*}+n_{31}{e}_1\otimes e_3^{*}-e_2^{*}\otimes n_{21}{e}_1-e_3^{*}\otimes n_{31}{e}_1,n_{21}\ne 0.$
• $r_5=n_{12}{e}_2\otimes e_1^{*}+n_{32}{e}_2\otimes e_3^{*}-e_1^{*}\otimes n_{12}{e}_2-e_3^{*}\otimes n_{32}{e}_2,n_{12}\ne 0.$
• $r_6=(n_{21}{e}_1-n_{31}{e}_2+n_{21}{e}_3)\otimes e_2^{*}+(n_{31}{e}_1-{\displaystyle \frac{{n_{31}}^2}{n_{21}}}{e}_2+n_{31}{e}_3)\otimes e_3^{*}-e_2^{*}\otimes (n_{21}{e}_1-n_{31}{e}_2+n_{21}{e}_3)-e_3^{*}\otimes (n_{31}{e}_1-{\displaystyle \frac{{n_{31}}^2}{n_{21}}}{e}_2+n_{31}{e}_3),n_{21}\ne 0.$
• $r_7=(n_{21}{e}_1-n_{33}{e}_2-n_{21}{e}_3)\otimes e_2^{*}+(-n_{33}{e}_1+{\displaystyle \frac{{n_{33}}^2}{n_{21}}}{e}_2+n_{33}{e}_3)\otimes e_3^{*}-e_2^{*}\otimes (n_{21}{e}_1-n_{33}{e}_2-n_{21}{e}_3)-e_3^{*}\otimes (-n_{33}{e}_1+{\displaystyle \frac{{n_{33}}^2}{n_{21}}}{e}_2+n_{33}{e}_3),n_{21}\ne 0.$

Proposition 18.

Let $\{e_1,e_2,e_3\}$ be a basis of ${\mathfrak{g}}_4$ and $\{e_1^{*},e_2^{*},e_3^{*}\}$ be its dual basis. Then the Lie algebra ${\mathfrak{g}}_4{⋉}_{{\mathrm{ad}}^{*}}{{\mathfrak{g}}_4}^{*}$ has a basis $\{e_1,e_2,e_3,e_1^{*},e_2^{*},e_3^{*}\}$, with the non-zero Lie brackets:

$$[e_2,e_3]=e_2,[e_2,e_2^{*}]=-e_3^{*},[e_3,e_2^{*}]=e_2^{*}.$$

The classical Yang–Baxter equation on this Lie algebra has solutions as follows:

• $r_1=n_{32}{e}_2\otimes e_3^{*}-e_3^{*}\otimes n_{32}{e}_2.$
• $r_2=(n_{22}{e}_2+n_{23}{e}_3)\otimes e_2^{*}-({\displaystyle \frac{{n_{22}}^2}{n_{23}}}{e}_2+n_{22}{e}_3)\otimes e_3^{*}-e_2^{*}\otimes (n_{22}{e}_2+n_{23}{e}_3)+e_3^{*}\otimes ({\displaystyle \frac{{n_{22}}^2}{n_{23}}}{e}_2+n_{22}{e}_3),n_{23}\ne 0.$
• $r_3=(n_{31}{e}_1+n_{32}{e}_2)\otimes e_3^{*}-e_3^{*}\otimes (n_{31}{e}_1+n_{32}{e}_2),n_{31}\ne 0.$
• $r_4=n_{21}{e}_1\otimes e_2^{*}+n_{31}{e}_1\otimes e_3^{*}-e_2^{*}\otimes n_{21}{e}_1-e_3^{*}\otimes n_{31}{e}_1,n_{21}\ne 0.$
• $r_5=n_{12}{e}_2\otimes e_1^{*}+n_{32}{e}_2\otimes e_3^{*}-e_1^{*}\otimes n_{12}{e}_2-e_3^{*}\otimes n_{32}{e}_2,n_{12}\ne 0.$
• $r_6=(n_{21}{e}_1-n_{31}{e}_2+n_{21}{e}_3)\otimes e_2^{*}+(n_{31}{e}_1-{\displaystyle \frac{{n_{31}}^2}{n_{21}}}{e}_2+n_{31}{e}_3)\otimes e_3^{*}-e_2^{*}\otimes (n_{21}{e}_1-n_{31}{e}_2+n_{21}{e}_3)-e_3^{*}\otimes (n_{31}{e}_1-{\displaystyle \frac{{n_{31}}^2}{n_{21}}}{e}_2+n_{31}{e}_3),n_{21}\ne 0.$

Proposition 19.

Let $\{e_1,e_2,e_3\}$ be a basis of $\mathfrak{g}\left(D_{12}\right)$, and let $\{e_1^{*},e_2^{*},e_3^{*}\}$ be its dual basis. Then the Lie algebra $\mathfrak{g}\left(D_{12}\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(D_{12}\right)}^{*}$ has a basis $\{e_1,e_2,e_3,e_1^{*},e_2^{*},e_3^{*}\}$, with the non-zero Lie brackets:

$$[e_1,e_3]=-e_1,[e_1,e_1^{*}]=e_3^{*},[e_2,e_3]=e_2,[e_2,e_2^{*}]=-e_3^{*},[e_3,e_1^{*}]=-e_1^{*},[e_3,e_2^{*}]=e_2^{*}.$$

The classical Yang–Baxter equation on this Lie algebra has solutions as follows:

• $r_1=(n_{31}{e}_1+n_{32}{e}_2)\otimes e_3^{*}-e_3^{*}\otimes (n_{31}{e}_1+n_{32}{e}_2).$
• $r_2=n_{21}{e}_1\otimes e_2^{*}+n_{31}{e}_1\otimes e_3^{*}-e_2^{*}\otimes n_{21}{e}_1-e_3^{*}\otimes n_{31}{e}_1,n_{21}\ne 0.$
• $r_3=n_{12}{e}_2\otimes e_1^{*}+n_{32}{e}_2\otimes e_3^{*}-e_1^{*}\otimes n_{12}{e}_2-e_3^{*}\otimes n_{32}{e}_2,n_{12}\ne 0.$
• $r_4=(n_{21}{e}_1-n_{33}{e}_2+n_{23}{e}_3)\otimes e_2^{*}+({\displaystyle \frac{n_{21}{n}_{33}}{n_{23}}}{e}_1-{\displaystyle \frac{{n_{33}}^2}{n_{23}}}{e}_2+n_{33}{e}_3)\otimes e_3^{*}-e_2^{*}\otimes (n_{21}{e}_1-n_{33}{e}_2+n_{23}{e}_3)-e_3^{*}\otimes ({\displaystyle \frac{n_{21}{n}_{33}}{n_{23}}}{e}_1-{\displaystyle \frac{{n_{33}}^2}{n_{23}}}{e}_2+n_{33}{e}_3),n_{23}\ne 0.$
• $r_5=(-n_{33}{e}_1+n_{12}{e}_2+n_{13}{e}_3)\otimes e_1^{*}+(-{\displaystyle \frac{{n_{33}}^2}{n_{13}}}{e}_1+{\displaystyle \frac{n_{12}{n}_{33}}{n_{13}}}{e}_2+n_{33}{e}_3)\otimes e_3^{*}-e_1^{*}\otimes (-n_{33}{e}_1+n_{12}{e}_2+n_{13}{e}_3)-e_3^{*}\otimes (-{\displaystyle \frac{{n_{33}}^2}{n_{13}}}{e}_1+{\displaystyle \frac{n_{12}{n}_{33}}{n_{13}}}{e}_2+n_{33}{e}_3),n_{13}\ne 0.$

Remark 9

(Relation to known CYBE solutions). Unlike the direct computation of Rota-Baxter operators on simple three-dimensional $\mathfrak{sl}\left(2\right)$ [31] or $\mathfrak{so}\left(3\right)$ [34], the solutions obtained in this subsection are constructed via Nijenhuis operators on three-dimensional non-commutative associative algebras $D_1$ to $D_{12}$. The sub-adjacent Lie algebras $\mathfrak{g}\left(D_1\right)$ to $\mathfrak{g}\left(D_{12}\right)$ are solvable three-dimensional Lie algebras (they are not isomorphic to the solvable 3-dimensional Lie algebra in [35]), from which we derive new Rota-Baxter operators and hence solutions to the classical Yang-Baxter equation on the corresponding semidirect products.

6. Conclusions

This work fully describes Nijenhuis operators on 2-dimensional pre-Lie algebras and 3-dimensional associative algebras, and systematically constructs the corresponding Rota-Baxter operators (from $N^2=0$) and solutions to the CYBE. We conclude with remarks on implications and future research directions.

(1) Classification suggests for higher dimensions. The systematic three-step approach demonstrated in dimensions 2 and 3 can be generalized to any finite dimension:

• Step 1: Classify the underlying high-dimensional algebras up to isomorphism (e.g., 4-dimensional nilpotent pre-Lie algebras [32]).
• Step 2: Derive equations for the coefficients satisfying the Nijenhuis operator identity using its definition. For dimensions $\ge 4$, symbolic computation combined with theoretical analysis is essential, as manual calculation becomes impractical (or solve the system of Equation (9) in Proposition 5).
• Step 3: Obtain the Rota-Baxter operator on the sub-adjacent Lie algebra from $N^2=0$. Following Section 2 in [30], the corresponding skew-symmetric solution of the Classical Yang-Baxter Equation (CYBE) on the Lie algebra $\mathfrak{g}\left(A\right){⋉}_{{\mathrm{ad}}^{*}}\mathfrak{g}{\left(A\right)}^{*}$ can be constructed.

(2) Computational verification of the Nijenhuis operators. All operator families presented in Theorems 1–4 were verified symbolically using Mathematica 14. For each Nijenhuis family, we confirmed that N satisfies the Nijenhuis equation:

$$N\left(x\right)·N\left(y\right)=N\left(N\right(x)·y+x·N(y)-N(x·y\left)\right),\forall x,y\in A.$$

The code checks all families for arbitrary parameter values and is available from the authors upon request. This computational verification prevents algebraic errors from manual calculations and ensures the correctness of the results.

(3) Future directions. Our Nijenhuis family lists are parametrically complete but do not classify up to algebra automorphisms. Several natural extensions remain open:

• A full orbit analysis under the automorphism group $\mathrm{Aut}\left(A\right)$ for each pre-Lie algebra A would simplify the Nijenhuis family lists. This is a natural but non-trivial next step.
• The Nijenhuis operators studied here can define para-complex structures on pre-Lie algebras [8], analogous to classical complex geometry. This enables the study of para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras, which may have rich geometric interpretations and connections to information geometry and statistical manifolds.
• The algebraic framework in this paper is independent of dimension and applies to infinite-dimensional pre-Lie algebras. Promising examples include those from vertex operator algebras (VOAs) in conformal field theory (Subsection 2.6, 2.7 in [36]). Such extensions could link our construction to theoretical physics and integrable systems.