约当三系上非零权的相对Rota-Baxter算子

Relative Rota-Baxter Operator of Non-Zero Weights on the Jordan Triple

摘要: 本文主要研究约当三系上关于非零权的相对Rota-Baxter算子。首先回顾了约当代数，约旦三系以及子系的概念，然后给出了约当三系的表示的定义，进而得出约当三系在另一个约当三系上作用的概念，以及约当三系上非零权的相对Rota-Baxter算子的定义，找到了在约当三系及其作用空间的直和上构造新约当三系的方法。最后，给出了在一个约当三系上定义与相对Rota-Baxter算子有关的新运算可得到新的约当三系的方法，并且构造了新的约当三系的表示。此外文中还给出了约当三系上构造相对Rota-Baxter算子的方法。

Abstract: In this paper, we mainly study the relative Rota-Baxter operators on Jordan triple systems of non-zero weights. First, we recall the concepts of Jordan algebra, Jordan triple systems, and subsystems, and then the definition of the representation of Jordan triple systems is given. Then, the concept of Jordan triple systems acting on another Jordan triple system is obtained, as well as the definition of relative Rota-Baxter operators of nonzero weights on Jordan triple systems. A method for con-structing a new Jordan triple system on the direct sum of Jordan triple systems and their action spaces is found. Finally, it is shown that by defining new operations related to the relative Ro-ta-Baxter operator on a Jordan triple system, a new Jordan triple system can be obtained, and a new representation of the Jordan triple system is constructed. In addition, a method for constructing relative Rota-Baxter operators is also presented.

1. 引言

约当三系的定义最早是1949年由Nathan Jacobson提出的 [1] ，通过约当代数可以构造出约当三系。而约当代数理论最早是有量子物理学的潜在研究所激发的，在物理，特别是量子力学中起着重要的作用。对于约当三系，可以通过约当三系给出一类对称流形的代数描述，并且，约当三系可以提供几种构造李代数的方法。对于约当三系的研究，前人已经得到了很多结果。例如，在 [2] 中作者利用TKK李代数的上同调研究了约当三系的上同调理论；在 [3] 中作者研究了约当三系的表示，证明了有限维约当三系的通用包络是有限维的，若约当三系是它自身的根基，则约当三系的通用包络是幂零的；在 [4] 中作者研究了特征不为2的域上中心非退化约当三系的非零中心及其等价形式。

相对Rota-Baxter算子是Rota-Baxter算子在结合代数上的相对推广，与O-算子有关，起源于经典Yang-Baxter方程的算子形式。在代数上，可以利用相对Rota-Baxte研究上同调理论，如，在 [5] 中作者研究了李代数上权为1的相对Rota-Baxter算子，构造了它们的上同调理论，并利用第二个上同调群来研究相对Rota-Baxter算子和r-矩阵的无穷小变形。在 [6] 中作者给出了在李三系上的相对Rota-Baxter算子以及上同调理论，并利用第一个上同调群对无穷小变形进行了分类。

2. 预备知识

定义1.1 [7] 设J为线性空间，J上有双线性的代数运算“ $\cdot$ ”： $J\times J\to J$ ，如果满足下列条件

$x\cdot y=y\cdot x$ ，

$\left(\left(x\cdot x\right)\cdot y\right)\cdot x=\left(x\cdot x\right)\cdot \left(y\cdot x\right)$ ，

$\forall x,y\in J$ ，则称 $\left(J,\cdot \right)$ 为约当代数。

定义1.2 [8] 设J为线性空间， $\left\{\cdot ,\cdot ,\cdot \right\}:J\times J\times J\to J$ 是三线性映射，如果满足下列条件

$\left\{x,y,z\right\}=\left\{z,y,x\right\}$ (1.1)

$\left\{x,y,\left\{z,u,v\right\}\right\}-\left\{z,u,\left\{x,y,v\right\}\right\}-\left\{\left\{x,y,z\right\},u,v\right\}+\left\{z,\left\{y,x,u\right\},v\right\}=0$ (1.2)

$\forall x,y,z,u,v\in J$ ，则称 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 为约当三系。

例1.1 [8] 设 $\left(J,\cdot \right)$ 为约当代数，在J上定义代数运算

$\left\{x,y,z\right\}=\left(x\cdot y\right)\cdot z+\left(z\cdot y\right)\cdot x-\left(x\cdot z\right)\cdot y$ ，

$\forall x,y,z\in J$ ，则 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 为约当三系。

定义1.3 [9] 设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系，H为J的子空间，如果H满足 $\left\{H,H,H\right\}\subseteq H$ ，则称H是J的子系。

3. 相对Rota-Baxter算子

定义2.1 设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系，V是线性空间， ${\theta }_{12},{\theta }_{13}:J\times J\to End\left(V\right)$ 是双线性映射，若 ${\theta }_{12},{\theta }_{13}$ 满足

${\theta }_{13}\left(a,b\right)={\theta }_{13}\left(b,a\right)$ ， (2.1)

${\theta }_{12}\left(x,y\right){\theta }_{12}\left(a,b\right)-{\theta }_{12}\left(a,b\right){\theta }_{12}\left(x,y\right)-{\theta }_{12}\left(\left\{x,y,a\right\},b\right)+{\theta }_{12}\left(a,\left\{y,x,b\right\}\right)=0$ ， (2.2)

${\theta }_{12}\left(x,y\right){\theta }_{13}\left(a,b\right)+{\theta }_{13}\left(a,b\right){\theta }_{12}\left(y,x\right)-{\theta }_{13}\left(a,{\left\{x,y,b\right\}}_1\right)-{\theta }_{13}\left({\left\{x,y,a\right\}}_1,b\right)=0$ ， (2.3)

${\theta }_{12}\left(\left\{x,a,b\right\},y\right)-{\theta }_{12}\left(x,a\right){\theta }_{12}\left(b,y\right)-{\theta }_{12}\left(b,a\right){\theta }_{12}\left(x,y\right)+{\theta }_{13}\left(x,b\right){\theta }_{13}\left(a,y\right)=0$ ， (2.4)

${\theta }_{13}\left(\left\{x,a,b\right\},y\right)-{\theta }_{12}\left(x,a\right){\theta }_{13}\left(b,y\right)-{\theta }_{12}\left(b,a\right){\theta }_{13}\left(x,y\right)+{\theta }_{13}\left(x,b\right){\theta }_{12}\left(a,y\right)=0$ ， (2.5)

$\forall a,b,x,y\in J$ ，则 $\left({\theta }_{12},{\theta }_{13},V\right)$ 称为J的表示。

例2.1 设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系，定义双线性映射 ${\sigma }_{12},{\sigma }_{13}:J\times J\to End\left(J\right)$ ，其中 ${\sigma }_{12}\left(x,y\right)z=\left\{x,y,z\right\}$ ， ${\sigma }_{13}\left(x,y\right)z=\left\{x,z,y\right\}\left(\forall x,y,z\in J\right)$ ，则 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是J的表示，称为伴随表示。

证：直接验证可知 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 满足(2.1)-(2.5)式，故 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是J的表示。

定义2.2 设 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系， $\left({\theta }_{12},{\theta }_{13},J_2\right)$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 的表示。若 $\forall x_1,y_1\in J_1,x_2,y_2,z_2\in J_2$ ，有

${\theta }_{12}\left(x_1,y_1\right)z_2\in C\left(J_2\right)$ ，

${\theta }_{13}\left(x_1,y_1\right)z_2\in C\left(J_2\right)$ ，

${\theta }_{12}\left(x_1,y_1\right){\left\{x_2,y_2,z_2\right\}}_2=0$ ，

${\theta }_{13}\left(x_1,y_1\right){\left\{x_2,y_2,z_2\right\}}_2=0$ ，

则称 ${\theta }_{12},{\theta }_{13}$ 为 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的一个作用。这里

$C\left(J_2\right)=\left\{u\in J_2|{\left\{u,v,w\right\}}_2=0,{\left\{v,u,w\right\}}_2=0,\forall v,w\in J_2\right\}$ 。

设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系，定义 $J^1=\left\{J,J,J\right\}$ 。

例2.2 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系，如果J满足 $J^1\subseteq C\left(J\right)$ ，则伴随表示 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是J在自身上的一个作用。

证：由例2.1可知，伴随表示 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是J的表示。因为 $J^1\subseteq C\left(J\right)$ ，所以 $\forall x_1,x_2,x_3,x_4,x_5\in J$ ，

${\sigma }_{12}\left(x_1,x_2\right)x_3=\left\{x_1,x_2,x_3\right\}\in C\left(J\right)$ ，

${\sigma }_{13}\left(x_1,x_2\right)x_3=\left\{x_1,x_3,x_2\right\}\in C\left(J\right)$ ，

${\sigma }_{12}\left(x_1,x_2\right)\left\{x_3,x_4,x_5\right\}=\left\{x_1,x_2,\left\{x_3,x_4,x_5\right\}\right\}=0$ ，

${\sigma }_{13}\left(x_1,x_2\right)\left\{x_3,x_4,x_5\right\}=\left\{x_1,\left\{x_3,x_4,x_5\right\},x_2\right\}=0$ ，

因此， ${\sigma }_{12},{\sigma }_{13}$ 是J在自身上的一个作用。

定义2.3 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系， ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用， $T:J_2\to J_1$ 是线性映射，如果T满足

${\left\{Tu,Tv,Tw\right\}}_1=T\left({\theta }_{12}\left(Tw,Tv\right)u+{\theta }_{13}\left(Tu,Tw\right)v+{\theta }_{12}\left(Tu,Tv\right)w+\lambda {\left\{u,v,w\right\}}_2\right)$ ， (2.6)

$\forall u,v,w\in J_2$ ，则T称为关于作用 ${\theta }_{12},{\theta }_{13}$ 的从 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 到 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 的权为 $\lambda$ 的相对Rota-Baxter算子。

命题2.1 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系， ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用。在 $J_1\oplus J_2$ 上定义运算 ${\left\{\cdot ,\cdot ,\cdot \right\}}_{\theta }$ ，其中

${\left\{x+u,y+v,z+w\right\}}_{\theta }={\left\{x,y,z\right\}}_1+{\theta }_{12}\left(z,y\right)u+{\theta }_{13}\left(x,z\right)v+{\theta }_{12}\left(x,y\right)w+\lambda {\left\{u,v,w\right\}}_2$ ，

$\forall x,y,z\in J_1,u,v,w\in J_2$ ，则 $\left(J_1\oplus J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_{\theta }\right)$ 是约当三系，称为 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 和 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 的半直积。

证：显然，在 $J_1\oplus J_2$ 上定义的运算对三个变量都是线性的，因此若要证 $\left(J_1\oplus J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_{\theta }\right)$ 是约当三系，只需在 $J_1\oplus J_2$ 上验证(1.1)-(1.2)式成立。 $\forall x_1,y_1,z_1,a_1,b_1\in J_1,x_2,y_2,z_2,a_2,b_2\in J_2$ ，直接计算(1.1)式左式得

$\begin{array}{l}{\left\{x_1+x_2,y_1+y_2,z_1+z_2\right\}}_{\theta }-{\left\{z_1+z_2,y_1+y_2,x_1+x_2\right\}}_{\theta }\\ ={\left\{x_1,y_1,z_1\right\}}_1+{\theta }_{13}\left(x_1,z_1\right)y_2+{\theta }_{12}\left(z_1,y_1\right)x_2+{\theta }_{12}\left(x_1,y_1\right)z_2+\lambda {\left\{x_2,y_2,z_2\right\}}_2\\ -{\left\{z_1,y_1,x_1\right\}}_1-{\theta }_{13}\left(z_1,x_1\right)y_2-{\theta }_{12}\left(x_1,y_1\right)z_2-{\theta }_{12}\left(z_1,y_1\right)x_2-\lambda {\left\{z_2,y_2,x_2\right\}}_2\end{array}$ ，

由于 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系且 ${\theta }_{13}$ 是对称的，故在 $J_1\oplus J_2$ 上(1.1)式成立。

$\begin{array}{l}{\left\{x_1+x_2,y_1+y_2,{\left\{z_1+z_2,a_1+a_2,b_1+b_2\right\}}_{\theta }\right\}}_{\theta }-{\left\{z_1+z_2,a_1+a_2,{\left\{x_1+x_2,y_1+y_2,b_1+b_2\right\}}_{\theta }\right\}}_{\theta }\\ -{\left\{{\left\{x_1+x_2,y_1+y_2,z_1+z_2\right\}}_{\theta },a_1+a_2,b_1+b_2\right\}}_{\theta }+{\left\{z_1+z_2,{\left\{y_1+y_2,x_1+x_2,a_1+a_2\right\}}_{\theta },b_1+b_2\right\}}_{\theta }\\ ={\left\{x_1+x_2,y_1+y_2,{\left\{z_1,a_1,b_1\right\}}_1+{\theta }_{13}\left(z_1,b_1\right)a_2+{\theta }_{12}\left(b_1,a_1\right)z_2+{\theta }_{12}\left(z_1,a_1\right)b_2+\lambda {\left\{z_2,a_2,b_2\right\}}_2\right\}}_{\theta }\\ -{\left\{z_1+z_2,a_1+a_2,{\left\{x_1,y_1,b_1\right\}}_1+{\theta }_{13}\left(x_1,b_1\right)y_2+{\theta }_{12}\left(b_1,y_1\right)x_2+{\theta }_{12}\left(x_1,y_1\right)b_2+\lambda {\left\{x_2,y_2,b_2\right\}}_2\right\}}_{\theta }\\ -{\left\{{\left\{x_1,y_1,z_1\right\}}_1+{\theta }_{13}\left(x_1,z_1\right)y_2+{\theta }_{12}\left(z_1,y_1\right)x_2+{\theta }_{12}\left(x_1,y_1\right)z_2+\lambda {\left\{x_2,y_2,z_2\right\}}_2,a_1+a_2,b_1+b_2\right\}}_{\theta }\end{array}$

$\begin{array}{l}+{\left\{z_1+z_2,{\left\{y_1,x_1,a_1\right\}}_1+{\theta }_{13}\left(y_1,a_1\right)x_2+{\theta }_{12}\left(a_1,x_1\right)y_2+{\theta }_{12}\left(y_1,x_1\right)a_2+\lambda {\left\{y_2,x_2,a_2\right\}}_2,b_1+b_2\right\}}_{\theta }\\ ={\left\{x_1,y_1,{\left\{z_1,a_1,b_1\right\}}_1\right\}}_1+{\theta }_{13}\left(x_1,{\left\{z_1,a_1,b_1\right\}}_1\right)y_2+{\theta }_{12}\left({\left\{z_1,a_1,b_1\right\}}_1,y_1\right)x_2+{\theta }_{12}\left(x_1,y_1\right){\theta }_{13}\left(z_1,b_1\right)a_2\\ +\lambda {\left\{x_2,y_2,{\theta }_{13}\left(z_1,b_1\right)a_2\right\}}_2+{\theta }_{12}\left(x_1,y_1\right){\theta }_{12}\left(b_1,a_1\right)z_2+\lambda {\left\{x_2,y_2,{\theta }_{12}\left(b_1,a_1\right)z_2\right\}}_2\\ +{\theta }_{12}\left(x_1,y_1\right){\theta }_{12}\left(z_1,a_1\right)b_2+\lambda {\left\{x_2,y_2,{\theta }_{12}\left(z_1,a_1\right)b_2\right\}}_2+\lambda {\theta }_{12}\left(x_1,y_1\right){\left\{z_2,a_2,b_2\right\}}_2\\ +\lambda {\left\{x_2,y_2,\lambda {\left\{z_2,a_2,b_2\right\}}_2\right\}}_2-{\left\{z_1,a_1,{\left\{x_1,y_1,b_1\right\}}_1\right\}}_1-{\theta }_{13}\left(z_1,{\left\{x_1,y_1,b_1\right\}}_1\right)a_2\end{array}$

$\begin{array}{l}-{\theta }_{12}\left({\left\{x_1,y_1,b_1\right\}}_1,a_1\right)z_2-{\theta }_{12}\left(z_1,a_1\right){\theta }_{13}\left(x_1,b_1\right)y_2-\lambda {\left\{z_2,a_2,{\theta }_{13}\left(x_1,b_1\right)y_2\right\}}_2\\ -{\theta }_{12}\left(z_1,a_1\right){\theta }_{12}\left(b_1,y_1\right)x_2-\lambda {\left\{z_2,a_2,{\theta }_{12}\left(b_1,y_1\right)x_2\right\}}_2-{\theta }_{12}\left(z_1,a_1\right){\theta }_{12}\left(x_1,y_1\right)b_2\\ -\lambda {\left\{z_2,a_2,{\theta }_{12}\left(x_1,y_1\right)b_2\right\}}_2-\lambda {\theta }_{12}\left(z_1,a_1\right){\left\{x_2,y_2,b_2\right\}}_2-\lambda {\left\{z_2,a_2,\lambda {\left\{x_2,y_2,b_2\right\}}_2\right\}}_2\\ -{\left\{{\left\{x_1,y_1,z_1\right\}}_1,a_1,b_1\right\}}_1-{\theta }_{13}\left({\left\{x_1,y_1,z_1\right\}}_1,b_1\right)a_2-{\theta }_{12}\left({\left\{x_1,y_1,z_1\right\}}_1,a_1\right)b_2\\ -{\theta }_{12}\left(b_1,a_1\right){\theta }_{13}\left(x_1,z_1\right)y_2-\lambda {\left\{{\theta }_{13}\left(x_1,z_1\right)y_2,a_2,b_2\right\}}_2-{\theta }_{12}\left(b_1,a_1\right){\theta }_{12}\left(z_1,y_1\right)x_2\end{array}$

$\begin{array}{l}-\lambda {\left\{{\theta }_{12}\left(z_1,y_1\right)x_2,a_2,b_2\right\}}_2-{\theta }_{12}\left(b_1,a_1\right){\theta }_{12}\left(x_1,y_1\right)z_2-\lambda {\left\{{\theta }_{12}\left(x_1,y_1\right)z_2,a_2,b_2\right\}}_2\\ -\lambda {\theta }_{12}\left(b_1,a_1\right){\left\{x_2,y_2,z_2\right\}}_2-\lambda {\left\{\lambda {\left\{x_2,y_2,z_2\right\}}_2,a_2,b_2\right\}}_2+{\left\{z_1,{\left\{y_1,x_1,a_1\right\}}_1,b_1\right\}}_1\\ +{\theta }_{12}\left(b_1,{\left\{y_1,x_1,a_1\right\}}_1\right)z_2+{\theta }_{12}\left(z_1,{\left\{y_1,x_1,a_1\right\}}_1\right)b_2+{\theta }_{13}\left(z_1,b_1\right){\theta }_{13}\left(y_1,a_1\right)x_2\\ +\lambda {\left\{z_2,{\theta }_{13}\left(y_1,a_1\right)x_2,b_2\right\}}_2+{\theta }_{13}\left(z_1,b_1\right){\theta }_{12}\left(a_1,x_1\right)y_2+\lambda {\left\{z_2,{\theta }_{12}\left(a_1,x_1\right),b_2\right\}}_2\\ +{\theta }_{13}\left(z_1,b_1\right){\theta }_{12}\left(y_1,x_1\right)a_2+\lambda {\left\{z_2,{\theta }_{12}\left(y_1,x_1\right)a_2,b_2\right\}}_2+\lambda {\theta }_{13}\left(z_1,b_1\right){\left\{y_2,x_2,a_2\right\}}_2+\lambda {\left\{z_2,\lambda {\left\{y_2,x_2,a_2\right\}}_2,b_2\right\}}_2\end{array}$

$\begin{array}{l}=\left({\left\{x_1,y_1,{\left\{z_1,a_1,b_1\right\}}_1\right\}}_1-{\left\{z_1,a_1,{\left\{x_1,y_1,b_1\right\}}_1\right\}}_1-{\left\{{\left\{x_1,y_1,z_1\right\}}_1,a_1,b_1\right\}}_1+{\left\{z_1,{\left\{y_1,x_1,a_1\right\}}_1,b_1\right\}}_1\right)\\ +{\lambda }^2({\left\{x_2,y_2,{\left\{z_2,a_2,b_2\right\}}_2\right\}}_2-{\left\{z_2,a_2,{\left\{x_2,y_2,b_2\right\}}_2\right\}}_2-{\left\{{\left\{x_2,y_2,z_2\right\}}_2,a_2,b_2\right\}}_2\\ +{\left\{z_2,{\left\{y_2,x_2,a_2\right\}}_2,b_2\right\}}_2)+A_1{x}_2+A_2{y}_2+A_3{z}_2+A_4{a}_2+A_5{b}_2+\lambda A_6\end{array}$

其中，

$A_1={\theta }_{12}\left({\left\{z_1,a_1,b_1\right\}}_1,y_1\right)-{\theta }_{12}\left(z_1,a_1\right){\theta }_{12}\left(b_1,y_1\right)-{\theta }_{12}\left(b_1,a_1\right){\theta }_{12}\left(z_1,y_1\right)+{\theta }_{13}\left(z_1,b_1\right){\theta }_{13}\left(y_1,a_1\right)$ ，

$A_2={\theta }_{13}\left(x_1,{\left\{z_1,a_1,b_1\right\}}_1\right)-{\theta }_{12}\left(z_1,a_1\right){\theta }_{13}\left(x_1,b_1\right)-{\theta }_{12}\left(b_1,a_1\right){\theta }_{13}\left(x_1,z_1\right)+{\theta }_{13}\left(z_1,b_1\right){\theta }_{12}\left(a_1,x_1\right)$ ，

$A_3={\theta }_{12}\left(x_1,y_1\right){\theta }_{12}\left(b_1,a_1\right)-{\theta }_{12}\left({\left\{x_1,y_1,b_1\right\}}_1,a_1\right)-{\theta }_{12}\left(b_1,a_1\right){\theta }_{12}\left(x_1,y_1\right)+{\theta }_{12}\left(b_1,{\left\{y_1,x_1,a_1\right\}}_1\right)$ ，

$A_4={\theta }_{12}\left(x_1,y_1\right){\theta }_{13}\left(z_1,b_1\right)-{\theta }_{13}\left(z_1,{\left\{x_1,y_1,b_1\right\}}_1\right)-{\theta }_{13}\left({\left\{x_1,y_1,z_1\right\}}_1,b_1\right)+{\theta }_{13}\left(z_1,b_1\right){\theta }_{12}\left(y_1,x_1\right)$ ，

$A_5={\theta }_{12}\left(x_1,y_1\right){\theta }_{12}\left(z_1,a_1\right)-{\theta }_{12}\left(z_1,a_1\right){\theta }_{12}\left(x_1,y_1\right)-{\theta }_{12}\left({\left\{x_1,y_1,z_1\right\}}_1,a_1\right)+{\theta }_{12}\left(z_1,{\left\{y_1,x_1,a_1\right\}}_1\right)$ ，

$\begin{array}{c}{A}_6={\left\{x_2,y_2,{\theta }_{13}\left(z_1,b_1\right)a_2\right\}}_2+{\left\{x_2,y_2,{\theta }_{12}\left(b_1,a_1\right)z_2\right\}}_2+{\left\{x_2,y_2,{\theta }_{12}\left(z_1,a_1\right)b_2\right\}}_2+{\theta }_{12}\left(x_1,y_1\right){\left\{z_2,a_2,b_2\right\}}_2\\ -{\left\{z_2,a_2,{\theta }_{13}\left(x_1,b_1\right)y_2\right\}}_2-{\left\{z_2,a_2,{\theta }_{12}\left(b_1,y_1\right)x_2\right\}}_2-{\left\{z_2,a_2,{\theta }_{12}\left(x_1,y_1\right)b_2\right\}}_2-{\theta }_{12}\left(z_1,a_1\right){\left\{x_2,y_2,b_2\right\}}_2\\ -{\left\{{\theta }_{13}\left(x_1,z_1\right)y_2,a_2,b_2\right\}}_2-{\left\{{\theta }_{12}\left(z_1,y_1\right)x_2,a_2,b_2\right\}}_2-{\left\{{\theta }_{12}\left(x_1,y_1\right)z_2,a_2,b_2\right\}}_2-{\theta }_{12}\left(b_1,a_1\right){\left\{x_2,y_2,z_2\right\}}_2\\ +{\left\{z_2,{\theta }_{13}\left(y_1,a_1\right)x_2,b_2\right\}}_2+{\left\{z_2,{\theta }_{12}\left(a_1,x_1\right)y_2,b_2\right\}}_2+{\left\{z_2,{\theta }_{12}\left(y_1,x_1\right)a_2,b_2\right\}}_2+{\theta }_{13}\left(z_1,b_1\right){\left\{y_2,x_2,a_2\right\}}_2\end{array}$ ，

由 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系知(1.2)式成立，由(2.4)式成立知A₁为零，由(2.5)式成立知A₂为零，由(2.2)式成立知A₃为零，由(2.3)式成立知A₄为零，由(2.2)式成立知A₅为零。由于 ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，故有 ${\left\{x_2,y_2,{\theta }_{13}\left(z_1,b_1\right)a_2\right\}}_2=0$ ， ${\theta }_{12}\left(y_1,x_1\right){\left\{z_2,a_2,b_2\right\}}_2=0$ ，类似可知A₆为零。所以在 $J_1\oplus J_2$ 上(1.2)式成立。综上， $\left(J_1\oplus J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_{\theta }\right)$ 是一个约当三系。

定理2.1 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系， ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，则线性映射 $T:J_2\to J_1$ 是关于作用 ${\theta }_{12},{\theta }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子当且仅当图 $Gr\left(T\right)=\left\{Tu+u|u\in J_2\right\}$ 是 $\left(J_1\oplus J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_{\theta }\right)$ 的子系。

证明：由于 $T:J_2\to J_1$ 是线性映射， $\forall u,v,w\in J_2$ ，由命题2.1有

${\left\{Tu+u,Tv+v,Tw+w\right\}}_{\theta }={\left\{Tu,Tv,Tw\right\}}_1+{\theta }_{13}\left(Tu,Tw\right)v+{\theta }_{12}\left(Tw,Tv\right)u+{\theta }_{12}\left(Tu,Tv\right)w+\lambda {\left\{u,v,w\right\}}_2$ ，

这表明，图 $Gr\left(T\right)$ 是 $\left(J_1\oplus J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_{\theta }\right)$ 的子系当且仅当T满足(2.6)式，所以结论成立。

4. 相对Rota-Baxter算子的应用

命题4.1 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系，线性映射 $T:J_2\to J_1$ 是关于作用 ${\theta }_{12},{\theta }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子，在 $J_2$ 上定义新运算 ${\left\{\cdot ,\cdot ,\cdot \right\}}_T$ ，其中

${\left\{u,v,w\right\}}_T={\theta }_{12}\left(Tw,Tv\right)u+{\theta }_{13}\left(Tu,Tw\right)v+{\theta }_{12}\left(Tu,Tv\right)w+\lambda {\left\{u,v,w\right\}}_2$ ，

$\forall u,v,w\in J_2$ ，则 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 也是约当三系，并且 是从约当三系 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 到约当三系 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 的同态。

证明：显然，在 $J_2$ 上定义的新代数运算 ${\left\{\cdot ,\cdot ,\cdot \right\}}_T$ 对三个变量都是线性的，若要证明 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 是一个约当三系，只需在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 上验证(1.1)~(1.2)式成立。 $\forall u_1,u_2,u_3,u_4,u_5\in J_2$ ，

$\begin{array}{l}{\left\{u_1,u_2,u_3\right\}}_T-{\left\{u_3,u_2,u_1\right\}}_T\\ ={\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1+{\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2+{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3+\lambda {\left\{u_1,u_2,u_3\right\}}_2\\ -{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3-{\theta }_{13}\left(T{u}_3,T{u}_1\right)u_2-{\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1-\lambda {\left\{u_3,u_2,u_1\right\}}_2\end{array}$ ，

由于 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系且 ${\theta }_{13}$ 是对称的，所以在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 上(1.1)式成立。

$\begin{array}{l}{\left\{u_1,u_2,{\left\{u_3,u_4,u_5\right\}}_T\right\}}_T-{\left\{u_3,u_4,{\left\{u_1,u_2,u_5\right\}}_T\right\}}_T-{\left\{{\left\{u_1,u_2,u_3\right\}}_T,u_4,u_5\right\}}_T+{\left\{u_3,{\left\{u_2,u_1,u_4\right\}}_T,u_5\right\}}_T\\ ={\left\{u_1,u_2,{\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3+{\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4+{\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5+\lambda {\left\{u_3,u_4,u_5\right\}}_2\right\}}_T\\ -{\left\{u_3,u_4,{\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1+{\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2+{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5+\lambda {\left\{u_1,u_2,u_5\right\}}_2\right\}}_T\\ -{\left\{{\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1+{\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2+{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3+\lambda {\left\{u_1,u_2,u_3\right\}}_2,u_4,u_5\right\}}_T\\ +{\left\{u_3,{\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2+{\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1+{\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4+\lambda {\left\{u_2,u_1,u_4\right\}}_2,u_5\right\}}_T\end{array}$

$\begin{array}{l}={\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right),T{u}_2\right)u_1+{\theta }_{13}\left(T{u}_1,T\left({\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right)\right)u_2+{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\\ +\lambda {\left\{u_1,u_2,{\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right\}}_2+{\theta }_{12}\left(T\left({\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4\right),T{u}_2\right)u_1+{\theta }_{13}\left(T{u}_1,T\left({\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4\right)\right)u_2\\ +{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4+\lambda {\left\{u_1,u_2,{\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4\right\}}_2+{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5\right),T{u}_2\right)u_1\\ +{\theta }_{13}\left(T{u}_1,T\left({\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5\right)\right)u_2+{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5+\lambda {\left\{u_1,u_2,{\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5\right\}}_2\\ +{\theta }_{12}\left(T\left(\lambda {\left\{u_3,u_4,u_5\right\}}_2\right),T{u}_2\right)u_1+{\theta }_{13}\left(T{u}_1,T\left(\lambda {\left\{u_3,u_4,u_5\right\}}_2\right)\right)u_2+\lambda {\theta }_{12}\left(T{u}_1,T{u}_2\right){\left\{u_3,u_4,u_5\right\}}_2\end{array}$

$\begin{array}{l}+\lambda {\left\{u_1,u_2,\lambda {\left\{u_3,u_4,u_5\right\}}_2\right\}}_2-{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1\right),T{u}_4\right)u_3-{\theta }_{13}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1\right)\right)u_4\\ -{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1-\lambda {\left\{u_3,u_4,{\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1\right\}}_2-{\theta }_{12}\left(T\left({\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2\right),T{u}_4\right)u_3\\ -{\theta }_{13}\left(T{u}_3,T\left({\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2\right)\right)u_4-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2-\lambda {\left\{u_3,u_4,{\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2\right\}}_2\\ -{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\right),T{u}_4\right)u_3-{\theta }_{13}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\right)\right)u_4-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\\ -\lambda {\left\{u_3,u_4,{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\right\}}_2-{\theta }_{12}\left(T\left(\lambda {\left\{u_1,u_2,u_5\right\}}_2\right),T{u}_4\right)u_3-{\theta }_{13}\left(T{u}_3,T\left(\lambda {\left\{u_1,u_2,u_5\right\}}_2\right)\right)u_4\end{array}$

$\begin{array}{l}-\lambda {\theta }_{12}\left(T{u}_3,T{u}_4\right){\left\{u_1,u_2,u_5\right\}}_2-\lambda {\left\{u_3,u_4,\lambda {\left\{u_1,u_2,u_5\right\}}_2\right\}}_2-{\theta }_{12}\left(T{u}_5,T{u}_4\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\\ -{\theta }_{13}\left(T\left({\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\right),T{u}_5\right)u_4-{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\right),T{u}_4\right)u_5-\lambda {\left\{{\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1,u_4,u_5\right\}}_2\\ -{\theta }_{12}\left(T{u}_5,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2-{\theta }_{13}\left(T\left({\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2\right),T{u}_5\right)u_4-{\theta }_{12}\left(T\left({\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2\right),T{u}_4\right)u_5\\ -\lambda {\left\{{\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2,u_4,u_5\right\}}_2-{\theta }_{12}\left(T{u}_5,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3-{\theta }_{13}\left(T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3\right),T{u}_5\right)u_4\\ -{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3\right),T{u}_4\right)u_5-\lambda {\left\{{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3,u_4,u_5\right\}}_2-\lambda {\theta }_{12}\left(T{u}_5,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2\end{array}$

$\begin{array}{l}-{\theta }_{13}\left(T\left(\lambda {\left\{u_1,u_2,u_3\right\}}_2\right),T{u}_5\right)u_4-{\theta }_{12}\left(T\left(\lambda {\left\{u_1,u_2,u_3\right\}}_2\right),T{u}_4\right)u_5-\lambda {\left\{\lambda {\left\{u_1,u_2,u_3\right\}}_2,u_4,u_5\right\}}_2\\ +{\theta }_{12}\left(T{u}_5,T\left({\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2\right)\right)u_3+{\theta }_{13}\left(T{u}_3,T{u}_5\right){\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2+{\theta }_{12}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2\right)\right)u_5\\ +\lambda {\left\{u_3,{\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2,u_5\right\}}_2+{\theta }_{12}\left(T{u}_5,T\left({\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1\right)\right)u_3+{\theta }_{13}\left(T{u}_3,T{u}_5\right){\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1\\ +{\theta }_{12}\left(T{u}_3,T\left({\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1\right)\right)u_5+\lambda {\left\{u_3,{\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1,u_5\right\}}_2+{\theta }_{12}\left(T{u}_5,T\left({\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4\right)\right)u_3\\ +{\theta }_{13}\left(T{u}_3,T{u}_5\right){\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4+{\theta }_{12}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4\right)\right)u_5+\lambda {\left\{u_3,{\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4,u_5\right\}}_2\end{array}$

$\begin{array}{l}+{\theta }_{12}\left(T{u}_5,T\left(\lambda {\left\{u_2,u_1,u_4\right\}}_2\right)\right)u_3+\lambda {\theta }_{13}\left(T{u}_3,T{u}_5\right){\left\{u_2,u_1,u_4\right\}}_2+{\theta }_{12}\left(T{u}_3,T\left(\lambda {\left\{u_2,u_1,u_4\right\}}_2\right)\right)u_5\\ +\lambda {\left\{u_3,\lambda {\left\{u_2,u_1,u_4\right\}}_2,u_5\right\}}_2\\ ={\lambda }^2({\left\{u_1,u_2,{\left\{u_3,u_4,u_5\right\}}_2\right\}}_2-{\left\{u_3,u_4,{\left\{u_1,u_2,u_5\right\}}_2\right\}}_2-{\left\{{\left\{u_1,u_2,u_3\right\}}_2,u_4,u_5\right\}}_2+{\left\{u_3,{\left\{u_2,u_1,u_4\right\}}_2,u_5\right\}}_2)\\ +B_1{u}_1+B_2{u}_2+B_3{u}_3+B_4{u}_4+B_5{u}_5+\lambda B_6\end{array}$

其中，

$\begin{array}{c}{B}_1={\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right),T{u}_2\right)+{\theta }_{12}\left(T\left({\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4\right),T{u}_2\right)+{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5\right),T{u}_2\right)\\ +{\theta }_{12}\left(T\left(\lambda {\left\{u_3,u_4,u_5\right\}}_2\right),T{u}_2\right)-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{12}\left(T{u}_5,T{u}_2\right)-{\theta }_{12}\left(T{u}_5,T{u}_4\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)\\ +{\theta }_{13}\left(T{u}_3,T{u}_5\right){\theta }_{13}\left(T{u}_2,T{u}_4\right)\end{array}$ ，

$\begin{array}{c}{B}_2={\theta }_{13}\left(T{u}_1,T\left({\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right)\right)+{\theta }_{13}\left(T{u}_1,T\left({\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4\right)\right)+{\theta }_{13}\left(T{u}_1,T\left({\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5\right)\right)\\ +{\theta }_{13}\left(T{u}_1,T\left(\lambda {\left\{u_3,u_4,u_5\right\}}_2\right)\right)-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_5\right)-{\theta }_{12}\left(T{u}_5,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)\\ +{\theta }_{13}\left(T{u}_3,T{u}_5\right){\theta }_{12}\left(T{u}_4,T{u}_1\right)\end{array}$ ，

$\begin{array}{c}{B}_3={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_5,T{u}_4\right)-{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1\right),T{u}_4\right)-{\theta }_{12}\left(T\left({\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2\right),T{u}_4\right)\\ -{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\right),T{u}_4\right)-{\theta }_{12}\left(T\left(\lambda {\left\{u_1,u_2,u_5\right\}}_2\right),T{u}_4\right)-{\theta }_{12}\left(T{u}_5,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)\\ +{\theta }_{12}\left(T{u}_5,T\left({\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2\right)\right)+{\theta }_{12}\left(T{u}_5,T\left({\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1\right)\right)+{\theta }_{12}\left(T{u}_5,T\left({\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4\right)\right)\\ +{\theta }_{12}\left(T{u}_5,T\left(\lambda {\left\{u_2,u_1,u_4\right\}}_2\right)\right)\end{array}$ ，

$\begin{array}{c}{B}_4={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{13}\left(T{u}_3,T{u}_5\right)-{\theta }_{13}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1\right)\right)-{\theta }_{13}\left(T{u}_3,T\left({\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2\right)\right)\\ -{\theta }_{13}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\right)\right)-{\theta }_{13}\left(T{u}_3,T\left(\lambda {\left\{u_1,u_2,u_5\right\}}_2\right)\right)-{\theta }_{13}\left(T\left({\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\right),T{u}_5\right)\\ -{\theta }_{13}\left(T\left({\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2\right),T{u}_5\right)-{\theta }_{13}\left(T({\theta }_{12}(T{u}_1,T{u}_2)u_3),T{u}_5\right)-{\theta }_{13}\left(T\left(\lambda {\left\{u_1,u_2,u_3\right\}}_2\right),T{u}_5\right)\\ +{\theta }_{13}\left(T{u}_3,T{u}_5\right){\theta }_{12}\left(T{u}_2,T{u}_1\right)\end{array}$ ，

$\begin{array}{c}{B}_5={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_3,T{u}_4\right)-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)-{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\right),T{u}_4\right)\\ -{\theta }_{12}\left(T\left({\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2\right),T{u}_4\right)-{\theta }_{12}\left(T\left({\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3\right),T{u}_4\right)-{\theta }_{12}\left(T\left(\lambda {\left\{u_1,u_2,u_3\right\}}_2\right),T{u}_4\right)\\ +{\theta }_{12}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2\right)\right)+{\theta }_{12}\left(T{u}_3,T\left({\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1\right)\right)+{\theta }_{12}\left(T{u}_3,T\left({\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4\right)\right)\\ +{\theta }_{12}\left(T{u}_3,T\left(\lambda {\left\{u_2,u_1,u_4\right\}}_2\right)\right)\end{array}$ ，

$\begin{array}{c}{B}_6={\left\{u_1,u_2,{\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right\}}_2+{\left\{u_1,u_2,{\theta }_{13}\left(T{u}_3,T{u}_5\right)u_4\right\}}_2+{\left\{u_1,u_2,{\theta }_{12}\left(T{u}_3,T{u}_4\right)u_5\right\}}_2\\ +{\theta }_{12}\left(T{u}_1,T{u}_2\right){\left\{u_3,u_4,u_5\right\}}_2-{\left\{u_3,u_4,{\theta }_{12}\left(T{u}_5,T{u}_2\right)u_1\right\}}_2-{\left\{u_3,u_4,{\theta }_{13}\left(T{u}_1,T{u}_5\right)u_2\right\}}_2\\ -{\left\{u_3,u_4,{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_5\right\}}_2-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\left\{u_1,u_2,u_5\right\}}_2-{\left\{{\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1,u_4,u_5\right\}}_2\\ -{\left\{{\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2,u_4,u_5\right\}}_2-{\left\{{\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3,u_4,u_5\right\}}_2-{\theta }_{12}\left(T{u}_5,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2\\ +{\left\{u_3,{\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2,u_5\right\}}_2+{\left\{u_3,{\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1,u_5\right\}}_2+{\left\{u_3,{\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4,u_5\right\}}_2\\ +{\theta }_{13}\left(T{u}_3,T{u}_5\right){\left\{u_2,u_1,u_4\right\}}_2\end{array}$ ，

由 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系知(1.2)式成立，由(2.4)式成立知B₁为零，由(2.5)式成立知B₂为零，由(2.2)式成立知B₃为零，由(2.3)式成立知B₄为零，由(2.2)式成立知B₅为零，由于 ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，故有 ${\left\{u_1,u_2,{\theta }_{12}\left(T{u}_5,T{u}_4\right)u_3\right\}}_2=0$ ， ${\theta }_{12}\left(T{u}_1,T{u}_2\right){\left\{u_3,u_4,u_5\right\}}_2=0$ ，类似可知B₆为零。故在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 上(1.2)式成立。综上， $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 也是约当三系。

下证T是从 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 到 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 的同态。由于线性映射 $T:J_2\to J_1$ 是关于作用 ${\theta }_{12},{\theta }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子，所以，

$T\left({\left\{u,v,w\right\}}_T\right)=T\left({\theta }_{12}\left(Tw,Tv\right)u+{\theta }_{13}\left(Tu,Tw\right)v+{\theta }_{12}\left(Tu,Tv\right)w+\lambda {\left\{u,v,w\right\}}_2\right)={\left\{Tu,Tv,Tw\right\}}_1$ ，

$\forall u,v,w\in J_2$ ，因此，T是从 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 到 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 的代数同态。

命题3.2 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right),\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 是约当三系，线性映射 $T:J_2\to J_1$ 是关于作用 ${\theta }_{12},{\theta }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子，定义双线性映射 $L_T,M_T:J_2\times J_2\to End\left(J_1\right)$ ，其中

$L_T\left(u,v\right)x={\left\{Tu,Tv,x\right\}}_1-T\left({\theta }_{13}\left(Tu,x\right)v+{\theta }_{12}\left(x,Tv\right)u\right)$ ，

$M_T\left(u,v\right)x={\left\{Tu,x,Tv\right\}}_1-T\left({\theta }_{12}\left(Tv,x\right)u+{\theta }_{12}\left(Tu,x\right)v\right)$ ，

$x\in J_1,u,v\in J_2$ ，则 $\left(L_T,M_T,J_1\right)$ 是 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 的表示。

证：显然， $L_T,M_T$ 是线性映射， $\forall u_1,u_2,u_3,u_4\in J_2,x\in J_1$ ，计算(2.1)式左边可得

$\begin{array}{l}\left(M_T\left(u_1,u_2\right)-M_T\left(u_2,u_1\right)\right)x\\ ={\left\{T{u}_1,x,T{u}_2\right\}}_1-T\left({\theta }_{12}\left(T{u}_2,x\right)u_1+{\theta }_{12}\left(T{u}_1,x\right)u_2\right)-{\left\{T{u}_2,x,T{u}_1\right\}}_1+T\left({\theta }_{12}\left(T{u}_1,x\right)u_2+{\theta }_{12}\left(T{u}_2,x\right)u_1\right)\end{array}$ ，

由于 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 是约当三系，故(2.1)式成立。

计算(2.2)式左边可得

$\begin{array}{l}\left(L_T\left(u_1,u_2\right)L_T\left(u_3,u_4\right)-L_T\left(u_3,u_4\right)L_T\left(u_1,u_2\right)-L_T\left({\left\{u_1,u_2,u_3\right\}}_T,u_4\right)+L_T\left(u_3,{\left\{u_2,u_1,u_4\right\}}_T\right)\right)x\\ =L_T\left(u_1,u_2\right){\left\{T{u}_3,T{u}_4,x\right\}}_1-L_T\left(u_1,u_2\right)T{\theta }_{12}\left(x,T{u}_4\right)u_3-L_T\left(u_1,u_2\right)T{\theta }_{13}\left(T{u}_3,x\right)u_4\\ -L_T\left(u_3,u_4\right){\left\{T{u}_1,T{u}_2,x\right\}}_1+L_T\left(u_3,u_4\right)T{\theta }_{12}\left(x,T{u}_2\right)u_1+L_T\left(u_3,u_4\right)T{\theta }_{13}\left(T{u}_1,x\right)u_2\\ -{\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,T{u}_4,x\right\}}_1+T{\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_T+T{\theta }_{13}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4\\ +{\left\{T{u}_3,{\left\{T{u}_2,T{u}_1,T{u}_4\right\}}_1,x\right\}}_1-T{\theta }_{12}\left(x,{\left\{T{u}_2,T{u}_1,T{u}_4\right\}}_1\right)u_3-T{\theta }_{13}\left(T{u}_3,x\right){\left\{u_2,u_1,u_4\right\}}_T\end{array}$

$\begin{array}{l}={\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right\}}_1-T{\theta }_{12}\left({\left\{T{u}_3,T{u}_4,x\right\}}_1,T{u}_2\right)u_1-T{\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right)u_2\\ -{\left\{T{u}_1,T{u}_2,T{\theta }_{12}\left(x,T{u}_4\right)u_3\right\}}_1+T{\theta }_{12}\left(T{\theta }_{12}\left(x,T{u}_4\right)u_3,T{u}_2\right)u_1+T{\theta }_{13}\left(T{u}_1,T{\theta }_{12}\left(x,T{u}_4\right)u_3\right)u_2\\ -{\left\{T{u}_1,T{u}_2,T{\theta }_{13}\left(T{u}_3,x\right)u_4\right\}}_1+T{\theta }_{12}\left(T{\theta }_{13}\left(T{u}_3,x\right)u_4,T{u}_2\right)u_1+T{\theta }_{13}\left(T{u}_1,T{\theta }_{13}\left(T{u}_3,x\right)u_4\right)u_2\\ -{\left\{T{u}_3,T{u}_4,{\left\{T{u}_1,T{u}_2,x\right\}}_1\right\}}_1+T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,x\right\}}_1,T{u}_4\right)u_3+T{\theta }_{13}\left(T{u}_3,{\left\{T{u}_1,T{u}_2,x\right\}}_1\right)u_4\\ +{\left\{T{u}_3,T{u}_4,T{\theta }_{12}\left(x,T{u}_2\right)u_1\right\}}_1-T{\theta }_{12}\left(T{\theta }_{12}\left(x,T{u}_2\right)u_1,T{u}_4\right)u_3-T{\theta }_{13}\left(T{u}_3,T{\theta }_{12}\left(x,T{u}_2\right)u_1\right)u_4\end{array}$

$\begin{array}{l}+{\left\{T{u}_3,T{u}_4,T{\theta }_{13}\left(T{u}_1,x\right)u_2\right\}}_1-T{\theta }_{12}\left(T{\theta }_{13}\left(T{u}_1,x\right)u_2,T{u}_4\right)u_3-T{\theta }_{13}\left(T{u}_3,T{\theta }_{13}\left(T{u}_1,x\right)u_2\right)u_4\\ -{\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,T{u}_4,x\right\}}_1+T{\theta }_{13}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4+T{\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\\ +T{\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3+T{\theta }_{12}\left(x,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2+\lambda T{\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2\\ +{\left\{T{u}_3,{\left\{T{u}_2,T{u}_1,T{u}_4\right\}}_1,x\right\}}_1-T{\theta }_{12}\left(x,{\left\{T{u}_2,T{u}_1,T{u}_4\right\}}_1\right)u_3-T{\theta }_{13}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_4,T{u}_1\right)u_2\\ -T{\theta }_{13}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_2,T{u}_1\right)u_4-T{\theta }_{13}\left(T{u}_3,x\right){\theta }_{13}\left(T{u}_2,T{u}_4\right)u_1-\lambda T{\theta }_{13}\left(T{u}_3,x\right){\left\{u_2,u_1,u_4\right\}}_2\end{array}$

$\begin{array}{l}=({\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right\}}_1-{\left\{T{u}_3,T{u}_4,{\left\{T{u}_1,T{u}_2,x\right\}}_1\right\}}_1-{\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,T{u}_4,x\right\}}_1\\ +{\left\{T{u}_3,{\left\{T{u}_2,T{u}_1,T{u}_4\right\}}_1,x\right\}}_1)-T\left(C_{11}{u}_1+C_{12}{u}_2+C_{13}{u}_3+C_{14}{u}_4+\lambda C_{15}\right)\end{array}$

其中，

$C_{11}={\theta }_{12}\left({\left\{T{u}_3,T{u}_4,x\right\}}_1,T{u}_2\right)-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{12}\left(x,T{u}_2\right)-{\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)+{\theta }_{13}\left(T{u}_3,x\right){\theta }_{13}\left(T{u}_2,T{u}_4\right)$ ，

$C_{12}={\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right)-{\theta }_{12}\left(T{u}_3,T{u}_4\right){\theta }_{13}\left(T{u}_1,x\right)-{\theta }_{12}\left(x,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)+{\theta }_{13}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_4,T{u}_1\right)$ ，

$C_{13}={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(x,T{u}_4\right)-{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,x\right\}}_1,T{u}_4\right)-{\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)+{\theta }_{12}\left(x,{\left\{T{u}_2,T{u}_1,T{u}_4\right\}}_1\right)$ ，

$C_{14}={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{13}\left(T{u}_3,x\right)-{\theta }_{13}\left(T{u}_3,{\left\{T{u}_1,T{u}_2,x\right\}}_1\right)-{\theta }_{13}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)+{\theta }_{13}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_2,T{u}_1\right)$ ，

$\begin{array}{c}{C}_{15}={\left\{u_1,u_2,{\theta }_{12}\left(x,T{u}_4\right)u_3\right\}}_2+{\left\{u_1,u_2,{\theta }_{13}\left(T{u}_3,x\right)\right\}}_2-{\left\{u_3,u_4,{\theta }_{12}\left(x,T{u}_2\right)u_1\right\}}_2-{\left\{u_3,u_4,{\theta }_{13}\left(T{u}_1,x\right)\right\}}_2\\ -{\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2+{\theta }_{13}\left(T{u}_3,x\right){\left\{u_2,u_1,u_4\right\}}_2\end{array}$ ，

由于 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 是约当三系，故(1.2)式成立，由(2.4)式成立知C₁₁为零，由(2.5)式成立知C₁₂为零，由(2.2)式成立知C₁₃为零，由(2.3)式成立知C₁₄为零，由于 ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，故有 ${\left\{u_1,u_2,{\theta }_{12}\left(x,T{u}_4\right)u_3\right\}}_2=0$ ， ${\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2=0$ ，类似可知C₁₅为零，故(2.2)式成立。

计算(2.3)式左边可得

$\begin{array}{l}\left(L_T\left(u_1,u_2\right)M_T\left(u_3,u_4\right)+M_T\left(u_3,u_4\right)L_T\left(u_2,u_1\right)-M_T\left(u_3,{\left\{u_1,u_2,u_4\right\}}_T\right)-M_T\left({\left\{u_1,u_2,u_3\right\}}_T,u_4\right)\right)x\\ =L_T\left(u_1,u_2\right){\left\{T{u}_3,x,T{u}_4\right\}}_1-L_T\left(u_1,u_2\right)T{\theta }_{12}\left(T{u}_4,x\right)u_3-L_T\left(u_1,u_2\right)T{\theta }_{12}\left(T{u}_3,x\right)u_4\\ +M_T\left(u_3,u_4\right){\left\{T{u}_2,T{u}_1,x\right\}}_1-M_T\left(u_3,u_4\right)T{\theta }_{13}\left(T{u}_2,x\right)u_1-M_T\left(u_3,u_4\right)T{\theta }_{12}\left(x,T{u}_1\right)u_2\\ -{\left\{T{u}_3,x,{\left\{T{u}_1,T{u}_2,T{u}_4\right\}}_1\right\}}_1+T{\theta }_{12}\left(T{u}_3,x\right){\left\{u_1,u_2,u_4\right\}}_T+T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_4\right\}}_1,x\right)u_3\\ -{\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x,T{u}_4\right\}}_1+T{\theta }_{12}\left(T{u}_4,x\right){\left\{u_1,u_2,u_3\right\}}_T+T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4\end{array}$

$\begin{array}{l}={\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right\}}_1-T{\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right)u_2-T{\theta }_{12}\left({\left\{T{u}_3,x,T{u}_4\right\}}_1,T{u}_2\right)u_1\\ -{\left\{T{u}_1,T{u}_2,T{\theta }_{12}\left(T{u}_4,x\right)u_3\right\}}_1+T{\theta }_{13}\left(T{u}_1,T{\theta }_{12}\left(T{u}_4,x\right)u_3\right)u_2+T{\theta }_{12}\left(T{\theta }_{12}\left(T{u}_4,x\right)u_3,T{u}_2\right)u_1\\ -{\left\{T{u}_1,T{u}_2,T{\theta }_{12}\left(T{u}_3,x\right)u_4\right\}}_1+T{\theta }_{13}\left(T{u}_1,T{\theta }_{12}\left(T{u}_3,x\right)u_4\right)u_2+T{\theta }_{12}\left(T{\theta }_{12}\left(T{u}_3,x\right)u_4,T{u}_2\right)u_1\\ +{\left\{T{u}_3,{\left\{T{u}_2,T{u}_1,x\right\}}_1,T{u}_4\right\}}_1-T{\theta }_{12}\left(T{u}_4,{\left\{T{u}_2,T{u}_1,x\right\}}_1\right)u_3-T{\theta }_{12}\left(T{u}_3,{\left\{T{u}_2,T{u}_1,x\right\}}_1\right)u_4\\ -{\left\{T{u}_3,T{\theta }_{13}\left(T{u}_2,x\right)u_1,T{u}_4\right\}}_1+T{\theta }_{12}\left(T{u}_4,T{\theta }_{13}\left(T{u}_2,x\right)u_1\right)u_3+T{\theta }_{12}\left(T{u}_3,T{\theta }_{13}\left(T{u}_2,x\right)u_1\right)u_4\end{array}$

$\begin{array}{l}-{\left\{T{u}_3,T{\theta }_{12}\left(x,T{u}_1\right)u_2,T{u}_4\right\}}_1+T{\theta }_{12}\left(T{u}_4,T{\theta }_{12}\left(x,T{u}_1\right)u_2\right)u_3+T{\theta }_{12}\left(T{u}_3,T{\theta }_{12}\left(x,T{u}_1\right)u_2\right)u_4\\ -{\left\{T{u}_3,x,{\left\{T{u}_1,T{u}_2,T{u}_4\right\}}_1\right\}}_1+T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_4\right\}}_1,x\right)u_3+T{\theta }_{12}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_4,T{u}_2\right)u_1\\ +T{\theta }_{12}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_4+T{\theta }_{12}\left(T{u}_3,x\right){\theta }_{13}\left(T{u}_1,T{u}_4\right)u_2+\lambda T{\theta }_{12}\left(T{u}_3,x\right){\left\{u_1,u_2,u_4\right\}}_2\\ -{\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x,T{u}_4\right\}}_1+T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4+T{\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\\ +T{\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3+T{\theta }_{12}\left(T{u}_4,x\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2+\lambda T\end{array}$

$\begin{array}{l}=({\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right\}}_1-{\left\{T{u}_3,x,{\left\{T{u}_1,T{u}_2,T{u}_4\right\}}_1\right\}}_1-{\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x,T{u}_4\right\}}_1\\ +{\left\{T{u}_3,{\left\{T{u}_2,T{u}_1,x\right\}}_1,T{u}_4\right\}}_1)-T\left(C_{21}{u}_1+C_{22}{u}_2+C_{23}{u}_3+C_{24}{u}_4+\lambda C_{25}\right)\end{array}$

其中，

$C_{21}={\theta }_{12}\left({\left\{T{u}_3,x,T{u}_4\right\}}_1,T{u}_2\right)+{\theta }_{13}\left(T{u}_3,T{u}_4\right){\theta }_{13}\left(T{u}_2,x\right)-{\theta }_{12}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_4,T{u}_2\right)-{\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)$ ，

$C_{22}={\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right)+{\theta }_{13}\left(T{u}_3,T{u}_4\right){\theta }_{12}\left(x,T{u}_1\right)-{\theta }_{12}\left(T{u}_3,x\right){\theta }_{13}\left(T{u}_1,T{u}_4\right)-{\theta }_{12}\left(T{u}_4,x\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)$ ，

$C_{23}={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_4,x\right)+{\theta }_{12}\left(T{u}_4,{\left\{T{u}_2,T{u}_1,x\right\}}_1\right)-{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_4\right\}}_1,x\right)-{\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)$ ，

$C_{24}={\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_3,x\right)+{\theta }_{12}\left(T{u}_3,{\left\{T{u}_2,T{u}_1,x\right\}}_1\right)-{\theta }_{12}\left(T{u}_3,x\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)-{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)$ ，

$\begin{array}{c}{C}_{25}={\left\{u_1,u_2,{\theta }_{12}\left(T{u}_4,x\right)u_3\right\}}_2+{\left\{u_1,u_2,{\theta }_{12}\left(T{u}_3,x\right)u_4\right\}}_1-{\theta }_{12}\left(T{u}_4,x\right){\left\{u_1,u_2,u_3\right\}}_2\\ +{\left\{u_3,{\theta }_{13}\left(T{u}_2,x\right)u_1,u_4\right\}}_1+{\left\{u_3,{\theta }_{12}\left(x,T{u}_1\right)u_2,u_4\right\}}_1-{\theta }_{12}\left(T{u}_3,x\right){\left\{u_1,u_2,u_4\right\}}_2\end{array}$ ，

由于 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 是约当三系，故(1.2)式成立，由(2.4)式成立知C₂₁为零，由(2.5)式成立知C₂₂为零，由(2.2)式成立知C₂₃为零，由(2.2)式成立知C₂₄为零，由于 ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，故有 ${\left\{u_3,{\theta }_{13}\left(x,T{u}_2\right)u_1,u_4\right\}}_2=0$ ， ${\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2=0$ ，类似可知C₂₅为零，故(2.3)式成立。

计算(2.4)式左边可得

$\begin{array}{l}\left(L_T\left({\left\{u_1,u_2,u_3\right\}}_T,u_4\right)-L_T\left(u_1,u_2\right)L_T\left(u_3,u_4\right)-L_T\left(u_3,u_2\right)L_T\left(u_1,u_4\right)+M_T\left(u_1,u_3\right)M_T\left(u_2,u_4\right)\right)x\\ ={\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,T{u}_4,x\right\}}_1-T{\theta }_{13}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4-T{\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_T\\ -L_T\left(u_1,u_2\right){\left\{T{u}_3,T{u}_4,x\right\}}_1+L_T\left(u_1,u_2\right)T{\theta }_{13}\left(T{u}_3,x\right)u_4+L_T\left(u_1,u_2\right)T{\theta }_{12}\left(x,T{u}_4\right)u_3\\ -L_T\left(u_3,u_2\right){\left\{T{u}_1,T{u}_4,x\right\}}_1+L_T\left(u_3,u_2\right)T{\theta }_{13}\left(T{u}_1,x\right)u_4+L_T\left(u_3,u_2\right)T{\theta }_{12}\left(x,T{u}_4\right)u_1\\ +M_T\left(u_1,u_3\right){\left\{T{u}_2,x,T{u}_4\right\}}_1-M_T\left(u_1,u_3\right)T{\theta }_{12}\left(T{u}_4,x\right)u_2-M_T\left(u_1,u_3\right)T{\theta }_{12}\left(T{u}_2,x\right)u_4\end{array}$

$\begin{array}{l}={\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,T{u}_4,x\right\}}_1-T{\theta }_{13}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4-T{\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\\ -T{\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3-T{\theta }_{12}\left(x,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2-\lambda T{\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2\\ -{\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right\}}_1+T{\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right)u_2+T{\theta }_{12}\left({\left\{T{u}_3,T{u}_4,x\right\}}_1,T{u}_2\right)u_1\\ +{\left\{T{u}_1,T{u}_2,T{\theta }_{13}\left(T{u}_3,x\right)u_4\right\}}_1-T{\theta }_{13}\left(T{u}_1,T{\theta }_{13}\left(T{u}_3,x\right)u_4\right)u_2-T{\theta }_{12}\left(T{\theta }_{13}\left(T{u}_3,x\right)u_4,T{u}_2\right)u_1\\ +{\left\{T{u}_1,T{u}_2,T{\theta }_{12}\left(x,T{u}_4\right)u_3\right\}}_1-T{\theta }_{13}\left(T{u}_1,T{\theta }_{12}\left(x,T{u}_4\right)u_3\right)u_2-T{\theta }_{12}\left(T{\theta }_{12}\left(x,T{u}_4\right)u_3,T{u}_2\right)u_1\end{array}$

$\begin{array}{l}-{\left\{T{u}_3,T{u}_2,{\left\{T{u}_1,T{u}_4,x\right\}}_1\right\}}_1+T{\theta }_{13}\left(T{u}_3,{\left\{T{u}_1,T{u}_4,x\right\}}_1\right)u_2+T{\theta }_{12}\left({\left\{T{u}_1,T{u}_4,x\right\}}_1,T{u}_2\right)u_3\\ +{\left\{T{u}_3,T{u}_2,T{\theta }_{13}\left(T{u}_1,x\right)u_4\right\}}_1-T{\theta }_{13}\left(T{u}_3,T{\theta }_{13}\left(T{u}_1,x\right)u_4\right)u_2-T{\theta }_{12}\left(T{\theta }_{13}\left(T{u}_1,x\right)u_4,T{u}_2\right)u_3\\ +{\left\{T{u}_3,T{u}_2,T{\theta }_{12}\left(x,T{u}_4\right)u_1\right\}}_1-T{\theta }_{13}\left(T{u}_3,T{\theta }_{12}\left(x,T{u}_4\right)u_1\right)u_2-T{\theta }_{12}\left(T{\theta }_{12}\left(x,T{u}_4\right)u_1,T{u}_2\right)u_3\\ +{\left\{T{u}_1,{\left\{T{u}_2,x,T{u}_4\right\}}_1,T{u}_3\right\}}_1-T{\theta }_{12}\left(T{u}_3,{\left\{T{u}_2,x,T{u}_4\right\}}_1\right)u_1-T{\theta }_{12}\left(T{u}_1,{\left\{T{u}_2,x,T{u}_4\right\}}_1\right)u_3\end{array}$

$\begin{array}{l}-{\left\{T{u}_1,T{\theta }_{12}\left(T{u}_4,x\right)u_2,T{u}_3\right\}}_1+T{\theta }_{12}\left(T{u}_3,T{\theta }_{12}\left(T{u}_4,x\right)u_2\right)u_1+T{\theta }_{12}\left(T{u}_1,T{\theta }_{12}\left(T{u}_4,x\right)u_2\right)u_3\\ -{\left\{T{u}_1,T{\theta }_{12}\left(T{u}_2,x\right)u_4,T{u}_3\right\}}_1+T{\theta }_{12}\left(T{u}_3,T{\theta }_{12}\left(T{u}_2,x\right)u_4\right)u_1+T{\theta }_{12}\left(T{u}_1,T{\theta }_{12}\left(T{u}_2,x\right)u_4\right)u_3\\ ={\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,T{u}_4,x\right\}}_1-{\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right\}}_1-{\left\{T{u}_3,T{u}_2,{\left\{T{u}_1,T{u}_4,x\right\}}_1\right\}}_1\\ +{\left\{T{u}_1,{\left\{T{u}_2,x,T{u}_4\right\}}_1,T{u}_3\right\}}_1-T\left(C_{31}{u}_1+C_{32}{u}_2+C_{33}{u}_3+C_{34}{u}_4+\lambda C_{35}\right)\end{array}$

其中，

$C_{31}={\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)-{\theta }_{12}\left({\left\{T{u}_3,T{u}_4,x\right\}}_1,T{u}_2\right)-{\theta }_{12}\left(T{u}_3,T{u}_2\right){\theta }_{12}\left(x,T{u}_4\right)+{\theta }_{12}\left(T{u}_3,{\left\{T{u}_2,x,T{u}_4\right\}}_1\right)$ ，

$C_{32}={\theta }_{12}\left(x,T{u}_4\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)-{\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,T{u}_4,x\right\}}_1\right)-{\theta }_{13}\left(T{u}_3,{\left\{T{u}_1,T{u}_4,x\right\}}_1\right)+{\theta }_{13}\left(T{u}_1,T{u}_3\right){\theta }_{12}\left(T{u}_4,x\right)$ ，

$C_{33}={\theta }_{12}\left(x,T{u}_4\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)-{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(x,T{u}_4\right)-{\theta }_{12}\left({\left\{T{u}_1,T{u}_4,x\right\}}_1,T{u}_2\right)+{\theta }_{12}\left(T{u}_1,{\left\{T{u}_2,x,T{u}_4\right\}}_1\right)$ ，

$C_{34}={\theta }_{13}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)-{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{13}\left(T{u}_3,x\right)-{\theta }_{12}\left(T{u}_3,T{u}_2\right){\theta }_{13}\left(T{u}_1,x\right)+{\theta }_{13}\left(T{u}_1,T{u}_3\right){\theta }_{12}\left(T{u}_2,x\right)$ ，

$\begin{array}{c}{C}_{35}={\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2-{\left\{u_1,u_2,{\theta }_{13}\left(T{u}_3,x\right)u_4\right\}}_2-{\left\{u_1,u_2,{\theta }_{12}\left(x,T{u}_4\right)u_3\right\}}_2-{\left\{u_3,u_2,{\theta }_{13}\left(T{u}_1,x\right)u_4\right\}}_2\\ -{\left\{u_3,u_2,{\theta }_{12}\left(x,T{u}_4\right)u_1\right\}}_2+{\left\{u_1,{\theta }_{12}\left(T{u}_4,x\right)u_2,u_3\right\}}_2+{\left\{u_1,{\theta }_{12}\left(T{u}_2,x\right)u_4,u_3\right\}}_2\end{array}$ ，

由于 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 是约当三系，故(1.2)式成立，由(2.2)式成立知C₃₁为零，由(2.3)式成立知C₃₂为零，由(2.2)式成立知C₃₃为零，由(2.5)式成立知C₃₄为零，由于 ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，故有 ${\left\{u_1,u_2,{\theta }_{13}\left(T{u}_3,x\right)u_4\right\}}_2=0$ ， ${\theta }_{12}\left(x,T{u}_4\right){\left\{u_1,u_2,u_3\right\}}_2=0$ ，类似可知C₃₅为零，故(2.4)式成立。

计算(2.5)式左边可得

$\begin{array}{l}\left(M_T\left({\left\{u_1,u_2,u_3\right\}}_T,u_4\right)-L_T\left(u_1,u_2\right)M_T\left(u_3,u_4\right)-L_T\left(u_3,u_2\right)M_T\left(u_1,u_4\right)+M_T\left(u_1,u_3\right)L_T\left(u_2,u_4\right)\right)x\\ ={\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x,T{u}_4\right\}}_1-T{\theta }_{12}\left(T{u}_4,x\right){\left\{u_1,u_2,u_3\right\}}_T-T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4\\ -L_T\left(u_1,u_2\right){\left\{T{u}_3,x,T{u}_4\right\}}_1+L_T\left(u_1,u_2\right)T{\theta }_{12}\left(T{u}_4,x\right)u_3+L_T\left(u_1,u_2\right)T{\theta }_{12}\left(T{u}_3,x\right)u_4\\ -L_T\left(u_3,u_2\right){\left\{T{u}_1,x,T{u}_4\right\}}_1+L_T\left(u_3,u_2\right)T{\theta }_{12}\left(T{u}_4,x\right)u_1+L_T\left(u_3,u_2\right)T{\theta }_{12}\left(T{u}_1,x\right)u_4\\ +M_T\left(u_1,u_3\right){\left\{T{u}_2,T{u}_4,x\right\}}_1-M_T\left(u_1,u_3\right)T{\theta }_{13}\left(T{u}_2,x\right)u_4-M_T\left(u_1,u_3\right)T{\theta }_{12}\left(x,T{u}_4\right)u_2\end{array}$

$\begin{array}{l}={\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x,T{u}_4\right\}}_1-T{\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)u_4-T{\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)u_1\\ -T{\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)u_3-T{\theta }_{12}\left(T{u}_4,x\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)u_2-\lambda T{\theta }_{12}\left(T{u}_4,x\right){\left\{u_1,u_2,u_3\right\}}_2\\ -{\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right\}}_1+T{\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right)u_2+T{\theta }_{12}\left({\left\{T{u}_3,x,T{u}_4\right\}}_1,T{u}_2\right)u_1\\ +{\left\{T{u}_1,T{u}_2,T{\theta }_{12}\left(T{u}_4,x\right)u_3\right\}}_1-T{\theta }_{13}\left(T{u}_1,T{\theta }_{12}\left(T{u}_4,x\right)u_3\right)u_2-T{\theta }_{12}\left(T{\theta }_{12}\left(T{u}_4,x\right)u_3,T{u}_2\right)u_1\\ +{\left\{T{u}_1,T{u}_2,T{\theta }_{12}\left(T{u}_3,x\right)u_4\right\}}_1-T{\theta }_{13}\left(T{u}_1,T{\theta }_{12}\left(T{u}_3,x\right)u_4\right)u_2-T{\theta }_{12}\left(T{\theta }_{12}\left(T{u}_3,x\right)u_4,T{u}_2\right)u_1\end{array}$

$\begin{array}{l}-{\left\{T{u}_3,T{u}_2,{\left\{T{u}_1,x,T{u}_4\right\}}_1\right\}}_1+T{\theta }_{13}\left(T{u}_3,{\left\{T{u}_1,x,T{u}_4\right\}}_1\right)u_2+T{\theta }_{12}\left({\left\{T{u}_1,x,T{u}_4\right\}}_1,T{u}_2\right)u_3\\ +{\left\{T{u}_3,T{u}_2,T{\theta }_{12}\left(T{u}_4,x\right)u_1\right\}}_1-T{\theta }_{13}\left(T{u}_3,T{\theta }_{12}\left(T{u}_4,x\right)u_1\right)u_2-T{\theta }_{12}\left(T{\theta }_{12}\left(T{u}_4,x\right)u_1,T{u}_2\right)u_3\\ +{\left\{T{u}_3,T{u}_2,T{\theta }_{12}\left(T{u}_1,x\right)u_4\right\}}_1-T{\theta }_{13}\left(T{u}_3,T{\theta }_{12}\left(T{u}_1,x\right)u_4\right)u_2-T{\theta }_{12}\left(T{\theta }_{12}\left(T{u}_1,x\right)u_4,T{u}_2\right)u_3\\ +{\left\{T{u}_1,{\left\{T{u}_2,T{u}_4,x\right\}}_1,T{u}_3\right\}}_1-T{\theta }_{12}\left(T{u}_3,{\left\{T{u}_2,T{u}_4,x\right\}}_1{u}_1\right)-T{\theta }_{12}\left(T{u}_1,{\left\{T{u}_2,T{u}_4,x\right\}}_1\right)u_3\end{array}$

$\begin{array}{l}-{\left\{T{u}_1,T{\theta }_{13}\left(T{u}_2,x\right)u_4,T{u}_3\right\}}_1+T{\theta }_{12}\left(T{u}_3,T{\theta }_{13}\left(T{u}_2,x\right)u_4\right)u_1+T{\theta }_{12}\left(T{u}_1,T{\theta }_{13}\left(T{u}_2,x\right)u_4\right)u_3\\ -{\left\{T{u}_1,T{\theta }_{12}\left(x,T{u}_4\right)u_2,T{u}_3\right\}}_1+T{\theta }_{12}\left(T{u}_3,x\right)T{\theta }_{12}\left(x,T{u}_4\right)u_2{u}_1+T{\theta }_{12}\left(T{u}_1,T{\theta }_{12}\left(x,T{u}_4\right)u_2\right)u_3\\ ={\left\{{\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x,T{u}_4\right\}}_1-{\left\{T{u}_1,T{u}_2,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right\}}_1-{\left\{T{u}_3,T{u}_2,{\left\{T{u}_1,x,T{u}_4\right\}}_1\right\}}_1\\ +\left\{T{u}_1,{\left\{T{u}_2,T{u}_4,x\right\}}_1,T{u}_3\right\}-T\left(C_{41}{u}_1+C_{42}{u}_2+C_{43}{u}_3+C_{44}{u}_4+\lambda C_{45}\right)\end{array}$

其中，

$C_{41}={\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_3,T{u}_2\right)-{\theta }_{12}\left({\left\{T{u}_3,x,T{u}_4\right\}}_1,T{u}_2\right)-{\theta }_{12}\left(T{u}_3,T{u}_2\right){\theta }_{12}\left(T{u}_4,x\right)+{\theta }_{12}\left(T{u}_3,{\left\{T{u}_2,T{u}_4,x\right\}}_1\right)$ ，

$C_{42}={\theta }_{12}\left(T{u}_4,x\right){\theta }_{13}\left(T{u}_1,T{u}_3\right)-{\theta }_{13}\left(T{u}_1,{\left\{T{u}_3,x,T{u}_4\right\}}_1\right)-{\theta }_{13}\left(T{u}_3,{\left\{T{u}_1,x,T{u}_4\right\}}_1\right)+{\theta }_{13}\left(T{u}_1,T{u}_3\right){\theta }_{12}\left(x,T{u}_4\right)$ ，

$C_{43}={\theta }_{12}\left(T{u}_4,x\right){\theta }_{12}\left(T{u}_1,T{u}_2\right)-{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_4,x\right)-{\theta }_{12}\left({\left\{T{u}_1,x,T{u}_4\right\}}_1,T{u}_2\right)+{\theta }_{12}\left(T{u}_1,{\left\{T{u}_2,T{u}_4,x\right\}}_1\right)$ ，

$C_{44}={\theta }_{12}\left({\left\{T{u}_1,T{u}_2,T{u}_3\right\}}_1,x\right)-{\theta }_{12}\left(T{u}_1,T{u}_2\right){\theta }_{12}\left(T{u}_3,x\right)-{\theta }_{12}\left(T{u}_3,T{u}_2\right){\theta }_{12}\left(T{u}_1,x\right)+{\theta }_{13}\left(T{u}_1,T{u}_3\right){\theta }_{13}\left(T{u}_2,x\right)$ ，

$\begin{array}{c}{C}_{45}={\theta }_{12}\left(T{u}_4,x\right){\left\{u_1,u_2,u_3\right\}}_2-{\left\{u_1,u_2,{\theta }_{12}\left(T{u}_4,x\right)u_3\right\}}_2-{\left\{u_1,u_2,{\theta }_{12}\left(T{u}_3,x\right)u_4\right\}}_2-{\left\{u_3,u_2,{\theta }_{12}\left(T{u}_4,x\right)u_1\right\}}_2\\ -{\left\{u_3,u_2,{\theta }_{12}\left(T{u}_1,x\right)u_4\right\}}_2+{\left\{u_1,{\theta }_{12}\left(x,T{u}_4\right)u_2,u_3\right\}}_2+{\left\{u_1,{\theta }_{13}\left(T{u}_2,x\right)u_4,u_3\right\}}_2\end{array}$ ，

由于 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 是约当三系，故(1.2)式成立，由(2.2)式成立知C₄₁为零，由(2.3)式成立知C₄₂为零，由(2.2)式成立知C₄₃为零，由(2.4)式成立知C₄₄为零，由于 ${\theta }_{12},{\theta }_{13}$ 是 $\left(J_1,{\left\{\cdot ,\cdot ,\cdot \right\}}_1\right)$ 在 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_2\right)$ 上的作用，故有 ${\theta }_{12}\left(T{u}_4,x\right){\left\{u_1,u_2,u_3\right\}}_2=0$ ， ${\left\{u_1,u_2,{\theta }_{12}\left(T{u}_4,x\right)u_3\right\}}_2=0$ ，类似可知C₄₅为零，故(2.5)式成立。

综上， $\left(L_T,M_T,J_1\right)$ 是 $\left(J_2,{\left\{\cdot ,\cdot ,\cdot \right\}}_T\right)$ 的表示。

5. 相对Rota-Baxter算子的构造

命题5.1 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系， $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 的伴随表示， $J_0$ 是J的子系且满足 $J_0^1=\left\{0\right\}$ ， $J^1\cap J_0=\left\{0\right\}$ ， $\left\{J_0,J_0,J\right\}=\left\{0\right\}$ ， $\left\{J_0,J,J_0\right\}=\left\{0\right\}$ 。设H是 $J_0$ 的补空间，使 $J=H\oplus J_0$ ，P是J在 $J_0$ 上的自然投影，即

$P\left(x\right)=x_0,x\in J,x_0\in J_0$ ，

则P是关于作用 ${\sigma }_{12},{\sigma }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子。

证：由于 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 的伴随表示，故由例2.2知， ${\sigma }_{12},{\sigma }_{13}$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 在自身上的作用， $\forall x,y,z\in J$ ，记 $x,y,z$ 在投影P下的像为 $x_0,y_0,z_0$ ，

$\begin{array}{l}\left\{Px,Py,Pz\right\}-P\left({\sigma }_{12}\left(Pz,Py\right)x+{\sigma }_{13}\left(Px,Pz\right)y+{\sigma }_{12}\left(Px,Py\right)z+\lambda \left\{x,y,z\right\}\right)\\ =\left\{x_0,y_0,z_0\right\}-P\left({\sigma }_{12}\left(z_0,y_0\right)x+{\sigma }_{13}\left(x_0,z_0\right)y+{\sigma }_{12}\left(x_0,y_0\right)z+\lambda \left\{x,y,z\right\}\right)\\ =0\end{array}$ ，

所以，P是关于作用 ${\sigma }_{12},{\sigma }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子。

例5.1 设 $\left(J,\cdot \right)$ 是三维约当代数， $n_1,n_2,n_3$ 是J的一组基，J中运算为

$n_1\cdot n_1=n_2,n_2\cdot n_2=0,n_3\cdot n_3=0,n_1\cdot n_2=n_2\cdot n_1=n_3,n_1\cdot n_3=n_3\cdot n_1=0,n_2\cdot n_3=n_3\cdot n_2=0$ 。

由例1.1知，在J上定义运算

$\left\{x,y,z\right\}=\left(x\cdot y\right)\cdot z+\left(y\cdot z\right)\cdot x-\left(x\cdot z\right)\cdot y$ ，

则 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 是约当三系，且基中元素的非零代数运算为

$\left\{n_1,n_1,n_1\right\}=n_3$ ，

$\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 的 $C\left(J\right)$ 是由 $\left\{n_2,n_3\right\}$ 生成的子系，则伴随表示 $\left({\sigma }_{12},{\sigma }_{13},J\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\}\right)$ 在自身上的一个作用。取 $J_0$ 为由 $\left\{n_2\right\}$ 所生成的子系，根据命题4.1，由

$P\left(n_1\right)=0,P\left(n_2\right)=n_2,P\left(n_3\right)=0$

所决定的投影 $P:J\to J_0$ 是关于作用 ${\sigma }_{12},{\sigma }_{13}$ 的权为 $\lambda$ 的相对Rota-Baxter算子。