泊松代数模的无穷小形变

Infinitesimal Deformation of the Module of Poisson Algebras

摘要: 本文主要研究泊松代数模的无穷小形变。构造了泊松代数的形变，以及泊松代数模的无穷小形变，讨论两个无穷小形变的等价条件和模的无穷小形变是平凡的条件。文章的最后给出了泊松代数上Nijenhuis算子的定义，并且找到泊松代数模的平凡形变与Nijenhuis算子结构的关系。

Abstract: In this paper, we study the infinitesimal deformation of the module of Poisson algebra. The deformation of Poisson algebra is constructed, the infinitesimal deformation of the module of Poisson algebra and the equivalent condition of two infinitesimal deformations are discussed. At the end of the paper, the Nijenhuis operators of Poisson algebras are given, and the relationship between trivial deformation and Nijenhuis operator structure is also found.

1. 前言

泊松代数是李代数与交换结合代数的复合结构。泊松代数在数学和物理的许多领域中起着重要的作用，如泊松几何、可积系统、非交换(代数或微分)几何等 [1]。由Drinfeld引入的李代数现在被公认为量子群的无穷小化 [2]。Joni和Rota引入无穷小代数的概念解释了微积分在代数学中的差异性 [3]。泊松代数的ON-结构和O-算子值得进一步研究。

2. 一类特殊泊松代数的构造

定义2.1 [4] 设P是域F上的线性空间，P中有双线性代数运算 $[,],\circ :P\otimes P\to P$，如果 $\left(P,[,]\right)$ 为李代数， $(P,\circ )$ 为交换结合代数，并且

$\left[x,y\circ z\right]=\left[x,y\right]\circ z+y\circ \left[x,z\right],\forall x,y,z\in P,$ (2.1)

则称 $\left(P,[,],\circ \right)$ 为泊松代数。

定理2.1 设 $\left(P,[,],\circ \right)$ 是泊松代数， ${\omega }_1,{\omega }_2:P\otimes P\to P$ 为双线性映射，定义

${\left[x,y\right]}_t=\left[x,y\right]+t{\omega }_1\left(x,y\right),$

$x{\circ }_{t}y=x\circ y+t{\omega }_2\left(x,y\right),$

$\forall x,y\in P$，t为参数，则 $\left(P,{[,]}_t,{\circ }_t\right)$ 是泊松代数当且仅当对 $\forall x,y,z\in P$， $k_1,k_2\in F$ 满足下列方程

${\omega }_{\text{1}}\left(x,y\right)+{\omega }_1\left(y,x\right)=0,$ (2.2)

$\left[x,{\omega }_1\left(y,z\right)\right]+{\omega }_1\left(x,\left[y,z\right]\right)+\left[y,{\omega }_1\left(z,x\right)\right]+{\omega }_1\left(y,\left[z,x\right]\right)+\left[z,{\omega }_1\left(x,y\right)\right]+{\omega }_1\left(z,\left[x,y\right]\right)=0,$ (2.3)

${\omega }_{\text{1}}\left(x,{\omega }_{\text{1}}\left(y,z\right)\right)+{\omega }_{\text{1}}\left(y,{\omega }_{\text{1}}\left(z,x\right)\right)+{\omega }_{\text{1}}\left(z,{\omega }_{\text{1}}\left(x,y\right)\right)=0,$ (2.4)

$x\circ {\omega }_2\left(y,z\right)+{\omega }_2\left(x,y\circ z\right)-{\omega }_2\left(x,y\right)\circ z-{\omega }_2\left(x\circ y,z\right)=0,$ (2.5)

${\omega }_{\text{2}}\left({\omega }_2\left(x,y\right),z\right)-{\omega }_{\text{2}}\left(x,{\omega }_2\left(y,z\right)\right)=0,$ (2.6)

${\omega }_{\text{2}}\left(x,y\right)-{\omega }_2\left(y,x\right)=0,$ (2.7)

$\left[x,{\omega }_2\left(y,z\right)\right]+{\omega }_1\left(x,y\circ z\right)-{\omega }_1\left(x,y\right)\circ z-{\omega }_2\left(\left[x,y\right],z\right)-y\circ {\omega }_1\left(x,z\right)-{\omega }_2\left(y,\left[x,z\right]\right)=0,$ (2.8)

${\omega }_{\text{1}}\left(x,{\omega }_2\left(y,z\right)\right)-{\omega }_2\left({\omega }_1\left(x,y\right),z\right)-{\omega }_2\left(y,{\omega }_1\left(x,z\right)\right)=0.$ (2.9)

此时称 $\left(P,{[,]}_t,{\circ }_t\right)$ 为泊松代数 $\left(P,[,],\circ \right)$ 的形变。

证明：由于 $[,]$ 双线性且 ${\omega }_{\text{1}}$ 双线性，显然 ${[,]}_t$ 双线性。同理可知，P上的代数运算 ${\circ }_t$ 也是双线性的。对 $\forall x,y,z\in P$， $k_1,k_2\in F$，由于

${\left[x,y\right]}_t+{\left[y,x\right]}_t=t\left({\omega }_1\left(x,y\right)+{\omega }_1\left(y,x\right)\right),$

因此 ${[,]}_t$ 反对称当且仅当 ${\omega }_{\text{1}}$ 反对称，即 ${\omega }_{\text{1}}$ 满足(2.2)。由于

$\begin{array}{l}{\left[x,{\left[y,z\right]}_t\right]}_t+{\left[y,{\left[z,x\right]}_t\right]}_t+{\left[z,{\left[x,y\right]}_t\right]}_t\\ =t\left(\left[x,{\omega }_1\left(y,z\right)\right]+{\omega }_1\left(x,\left[y,z\right]\right)+\left[y,{\omega }_1\left(z,x\right)\right]+{\omega }_1\left(y,\left[z,x\right]\right)+\left[z,{\omega }_1\left(x,y\right)\right]+{\omega }_1\left(z,\left[x,y\right]\right)\right)\\ +t^2\left({\omega }_1\left(x,{\omega }_1\left(y,z\right)\right)+{\omega }_1\left(y,{\omega }_1\left(z,x\right)\right)+{\omega }_1\left(z,{\omega }_1\left(x,y\right)\right)\right),\end{array}$

因此 ${[,]}_t$ 满足Jacobi恒等式当且仅当 ${\omega }_{\text{1}}$ 满足(2.3)、(2.4)。由于

$\begin{array}{l}\left(x{\circ }_{t}y\right){\circ }_{t}z-x{\circ }_t\left(y{\circ }_{t}z\right)\\ =t\left({\omega }_2\left(x,y\right)\circ z+{\omega }_2\left(x\circ y,z\right)-x\circ {\omega }_2\left(y,z\right)-{\omega }_2\left(x,y\circ z\right)\right)\\ +t^2\left({\omega }_2\left({\omega }_2\left(x,y\right),z\right)-{\omega }_2\left(x,{\omega }_2\left(y,z\right)\right)\right),\end{array}$

因此 ${\circ }_t$ 满足结合律当且仅当 ${\omega }_{\text{2}}$ 满足(2.5)、(2.6)。由于

$x{\circ }_{t}y-y{\circ }_{t}x=t\left({\omega }_2\left(x,y\right)-{\omega }_2\left(y,x\right)\right),$

因此 ${\circ }_t$ 满足交换律当且仅当 ${\omega }_{\text{2}}$ 满足(2.7)。由于

$\begin{array}{l}{\left[x,y{\circ }_{t}z\right]}_t-{\left[x,y\right]}_t{\circ }_{t}z-y{\circ }_t{\left[x,z\right]}_t\\ =t\left(\left[x,{\omega }_2\left(y,z\right)\right]+{\omega }_1\left(x,y\circ z\right)-{\omega }_1\left(x,y\right)\circ z-{\omega }_2\left(\left[x,y\right],z\right)-y\circ {\omega }_1\left(x,z\right)-{\omega }_2\left(y,\left[x,z\right]\right)\right)\\ +t^2\left({\omega }_1\left(x,{\omega }_2\left(y,z\right)\right)-{\omega }_2\left({\omega }_1\left(x,y\right),z\right)-{\omega }_2\left(y,{\omega }_1\left(x,z\right)\right)\right),\end{array}$

因此 ${[,]}_t,{\circ }_t$ 满足等式(2.1)当且仅当 ${\omega }_{\text{1}},{\omega }_2$ 满足(2.8)、(2.9)。综上可知，结论成立。

3. 泊松代数模及形变

定义3.1 [4] 设 $\left(P,\circ \right)$ 是交换结合代数，V是域F上的线性空间， $S:P\to End\left(V\right)$ 为线性映射，如果满足

$S\left(x\circ y\right)=S\left(x\right)S\left(y\right),\forall x,y\in P,$ (3.1)

则称 $\left(V,S\right)$ 是交换结合代数 $\left(P,\circ \right)$ 的模。

定义3.2 [4] 设 $\left(P,[,],\circ \right)$ 是泊松代数，V是域F上的线性空间， $S_{[,]},S_{\circ }:P\to End\left(V\right)$ 为线性映射，如果 $\left(V,S_{[,]}\right)$ 是李代数 $\left(P,[,]\right)$ 的表示， $\left(V,S_{\circ }\right)$ 是交换结合代数 $\left(P,\circ \right)$ 的模，并且满足

$S_{[,]}\left(x\circ y\right)=S_{\circ }\left(y\right)S_{[,]}\left(x\right)+S_{\circ }\left(x\right)S_{[,]}\left(y\right),$ (3.2)

$S_{\circ }\left(\left[x,y\right]\right)=S_{[,]}\left(x\right)S_{\circ }\left(y\right)-S_{\circ }\left(y\right)S_{[,]}\left(x\right),$ (3.3)

$\forall x,y\in P$，则称 $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模，V简称P-模V。

定理3.1 设 $\left(P,[,],\circ \right)$ 为泊松代数， $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模， $\sigma ,\tau :P\to End\left(V\right)$ 为线性映射，定义 $S_{{[,]}_t},S_{{\circ }_t}:P\to End\left(V\right)$，其中

$S_{{[,]}_t}\left(x\right)=S_{[,]}\left(x\right)+t\sigma \left(x\right),$ (3.4)

$S_{{\circ }_t}\left(x\right)=S_{\circ }\left(x\right)+t\tau \left(x\right),$ (3.5)

$\forall x\in P,t$ 为参数，则 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 是泊松代数 $\left(P,{[,]}_t,{\circ }_t\right)$ 的模当且仅当满足下列方程

$S_{[,]}\left({\omega }_1\left(x,y\right)\right)+\sigma \left(\left[x,y\right]\right)-S_{[,]}\left(x\right)\sigma \left(y\right)-\sigma \left(x\right)S_{[,]}\left(y\right)+S_{[,]}\left(y\right)\sigma \left(x\right)+\sigma \left(y\right)S_{[,]}\left(x\right)=0,$ (3.6)

$\sigma \left({\omega }_1\left(x,y\right)\right)-\sigma \left(x\right)\sigma \left(y\right)+\sigma \left(y\right)\sigma \left(x\right)=0,$ (3.7)

$S_{\circ }\left({\omega }_2\left(x,y\right)\right)+\tau \left(x\circ y\right)-S_{\circ }\left(x\right)\tau \left(y\right)-\tau \left(x\right)S_{\circ }\left(y\right)=0,$ (3.8)

$\tau \left({\omega }_2\left(x,y\right)\right)-\tau \left(x\right)\tau \left(y\right)=0,$ (3.9)

$S_{[,]}\left({\omega }_2\left(x,y\right)\right)+\sigma \left(x\circ y\right)-S_{\circ }\left(y\right)\sigma \left(x\right)-\tau \left(y\right)S_{[,]}\left(x\right)-S_{\circ }\left(x\right)\sigma \left(y\right)-\tau \left(x\right)S_{[,]}\left(y\right)=0,$ (3.10)

$\sigma \left({\omega }_2\left(x,y\right)\right)-\tau \left(y\right)\sigma \left(x\right)-\tau \left(x\right)\sigma \left(y\right)=0,$ (3.11)

$S_{\circ }\left({\omega }_1\left(x,y\right)\right)+\tau \left(\left[x,y\right]\right)-S_{[,]}\left(x\right)\tau \left(y\right)-\sigma \left(x\right)S_{\circ }\left(y\right)+S_{\circ }\left(y\right)\sigma \left(x\right)+\tau \left(y\right)S_{[,]}\left(x\right)=0,$ (3.12)

$\tau \left({\omega }_1\left(x,y\right)\right)-\sigma \left(x\right)\tau \left(y\right)+\tau \left(y\right)\sigma \left(x\right)=0,$ (3.13)

$\forall x,y\in P$.

证明：显然由 $S_{{[,]}_t},S_{{\circ }_t}$ 的定义知 $S_{{[,]}_t},S_{{\circ }_t}$ 是线性的。 $\forall x,y\in P$，由于

$\begin{array}{l}{S}_{{[,]}_t}\left({\left[x,y\right]}_t\right)-\left[S_{{[,]}_t}\left(x\right),S_{{[,]}_t}\left(y\right)\right]\\ =t\left(S_{[,]}\left({\omega }_1\left(x,y\right)\right)+\sigma \left(\left[x,y\right]\right)-S_{[,]}\left(x\right)\sigma \left(y\right)-\sigma \left(x\right)S_{[,]}\left(y\right)+S_{[,]}\left(y\right)\sigma \left(x\right)+\sigma \left(y\right)S_{[,]}\left(x\right)\right)\\ +t^2\left(\sigma \left({\omega }_1\left(x,y\right)\right)-\sigma \left(x\right)\sigma \left(y\right)+\sigma \left(y\right)\sigma \left(x\right)\right),\end{array}$

因此 $\left(V,S_{{[,]}_t}\right)$ 是 $\left(P,{[,]}_t\right)$ 的表示当且仅当(3.6)、(3.7)成立。由于

$\begin{array}{l}{S}_{{\circ }_t}\left(x{\circ }_{t}y\right)-S_{{\circ }_t}\left(x\right)S_{{\circ }_t}\left(y\right)\\ =t\left(S_{\circ }\left({\omega }_2\left(x,y\right)\right)+\tau \left(x\circ y\right)-S_{\circ }\left(x\right)\tau \left(y\right)-\tau \left(x\right)S_{\circ }\left(y\right)\right)\\ +t^2\left(\tau \left({\omega }_2\left(x,y\right)\right)-\tau \left(x\right)\tau \left(y\right)\right),\end{array}$

因此 $\left(V,S_{{\circ }_t}\right)$ 是 $\left(P,{\circ }_t\right)$ 的模当且仅当(3.8)、(3.9)成立。由于

$\begin{array}{l}{S}_{{[,]}_t}\left(x{\circ }_{t}y\right)-S_{{\circ }_t}\left(y\right)S_{{[,]}_t}\left(x\right)-S_{{\circ }_t}\left(x\right)S_{{[,]}_t}\left(y\right)\\ =S_{[,]}\left(x\circ y\right)-S_{\circ }\left(y\right)S_{[,]}\left(x\right)+t^2\left(\sigma \left({\omega }_2\left(x,y\right)\right)-\tau \left(y\right)\sigma \left(y\right)-\tau \left(x\right)\sigma \left(y\right)\right)\\ +t\left(S_{[,]}\left({\omega }_2\left(x,y\right)\right)+\sigma \left(x\circ y\right)-S_{\circ }\left(y\right)\sigma \left(x\right)-\tau \left(y\right)S_{[,]}\left(x\right)-S_{\circ }\left(x\right)\sigma \left(y\right)-\tau \left(x\right)S_{[,]}\left(y\right)\right),\end{array}$

因此对于 $S_{{[,]}_t},S_{{\circ }_t}$ (3.2)成立当且仅当(3.10)、(3.11)成立。同理对于 $S_{{[,]}_t},S_{{\circ }_t}$ (3.3)成立当且仅当(3.12)、(3.13)成立。综上可知，结论成立。

定义3.3 设 $\left(P,{[,]}_t,{\circ }_t\right)$ 为泊松代数 $\left(P,[,],\circ \right)$ 的形变， $\left(V,S_{[,]},S_{\circ }\right)$ 为 $\left(P,[,],\circ \right)$ 的模，若 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 是泊松代数 $\left(P,{[,]}_t,{\circ }_t\right)$ 的模，则称 $\left({\omega }_1,{\omega }_2,\sigma ,\tau \right)$ 是模 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 的无穷小形变。

由定理3.1可知泊松代数 $\left(P,{[,]}_t,{\circ }_t\right)$ 的模 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 是P-模V的无穷小形变当且仅当 ${\omega }_{\text{1}},{\omega }_2$ 满足等式(2.2)~(2.9)且 $\sigma ,\tau$ 满足等式(3.6)~(3.13)。

定义3.4 设 $\left(V,S_{[,]},S_{\circ }\right)$ 和 $\left(V^{\prime },S_{[,{]}^{\prime }},S_{{\circ }^{\prime }}\right)$ 分别是泊松代数 $\left(P,[,],\circ \right)$ 和 $\left(P,[,]\text{'},{\circ }^{\prime }\right)$ 的模，如果存在泊松代数的同态映射 ${\phi }_{P^{\prime }}:P^{\prime }\to P$ 和线性映射 ${\phi }_{V^{\prime }}:V^{\prime }\to V$ 且满足下列条件

${\phi }_{V^{\prime }}\left(S_{{\circ }^{\prime }}\left(x\right)\right)=S_{\circ }\left({\phi }_{P^{\prime }}\left(x\right)\right){\phi }_{V^{\prime }},{\phi }_{V^{\prime }}\left(S_{[,{]}^{\prime }}\left(x\right)\right)=S_{[,]}\left({\phi }_{P^{\prime }}\left(x\right)\right){\phi }_{V^{\prime }},$ (3.14)

$\forall x\in P^{\prime }$，则称 ${\phi }_{P^{\prime }},{\phi }_{V^{\prime }}$ 为模 $\left(V^{\prime },S_{[,{]}^{\prime }},S_{{\circ }^{\prime }}\right)$ 到 $\left(V,S_{[,]},S_{\circ }\right)$ 的同态。

定义3.5 设 $\left(P,[,],\circ \right)$ 为泊松代数， $\left(P,{[,]}_t,{\circ }_t\right)$ 与 $\left(P,{[,]}_{t^{\prime }},{\circ }_{t^{\prime }}\right)$ 分别为 $\left(P,[,],\circ \right)$ 的形变， $\left(V,S_{[,]},S_{\circ }\right)$ 为 $\left(P,[,],\circ \right)$ 的模，模 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 和模 $\left(V,S_{{[,{]}^{\prime }}_t},S_{{{\circ }^{\prime }}_t}\right)$ 分别是P-模V的两个无穷小形变，如果存在 $N\in End\left(P\right)$ 和 $M\in End\left(V\right)$，使得 $\left(I{d}_P+tN,I{d}_V+tM\right)$ 是模 $\left(V,S_{{[,{]}^{\prime }}_t},S_{{{\circ }^{\prime }}_t}\right)$ 到模 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 的同态，即满足下列方程

$\left(I{d}_P+tN\right)\left(\left[x,y\right]{\text{'}}_t\right)={\left[\left(I{d}_P+tN\right)\left(x\right),\left(I{d}_P+tN\right)\left(y\right)\right]}_t,$ (3.15)

$\left(I{d}_P+tN\right)\left(x{{\circ }^{\prime }}_{t}y\right)=\left(I{d}_P+tN\right)\left(x\right){\circ }_t\left(I{d}_P+tN\right)\left(y\right),$ (3.16)

$\left(I{d}_V+tM\right)S_{{[,{]}^{\prime }}_t}\left(x\right)=S_{{[,]}_t}\left(\left(I{d}_V+tM\right)\left(x\right)\right)\left(I{d}_V+tM\right),$ (3.17)

$\left(I{d}_V+tM\right)S_{{{\circ }^{\prime }}_t}\left(x\right)=S_{{\circ }_t}\left(\left(I{d}_V+tM\right)\left(x\right)\right)\left(I{d}_V+tM\right),$ (3.18)

$\forall x,y\in P$，则称两个无穷小形变是等价的。如果P-模V的一个无穷小形变和P-模V是等价的，则称该形变为平凡形变。

定理3.2 设 $\left(P,{[,]}_t,{\circ }_t\right)$ 为 $\left(P,[,],\circ \right)$ 的形变， $\left(V,S_{[,]},S_{\circ }\right)$ 为 $\left(P,[,],\circ \right)$ 的模，则泊松代数 $\left(P,{[,]}_t,{\circ }_t\right)$ 的模 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 是P-模V的平凡形变当且仅当存在 $N\in End\left(P\right),M\in End\left(V\right)$ 使得

${\omega }_{\text{1}}\left(x,y\right)=\left[x,N\left(y\right)\right]+\left[N\left(x\right),y\right]-N\left(\left[x,y\right]\right),$ (3.19)

$N\left({\omega }_1\left(x,y\right)\right)=\left[N\left(x\right),N\left(y\right)\right],$ (3.20)

${\omega }_{\text{2}}\left(x,y\right)=x\circ N\left(y\right)+N\left(x\right)\circ y-N\left(x\circ y\right),$ (3.21)

$N\left({\omega }_2\left(x,y\right)\right)=N\left(x\right)\circ N\left(y\right),$ (3.22)

$\sigma \left(x\right)=S_{[,]}\left(N\left(x\right)\right)+S_{[,]}\left(x\right)M-M{S}_{[,]}\left(x\right),$ (3.23)

$S_{[,]}\left(N\left(x\right)\right)M=M\sigma \left(x\right),$ (3.24)

$\tau \left(x\right)=S_{\circ }\left(N\left(x\right)\right)+S_{\circ }\left(x\right)M-M{S}_{\circ }\left(x\right),$ (3.25)

$S_{\circ }\left(N\left(x\right)\right)M=M\tau \left(x\right),$ (3.26)

$\forall x,y\in P.$

证明: $\forall x,y\in P$，由定义3.5知，泊松代数 $\left(P,{[,]}_t,{\circ }_t\right)$ 的模 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 是P-模V的平凡形变当且仅当存在 $N\in End\left(P\right),M\in End\left(V\right)$ 使得

$\left(I{d}_P+tN\right)\left({\left[x,y\right]}_t\right)=\left[\left(I{d}_P+tN\right)\left(x\right),\left(I{d}_P+tN\right)\left(y\right)\right],$ (3.27)

$\left(I{d}_P+tN\right)\left(x{\circ }_{t}y\right)=\left(I{d}_P+tN\right)\left(x\right)\circ \left(I{d}_P+tN\right)\left(y\right),$ (3.28)

$\left(I{d}_V+tM\right)S_{{[,]}_t}\left(x\right)=S_{[,]}\left(\left(I{d}_V+tM\right)\left(x\right)\right)\left(I{d}_V+tM\right),$ (3.29)

$\left(I{d}_V+tM\right)S_{{\circ }_t}\left(x\right)=S_{\circ }\left(\left(I{d}_V+tM\right)\left(x\right)\right)\left(I{d}_V+tM\right),$ (3.30)

由于

$\begin{array}{l}\left(I{d}_P+tN\right)\left({\left[x,y\right]}_t\right)-\left[\left(I{d}_P+tN\right)\left(x\right),\left(I{d}_P+tN\right)\left(y\right)\right]\\ =\left(\left[x,y\right]+t{\omega }_1\left(x,y\right)\right)+tN\left(\left[x,y\right]+t{\omega }_1\left(x,y\right)\right)-\left(\left[x+tN\left(x\right),y+tN\left(y\right)\right]\right)\\ =t\left({\omega }_1\left(x,y\right)-\left[x,N\left(y\right)\right]-\left[N\left(x\right),y\right]+N\left(\left[x,y\right]\right)\right)+t^2\left(N\left({\omega }_1\left(x,y\right)\right)-\left[N\left(x\right),N\left(y\right)\right]\right),\end{array}$

因此(3.27)式成立当且仅当(3.19)、(3.20)成立。由于

$\begin{array}{l}\left(I{d}_P+tN\right)\left(x{\circ }_{t}y\right)-\left(\left(I{d}_P+tN\right)\left(x\right)\circ \left(I{d}_P+tN\right)\left(y\right)\right)\\ =\left(x\circ y+t{\omega }_2\left(x,y\right)\right)+tN\left(\left(x\circ y\right)+t{\omega }_2\left(x,y\right)\right)-\left(\left(x+tN\left(x\right)\right)\circ \left(y+tN\left(y\right)\right)\right)\\ =t\left({\omega }_2\left(x,y\right)-x\circ N\left(y\right)-N\left(x\right)\circ y+N\left(x\circ y\right)\right)+t^2\left(N\left({\omega }_2\left(x,y\right)\right)-N\left(x\right)\circ N\left(y\right)\right),\end{array}$

因此(3.28)式成立当且仅当(3.21)、(3.22)成立。由于

$\begin{array}{l}\left(I{d}_V+tM\right)S_{{[,]}_t}\left(x\right)-\left(S_{[,]}\left(\left(I{d}_V+tM\right)\left(x\right)\right)\left(I{d}_V+tM\right)\right)\\ =\left(S_{[,]}\left(x\right)+t\sigma \left(x\right)\right)+tM\left(S_{[,]}\left(x\right)+t\sigma \left(x\right)\right)-S_{[,]}\left(x+tN\left(x\right)\right)-t{S}_{[,]}\left(x+tN\left(x\right)\right)M\\ =t\left(\sigma \left(x\right)-S_{[,]}\left(N\left(x\right)\right)-S_{[,]}\left(x\right)M+M{S}_{[,]}\left(x\right)\right)+t^2\left(S_{[,]}\left(N\left(x\right)\right)M-M\sigma \left(x\right)\right),\end{array}$

因此(3.29)成立当且仅当(3.23)、(3.24)成立。由于

$\begin{array}{l}\left(I{d}_V+tM\right)S_{{\circ }_t}\left(x\right)-\left(S_{\circ }\left(\left(I{d}_V+tM\right)\left(x\right)\right)\left(I{d}_V+tM\right)\right)\\ =\left(S_{\circ }\left(x\right)+t\tau \left(x\right)\right)+tM\left(S_{\circ }\left(x\right)+t\tau \left(x\right)\right)-S_{\circ }\left(x+tN\left(x\right)\right)-t{S}_{\circ }\left(x+tN\left(x\right)\right)M\\ =t\left(\tau \left(x\right)-S_{\circ }\left(N\left(x\right)\right)-S_{\circ }\left(x\right)M+M{S}_{\circ }\left(x\right)\right)+t^2\left(S_{\circ }\left(N\left(x\right)\right)M-M\tau \left(x\right)\right),\end{array}$

因此(3.30)成立当且仅当(3.25)、(3.26)成立。综上，结论成立。

定义3.6 如果泊松代数 $\left(P,[,],\circ \right)$ 上线性变换N满足

$\left[N\left(x\right),N\left(y\right)\right]=N\left(\left[N\left(x\right),y\right]+\left[x,N\left(y\right)\right]-N\left(\left[x,y\right]\right)\right),$ (3.31)

$N\left(x\right)\circ N\left(y\right)=N\left(N\left(x\right)\circ y+x\circ N\left(y\right)-N\left(x\circ y\right)\right),$ (3.32)

$\forall x,y\in P$，则称N是泊松代数 $\left(P,[,],\circ \right)$ 的Nijenhuis算子。

定理3.3 设 $\left(P,[,],\circ \right)$ 为泊松代数， $\left(V,S_{[,]},S_{\circ }\right)$ 为 $\left(P,[,],\circ \right)$ 的模， $N\in End\left(P\right)$， $M\in End\left(V\right)$， $\sigma ,\tau :P\to End\left(V\right)$ 为线性映射，如果 $N,M,\sigma ,\tau$ 满足等式(3.19)~(3.26)，则N是泊松代数 $\left(P,[,],\circ \right)$ 的Nijenhuis算子且下面两个等式成立:

$S_{[,]}\left(N\left(x\right)\right)M\left(v\right)=M\left(S_{[,]}\left(N\left(x\right)\right)v+S_{[,]}\left(x\right)M\left(v\right)-M\left(S_{[,]}\left(x\right)v\right)\right),\forall x\in P,v\in V.$ (3.33)

$S_{\circ }\left(N\left(x\right)\right)M\left(v\right)=M\left(S_{\circ }\left(N\left(x\right)\right)v+S_{\circ }\left(x\right)M\left(v\right)-M\left(S_{\circ }\left(x\right)v\right)\right),\forall x\in P,v\in V.$ (3.34)

证明：将(3.19)代入(3.20)中，有

$\left[N\left(x\right),N\left(y\right)\right]=N\left(\left[N\left(x\right),y\right]+\left[x,N\left(y\right)\right]-N\left(\left[x,y\right]\right)\right),$

同理将(3.21)代入3.22)中，有

$N\left(x\right)\circ N\left(y\right)=N\left(N\left(x\right)\circ y+x\circ N\left(y\right)-N\left(x\circ y\right)\right),$

即N是泊松代数 $\left(P,[,],\circ \right)$ 的Nijenhuis算子。将(3.23)代入(3.24)中有(3.33)成立。将(3.25)代入(3.26)中有(3.34)成立。综上可知，结论成立。

定理3.4 设 $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模， $M\in End\left(V\right)$，N是泊松代数 $\left(P,[,],\circ \right)$ 的Nijenhuis算子，并且M满足等式(3.33)、(3.34)，设 ${\omega }_{\text{1}},{\omega }_2:P\otimes P\to P$， $\sigma ,\tau \in End\left(V\right)$ 使得

${\omega }_{\text{1}}\left(x,y\right)=\left[N\left(x\right),y\right]+\left[x,N\left(y\right)\right]-N\left(\left[x,y\right]\right),$ (3.35)

${\omega }_{\text{2}}\left(x,y\right)=N\left(x\right)\circ y+x\circ N\left(y\right)-N\left(x\circ y\right),$ (3.36)

$\sigma \left(x\right)=S_{[,]}\left(N\left(x\right)\right)+S_{[,]}\left(x\right)M-M{S}_{[,]}\left(x\right),$ (3.37)

$\tau \left(x\right)=S_{\circ }\left(N\left(x\right)\right)+S_{\circ }\left(x\right)M-M{S}_{\circ }\left(x\right),$ (3.38)

$\forall x,y\in P$，则 $\left({\omega }_1,{\omega }_2,\sigma ,\tau \right)$ 生成P-模V的平凡无穷小形变。

证明：要证 $\left({\omega }_1,{\omega }_2,\sigma ,\tau \right)$ 生成P-模V的无穷小形变，须证 $\left(P,{[,]}_t,{\circ }_t\right)$ 是泊松代数，且 $\left(V,S_{{[,]}_t},S_{{\circ }_t}\right)$ 是 $\left(P,{[,]}_t,{\circ }_t\right)$ 的模，即证所设 $\left({\omega }_1,{\omega }_2,\sigma ,\tau \right)$ 满足等式(2.2)~(2.9)和(3.6)~(3.13)。

$\forall x_1,x_2,y\in P$， $k_1,k_2\in F$，由 ${\omega }_{\text{1}}$ 满足等式(3.35)及 $[,]$ 双线性可知，

$\begin{array}{l}{\omega }_1\left(k_1{x}_1+k_2{x}_2,y\right)-k_1{\omega }_1\left(x_1,y\right)-k_2{\omega }_{\text{1}}\left(x_2,y\right)\\ =k_1\left(\left[N\left(x_1\right),y\right]-\left[N\left(x_1\right),y\right]+\left[x_1,N\left(y\right)\right]-\left[x_1,N\left(y\right)\right]+N\left(\left[x_1,y\right]\right)-N\left(\left[x_1,y\right]\right)\right)\\ +k_2\left(\left[N\left(x_2\right),y\right]-\left[N\left(x_2\right),y\right]+\left[x_2,N\left(y\right)\right]-\left[x_2,N\left(y\right)\right]+N\left(\left[x_2,y\right]\right)-N\left(\left[x_2,y\right]\right)\right)\\ =0,\end{array}$

因此 ${\omega }_{\text{1}}$ 双线性。同理可知 ${\omega }_{\text{2}}$ 双线性。由 ${\omega }_{\text{1}}$ 满足等式(3.35)及 $[,]$ 反对称性知

${\omega }_{\text{1}}\left(x,y\right)+{\omega }_1\left(y,x\right)=\left[N\left(x\right),y\right]+\left[y,N\left(x\right)\right]+\left[x,N\left(y\right)\right]+\left[N\left(y\right),x\right]-N\left(\left[x,y\right]\right)-N\left(\left[y,x\right]\right)=0,$

因此(2.2)式成立。由 ${\omega }_{\text{1}}$ 满足等式(3.35)、 $[,]$ Jacobi恒等式及N是 $\left(P,[,],\circ \right)$ 上的Nijenhuis算子知

$\begin{array}{l}\left[x,{\omega }_1\left(y,z\right)\right]+{\omega }_{\text{1}}\left(x,\left[y,z\right]\right)+\left[y,{\omega }_1\left(z,x\right)\right]+{\omega }_{\text{1}}\left(y,\left[z,x\right]\right)+\left[z,{\omega }_1\left(y,x\right)\right]+{\omega }_{\text{1}}\left(z,\left[y,x\right]\right)\\ =\left[x,\left[N\left(y\right),z\right]\right]-\left[x,\left[N\left(y\right),z\right]\right]+\left[x,\left[y,N\left(z\right)\right]\right]-\left[x,\left[y,N\left(z\right)\right]\right]\\ +\left[N\left(x\right),\left[y,z\right]\right]+\left[x,N\left(\left[y,z\right]\right)\right]-N\left(\left[x,\left[y,z\right]\right]\right)\\ +\left[y,\left[N\left(z\right),x\right]\right]-\left[y,\left[N\left(z\right),x\right]\right]+\left[y,\left[z,N\left(x\right)\right]\right]-\left[y,\left[z,N\left(x\right)\right]\right]\\ +\left[N\left(y\right),\left[z,x\right]\right]+\left[y,N\left(\left[z,x\right]\right)\right]-N\left(\left[y,\left[z,x\right]\right]\right)\end{array}$

$\begin{array}{l}+\left[z,\left[N\left(x\right),y\right]\right]-\left[z,\left[N\left(x\right),y\right]\right]+\left[z,\left[x,N\left(y\right)\right]\right]-\left[z,\left[x,N\left(y\right)\right]\right]\\ +\left[N\left(z\right),\left[x,y\right]\right]+\left[z,N\left(\left[x,y\right]\right)\right]-N\left(\left[z,\left[x,y\right]\right]\right)\\ +\text{2}\left(\left[x,\left[y,z\right]\right]+\left[y,\left[z,x\right]\right]+\left[z,\left[x,y\right]\right]\right)\\ =0,\end{array}$

因此(2.3)式成立。同理将 ${\omega }_{\text{1}},{\omega }_2,\sigma ,\tau$ 代入等式(2.4)~(2.9)和(3.6)~(3.13)，利用已知条件，得以上等式均成立。故 $\left({\omega }_1,{\omega }_2,\sigma ,\tau \right)$ 生成P-模V的无穷小形变。由N是泊松代数 $\left(P,[,],\circ \right)$ 的Nijenhuis算子和等式(3.35)、(3.36)知(3.19)~(3.22)成立，由等式(3.33)、(3.37)知(3.23)、(3.24)成立，同理由等式(3.34)、(3.38)知(3.25)、(3.26)成立，所以此无穷小形变也满足条件(3.19)~(3.26)，故该形变是平凡形变。

4. 泊松代数半直积的Nijenhuis算子

命题4.1 [5] 设 $\left(P,[,],\circ \right)$ 是泊松代数，V是域F上的线性空间， $S_{[,]},S_{\circ }:P\to End\left(V\right)$ 为线性映射，在 $P\oplus V$ 上定义

$\left[x_1+v_1,x_2+v_2\right]=\left[x_1,x_2\right]+S_{[,]}\left(x_1\right)v_2-S_{[,]}\left(x_2\right)v_1,$ (4.1)

$\left(x_1+v_1\right)\circ \left(x_2+v_2\right)=x_1\circ x_2+S_{\circ }\left(x_1\right)v_2+S_{\circ }\left(x_2\right)v_1,$ (4.2)

$\forall x_1,x_2\in P,v_1,v_2\in V$，则 $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模当且仅当 $\left(P\oplus V,[,],\circ \right)$ 是泊松代数，简记为 $P{⋉}_{S_{[,]},S_{°}}V$。

定理4.2 设 $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模， $N\in End\left(P\right)$， $M\in End\left(V\right)$，则N是泊松代数 $\left(P,[,],\circ \right)$ 上的Nijenhuis算子且M满足等式(3.33)、(3.34)当且仅当线性变换 $N+M$ 是 $P{⋉}_{S_{[,]},S_{°}}V$ 上的Nijenhuis算子。

证明：必要性。 $\forall x_1,x_2,P,v_1,v_2\in V$，由等式(3.33)及N是 $\left(P,[,],\circ \right)$ 上的Nijenhuis算子知，

$\begin{array}{l}\left[\left(N+M\right)\left(x_1+v_1\right),\left(N+M\right)\left(x_2+v_2\right)\right]\\ =\left[N\left(x_1\right)+M\left(v_1\right),N\left(x_2\right)+M\left(v_2\right)\right]\\ =\left[N\left(x_1\right),N\left(x_2\right)\right]+S_{[,]}\left(N\left(x_1\right)\right)M\left(v_2\right)-S_{[,]}\left(N\left(x_2\right)\right)M\left(v_1\right),\end{array}$

$\begin{array}{l}\left(N+M\right)\left[\left(N+M\right)\left(x_1+v_1\right),x_2+v_2\right]+\left[x_1+v_1,\left(N+M\right)\left(x_2+v_2\right)\right]-\left(N+M\right)\left(\left[x_1+v_1,x_2+v_2\right]\right))\\ =\left(N+M\right)(\left[N\left(x_1\right)+M\left(v_1\right),x_2+v_2\right]+\left[x_1+v_1,N\left(x_2\right)+M\left(v_2\right)\right]\\ -\left(N+M\right)\left(\left[x_1,x_2\right]+S_{[,]}\left(x_1\right)v_2-S_{[,]}\left(x_2\right)v_1\right))\\ =N\left(\left[N\left(x_1\right),x_2\right]+\left[x_1,N\left(x_2\right)\right]-N\left(\left[x_1,x_2\right]\right)\right)+M\left(S_{[,]}\left(N\left(x_1\right)\right)v_2+S_{[,]}\left(x_1\right)M\left(v_2\right)-M\left(S_{[,]}\left(x_1\right)v_2\right)\right)\\ -M\left(S_{[,]}\left(N\left(x_2\right)\right)v_1+S_{[,]}\left(x_2\right)M\left(v_1\right)-M\left(S_{[,]}\left(x_2\right)v_1\right)\right)\\ =\left[N\left(x_1\right),N\left(x_2\right)\right]+S_{[,]}\left(N\left(x_1\right)\right)M\left(v_2\right)-S_{[,]}\left(N\left(x_2\right)\right)M\left(v_1\right),\end{array}$

比较上述两式知线性变换 $N+M$ 满足等式(3.31)。由等式(3.34)及N是 $\left(P,[,],\circ \right)$ 上的Nijenhuis算子知，

$\begin{array}{l}\left(N+M\right)\left(x_1+v_1\right)\circ \left(N+M\right)\left(x_2+v_2\right)\\ =\left(N\left(x_1\right)+M\left(v_1\right)\right)\circ \left(N\left(x_2\right)+M\left(v_2\right)\right)\\ =N\left(x_1\right)\circ N\left(x_2\right)+S_{\circ }\left(N\left(x_1\right)\right)M\left(v_2\right)+S_{\circ }\left(N\left(x_2\right)\right)M\left(v_1\right),\end{array}$

$\begin{array}{l}\left(N+M\right)\left(\left(N+M\right)\left(x_1+v_1\right)\circ \left(x_2+v_2\right)+\left(x_1+v_1\right)\circ \left(N+M\right)\left(x_2+v_2\right)-\left(N+M\right)\left(\left(x_1+v_1\right)\circ \left(x_2+v_2\right)\right)\right)\\ =\left(N+M\right)(\left(N\left(x_1\right)+M\left(v_1\right)\right)\circ \left(x_2+v_2\right)+\left(x_1+v_1\right)\circ \left(N\left(x_2\right)+M\left(v_2\right)\right)\\ -\left(N+M\right)\left(\left(x_1\circ x_2\right)+S_{\circ }\left(x_1\right)v_2-S_{\circ }\left(x_2\right)v_1\right))\\ =N\left(N\left(x_1\right)\circ x_2+x_1\circ N\left(x_2\right)-N\left(x_1\circ x_2\right)\right)+M\left(S_{\circ }\left(N\left(x_1\right)\right)v_2+S_{\circ }\left(x_1\right)M\left(v_2\right)-M\left(S_{\circ }\left(x_1\right)v_2\right)\right)\\ +M\left(S_{\circ }\left(N\left(x_2\right)\right)v_1+S_{\circ }\left(x_2\right)M\left(v_1\right)-M\left(S_{\circ }\left(x_2\right)v_1\right)\right)\\ =N\left(x_1\right)\circ N\left(x_2\right)+S_{\circ }\left(N\left(x_1\right)\right)M\left(v_2\right)-S_{\circ }\left(N\left(x_2\right)\right)M\left(v_1\right),\end{array}$

比较上述两式知线性变换 $N+M$ 满足等式(3.32)。所以 $N+M$ 是泊松代数半直积 $P{⋉}_{S_{[,]},S_{°}}V$ 上的Nijenhuis算子。

充分性。设 $N+M$ 是半直积泊松代数 $P{⋉}_{S_{[,]},S_{°}}V$ 上的Nijenhuis算子，则

$\begin{array}{l}\left(N+M\right)(\left[\left(N+M\right)\left(x_1+v_1\right),x_2+v_2\right]+\left[x_1+v_1,\left(N+M\right)\left(x_2+v_2\right)\right]\\ -\left(N+M\right)\left(\left[x_1+v_1,x_2+v_2\right]\right))-\left(\left[\left(N+M\right)\left(x_1+v_1\right),\left(N+M\right)\left(x_2+v_2\right)\right]\right)\\ =\left(N\left(\left[N\left(x_1\right),x_2\right]+\left[x_1,N\left(x_2\right)\right]-N\left(\left[x_1,x_2\right]\right)\right)-\left(\left[N\left(x_1\right),N\left(x_{\text{2}}\right)\right]\right)\right)\\ +\left(-S_{[,]}\left(N\left(x_{\text{1}}\right)\right)M\left(v_{\text{2}}\right)+M\left(S_{[,]}\left(N\left(x_1\right)\right)v_2+S_{[,]}\left(x_1\right)M\left(v_2\right)-M\left(S_{[,]}\left(x_1\right)v_2\right)\right)\right)\\ +\left(S_{[,]}\left(N\left(x_2\right)\right)M\left(v_1\right)-M\left(S_{[,]}\left(N\left(x_2\right)\right)v_1+S_{[,]}\left(x_2\right)M\left(v_1\right)-M\left(S_{[,]}\left(x_2\right)v_1\right)\right)\right)\\ =0,\end{array}$

故线性变换N满足等式(3.30)且M满足等式(3.33)。又

$\begin{array}{l}\left(N+M\right)(\left(N+M\right)\left(x_1+v_1\right)\circ \left(x_2+v_2\right)+\left(x_1+v_1\right)\circ \left(N+M\right)\left(x_2+v_2\right)\\ -\left(N+M\right)\left(\left(x_1+v_1\right)\circ \left(x_2+v_2\right)\right))-\left(\left(N+M\right)\left(x_1+v_1\right)\circ \left(N+M\right)\left(x_2+v_2\right)\right)\\ =\left(N\left(N\left(x_1\right)\circ x_2+x_1\circ N\left(x_2\right)-N\left(x_1\circ x_2\right)\right)-\left(N\left(x_1\right)\circ N\left(x_2\right)\right)\right)\\ +\left(-S_{[,]}\left(N\left(x_{\text{1}}\right)\right)M(v_{\text{2}})+M\left(S_{\circ }\left(N\left(x_{\text{1}}\right)\right)v_{\text{2}}+S_{\circ }\left(x_{\text{1}}\right)M\left(v_{\text{2}}\right)-M\left(S_{\circ }\left(x_{\text{1}}\right)v_{\text{2}}\right)\right)\right)\\ +\left(S_{[,]}\left(N\left(x_2\right)\right)M\left(v_1\right)-M\left(S_{\circ }\left(N\left(x_2\right)\right)v_1+S_{\circ }\left(x_2\right)M\left(v_1\right)-M\left(S_{\circ }\left(x_2\right)v_1\right)\right)\right)\\ =\text{0},\end{array}$

故线性变换N满足等式(3.31)且M满足等式(3.34)。所以N是泊松代数 $\left(P,[,],\circ \right)$ 上的Nijenhuis算子且M满足等式(3.33)、(3.34)。综上，结论成立。

定理4.3 [4] 设 $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模，则 $\left(V^{\ast },S_{[,]}^{\ast },S_{\circ }^{\ast }\right)$ 是 $\left(P,[,],\circ \right)$ 的模。

定义4.1 设 $\left(V,S_{[,]},S_{\circ }\right)$ 是泊松代数 $\left(P,[,],\circ \right)$ 的模， $N\in End\left(P\right),M\in End\left(V\right)$。如果 $N,M^{\ast }$ 生成 $\left(V^{\ast },S_{[,]}^{\ast },S_{\circ }^{\ast }\right)$ 的平凡形变，则称 $\left(N,M\right)$ 是P-模V的Nijenhuis结构。

定理4.4 设 $\left(V,S_{[,]},S_{\circ }\right)$ 是 $\left(P,[,],\circ \right)$ 的模，则 $\left(N,M\right)$ 是P-模V的Nijenhuis结构等价于N是 $\left(P,[,],\circ \right)$ 的Nijenhuis算子且满足

$S_{[,]}\left(N\left(x\right)\right)M\left(v\right)=M\left(S_{[,]}\left(N\left(x\right)\right)v\right)+S_{[,]}\left(x\right)M^2\left(v\right)-M\left(S_{[,]}\left(x\right)M\left(v\right)\right),$ (4.3)

$S_{\circ }\left(N\left(x\right)\right)M\left(v\right)=M\left(S_{\circ }\left(N\left(x\right)\right)v\right)+S_{\circ }\left(x\right)M^2\left(v\right)-M\left(S_{\circ }\left(x\right)M\left(v\right)\right).$ (4.4)

$\forall x\in P,v\in V.$

证明：由定理4.2知，N和 $M^{\ast }$ 生成泊松代数 $\left(P,[,],\circ \right)$ 的模 $\left(V^{\ast },S_{[,]}^{\ast },S_{\circ }^{\ast }\right)$ 的无穷小形变当且仅当N是泊松代数 $\left(P,[,],\circ \right)$ 的Nijenhuis算子，且满足下列方程

$S_{[,]}^{\ast }\left(N\left(x\right)\right)M^{\ast }\left(v^{\ast }\right)=M^{\ast }\left(S_{[,]}^{\ast }\left(N\left(x\right)\right)v^{\ast }+S_{[,]}^{\ast }\left(x\right)M^{\ast }\left(v\right)-M^{\ast }\left(S_{[,]}^{\ast }\left(x\right)v^{\ast }\right)\right),$ (4.5)

$S_{\circ }^{\ast }\left(N\left(x\right)\right)M^{\ast }\left(v^{\ast }\right)=M^{\ast }\left(S_{\circ }^{\ast }\left(N\left(x\right)\right)v^{\ast }+S_{\circ }^{\ast }\left(x\right)M^{\ast }\left(v\right)-M^{\ast }\left(S_{\circ }^{\ast }\left(x\right)v^{\ast }\right)\right).$ (4.6)

$\forall x\in P,v\in V,v^{\ast }\in V$。由于

$\begin{array}{l}\langle S_{[,]}^{\ast }\left(N\left(x\right)\right)M^{\ast }\left(v^{\ast }\right)-M^{\ast }\left(S_{[,]}^{\ast }\left(N\left(x\right)\right)v^{\ast }\right)-M^{\ast }\left(S_{[,]}^{\ast }\left(x\right)M^{\ast }\left(v\right)\right)+M^{\ast }\left(M^{\ast }\left(S_{[,]}^{\ast }\left(x\right)v^{\ast }\right)\right),v\rangle \\ =\langle v^{\ast },S_{[,]}\left(N\left(x\right)\right)M\left(v\right)-M\left(S_{[,]}\left(N\left(x\right)\right)v\right)-S_{[,]}\left(x\right)M^2\left(v\right)+M\left(S_{[,]}\left(x\right)M\left(v\right)\right)\rangle ,\end{array}$

所以等式(4.5)成立当且仅当等式(4.3)成立。由于

$\begin{array}{l}\langle S_{[,]}^{\ast }\left(N\left(x\right)\right)M^{\ast }\left(v^{\ast }\right)-M^{\ast }\left(S_{[,]}^{\ast }\left(N\left(x\right)\right)v^{\ast }\right)-M^{\ast }\left(S_{[,]}^{\ast }\left(x\right)M^{\ast }\left(v\right)\right)+M^{\ast }\left(M^{\ast }\left(S_{[,]}^{\ast }\left(x\right)v^{\ast }\right)\right),v\rangle \\ =\langle v^{\ast },S_{[,]}\left(N\left(x\right)\right)M\left(v\right)-M\left(S_{[,]}\left(N\left(x\right)\right)v\right)-S_{[,]}\left(x\right)M^2\left(v\right)+M\left(S_{[,]}\left(x\right)M\left(v\right)\right)\rangle ,\end{array}$

所以等式(4.6)成立当且仅当等式(4.4)成立。综上可知，结论成立。

5. 结束语

本文给出了一类特殊泊松代数的构造方法，并且由此为基础进一步研究泊松代数的模、无穷小形变以及泊松代数半直积的Nijenhuis算子。对于这类特殊的泊松代数的研究具有一定的意义。