相容BiHom-李代数的表示及BiHom-李代数的形变

Representation of Compatible BiHom-Lie Algebra and Deformation of BiHom-Lie Algebra

摘要: 本文主要研究相容BiHom-李代数的表示与BiHom-李代数的形变。首先给出了相容BiHom-李代数的定义，找到判断两个BiHom-李代数是相容的方法。其次定义了相容BiHom-李代数的表示，得到了相容BiHom-李代数与其表示空间的直和上存在相容BiHom-李代数结构的条件，并给出了相容BiHom-李代数表示的例子。然后讨论了相容BiHom-李代数表示的对偶映射同样是该相容BiHom-李代数表示所满足的条件，并构造出了对偶表示。最后构造了BiHom-李代数的形变。

Abstract: In this paper, we mainly discuss the representation of BiHom-Lie algebra and the deformation of BiHom-Lie algebra. Firstly, we give the definition of compatible BiHom-Lie algebra and the method of judging the compatibility of two BiHom-Lie algebras. Secondly, we define the representation of compatible BiHom-Lie algebra, and get the conditions for the existence of compatible BiHom-Lie al-gebra structure on the direct sum of the compatible BiHom-Lie algebra and its representation space. We also give an example of the representation of compatible BiHom-Lie algebra. Then we give the conditions when the dual mapping of the representation of compatible BiHom-Lie algebra is also the representation of compatible BiHom-Lie algebra and constructs the dual representation of the representation of compatible BiHom-Lie algebra. Finally, we introduce the deformation of Bi-Hom-Lie algebra.

1. 引言

BiHom-李代数是一类非常重要的代数，与李代数、Hom-李代数有非常密切的联系。当BiHom-李代数的两个扭曲映射为恒等映射时，BiHom-李代数即为李代数；当BiHom-李代数的两个扭曲映射相等时即为Hom-李代数。BiHom-李代数是2015年Graziani G，Makhlouf A，Menini C在文献 [1] 中提出的，之后一些学者对BiHom-李代数作了进一步研究。例如，文献 [2] 中作者研究了BiHom-李代数的分裂扩张和相应的上同调，并且建立了一个由交换BiHom-李代数V扩张的BiHom-李代数L的等价类与它的第二上同调群之间的对应关系。文献 [3] 中作者研究了BiHom-李代数关于表示、平凡表示、伴随表示的上同调。对于李代数和Hom-李代数，文献 [4] [5] 研究了相容李代数和相容Hom-李代数的上同调。本文主要研究相容BiHom-李代数的表示和BiHom-李代数的形变。

2. 预备知识

定义2.1 ( [1])若g是域K上的线性空间， $\alpha ,\beta \in End\left(g\right)$， $\left[-,-\right]:g\otimes g\to g$ 是双线性映射， $\alpha ,\beta :g\to g$ 是线性映射，如果 $\forall a,b,c\in g$，恒有

$\alpha \beta =\beta \alpha$， (2.1)

$\left[\beta \left(a\right),\alpha \left(b\right)\right]=-\left[\beta \left(b\right),\alpha \left(a\right)\right]$， (2.2)

$\left[{\beta }^{\text{2}}\left(a\right),\left[\beta \left(b\right),\alpha \left(c\right)\right]\right]+\left[{\beta }^2\left(b\right),\left[\beta \left(c\right),\alpha \left(a\right)\right]\right]+\left[{\beta }^2\left(c\right),\left[\beta \left(a\right),\alpha \left(b\right)\right]\right]=0$， (2.3)

$\alpha \left[a,b\right]=\left[\alpha \left(a\right),\alpha \left(b\right)\right]$， $\beta \left[a,b\right]=\left[\beta \left(a\right),\beta \left(b\right)\right]$， (2.4)

则称 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数。

注记2.1 ( [3]) 设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数，若 $\alpha ,\beta$ 是双射，则称 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 为正则BiHom-李代数。

定义2.2 ( [1])设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数，V是线性空间， ${\alpha }_V,{\beta }_V\in End\left(V\right)$ 且 ${\alpha }_V{\beta }_V={\beta }_V{\alpha }_V$， $\rho :g\to End\left(V\right)$ 是线性映射，如果对于 $\forall x,y\in g$ 恒有

$\rho \left(\alpha \left(x\right)\right)\circ {\alpha }_V={\alpha }_V\circ \rho \left(x\right)$， $\rho \left(\beta \left(x\right)\right)\circ {\beta }_V={\beta }_V\circ \rho \left(x\right)$， (2.5)

$\rho \left(\left[\beta \left(x\right),y\right]\right)\circ {\beta }_V=\rho \left(\alpha \beta \left(x\right)\right)\circ \rho \left(y\right)-\rho \left(\beta \left(y\right)\right)\circ \rho \left(\alpha \left(x\right)\right)$，(2.6)

则称 $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 是BiHom-李代数 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示。

命题2.1 ( [1])设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数， $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示， $\alpha ,{\beta }_V$ 可逆，在 $g\oplus V$ 上定义

$\left(\alpha +{\alpha }_V\right)\left(x+u\right)=\alpha \left(x\right)+{\alpha }_V\left(u\right)$， $\left(\beta +{\beta }_V\right)\left(x+u\right)=\beta \left(x\right)+{\beta }_V\left(u\right)$，

$\left\{x+u,y+v\right\}=\left[x,y\right]+\rho \left(x\right)v-\rho \left({\alpha }^{-1}\beta \left(y\right)\right)\left({\alpha }_V{\beta }_V^{-1}\left(u\right)\right)$，

其中 $x,y\in g,u,v\in V$，则 $\left(g\oplus V,\left\{-,-\right\},\alpha +{\alpha }_V,\beta +{\beta }_V\right)$ 是BiHom-李代数。

注记2.2 通过直接计算可知，当 $\left(g\oplus V,\left\{-,-\right\},\alpha +{\alpha }_V,\beta +{\beta }_V\right)$ 是BiHom-李代数时， $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示。

命题2.2 ( [1])设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是正则BiHom-李代数，定义 $ad:g\to End\left(g\right)$，其中 $ad\left(x\right)y=\left[x,y\right]\left(\forall x,y\in g\right)$，则 $\left(g,ad,\alpha ,\beta \right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示，称为伴随表示。

命题2.3 ( [6])设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数， $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 是它的表示，定义 ${\rho }^{*}:g\to End\left(V^{*}\right)$，其中 ${\rho }^{*}\left(x\right)\left(f\right)=-f\circ \rho \left(x\right)$ $\left(\forall f\in V^{*},x\in g\right)$，则 $\left(V^{*},{\rho }^{*},{\alpha }_V^{*},{\beta }_V^{*}\right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示当且仅当对于 $\forall x,y\in g$ 满足

$\rho \left(x\right)\circ {\alpha }_V={\alpha }_V\circ \rho \left(\alpha \left(x\right)\right)$， $\rho \left(x\right)\circ {\beta }_V={\beta }_V\circ \rho \left(\beta \left(x\right)\right)$，(2.7)

${\beta }_V\circ \rho \left(\left[\beta \left(x\right),y\right]\right)=\rho \left(\alpha \left(x\right)\right)\circ \rho \left(\beta \left(y\right)\right)-\rho \left(y\right)\circ \rho \left(\alpha \beta \left(x\right)\right)$。(2.8)

命题2.4 ( [6])设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数， $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示， ${\alpha }_V,{\beta }_V$ 可逆，定义 ${\rho }^{\circ }:g\to End\left(V^{*}\right)$，其中

${\rho }^{\circ }\left(x\right)={\rho }^{\ast }\left(\alpha \beta \left(x\right)\right){\left({\beta }_V^{-2}\right)}^{\ast }$ $\left(\forall x\in g\right)$， (2.9)

则 $\left(V^{*},{\rho }^{\circ },{\left({\alpha }_V^{-1}\right)}^{*},{\left({\beta }_V^{-1}\right)}^{*}\right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示，称为 $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 的对偶表示。

3. 相容BiHom-李代数的表示

定义3.1 设 $\left(g,{\left[-,-\right]}_1,\alpha ,\beta \right)$， $\left(g,{\left[-,-\right]}_{\text{2}},\alpha ,\beta \right)$ 是BiHom-李代数，如果对于 $\forall k_1,k_2\in K$， $\left(g,k_1{\left[-,-\right]}_1+k_2{\left[-,-\right]}_2,\alpha ,\beta \right)$ 恒为BiHom-李代数，则称 $\left(g,{\left[-,-\right]}_1,\alpha ,\beta \right)$ 与 $\left(g,{\left[-,-\right]}_{\text{2}},\alpha ,\beta \right)$ 是相容的，用 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 表示。

定理3.1 两个BiHom-李代数 $\left(g,{\left[-,-\right]}_1,\alpha ,\beta \right)$ 和 $\left(g,{\left[-,-\right]}_{\text{2}},\alpha ,\beta \right)$ 是相容的当且仅当

$\begin{array}{l}{\left[{\beta }^2\left(x\right),{\left[\beta \left(y\right),\alpha \left(z\right)\right]}_1\right]}_2+{\left[{\beta }^2\left(x\right),{\left[\beta \left(y\right),\alpha \left(z\right)\right]}_{\text{2}}\right]}_{\text{1}}+{\left[{\beta }^2\left(y\right),{\left[\beta \left(z\right),\alpha \left(x\right)\right]}_1\right]}_2\\ +{\left[{\beta }^2\left(y\right),{\left[\beta \left(z\right),\alpha \left(x\right)\right]}_2\right]}_1+{\left[{\beta }^2\left(z\right),{\left[\beta \left(x\right),\alpha \left(y\right)\right]}_1\right]}_2+{\left[{\beta }^2\left(z\right),{\left[\beta \left(x\right),\alpha \left(y\right)\right]}_2\right]}_1=0\end{array}$， (3.1)

其中 $\forall x,y,z\in g$。

证明由定义2.1， $\left(g,{\left[-,-\right]}_1,\alpha ,\beta \right)$ 和 $\left(g,{\left[-,-\right]}_{\text{2}},\alpha ,\beta \right)$ 是相容的当且仅当 $\left[-,-\right]=k_1[-,{-]}_1+k_2[-,{-]}_2$ 满足等式(2.1)~(2.4)。由已知 $\alpha ,\beta$ 是可交换的。由于 $\alpha ,\beta$ 关于 ${\left[-,-\right]}_1,{\left[-,-\right]}_2$ 满足等式(2.2)，因此有 $\left[\beta \left(x\right),\alpha \left(y\right)\right]=k_1{\left[\beta \left(x\right),\alpha \left(y\right)\right]}_1+k_2{\left[\beta \left(x\right),\alpha \left(y\right)\right]}_2=-k_1{\left[\beta \left(y\right),\alpha \left(x\right)\right]}_1-k_2{\left[\beta \left(y\right),\alpha \left(x\right)\right]}_2=-\left[\beta \left(y\right),\alpha \left(x\right)\right]$，所以 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 满足等式(2.2)。由于 ${\left[-,-\right]}_1,{\left[-,-\right]}_2$ 满足等式(2.3)，所以

$\begin{array}{l}\left[{\beta }^2\left(x\right),\left[\beta \left(y\right),\alpha \left(z\right)\right]\right]+\left[{\beta }^2\left(y\right),\left[\beta \left(z\right),\alpha \left(x\right)\right]\right]+\left[{\beta }^2\left(z\right),\left[\beta \left(x\right),\alpha \left(y\right)\right]\right]\\ =k_1{k}_2({\left[{\beta }^2\left(x\right),{\left[\beta \left(y\right),\alpha \left(z\right)\right]}_1\right]}_2+{\left[{\beta }^2\left(x\right),{\left[\beta \left(y\right),\alpha \left(z\right)\right]}_2\right]}_1+{\left[{\beta }^2\left(y\right),{\left[\beta \left(z\right),\alpha \left(x\right)\right]}_1\right]}_2\\ +{\left[{\beta }^2\left(y\right),{\left[\beta \left(z\right),\alpha \left(x\right)\right]}_2\right]}_1+{\left[{\beta }^2\left(z\right),{\left[\beta \left(x\right),\alpha \left(y\right)\right]}_1\right]}_2+{\left[{\beta }^2\left(z\right),{\left[\beta \left(x\right),\alpha \left(y\right)\right]}_2\right]}_1)\end{array}$，

因此 $\left[-,-\right]$ 满足等式(2.3)当且仅当 ${\left[-,-\right]}_1,{\left[-,-\right]}_2$ 满足等式(3.1)。由于 $\alpha ,\beta$ 关于 ${\left[-,-\right]}_1,{\left[-,-\right]}_2$ 满足等式(2.4)，所以显然有 $\alpha ,\beta$ 关于 $\left[-,-\right]$ 满足等式(2.4)。证毕。

定义3.2 设 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容的BiHom-李代数，如果 $\left(V,{\rho }_{\text{1}},{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_1,\alpha ,\beta \right)$ 的表示， $\left(V,{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_{\text{2}},\alpha ,\beta \right)$ 的表示，并且对于 $\forall x,y\in g$ 恒有

$\begin{array}{l}{\rho }_{\text{1}}\left({\left[\beta \left(x\right),y\right]}_2\right){\beta }_V+{\rho }_2\left({\left[\beta \left(x\right),y\right]}_1\right){\beta }_V+{\rho }_1\left(\beta \left(y\right)\right){\rho }_2\left(\alpha \left(x\right)\right)\\ +{\rho }_2\left(\beta \left(y\right)\right){\rho }_1\left(\alpha \left(x\right)\right)-{\rho }_1\left(\alpha \beta \left(x\right)\right){\rho }_2\left(y\right)-{\rho }_2\left(\alpha \beta \left(x\right)\right){\rho }_1\left(y\right)=0\end{array}$， (3.2)

则称 $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 是相容BiHom-李代数 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示。

定理3.2 设 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容的BiHom-李代数，V是线性空间， ${\rho }_{\text{1}},{\rho }_{\text{2}}:g\to End\left(V\right)$ 为线性映射， ${\alpha }_V,{\beta }_V\in End\left(V\right)$， $\alpha ,{\beta }_V$ 是双射，在 $g\oplus V$ 上定义

$\left(\alpha +{\alpha }_V\right)\left(x+u\right)=\alpha \left(x\right)+{\alpha }_V\left(u\right)$， $\left(\beta +{\beta }_V\right)\left(x+u\right)=\beta \left(x\right)+{\beta }_V\left(u\right)$，

${\left\{x+u,y+v\right\}}_i={\left[x,y\right]}_i+{\rho }_i\left(x\right)v-{\rho }_i\left({\alpha }^{-1}\beta \left(y\right)\right)\left({\alpha }_V{\beta }_V^{-1}\left(u\right)\right)$，

其中 $i=1,2$， $x,y\in g$， $u,v\in V$，则 $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示当且仅当 $\left(g\oplus V,{\left\{-,-\right\}}_1,{\left\{-,-\right\}}_2,\alpha +{\alpha }_V,\beta +{\beta }_V\right)$ 是相容的BiHom-李代数。

证明 由命题2.1可知， $\left(V,{\rho }_i,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_i,\alpha ,\beta \right)\left(i=1,2\right)$ 的表示当且仅当 $\left(g\oplus V,{\left\{-,-\right\}}_i,\alpha +{\alpha }_V,\beta +{\beta }_V\right)\left(i=1,2\right)$ 是BiHom-李代数，因此只需证明 $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 满足等式(3.2)当且仅当 $\left(g\oplus V,{\left\{-,-\right\}}_1,{\left\{-,-\right\}}_2,\alpha +{\alpha }_V,\beta +{\beta }_V\right)$ 满足等式(3.1)。由于

${\left\{{\left(\beta +\beta {}_V\right)}^2\left(x+u\right),{\left\{\left(\beta +{\beta }_V\right)\left(y+v\right),\left(\alpha +{\alpha }_V\right)\left(z+w\right)\right\}}_1\right\}}_2+\{{\left(\beta +{\beta }_V\right)}^2\left(x+u\right),\{\left(\beta +{\beta }_V\right)\left(y+v\right)$，

$\begin{array}{l}{{\left(\alpha +{\alpha }_V\right)\left(z+w\right)\}}_2\}}_1+{\left\{{\left(\beta +{\beta }_V\right)}^2\left(y+v\right),{\left\{\left(\beta +{\beta }_V\right)\left(z+w\right),\left(\alpha +{\alpha }_V\right)\left(x+u\right)\right\}}_1\right\}}_2\\ +{\left\{{\left(\beta +{\beta }_V\right)}^2\left(y+v\right),{\left\{\left(\beta +{\beta }_V\right)\left(z+w\right),\left(\alpha +{\alpha }_V\right)\left(x+u\right)\right\}}_2\right\}}_1+\{{\left(\beta +{\beta }_V\right)}^2\left(z+w\right),\\ {{\{\left(\beta +{\beta }_V\right)\left(x+u\right),\left(\alpha +{\alpha }_V\right)\left(y+z\right)\}}_1\}}_2+{\left\{{\left(\beta +{\beta }_V\right)}^2\left(z+w\right),{\left\{\left(\beta +{\beta }_V\right)\left(x+u\right),\left(\alpha +{\alpha }_V\right)\left(y+v\right)\right\}}_2\right\}}_1\\ =(-{\rho }_2\left({\left[{\alpha }^{-1}{\beta }^2\left(y\right),\beta \left(z\right)\right]}_1\right)\left({\alpha }_V{\beta }_V\right)-{\rho }_1\left({\left[{\alpha }^{-1}{\beta }^2\left(y\right),\beta \left(z\right)\right]}_2\right)\left({\alpha }_V{\beta }_V\right)+{\rho }_2\left({\beta }^2\left(y\right)\right)\left({\rho }_1\left(\beta \left(z\right)\right)\left({\alpha }_V\right)\right)\\ \begin{array}{c}\end{array}+{\rho }_1\left({\beta }^2\left(y\right)\right)\left({\rho }_2\left(\beta \left(z\right)\right)\left({\alpha }_V\right)\right)-{\rho }_2\left({\beta }^2\left(z\right)\right)\left({\rho }_1\left(\beta \left(y\right)\right)\left({\alpha }_V\right)\right)-{\rho }_1\left({\beta }^2\left(z\right)\right)\left({\rho }_2\left(\beta \left(y\right)\right)\left({\alpha }_V\right)\right))(\; u\; )\end{array}$

$\begin{array}{l}+(-{\rho }_2\left({\left[{\alpha }^{-1}{\beta }^2\left(z\right),\beta \left(x\right)\right]}_1\right)\left({\alpha }_V{\beta }_V\right)-{\rho }_1\left({\left[{\alpha }^{-1}{\beta }^2\left(z\right),\beta \left(x\right)\right]}_2\right)\left({\alpha }_V{\beta }_V\right)+\rho {}_2\left({\beta }^2\left(z\right)\right)\left({\rho }_1\left(\beta \left(x\right)\right)\left({\alpha }_V\right)\right)\\ +{\rho }_1\left({\beta }^2\left(z\right)\right)\left({\rho }_2\left(\beta \left(x\right)\right)\left({\alpha }_V\right)\right)-{\rho }_2\left({\beta }^2\left(x\right)\right)\left({\rho }_1\left(\beta \left(z\right)\right)\left({\alpha }_V\right)\right)-{\rho }_1\left({\beta }^2\left(x\right)\right)\left({\rho }_2\left(\beta \left(z\right)\right)\left({\alpha }_V\right)\right))\left(v\right)\\ +(-{\rho }_2\left({\left[{\alpha }^{-1}{\beta }^2\left(x\right),\beta \left(y\right)\right]}_1\right)\left({\alpha }_V{\beta }_V\right)-{\rho }_1\left({\left[{\alpha }^{-1}{\beta }^2\left(x\right),\beta \left(y\right)\right]}_2\right)\left({\alpha }_V{\beta }_V\right)+{\rho }_2\left({\beta }^2\left(x\right)\right)\left({\rho }_1\left(\beta \left(y\right)\right)\left({\alpha }_V\right)\right)\\ +{\rho }_1\left({\beta }^2\left(x\right)\right)\left({\rho }_2\left(\beta \left(y\right)\right)\left({\alpha }_V\right)\right)-{\rho }_2\left({\beta }^2\left(y\right)\right)\left({\rho }_1\left(\beta \left(x\right)\right)\left({\alpha }_V\right)\right)-{\rho }_1\left({\beta }^2\left(y\right)\right)\left({\rho }_2\left(\beta \left(x\right)\right)\left({\alpha }_V\right)\right))(\; w\; )\end{array}$

令 $a={\alpha }^{-1}\beta \left(x\right),b=\beta \left(y\right),c={\alpha }^{-1}\beta \left(y\right),d=\beta \left(z\right),r={\alpha }^{-1}\beta \left(z\right),s=\beta \left(x\right)$，因为 ${\alpha }_V{\beta }_V={\beta }_V{\alpha }_V$，所以上式等于

$\begin{array}{l}(-{\rho }_2\left({\left[\beta \left(c\right),d\right]}_1\right){\beta }_V-{\rho }_1\left({\left[\beta \left(c\right),d\right]}_2\right){\beta }_V+{\rho }_2\left(\alpha \beta \left(c\right)\right){\rho }_1\left(d\right)\\ +{\rho }_1\left(\alpha \beta \left(c\right)\right){\rho }_2\left(d\right)-{\rho }_2\left(\beta \left(d\right)\right){\rho }_1\left(\alpha \left(c\right)\right)-{\rho }_1\left(\beta \left(d\right)\right){\rho }_2\left(\alpha \left(c\right)\right))\left(u\right)\\ +(-{\rho }_2\left({\left[\beta \left(r\right),s\right]}_1\right){\beta }_V-{\rho }_1\left({\left[\beta \left(r\right),s\right]}_2\right){\beta }_V+\rho {}_2\left(\alpha \beta \left(r\right)\right){\rho }_1\left(s\right)\\ +{\rho }_1\left(\alpha \beta \left(r\right)\right){\rho }_2\left(s\right)-{\rho }_2\left(\beta \left(s\right)\right){\rho }_1\left(\alpha \left(r\right)\right)-{\rho }_1\left(\beta \left(s\right)\right){\rho }_2\left(\alpha \left(r\right)\right))\left(v\right)\\ +(-{\rho }_2\left({\left[\beta \left(a\right),b\right]}_1\right){\beta }_V-{\rho }_1\left({\left[\beta \left(a\right),b\right]}_2\right){\beta }_V+{\rho }_2\left(\alpha \beta \left(a\right)\right){\rho }_1\left(b\right)\\ +{\rho }_1\left(\alpha \beta \left(a\right)\right){\rho }_2\left(b\right)-{\rho }_2\left(\beta \left(b\right)\right){\rho }_1\left(\alpha \left(a\right)\right)-{\rho }_1\left(\beta \left(b\right)\right){\rho }_2\left(\alpha \left(a\right)\right))\left(w\right)=0\end{array}$

由 $u,v,w$ 的任意性， $\left(g\oplus V,{\left\{-,-\right\}}_1,{\left\{-.-\right\}}_2,\alpha +{\alpha }_V,\beta +{\beta }_V\right)$ 满足等式(3.1)当且仅当 $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 满足等式(3.2)。证毕。

例3.1 设 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容正则BiHom-李代数，则 $\left(g,a{d}_1,a{d}_2,\alpha ,\beta \right)$ 是 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2\alpha ,\beta \right)$ 的表示。

证明 由命题2.2知 $\left(g,a{d}_i,\alpha ,\beta \right)$ 是正则BiHom-李代数 $\left(g,{\left[-,-\right]}_i,\alpha ,\beta \right)\left(i=1,2\right)$ 的表示。 $\forall x,y,z\in g$，

$\begin{array}{l}(a{d}_{\text{1}}\left({\left[\beta \left(x\right),y\right]}_2\right)\beta +a{d}_2\left({\left[\beta \left(x\right),y\right]}_1\right)\beta +a{d}_1\left(\beta \left(y\right)\right)\left(a{d}_2\left(\alpha \left(x\right)\right)\right)\\ +a{d}_2\left(\beta \left(y\right)\right)a{d}_1\left(\alpha \left(x\right)\right)-a{d}_1\left(\alpha \beta \left(x\right)\right)a{d}_2\left(y\right)-a{d}_2\left(\alpha \beta \left(x\right)\right)a{d}_1\left(y\right))\left(z\right)\\ =-{\left[{\beta }^2\left({\alpha }^{-1}\left(z\right)\right),{\left[\beta \left({\beta }^{-1}\alpha \left(x\right)\right),\alpha \left({\beta }^{-1}\left(y\right)\right)\right]}_2\right]}_1-{\left[{\beta }^2\left({\alpha }^{-1}\left(z\right)\right),{\left[\beta \left({\beta }^{-1}\alpha \left(x\right)\right),\alpha \left({\beta }^{-1}\left(y\right)\right)\right]}_1\right]}_2\\ \begin{array}{c}\end{array}-{\left[{\beta }^2\left({\beta }^{-1}\left(y\right)\right),{\left[\beta \left({\alpha }^{-1}\left(z\right)\right),\alpha \left({\beta }^{-1}\alpha \left(x\right)\right)\right]}_2\right]}_1-{\left[{\beta }^2\left({\beta }^{-1}\left(y\right)\right),{\left[\beta \left({\alpha }^{-1}\left(z\right)\right),\alpha \left({\beta }^{-1}\alpha \left(x\right)\right)\right]}_1\right]}_2\\ \begin{array}{c}\end{array}-{\left[{\beta }^2\left({\beta }^{-1}\alpha \left(x\right)\right),{\left[\beta \left({\beta }^{-1}\left(y\right)\right),\alpha \left({\alpha }^{-1}\left(z\right)\right)\right]}_2\right]}_1-{\left[{\beta }^2\left({\beta }^{-1}\alpha \left(x\right)\right),{\left[\beta \left({\beta }^{-1}\left(y\right)\right),\alpha \left({\alpha }^{-1}\left(z\right)\right)\right]}_1\right]}_2\end{array}$，

令 $a={\alpha }^{-1}\left(z\right),b={\beta }^{-1}\alpha \left(x\right),b={\beta }^{-1}\left(y\right)$，由于 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容正则BiHom-李代数，因此 ${\left[-,-\right]}_1,{\left[-,-\right]}_2$ 满足等式(3.1)，即上式得0，所以 $\left(g,a{d}_1,a{d}_2,\alpha ,\beta \right)$ 满足等式(3.2)，所以 $\left(g,a{d}_1,a{d}_2,\alpha ,\beta \right)$ 是 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示。证毕。

定理3.3 设 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容的BiHom-李代数， $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示，定义 ${\rho }_i^{\text{*}}:g\to End\left(V^{*}\right)$，其中 ${\rho }_i^{\text{*}}\left(x\right)\left(f\right)=-f\circ {\rho }_i\left(x\right)\left(i=1,2,\forall f\in V^{*},x\in g\right)$，则 $\left(V^{\text{*}},{\rho }_1^{\text{*}},{\rho }_2^{*},{\alpha }_V^{*},{\beta }_V^{*}\right)$ 是 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示当且仅当 ${\rho }_1,{\rho }_2$ 满足等式(2.7)~(2.8)，且 $\forall x,y\in g$，

$\begin{array}{l}{\beta }_V\circ {\rho }_{\text{1}}\left({\left[\beta \left(x\right),y\right]}_2\right)+{\beta }_V\circ {\rho }_2\left({\left[\beta \left(x\right),y\right]}_1\right)+{\rho }_2\left(y\right){\rho }_1\left(\alpha \beta \left(x\right)\right)\\ +{\rho }_1\left(y\right){\rho }_2\left(\alpha \beta \left(x\right)\right)-{\rho }_2\left(\alpha \left(x\right)\right){\rho }_1\left(\beta \left(y\right)\right)-{\rho }_1\left(\alpha \left(x\right)\right){\rho }_2\left(\beta \left(y\right)\right)=0\end{array}$。 (3.3)

证明 由定义3.2， $\left(V^{\text{*}},{\rho }_1^{\text{*}},{\rho }_2^{*},{\alpha }_V^{*},{\beta }_V^{*}\right)$ 是 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示当且仅当 $\left(V^{\text{*}},{\rho }_i^{\text{*}},{\alpha }_V^{*},{\beta }_V^{*}\right)$ 是 $\left(g,{\left[-,-\right]}_i,\alpha ,\beta \right)\left(i=1,2\right)$ 的表示，并且 ${\rho }_1^{\text{*}},{\rho }_2^{*},{\alpha }_V^{*},{\beta }_V^{*}$ 满足等式(3.2)。由命题2.3得 $\left(V^{\text{*}},{\rho }_i^{\text{*}},{\alpha }_V^{*},{\beta }_V^{*}\right)$ 是 $\left(g,{\left[-,-\right]}_i,\alpha ,\beta \right)\left(i=1,2\right)$ 的表示当且仅当 ${\rho }_1,{\rho }_2$ 满足等式(2.7)~(2.8)。由于 $\forall x,y\in g,f\in V^{*},u\in V$，

$\begin{array}{l}\langle ({\rho }_1^{*}\left({\left[\beta \left(x\right),y\right]}_2\right){\beta }_V^{*}+{\rho }_2^{*}\left({\left[\beta \left(x\right),y\right]}_1\right){\beta }_V^{*}+{\rho }_1^{*}\left(\beta \left(y\right)\right){\rho }_2^{*}\left(\alpha \left(x\right)\right)\\ +{\rho }_2^{*}\left(\beta \left(y\right)\right){\rho }_1^{*}\left(\alpha \left(x\right)\right)-{\rho }_1^{*}\left(\alpha \beta \left(x\right)\right){\rho }_2^{*}\left(y\right)-{\rho }_2^{*}\left(\alpha \beta \left(x\right)\right){\rho }_1^{*}\left(y\right))f,u\rangle \\ =\langle f,\left(-{\beta }_V\circ {\rho }_1\left({\left[\beta \left(x\right),y\right]}_2\right)-{\beta }_V\circ {\rho }_2\left({\left[\beta \left(x\right),y\right]}_1\right)+{\rho }_2\left(\alpha \left(x\right)\right){\rho }_1\left(\beta \left(y\right)\right)\right)\\ \begin{array}{c}\end{array}+{\rho }_1\left(\alpha \left(x\right)\right){\rho }_2\left(\beta \left(y\right)\right)-{\rho }_2\left(y\right){\rho }_1\left(\alpha \beta \left(x\right)\right)-{\rho }_1\left(y\right){\rho }_2\left(\alpha \beta \left(x\right)\right))\left(u\right)\rangle \end{array}$，

所以 ${\rho }_1^{\text{*}},{\rho }_2^{*},{\alpha }_V^{*},{\beta }_V^{*}$ 满足等式(3.2)当且仅当 ${\rho }_1,{\rho }_2,{\alpha }_V,{\beta }_V$ 满足等式(3.3)。证毕。

定理3.4 设 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容的BiHom-李代数， $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 是它的表示， ${\alpha }_V,{\beta }_V$ 可逆，定义 ${\rho }_i^{\circ }:g\to End\left(V^{\ast }\right)$，其中

${\rho }_i^{\circ }\left(x\right)={\rho }_i^{\ast }\left(\alpha \beta \left(x\right)\right){\left({\beta }_V{}^{-2}\right)}^{\ast }\left(i=1,2,x\in g\right)$，

则 $\left(V^{\text{*}},{\rho }_{\text{1}}^{\circ },{\rho }_{\text{2}}^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}\right)$ 是 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示，称为 $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 的对偶表示。

证明 由定义3.2， $\left(V^{\text{*}},{\rho }_{\text{1}}^{\circ },{\rho }_{\text{2}}^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}\right)$ 是相容BiHom-李代数 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示当且仅当 $\left(V^{\text{*}},{\rho }_i^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}\right)$ 是 $\left(g,{\left[-,-\right]}_i,\alpha ,\beta \right)\left(i=1,2\right)$ 的表示，并且 ${\rho }_{\text{1}}^{\circ },{\rho }_{\text{2}}^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}$ 满足等式(3.2)。由命题2.4知 $\left(V^{\text{*}},{\rho }_i^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}\right)$ 是 $\left(g,{\left[-,-\right]}_i,\alpha ,\beta \right)\left(i=1,2\right)$ 的表示。因为 ${\rho }_1,{\rho }_2,{\alpha }_V,{\beta }_V$ 满足等式(2.5)，且 ${\beta }_V$ 可逆，所以 $\forall x,y\in g,f\in V^{*},u\in V$，

$\begin{array}{l}\langle ({\rho }_{\text{1}}^{\circ }\left({\left[\beta \left(x\right),y\right]}_2\right){\left({\beta }_V^{-1}\right)}^{*}+{\rho }_2^{\circ }\left({\left[\beta \left(x\right),y\right]}_1\right){\left({\beta }_V^{-1}\right)}^{*}+{\rho }_1^{\circ }\left(\beta \left(y\right)\right){\rho }_2^{\circ }\left(\alpha \left(x\right)\right)\\ +{\rho }_2^{\circ }\left(\beta \left(y\right)\right){\rho }_1^{\circ }\left(\alpha \left(x\right)\right)-{\rho }_1^{\circ }\left(\alpha \beta \left(x\right)\right){\rho }_2^{\circ }\left(y\right)-{\rho }_2^{\circ }\left(\alpha \beta \left(x\right)\right){\rho }_1^{\circ }\left(y\right))f,u\rangle \\ ={\beta }_V^{-3}(-{\rho }_1{\left[\alpha {\beta }^2\left(x\right),\alpha \beta \left(y\right)\right]}_2{\beta }_V-{\rho }_2{\left[\alpha {\beta }^2\left(x\right),\alpha \beta \left(y\right)\right]}_1{\beta }_V+{\rho }_2\left({\alpha }^2{\beta }^2\left(x\right)\right){\rho }_1\left(\alpha \beta \left(y\right)\right)\\ \begin{array}{c}\end{array}+{\rho }_1\left({\alpha }^2{\beta }^2\left(x\right)\right){\rho }_2\left(\alpha \beta \left(y\right)\right)-{\rho }_2\left(\alpha {\beta }^2\left(y\right)\right){\rho }_1\left({\alpha }^2\beta \left(x\right)\right)-{\rho }_1\left(\alpha {\beta }^2\left(y\right)\right){\rho }_2\left({\alpha }^2\beta \left(x\right)\right)){\beta }_V^{-1}\end{array}$，

令 $a=\alpha \beta \left(x\right),b=\alpha \beta \left(y\right)$，因为 $\left(V,{\rho }_{\text{1}},{\rho }_{\text{2}},{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示，所以 ${\rho }_1,{\rho }_2$ 满足等式(3.2)，所以上式等于0，即 ${\rho }_{\text{1}}^{\circ },{\rho }_{\text{2}}^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}$ 满足等式(3.2)。因此 $\left(V^{\text{*}},{\rho }_{\text{1}}^{\circ },{\rho }_{\text{2}}^{\circ },{\left({\alpha }_V^{-1}\right)}^{\text{*}},{\left({\beta }_V^{-1}\right)}^{\text{*}}\right)$ 是相容BiHom-李代数 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示。

推论3.1 设 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 是相容的正则BiHom-李代数，则 $\left(g^{*},a{d}_1^{\circ },a{d}_2^{\circ },{\left({\alpha }^{-1}\right)}^{*},{\left({\beta }^{-1}\right)}^{\ast }\right)$ 是 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示。

证明 由例3.1知 $\left(g,a{d}_1,a{d}_2,\alpha ,\beta \right)$ 是相容正则BiHom-李代数 $\left(g,{\left[-,-\right]}_1,{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示，所以由定理3.4知 $\left(g^{*},a{d}_1^{\circ },a{d}_2^{\circ },{\left({\alpha }^{-1}\right)}^{*},{\left({\beta }^{-1}\right)}^{\ast }\right)$ 是 $\left(g,{\left[-,-\right]}_{\text{1}},{\left[-,-\right]}_2,\alpha ,\beta \right)$ 的表示。

4. BiHom-李代数的形变

定理4.1 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 为BiHom-李代数， $\omega :g\otimes g\to g$ 为双线性映射，定义

${\left[x,y\right]}_t=\left[x,y\right]+t\omega \left(x,y\right)\left(\forall x,y\in g\right)$， (4.1)

t为参数，则 $\left(g,{\left[-,-\right]}_t,\alpha ,\beta \right)$ 是BiHom-李代数当且仅当对于 $\forall x,y,z\in g$ 有

$\alpha \omega \left(x,y\right)=\omega \left(\alpha \left(x\right),\alpha \left(y\right)\right)$， $\beta \omega \left(x,y\right)=\omega \left(\beta \left(x\right),\beta \left(y\right)\right)$， (4.2)

$\omega \left(\beta \left(x\right),\alpha \left(y\right)\right)=-\omega \left(\beta \left(y\right),\alpha \left(x\right)\right)$， (4.3)

$\omega \left({\beta }^2\left(x\right),\omega \left(\beta \left(y\right),\alpha \left(z\right)\right)\right)+\omega \left({\beta }^2\left(y\right),\omega \left(\beta \left(z\right),\alpha \left(x\right)\right)\right)+\omega \left({\beta }^2\left(z\right),\omega \left(\beta \left(x\right),\alpha \left(y\right)\right)\right)=0$， (4.4)

$\begin{array}{l}\left[{\beta }^{\text{2}}\left(x\right),\omega \left(\beta \left(y\right),\alpha \left(z\right)\right)\right]+\omega \left({\beta }^2\left(x\right),\left[\beta \left(y\right),\alpha \left(z\right)\right]\right)+\left[{\beta }^2\left(y\right),\omega \left(\beta \left(z\right),\alpha \left(x\right)\right)\right]\\ +\omega \left({\beta }^2\left(y\right),\left[\beta \left(z\right),\alpha \left(x\right)\right]\right)+\left[{\beta }^2\left(z\right),\omega \left(\beta \left(x\right),\alpha \left(y\right)\right)\right]+\omega \left({\beta }^2\left(z\right),\left[\beta \left(x\right),\alpha \left(y\right)\right]\right)=0\end{array}$， (4.5)

此时称 $\left(g,{\left[-,-\right]}_t,\alpha ,\beta \right)$ 为BiHom-李代数 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的形变。

证明 由 ${\left[-,-\right]}_t$ 的定义知 ${\left[-,-\right]}_t$ 是双线性的。由定义2.1， $\left(g,{\left[-,-\right]}_t,\alpha ,\beta \right)$ 是BiHom-李代数当且仅当 ${\left[-,-\right]}_t$ 满足等式(2.1)~(2.4)。 $\alpha \beta =\beta \alpha$ 显然成立。由于 $\left[-,-\right]$ 满足等式(2.4)，所以 $\forall x,y\in g$，

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

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

因此 ${\left[-,-\right]}_t$ 满足等式(2.4)当且仅当等式(4.2)成立。由于 $\left[-,-\right]$ 满足等式(2.2)，所以

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

因此 ${\left[-,-\right]}_t$ 满足等式(2.2)当且仅当等式(4.3)成立。因为 $\forall x,y,z\in g$， $\left[-,-\right]$ 满足等式(2.3)，因此

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

由t的任意性， ${\left[-,-\right]}_t$ 满足等式(2.3)当且仅当等式(4.4)~(4.5)成立。证毕。

定理4.2 设 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 是BiHom-李代数， $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 的表示， $\omega :g\otimes g\to g$ 为双线性映射且满足等式(4.1)~(4.5)， $\sigma :g\to End\left(V\right)$ 为线性映射，定义 ${\rho }^t:g\to End\left(V\right)$，其中

${\rho }^t\left(x\right)=\rho \left(x\right)+t\sigma \left(x\right)\left(\forall x\in g\right)$，

t为参数，则对于(4.1)定义的 ${\left[-,-\right]}_t$， $\left(V,{\rho }^t,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_t,\alpha ,\beta \right)$ 的表示当且仅当对于 $\forall x,y\in g$ 恒有

$\sigma \left(\alpha \left(x\right)\right)\circ {\alpha }_V={\alpha }_V\circ \sigma \left(x\right)$， $\sigma \left(\beta \left(x\right)\right)\circ {\beta }_V={\beta }_V\circ \sigma \left(x\right)$， (4.6)

$\sigma \left(\omega \left(\beta \left(x\right),y\right)\right)\circ {\beta }_V=\sigma \left(\alpha \beta \left(x\right)\right)\sigma \left(y\right)-\sigma \left(\beta \left(y\right)\right)\sigma \left(\alpha \left(x\right)\right)$， (4.7)

$\begin{array}{l}\rho \omega \left(\beta \left(x\right),y\right)\circ {\beta }_V+\sigma \left(\left[\beta \left(x\right),y\right]\right)\circ {\beta }_V\\ =\rho \left(\alpha \beta \left(x\right)\right)\sigma \left(y\right)+\sigma \left(\alpha \beta \left(x\right)\right)\rho \left(y\right)-\rho \left(\beta \left(y\right)\right)\sigma \left(\alpha \left(x\right)\right)-\sigma \left(\beta \left(y\right)\right)\rho \left(\alpha \left(x\right)\right)\end{array}$， (4.8)

此时称 $\left(V,{\rho }^t,{\alpha }_V,{\beta }_V\right)$ 是表示 $\left(V,\rho ,{\alpha }_V,{\beta }_V\right)$ 的形变。

证明 由定理4.1知 $\left(g,{\left[-,-\right]}_t,\alpha ,\beta \right)$ 是BiHom-李代数。 ${\alpha }_V,{\beta }_V$ 显然是可交换的。由定义2.2， $\left(V,{\rho }^t,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,{\left[-,-\right]}_t,\alpha ,\beta \right)$ 的表示当且仅当 ${\rho }^t,{\alpha }_V,{\beta }_V$ 满足等式(2.5)~(2.6)。 $\forall x\in g$，因为 $\rho ,{\alpha }_V,{\beta }_V$ 满足等式(2.5)，所以

${\rho }^t\left(\alpha \left(x\right)\right)\circ {\alpha }_V-{\alpha }_V\circ {\rho }^t\left(x\right)=t\sigma \left(\alpha \left(x\right)\right)\circ {\alpha }_V-t{\alpha }_V\circ \sigma \left(x\right)$，

${\rho }^t\left(\beta \left(x\right)\right)\circ {\beta }_V-{\beta }_V\circ {\rho }^t\left(x\right)=t\sigma \left(\beta \left(x\right)\right)\circ {\beta }_V-t{\beta }_V\circ \sigma \left(x\right)$，

因此 ${\rho }^t,{\alpha }_V,{\beta }_V$ 满足等式(2.5)当且仅当等式(4.6)成立。 $\forall x,y\in g$，由于 $\rho ,{\alpha }_V,{\beta }_V$ 满足等式(2.6)，因此

$\begin{array}{l}{\rho }^t\left({\left[\beta \left(x\right),y\right]}_t\right)\circ {\beta }_V-{\rho }^t\left(\alpha \beta \left(x\right)\right)\circ {\rho }^t\left(y\right)+{\rho }^t\left(\beta \left(y\right)\right)\circ {\rho }^t\left(\alpha \left(x\right)\right)\\ =t(\rho \omega \left(\beta \left(x\right),y\right)\circ {\beta }_V+\sigma \left(\left[\beta \left(x\right),y\right]\right)\circ {\beta }_V-\rho \left(\alpha \beta \left(x\right)\right)\sigma \left(y\right)-\sigma \left(\alpha \beta \left(x\right)\right)\rho \left(y\right)+\rho \left(\beta \left(y\right)\right)\sigma \left(\alpha \left(x\right)\right)\\ \begin{array}{c}\end{array}+\sigma \left(\beta \left(y\right)\right)\rho \left(\alpha \left(x\right)\right))+t^2\left(\sigma \left(\omega \left(\beta \left(x\right),y\right)\right)\circ {\beta }_V-\sigma \left(\alpha \beta \left(x\right)\right)\sigma \left(y\right)+\sigma \left(\beta \left(y\right)\right)\sigma \left(\alpha \left(y\right)\right)\right)\\ \end{array}$，

由t的任意性知， ${\rho }^t,{\alpha }_V,{\beta }_V$ 满足等式(2.6)当且仅当等式(4.7)~(4.8)成立。证毕。

注记4.1等式(4.2)~(4.4)说明 $\left(g,\omega ,\alpha ,\beta \right)$ 是BiHom-李代数；等式(4.5)说明 $\left(g,\left[-,-\right],\alpha ,\beta \right)$ 与 $\left(g,\omega ,\alpha ,\beta \right)$ 是相容的BiHom-李代数；等式(4.6)~(4.7)说明 $\left(V,\sigma ,{\alpha }_V,{\beta }_V\right)$ 是 $\left(g,\omega ,\alpha ,\beta \right)$ 的表示；等式(4.8)说明 $\left(V,\rho ,\sigma ,{\alpha }_V,{\beta }_V\right)$ 是相容BiHom-李代数 $\left(g,\left[-,-\right],\omega ,\alpha ,\beta \right)$ 的表示。

基金项目

辽宁师范大学教改项目LS202002。