李三系最初源于Cartan对黎曼几何的研究中, 但它的概念是由Jacobson在1949年引入的^[1], 用于研究在三元交换子[[u, v], w]下的封闭的结合代数的子空间. 之后赵冠华, 吴辰余在^[2]中研究了李三系的扩张问题, 且李三系的结构内容和广义导子等的研究见文献[3, 4]. Okubo首次提出了李超三系的定义, 李超三系的线性形变与阿贝尔扩张以及李超三系的上同调及NiJenhuis算子等内容见文献[5, 6]. 文献[7]研究了李$ \mathrm{color} $代数的定义, 进一步张健等人在^[8]中研究了李$ \mathrm{color} $三系的导子以及广义导子等的性质.

在文献[9]中研究了$ \mathrm{Hom} $-李代数的广义导子, 并且得到了$ \mathrm{Hom} $-李代数的拟导子代数可以嵌入到较大的$ \mathrm{Hom} $-李代数的导子中, $ \mathrm{Hom} $-李$ \mathrm{color} $代数的概念见文献[10]. 进一步, 文献[11]研究了分裂的正则双$ \mathrm{Hom} $-李$ \mathrm{color} $代数的结构等内容, 在^[12]中郭双建等人研究了$ \mathrm{BiHom} $-$ \mathrm{Lie} $共形代数的上同调与形变, 在^[13]中研究了$ \delta $-$ \mathrm{BiHom} $-$ \mathrm{Jordan} $李超代数的阿贝尔扩张的相关内容. 文献[14, 15]中分别研究了$ \mathrm{BiHom} $-李三系的广义导子和$ \delta $-$ \mathrm{Jordan} $-李三系上带有权$ \lambda $的$ \mathrm{K} $-阶广义导子. 本文给出了$ \mathrm{BiHom} $-李$ \mathrm{color} $三系定义, 进而研究保积$ \mathrm{BiHom} $-李$ \mathrm{color} $三系的广义导子的一些性质.

定义2.1[7]  设$ G $是交换群, $ \mathbb{F} $是任意域. 若对任意的$ \alpha, \beta, \gamma \in G $, 下列等式均成立$ : $

$ (1) $ $ \varepsilon(\alpha, \beta)\varepsilon(\beta, \alpha)=1 $,

$ (2) $ $ \varepsilon(\alpha, \beta+\gamma)=\varepsilon(\alpha, \beta)\varepsilon(\alpha, \gamma) $,

$ (3) $ $ \varepsilon(\alpha+\beta, \gamma)=\varepsilon(\alpha, \gamma)\varepsilon(\beta, \gamma) $.

则称映射$ \varepsilon: G\times G\rightarrow \mathbb{F}\backslash\{0\} $为$ G $的斜对称双特征标(或交换因子). 易知,

$ \varepsilon(\alpha, 0)=\varepsilon(0, \alpha)=1, \; \varepsilon(\alpha, \alpha)=\pm 1. $

如果存在$ V $的一簇子空间$ \{V_{\gamma}\}_{\gamma\in G} $, 满足$ V=\oplus_{\gamma\in G}V_{\gamma} $, 则称线性空间$ V $为$ G $-阶化; 如果$ x\in V_{\gamma}(\gamma\in G) $, 则称$ x $为$ \gamma $次齐次元素. 如果$ x $, $ y $, $ z $是$ G $-阶化向量空间中的齐次元, 用$ |x|, |y|, |z|\in G $表示它们的次数. 为方便, 用$ \varepsilon(x, y) $表示$ \varepsilon(|x|, |y|) $, 用$ \varepsilon(x, y+z) $表示$ \varepsilon(|x|, |y|+|z|) $, 以此类推. 此外, $ \varepsilon(x, y) $出现即表明其中的$ x, y $是齐次元. 在本文中, 用$ hg(V) $表示$ V $中所有齐次元素.

设$ V, W $是两个$ G $-阶化的线性空间, 如果对于任意的$ x\in V_{\gamma} $, 都有$ f(x)\in W_{\gamma+\theta} $, 则称线性映射$ f:V\rightarrow W $为$ \theta $次. 若$ f $是零次的, 即$ f(V_{\gamma})\subseteq W_{\gamma} $, 则称$ f $是偶的.

若对任意的$ \theta, \mu\in G $, 如果$ T $是$ G $-阶化线性空间, 即$ T=\oplus_{\gamma\in G}T_{\gamma} $, 并且$ T_{\theta}T_{\mu}\subseteq T_{\theta+\mu} $, 则$ T $称为$ G $-阶化的代数, 如果$ \alpha(A_{\gamma})\subseteq B_{\gamma} $, 称同态$ \alpha:A\rightarrow B $是偶的.

定义2.2[7]  李$ \mathrm{color} $代数是一个三元组($ T, [\cdot, \cdot], \varepsilon $), 其中$ T=\oplus_{g\in G}T_g $是域$ \mathbb{F} $上的一个$ G $-阶化向量空间, 如果存在双线性映射$ [\cdot, \cdot]: T\times T\rightarrow T $$ ( $对任意的$ g, g'\in G $, 有$ [T_g, T_{g'}]\subseteq T_{g+g'} $$ ) $和$ G $上的一个斜对称双特征标$ \varepsilon:G\times G\rightarrow \mathbb{F}\backslash\{0\} $, 满足

$ (1) $ $ [x, y]=-\varepsilon(x, y)[y, x] $,

$ (2) $ $ \varepsilon(z, x)[x, [y, z]]+\varepsilon(x, y)[y, [z, x]]+\varepsilon(y, z)[z, [x, y]]=0 $.

对任意的$ x, y, z\in hg(T) $.

当$ G=\mathbb{Z}_{2} $且$ \varepsilon(x, y)=(-1)^{|x||y|} $时, 李$ \mathrm{color} $代数成为李超代数$ ; $当$ \varepsilon(x, y)\equiv 1 $时, 李$ \mathrm{color} $代数成为李代数. 因此, 李$ \mathrm{color} $代数是一类包含李代数和李超代数的更广泛的代数结构.

定义2.3[10]  $ \mathrm{Hom} $-李$ \mathrm{color} $代数是一个四元组($ T, [\cdot, \cdot], \alpha, \varepsilon $), 其中$ T=\oplus_{g\in G}T_g $是域$ \mathbb{F} $上的一个$ G $-阶化向量空间, 如果$ T $上有偶的双线性映射$ [\cdot, \cdot]:T \times T \rightarrow T $, 偶的同态$ \alpha:T \rightarrow T $和$ G $上的一个斜对称双特征标$ \varepsilon:G\times G\rightarrow \mathbb{F}\backslash\{0\} $, 满足

$ (1) $ $ [x, y]=-\varepsilon(x, y)[y, x] $, $ (\varepsilon $-反对称性$ ) $,

$ (2) $ $ \varepsilon(z, x)[\alpha(x), [y, z]]+ \varepsilon(x, y)[\alpha(y), [z, x]]+\varepsilon(y, z)[\alpha(z), [x, y]]=0 $. ($ \mathrm{Hom} $-$ \mathrm{Jacobi} $等式)

对任意的$ x, y, z\in hg(T) $.

定义2.4[11]  $ \mathrm{BiHom} $-李$ \mathrm{color} $代数是一个五元组($ T, [\cdot, \cdot], \alpha, \beta, \varepsilon $), 其中$ T=\oplus_{g\in G}T_g $是域$ \mathbb{F} $上的一个$ G $-阶化向量空间, $ T $上有一个偶的双线性映射$ [\cdot, \cdot]: T\times T\rightarrow T $(对任意的$ g, h\in G $, 有$ [T_g, T_h]\subset T_{g+h} $)及两个偶自同态$ \alpha, \beta:T\rightarrow T $和$ G $上的一个斜对称双特征标$ \varepsilon:G\times G\rightarrow \mathbb{F}\backslash\{0\} $, 满足

$ (1) $ $ \alpha\circ\beta=\beta\circ\alpha $,

$ (2) $ $ [\beta(x), \alpha(y)]=-\varepsilon(x, y)[\beta(y), \alpha(x)] $, $ ( $$ \varepsilon $-反对称性$ ) $,

$ (3) $ $ \varepsilon(z, x)[\beta^{2}(x), [\beta(y), \alpha(z)]]$$+ \varepsilon(x, y)[\beta^{2}(y), [\beta(z), \alpha(x)]]+\varepsilon(y, z)[\beta^{2}(z), [\beta(x), \alpha(y)]]=0 $. ($ \mathrm{BiHom} $-$ \mathrm{Jacobi} $等式).

对任意的$ x, y, z \in hg(T) $.

定义2.5[7]  李$ \mathrm{color} $三系一个三元组($ T, [\cdot, \cdot, \cdot], \varepsilon $), 其中$ T=\oplus_{g\in G}T_g $是域$ \mathbb{F} $上的一个$ G $-阶化向量空间, 如果$ T $上有三元运算$ [\cdot, \cdot, \cdot]:T \times T \times T \rightarrow T $和$ G $上的一个斜对称双特征标$ \varepsilon:G\times G\rightarrow \mathbb{F}\backslash\{0\} $, 满足

$ (1) $ $ [x, y, z]=-\varepsilon(x, y)[y, x, z] $,

$ (2) $ $ \varepsilon(z, x)[x, y, z]+ \varepsilon(x, y)[y, z, x]+\varepsilon(y, z)[z, x, y]=0 $,

$ (3) $ $ [u, v, [x, y, z]]=[[u, v, x], y, z]+\varepsilon(u+v, x)$$[x, [u, v, y], z]+\varepsilon(u+v, x+y)[x, y, [u, v, z]] $.

对任意的$ x, y, z, u, v\in hg(T) $.

当$ G=\mathbb{Z}_{2} $且$ \varepsilon(x, y)=(-1)^{|x||y|} $时, 李$ \mathrm{color} $三系成为李超三系$ ; $当$ \varepsilon(x, y)\equiv 1 $时, 李$ \mathrm{color} $三系成为李三系. 因此, 李$ \mathrm{color} $三系是一类包含李三系和李超三系的更广泛的代数结构.

定义2.6  $ \mathrm{Hom} $-李$ \mathrm{color} $三系是一个四元组($ T, [\cdot, \cdot, \cdot], \alpha, \varepsilon $), 其中$ T=\oplus_{g\in G}T_g $是域$ \mathbb{F} $上的一个$ G $-阶化向量空间, 如果$ T $上有三元运算$ [\cdot, \cdot, \cdot]:T \times T \times T \rightarrow T $, 偶自同态$ \alpha:T\rightarrow T $和$ G $上的一个斜对称双特征标$ \varepsilon:G\times G\rightarrow \mathbb{F}\backslash\{0\} $, 满足

$ (1) $ $ [x, y, z]=-\varepsilon(x, y)[y, x, z] $,

$ (2) $ $ \varepsilon(z, x)[x, y, z]+ \varepsilon(x, y)[y, z, x]+\varepsilon(y, z)[z, x, y]=0 $,

$ (3) $ $ [\alpha(u), \alpha(v), [x, y, z]]=[[u, v, x], \alpha(y), \alpha(z)]+\varepsilon(u+v, x)$$[\alpha(x), [u, v, y], \alpha(z)] +\varepsilon(u+v, x+y)[\alpha(x), \alpha(y), [u, v, z]] $.

对任意的$ x, y, z, u, v\in hg(T) $. 特别地, 若还满足

$ \alpha([x, y, z])=[\alpha(x), \alpha(y), \alpha(z)]. $

则称$ T $是保积的$ \mathrm{Hom} $-李$ \mathrm{color} $三系.

定义2.7  $ \mathrm{BiHom} $-李$ \mathrm{color} $三系是一个五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $), 其中$ T=\oplus_{g\in G}T_g $是域$ \mathbb{F} $上的一个$ G $-阶化向量空间, 设$ T $具有三元运算$ [\cdot, \cdot, \cdot]: T\times T\times T\rightarrow T $, 及两个偶自同态$ \alpha, \beta: T\rightarrow T $和$ G $上的一个斜对称双特征标$ \varepsilon:G\times G\rightarrow \mathbb{F}\backslash\{0\} $, 满足

$ \rm(1) $ $ \alpha\circ \beta=\beta \circ \alpha $,

$ \rm(2) $ $ [x, y, z]=-\varepsilon(x, y)[y, x, z] $,

$ \rm(3) $ $ \varepsilon(z, x)[x, y, z]+\varepsilon(x, y)[y, z, x]+\varepsilon(y, z)[z, x, y]=0 $,

$ \rm(4) $ $ [\beta^{2}(u), \beta^{2}(v), [\beta(x), \beta(y), \alpha(z)]]=[[\beta(u), \beta(v), \alpha(x)], \beta^{2}(y), \beta^{2}(z)] $ $ +\varepsilon(u+v, x)[\beta^{2}(x), [\beta(u), \beta(v), \alpha(y)], \beta^{2}(z)]$$+\varepsilon(u+v, x+y)[\beta^{2}(x), \beta^{2}(y), [\beta(u), \beta(v), \alpha(z)]] $.

对任意的$ x, y, z, u, v\in hg(T) $. 特别地, 若还满足

$ \alpha([x, y, z])=[\alpha(x), \alpha(y), \alpha(z)], \quad \beta([x, y, z])=[\beta(x), \beta(y), \beta(z)], $

则称$ T $是保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系.

命题2.8  设五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系, $ \alpha, \beta $是$ T $上的偶同态, $ \mathrm{End}(T) $表示$ T $的所有线性变换构成的线性空间. 令$ \mho:=\{D\in \mathrm{End}(T)\mid D\alpha=\alpha D, \; \; D\beta=\beta D \}. $则五元组($ \mho, [\cdot, \cdot], \widetilde{\alpha}, \widetilde{\beta}, \varepsilon $)是$ \mathrm{BiHom} $-李$ \mathrm{color} $代数, 其中李$ \mathrm{color} $扩积为

$ [D_{\theta}, D_{\mu}]=D_{\theta}D_{\mu}-\varepsilon(\theta, \mu)D_{\mu}D_{\theta}. $

且同态$ \widetilde{\alpha}, \widetilde{\beta}:\mathrm{End}(T)\rightarrow \mathrm{End}(T) $是偶的, 满足$ \widetilde{\alpha}(D)=\alpha D, \; \; \widetilde{\beta}(D)=\beta D $.

证  对任意的$ D_{\theta}, D_{\mu}, D_{\eta}\in hg(\mathrm{End}(T)) $, 可得

$ \widetilde{\alpha}\widetilde{\beta} (D_{\theta})=\widetilde{\alpha}(\beta D_{\theta})=\alpha \beta D_{\theta}=\beta \alpha D_{\theta}= \beta\widetilde{\alpha}(D_{\theta})=\widetilde{\beta}(\widetilde{\alpha}(D_{\theta}))=\widetilde{\beta}\widetilde{\alpha}(D_{\theta}). $

其次,

$ \begin{equation*} \begin{split} [\widetilde{\beta}(D_{\mu}), \widetilde{\alpha}(D_{\eta}) ]=&[\beta D_{\mu}, \alpha D_{\eta}]\\ =&(\beta D_{\mu}) (\alpha D_{\eta})-\varepsilon(\mu, \eta)(\alpha D_{\eta}) (\beta D_{\mu})\\ =&\alpha \beta D_{\mu} D_{\eta}-\varepsilon(\mu, \eta)\alpha \beta D_{\eta} D_{\mu}\\ =& -\varepsilon(\mu, \eta)[\beta D_{\eta}, \alpha D_{\mu}]\\ =& -\varepsilon(\mu, \eta)[\widetilde{\beta} (D_{\eta}), \widetilde{\alpha} (D_{\mu})].\\ \end{split} \end{equation*} $

再利用李$ \mathrm{color} $括积运算, 对任意的$ D_{\theta}, D_{\mu}, D_{\eta}\in hg(\mathrm{End}(T)) $, 有

$ \begin{eqnarray*} \varepsilon(\eta, \theta)[\widetilde{\beta}^{2}(D_{\theta}), [\widetilde{\beta} (D_{\mu}), \widetilde{\alpha} (D_{\eta})]] &=&\varepsilon(\eta, \theta)[\widetilde{\beta}(D_{\theta}\beta), [\widetilde{\beta} (D_{\mu}), \widetilde{\alpha} (D_{\eta})]]\\ &=&\varepsilon(\eta, \theta)[\beta^{2}D_{\theta}, \beta D_{\mu}\alpha D_{\eta}-\varepsilon(\mu, \eta)\alpha D_{\eta}\beta D_{\mu} ]\\ &=&\varepsilon(\eta, \theta)[\beta^{2}D_{\theta}, \beta D_{\mu}\alpha D_{\eta}]-\varepsilon(\eta, \theta)\varepsilon(\mu, \eta)[\beta^{2}D_{\theta}, \alpha D_{\eta}\beta D_{\mu}]\\ &=&\varepsilon(\eta, \theta)(\beta^{2}D_{\theta}\beta D_{\mu}\alpha D_{\eta}-\varepsilon(\theta, \mu+\eta)\beta D_{\mu}\alpha D_{\eta}\beta^{2}D_{\theta})\\ &&-\varepsilon(\eta, \theta)\varepsilon(\mu, \eta)(\beta^{2}D_{\theta}\alpha D_{\eta}\beta D_{\mu} -\varepsilon(\theta, \eta+\mu)\alpha D_{\eta}\beta D_{\mu}\beta^{2}D_{\theta})\\ &=&\varepsilon(\eta, \theta)\beta^{2}D_{\theta}\beta D_{\mu}\alpha D_{\eta}-\varepsilon(\theta, \mu)\beta D_{\mu}\alpha D_{\eta}\beta^{2}D_{\theta}\\ &&-\varepsilon(\eta, \theta)\varepsilon(\mu, \eta)\beta^{2}D_{\theta}\alpha D_{\eta}\beta D_{\mu} +\varepsilon(\mu, \eta)\varepsilon(\theta, \mu)\alpha D_{\eta}\beta D_{\mu}\beta^{2}D_{\theta}. \end{eqnarray*} $

类似地, 有

$ \begin{eqnarray*} \varepsilon(\theta, \mu)[\widetilde{\beta}^{2}(D_{\mu}), [\widetilde{\beta} (D_{\eta}), \widetilde{\alpha} (D_{\theta})]] &=&\varepsilon(\theta, \mu)[\widetilde{\beta}(D_{\mu}\beta), [\widetilde{\beta}(D_{\eta}), \widetilde{\alpha} (D_{\theta})]]\\ &=&\varepsilon(\theta, \mu)[\beta^{2}D_{\mu}, \beta D_{\eta}\alpha D_{\theta}-\varepsilon(\eta, \theta)\alpha D_{\theta}\beta D_{\eta} ]\\ &=&\varepsilon(\theta, \mu)[\beta^{2}D_{\mu}, \beta D_{\eta}\alpha D_{\theta}]-\varepsilon(\theta, \mu)\varepsilon(\eta, \theta)[\beta^{2}D_{\mu}, \alpha D_{\theta} \beta D_{\eta}]\\ &=&\varepsilon(\theta, \mu)(\beta^{2}D_{\mu}\beta D_{\eta}\alpha D_{\theta}- \varepsilon(\mu, \eta+\theta)\beta D_{\eta}\alpha D_{\theta}\beta^{2}D_{\mu})\\ &&-\varepsilon(\theta, \mu)\varepsilon(\eta, \theta)(\beta^{2}D_{\mu}\alpha D_{\theta}\beta D_{\eta} -\varepsilon(\mu, \theta+\eta) \alpha D_{\theta}\beta D_{\eta}\beta^{2}D_{\mu})\\ &=&\varepsilon(\theta, \mu)\beta^{2}D_{\mu}\beta D_{\eta}\alpha D_{\theta}-\varepsilon(\mu, \eta)\beta D_{\eta}\alpha D_{\theta}\beta^{2}D_{\mu}\\ &&-\varepsilon(\theta, \mu)\varepsilon(\eta, \theta)\beta^{2}D_{\mu}\alpha D_{\theta}\beta D_{\eta} +\varepsilon(\eta, \theta)\varepsilon(\mu, \eta) \alpha D_{\theta}\beta D_{\eta}\beta^{2}D_{\mu}, \end{eqnarray*} $

和

$ \begin{eqnarray*} \varepsilon(\mu, \eta)[\widetilde{\beta}^{2}(D_{\eta}), [\widetilde{\beta} (D_{\theta}), \widetilde{\alpha} (D_{\mu})]] &=&\varepsilon(\mu, \eta)[\widetilde{\beta}(D_{\eta}\beta), [\widetilde{\beta}(D_{\theta}), \widetilde{\alpha}(D_{\mu})]]\\ &=&\varepsilon(\mu, \eta)[\beta^{2}D_{\eta}, \beta D_{\theta}\alpha D_{\mu}-\varepsilon(\theta, \mu)\alpha D_{\mu}\beta D_{\theta} ]\\ &=&\varepsilon(\mu, \eta)[\beta^{2}D_{\eta}, \beta D_{\theta}\alpha D_{\mu}]-\varepsilon(\mu, \eta)\varepsilon(\theta, \mu)[\beta^{2}D_{\eta}, \alpha D_{\mu} \beta D_{\theta}]\\ &=&\varepsilon(\mu, \eta)(\beta^{2}D_{\eta}\beta D_{\theta}\alpha D_{\mu}-\varepsilon(\eta, \theta+\mu)\beta D_{\theta}\alpha D_{\mu}\beta^{2}D_{\eta})\\ &&-\varepsilon(\mu, \eta)\varepsilon(\theta, \mu)(\beta^{2}D_{\eta}\alpha D_{\mu}\beta D_{\theta} -\varepsilon(\eta, \mu+\theta) \alpha D_{\mu}\beta D_{\theta}\beta^{2}D_{\eta})\\ &=&\varepsilon(\mu, \eta)\beta^{2}D_{\eta}\beta D_{\theta}\alpha D_{\mu}-\varepsilon(\eta, \theta)\beta D_{\theta}\alpha D_{\mu}\beta^{2}D_{\eta}\\ &&-\varepsilon(\mu, \eta)\varepsilon(\theta, \mu)\beta^{2}D_{\eta}\alpha D_{\mu}\beta D_{\theta} +\varepsilon(\eta, \theta)\varepsilon(\theta, \mu) \alpha D_{\mu}\beta D_{\theta}\beta^{2}D_{\eta}. \end{eqnarray*} $

将以上三式相加, 可得$ \varepsilon(\eta, \theta)[\widetilde{\beta}^{2}(D_{\theta}), [\widetilde{\beta} (D_{\mu}), \widetilde{\alpha} (D_{\eta})]] +\varepsilon(\theta, \mu)[\widetilde{\beta}^{2}(D_{\mu}), [\widetilde{\beta} (D_{\eta}), \widetilde{\alpha} (D_{\theta})]]\\ +\varepsilon(\mu, \eta)[\widetilde{\beta}^{2}(D_{\eta}), [\widetilde{\beta} (D_{\theta}), \widetilde{\alpha} (D_{\mu})]]=0 $.

综上, 五元组($ \mho, [\cdot, \cdot], \widetilde{\alpha}, \widetilde{\beta}, \varepsilon $)是$ \mathrm{BiHom} $-李$ \mathrm{color} $代数.

任取$ \theta\in G $, 对任意的$ \mu \in G $, 令$ \mathrm{End}_{\theta}(T)=\{D_{\theta}\in \mathrm{End}(T)\mid D(T_{\mu})\subseteq T_{\theta+\mu}\} $.

定义2.9  设五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系. $ D\in \mathrm{End}_{\theta}(T) $, 其中$ \theta\in G $,

$ \bullet $对任意的$ x, y, z\in hg(T) $, 满足

$ [D, \alpha]=0, \; \; [D, \beta]=0, $

$ \begin{equation*} \begin{split} &D([x, y, z])=[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]+\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &+\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)]. \end{split} \end{equation*} $

则称$ D $为$ T $的$ \theta $次$ \alpha^{k}\beta^{l} $-导子.

$ \bullet $对任意的$ x, y, z\in hg(T) $, 如果存在$ D^{'}, D^{''}, D^{'''}\in \mathrm{End}_{\theta}(T) $, 满足

$ [D, \alpha]=[D^{'}, \alpha]=[D^{''}, \alpha]=[D^{'''}, \alpha]=0, \quad [D, \beta]=[D^{'}, \beta]=[D^{''}, \beta]=[D^{'''}, \beta]=0, $

$ \begin{equation*} \begin{split} &D^{'''}([x, y, z])=[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]+\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D^{'}(y), \alpha^{k}\beta^{l}(z)]\\ &+\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D^{''}(z)]. \end{split} \end{equation*} $

则称$ D $为$ T $的$ \theta $次$ \alpha^{k}\beta^{l} $-广义导子.

$ \bullet $对任意的$ x, y, z\in hg(T) $, 如果存在$ D^{'}\in \mathrm{End}_{\theta}(T) $, 满足

$ [D, \alpha]=[D^{'}, \alpha]=0, \quad [D, \beta]=[D^{'}, \beta]=0, $

$ \begin{equation*} \label{cao} \begin{split} &D^{'}([x, y, z])=[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]+\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &+\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)]. \end{split} \end{equation*} $

则称$ D $为$ T $的$ \theta $次$ \alpha^{k}\beta^{l} $-拟导子.

$ \bullet $对任意的$ x, y, z\in hg(T) $, 若满足

$ [D, \alpha]=0, \; \; [D, \beta]=0, $

$ \begin{equation*} \begin{split} \begin{aligned} D([x, y, z])&=[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]=\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &=\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)].\\ \end{aligned} \end{split} \end{equation*} $

则称$ D $为$ T $的$ \theta $次$ \alpha^{k}\beta^{l} $-型心.

$ \bullet $对任意的$ x, y, z\in hg(T) $, 若满足

$ [D, \alpha]=0, \; \; [D, \beta]=0, $

$ \begin{equation*} \begin{split} &[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]=\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &=\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)]. \end{split} \end{equation*} $

则称$ D $为$ T $的$ \theta $次$ \alpha^{k}\beta^{l} $-拟型心.

$ \bullet $对任意的$ x, y, z\in hg(T) $, 若满足

$ [D, \alpha]=0, \; \; [D, \beta]=0, $

$ \begin{equation*} \begin{split} &D([x, y, z])=[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]=0. \end{split} \end{equation*} $

则称$ D $为$ T $的$ \theta $次$ \alpha^{k}\beta^{l} $-中心导子.

用$ \mathrm{Der}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{GDer}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{QDer}_{\alpha^{k}\beta^{l}}^{\theta}(T) $, $ \mathrm{C}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{QC}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{ZDer}^{\theta}_{\alpha^{k}\beta^{l}}(T) $分别表示五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)的$ \theta $次$ \alpha^{k}\beta^{l} $-导子, $ \theta $次$ \alpha^{k}\beta^{l} $-广义导子, $ \theta $次$ \alpha^{k}\beta^{l} $-拟导子, $ \theta $次$ \alpha^{k}\beta^{l} $-型心, $ \theta $次$ \alpha^{k}\beta^{l} $-拟型心, $ \theta $次$ \alpha^{k}\beta^{l} $-中心导子构成的全体. 令

$ \mathrm{Der}(T)=\oplus_{k\geq0, l\geq0}\mathrm{Der}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{GDer}(T)=\oplus_{k\geq0, l\geq0}\mathrm{GDer}_{\alpha^{k}\beta^{l}}(T) $,

$ \mathrm{QDer}(T)=\oplus_{k\geq0, l\geq0}\mathrm{QDer}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{C}(T)=\oplus_{k\geq0, l\geq0}\mathrm{C}_{\alpha^{k}\beta^{l}}(T) $,

$ \mathrm{QC}(T)=\oplus_{k\geq0, l\geq0}\mathrm{QC}_{\alpha^{k}\beta^{l}}(T) $, $ \mathrm{ZDer}(T)=\oplus_{k\geq0, l\geq0}\mathrm{ZDer}_{\alpha^{k}\beta^{l}}(T) $.

称$ \mathrm{Der}(T)=\oplus_{k\geq0, l\geq0}\mathrm{Der}_{\alpha^{k}\beta^{l}}(T) $为$ T $的导子, 其中$ \mathrm{Der}_{\alpha^{k}\beta^{l}}(T) $是$ G $-阶化的, 即

$ \mathrm{Der}_{\alpha^{k}\beta^{l}}(T)=\oplus_{\theta\in G}\mathrm{Der}_{\alpha^{k}\beta^{l}}^{\theta}(T). $

称$ \mathrm{GDer}(T)=\oplus_{k\geq0, l\geq0}\mathrm{GDer}_{\alpha^{k}\beta^{l}}(T) $为$ T $的广义导子, 其中$ \mathrm{GDer}_{\alpha^{k}\beta^{l}}(T) $是$ G $-阶化的, 即

$ \mathrm{GDer}_{\alpha^{k}\beta^{l}}(T)=\oplus_{\theta\in G}\mathrm{GDer}_{\alpha^{k}\beta^{l}}^{\theta}(T). $

称$ \mathrm{QDer}(T)=\oplus_{k\geq0, l\geq0}\mathrm{QDer}_{\alpha^{k}\beta^{l}}(T) $为$ T $的拟导子, 其中$ \mathrm{QDer}_{\alpha^{k}\beta^{l}}(T) $是$ G $-阶化的, 即

$ \mathrm{QDer}_{\alpha^{k}\beta^{l}}(T)=\oplus_{\theta\in G}\mathrm{QDer}_{\alpha^{k}\beta^{l}}^{\theta}(T). $

称$ \mathrm{C}(T)=\oplus_{k\geq0, l\geq0}\mathrm{C}_{\alpha^{k}\beta^{l}}(T) $为$ T $的型心, 其中$ \mathrm{C}_{\alpha^{k}\beta^{l}}(T) $是$ G $-阶化的, 即

$ \mathrm{C}_{\alpha^{k}\beta^{l}}(T)=\oplus_{\theta\in G}\mathrm{C}_{\alpha^{k}\beta^{l}}^{\theta}(T). $

称$ \mathrm{QC}(T)=\oplus_{k\geq0, l\geq0}\mathrm{QC}_{\alpha^{k}\beta^{l}}(T) $为$ T $的拟型心, 其中$ \mathrm{QC}_{\alpha^{k}\beta^{l}}(T) $是$ G $-阶化的, 即

$ \mathrm{QC}_{\alpha^{k}\beta^{l}}(T)=\oplus_{\theta\in G}\mathrm{QC}_{\alpha^{k}\beta^{l}}^{\theta}(T). $

称$ \mathrm{ZDer}(T)=\oplus_{k\geq0, l\geq0} \mathrm{ZDer}_{\alpha^{k}\beta^{l}}(T) $为$ T $的中心导子, 其中$ \mathrm{ZDer}_{\alpha^{k}\beta^{l}}(T) $是$ G $-阶化的, 即

$ \mathrm{ZDer}_{\alpha^{k}\beta^{l}}(T)=\oplus_{\theta\in G}\mathrm{ZDer}_{\alpha^{k}\beta^{l}}^{\theta}(T). $

根据以上定义, 易得

$ \mathrm{ZDer}(T) \subseteq \mathrm{Der}(T) \subseteq \mathrm{QDer}(T) \subseteq \mathrm{GDer}(T) \subseteq \mathrm{End}(T), $

$ \mathrm{C}(T) \subseteq \mathrm{QC}(T) . $

定义2.10  设五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系. 如果对任意的$ y, z\in T $, 满足

$ {\rm Z}(T)=\{x\in hg(T)|[x, y, z]=0\}. $

那么$ {\rm Z}(T) $为$ T $的中心.

下面给出$ \mathrm{BiHom} $-李$ \mathrm{color} $代数的子代数、$ \mathrm{BiHom} $-子代数、理想以及$ \mathrm{BiHom} $-理想的定义.

定义2.11  设五元组($ T, [\cdot, \cdot], \alpha, \beta, \varepsilon $)是$ \mathrm{BiHom} $-李$ \mathrm{color} $代数, $ M $和$ I $是$ T $的$ G $-阶化子空间, 如果$ [M, M]\subseteq M $, 则称$ M $是$ T $的子代数$ ; $如果$ [T, I]\subseteq I $, 则称$ I $是$ T $的理想.

定义2.12  设五元组($ T, [\cdot, \cdot], \alpha, \beta, \varepsilon $)是$ \mathrm{BiHom} $-李$ \mathrm{color} $代数, 若$ M $是$ T $的子代数, 还满足$ \alpha(M)\subseteq M $, $ \beta(M)\subseteq M $, 则称$ M $是$ T $的$ \mathrm{BiHom} $-子代数$ ; $若$ I $是$ T $的理想, 还满足$ \alpha(I)\subseteq I $, $ \beta(I)\subseteq I $, 则称$ I $是$ T $的$ \mathrm{BiHom} $-理想.

命题3.1  若五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是一个保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系, 则下列成立$ : $

$ (\rm 1) $ $ \mathrm{GDer}(T) $, $ \mathrm{QDer}(T) $和$ \mathrm{C}(T) $是$ \mho $的$ \mathrm{BiHom} $-子代数,

$ (\rm 2) $ $ \mathrm{ZDer}(T) $是$ \mathrm{Der}(T) $的$ \mathrm{BiHom} $-理想.

证  (1) 假设$ D_{1} \in \mathrm{GDer}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ D_{2} \in \mathrm{GDer}^{\eta}_{\alpha^{s}\beta^{t}} (T ) $. 对任意的$ x, y, z\in hg(T) $. 有

$ \begin{align*} &\quad[(\widetilde{\alpha}(D_{1})(x)), \alpha^{k+1}\beta^{l}(y), \alpha^{k+1}\beta^{l}(z)]\\ &= [D_{1}\circ\alpha(x), \alpha^{k+1}\beta^{l}(y), \alpha^{k+1}\beta^{l}(z)]\\ &= \alpha([D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)])\\ &= \alpha([D_{1}^{'''}([x, y, z])-\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D_{1}^{'}(y), \alpha^{k}\beta^{l}(z)]\\ &\quad-\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D_{1}^{''}(z)])\\ &=\widetilde{\alpha}(D_{1}^{'''}[x, y, z])-\varepsilon(\theta, x)[\alpha^{k+1}\beta^{l}(x), \widetilde{\alpha}(D_{1}^{'})(y), \alpha^{k+1}\beta^{l}(z)]\\ &\quad-\varepsilon(\theta, x+y)[\alpha^{k+1}\beta^{l}(x), \alpha^{k+1}\beta^{l}(y), \widetilde{\alpha}(D_{1}^{''})(z)]. \end{align*} $

因为$ \widetilde{\alpha}(D_{1}^{'''}) $, $ \widetilde{\alpha}(D_{1}^{''}) $和$ \widetilde{\alpha}(D_{1}^{'}) $全部属于$ \mathrm{End}(T) $, 因此$ \widetilde{\alpha}(D_{1})\in \mathrm{GDer}_{\alpha^{k+1}\beta^{l}}^{\theta}(T ) $. 类似可得$ \widetilde{\beta}(D_{1})\in \mathrm{GDer}_{\alpha^{k}\beta^{l+1}}^{\theta}(T) $. 又有

$ \begin{align*} &\quad [D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &= D_{1}^{'''}([D_{2}(x), \alpha^{s}\beta^{t}(y), \alpha^{s}\beta^{t}(z)])\\ &\quad-\varepsilon(\theta, \eta+x)[\alpha^{k}\beta^{l}(D_{2}(x)), D_{1}^{'}(\alpha^{s}\beta^{t}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\theta, \eta+x+y)[\alpha^{k}\beta^{l}(D_{2}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{1}^{''}(\alpha^{s}\beta^{t}(z))]\\ &= D_{1}^{'''}(D_{2}^{'''}([x, y, z])-\varepsilon(\eta, x)[\alpha^{s}\beta^{t}(x), D_{2}^{'}(y), \alpha^{s}\beta^{t}(z)]\\ &\quad-\varepsilon(\eta, x+y)[\alpha^{s}\beta^{t}(x), \alpha^{s}\beta^{t}(y), D_{2}^{''}(z)])\\ &\quad-\varepsilon(\theta, \eta+x)[\alpha^{k}\beta^{l}(D_{2}(x)), D_{1}^{'}(\alpha^{s}\beta^{t}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\theta, \eta+x+y)[\alpha^{k}\beta^{l}(D_{2}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{1}^{''}(\alpha^{s}\beta^{t}(z))]\\ &= D_{1}^{'''}(D_{2}^{'''}([x, y, z])-\varepsilon(\eta, x)D_{1}^{'''}([\alpha^{s}\beta^{t}(x), D_{2}^{'}(y), \alpha^{s}\beta^{t}(z)])\\ &\quad-\varepsilon(\eta, x+y)D_{1}^{'''}([\alpha^{s}\beta^{t}(x), \alpha^{s}\beta^{t}(y), D_{2}^{''}(z)])\\ &\quad-\varepsilon(\theta, \eta+x)[\alpha^{k}\beta^{l}(D_{2}(x)), D_{1}^{'}(\alpha^{s}\beta^{t}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\theta, \eta+x+y)[\alpha^{k}\beta^{l}(D_{2}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{1}^{''}(\alpha^{s}\beta^{t}(z))]\\ \end{align*} $

$ \begin{align*} &= D_{1}^{'''}(D_{2}^{'''}([x, y, z]))-\varepsilon(\eta, x)[D_{1}(\alpha^{s}\beta^{t}(x)), \alpha^{k}\beta^{l}(D_{2}^{'}(y)), \alpha^{k+s}\beta^{t+l}(z)]\\ &\quad-\varepsilon(\eta, x)\varepsilon(\theta, x)[\alpha^{k+s}\beta^{t+l}(x), D_{1}^{'}D_{2}^{'}(y), \alpha^{k+s}\beta^{t+l}(z)]\\ &\quad-\varepsilon(\eta, x)\varepsilon(\theta, x+y+\eta)[\alpha^{k+s}\beta^{t+l}(x), \alpha^{k}\beta^{l}(D_{2}^{'}(y)), D_{1}^{''}(\alpha^{s}\beta^{t}(z))]\\ &\quad-\varepsilon(\eta, x+y)[D_{1}(\alpha^{s}\beta^{t}(x)), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k}\beta^{l}(D_{2}^{''}(z))]\\ &\quad-\varepsilon(\eta, x+y)\varepsilon(\theta, x)[\alpha^{k+s}\beta^{t+l}(x), D_{1}^{'}(\alpha^{s}\beta^{t}(y)), \alpha^{k}\beta^{l}(D_{2}^{''}(z))]\\ &\quad-\varepsilon(\eta, x+y)\varepsilon(\theta, x+y)[\alpha^{k+s}\beta^{t+l}(x), \alpha^{k+s}\beta^{t+l}(y), D_{1}^{''}D_{2}^{''}(z)]\\ &\quad-\varepsilon(\theta, \eta+x)[\alpha^{k}\beta^{l}(D_{2}(x)), D_{1}^{'}(\alpha^{s}\beta^{t}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\theta, \eta+x+y)[\alpha^{k}\beta^{l}(D_{2}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{1}^{''}(\alpha^{s}\beta^{t}(z))], \end{align*} $

和

$ \begin{align*} &\quad [D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &= D_{2}^{'''}([D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)])\\ &\quad-\varepsilon(\eta, \theta+x)[\alpha^{s}\beta^{t}(D_{1}(x)), D_{2}^{'}(\alpha^{k}\beta^{l}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\eta, \theta+x+y)[\alpha^{s}\beta^{t}(D_{1}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{2}^{''}(\alpha^{k}\beta^{l}(z))]\\ &= D_{2}^{'''}\big(D_{1}^{'''}([x, y, z])-\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D_{1}^{'}(y), \alpha^{k}\beta^{l}(z)]\\ &\quad-\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D_{1}^{''}(z)]\big)\\ &\quad-\varepsilon(\eta, \theta+x)[\alpha^{s}\beta^{t}(D_{1}(x)), D_{2}^{'}(\alpha^{k}\beta^{l}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\eta, \theta+x+y)[\alpha^{s}\beta^{t}(D_{1}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{2}^{''}(\alpha^{k}\beta^{l}(z))]\\ &= D_{2}^{'''}D_{1}^{'''}([x, y, z])-\varepsilon(\theta, x)D_{2}^{'''}([\alpha^{k}\beta^{l}(x), D_{1}^{'}(y), \alpha^{k}\beta^{l}(z)])\\ &\quad-\varepsilon(\theta, x+y)D_{2}^{'''}([\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D_{1}^{''}(z)])\\ &\quad-\varepsilon(\eta, \theta+x)[\alpha^{s}\beta^{t}(D_{1}(x)), D_{2}^{'}(\alpha^{k}\beta^{l}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\eta, \theta+x+y)[\alpha^{s}\beta^{t}(D_{1}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{2}^{''}(\alpha^{k}\beta^{l}(z))]\\ &= D_{2}^{'''}D_{1}^{'''}([x, y, z])-\varepsilon(\theta, x)[D_{2}(\alpha^{k}\beta^{l}(x)), \alpha^{s}\beta^{t}(D_{1}^{'}(y)), \alpha^{k+s}\beta^{t+l}(z)]\\ &\quad-\varepsilon(\theta, x)\varepsilon(\eta, x)[\alpha^{k+s}\beta^{l+t}(x), D_{2}^{'}(D_{1}^{'}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\theta, x)\varepsilon(\eta, x+y+\theta)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{s}\beta^{t}(D_{1}^{'}(y)), D_{2}^{''}(\alpha^{k}\beta^{l}(z))]\\ &\quad-\varepsilon(\theta, x+y)[D_{2}(\alpha^{k}\beta^{l}(x)), \alpha^{k+s}\beta^{l+t}(y), \alpha^{s}\beta^{t}(D_{1}^{''}(z))]\\ &\quad-\varepsilon(\theta, x+y)\varepsilon(\eta, x)[\alpha^{k+s}\beta^{l+t}(x), D_{2}^{'}(\alpha^{k}\beta^{l}(y)), \alpha^{s}\beta^{t}(D_{1}^{''}(z))]\\ &\quad-\varepsilon(\theta, x+y)\varepsilon(\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), D_{2}^{''}D_{1}^{''}(z)]\\ &\quad-\varepsilon(\eta, \theta+x)[\alpha^{s}\beta^{t}(D_{1}(x)), D_{2}^{'}(\alpha^{k}\beta^{l}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\eta, \theta+x+y)[\alpha^{s}\beta^{t}(D_{1}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{2}^{''}(\alpha^{k}\beta^{l}(z))]. \end{align*} $

从而对于任意的$ x, y, z \in hg(T) $, 有

$ \begin{align*} [[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]=& [D_{1}^{'''}, D_{2}^{'''}]([x, y, z])\\ &-\varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), [D_{1}^{'}, D_{2}^{'}](y), \alpha^{k+s}\beta^{l+t}(z)]\\ &-\varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), [D_{1}^{''}, D_{2}^{''}](z)]. \end{align*} $

显然$ [D_{1}^{'}, D_{2}^{'}] $, $ [D_{1}^{''}, D_{2}^{''}] $和$ [D_{1}^{'''}, D_{2}^{'''}] $都属于$ \mathrm{End}(T) $, 所以$ [D_{1}, D_{2}] \in {\rm GDer}_{\alpha^{k+s}\beta^{l+t}}^{\theta+\eta}(T ) $, 对任意的$ x, y, z \in T $, 所以$ \mathrm{GDer}(T ) $是$ \mho $的$ \mathrm{BiHom} $-子代数.

同理可证$ \mathrm{QDer}(T ) $是$ \mho $的$ \mathrm{BiHom} $-子代数.

下面证明$ \mathrm{C}(T) $是$ \mho $的$ \mathrm{BiHom} $-子代数. 假设$ D_{1} \in \mathrm{C}_{\alpha^{k}\beta^{l}}^{\theta}(T) $, $ D_{2} \in \mathrm{C}_{\alpha^{s}\beta^{t}}^{\eta} (T ) $, 对任意的$ x, y, z\in hg(T) $, 我们有

$ \begin{align*} \widetilde{\alpha}(D_{1})([x, y, z]) &= \alpha\circ D_{1}([x, y, z])\\ &= D_{1}\circ\alpha([x, y, z])\\ &= D_{1}[\alpha(x), \alpha(y), \alpha(z)] \\ &=[D_{1}(\alpha(x)), \alpha^{k+1}\beta^{l}(y), \alpha^{k+1}\beta^{l}(z)]\\ &=\varepsilon(\theta, x)[\alpha^{k+1}\beta^{l}(x), D_{1}(\alpha(y)), \alpha^{k+1}\beta^{l}(z)]\\ &=\varepsilon(\theta, x)[\alpha^{k+1}\beta^{l}(x), \widetilde{\alpha}(D_{1}(y)), \alpha^{k+1}\beta^{l}(z)]. \end{align*} $

类似可得$ \widetilde{\beta}(D_{1})([x, y, z])=\varepsilon(\theta, x)[\alpha^{k}\beta^{l+1}(x), \widetilde{\beta}(D_{1}(y)), \alpha^{k}\beta^{l+1}(z)]. $

所以, $ \widetilde{\alpha}(D_{1})\in \mathrm{C}_{\alpha^{k+1}\beta^{l}}^{\theta} (T) $和$ \widetilde{\beta}(D_{1})\in \mathrm{C}_{\alpha^{k}\beta^{l+1}}^{\theta} (T) $, 注意到

$ \begin{align*} &\quad [[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]-\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=[D_{1}(D_{2}(x)), \alpha^{k}\beta^{l}(\alpha^{s}\beta^{t}(y)), \alpha^{k}\beta^{l}(\alpha^{s}\beta^{t}(z))]\\ &\quad -\varepsilon(\theta, \eta)[D_{2}(D_{1}(x)), \alpha^{s}\beta^{t}(\alpha^{k}\beta^{l}(y)), \alpha^{s}\beta^{t}(\alpha^{k}\beta^{l}(z))]\\ &=D_{1}[D_{2}(x), \alpha^{s}\beta^{t}(y), \alpha^{s}\beta^{t}(z)]-\varepsilon(\theta, \eta)D_{2}[D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]\\ &=D_{1}D_{2}([x, y, z])-\varepsilon(\theta, \eta)D_{2}D_{1}([x, y, z])=[D_{1}, D_{2}]([x, y, z]). \end{align*} $

同理可得

$ \varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), [D_{1}, D_{2}](y), \alpha^{k+s}\beta^{l+t}(z)]=[D_{1}, D_{2}]([x, y, z]), $

和

$ \varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), [D_{1}, D_{2}](z)]=[D_{1}, D_{2}]([x, y, z]). $

因此$ [D_{1}, D_{2}] \in \mathrm{C}_{\alpha^{k+s}\beta^{l+t}}^{\theta+\eta}(T) $, 即证$ \mathrm{C}(T ) $是$ \mho $的$ \mathrm{BiHom} $-子代数.

(2) 假设$ D_{1} \in \mathrm{ZDer}^{\theta}_{\alpha^{k}\beta^{l}} (T ) $, $ D_{2} \in \mathrm{Der}^{\eta}_{\alpha^{s}\beta^{t}}(T) $. 对任意的$ x, y, z \in T $, 则有

$ \begin{align*} [\widetilde{\alpha}(D_{1})(x), \alpha^{k+1}\beta^{l}(y), \alpha^{k+1}\beta^{l}(z)] &=\alpha([D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)])\\ &=\alpha\circ D_{1}([x, y, z])\\ &=\widetilde{\alpha}(D_{1})([x, y, z])=0, \end{align*} $

和

$ \begin{align*} [\widetilde{\beta}(D_{1})(x), \alpha^{k}\beta^{l+1}(y), \alpha^{k}\beta^{l+1}(z)] &=\beta([D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)])\\ &=\beta\circ D_{1}([x, y, z])\\ &=\widetilde{\beta}(D_{1})([x, y, z])=0. \end{align*} $

因此$ \widetilde{\alpha}(D_{1})\in {\rm ZDer}_{\alpha^{k+1}\beta^{l}}^{\theta} (T ) $和$ \widetilde{\beta}(D_{1})\in {\rm ZDer}_{\alpha^{k}\beta^{l+1}}^{\theta}(T) $. 注意到

$ \begin{align*} \quad [[D_{1}, D_{2}]([x, y, z])]&=D_{1}D_{2}([x, y, z])-\varepsilon(\theta, \eta)D_{2}D_{1}([x, y, z])\\ &=D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y), \alpha^{s}\beta^{t}(z)])+\varepsilon(\eta, x)D_{1}([\alpha^{s}\beta^{t}(x), D_{2}(y), \alpha^{s}\beta^{t}(z)])\\ &\quad +\varepsilon(\eta, x+y)D_{1}([\alpha^{s}\beta^{t}(x), \alpha^{s}\beta^{t}(y), D_{2}(z)])-0\\ &=0, \end{align*} $

且

$ \begin{align*} &\quad [[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad-\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y), \alpha^{s}\beta^{t}(z)])\\ &\quad-\varepsilon(\theta, \eta) D_{2}([D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)])+\varepsilon(\eta, \theta+x)[\alpha^{s}\beta^{t}(D_{1}(x)), D_{2}(\alpha^{k}\beta^{l}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &\quad +\varepsilon(\eta, \theta+x+y)[\alpha^{s}\beta^{t}(D_{1}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{2}(\alpha^{k}\beta^{l}(z))]\\ &=0. \end{align*} $

则$ [D_{1}, D_{2}]\in \mathrm{ZDer}_{\alpha^{k+s}\beta^{l+t}}^{\theta+\eta} (T ) $. 于是, $ \mathrm{ZDer(T)} $是$ \mathrm{Der}(T ) $的$ \mathrm{BiHom} $-理想.

引理3.2  若五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是特征不等于$ 3 $的域$ \mathbb{F} $上保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系. 则

$ (1) $ $ [\mathrm{Der}(T), \mathrm{C}(T)]\subseteq \mathrm{C}(T) $,

$ (2) $ $ [\mathrm{QDer}(T), \mathrm{QC}(T)]\subseteq \mathrm{QC}(T) $,

$ (3) $ $ [\mathrm{QC}(T), \mathrm{QC}(T)]\subseteq \mathrm{QDer}(T) $,

$ (4) $ $ \mathrm{C}(T)\subseteq \mathrm{QDer}(T) $.

证   $ (1) $假设$ D_{1}\in \mathrm{Der}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ D_{2}\in \mathrm{C}^{\eta}_{\alpha^{s}\beta^{t}}(T) $. 对任意的$ x, y, z\in hg(T) $, 则有

$ \begin{eqnarray*} &&[D_{1}, D_{2}]([x, y, z])\\ &=&D_{1}D_{2}([x, y, z])-\varepsilon(\theta, \eta)D_{2}D_{1}([x, y, z])\\ &=&D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y), \alpha^{s}\beta^{t}(z)])\\ &&-\varepsilon(\theta, \eta)D_{2}\big([D_{1}(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]\\ &&+\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D_{1}(y), \alpha^{k}\beta^{l}(z)]+\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D_{1}(z)]\big)\\ &=&[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&+\varepsilon(\theta, \eta+x)[\alpha^{k}\beta^{l}(D_{2}(x)), D_{1}(\alpha^{s}\beta^{t}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &&+\varepsilon(\theta, \eta+x+y)[\alpha^{k}\beta^{l}(D_{2}(x)), \alpha^{k+s}\beta^{l+t}(y), D_{1}(\alpha^{s}\beta^{t}(z))]\\ &&-\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&-\varepsilon(\theta, \eta)\varepsilon(\theta, x)[(\alpha^{k}\beta^{l}(D_{2}(x)), \alpha^{s}\beta^{t}(D_{1}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &&-\varepsilon(\theta, \eta)\varepsilon(\theta, x+y)[D_{2}(\alpha^{k}\beta^{l}(x)), \alpha^{k+s}\beta^{l+t}(y), \alpha^{s}\beta^{t}(D_{1}(z))]\\ &=&[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]-\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=&[[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]. \end{eqnarray*} $

同理可得

$ [D_{1}, D_{2}]([x, y, z])=\varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), [D_{1}, D_{2}](y), \alpha^{k+s}\beta^{l+t}(z)], $

且

$ [D_{1}, D_{2}]([x, y, z])=\varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), [D_{1}, D_{2}](z)]. $

则$ [D_{1}, D_{2}]([x, y, z])\in \mathrm{C}_{\alpha^{k+s}\beta^{l+t}}^{\theta+\eta}(T) $. 因此$ [\mathrm{Der}(T), \mathrm{C}(T)]\subseteq \mathrm{C}(T) $.

$ (2) $类似于$ (1) $的证明.

$ (3) $假设$ D_{1}\in \mathrm{QC}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ D_{2}\in \mathrm{QC}^{\eta}_{\alpha^{s}\beta^{t}}(T) $. 对任意的$ x, y, z\in hg(T) $, 则有

$ \begin{eqnarray*} &&[[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&+\varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), [D_{1}, D_{2}](y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&+\varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), [D_{1}, D_{2}](z)]\\ &=&[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&+\varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), D_{1}D_{2}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&+\varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), D_{1}D_{2}(z)]\\ &&-\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&-\varepsilon(\theta+\eta, x)\varepsilon(\theta, \eta)[\alpha^{k+s}\beta^{l+t}(x), D_{2}D_{1}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &&-\varepsilon(\theta+\eta, x+y)\varepsilon(\theta, \eta)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), D_{2}D_{1}(z)]. \end{eqnarray*} $

很容易验证

$ \begin{eqnarray*} &&[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=&[D_{1}(D_{2}(x)), \alpha^{k}\beta^{l}(\alpha^{s}\beta^{t}(y)), \alpha^{k}\beta^{l}(\alpha^{s}\beta^{t}(z))]\\ &=&\varepsilon(\theta, \eta+x)[\alpha^{k}\beta^{l}(D_{2}(x)), D_{1}(\alpha^{s}\beta^{t}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &=&\varepsilon(\theta, \eta+x)\varepsilon(\eta, x)[\alpha^{k+s}\beta^{l+t}(x), D_{2}(D_{1}(y)), \alpha^{k+s}\beta^{l+t}(z)]\\ &=&\varepsilon(\theta+\eta, x)\varepsilon(\theta, \eta)[\alpha^{k+s}\beta^{l+t}(x), D_{2}D_{1}(y), \alpha^{k+s}\beta^{l+t}(z)], \end{eqnarray*} $

且

$ \begin{eqnarray*} &&\varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), D_{1}D_{2}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=&\varepsilon(\theta+\eta, x)[\alpha^{k}\beta^{l}(\alpha^{s}\beta^{t}(x)), D_{1}(D_{2}(y)), \alpha^{k}\beta^{l}(\alpha^{s}\beta^{t}(z))]\\ &=&\varepsilon(\theta+\eta, x)\varepsilon(\theta, \eta+y)[\alpha^{k+s}\beta^{l+t}(x), (\alpha^{k}\beta^{l}(D_{2}(y)), D_{1}(\alpha^{s}\beta^{t}(z))]\\ &=&\varepsilon(\theta+\eta, x)\varepsilon(\theta, \eta+y)\varepsilon(\eta, y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), D_{2}D_{1}(z)]\\ &=&\varepsilon(\theta+\eta, x+y)\varepsilon(\theta, \eta)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), D_{2}D_{1}(z)], \end{eqnarray*} $

且

$ \begin{eqnarray*} &&\varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), D_{1}D_{2}(z)]\\ &=&\varepsilon(\theta+\eta, x+y)\varepsilon(x+y, \theta)[D_{1}(\alpha^{s}\beta^{t}(x)), (\alpha^{k+s}\beta^{l+t}(y)), \alpha^{k}\beta^{l}D_{2}(z)]\\ &=&\varepsilon(\theta+\eta, x+y)\varepsilon(x+y, \theta)\varepsilon(\theta+x+y, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]\\ &=&\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]. \end{eqnarray*} $

因此, $ [[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y), \alpha^{k+s}\beta^{l+t}(z)]+\varepsilon(\theta+\eta, x)[\alpha^{k+s}\beta^{l+t}(x), [D_{1}, D_{2}](y), \alpha^{k+s}\beta^{l+t}(z)] $ $ +\varepsilon(\theta+\eta, x+y)[\alpha^{k+s}\beta^{l+t}(x), \alpha^{k+s}\beta^{l+t}(y), [D_{1}, D_{2}](z)]=0 $, 所以$ [D_{1}, D_{2}]\in \mathrm{QDer}_{\alpha^{k+s}\beta^{l+t}}^{\theta+\eta}(T) $. 即证$ [\mathrm{QC}(T), \mathrm{QC}(T)]\subseteq \mathrm{QDer}(T) $.

$ (4) $假设$ D\in \mathrm{C}^{\theta}_{\alpha^{k}\beta^{l}}(T) $. 则对任意的$ x, y, z\in hg(T) $, 有

$ \begin{eqnarray*} D([x, y, z])&=&[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]=\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &=&\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)]. \end{eqnarray*} $

因此

$ \begin{eqnarray*} 3D([x, y, z])&=&[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]+\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &&+\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)]. \end{eqnarray*} $

这就是说$ D^{'}=3D\in \mathrm{C}^{\theta}_{\alpha^{k}\beta^{l}}(T) $. 故$ \mathrm{C}(T)\subseteq \mathrm{QDer}(T) $.

定理3.3  设五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是一个保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系, $ \alpha, \beta $是满射, 则$ [\mathrm{C}(T), \mathrm{QC}(T)]\subseteq \mathrm{End}(T, {\rm Z}(T)) $. 特别地, 若$ {\rm Z}(T)=\{0\} $, 则$ [{\rm C}(T), \mathrm{QC}(T)]=\{0\} $.

证  假设$ D_{1}\in \mathrm{C}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, $ D_{2}\in \mathrm{QC}^{\eta}_{\alpha^{s}\beta^{t}}(T) $. 对任意的$ x\in hg(T) $. 因为$ \alpha $和$ \beta $是满射, 对任意的$ y, z\in hg(T) $, 存在$ y^{'}, z^{'}\in hg(T) $, 使得$ y=\alpha^{k+s}\beta^{l+t}(y^{'}), z=\alpha^{k+s}\beta^{l+t}(z^{'}) $, 则

$ \begin{eqnarray*} [[D_{1}, D_{2}](x), y, z] &=&[[D_{1}, D_{2}](x), \alpha^{k+s}\beta^{l+t}(y^{'}), \alpha^{k+s}\beta^{l+t}(z^{'})]\\ &=&[D_{1}D_{2}(x), \alpha^{k+s}\beta^{l+t}(y^{'}), \alpha^{k+s}\beta^{l+t}(z^{'})]\\ &&-\varepsilon(\theta, \eta)[D_{2}D_{1}(x), \alpha^{k+s}\beta^{l+t}(y^{'}), \alpha^{k+s}\beta^{l+t}(z^{'})]\\ &=&D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y^{'}), \alpha^{s}\beta^{t}(z^{'})])\\ &&-\varepsilon(\theta, \eta)\varepsilon(\eta, \theta+x)[\alpha^{s}\beta^{t}(D_{1}(x)), D_{2}(\alpha^{k}\beta^{l}(y^{'})), \alpha^{k+s}\beta^{l+t}(z^{'})]\\ &=&D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y^{'}), \alpha^{s}\beta^{t}(z^{'})])\\ &&-\varepsilon(\eta, x)D_{1}([\alpha^{s}\beta^{t}(x), D_{2}(y^{'}), \alpha^{s}\beta^{t}(z^{'})])\\ &=&D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y^{'}), \alpha^{s}\beta^{t}(z^{'})])\\ &&-D_{1}([D_{2}(x), \alpha^{s}\beta^{t}(y^{'}), \alpha^{s}\beta^{t}(z^{'})])\\ &=&0. \end{eqnarray*} $

因此$ [D_{1}, D_{2}](x)\in {\rm Z}(T) $, 从而$ [D_{1}, D_{2}]\in \mathrm{End}(T, {\rm Z}(T)) $. 特别地, 若$ {\rm Z}(T)=\{0\} $, 显然有

$ [{\rm C}(T), \mathrm{QC}(T)]=\{0\} $.

定理3.4  如果五元组($ T, [\cdot, \cdot, \cdot], \alpha, \beta, \varepsilon $)是在特征不为$ 2 $的域$ \mathbb{F} $上的一个保积的$ \mathrm{BiHom} $-李$ \mathrm{color} $三系, 则有

$ \mathrm{ZDer}(T)={\rm C}(T)\cap \mathrm{Der}(T). $

证  假设$ D\in {\rm C}^{\theta}_{\alpha^{k}\beta^{l}}(T)\cap \mathrm{Der}^{\theta}_{\alpha^{k}\beta^{l}}(T) $. 对任意的$ x, y, z\in hg(T) $, 则有

$ \begin{eqnarray*} D([x, y, z]) &=&[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]+\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &&+\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)], \end{eqnarray*} $

和

$ \begin{eqnarray*} D([x, y, z]) &=&[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]=\varepsilon(\theta, x)[\alpha^{k}\beta^{l}(x), D(y), \alpha^{k}\beta^{l}(z)]\\ &=&\varepsilon(\theta, x+y)[\alpha^{k}\beta^{l}(x), \alpha^{k}\beta^{l}(y), D(z)]. \end{eqnarray*} $

则$ 2D([x, y, z])=0 $, 因为数域$ \mathbb{F} $的特征不等于$ 2 $, 因此$ D([x, y, z])=0 $. 因此$ D \in \mathrm{ZDer}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, 从而$ {\rm C}(T)\cap \mathrm{Der}(T)\subseteq \mathrm{ZDer}(T) $.

另一方面, 假设$ D \in \mathrm{ZDer}^{\theta}_{\alpha^{k}\beta^{l}}(T) $, 对任意的$ x, y, z\in hg(T) $, 我们有

$ D([x, y, z])=[D(x), \alpha^{k}\beta^{l}(y), \alpha^{k}\beta^{l}(z)]=0. $

很容易验证$ D \in {\rm C}^{\theta}_{\alpha^{k}\beta^{l}}(T)\cap \mathrm{Der}^{\theta}_{\alpha^{k}\beta^{l}}(T) $. 从而$ \mathrm{ZDer}(T)\subseteq {\rm C}(T)\cap \mathrm{Der}(T) $.

综上$ \mathrm{ZDer}(T)={\rm C}(T)\cap \mathrm{Der}(T) $.