作为李三系和李超三系的推广, 李$ \mathrm{color} $三系的性质得到了广泛研究. 2007年, 文献[1] 给出李$ \mathrm{color} $三系的概念. 文献[2] 研究了李$ \mathrm{color} $三系的幂零理想. 文献[3] 探究了李$ \mathrm{color} $三系的$ \mathrm{Frattini} $子系的定义和性质. 文献[4] 讨论了李$ \mathrm{color} $三系的型心. 文献[5, 6] 探讨了李$ \mathrm{color} $三系的导子、广义导子和拟导子. 文献[7] 研究了分裂李$ \mathrm{color} $三系.

Yamaguti在文献[8] 中提出了李三系的表示与上同调理论. 文献[9] 利用上同调研究了李三系的形变和扩张理论. 目前, 文献[10, 11] 讨论了$ \mathrm{\delta} $-$ \mathrm{Jordan} $李三系的上同调和Hom-李三系的$ \mathrm{Nijenhuis} $算子. 文献[12] 刻画了李超三系的上同调和$ \mathrm{Nijenhuis} $算子. 文献[13]探究了Hom-李超三系的上同调和形变. 于是想到将上同调理论推广到李$ \mathrm{color} $三系上. 本文研究李$ \mathrm{color} $三系的上同调和$ \mathrm{Nijenhuis} $算子, 并讨论其形变.

定义2.1[1]    设$ G $是交换群, $ \mathbb{K} $是任意域. 若$ \forall\; \alpha, \beta, \gamma \in G $, 下列等式均成立$ : $

$ \varepsilon(\alpha,\beta)\varepsilon(\beta,\alpha)=1, \; \; \varepsilon(\alpha,\beta+\gamma)=\varepsilon(\alpha,\beta)\varepsilon(\alpha,\gamma), \; \; \varepsilon(\alpha+\beta,\gamma)=\varepsilon(\alpha,\gamma)\varepsilon(\beta,\gamma), $

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

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 $-阶化的. 对于$ G $-阶化向量空间中的齐次元素$ a $, $ b $, $ c $, 用$ |a|, |b|, |c| \in G $表示它们的次数. 为简便, 用$ \varepsilon(a,b) $代表$ \varepsilon(|a|,|b|) $, 用$ \varepsilon(a, b+c) $代表$ \varepsilon(|a|,|b|+|c|) $, 以此类推. 此外, 符号$ \varepsilon(a,b) $若出现均默认$ a, b $是$ V $的齐次元素.

定义2.2[1]   设$ T=\oplus_{g\in G}T_g $是域$ \mathbb{K} $上的一个$ G $-阶化向量空间. 若$ T $上有三元运算$ [\cdot,\cdot,\cdot]: T\times T\times T \rightarrow T $满足$ : $

$ \begin{equation} \begin{aligned} [a,b,c]=-\varepsilon(a,b)[b,a,c], \end{aligned} \end{equation} $

(2.1)

$ \begin{equation} \begin{aligned} \varepsilon(c,a)[a,b,c]+ \varepsilon(a,b)[b,c,a]+\varepsilon(b,c)[c,a,b]=0, \end{aligned} \end{equation} $

(2.2)

$ \begin{equation} \begin{aligned} [a,b,[c,d,e]]=[[a,b,c],d,e]+\varepsilon(a+b,c)[c,[a,b,d],e]+\varepsilon(a+b,c+d)[c,d,[a,b,e]], \end{aligned} \end{equation} $

(2.3)

$ \forall $ $ a,b,c,d,e\in T $, 则称$ T $是李$ \mathrm{color} $三系.

如果三线性映射$ f:(T,[\cdot,\cdot,\cdot])\rightarrow (T^{'},[\cdot,\cdot,\cdot]^{'}) $是$ G $-阶化向量空间上的映射, 且满足$ f([a,b,c])=[f(a),f(b),f(c)]^{'} $, 则称$ f $是李$ \mathrm{color} $三系的同态.

定义2.3   设$ T $是李$ \mathrm{color} $三系, $ V $为域$ \mathbb{K} $上的$ G $-阶化向量空间. 若有双线性映射$ \theta: T\otimes T \rightarrow \mathrm{End}(V) $, $ \forall $ $ a,b,c,d\in T $满足$ : $

$ \begin{equation} \begin{aligned} &\varepsilon(a+b,c+d)\theta(c,d)\theta(a,b)-\varepsilon(a,b)\varepsilon(a+c,d)\theta(b,d)\theta(a,c)\\ &-\theta(a,[b,c,d]) +\varepsilon(a,b+c)D(b,c)\theta(a,d)=0, \end{aligned} \end{equation} $

(2.4)

$ \begin{equation} \begin{aligned} &\varepsilon(a+b,c+d)\theta(c,d)D(a,b)-D(a,b)\theta(c,d)\\&+\theta([a,b,c],d)+\varepsilon(a+b,c)\theta(c,[a,b,d])=0, \end{aligned} \end{equation} $

(2.5)

$ \begin{equation} \begin{aligned} &D([a,b,c],d)+\varepsilon(a+b,c)D(c,[a,b,d]) -D(a,b)D(c,d)\\&+\varepsilon(a+b,c+d)D(c,d)D(a,b)=0, \end{aligned} \end{equation} $

(2.6)

其中$ D(a,b)=\varepsilon(a,b)\theta(b,a)-\theta(a,b) $, 则称$ (V,\theta) $是$ T $的表示, $ V $为$ T $-模.

例2.4    设$ T $是李$ \mathrm{color} $三系. 定义$ \theta: T\otimes T \rightarrow \mathrm{End}(T) $为

$ \theta(a,b)(x)=\varepsilon(a+b,x)[x,a,b],\; \; D(a,b)=\varepsilon(a,b)\theta(b,a)-\theta(a,b), $

易证$ D(a,b)(x)=[a,b,x] $, $ T $为$ T $-模, 称$ (T,\theta) $是$ T $的伴随表示.

命题2.5   设$ T $是李$ \mathrm{color} $三系, $ (V,\theta) $是$ T $的一个表示. 则$ T\oplus V $是李$ \mathrm{color} $三系.

证  定义三线性积$ [\cdot,\cdot,\cdot]:(T\oplus V)\otimes (T\oplus V)\otimes (T\oplus V)\rightarrow T\oplus V $为

$ [(a,u),(b,v),(c,w)]=([a,b,c],\varepsilon(a,b+c)\theta(b,c)(u)-\varepsilon(b,c)\theta(a,c)(v)+D(a,b)(w)), $

$ \forall $ $ (a,u), (b,v), (c,w)\in T\oplus V $, 其中$ |(a,0)|=|a|,\; |(0,u)|=|u| $, $ D(a,b)=\varepsilon(a,b)\theta(b,a)-\theta(a,b) $.

由于$ T $是李$ \mathrm{color} $三系, 易证($ 2.1 $) 式成立. $ \forall $ $ (a,u), (b,v), (c,w)\in T\oplus V $, 有如下计算

$ \begin{eqnarray*} &&-\varepsilon(a,b)[(b,v),(a,u),(c,w)]\\ &=&-\varepsilon(a,b)([b,a,c],\varepsilon(b,a+c)\theta(a,c)(v)-\varepsilon(a,c)\theta(b,c)(u)+D(b,a)(w))\\ &=&(-\varepsilon(a,b)[b,a,c],-\varepsilon(b,c)\theta(a,c)(v)+\varepsilon(a,b+c)\theta(b,c)(u)-\varepsilon(a,b)D(b,a)(w))\\ &=&([a,b,c],\varepsilon(a,b+c)\theta(b,c)(u)-\varepsilon(b,c)\theta(a,c)(v)+D(a,b)(w))\\ &=&[(a,u), (b,v), (c,w)], \end{eqnarray*} $

所以($ 2.1 $) 式成立.

利用($ 2.2 $) 式, 有以下计算

$ \begin{eqnarray*} &&\varepsilon(c,a)[(a,u),(b,v),(c,w)]+\varepsilon(a,b)[(b,v),(c,w),(a,u)]+\varepsilon(b,c)[(c,w),(a,u),(b,v)]\\ &=&(\varepsilon(c,a)[a,b,c]+\varepsilon(a,b)[b,c,a]+\varepsilon(b,c)[c,a,b],\Omega), \end{eqnarray*} $

其中:

$ \begin{eqnarray*} \Omega&=&\varepsilon(a,b)\theta(b,c)(u)-\varepsilon(c,a)\varepsilon(b,c)\theta(a,c)(v)+\varepsilon(c,a)D(a,b)(w)\\ &&+\varepsilon(b,c)\theta(c,a)(v)-\varepsilon(a,b)\varepsilon(c,a)\theta(b,a)(w)+\varepsilon(a,b)D(b,c)(u)\\ &&+\varepsilon(c,a)\theta(a,b)(w)-\varepsilon(b,c)\varepsilon(a,b)\theta(c,b)(u)+\varepsilon(b,c)D(c,a)(v)\\ &=&\varepsilon(a,b)\theta(b,c)(u)+(\varepsilon(a,b)D(b,c)(u)-\varepsilon(b,c)\varepsilon(a,b)\theta(c,b)(u))\\ &&+\varepsilon(b,c)\theta(c,a)(v)+(\varepsilon(b,c)D(c,a)(v)-\varepsilon(c,a)\varepsilon(b,c)\theta(a,c)(v))\\ &&+\varepsilon(c,a)\theta(a,b)(w)+(\varepsilon(c,a)D(a,b)(w)-\varepsilon(a,b)\varepsilon(c,a)\theta(b,a)(w))\\ &=&\varepsilon(a,b)\theta(b,c)(u)-\varepsilon(a,b)\theta(b,c)(u)+\varepsilon(b,c)\theta(c,a)(v)-\varepsilon(b,c)\theta(c,a)(v)\\ &&+\varepsilon(c,a)\theta(a,b)(w)-\varepsilon(c,a)\theta(a,b)(w)\\ &=&0, \end{eqnarray*} $

所以

$ \begin{eqnarray*} &&\varepsilon(c,a)[(a,u),(b,v),(c,w)]+\varepsilon(a,b)[(b,v),(c,w),(a,u)]+\varepsilon(b,c)[(c,w),(a,u),(b,v)]\\ &=&(0,0). \end{eqnarray*} $

利用($ 2.3 $) 式, $ \forall $ $ (a,u), (b,v), (c,w), (d,m), (e,n)\in T\oplus V $. 计算如下

$ \begin{eqnarray*} &&[[(a,u),(b,v),(c,w)],(d,m),(e,n)]+\varepsilon(a+b,c)[(c,w),[(a,u),(b,v),(d,m)],(e,n)]\\&&+\varepsilon(a+b,c+d)[(c,w),(d,m),[(a,u), (b,v),(e,n)]]\\ &=&([[a,b,c],d,e]+\varepsilon(a+b,c)[c,[a,b,d],e]+\varepsilon(a+b,c+d)[c,d,[a,b,e]],\Pi_{1}+\Pi_{2}+\Pi_{3})\\ &=&([a,b,[c,d,e]],\Pi_{4})=[(a,u),(b,v),[(c,w),(d,m),(e,n)]]. \end{eqnarray*} $

其中:

$ \begin{eqnarray*} \Pi_{1}&=&\varepsilon(a+b+c,d+e)\varepsilon(a,b+c)\theta(d,e)\theta(b,c)(u) -\varepsilon(a+b+c,d+e)\varepsilon(b,c)\theta(d,e)\theta(a,c)(v)\\ &&+\varepsilon(a+b+c,d+e)\theta(d,e)D(a,b)(w) -\varepsilon(d,e)\theta([a,b,c],e)(m)+D([a,b,c],d)(n);\\ \Pi_{2}&=&-\varepsilon(a+b+d,e)\varepsilon(a,b+d)\varepsilon(a+b,c)\theta(c,e)\theta(b,d)(u)\\ &&+\varepsilon(a+b+d,e)\varepsilon(b,d)\varepsilon(a+b,c)\theta(c,e)\theta(a,d)(v)\\ &&+\varepsilon(c,a+b+d+e)\varepsilon(a+b,c)\theta([a,b,d],e)(w)\\ &&-\varepsilon(a+b+d,e)\varepsilon(a+b,c)\theta(c,e)D(a,b)(m)+\varepsilon(a+b,c)D(c,[a,b,d])(n);\\ \Pi_{3}&=&\varepsilon(a+b,c+d)\varepsilon(a,b+e)D(c,d)\theta(b,e)(u)-\varepsilon(a+b,c+d)\varepsilon(b,e)D(c,d)\theta(a,e)(v)\\ &&+\varepsilon(a+b,c+d)\varepsilon(c,a+b+d+e)\theta(d,[a,b,e])(w)\\ &&-\varepsilon(a+b,c+d)\varepsilon(d,a+b+e)\theta(c,[a,b,e])(m)+\varepsilon(a+b,c+d)D(c,d)D(a,b)(n);\\ \Pi_{4}&=&\varepsilon(a,b+c+d+e)\theta(b,[c,d,e])(u)-\varepsilon(b,c+d+e)\theta(a,[c,d,e])(v)\\ &&+\varepsilon(c,d+e)D(a,b)\theta(d,e)(w)-\varepsilon(d,e)D(a,b)\theta(c,e)(m)+D(a,b)D(c,d)(n). \end{eqnarray*} $

定义2.6    设$ T $是李$ \mathrm{color} $三系, $ V $为$ T $-模. 若$ n $-线性映射$ f:T\times T\times\cdot\cdot\cdot \times T\rightarrow T $满足$ : $

$ (1)\; \; $ $ f(x_{1},x_{2},\cdot\cdot\cdot,x,y,\cdot\cdot\cdot,x_{n}) =-\varepsilon(x,y)f(x_{1},x_{2},\cdot\cdot\cdot,y,x,\cdot\cdot\cdot,x_{n}) $,

$ (2)\; \; $ $ \varepsilon(z,x)f(x_{1},x_{2},\cdot\cdot\cdot,x_{n-3},x,y,z)+\varepsilon(x,y)f(x_{1},x_{2},\cdot\cdot\cdot,x_{n-3},y,z,x) +\varepsilon(y,z)f(x_{1},x_{2},\cdot\cdot\cdot,x_{n-3},z,x,y)=0 $,

则称$ f $为$ T $的$ n $-上链. 记$ C^{n}(T,V) $是全体$ n $-上链的集合, $ \forall\; n\geq1 $.

定义2.7    设$ T $是李$ \mathrm{color} $三系, $ V $为$ T $-模. 对于$ n=1, 2, 3, 4 $, 上边界算子$ d^{n}:C^{n}(T,V)\rightarrow C^{n+2}(T,V) $的定义如下:

$ \bullet $如果$ f\in C^{1}(T,V) $, 则

$ \begin{eqnarray*} &&d^{1}f(x_{1},x_{2},x_{3})\\ &=&\varepsilon(f+x_{1},x_{2}+x_{3})\theta(x_{2},x_{3})f(x_{1})-f([x_{1},x_{2},x_{3}])\\ &&-\varepsilon(f,x_{1}+x_{3})\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})f(x_{2}) +\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})f(x_{3}). \end{eqnarray*} $

$ \bullet $如果$ f\in C^{2}(T,V) $, 则

$ \begin{eqnarray*} &&d^{2}f(y,x_{1},x_{2},x_{3})\\ &=&\varepsilon(f+y+x_{1},x_{2}+x_{3})\theta(x_{2},x_{3})f(y,x_{1})-f(y,[x_{1},x_{2},x_{3}])\\ &&-\varepsilon(f+y,x_{1}+x_{3})\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})f(y,x_{2}) +\varepsilon(f+y,x_{1}+x_{2})D(x_{1},x_{2})f(y,x_{3}). \end{eqnarray*} $

$ \bullet $如果$ f\in C^{3}(T,V) $, 则

$ \begin{eqnarray*} &&d^{3}f(x_{1},x_{2},x_{3},x_{4},x_{5})\\ &=&\varepsilon(f+x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})f(x_{1},x_{2},x_{3})\\ &&-\varepsilon(f+x_{1}+x_{2},x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})f(x_{1},x_{2},x_{4})\\ &&-\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})f(x_{3},x_{4},x_{5}) +\varepsilon(f+x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})f(x_{1},x_{2},x_{5})\\ &&+f([x_{1},x_{2},x_{3}],x_{4},x_{5})-f(x_{1},x_{2},[x_{3},x_{4},x_{5}]) +\varepsilon(x_{1}+x_{2},x_{3})f(x_{3},[x_{1},x_{2},x_{4}],x_{5})\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})f(x_{3},x_{4},[x_{1},x_{2},x_{5}]). \end{eqnarray*} $

$ \bullet $如果$ f\in C^{4}(T,V) $, 则

$ \begin{eqnarray*} &&d^{4}f(y,x_{1},x_{2},x_{3},x_{4},x_{5})\\ &=&\varepsilon(f+y+x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})f(y,x_{1},x_{2},x_{3})\\ &&-\varepsilon(f+y+x_{1}+x_{2},x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})f(y,x_{1},x_{2},x_{4})\\ &&-\varepsilon(f+y,x_{1}+x_{2})D(x_{1},x_{2})f(y,x_{3},x_{4},x_{5})\\ &&+\varepsilon(f+y+x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})f(y,x_{1},x_{2},x_{5})+f(y,[x_{1},x_{2},x_{3}],x_{4},x_{5})\\ &&-f(y,x_{1},x_{2},[x_{3},x_{4},x_{5}])+\varepsilon(x_{1}+x_{2},x_{3})f(y,x_{3},[x_{1},x_{2},x_{4}],x_{5})\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})f(y,x_{3},x_{4},[x_{1},x_{2},x_{5}]). \end{eqnarray*} $

定理2.8   设$ {T} $是李$ \mathrm{color} $三系, $ V $为$ T $-模. 则上边界算子$ d^{n} $满足$ d^{n+2}d^{n}=0 $, $ n=1,2 $.

证  由上边界算子的定义知, $ d^{3}d^{1}=0 $可得到$ d^{4}d^{2}=0 $. 于是只需验证$ d^{3}d^{1}=0 $.

$ \begin{eqnarray*} &&d^{3}(d^{1}f)(x_{1},x_{2},x_{3},x_{4},x_{5})\\ &=&\varepsilon(f+x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})(d^{1}f)(x_{1},x_{2},x_{3})\\ &&-\varepsilon(f+x_{1}+x_{2},x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})(d^{1}f)(x_{1},x_{2},x_{4})\\ &&-\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})(d^{1}f)(x_{3},x_{4},x_{5})\\ &&+\varepsilon(f+x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})(d^{1}f)(x_{1},x_{2},x_{5})+(d^{1}f)([x_{1},x_{2},x_{3}],x_{4},x_{5})\\ &&-(d^{1}f)(x_{1},x_{2},[x_{3},x_{4},x_{5}])+\varepsilon(x_{1}+x_{2},x_{3})(d^{1}f)(x_{3},[x_{1},x_{2},x_{4}],x_{5})\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})(d^{1}f)(x_{3},x_{4},[x_{1},x_{2},x_{5}])\\ &=&\varepsilon(f+x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})(\varepsilon(f+x_{1},x_{2}+x_{3})\theta(x_{2},x_{3})f(x_{1})\\ &&-\varepsilon(f,x_{1}+x_{3})\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})f(x_{2}) +\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})f(x_{3})-f([x_{1},x_{2},x_{3}]))\\ &&-\varepsilon(f+x_{1}+x_{2},x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})( \varepsilon(f+x_{1},x_{2}+x_{4})\theta(x_{2},x_{4})f(x_{1})\\ &&-\varepsilon(f,x_{1}+x_{4})\varepsilon(x_{2},x_{4})\theta(x_{1},x_{4})f(x_{2})+\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})f(x_{4}) -f([x_{1},x_{2},x_{4}]))\\ &&-\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})(\varepsilon(f+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})f(x_{3}) -f([x_{3},x_{4},x_{5}])\\ &&-\varepsilon(f,x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})f(x_{4}) +\varepsilon(f,x_{3}+x_{4})D(x_{3},x_{4})f(x_{5}))\\ &&+\varepsilon(f+x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})(\varepsilon(f+x_{1},x_{2}+x_{5})\theta(x_{2},x_{5})f(x_{1})\\ &&-\varepsilon(f,x_{1}+x_{5})\varepsilon(x_{2},x_{5})\theta(x_{1},x_{5})f(x_{2})-f([x_{1},x_{2},x_{5}]) +\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})f(x_{5}))\\ &&+\varepsilon(f+x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})f([x_{1},x_{2},x_{3}])-f([[x_{1},x_{2},x_{3}],x_{4},x_{5}])\\ &&-\varepsilon(f,x_{1}+x_{2}+x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta([x_{1},x_{2},x_{3}],x_{5})f(x_{4})\\ &&+\varepsilon(f,x_{1}+x_{2}+x_{3}+x_{4})D([x_{1},x_{2},x_{3}],x_{4})f(x_{5})-(-f([x_{1},x_{2},[x_{3},x_{4},x_{5}]])\\ &&+\varepsilon(f+x_{1},x_{2}+x_{3}+x_{4}+x_{5})\theta(x_{2},[x_{3},x_{4},x_{5}])f(x_{1}) +\varepsilon(f,x_{1}+x_{2})D(x_{1},x_{2})f([x_{3},x_{4},x_{5}])\\ &&-\varepsilon(f,x_{1}+x_{3}+x_{4}+x_{5})\varepsilon(x_{2},x_{3}+x_{4}+x_{5})\theta(x_{1},[x_{3},x_{4},x_{5}])f(x_{2}))\\ &&+\varepsilon(x_{1}+x_{2},x_{3})(\varepsilon(f+x_{3},x_{1}+x_{2}+x_{4}+x_{5})\theta([x_{1},x_{2},x_{4}],x_{5})f(x_{3}) -f([x_{3},[x_{1},x_{2},x_{4}],x_{5}])\\ &&-\varepsilon(f,x_{3}+x_{5})\varepsilon(x_{1}+x_{2}+x_{4},x_{5})\theta(x_{3},x_{5})f([x_{1},x_{2},x_{4}])\\ &&+\varepsilon(f,x_{1}+x_{2}+x_{3}+x_{4})D(x_{3},[x_{1},x_{2},x_{4}])f(x_{5}))\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})(\varepsilon(f+x_{3},x_{1}+x_{2}+x_{4}+x_{5})\theta(x_{4},[x_{1},x_{2},x_{5}])f(x_{3})\\ &&-\varepsilon(f,x_{1}+x_{2}+x_{3}+x_{5})\varepsilon(x_{4},x_{1}+x_{2}+x_{5})\theta(x_{3},[x_{1},x_{2},x_{5}])f(x_{4})\\ &&+\varepsilon(f,x_{3}+x_{4})D(x_{3},x_{4})f([x_{1},x_{2},x_{5}])-f([x_{3},x_{4},[x_{1},x_{2},x_{5}]]))\\ &=&\varepsilon(f+x_{1},x_{2}+x_{3}+x_{4}+x_{5})\Lambda_{1}f(x_{1}) -\varepsilon(f+x_{2},x_{3}+x_{4}+x_{5})\varepsilon(f,x_{1})\Lambda_{2}f(x_{2})\\ &&+\varepsilon(f,x_{1}+x_{2}+x_{4}+x_{5})\varepsilon(x_{3},x_{4}+x_{5})\Lambda_{3}f(x_{3}) -\varepsilon(f,x_{1}+x_{2}+x_{3}+x_{5})\varepsilon(x_{4},x_{5})\Lambda_{4}f(x_{4})\\ &&+\varepsilon(f,x_{1}+x_{2}+x_{3}+x_{4})\Lambda_{5}f(x_{5}). \end{eqnarray*} $

其中:

$ \begin{eqnarray*} \Lambda_{1}&=&\varepsilon(x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})\theta(x_{2},x_{3}) -\varepsilon(x_{2},x_{3})\varepsilon(x_{2}+x_{4},x_{5})\theta(x_{3},x_{5})\theta(x_{2},x_{4})\\ &&-\theta(x_{2},[x_{3},x_{4},x_{5}])+\varepsilon(x_{2},x_{3}+x_{4})D(x_{3},x_{4})\theta(x_{2},x_{5})=0; \end{eqnarray*} $

$ \begin{eqnarray*} \Lambda_{2}&=&\varepsilon(x_{1}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})\theta(x_{1},x_{3}) -\varepsilon(x_{1},x_{3})\varepsilon(x_{1}+x_{4},x_{5})\theta(x_{3},x_{5})\theta(x_{1},x_{4})\\ &&-\theta(x_{1},[x_{3},x_{4},x_{5}])+\varepsilon(x_{1},x_{3}+x_{4})D(x_{3},x_{4})\theta(x_{1},x_{5})=0;\\ \Lambda_{3}&=&\varepsilon(x_{1}+x_{2},x_{4}+x_{5})\theta(x_{4},x_{5})D(x_{1},x_{2})-D(x_{1},x_{2})\theta(x_{4},x_{5})\\ &&+\theta([x_{1},x_{2},x_{4}],x_{5})+\varepsilon(x_{1}+x_{2},x_{4})\theta(x_{4},[x_{1},x_{2},x_{5}])=0;\\ \Lambda_{4}&=&\varepsilon(x_{1}+x_{2},x_{3}+x_{5})\theta(x_{3},x_{5})D(x_{1},x_{2})-D(x_{1},x_{2})\theta(x_{3},x_{5})\\ &&+\theta([x_{1},x_{2},x_{3}],x_{5})+\varepsilon(x_{1}+x_{2},x_{3})\theta(x_{3},[x_{1},x_{2},x_{5}])=0;\\ \Lambda_{5}&=&D([x_{1},x_{2},x_{3}],x_{4})+\varepsilon(x_{1}+x_{2},x_{3})D(x_{3},[x_{1},x_{2},x_{4}])\\ &&-D(x_{1},x_{2})D(x_{3},x_{4})+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})D(x_{1},x_{2})=0. \end{eqnarray*} $

由此可得$ d^{3}(d^{1}f)(x_{1},x_{2},x_{3},x_{4},x_{5})=0 $, 即$ d^{3}d^{1}=0 $.

若$ d^{n}f=0 $, $ n=1, 2, 3,\cdot\cdot\cdot $, 则称$ f\in C^{n}(T,V) $为$ n $-余循环, 记$ Z^{n}(T,V) $是$ n $-余循环构成的子空间, $ B^{n}(T,V)=d^{n-2}C^{n-2}(T,V) $, 其中$ n\geq3 $. 由$ d^{n+2}d^{n}=0 $知, $ B^{n}(T,V) $是$ Z^{n}(T,V) $的子空间, 于是可定义李$ \mathrm{color} $三系$ {T} $的$ n $-阶上同调为$ H^{n}(T,V)=Z^{n}(T,V)/B^{n}(T,V) $.

设$ {T} $是李$ \mathrm{color} $三系, $ \mathbb{K}[[t]] $是以$ t $为变量的形式幂级数环. 假设$ T[[t]] $是$ {T} $上的一组形式级数.

定义3.1   设$ {T} $是李$ \mathrm{color} $三系, 则$ T $的单参数形式形变是一组幂级数$ f_{t}:T\times T\times T\rightarrow T[[t]] $为

$ f_{t}(x_{1},x_{2},x_{3})=\sum\limits_{i\geq0}F_{i}(x_{1},x_{2},x_{3})t^{i}=F_{0}(x_{1},x_{2},x_{3})+F_{1}(x_{1},x_{2},x_{3})t +F_{2}(x_{1},x_{2},x_{3})t^{2}+\cdot\cdot\cdot, $

其中每个$ F_{i} $是$ \mathbb{K} $-三线性映射, $ F_{i}:T\times T\times T\rightarrow T $, $ F_{0}(x_{1},x_{2},x_{3})=[x_{1},x_{2},x_{3}] $, 满足以下条件$ : $

$ \begin{equation} \begin{aligned} f_{t}(x_{1},x_{2},x_{3})=-\varepsilon(x_{1},x_{2})f_{t}(x_{2},x_{1},x_{3}), \end{aligned} \end{equation} $

(3.1)

$ \begin{equation} \begin{aligned} \varepsilon(x_{3},x_{1})f_{t}(x_{1},x_{2},x_{3})+\varepsilon(x_{1},x_{2})f_{t}(x_{2},x_{3},x_{1}) +\varepsilon(x_{2},x_{3})f_{t}(x_{3},x_{1},x_{2})=0, \end{aligned} \end{equation} $

(3.2)

$ \begin{equation} \begin{aligned} f_{t}(x_{1},x_{2},f_{t}(x_{3},x_{4},x_{5}))=&f_{t}(f_{t}(x_{1},x_{2},x_{3}),x_{4},x_{5})\\ &+\varepsilon(x_{1}+x_{2},x_{3})f_{t}(x_{3},f_{t}(x_{1},x_{2},x_{4}),x_{5})\\ &+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})f_{t}(x_{3},x_{4},f_{t}(x_{1},x_{2},x_{5})). \end{aligned} \end{equation} $

(3.3)

注   式$ (3.1) $—$ (3.3) $等价于$ (i,j\leq n, \; n=0,1,2,\cdot\cdot\cdot) $

$ \begin{equation} \begin{aligned} F_{i}(x_{1},x_{2},x_{3})=-\varepsilon(x_{1},x_{2})F_{i}(x_{2},x_{1},x_{3}), \end{aligned} \end{equation} $

(3.4)

$ \begin{equation} \begin{aligned} \varepsilon(x_{3},x_{1})F_{i}(x_{1},x_{2},x_{3})+\varepsilon(x_{1},x_{2})F_{i}(x_{2},x_{3},x_{1}) +\varepsilon(x_{2},x_{3})F_{i}(x_{3},x_{1},x_{2})=0, \end{aligned} \end{equation} $

(3.5)

$ \begin{equation} \begin{aligned} \sum\limits_{i+j=n}F_{i}(x_{1},x_{2},F_{j}(x_{3},x_{4},x_{5})) =&\sum\limits_{i+j=n}(F_{i}(F_{j}(x_{1},x_{2},x_{3}),x_{4},x_{5})\\ &+\varepsilon(x_{1}+x_{2},x_{3})F_{i}(x_{3},F_{j}(x_{1},x_{2},x_{4}),x_{5})\\ &+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})F_{i}(x_{3},x_{4},F_{j}(x_{1},x_{2},x_{5})). \end{aligned} \end{equation} $

(3.6)

进一步, 式$ (3.6) $等价于$ \sum_{i+j=n}F_{i}F_{j}=0 $, 其中

$ \begin{eqnarray*} &&F_{i}F_{j}(x_{1},x_{2},x_{3},x_{4},x_{5})\\ &=&-F_{i}(x_{1},x_{2},F_{j}(x_{3},x_{4},x_{5}))+\varepsilon(x_{1}+x_{2},x_{3})F_{i}(x_{3},F_{j}(x_{1},x_{2},x_{4}),x_{5})\\ &&+F_{i}(F_{j}(x_{1},x_{2},x_{3}),x_{4},x_{5})+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})F_{i}(x_{3},x_{4},F_{j}(x_{1},x_{2},x_{5})). \end{eqnarray*} $

当$ n=1 $, 式$ (3.6) $等价于$ F_{0}F_{1}+F_{1}F_{0}=0 $;

当$ n\geq2 $, 式$ (3.6) $等价于$ -(F_{0}F_{n}+F_{n}F_{0})=F_{1}F_{n-1}+F_{2}F_{n-2}+\cdot\cdot\cdot+F_{n-1}F_{1} $.

由$ F_{0}(x_{1},x_{2},x_{3})=[x_{1},x_{2},x_{3}] $, 则有

$ \begin{eqnarray*} &&F_{0}F_{1}(x_{1},x_{2},x_{3},x_{4},x_{5})\\ &=&-F_{0}(x_{1},x_{2},F_{1}(x_{3},x_{4},x_{5}))+\varepsilon(x_{1}+x_{2},x_{3})F_{0}(x_{3},F_{1}(x_{1},x_{2},x_{4}),x_{5})\\ &&+F_{0}(F_{1}(x_{1},x_{2},x_{3}),x_{4},x_{5})+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})F_{0}(x_{3},x_{4},F_{1}(x_{1},x_{2},x_{5}))\\ &=&-[x_{1},x_{2},F_{1}(x_{3},x_{4},x_{5})]+\varepsilon(x_{1}+x_{2},x_{3})[x_{3},F_{1}(x_{1},x_{2},x_{4}),x_{5}]\\ &&+[F_{1}(x_{1},x_{2},x_{3}),x_{4},x_{5}]+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})[x_{3},x_{4},F_{1}(x_{1},x_{2},x_{5})]\\ &=&-D(x_{1},x_{2})F_{1}(x_{3},x_{4},x_{5}) -\varepsilon(x_{1}+x_{2},x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})F_{1}(x_{1},x_{2},x_{4})\\ &&+\varepsilon(x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})F_{1}(x_{1},x_{2},x_{3})\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})F_{1}(x_{1},x_{2},x_{5}). \end{eqnarray*} $

同理, 有

$ \begin{eqnarray*} &&F_{1}F_{0}(x_{1},x_{2},x_{3},x_{4},x_{5})\\ &=&-F_{1}(x_{1},x_{2},F_{0}(x_{3},x_{4},x_{5}))+\varepsilon(x_{1}+x_{2},x_{3})F_{1}(x_{3},F_{0}(x_{1},x_{2},x_{4}),x_{5})\\ &&+F_{1}(F_{0}(x_{1},x_{2},x_{3}),x_{4},x_{5})+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})F_{1}(x_{3},x_{4},F_{0}(x_{1},x_{2},x_{5}))\\ &=&-F_{1}(x_{1},x_{2},[x_{3},x_{4},x_{5}])+\varepsilon(x_{1}+x_{2},x_{3})F_{1}(x_{3},[x_{1},x_{2},x_{4}],x_{5})\\ &&+F_{1}([x_{1},x_{2},x_{3}],x_{4},x_{5})+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})F_{1}(x_{3},x_{4},[x_{1},x_{2},x_{5}]). \end{eqnarray*} $

于是

$ (F_{0}F_{1}+F_{1}F_{0})(x_{1},x_{2},x_{3},x_{4},x_{5})=d^{3}F_{1}(x_{1},x_{2},x_{3},x_{4},x_{5}). $

由$ F_{0}F_{1}+F_{1}F_{0}=0 $可得$ d^{3}F_{1}=0 $, 也可得$ -d^{3}F_{n}=F_{1}F_{n-1}+F_{2}F_{n-2}+\cdot\cdot\cdot+F_{n-1}F_{1} $. 此时称$ F_{1} $为$ f_{t} $的无穷小形变.

定义3.2    设$ {T} $是李$ \mathrm{color} $三系, 若存在$ \mathbb{K}[[t]] $-模形式同构

$ \phi_{t}(x)=\sum\limits_{i\geq0}\phi_{i}(x)t^{i}:(T[[t]],f_{t})\rightarrow (T[[t]],f_{t}^{'}), $

其中$ \phi_{0}=\mathrm{id}_{T} $, $ \phi_{i}:T\rightarrow T $为$ \mathbb{K} $-线性映射, 满足

$ \phi_{t}f_{t}(x_{1},x_{2},x_{3})=f_{t}^{'}(\phi_{t}(x_{1}),\phi_{t}(x_{2}),\phi_{t}(x_{3})), \; \forall\; x_{1},x_{2},x_{3}\in T, $

则称$ {T} $的两个单参数形式形变$ f_{t} $和$ f_{t}^{'} $是等价的, 记为$ f_{t}\sim f_{t}^{'} $.

特别地, 如果$ F_{1}=F_{2}=\cdot\cdot\cdot=0 $, 则称$ f_{t}=F_{0} $为零形变. 如果$ f_{t}\sim F_{0} $, 则称$ f_{t} $为微小形变. 如果每一个单参数形式形变$ f_{t} $均为微小的, 则称$ {T} $为解析刚性李$ \mathrm{color} $三系.

定理3.3    设$ f_{t}=\sum_{i\geq0}F_{i}(x_{1},x_{2},x_{3})t^{i} $与$ f_{t}^{'}=\sum_{i\geq0}F_{i}^{'}(x_{1},x_{2},x_{3})t^{i} $为$ {T} $的两个等价的单参数形式形变. 则微小形变$ F_{1} $和$ F_{1}^{'} $属于同一个上同调$ H^{3}(T,T) $.

证  假设$ F_{1} $和$ F_{1}^{'} $等价, 则存在$ \mathbb{K}[[t]] $-模同构$ \phi_{t}(x)=\sum_{i\geq0}\phi_{i}(x)t^{i} $使得

$ \sum\limits_{i\geq0}\phi_{i}\left(\sum\limits_{j\geq0}F_{j}(x_{1},x_{2},x_{3})t^{j}\right)t^{i} =\sum\limits_{i\geq0}F_{i}^{'}\left(\sum\limits_{k\geq0}\phi_{k}(x_{1})t^{k},\sum\limits_{l\geq0}\phi_{l}(x_{2})t^{l}, \sum\limits_{m\geq0}\phi_{m}(x_{3})t^{m}\right)t^{i}, $

进一步, 有

$ \sum\limits_{i+j=n}\phi_{i}(F_{j}(x_{1},x_{2},x_{3}))t^{i+j} =\sum\limits_{i+k+l+m=n}F_{i}^{'}(\phi_{k}(x_{1}),\phi_{l}(x_{2}),\phi_{m}(x_{3}))t^{i+k+l+m}. $

特别地

$ \sum\limits_{i+j=1}\phi_{i}(F_{j}(x_{1},x_{2},x_{3}))t =\sum\limits_{i+k+l+m=1}F_{i}^{'}(\phi_{k}(x_{1}),\phi_{l}(x_{2}),\phi_{m}(x_{3}))t. $

比较上式两边的系数得

$ \begin{eqnarray*} &&F_{1}(x_{1},x_{2},x_{3})+\phi_{1}([x_{1},x_{2},x_{3}])\\ &=&F_{1}^{'}(x_{1},x_{2},x_{3})+[\phi_{1}(x_{1}),x_{2},x_{3}] +[x_{1},\phi_{1}(x_{2}),x_{3}]+[x_{1},x_{2},\phi_{1}(x_{3})], \end{eqnarray*} $

整理得

$ \begin{eqnarray*} &&F_{1}(x_{1},x_{2},x_{3})-F_{1}^{'}(x_{1},x_{2},x_{3})\\ &=&[\phi_{1}(x_{1}),x_{2},x_{3}]+[x_{1},\phi_{1}(x_{2}),x_{3}]+[x_{1},x_{2},\phi_{1}(x_{3})]-\phi_{1}([x_{1},x_{2},x_{3}])\\ &=&\varepsilon(x_{1},x_{2}+x_{3})\theta(x_{2},x_{3})\phi_{1}(x_{1})-\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})\phi_{1}(x_{2}) +D(x_{1},x_{2})\phi_{1}(x_{3})-\phi_{1}([x_{1},x_{2},x_{3}])\\ &=&d^{1}\phi_{1}(x_{1},x_{2},x_{3}). \end{eqnarray*} $

因此$ F_{1}-F_{1}^{'}=d^{1}\phi_{1}\in B^{3}(T,T) $.

定理3.4   设$ T $是李$ \mathrm{color} $三系, 若$ H^{3}(T,T)=0 $, 则$ T $是解析刚性的.

证  设$ f_{t} $是$ T $的单参数形变, 假设$ f_{t}=F_{0}+\sum_{i\geq n}F_{i}t^{i} $. 则$ -d^{3}F_{n}=F_{1}F_{n-1}+F_{2}F_{n-2}+\cdot\cdot\cdot+F_{n-1}F_{1}=0 $. 由$ H^{3}(T,T)=0 $知, $ F_{n}\in Z^{3}(T,T)=B^{3}(T,T) $, 即存在$ g_{n}\in C^{1}(T,T) $使得$ F_{n}=d^{1}g_{n} $.

令$ \phi_{t}=\mathrm{id}_{T}-g_{n}t^{n} $, 则

$ \begin{eqnarray*} &&\phi_{t}(\mathrm{id}_{T}+g_{n}t^{n}+g_{n}^{2}t^{2n}+g_{n}^{3}t^{3n}+\cdot\cdot\cdot)\\ &=&(\mathrm{id}_{T}-g_{n}t^{n})(\mathrm{id}_{T}+g_{n}t^{n}+g_{n}^{2}t^{2n}+g_{n}^{3}t^{3n}+\cdot\cdot\cdot)\\ &=&(\mathrm{id}_{T}+g_{n}t^{n}+g_{n}^{2}t^{2n}+g_{n}^{3}t^{3n}+\cdot\cdot\cdot) -(g_{n}t^{n}+g_{n}^{2}t^{2n}+g_{n}^{3}t^{3n}+\cdot\cdot\cdot)\\ &=&\mathrm{id}_{T}. \end{eqnarray*} $

同理可证$ (\mathrm{id}_{T}+g_{n}t^{n}+g_{n}^{2}t^{2n}+g_{n}^{3}t^{3n}+\cdot\cdot\cdot)\phi_{t}=\mathrm{id}_{T} $. 因此$ \phi_{t}:(T[[t]],f_{t})\rightarrow (T[[t]],f_{t}^{'}) $是线性同构. 考虑另一个单参数形变

$ f_{t}^{'}(x_{1},x_{2},x_{3})=\phi_{t}^{-1}f_{t}(\phi_{t}(x_{1}),\phi_{t}(x_{2}),\phi_{t}(x_{3})). $

显然, $ f_{t}\sim f_{t}^{'} $. 假设$ f_{t}^{'}=\sum_{i\geq 0}F_{i}^{'}t^{i} $. 则

$ \begin{eqnarray*} (\mathrm{id}_{T}-g_{n}t^{n})\left(\sum\limits_{i\geq 0}F_{i}^{'}(x_{1},x_{2},x_{3})t^{i}\right) =\left(F_{0}+\sum\limits_{i\geq n}F_{i}t^{i}\right)(x_{1}-g_{n}(x_{1})t^{n},x_{2}-g_{n}(x_{2})t^{n},x_{3}-g_{n}(x_{3})t^{n}), \end{eqnarray*} $

即

$ \begin{eqnarray*} &&\sum\limits_{i\geq 0}F_{i}^{'}(x_{1},x_{2},x_{3})t^{i}-\sum\limits_{i\geq 0}g_{n}F_{i}^{'}(x_{1},x_{2},x_{3})t^{i+n}\\ &=&F_{0}(x_{1},x_{2},x_{3})-\{F_{0}(g_{n}(x_{1}),x_{2},x_{3})+F_{0}(x_{1},g_{n}(x_{2}),x_{3}) +F_{0}(x_{1},x_{2},g_{n}(x_{3}))\}t^{n}\\ &&+\{F_{0}(g_{n}(x_{1}),g_{n}(x_{2}),x_{3})+F_{0}(x_{1},g_{n}(x_{2}),g_{n}(x_{3}))+F_{0}(g_{n}(x_{1}),x_{2},g_{n}(x_{3}))\}t^{2n}\\ &&+F_{0}(g_{n}(x_{1}),g_{n}(x_{2}),g_{n}(x_{3}))t^{3n}+\sum\limits_{i\geq n}F_{i}(x_{1},x_{2},x_{3})t^{i}\\ &&-\sum\limits_{i\geq n}\{F_{i}(g_{n}(x_{1}),x_{2},x_{3})+F_{i}(x_{1},g_{n}(x_{2}),x_{3}) +F_{i}(x_{1},x_{2},g_{n}(x_{3}))\}t^{i+n}\\ &&+\sum\limits_{i\geq n}\{F_{i}(g_{n}(x_{1}),g_{n}(x_{2}),x_{3})+F_{i}(x_{1},g_{n}(x_{2}),g_{n}(x_{3})) +F_{i}(g_{n}(x_{1}),x_{2},g_{n}(x_{3}))\}t^{i+2n}\\ &&-\sum\limits_{i\geq n}F_{i}(g_{n}(x_{1}),g_{n}(x_{2}),g_{n}(x_{3}))t^{i+3n}. \end{eqnarray*} $

于是

$ F_{0}^{'}(x_{1},x_{2},x_{3})=F_{0}(x_{1},x_{2},x_{3})=[x_{1},x_{2},x_{3}], $

$ F_{1}^{'}(x_{1},x_{2},x_{3})=F_{2}^{'}(x_{1},x_{2},x_{3})=\cdot\cdot\cdot=F_{n-1}^{'}(x_{1},x_{2},x_{3})=0, $

和

$ \begin{eqnarray*} &&F_{n}^{'}(x_{1},x_{2},x_{3})-g_{n}([x_{1},x_{2},x_{3}])\\ &=&F_{n}(x_{1},x_{2},x_{3})-[g_{n}(x_{1}),x_{2},x_{3}]-[x_{1},g_{n}(x_{2}),x_{3}]-[x_{1},x_{2},g_{n}(x_{3})]\\ &=&F_{n}(x_{1},x_{2},x_{3})-\varepsilon(x_{1},x_{2}+x_{3})\theta(x_{2},x_{3})g_{n}(x_{1}) +\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})g_{n}(x_{2}) -D(x_{1},x_{2})g_{n}(x_{3}). \end{eqnarray*} $

于是

$ F_{n}^{'}(x_{1},x_{2},x_{3})=F_{n}(x_{1},x_{2},x_{3})-d^{1}g_{n}(x_{1},x_{2},x_{3})=0. $

可得$ f_{t}^{'}=F_{0}+\Sigma_{i\geq n+1}F_{i}^{'}t^{i} $. 由归纳法, 有$ f_{t}\sim F_{0} $, 故$ T $是解析刚性的.

设$ T $是李$ \mathrm{color} $三系, $ \psi:T\times T\times T\rightarrow T $是三线性映射. 考虑线性算子的$ \lambda $-参数簇:

$ \begin{equation} \begin{aligned} [x_{1},x_{2},x_{3}]_{\lambda}=[x_{1},x_{2},x_{3}]+\lambda\psi(x_{1},x_{2},x_{3}), \end{aligned} \end{equation} $

(4.1)

其中$ x_{1},x_{2},x_{3}\in T $, $ \lambda $为变量.

若$ T $对式(4.1) 定义的运算$ [\cdot,\cdot,\cdot]_{\lambda} $构成李$ \mathrm{color} $三系, 记为$ T_{\lambda} $, 则称$ \psi $生成$ \lambda $-参数李$ \mathrm{color} $三系$ T $的无穷小形变.

定理4.1    设$ T $是李$ \mathrm{color} $三系, 则$ T_{\lambda} $是李$ \mathrm{color} $三系当且仅当

$ (1) $ $ \psi $在$ T $上定义李$ \mathrm{color} $三系$ ; $

$ (2) $ $ \psi $是$ T $的$ 3 $-余循环.

证  $ (\Rightarrow ) $若$ T_{\lambda} $是李$ \mathrm{color} $三系, 则$ \forall\; x_{1}, x_{2}, x_{3}\in T $, 有

$ \begin{align*} [x_{1}, x_{2}, x_{3}]_{\lambda}&=[x_{1}, x_{2}, x_{3}]+\lambda\psi(x_{1}, x_{2}, x_{3})=-\varepsilon(x_{1}, x_{2})[x_{2}, x_{1}, x_{3}]+\lambda\psi(x_{1}, x_{2}, x_{3}),\\ [x_{1},x_{2},x_{3}]_{\lambda}&=-\varepsilon(x_{1}, x_{2})[x_{2}, x_{1}, x_{3}]_{\lambda}=-\varepsilon(x_{1}, x_{2})[x_{2}, x_{1}, x_{3}]-\varepsilon(x_{1}, x_{2})\lambda\psi(x_{2}, x_{1}, x_{3}). \end{align*} $

于是

$ \begin{equation} \begin{aligned} \psi(x_{1}, x_{2}, x_{3})=-\varepsilon(x_{1}, x_{2})\psi(x_{2}, x_{1}, x_{3}). \end{aligned} \end{equation} $

(4.2)

由于

$ \begin{eqnarray*} 0&=&\varepsilon(x_{3},x_{1})[x_{1}, x_{2}, x_{3}]_{\lambda}+\varepsilon(x_{1},x_{2})[x_{2}, x_{3}, x_{1}]_{\lambda}+\varepsilon(x_{2},x_{3})[x_{3}, x_{1}, x_{2}]_{\lambda}\\ &=&\varepsilon(x_{3},x_{1})[x_{1}, x_{2}, x_{3}]+\varepsilon(x_{3},x_{1})\lambda\psi(x_{1}, x_{2}, x_{3})+\varepsilon(x_{1},x_{2})[x_{2}, x_{3}, x_{1}]+\varepsilon(x_{1},x_{2})\lambda\psi(x_{2}, x_{3}, x_{1})\\ &&+\varepsilon(x_{2},x_{3})[x_{3}, x_{1}, x_{2}]+\varepsilon(x_{2},x_{3})\lambda\psi(x_{3}, x_{1}, x_{2})\\ &=&\varepsilon(x_{3},x_{1})[x_{1}, x_{2}, x_{3}]+\varepsilon(x_{1},x_{2})[x_{2}, x_{3}, x_{1}]+\varepsilon(x_{2},x_{3})[x_{3}, x_{1}, x_{2}]\\ &&+\lambda(\varepsilon(x_{3},x_{1})\psi(x_{1}, x_{2}, x_{3})+\varepsilon(x_{1},x_{2})\psi(x_{2}, x_{3}, x_{1})+\varepsilon(x_{2},x_{3})\psi(x_{3}, x_{1}, x_{2})), \end{eqnarray*} $

可得

$ \begin{equation} \begin{aligned} \varepsilon(x_{3},x_{1})\psi(x_{1}, x_{2}, x_{3})+\varepsilon(x_{1},x_{2})\psi(x_{2}, x_{3}, x_{1})+\varepsilon(x_{2},x_{3})\psi(x_{3}, x_{1}, x_{2})=0. \end{aligned} \end{equation} $

(4.3)

令$ x_{1}, x_{2}, x_{3},x_{4},x_{5}\in T $, 考虑下面等式

$ \begin{equation} \begin{aligned} [x_{1},x_{2},[x_{3},x_{4},x_{5}]_{\lambda}]_{\lambda}=&[[x_{1},x_{2},x_{3}]_{\lambda},x_{4},x_{5}]_{\lambda} +\varepsilon(x_{1}+x_{2},x_{3})[x_{3},[x_{1},x_{2},x_{4}]_{\lambda},x_{5}]_{\lambda}\\ &+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})[x_{3},x_{4},[x_{1},x_{2},x_{5}]_{\lambda}]_{\lambda}, \end{aligned} \end{equation} $

(4.4)

式(4.4) 左端等价于

$ \begin{eqnarray*} &&[x_{1},x_{2},[x_{3},x_{4},x_{5}]+\lambda\psi(x_{3},x_{4},x_{5})]_{\lambda}\\ &=&[x_{1},x_{2},[x_{3},x_{4},x_{5}]+\lambda\psi(x_{3},x_{4},x_{5})] +\lambda\psi(x_{1},x_{2},[x_{3},x_{4},x_{5}]+\lambda\psi(x_{3},x_{4},x_{5}))\\ &=&[x_{1},x_{2},[x_{3},x_{4},x_{5}]]+\lambda([x_{1},x_{2},\psi(x_{3},x_{4},x_{5})]+\psi(x_{1},x_{2},[x_{3},x_{4},x_{5}]))\\ &&+\lambda^{2}\psi(x_{1},x_{2},\psi(x_{3},x_{4},x_{5})), \end{eqnarray*} $

式(4.4) 右端等价于

$ \begin{eqnarray*} &&[[x_{1},x_{2},x_{3}]+\lambda\psi(x_{1},x_{2},x_{3}),x_{4},x_{5}]_{\lambda} +\varepsilon(x_{1}+x_{2},x_{3})[x_{3},[x_{1},x_{2},x_{4}]+\lambda\psi(x_{1},x_{2},x_{4}),x_{5}]_{\lambda}\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})[x_{3},x_{4},[x_{1},x_{2},x_{5}]+\lambda\psi(x_{1},x_{2},x_{5})]_{\lambda}\\ &=&[[x_{1},x_{2},x_{3}],x_{4},x_{5}]+\varepsilon(x_{1}+x_{2},x_{3})[x_{3},[x_{1},x_{2},x_{4}],x_{5}]\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})[x_{3},x_{4},[x_{1},x_{2},x_{5}]]\\ &&+\lambda\{[\psi(x_{1},x_{2},x_{3}),x_{4},x_{5}]+\psi([x_{1},x_{2},x_{3}],x_{4},x_{5}) +\varepsilon(x_{1}+x_{2},x_{3})[x_{3},\psi(x_{1},x_{2},x_{4}),x_{5}]\\ &&+\varepsilon(x_{1}+x_{2},x_{3})\psi(x_{3},[x_{1},x_{2},x_{4}],x_{5}) +\varepsilon(x_{1}+x_{2},x_{3}+x_{4})[x_{3},x_{4},\psi(x_{1},x_{2},x_{5})]\\ &&+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})\psi(x_{3},x_{4},[x_{1},x_{2},x_{5}])\} +\lambda^{2}\{\psi(\psi(x_{1},x_{2},x_{3}),x_{4},x_{5})\\ &&+\varepsilon(x_{1}+x_{2},x_{3})\psi(x_{3},\psi(x_{1},x_{2},x_{4}),x_{5}) +\varepsilon(x_{1}+x_{2},x_{3}+x_{4})\psi(x_{3},x_{4},\psi(x_{1},x_{2},x_{5}))\}. \end{eqnarray*} $

于是, 有

$ \begin{equation} \begin{aligned} &[x_{1},x_{2},\psi(x_{3},x_{4},x_{5})]+\psi(x_{1},x_{2},[x_{3},x_{4},x_{5}])\\ =&[\psi(x_{1},x_{2},x_{3}),x_{4},x_{5}]+\psi([x_{1},x_{2},x_{3}],x_{4},x_{5}) +\varepsilon(x_{1}+x_{2},x_{3})[x_{3},\psi(x_{1},x_{2},x_{4}),x_{5}]\\ &+\varepsilon(x_{1}+x_{2},x_{3})\psi(x_{3},[x_{1},x_{2},x_{4}],x_{5}) +\varepsilon(x_{1}+x_{2},x_{3}+x_{4})[x_{3},x_{4},\psi(x_{1},x_{2},x_{5})]\\ &+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})\psi(x_{3},x_{4},[x_{1},x_{2},x_{5}]), \end{aligned} \end{equation} $

(4.5)

和

$ \begin{equation} \begin{aligned} &\psi(x_{1},x_{2},\psi(x_{3},x_{4},x_{5}))\\ =&\psi(\psi(x_{1},x_{2},x_{3}),x_{4},x_{5})+\varepsilon(x_{1}+x_{2},x_{3})\psi(x_{3},\psi(x_{1},x_{2},x_{4}),x_{5})\\ &+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})\psi(x_{3},x_{4},\psi(x_{1},x_{2},x_{5})). \end{aligned} \end{equation} $

(4.6)

由式(4.2), 式(4.3) 和式(4.6) 知$ \psi $在$ T $上定义李$ \mathrm{color} $三系.

利用$ \theta(a,b)(x)=\varepsilon(a+b,x)[x,a,b] $和$ D(a,b)(x)=[a,b,x] $, 式(4.5) 可写为

$ \begin{equation} \nonumber \begin{aligned} 0=&-D(x_{1},x_{2})\psi(x_{3},x_{4},x_{5})-\psi(x_{1},x_{2},[x_{3},x_{4},x_{5}])\\ &+\varepsilon(x_{1}+x_{2}+x_{3},x_{4}+x_{5})\theta(x_{4},x_{5})\psi(x_{1},x_{2},x_{3})+\psi([x_{1},x_{2},x_{3}],x_{4},x_{5})\\ &-\varepsilon(x_{1}+x_{2},x_{3}+x_{5})\varepsilon(x_{4},x_{5})\theta(x_{3},x_{5})\psi(x_{1},x_{2},x_{4}) +\varepsilon(x_{1}+x_{2},x_{3})\psi(x_{3},[x_{1},x_{2},x_{4}],x_{5})\\ &+\varepsilon(x_{1}+x_{2},x_{3}+x_{4})D(x_{3},x_{4})\psi(x_{1},x_{2},x_{5}) +\varepsilon(x_{1}+x_{2},x_{3}+x_{4})\psi(x_{3},x_{4},[x_{1},x_{2},x_{5}])\\ =&d^{3}\psi(x_{1}, x_{2}, x_{3},x_{4},x_{5}). \end{aligned} \end{equation} $

因此$ d^{3}\psi=0 $.

($ \Leftarrow $) 若$ \psi $满足条件$ (1) $和$ (2) $, 易证$ T_{\lambda} $是李$ \mathrm{color} $三系.

定义4.2   设$ {T} $是李$ \mathrm{color} $三系, 若存在线性映射$ N:T\rightarrow T $, 使得$ \varphi_{\lambda}=\mathrm{id}+\lambda N:T_{\lambda}\rightarrow T $满足

$ \begin{equation} \begin{aligned} \varphi_{\lambda}[x_{1}, x_{2}, x_{3}]_{\lambda}=[\varphi_{\lambda}x_{1},\varphi_{\lambda}x_{2},\varphi_{\lambda}x_{3}], \end{aligned} \end{equation} $

(4.7)

则称形变$ N $是平凡的. 其中$ x_{1},x_{2},x_{3}\in T $, $ \lambda $为变量.

易知, 等式$ (4.7) $左端等价于

$ \begin{eqnarray*} [x_{1}, x_{2}, x_{3}]+\lambda\{\psi(x_{1}, x_{2}, x_{3})+N[x_{1}, x_{2}, x_{3}]\}+\lambda^{2}N\psi(x_{1}, x_{2}, x_{3}), \end{eqnarray*} $

等式$ (4.7) $右端等价于

$ \begin{eqnarray*} &&[x_{1}, x_{2}, x_{3}]+\lambda\{[Nx_{1},x_{2},x_{3}]+[x_{1},Nx_{2},x_{3}]+[x_{1},x_{2},Nx_{3}]\}\\ &&+\lambda^{2}\{[Nx_{1},Nx_{2},x_{3}]+[x_{1},Nx_{2},Nx_{3}]+[Nx_{1},x_{2},Nx_{3}]\}+\lambda^{3}[Nx_{1},Nx_{2},Nx_{3}], \end{eqnarray*} $

于是

$ \begin{equation} \begin{aligned} &\psi(x_{1}, x_{2}, x_{3})\\=&[Nx_{1},x_{2},x_{3}]+[x_{1},Nx_{2},x_{3}]+[x_{1},x_{2},Nx_{3}]-N[x_{1}, x_{2}, x_{3}]\\ =&\varepsilon(x_{1}, x_{2}+x_{3})\theta(x_{2},x_{3})Nx_{1}-\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})Nx_{2} +D(x_{1}, x_{2})Nx_{3}-N[x_{1}, x_{2}, x_{3}]\\ =&d^{1}N(x_{1}, x_{2}, x_{3}), \end{aligned} \end{equation} $

(4.8)

$ \begin{equation} \begin{aligned} N\psi(x_{1}, x_{2}, x_{3})=[Nx_{1},Nx_{2},x_{3}]+[x_{1},Nx_{2},Nx_{3}]+[Nx_{1},x_{2},Nx_{3}], \end{aligned} \end{equation} $

(4.9)

$ \begin{equation} \begin{aligned} \left[ {Nx_{1},Nx_{2},Nx_{3}} \right]=0. \end{aligned} \end{equation} $

(4.10)

由式$ (4.8) $和式$ (4.9) $, 可得

$ \begin{equation} \begin{aligned} N^{2}[x_{1}, x_{2}, x_{3}]=&N[Nx_{1},x_{2},x_{3}]+N[x_{1},Nx_{2},x_{3}]+N[x_{1},x_{2},Nx_{3}]\\ &-[Nx_{1},Nx_{2},x_{3}]-[x_{1},Nx_{2},Nx_{3}]-[Nx_{1},x_{2},Nx_{3}]. \end{aligned} \end{equation} $

(4.11)

记

$ \begin{equation} \begin{aligned} \psi(x_{1}, x_{2}, x_{3})=[x_{1}, x_{2}, x_{3}]_{N}, \end{aligned} \end{equation} $

(4.12)

则式$ (4.9) $等价于

$ \begin{equation} \begin{aligned} N[x_{1}, x_{2}, x_{3}]_{N}=[Nx_{1},Nx_{2},x_{3}]+[x_{1},Nx_{2},Nx_{3}]+[Nx_{1},x_{2},Nx_{3}]. \end{aligned} \end{equation} $

(4.13)

定义4.3   若式$ (4.10) $和式$ (4.11) $成立, 则称线性算子$ N:T\rightarrow T $为$ \mathrm{Nijenhuis} $算子.

定理4.4    设$ N $是$ T $的$ \mathrm{Nijenhuis} $算子, 则$ T $的形变可由以下定义得到

$ \psi(x_{1}, x_{2}, x_{3})=\varepsilon(x_{1}, x_{2}+x_{3})\theta(x_{2},x_{3})Nx_{1}-\varepsilon(x_{2},x_{3})\theta(x_{1},x_{3})Nx_{2} +D(x_{1}, x_{2})Nx_{3}-N[x_{1}, x_{2}, x_{3}]. $

并且, 该形变是平凡的.

证  显然$ \psi=dN $且$ d\psi=d^{2}N=0 $, 因此$ \psi $是$ T $的$ 3 $-余循环. 下面证明式$ (2.3) $成立. 考虑式$ (4.8) $, 式$ (4.12) $和式$ (4.13) $, 推得

$ \begin{eqnarray*} &&\psi(x_{1},x_{2},\psi(y_{1},y_{2},y_{3}))\\ &=&[x_{1},x_{2},[Ny_{1},y_{2},y_{3}]+[y_{1},Ny_{2},y_{3}]+[y_{1},y_{2},Ny_{3}]-N[y_{1},y_{2},y_{3}]]_{N}\\ &=&[x_{1},x_{2},[Ny_{1},y_{2},y_{3}]]_{N}+[x_{1},x_{2},[y_{1},Ny_{2},y_{3}]]_{N}+[x_{1},x_{2},[y_{1},y_{2},Ny_{3}]]_{N}\\ &&-[x_{1},x_{2},N[y_{1},y_{2},y_{3}]]_{N}\\ &=&[Nx_{1},x_{2},[Ny_{1},y_{2},y_{3}]]+[x_{1},Nx_{2},[Ny_{1},y_{2},y_{3}]]+[x_{1},x_{2},N[Ny_{1},y_{2},y_{3}]]\\ &&-N[x_{1},x_{2},[Ny_{1},y_{2},y_{3}]]+[Nx_{1},x_{2},[y_{1},Ny_{2},y_{3}]]+[x_{1},Nx_{2},[y_{1},Ny_{2},y_{3}]]\\ &&+[x_{1},x_{2},N[y_{1},Ny_{2},y_{3}]]-N[x_{1},x_{2},[y_{1},Ny_{2},y_{3}]]+[Nx_{1},x_{2},[y_{1},y_{2},Ny_{3}]]\\ &&+[x_{1},Nx_{2},[y_{1},y_{2},Ny_{3}]]+[x_{1},x_{2},N[y_{1},y_{2},Ny_{3}]]-N[x_{1},x_{2},[y_{1},y_{2},Ny_{3}]]\\ &&-[Nx_{1},x_{2},N[y_{1},y_{2},y_{3}]]-[x_{1},Nx_{2},N[y_{1},y_{2},y_{3}]]-[x_{1},x_{2},N^{2}[y_{1},y_{2},y_{3}]]\\ &&+N[x_{1},x_{2},N[y_{1},y_{2},y_{3}]]\\ &=&[Nx_{1},x_{2},[Ny_{1},y_{2},y_{3}]]+[x_{1},Nx_{2},[Ny_{1},y_{2},y_{3}]]-N[x_{1},x_{2},[Ny_{1},y_{2},y_{3}]]\\ &&+[Nx_{1},x_{2},[y_{1},Ny_{2},y_{3}]]+[x_{1},Nx_{2},[y_{1},Ny_{2},y_{3}]]-N[x_{1},x_{2},[y_{1},Ny_{2},y_{3}]]\\ &&+[Nx_{1},x_{2},[y_{1},y_{2},Ny_{3}]]+[x_{1},Nx_{2},[y_{1},y_{2},Ny_{3}]]-N[x_{1},x_{2},[y_{1},y_{2},Ny_{3}]]\\ &&-[Nx_{1},x_{2},N[y_{1},y_{2},y_{3}]]-[x_{1},Nx_{2},N[y_{1},y_{2},y_{3}]]+N[x_{1},x_{2},N[y_{1},y_{2},y_{3}]]\\ &&+[x_{1},x_{2},[Ny_{1},Ny_{2},y_{3}]]+[x_{1},x_{2},[Ny_{1},y_{2},Ny_{3}]]+[x_{1},x_{2},[y_{1},Ny_{2},Ny_{3}]], \end{eqnarray*} $

同理可得

$ \begin{eqnarray*} &&\psi(\psi(x_{1},x_{2},y_{1}),y_{2},y_{3})\\ &=&[[Nx_{1},x_{2},y_{1}],Ny_{2},y_{3}]+[[Nx_{1},x_{2},y_{1}],y_{2},Ny_{3}]-N[[Nx_{1},x_{2},y_{1}],y_{2},y_{3}]\\ &&+[[x_{1},Nx_{2},y_{1}],Ny_{2},y_{3}]+[[x_{1},Nx_{2},y_{1}],y_{2},Ny_{3}]-N[[x_{1},Nx_{2},y_{1}],y_{2},y_{3}]\\ &&+[[x_{1},x_{2},Ny_{1}],Ny_{2},y_{3}]+[[x_{1},x_{2},Ny_{1}],y_{2},Ny_{3}]-N[[x_{1},x_{2},Ny_{1}],y_{2},y_{3}]\\ &&-[N[x_{1},x_{2},y_{1}],Ny_{2},y_{3}]-[N[x_{1},x_{2},y_{1}],y_{2},Ny_{3}]+N[N[x_{1},x_{2},y_{1}],y_{2},y_{3}]\\ &&+[[Nx_{1},Nx_{2},y_{1}],y_{2},y_{3}]+[[Nx_{1},x_{2},Ny_{1}],y_{2},y_{3}]+[[x_{1},Nx_{2},Ny_{1}],y_{2},y_{3}], \end{eqnarray*} $

和

$ \begin{eqnarray*} &&\psi(y_{1},\psi(x_{1},x_{2},y_{2}),y_{3})\\ &=&[Ny_{1},[Nx_{1},x_{2},y_{2}],y_{3}]+[y_{1},[Nx_{1},x_{2},y_{2}],Ny_{3}]-N[y_{1},[Nx_{1},x_{2},y_{2}],y_{3}]\\ &&+[Ny_{1},[x_{1},Nx_{2},y_{2}],y_{3}]+[y_{1},[x_{1},Nx_{2},y_{2}],Ny_{3}]-N[y_{1},[x_{1},Nx_{2},y_{2}],y_{3}]\\ &&+[Ny_{1},[x_{1},x_{2},Ny_{2}],y_{3}]+[y_{1},[x_{1},x_{2},Ny_{2}],Ny_{3}]-N[y_{1},[x_{1},x_{2},Ny_{2}],y_{3}]\\ &&-[Ny_{1},N[x_{1},x_{2},y_{2}],y_{3}]-[y_{1},N[x_{1},x_{2},y_{2}],Ny_{3}]+N[y_{1},N[x_{1},x_{2},y_{2}],y_{3}]\\ &&+[y_{1},[Nx_{1},Nx_{2},y_{2}],y_{3}]+[y_{1},[Nx_{1},x_{2},Ny_{2}],y_{3}]+[y_{1},[x_{1},Nx_{2},Ny_{2}],y_{3}], \end{eqnarray*} $

和

$ \begin{eqnarray*} &&\psi(y_{1},y_{2},\psi(x_{1},x_{2},y_{3}))\\ &=&[Ny_{1},y_{2},[Nx_{1},x_{2},y_{3}]]+[y_{1},Ny_{2},[Nx_{1},x_{2},y_{3}]]-N[y_{1},y_{2},[Nx_{1},x_{2},y_{3}]]\\ &&+[Ny_{1},y_{2},[x_{1},Nx_{2},y_{3}]]+[y_{1},Ny_{2},[x_{1},Nx_{2},y_{3}]]-N[y_{1},y_{2},[x_{1},Nx_{2},y_{3}]]\\ &&+[Ny_{1},y_{2},[x_{1},x_{2},Ny_{3}]]+[y_{1},Ny_{2},[x_{1},x_{2},Ny_{3}]]-N[y_{1},y_{2},[x_{1},x_{2},Ny_{3}]]\\ &&-[Ny_{1},y_{2},N[x_{1},x_{2},y_{3}]]-[y_{1},Ny_{2},N[x_{1},x_{2},y_{3}]]+N[y_{1},y_{2},N[x_{1},x_{2},y_{3}]]\\ &&+[y_{1},y_{2},[Nx_{1},Nx_{2},y_{3}]]+[y_{1},y_{2},[Nx_{1},x_{2},Ny_{3}]]+[y_{1},y_{2},[x_{1},Nx_{2},Ny_{3}]], \end{eqnarray*} $

易知$ N\in C^{1}(T) $, 由式$ (2.3) $, 式$ (4.13) $和定理$ 4.1 $可得

$ \begin{eqnarray*} &&\psi(x_{1},x_{2},\psi(y_{1},y_{2},y_{3}))-\psi(\psi(x_{1},x_{2},y_{1}),y_{2},y_{3})\\ &&-\varepsilon(x_{1}+x_{2},y_{1})\psi(y_{1},\psi(x_{1},x_{2},y_{2}),y_{3}) -\varepsilon(x_{1}+x_{2},y_{1}+y_{2})\psi(y_{1},y_{2},\psi(x_{1},x_{2},y_{3}))\\ &=&0. \end{eqnarray*} $