李余代数

定义

E k 上一个向量空间，上有一个线性映射 ${\displaystyle d\colon E\to E\wedge E}$EE 与自身的外积。可将 d 惟一扩张成 E外代数上一个度数为 1 的分次导子[1]

${\displaystyle d\colon \bigwedge ^{\bullet }E\rightarrow \bigwedge ^{\bullet +1}E.}$

${\displaystyle E\ \rightarrow ^{\!\!\!\!\!\!d}\ E\wedge E\ \rightarrow ^{\!\!\!\!\!\!d}\ \bigwedge ^{3}E\rightarrow ^{\!\!\!\!\!\!d}\ \dots .}$

对偶的李代数

${\displaystyle [\cdot ,\cdot ]^{*}\colon {\mathfrak {g}}^{*}\to ({\mathfrak {g}}\wedge {\mathfrak {g}})^{*}\cong {\mathfrak {g}}^{*}\wedge {\mathfrak {g}}^{*}}$

α([x, y]) = dα(xy)，对所有 α ∈ Ex,yE*

${\displaystyle d^{2}\alpha (x\wedge y\wedge z)={\frac {1}{3}}d^{2}\alpha (x\wedge y\wedge z+y\wedge z\wedge x+z\wedge x\wedge y)={\frac {1}{3}}\left(d\alpha ([x,y]\wedge z)+d\alpha ([y,z]\wedge x)+d\alpha ([z,x]\wedge y)\right),}$

${\displaystyle d^{2}\alpha (x\wedge y\wedge z)={\frac {1}{3}}\left(\alpha ([[x,y],z])+\alpha ([[y,z],x])+\alpha ([[z,x],y])\right).}$

d2 = 0，从而

${\displaystyle \alpha ([[x,y],z]+[[y,z],x]+[[z,x],y])=0}$ 对任意 α, x, y, 与 z

注释

1. ^ 这意味着，对任何齐次元素 a, bE${\displaystyle d(a\wedge b)=(da)\wedge b+(-1)^{\operatorname {deg} a}a\wedge (db)}$