# 弗勒利歇尔－奈恩黑斯括号

## 定义

${\displaystyle \Omega ^{*}(M)=\bigoplus _{k=0}^{\infty }\Omega ^{k}(M).}$

${\displaystyle D:\Omega ^{*}(M)\to \Omega ^{*+l}(M)}$

${\displaystyle D(\alpha \wedge \beta )=D(\alpha )\wedge \beta +(-1)^{\ell \deg(\alpha )}\alpha \wedge D(\beta ).}$

${\displaystyle \mathrm {Der} \,\Omega ^{*}(M)=\bigoplus _{k=-\infty }^{\infty }\mathrm {Der} _{k}\,\Omega ^{*}(M).}$

${\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-(-1)^{d_{1}d_{2}}D_{2}\circ D_{1}.}$

${\displaystyle i_{K}\,\omega (X_{1},\dots ,X_{k+\ell -1})={\frac {1}{k!(\ell -1)!}}\sum _{\sigma \in {S}_{k+\ell -1}}{\textrm {sign}}\,\sigma \cdot \omega (K(X_{\sigma (1)},\dots ,X_{\sigma (k)}),X_{\sigma (k+1)},\dots ,X_{\sigma (k+\ell -1)})}$

${\displaystyle {\mathcal {L}}_{K}=[d,i_{K}]=d\,{\circ }\,i_{K}-(-1)^{k-1}i_{K}{\circ }\,d,}$

${\displaystyle [\cdot ,\cdot ]:\Omega ^{k}(M,\mathrm {T} M)\times \Omega ^{\ell }(M,\mathrm {T} M)\to \Omega ^{k+\ell }(M,\mathrm {T} M):(K,L)\mapsto [K,L]}$

${\displaystyle {\mathcal {L}}_{[K,L]}=[{\mathcal {L}}_{K},{\mathcal {L}}_{L}].}$

${\displaystyle {\mathcal {L}}_{K}=[d,i_{K}]=d\,{\circ }\,i_{K}+i_{K}\,{\circ }\,d.}$

${\displaystyle \phi \otimes X}$${\displaystyle \psi \otimes Y}$（这里 φ 与 ψ 是形式，XY 是向量场）的弗勒利歇尔－奈恩黑斯括号的明确表达式为

${\displaystyle \left.\right.[\phi \otimes X,\psi \otimes Y]=\phi \wedge \psi \otimes [X,Y]+\phi \wedge {\mathcal {L}}_{X}\psi \otimes Y-{\mathcal {L}}_{Y}\phi \wedge \psi \otimes X+(-1)^{\deg(\phi )}(d\phi \wedge i_{X}(\psi )\otimes Y+i_{Y}(\phi )\wedge d\psi \otimes X).}$

## 形式环的导子

Ω*(M) 上任何导子，存在惟一元素 KL 属于 Ω*(M, TM) 使得

${\displaystyle i_{L}+{\mathcal {L}}_{K}.\,}$

• 形为 ${\displaystyle {\mathcal {L}}_{K}}$ 的导子组成与所有 d 可交换的李超代数。其括号为：
${\displaystyle [{\mathcal {L}}_{K_{1}},{\mathcal {L}}_{K_{2}}]={\mathcal {L}}_{[K_{1},K_{2}]}}$

• 形为 ${\displaystyle i_{L}}$ 的导子组成在函数 Ω0(M) 上消没的李超代数。其括号为
${\displaystyle [i_{L_{1}},i_{L_{2}}]=i_{[L_{1},L_{2}]^{\land }}}$

• 不同类型的导子之括号为
${\displaystyle [{\mathcal {L}}_{K},i_{L}]=i_{[K,L]}-(-1)^{kl}{\mathcal {L}}_{i_{L}K}}$