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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202301.png" /> be a smooth manifold (cf. also [[Differentiable manifold|Differentiable manifold]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202302.png" />. One calls
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202303.png" /></td> </tr></table>
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the space of all vector-valued differential forms. The Frölicher–Nijenhuis bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202304.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202305.png" />-graded [[Lie bracket|Lie bracket]]:
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Let $M$ be a smooth manifold (cf. also [[Differentiable manifold|Differentiable manifold]]) and let $\Omega ^ { k } ( M ; T M ) = \Gamma ( \wedge ^ { k } T ^ { * } M \otimes T M )$. One calls
  
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\begin{equation*} \Omega ( M , T M ) = \oplus _ { k = 0 } ^ { \operatorname { dim } M } \Omega ^ { k } ( M , T M ) \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202307.png" /></td> </tr></table>
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the space of all vector-valued differential forms. The Frölicher–Nijenhuis bracket $[ \cdot , \cdot ] : \Omega ^ { k } ( M ; T M ) \times \Omega ^ { l } ( M ; T M ) \rightarrow \Omega ^ { k + l } ( M ; T M )$ is a $\bf Z$-graded [[Lie bracket|Lie bracket]]:
  
It extends the Lie bracket of smooth vector fields, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202308.png" />. The identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f1202309.png" /> generates the one-dimensional centre. It is called the Frölicher–Nijenhuis bracket since it appeared with its full properties for the first time in [[#References|[a1]]], after some indication in [[#References|[a8]]]. One formula for it is:
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\begin{equation*} [ K , L ] = - ( - 1 ) ^ { k l } [ L , K ], \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023010.png" /></td> </tr></table>
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\begin{equation*} [ K _ { 1 } , [ K _ { 2 } , K _ { 3 } ] ] = [ [ K _ { 1 } , K _ { 2 } ] , K _ { 3 } ] + ( - 1 ) ^ { k _ { 1 } k _ { 2 } } [ K _ { 2 } , [ K _ { 1 }, K _ { 3 }] ]. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023011.png" /></td> </tr></table>
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It extends the Lie bracket of smooth vector fields, since $\Omega ^ { 0 } ( M ; T M ) = \Gamma ( T M ) = \mathcal{X} ( M )$. The identity on $T M$ generates the one-dimensional centre. It is called the Frölicher–Nijenhuis bracket since it appeared with its full properties for the first time in [[#References|[a1]]], after some indication in [[#References|[a8]]]. One formula for it is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023012.png" /></td> </tr></table>
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\begin{equation*} [ \varphi \bigotimes x , \psi \bigotimes Y ] = \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023014.png" /> are vector fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023015.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023016.png" />-form, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023017.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023018.png" />-form. It is a bilinear differential operator of bi-degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023019.png" />.
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\begin{equation*} = \varphi \bigwedge \psi \bigotimes [ X , Y ] + \varphi \bigwedge {\cal L} _ { X } \psi \bigotimes Y - {\cal L} _ { Y } \varphi \bigwedge \psi \bigotimes X + \end{equation*}
  
The Frölicher–Nijenhuis bracket is natural in the same way as the Lie bracket for vector fields: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023020.png" /> is smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023021.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023022.png" />-related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023024.png" /> is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023025.png" />-related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023027.png" />.
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\begin{equation*} + ( - 1 ) ^ { k } \left( d \varphi \bigwedge i _ { X } \psi \bigotimes Y + i _{Y} \varphi \bigwedge d \psi \bigotimes X \right), \end{equation*}
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where $X$ and $Y$ are vector fields, $\varphi$ is a $k$-form, and $\psi$ is an $l$-form. It is a bilinear differential operator of bi-degree $( 1,1 )$.
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The Frölicher–Nijenhuis bracket is natural in the same way as the Lie bracket for vector fields: if $f : M \rightarrow N$ is smooth and $K _ { i } \in \Omega ^ { k _ { i } } ( M ; T M )$ are $f$-related to $L _ { i } \in \Omega ^ { l } ( N ; T N )$, then $[ K _ { 1 } , K _ { 2 } ]$ is also $f$-related to $L_1$, $L_{2}$.
  
 
==Details.==
 
==Details.==
 
A convenient source is [[#References|[a3]]], Sect. 8. The basic formulas of the calculus of differential forms extend naturally to include the Frölicher–Nijenhuis bracket: Let
 
A convenient source is [[#References|[a3]]], Sect. 8. The basic formulas of the calculus of differential forms extend naturally to include the Frölicher–Nijenhuis bracket: Let
  
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\begin{equation*} \Omega ( M ) = \bigoplus _ { k \geq 0 } \Omega ^ { k } ( M ) = \bigoplus _ { k = 0 } ^ { \operatorname { dim } M } \Gamma \left( \bigwedge^k T ^ { * } M \right) \end{equation*}
  
be the algebra of differential forms. One denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023029.png" /> the space of all (graded) derivations of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023030.png" />, i.e. all bounded linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023031.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023034.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023035.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023036.png" />-graded Lie algebra with the graded commutator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023037.png" /> as bracket.
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be the algebra of differential forms. One denotes by $\operatorname { Der } _ { k } \Omega ( M )$ the space of all (graded) derivations of degree $k$, i.e. all bounded linear mappings $D : \Omega ( M ) \rightarrow \Omega ( M )$ with $D ( \Omega ^ { l } ( M ) ) \subset \Omega ^ { k + l } ( M )$ and $D ( \varphi \wedge \psi ) = D ( \varphi ) \wedge \psi + ( - 1 ) ^ { k l } \varphi \wedge D ( \psi )$ for $\varphi \in \Omega ^ { l } ( M )$. The space $\operatorname { Der }\Omega ( M ) = \oplus _ { k } \operatorname { Der } _ { k } \Omega ( M )$ is a $\bf Z$-graded Lie algebra with the graded commutator $[ D _ { 1 } , D _ { 2 } ] = D _ { 1 } D _ { 2 } - ( - 1 ) ^ { k _ { 1 } k _ { 2 } } D _ { 2 } D _ { 1 }$ as bracket.
  
A derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023038.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023039.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023040.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023041.png" />, thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023042.png" /> is of tensorial character and induces a derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023043.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023044.png" />. It is uniquely determined by its restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023045.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023046.png" />, which can be viewed as an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023047.png" /> depending smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023048.png" />; this is expressed by writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023050.png" />, and one has
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A derivation $D \in \operatorname { Der } _ { k } \Omega ( M )$ with $D | _ { \Omega ^ { 0 } } ( M ) = 0$ satisfies $D ( f . \omega ) = f . D ( \omega )$ for $f \in C ^ { \infty } ( M , \mathbf{R} )$, thus $D$ is of tensorial character and induces a derivation $D _ { X } \in \operatorname { Der } _ { k } \wedge T _ { X } ^ { * } M$ for each $x \in M$. It is uniquely determined by its restriction to $1$-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023046.png"/>, which can be viewed as an element $K _ { x } \in \wedge ^ { k + 1 } T _ { x } ^ { * } M \otimes T _ { x } M$ depending smoothly on $x \in M$; this is expressed by writing $D = i _ { K }$, where $K \in C ^ { \infty } ( \wedge ^ { k + 1 } T ^ { * } M \otimes T M ) = \Omega ^ { k + 1 } ( M ; T M )$, and one has
  
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\begin{equation*} ( i _ { K } \omega ) ( X _ { 1 } , \dots , X _ { k + 1 } ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023052.png" /></td> </tr></table>
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\begin{equation*} =\frac { 1 } { ( k + 1 ) ! ( l - 1 ) ! } \times \times \sum _ { \sigma \in S _ { k + \text{l} } } \operatorname { sign } \sigma . \omega ( K ( X _ { \sigma 1 } , \ldots , X _ { \sigma ( k + 1 ) } ) , X _ { \sigma ( k + 2 ) } , \ldots ) \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023054.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023055.png" />).
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for $\omega \in \Omega ^ { 1 } ( M )$ and $X_i \in \mathcal{X} ( M )$ (or $T _ { x } M$).
  
By putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023056.png" />, one obtains a bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023058.png" /> which defines a graded Lie algebra structure with the grading as indicated, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023060.png" /> one has
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By putting $i ( [ K , L ] ^ { \wedge } ) = [ i _ { K } , i _ { L } ]$, one obtains a bracket $[ ., . ] ^ { \wedge }$ on $\Omega ^ { * + 1 } ( M , T M )$ which defines a graded Lie algebra structure with the grading as indicated, and for $K \in \Omega ^ { k + 1 } ( M , T M )$, $L \in \Omega ^ { \text{l} + 1 } ( M , T M )$ one has
  
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\begin{equation*} [ K , L ] \bigwedge = i _ { K } L - ( - 1 ) ^ { k \text{l}} i _ { L } K, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023062.png" />. The bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023063.png" /> is called the the Nijenhuis–Richardson bracket, see [[#References|[a6]]] and [[#References|[a7]]]. If viewed on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023064.png" />, it recognizes Lie algebra structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023065.png" />: A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023066.png" /> is a [[Lie bracket|Lie bracket]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023067.png" />. This can be used to study deformations of Lie algebra structures: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023068.png" /> is again a Lie bracket on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023069.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023070.png" />; this can be written in the form of a Maurer–Cartan equation (cf. also [[Maurer–Cartan form|Maurer–Cartan form]]) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023071.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023072.png" /> is the coboundary operator for the Chevalley cohomology (cf. also [[Cohomology|Cohomology]]) of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023073.png" /> with values in the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023074.png" />. See [[#References|[a4]]] for a multi-graded elaboration of this.
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where $i _ { K } ( \omega \otimes X ) = i _ { K } ( \omega ) \otimes X$. The bracket $[ ., . ] ^ { \wedge }$ is called the Nijenhuis–Richardson bracket, see [[#References|[a6]]] and [[#References|[a7]]]. If viewed on a vector space $V$, it recognizes Lie algebra structures on $V$: A mapping $P \in L ^ { 2 }_\text { skew } ( V ; V )$ is a [[Lie bracket|Lie bracket]] if and only if $[ P , P ] ^ { \wedge } = 0$. This can be used to study deformations of Lie algebra structures: $P + A$ is again a Lie bracket on $V$ if and only if $[ P + A , P + A ] ^ { \wedge } = 2 [ P , A ] ^ { \wedge } + [ A , A ] ^ { \wedge } = 0$; this can be written in the form of a Maurer–Cartan equation (cf. also [[Maurer–Cartan form|Maurer–Cartan form]]) as $\delta _ { P } ( A ) + [ A , A ] ^ { \wedge } / 2 = 0$, since $\delta _ { P } = [ P , . ] ^ { \wedge }$ is the coboundary operator for the Chevalley cohomology (cf. also [[Cohomology|Cohomology]]) of the Lie algebra $( V , P )$ with values in the adjoint representation $V$. See [[#References|[a4]]] for a multi-graded elaboration of this.
  
The exterior derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023075.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023076.png" />. In view of the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023077.png" /> for vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023078.png" />, one defines for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023079.png" /> the Lie derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023080.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023081.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023082.png" /> is injective. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023083.png" />.
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The exterior derivative $d$ is an element of $\operatorname { Der } _ { 1 } \Omega ( M )$. In view of the formula $\mathcal{L} _ { X } = [ i_{X} , d ] = i_{X} d + d i _{X}$ for vector fields $X$, one defines for $K \in \Omega ^ { k } ( M ; T M )$ the Lie derivation $\mathcal{L} _ { K } = \mathcal{L} ( K ) \in \operatorname { Der } _ { k } \Omega ( M )$ by $\mathcal{L} _ { K } = [ i _ { K } , d ]$. The mapping ${\cal L} : \Omega ( M , T M ) \rightarrow \operatorname { Der } \Omega ( M )$ is injective. One has $\mathcal{L} ( \operatorname { ld } _ { T M } ) = d$.
  
For any graded derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023084.png" /> there are unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023086.png" /> such that
+
For any graded derivation $D \in \operatorname { Der } _ { k } \Omega ( M )$ there are unique $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { k + 1 } ( M ; T M )$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023087.png" /></td> </tr></table>
+
\begin{equation*} D = \mathcal{L} _ { K } + i _ { L }. \end{equation*}
  
One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023088.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023089.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023090.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023091.png" />.
+
One has $L = 0$ if and only if $[ D , d ] = 0$. Moreover, $D | _ { \Omega ^ { 0 } ( M ) } = 0$ if and only if $K = 0$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023093.png" />. Then, obviously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023094.png" />, so
+
Let $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { 1 } ( M ; T M )$. Then, obviously, $[ [ \mathcal{L} _ { K } , \mathcal{L} _ { L } ] , d ] = 0$, so
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023095.png" /></td> </tr></table>
+
\begin{equation*} [ \mathcal{L} ( K ) , \mathcal{L} ( L ) ] = \mathcal{L} ( [ K , L ] ) \end{equation*}
  
for a uniquely defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023096.png" />. This vector-valued form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023097.png" /> is the Frölicher–Nijenhuis bracket of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f12023099.png" />.
+
for a uniquely defined $[ K , L ] \in \Omega ^ { k + 1 } ( M ; T M )$. This vector-valued form $[ K , L ]$ is the Frölicher–Nijenhuis bracket of $K$ and $L$.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230101.png" /> one has
+
For $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { \text{l} + 1 } ( M ; T M )$ one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230102.png" /></td> </tr></table>
+
\begin{equation*} [\mathcal{L} _ { K } , i _ { L } ] = i ( [ K , L ] ) - ( - 1 ) ^ { k \text{l} } \mathcal{L} ( i _ { L } K ). \end{equation*}
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230103.png" /> is a graded module over the graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230104.png" /> with the action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230105.png" />, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230106.png" /> is graded commutative. Let the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230107.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230108.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230109.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230110.png" />, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230111.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230112.png" />. Let the other degrees be as indicated. Then:
+
The space $\operatorname{Der} \Omega ( M )$ is a graded module over the graded algebra $\Omega ( M )$ with the action $( \omega \wedge D ) \varphi = \omega \wedge D ( \varphi )$, because $\Omega ( M )$ is graded commutative. Let the degree of $\omega$ be $q$, of $\varphi$ be $k$, and of $\psi$ be $\operatorname{l}$. Let the other degrees be as indicated. Then:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230113.png" /></td> </tr></table>
+
\begin{equation*} [ \omega \wedge D _ { 1 } , D _ { 2 } ] = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230114.png" /></td> </tr></table>
+
\begin{equation*} = \omega \bigwedge [ D _ { 1 } , D _ { 2 } ] - ( - 1 ) ^ { ( q + k _ { 1 } ) k _ { 2 } } D _ { 2 } ( \omega ) \bigwedge D _ { 1 } ,\, i ( \omega \bigwedge L ) = \omega \bigwedge i ( L ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230115.png" /></td> </tr></table>
+
\begin{equation*} \omega \bigwedge \mathcal{L} _ { K } = \mathcal{L} ( \omega \bigwedge K ) + ( - 1 ) ^ { q + k - 1 } i ( d \omega \bigwedge K ) ,\, [ \omega \bigwedge L _ { 1 } , L _ { 2 } ] ^ { \bigwedge } = \omega \bigwedge [ L _ { 1 } , L _ { 2 } ] + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230116.png" /></td> </tr></table>
+
\begin{equation*} - ( - 1 ) ^ { ( q + \operatorname{l} _ { 1 } - 1 ) ( \operatorname{l} _ { 2 } - 1 ) } i ( L _ { 2 } ) \omega \bigwedge L _ { 1 } , [ \omega \bigwedge K _ { 1 } , K _ { 2 } ] = \omega \bigwedge [ K _ { 1 } , K _ { 2 } ] + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230117.png" /></td> </tr></table>
+
\begin{equation*} - ( - 1 ) ^ { ( q + k _ { 1 } ) k _ { 2 } } \mathcal{L} ( K _ { 2 } ) \omega \bigwedge K _ { 1 } + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230118.png" /></td> </tr></table>
+
\begin{equation*} + ( - 1 ) ^ { q + k _ { 1 } } d \omega \bigwedge i ( K _ { 1 } ) K _ { 2 }. \end{equation*}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230120.png" />, the Lie derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230121.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230122.png" /> is given by:
+
For $K \in \Omega ^ { k } ( M ; T M )$ and $\omega \in \Omega ^ { 1 } ( M )$, the Lie derivative of $\omega$ along $K$ is given by:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230123.png" /></td> </tr></table>
+
\begin{equation*} ( \mathcal{L} _ { K } \omega ) ( X _ { 1 } , \dots , X _ { k + 1 } ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230124.png" /></td> </tr></table>
+
\begin{equation*} = \frac { 1 } { k ! \text{l} ! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times \mathcal{L} ( K(X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) ) ( \omega ( X _ { \sigma ( k + 1 ) } , \ldots , X _ { \sigma ( k + 1 ) } ) ) + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230125.png" /></td> </tr></table>
+
\begin{equation*} +\frac { - 1 } { k ! ( \text{l} - 1 ) ! } \times \times \sum _ { \sigma } \operatorname { sign } \sigma \omega ( [ K ( X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) , X _ { \sigma ( k + 1 ) } ] , X _ { \sigma ( k + 2 ) } , \ldots )+ \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230126.png" /></td> </tr></table>
+
\begin{equation*} + \frac { ( - 1 ) ^ { k - 1 } } { ( k - 1 ) ! ( 1 - 1 ) ! 2 ! } \times \times \sum _ { \sigma } \operatorname { sign } \ \sigma \ \omega ( K ( [ X _ { \sigma 1 } , X _ { \sigma 2 } ] , X _ { \sigma 3 } , \ldots ) , X _ { \sigma ( k + 2 ) } , \ldots ). \end{equation*}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230128.png" />, the Frölicher–Nijenhuis bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230129.png" /> is given by:
+
For $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { 1 } ( M ; T M )$, the Frölicher–Nijenhuis bracket $[ K , L ]$ is given by:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230130.png" /></td> </tr></table>
+
\begin{equation*} [ K , L ] ( X _ { 1 } , \dots , X _ { k + 1 } ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230131.png" /></td> </tr></table>
+
\begin{equation*} = \frac { 1 } { k ! l ! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times [ K ( X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) , L ( X _ { \sigma ( k + 1 ) } , \ldots , X _ { \sigma ( k + 1 ) } ) ] + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230132.png" /></td> </tr></table>
+
\begin{equation*} + \frac { - 1 } { k ! ( \operatorname {l} - 1 ) ! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times L ( [ K ( X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) , X _ { \sigma ( k + 1 ) } ] , X _ { \sigma ( k + 2 ) } , \ldots )+ \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230133.png" /></td> </tr></table>
+
\begin{equation*} + \frac { ( - 1 ) ^ { k \text{l} } } { ( k - 1 ) ! \text{l}! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times K ( [ L ( X _ { \sigma 1 } , \ldots , X _ { \sigma \text{l} } ) , X _ { \sigma ( \text{l} + 1 ) } ] , X _ { \sigma ( \text{l} + 2 ) } , \ldots ) + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230134.png" /></td> </tr></table>
+
\begin{equation*} + \frac { ( - 1 ) ^ { k - 1 } } { ( k - 1 ) ! ( 1 - 1 ) ! 2 ! } \times \times \sum _ { \sigma } \operatorname { sign } \sigma L ( K ( [ X _ { \sigma 1 } , X _ { \sigma 2 } ] , X _ { \sigma 3 } , \ldots ) , X _ { \sigma ( k + 2 ) } , \ldots ) + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230135.png" /></td> </tr></table>
+
\begin{equation*} +\frac { ( - 1 ) ^ { ( k - 1 ) \text{l} } } { ( k - 1 ) ! ( \text{l} - 1 ) ! 2 ! } \times \times \sum _ { \sigma } \operatorname { sign } \sigma\, K ( L ( [ X _ { \sigma 1 } , X _ { \sigma 2 } ] , X _ { \sigma 3 } , \ldots ) , X _ { \sigma ( \text{l} + 2 ) , \ldots } ). \end{equation*}
  
The Frölicher–Nijenhuis bracket expresses obstructions to integrability in many different situations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230136.png" /> is an almost-complex structure, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230137.png" /> is complex structure if and only if the Nijenhuis tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230138.png" /> vanishes (the Newlander–Nirenberg theorem, [[#References|[a5]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230139.png" /> is a fibre-wise projection on the tangent spaces of a fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230140.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230141.png" /> is a version of the curvature (see [[#References|[a3]]], Sects. 9; 10). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230142.png" /> is fibre-wise diagonalizable with all eigenvalues real and of constant multiplicity, then the eigenspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230143.png" /> are integrable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120230/f120230144.png" />.
+
The Frölicher–Nijenhuis bracket expresses obstructions to integrability in many different situations: If $J : T M \rightarrow T M$ is an almost-complex structure, then $J$ is complex structure if and only if the Nijenhuis tensor $[ J , J ]$ vanishes (the Newlander–Nirenberg theorem, [[#References|[a5]]]). If $P : T M \rightarrow T M$ is a fibre-wise projection on the tangent spaces of a fibre bundle $M \rightarrow B$, then $[ P , P ]$ is a version of the curvature (see [[#References|[a3]]], Sects. 9; 10). If $A : T M \rightarrow T M$ is fibre-wise diagonalizable with all eigenvalues real and of constant multiplicity, then the eigenspaces of $A$ are integrable if and only if $[ A , A ] = 0$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Frölicher,  A. Nijenhuis,  "Theory of vector valued differential forms. Part I."  ''Indag. Math.'' , '''18'''  (1956)  pp. 338–359</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Frölicher,  A. Nijenhuis,  "Invariance of vector form operations under mappings"  ''Comment. Math. Helvetici'' , '''34'''  (1960)  pp. 227–248</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Kolář,  Peter W. Michor,  J. Slovák,  "Natural operations in differential geometry" , Springer  (1993)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Pierre Lecomte,  Peter W. Michor,  Hubert Schicketanz,  "The multigraded Nijenhuis–Richardson Algebra, its universal property and application"  ''J. Pure Appl. Algebra'' , '''77'''  (1992)  pp. 87–102</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Newlander,  L. Nirenberg,  "Complex analytic coordinates in almost complex manifolds"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 391–404</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Nijenhuis,  R. Richardson,  "Cohomology and deformations in graded Lie algebras"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 1–29</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Nijenhuis,  R. Richardson,  "Deformation of Lie algebra structures"  ''J. Math. Mech.'' , '''17'''  (1967)  pp. 89–105</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.A. Schouten,  "Über Differentialkonkomitanten zweier kontravarianten Grössen"  ''Indag. Math.'' , '''2'''  (1940)  pp. 449–452</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Frölicher,  A. Nijenhuis,  "Theory of vector valued differential forms. Part I."  ''Indag. Math.'' , '''18'''  (1956)  pp. 338–359</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Frölicher,  A. Nijenhuis,  "Invariance of vector form operations under mappings"  ''Comment. Math. Helvetici'' , '''34'''  (1960)  pp. 227–248</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  I. Kolář,  Peter W. Michor,  J. Slovák,  "Natural operations in differential geometry" , Springer  (1993)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  Pierre Lecomte,  Peter W. Michor,  Hubert Schicketanz,  "The multigraded Nijenhuis–Richardson Algebra, its universal property and application"  ''J. Pure Appl. Algebra'' , '''77'''  (1992)  pp. 87–102</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Newlander,  L. Nirenberg,  "Complex analytic coordinates in almost complex manifolds"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 391–404</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Nijenhuis,  R. Richardson,  "Cohomology and deformations in graded Lie algebras"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 1–29</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A. Nijenhuis,  R. Richardson,  "Deformation of Lie algebra structures"  ''J. Math. Mech.'' , '''17'''  (1967)  pp. 89–105</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J.A. Schouten,  "Über Differentialkonkomitanten zweier kontravarianten Grössen"  ''Indag. Math.'' , '''2'''  (1940)  pp. 449–452</td></tr></table>

Latest revision as of 12:23, 12 December 2020

Let $M$ be a smooth manifold (cf. also Differentiable manifold) and let $\Omega ^ { k } ( M ; T M ) = \Gamma ( \wedge ^ { k } T ^ { * } M \otimes T M )$. One calls

\begin{equation*} \Omega ( M , T M ) = \oplus _ { k = 0 } ^ { \operatorname { dim } M } \Omega ^ { k } ( M , T M ) \end{equation*}

the space of all vector-valued differential forms. The Frölicher–Nijenhuis bracket $[ \cdot , \cdot ] : \Omega ^ { k } ( M ; T M ) \times \Omega ^ { l } ( M ; T M ) \rightarrow \Omega ^ { k + l } ( M ; T M )$ is a $\bf Z$-graded Lie bracket:

\begin{equation*} [ K , L ] = - ( - 1 ) ^ { k l } [ L , K ], \end{equation*}

\begin{equation*} [ K _ { 1 } , [ K _ { 2 } , K _ { 3 } ] ] = [ [ K _ { 1 } , K _ { 2 } ] , K _ { 3 } ] + ( - 1 ) ^ { k _ { 1 } k _ { 2 } } [ K _ { 2 } , [ K _ { 1 }, K _ { 3 }] ]. \end{equation*}

It extends the Lie bracket of smooth vector fields, since $\Omega ^ { 0 } ( M ; T M ) = \Gamma ( T M ) = \mathcal{X} ( M )$. The identity on $T M$ generates the one-dimensional centre. It is called the Frölicher–Nijenhuis bracket since it appeared with its full properties for the first time in [a1], after some indication in [a8]. One formula for it is:

\begin{equation*} [ \varphi \bigotimes x , \psi \bigotimes Y ] = \end{equation*}

\begin{equation*} = \varphi \bigwedge \psi \bigotimes [ X , Y ] + \varphi \bigwedge {\cal L} _ { X } \psi \bigotimes Y - {\cal L} _ { Y } \varphi \bigwedge \psi \bigotimes X + \end{equation*}

\begin{equation*} + ( - 1 ) ^ { k } \left( d \varphi \bigwedge i _ { X } \psi \bigotimes Y + i _{Y} \varphi \bigwedge d \psi \bigotimes X \right), \end{equation*}

where $X$ and $Y$ are vector fields, $\varphi$ is a $k$-form, and $\psi$ is an $l$-form. It is a bilinear differential operator of bi-degree $( 1,1 )$.

The Frölicher–Nijenhuis bracket is natural in the same way as the Lie bracket for vector fields: if $f : M \rightarrow N$ is smooth and $K _ { i } \in \Omega ^ { k _ { i } } ( M ; T M )$ are $f$-related to $L _ { i } \in \Omega ^ { l } ( N ; T N )$, then $[ K _ { 1 } , K _ { 2 } ]$ is also $f$-related to $L_1$, $L_{2}$.

Details.

A convenient source is [a3], Sect. 8. The basic formulas of the calculus of differential forms extend naturally to include the Frölicher–Nijenhuis bracket: Let

\begin{equation*} \Omega ( M ) = \bigoplus _ { k \geq 0 } \Omega ^ { k } ( M ) = \bigoplus _ { k = 0 } ^ { \operatorname { dim } M } \Gamma \left( \bigwedge^k T ^ { * } M \right) \end{equation*}

be the algebra of differential forms. One denotes by $\operatorname { Der } _ { k } \Omega ( M )$ the space of all (graded) derivations of degree $k$, i.e. all bounded linear mappings $D : \Omega ( M ) \rightarrow \Omega ( M )$ with $D ( \Omega ^ { l } ( M ) ) \subset \Omega ^ { k + l } ( M )$ and $D ( \varphi \wedge \psi ) = D ( \varphi ) \wedge \psi + ( - 1 ) ^ { k l } \varphi \wedge D ( \psi )$ for $\varphi \in \Omega ^ { l } ( M )$. The space $\operatorname { Der }\Omega ( M ) = \oplus _ { k } \operatorname { Der } _ { k } \Omega ( M )$ is a $\bf Z$-graded Lie algebra with the graded commutator $[ D _ { 1 } , D _ { 2 } ] = D _ { 1 } D _ { 2 } - ( - 1 ) ^ { k _ { 1 } k _ { 2 } } D _ { 2 } D _ { 1 }$ as bracket.

A derivation $D \in \operatorname { Der } _ { k } \Omega ( M )$ with $D | _ { \Omega ^ { 0 } } ( M ) = 0$ satisfies $D ( f . \omega ) = f . D ( \omega )$ for $f \in C ^ { \infty } ( M , \mathbf{R} )$, thus $D$ is of tensorial character and induces a derivation $D _ { X } \in \operatorname { Der } _ { k } \wedge T _ { X } ^ { * } M$ for each $x \in M$. It is uniquely determined by its restriction to $1$-forms , which can be viewed as an element $K _ { x } \in \wedge ^ { k + 1 } T _ { x } ^ { * } M \otimes T _ { x } M$ depending smoothly on $x \in M$; this is expressed by writing $D = i _ { K }$, where $K \in C ^ { \infty } ( \wedge ^ { k + 1 } T ^ { * } M \otimes T M ) = \Omega ^ { k + 1 } ( M ; T M )$, and one has

\begin{equation*} ( i _ { K } \omega ) ( X _ { 1 } , \dots , X _ { k + 1 } ) = \end{equation*}

\begin{equation*} =\frac { 1 } { ( k + 1 ) ! ( l - 1 ) ! } \times \times \sum _ { \sigma \in S _ { k + \text{l} } } \operatorname { sign } \sigma . \omega ( K ( X _ { \sigma 1 } , \ldots , X _ { \sigma ( k + 1 ) } ) , X _ { \sigma ( k + 2 ) } , \ldots ) \end{equation*}

for $\omega \in \Omega ^ { 1 } ( M )$ and $X_i \in \mathcal{X} ( M )$ (or $T _ { x } M$).

By putting $i ( [ K , L ] ^ { \wedge } ) = [ i _ { K } , i _ { L } ]$, one obtains a bracket $[ ., . ] ^ { \wedge }$ on $\Omega ^ { * + 1 } ( M , T M )$ which defines a graded Lie algebra structure with the grading as indicated, and for $K \in \Omega ^ { k + 1 } ( M , T M )$, $L \in \Omega ^ { \text{l} + 1 } ( M , T M )$ one has

\begin{equation*} [ K , L ] \bigwedge = i _ { K } L - ( - 1 ) ^ { k \text{l}} i _ { L } K, \end{equation*}

where $i _ { K } ( \omega \otimes X ) = i _ { K } ( \omega ) \otimes X$. The bracket $[ ., . ] ^ { \wedge }$ is called the Nijenhuis–Richardson bracket, see [a6] and [a7]. If viewed on a vector space $V$, it recognizes Lie algebra structures on $V$: A mapping $P \in L ^ { 2 }_\text { skew } ( V ; V )$ is a Lie bracket if and only if $[ P , P ] ^ { \wedge } = 0$. This can be used to study deformations of Lie algebra structures: $P + A$ is again a Lie bracket on $V$ if and only if $[ P + A , P + A ] ^ { \wedge } = 2 [ P , A ] ^ { \wedge } + [ A , A ] ^ { \wedge } = 0$; this can be written in the form of a Maurer–Cartan equation (cf. also Maurer–Cartan form) as $\delta _ { P } ( A ) + [ A , A ] ^ { \wedge } / 2 = 0$, since $\delta _ { P } = [ P , . ] ^ { \wedge }$ is the coboundary operator for the Chevalley cohomology (cf. also Cohomology) of the Lie algebra $( V , P )$ with values in the adjoint representation $V$. See [a4] for a multi-graded elaboration of this.

The exterior derivative $d$ is an element of $\operatorname { Der } _ { 1 } \Omega ( M )$. In view of the formula $\mathcal{L} _ { X } = [ i_{X} , d ] = i_{X} d + d i _{X}$ for vector fields $X$, one defines for $K \in \Omega ^ { k } ( M ; T M )$ the Lie derivation $\mathcal{L} _ { K } = \mathcal{L} ( K ) \in \operatorname { Der } _ { k } \Omega ( M )$ by $\mathcal{L} _ { K } = [ i _ { K } , d ]$. The mapping ${\cal L} : \Omega ( M , T M ) \rightarrow \operatorname { Der } \Omega ( M )$ is injective. One has $\mathcal{L} ( \operatorname { ld } _ { T M } ) = d$.

For any graded derivation $D \in \operatorname { Der } _ { k } \Omega ( M )$ there are unique $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { k + 1 } ( M ; T M )$ such that

\begin{equation*} D = \mathcal{L} _ { K } + i _ { L }. \end{equation*}

One has $L = 0$ if and only if $[ D , d ] = 0$. Moreover, $D | _ { \Omega ^ { 0 } ( M ) } = 0$ if and only if $K = 0$.

Let $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { 1 } ( M ; T M )$. Then, obviously, $[ [ \mathcal{L} _ { K } , \mathcal{L} _ { L } ] , d ] = 0$, so

\begin{equation*} [ \mathcal{L} ( K ) , \mathcal{L} ( L ) ] = \mathcal{L} ( [ K , L ] ) \end{equation*}

for a uniquely defined $[ K , L ] \in \Omega ^ { k + 1 } ( M ; T M )$. This vector-valued form $[ K , L ]$ is the Frölicher–Nijenhuis bracket of $K$ and $L$.

For $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { \text{l} + 1 } ( M ; T M )$ one has

\begin{equation*} [\mathcal{L} _ { K } , i _ { L } ] = i ( [ K , L ] ) - ( - 1 ) ^ { k \text{l} } \mathcal{L} ( i _ { L } K ). \end{equation*}

The space $\operatorname{Der} \Omega ( M )$ is a graded module over the graded algebra $\Omega ( M )$ with the action $( \omega \wedge D ) \varphi = \omega \wedge D ( \varphi )$, because $\Omega ( M )$ is graded commutative. Let the degree of $\omega$ be $q$, of $\varphi$ be $k$, and of $\psi$ be $\operatorname{l}$. Let the other degrees be as indicated. Then:

\begin{equation*} [ \omega \wedge D _ { 1 } , D _ { 2 } ] = \end{equation*}

\begin{equation*} = \omega \bigwedge [ D _ { 1 } , D _ { 2 } ] - ( - 1 ) ^ { ( q + k _ { 1 } ) k _ { 2 } } D _ { 2 } ( \omega ) \bigwedge D _ { 1 } ,\, i ( \omega \bigwedge L ) = \omega \bigwedge i ( L ), \end{equation*}

\begin{equation*} \omega \bigwedge \mathcal{L} _ { K } = \mathcal{L} ( \omega \bigwedge K ) + ( - 1 ) ^ { q + k - 1 } i ( d \omega \bigwedge K ) ,\, [ \omega \bigwedge L _ { 1 } , L _ { 2 } ] ^ { \bigwedge } = \omega \bigwedge [ L _ { 1 } , L _ { 2 } ] + \end{equation*}

\begin{equation*} - ( - 1 ) ^ { ( q + \operatorname{l} _ { 1 } - 1 ) ( \operatorname{l} _ { 2 } - 1 ) } i ( L _ { 2 } ) \omega \bigwedge L _ { 1 } , [ \omega \bigwedge K _ { 1 } , K _ { 2 } ] = \omega \bigwedge [ K _ { 1 } , K _ { 2 } ] + \end{equation*}

\begin{equation*} - ( - 1 ) ^ { ( q + k _ { 1 } ) k _ { 2 } } \mathcal{L} ( K _ { 2 } ) \omega \bigwedge K _ { 1 } + \end{equation*}

\begin{equation*} + ( - 1 ) ^ { q + k _ { 1 } } d \omega \bigwedge i ( K _ { 1 } ) K _ { 2 }. \end{equation*}

For $K \in \Omega ^ { k } ( M ; T M )$ and $\omega \in \Omega ^ { 1 } ( M )$, the Lie derivative of $\omega$ along $K$ is given by:

\begin{equation*} ( \mathcal{L} _ { K } \omega ) ( X _ { 1 } , \dots , X _ { k + 1 } ) = \end{equation*}

\begin{equation*} = \frac { 1 } { k ! \text{l} ! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times \mathcal{L} ( K(X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) ) ( \omega ( X _ { \sigma ( k + 1 ) } , \ldots , X _ { \sigma ( k + 1 ) } ) ) + \end{equation*}

\begin{equation*} +\frac { - 1 } { k ! ( \text{l} - 1 ) ! } \times \times \sum _ { \sigma } \operatorname { sign } \sigma \omega ( [ K ( X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) , X _ { \sigma ( k + 1 ) } ] , X _ { \sigma ( k + 2 ) } , \ldots )+ \end{equation*}

\begin{equation*} + \frac { ( - 1 ) ^ { k - 1 } } { ( k - 1 ) ! ( 1 - 1 ) ! 2 ! } \times \times \sum _ { \sigma } \operatorname { sign } \ \sigma \ \omega ( K ( [ X _ { \sigma 1 } , X _ { \sigma 2 } ] , X _ { \sigma 3 } , \ldots ) , X _ { \sigma ( k + 2 ) } , \ldots ). \end{equation*}

For $K \in \Omega ^ { k } ( M ; T M )$ and $L \in \Omega ^ { 1 } ( M ; T M )$, the Frölicher–Nijenhuis bracket $[ K , L ]$ is given by:

\begin{equation*} [ K , L ] ( X _ { 1 } , \dots , X _ { k + 1 } ) = \end{equation*}

\begin{equation*} = \frac { 1 } { k ! l ! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times [ K ( X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) , L ( X _ { \sigma ( k + 1 ) } , \ldots , X _ { \sigma ( k + 1 ) } ) ] + \end{equation*}

\begin{equation*} + \frac { - 1 } { k ! ( \operatorname {l} - 1 ) ! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times L ( [ K ( X _ { \sigma 1 } , \ldots , X _ { \sigma k } ) , X _ { \sigma ( k + 1 ) } ] , X _ { \sigma ( k + 2 ) } , \ldots )+ \end{equation*}

\begin{equation*} + \frac { ( - 1 ) ^ { k \text{l} } } { ( k - 1 ) ! \text{l}! } \sum _ { \sigma } \operatorname { sign } \sigma \times \times K ( [ L ( X _ { \sigma 1 } , \ldots , X _ { \sigma \text{l} } ) , X _ { \sigma ( \text{l} + 1 ) } ] , X _ { \sigma ( \text{l} + 2 ) } , \ldots ) + \end{equation*}

\begin{equation*} + \frac { ( - 1 ) ^ { k - 1 } } { ( k - 1 ) ! ( 1 - 1 ) ! 2 ! } \times \times \sum _ { \sigma } \operatorname { sign } \sigma L ( K ( [ X _ { \sigma 1 } , X _ { \sigma 2 } ] , X _ { \sigma 3 } , \ldots ) , X _ { \sigma ( k + 2 ) } , \ldots ) + \end{equation*}

\begin{equation*} +\frac { ( - 1 ) ^ { ( k - 1 ) \text{l} } } { ( k - 1 ) ! ( \text{l} - 1 ) ! 2 ! } \times \times \sum _ { \sigma } \operatorname { sign } \sigma\, K ( L ( [ X _ { \sigma 1 } , X _ { \sigma 2 } ] , X _ { \sigma 3 } , \ldots ) , X _ { \sigma ( \text{l} + 2 ) , \ldots } ). \end{equation*}

The Frölicher–Nijenhuis bracket expresses obstructions to integrability in many different situations: If $J : T M \rightarrow T M$ is an almost-complex structure, then $J$ is complex structure if and only if the Nijenhuis tensor $[ J , J ]$ vanishes (the Newlander–Nirenberg theorem, [a5]). If $P : T M \rightarrow T M$ is a fibre-wise projection on the tangent spaces of a fibre bundle $M \rightarrow B$, then $[ P , P ]$ is a version of the curvature (see [a3], Sects. 9; 10). If $A : T M \rightarrow T M$ is fibre-wise diagonalizable with all eigenvalues real and of constant multiplicity, then the eigenspaces of $A$ are integrable if and only if $[ A , A ] = 0$.

References

[a1] A. Frölicher, A. Nijenhuis, "Theory of vector valued differential forms. Part I." Indag. Math. , 18 (1956) pp. 338–359
[a2] A. Frölicher, A. Nijenhuis, "Invariance of vector form operations under mappings" Comment. Math. Helvetici , 34 (1960) pp. 227–248
[a3] I. Kolář, Peter W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993)
[a4] Pierre Lecomte, Peter W. Michor, Hubert Schicketanz, "The multigraded Nijenhuis–Richardson Algebra, its universal property and application" J. Pure Appl. Algebra , 77 (1992) pp. 87–102
[a5] A. Newlander, L. Nirenberg, "Complex analytic coordinates in almost complex manifolds" Ann. of Math. , 65 (1957) pp. 391–404
[a6] A. Nijenhuis, R. Richardson, "Cohomology and deformations in graded Lie algebras" Bull. Amer. Math. Soc. , 72 (1966) pp. 1–29
[a7] A. Nijenhuis, R. Richardson, "Deformation of Lie algebra structures" J. Math. Mech. , 17 (1967) pp. 89–105
[a8] J.A. Schouten, "Über Differentialkonkomitanten zweier kontravarianten Grössen" Indag. Math. , 2 (1940) pp. 449–452
How to Cite This Entry:
Frölicher-Nijenhuis bracket. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%B6licher-Nijenhuis_bracket&oldid=18865
This article was adapted from an original article by Peter W. Michor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article