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nLab > Latest Changes: Frölicher–Nijenhuis bracket

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- CommentRowNumber1.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 3rd 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexCreated: ## Definition Given a smooth manifold $M$ and differential forms $P\in\Omega^k(M,TM)$, $Q\in\Omega^l(M,TM)$ valued in the [[tangent bundle]] $TM$ of $M$, their __Frölicher–Nijenhuis bracket__ is a [[differential form]] $$[P,Q]\in\Omega^{k+l}(M,TM)$$ defined by the formula $$[P,Q](X_1,\ldots,X_{k+l})={1\over k!l!} \sum_\sigma (-1)^\sigma \left( [P(Y_1,\ldots,Y_k),Q(Y_{k+1},\ldots,Y_{k+l})] -l Q([P(Y_1,\ldots,Y_k),Y_{k+1}],Y_{k+2},\ldots) +(-1)^{k l}k P([Q(Y_1,\ldots,Y_k),Y_{k+1}],Y_{k+2},\ldots) +(-1)^{k-1}(k l/2) Q(P([Y_1,Y_2],Y_3,\ldots),Y_{k+2},\ldots) +(-1)^{(k-1)l}(k l/2) P(Q([Y_1,Y_2],Y_3,\ldots),Y_{k+2},\ldots),$$ where $Y_i=X_{\sigma(i)}$ and $(-1)^\sigma$ is the sign of the [[permutation]] $\sigma$. ## Related concepts * [[Schouten–Nijenhuis bracket]] <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
Created:

Definition

Given a smooth manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ and differential forms $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P\in\Omega^k(M,TM)</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q\in\Omega^l(M,TM)</annotation></semantics></math>$ valued in the tangent bundle $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>TM</mi></mrow><annotation encoding="application/x-tex">TM</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ , their Frölicher–Nijenhuis bracket is a differential form
[P,Q]∈Ωk+l(M,TM)

defined by the formula

where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Y</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>X</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Y_i=X_{\sigma(i)}</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>σ</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^\sigma</annotation></semantics></math>$ is the sign of the permutation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>$ .

Related concepts
- Schouten–Nijenhuis bracket
v1, current
- CommentRowNumber2.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 3rd 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexEnhanced: ## Definition Suppose $M$ is a [[smooth manifold]]. Recall (from the article [[Nijenhuis–Richardson bracket]]) that any differential $(k+1)$-form $K\in\Omega^{k+1}(M,TM)$ valued in the [[tangent bundle]] of $M$ gives rise to a [[graded derivation]] $\iota_K$ of degree $k$ on the algebra of [[differential forms]] on $M$: on 1-forms we have $\iota_K \omega=\omega\circ K$ and on higher forms we extend using the Leibniz identity. Concretely, $$\iota_K \omega(X_1,\ldots,X_{k+l})=1/((k+1)!(l-1)!)\sum_\sigma (-1)^\sigma \omega(K(Y_1,\ldots,Y_{k+1}),Y_{k+2},\ldots),$$ where $Y_i=X_{\sigma(i)}$. [[Cartan's magic formula]] $$L_X=[\iota_X,d]$$ makes it natural to define the [[Lie derivative]] with respect to $K\in\Omega^k(M,TM)$: $$L_K=[\iota_K,d].$$ The map $L$ defines an injective homomorphism of [[graded vector spaces]] from $\Omega(M,TM)$ to graded derivations of $\Omega(M)$. Its image comprises precisely those derivations $D$ such that $[D,d]=0$ and is closed under the commutator operation. Transferring the bracket to its source yields the __Frölicher–Nijenhuis bracket__: $$L_{[K,L]} = [L_K,L_L]$$ for a uniquely defined $[K,L]\in\Omega^{k+l}(M,TM)$. ## Classification of graded derivations of differential forms Taken together, the [[Frölicher–Nijenhuis bracket]] and the [[Nijenhuis–Richardson bracket]] allows us to fully classify the graded derivations of the algebra of [[differential forms]] on $M$: A graded dervation $D$ of degree $k$ on $\Omega(M)$ has a unique presentation of the form $$D=L_K + \iota_L,$$ where $K\in\Omega^k(M,TM)$, $L\in\Omega^{k+1}(M,TM)$. We have $L=0$ if and only if $[D,d]=0$ and $K=0$ if and only if $D$ vanishes on 0-forms. Finally, the graded commutator of graded derivations can be expressed in terms of the [[Frölicher–Nijenhuis bracket]] (for $K$) and the [[Nijenhuis–Richardson bracket]] (for $L$): $$[L_K,L_L]=L_{[K,L]},$$ $$[\iota_K,\iota_L]=\iota_{[K,L]^\wedge},$$ $$[L_K,\iota_L]=\iota_{[K,L]}-(-1)^{k l}L_{\iota_L K},$$ $$[\iota_K,L_L]=L_{\iota_L K}+(-1)^k \iota_{[L,K]}.$$ ## Explicit formula Given a smooth manifold $M$ and differential forms $P\in\Omega^k(M,TM)$, $Q\in\Omega^l(M,TM)$ valued in the [[tangent bundle]] $TM$ of $M$, their __Frölicher–Nijenhuis bracket__ is a [[differential form]] $$[P,Q]\in\Omega^{k+l}(M,TM)$$ defined by the formula $$[P,Q](X_1,\ldots,X_{k+l})=1/(k!l!) \sum_\sigma (-1)^\sigma \left( [P(Y_1,\ldots,Y_k),Q(Y_{k+1},\ldots,Y_{k+l})] -l Q([P(Y_1,\ldots,Y_k),Y_{k+1}],Y_{k+2},\ldots) +(-1)^{k l}k P([Q(Y_1,\ldots,Y_k),Y_{k+1}],Y_{k+2},\ldots) +(-1)^{k-1}(k l/2) Q(P([Y_1,Y_2],Y_3,\ldots),Y_{k+2},\ldots) +(-1)^{(k-1)l}(k l/2) P(Q([Y_1,Y_2],Y_3,\ldots),Y_{k+2},\ldots) \right),$$ where $Y_i=X_{\sigma(i)}$ and $(-1)^\sigma$ is the sign of the [[permutation]] $\sigma$. ## Related concepts * [[Schouten–Nijenhuis bracket]] * [[Nijenhuis–Richardson bracket]] <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
Enhanced:

Definition

Suppose $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ is a smooth manifold. Recall (from the article Nijenhuis–Richardson bracket) that any differential $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>$ -form $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K\in\Omega^{k+1}(M,TM)</annotation></semantics></math>$ valued in the tangent bundle of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ gives rise to a graded derivation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ι</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\iota_K</annotation></semantics></math>$ of degree $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>$ on the algebra of differential forms on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ : on 1-forms we have $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ι</mi> <mi>K</mi></msub><mi>ω</mi><mo>=</mo><mi>ω</mi><mo>∘</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\iota_K \omega=\omega\circ K</annotation></semantics></math>$ and on higher forms we extend using the Leibniz identity.

Concretely,
ιKω(X1,…,Xk+l)=1/((k+1)!(l−1)!)∑σ(−1)σω(K(Y1,…,Yk+1),Yk+2,…),

where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Y</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>X</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Y_i=X_{\sigma(i)}</annotation></semantics></math>$ .

Cartan’s magic formula
LX=[ιX,d]

makes it natural to define the Lie derivative with respect to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K\in\Omega^k(M,TM)</annotation></semantics></math>$ :
LK=[ιK,d].

The map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>$ defines an injective homomorphism of graded vector spaces from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(M,TM)</annotation></semantics></math>$ to graded derivations of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(M)</annotation></semantics></math>$ . Its image comprises precisely those derivations $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>$ such that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>d</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[D,d]=0</annotation></semantics></math>$ and is closed under the commutator operation. Transferring the bracket to its source yields the Frölicher–Nijenhuis bracket:
L[K,L]=[LK,LL]

for a uniquely defined $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>L</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi></mrow></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[K,L]\in\Omega^{k+l}(M,TM)</annotation></semantics></math>$ .

Classification of graded derivations of differential forms

Taken together, the Frölicher–Nijenhuis bracket and the Nijenhuis–Richardson bracket allows us to fully classify the graded derivations of the algebra of differential forms on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ :

A graded dervation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>$ of degree $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>$ on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(M)</annotation></semantics></math>$ has a unique presentation of the form
D=LK+ιL,

where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K\in\Omega^k(M,TM)</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L\in\Omega^{k+1}(M,TM)</annotation></semantics></math>$ .

We have $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">L=0</annotation></semantics></math>$ if and only if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>d</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[D,d]=0</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">K=0</annotation></semantics></math>$ if and only if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>$ vanishes on 0-forms.

Finally, the graded commutator of graded derivations can be expressed in terms of the Frölicher–Nijenhuis bracket (for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>$ ) and the Nijenhuis–Richardson bracket (for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>$ ):
[LK,LL]=L[K,L],
[ιK,ιL]=ι[K,L]∧,
[LK,ιL]=ι[K,L]−(−1)klLιLK,
[ιK,LL]=LιLK+(−1)kι[L,K].

Explicit formula

Given a smooth manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ and differential forms $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P\in\Omega^k(M,TM)</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q\in\Omega^l(M,TM)</annotation></semantics></math>$ valued in the tangent bundle $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>TM</mi></mrow><annotation encoding="application/x-tex">TM</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ , their Frölicher–Nijenhuis bracket is a differential form
[P,Q]∈Ωk+l(M,TM)

defined by the formula
[P,Q](X1,…,Xk+l)=1/(k!l!)∑σ(−1)σ([P(Y1,…,Yk),Q(Yk+1,…,Yk+l)]−lQ([P(Y1,…,Yk),Yk+1],Yk+2,…)+(−1)klkP([Q(Y1,…,Yk),Yk+1],Yk+2,…)+(−1)k−1(kl/2)Q(P([Y1,Y2],Y3,…),Yk+2,…)+(−1)(k−1)l(kl/2)P(Q([Y1,Y2],Y3,…),Yk+2,…)),

where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Y</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>X</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Y_i=X_{\sigma(i)}</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>σ</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^\sigma</annotation></semantics></math>$ is the sign of the permutation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>$ .

Related concepts
- Schouten–Nijenhuis bracket
- Nijenhuis–Richardson bracket
v1, current
- CommentRowNumber3.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 3rd 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded: ## Applications The [[Nijenhuis tensor]] of an [[almost complex structure]] $J\in\Omega^1(M,TM)$ is $[J,J]$. The explicit formula yields $$[J,J](X,Y)=2([JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]).$$ <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>

Added:

Applications

The Nijenhuis tensor of an almost complex structure $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>TM</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J\in\Omega^1(M,TM)</annotation></semantics></math>$ is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo>,</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J,J]</annotation></semantics></math>$ . The explicit formula yields
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo>,</mo><mi>J</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>JX</mi><mo>,</mo><mi>JY</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>−</mo><mi>J</mi><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>JY</mi><mo stretchy="false">]</mo><mo>−</mo><mi>J</mi><mo stretchy="false">[</mo><mi>JX</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">[J,J](X,Y)=2([JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]).</annotation></semantics></math>$
v1, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeMay 3rd 2023
- PermaLink
Author: Urs
Format: MarkdownItexhave touched the typesetting such as using `\frac`, breaking the lines in the big sum and making the permutations explicit in the indices <a href="https://ncatlab.org/nlab/revision/diff/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>

have touched the typesetting such as using \frac, breaking the lines in the big sum and making the permutations explicit in the indices

diff, v2, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeMay 3rd 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], p. 175 of: _[[Natural operations in differential geometry]]_, Springer (1993) [[webpage](https://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3), [pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)] <a href="https://ncatlab.org/nlab/revision/diff/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
added pointer to:
- Ivan Kolář, Peter Michor, Jan Slovák, p. 175 of: Natural operations in differential geometry, Springer (1993) [webpage, doi:10.1007/978-3-662-02950-3, pdf]
diff, v3, current
- CommentRowNumber6.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 3rd 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded: Original definition: * [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359. [doi:10.1016/s1385-7258(56)50046-7](10.1016/s1385-7258(56)50046-7) and [doi:10.1016/s1385-7258(56)50047-9](10.1016/s1385-7258(56)50047-9). Refinements for [[almost complex structures]]: * [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part II. Almost-complex structures_, Indagationes Mathematicae (Proceedings) 61 (1958), 414–421 and 422-429. [doi:10.1016/s1385-7258(58)50057-2](10.1016/s1385-7258(58)50057-2) and [doi:10.1016/s1385-7258(58)50058-4](10.1016/s1385-7258(58)50058-4). <a href="https://ncatlab.org/nlab/revision/diff/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
Added:

Original definition:
- Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359. doi:10.1016/s1385-7258(56)50046-7 and doi:10.1016/s1385-7258(56)50047-9.
Refinements for almost complex structures:
- Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part II. Almost-complex structures, Indagationes Mathematicae (Proceedings) 61 (1958), 414–421 and 422-429. doi:10.1016/s1385-7258(58)50057-2 and doi:10.1016/s1385-7258(58)50058-4.
diff, v4, current
- CommentRowNumber7.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 3rd 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded an earlier reference: The original definition, with an explicit formula is in Section 6 of * [[Albert Nijenhuis]], _Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II_, [[Indagationes Mathematicae]] (Proceedings) 58: (1955), 398-403. [doi][1] [1]: http://dx.doi.org/10.1016/s1385-7258(55)50055-2 Further development: * [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359. [doi:10.1016/s1385-7258(56)50046-7][1] and [doi:10.1016/s1385-7258(56)50047-9][2]. <a href="https://ncatlab.org/nlab/revision/diff/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
Added an earlier reference:

The original definition, with an explicit formula is in Section 6 of
- Albert Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II, Indagationes Mathematicae (Proceedings) 58: (1955), 398-403. doi
Further development:
- Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359. doi:10.1016/s1385-7258(56)50046-7 and [doi:10.1016/s1385-7258(56)50047-9][2].
diff, v5, current
- CommentRowNumber8.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 4th 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded: An expository account: * [[Peter W. Michor]], _Frölicher-Nijenhuis bracket_, [HTML](https://encyclopediaofmath.org/wiki/Frölicher-Nijenhuis_bracket), [PDF](https://www.mat.univie.ac.at/~michor/fn-rev.pdf). <a href="https://ncatlab.org/nlab/revision/diff/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
Added:

An expository account:
- Peter W. Michor, Frölicher-Nijenhuis bracket, HTML, PDF.
diff, v6, current
- CommentRowNumber9.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 4th 2023
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded: A textbook account: Chapter 16 of * [[Peter W. Michor]], _Topics in Differential Geometry_, Graduate Studies in Mathematics 93 (2008). [PDF](https://www.mat.univie.ac.at/~michor/dgbook.pdf). <a href="https://ncatlab.org/nlab/revision/diff/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Fr%C3%B6licher%E2%80%93Nijenhuis+bracket">current</a>
Added:

A textbook account: Chapter 16 of
- Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics 93 (2008). PDF.
diff, v6, current

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