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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 3rd 2023

Created:

## 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

where $Y_i=X_{\sigma(i)}$ and $(-1)^\sigma$ is the sign of the permutation $\sigma$.

## Related concepts

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 3rd 2023

Enhanced:

## 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

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 3rd 2023

## Applications

The Nijenhuis tensor of an almost complex structure $J\in\Omega^1(M,TM)$ is $[J,J]$. The explicit formula yields

$[J,J](X,Y)=2([JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]).$
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 3rd 2023

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 3rd 2023
• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 3rd 2023

Original definition:

Refinements for almost complex structures:

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 3rd 2023

The original definition, with an explicit formula is in Section 6 of

Further development:

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 4th 2023