# nLab Frölicher–Nijenhuis bracket

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

Suppose $M$ is a smooth manifold. Recall (e.g. from the article Nijenhuis–Richardson bracket) that any differential $(k+1)$-form $K\in\Omega^{k+1}(M,T M)$ valued in the tangent bundle of $M$ gives rise to a graded derivation $\iota_K$ of degree $k$ on the de Rham 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 rule.

Concretely,

$\iota_K \omega(X_1,\ldots,X_{k+l}) \;=\; \frac {1} {(k+1)!(l-1)!} \sum_\sigma (-1)^\sigma \omega\big( K(Y_{\sigma(1)} ,\, \ldots ,\, Y_{\sigma(k+1)}) ,\, Y_{\sigma(k+2)} ,\,\ldots \big) \,,$

where the sum is over all permutations $\sigma \in$ $Sym(k+l)$ and where $(-1)^\sigma$ denotes the sign of the permutation.

Cartan's magic formula

$L_X=[\iota_X,d]$

makes it natural to define the Lie derivative with respect to $K\in\Omega^k(M,\, T M)$:

$L_K=[\iota_K,d].$

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 de Rham 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

$\begin{array}{l} [P,Q](X_1,\ldots,X_{k+l}) \;\coloneqq\; \frac{1}{k! l!} \sum_\sigma (-1)^\sigma \Big( & \big[ P(Y_{\sigma(1)},\ldots, Y_{\sigma(k)}), \, Q(Y_{\sigma(k+1)},\ldots,Y_{\sigma(k+l)}) \big] \\ & -l \, Q\big( [P(Y_{\sigma(1)},\ldots,Y_{\sigma(k)}) ,\, Y_{\sigma(k+1)}] ,\,Y_{\sigma(k+2)}, \ldots \big) \\ & +(-1)^{k l} k \, P\big( [Q(Y_{\sigma(1)},\ldots,Y_{\sigma(k)}) ,\, Y_{\sigma(k+1)}], Y_{\sigma(k+2)},\ldots\big) \\ & + \frac{1}{2} (-1)^{k-1} k l \, Q\big( P([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)}, \ldots),Y_{\sigma(k+2)},\ldots \big) \\ & + \frac{1}{2} (-1)^{(k-1)l} k l \, P\big( Q([Y_{\sigma(1)},Y_{\sigma(2)}],Y_{\sigma(3)},\ldots),Y_{\sigma(k+2)}, \ldots \big) \Big) \,, \end{array}$

and $(-1)^\sigma$ is the sign of the permutation $\sigma$.

## Applications

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

$[J,J](X,Y) \;=\; 2\big( [J X, J Y] - [X,Y] - J[X, J Y] - J[J X,Y] \big).$

## References

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

Further development:

Refinements for almost complex structures:

Discussion as a natural operation:

An article in Encyclopedia of Mathematics:

A textbook account: Chapter 16 of

• Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics 93 (2008). PDF.

Last revised on May 4, 2023 at 00:50:10. See the history of this page for a list of all contributions to it.