# nLab Nijenhuis–Richardson 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 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 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 $Y_i=X_{\sigma(i)}$.

The map $\iota$ defined an injective homomorphism of graded vector spaces from $\Omega^{\bullet+1}(M,T M)$ to graded derivations of $\Omega(M)$. Its image comprises precisely those derivations that vanish on 0-forms and is closed under the commutator operation. Transferring the bracket to its source yields the Nijenhuis–Richardson bracket:

$[K,L]^\wedge = \iota_K L-(-1)^{k l}\iota_L K,$

where $\iota_K(\omega\otimes X)=\iota_K \omega\otimes X$.

## Classification of graded derivations of differential forms

The Nijenhuis–Richardson bracket is an important ingredient in the classification of graded derivations of differential forms.

## References

The formula for the NR bracket is made explicit in and thus derives its name from:

but the existence of the bracket is already implicit in:

and refined to almost complex structures in