nLab Nijenhuis–Richardson bracket



Differential geometry

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geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

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graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

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Cartan geometry (super, higher)



Suppose MM is a smooth manifold. Recall that any differential (k+1)(k+1)-form KΩ k+1(M,TM)K\in\Omega^{k+1}(M,T M) valued in the tangent bundle of MM gives rise to a graded derivation ι K\iota_K of degree kk on the algebra of differential forms on MM: on 1-forms we have ι Kω=ωK\iota_K \omega=\omega\circ K and on higher forms we extend using the Leibniz identity.


ι Kω(X 1,,X k+l)=1/((k+1)!(l1)!) σ(1) σω(K(Y 1,,Y k+1),Y k+2,),\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 σ(i)Y_i=X_{\sigma(i)}.

The map ι\iota defined an injective homomorphism of graded vector spaces from Ω +1(M,TM)\Omega^{\bullet+1}(M,T M) to graded derivations of Ω(M)\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] =ι KL(1) klι LK,[K,L]^\wedge = \iota_K L-(-1)^{k l}\iota_L K,

where ι K(ωX)=ι KωX\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.

See the article Frölicher–Nijenhuis bracket for more information.


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

See also:

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 00:50:50. See the history of this page for a list of all contributions to it.