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nLab > Latest Changes: Nijenhuis–Richardson bracket

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Created:

    Definition

    Suppose MM is a smooth manifold. Recall that any differential (k+1)(k+1)-form K∈Ω k+1(M,TM)K\in\Omega^{k+1}(M,TM) 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=ω∘K\iota_K \omega=\omega\circ K and on higher forms we extend using the Leibniz identity.

    Concretely,

    ι Kω(X 1,…,X k+l)=1/((k+1)!(l−1)!)∑ 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,TM) 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(ω⊗X)=ι Kω⊗X\iota_K(\omega\otimes X)=\iota_K \omega\otimes X.

    Properties

    Related concepts

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added references:

    Original definition:

    Refinements for almost complex structures:

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023

    So who is Richardson?

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023

    Or maybe you made a copy-and-paste error: The original references seem to be these here, instead:

    diff, v4, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023
    • (edited May 3rd 2023)

    No, the original references were correct. Frölicher and Nijenhuis computed the whole graded Lie algebra of graded derivations, and as explained in the article Frölicher–Nijenhuis bracket, both brackets participate in the description.

    The first paper you added appears not to mention the Nijenhuis–Richardson bracket at all. I am not sure why Richard Borcherds added it as a reference to the Wikipedia article.

    The second paper does explore it briefly in Section 5 and gives explicit formulas for it, unlike the Frölicher–Nijenhuis paper, which simply transfers the bracket along the injective homomorphism.

    However, it came out much later than the Frölicher–Nijenhuis paper.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023

    I see, so I have reworded to reflect this state of affairs.

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added an earlier reference:

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023

    So then it looks like on Wikipedia here they confused the two articles by Nijenhuis & Richardson which both start with “Cohomology and deformations…”

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 4th 2023

    Added:

    A textbook account: Chapter 16 of

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

    diff, v7, current

1 to 9 of 9

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