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nLab > Latest Changes: Frölicher–Nijenhuis bracket

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

    Created:

    Definition

    Given a smooth manifold MM and differential forms P∈Ω k(M,TM)P\in\Omega^k(M,TM), Q∈Ω l(M,TM)Q\in\Omega^l(M,TM) valued in the tangent bundle TMTM of MM, their Frölicher–Nijenhuis bracket is a differential form

    [P,Q]∈Ω k+l(M,TM)[P,Q]\in\Omega^{k+l}(M,TM)

    defined by the formula

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

    Related concepts

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Enhanced:

    Definition

    Suppose MM is a smooth manifold. Recall (from the article Nijenhuis–Richardson bracket) 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)}.

    Cartan’s magic formula

    L X=[ι X,d]L_X=[\iota_X,d]

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

    L K=[ι K,d].L_K=[\iota_K,d].

    The map LL defines an injective homomorphism of graded vector spaces from Ω(M,TM)\Omega(M,TM) to graded derivations of Ω(M)\Omega(M). Its image comprises precisely those derivations DD such that [D,d]=0[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]L_{[K,L]} = [L_K,L_L]

    for a uniquely defined [K,L]∈Ω k+l(M,TM)[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 MM:

    A graded dervation DD of degree kk on Ω(M)\Omega(M) has a unique presentation of the form

    D=L K+ι L,D=L_K + \iota_L,

    where K∈Ω k(M,TM)K\in\Omega^k(M,TM), L∈Ω k+1(M,TM)L\in\Omega^{k+1}(M,TM).

    We have L=0L=0 if and only if [D,d]=0[D,d]=0 and K=0K=0 if and only if DD vanishes on 0-forms.

    Finally, the graded commutator of graded derivations can be expressed in terms of the Frölicher–Nijenhuis bracket (for KK) and the Nijenhuis–Richardson bracket (for LL):

    [L K,L L]=L [K,L],[L_K,L_L]=L_{[K,L]}, [ι K,ι L]=ι [K,L] ∧,[\iota_K,\iota_L]=\iota_{[K,L]^\wedge}, [L K,ι L]=ι [K,L]−(−1) klL ι LK,[L_K,\iota_L]=\iota_{[K,L]}-(-1)^{k l}L_{\iota_L K}, [ι K,L L]=L ι LK+(−1) kι [L,K].[\iota_K,L_L]=L_{\iota_L K}+(-1)^k \iota_{[L,K]}.

    Explicit formula

    Given a smooth manifold MM and differential forms P∈Ω k(M,TM)P\in\Omega^k(M,TM), Q∈Ω l(M,TM)Q\in\Omega^l(M,TM) valued in the tangent bundle TMTM of MM, their Frölicher–Nijenhuis bracket is a differential form

    [P,Q]∈Ω k+l(M,TM)[P,Q]\in\Omega^{k+l}(M,TM)

    defined by the formula

    [P,Q](X 1,…,X k+l)=1/(k!l!)∑ σ(−1) σ([P(Y 1,…,Y k),Q(Y k+1,…,Y k+l)]−lQ([P(Y 1,…,Y k),Y k+1],Y k+2,…)+(−1) klkP([Q(Y 1,…,Y k),Y k+1],Y k+2,…)+(−1) k−1(kl/2)Q(P([Y 1,Y 2],Y 3,…),Y k+2,…)+(−1) (k−1)l(kl/2)P(Q([Y 1,Y 2],Y 3,…),Y k+2,…)),[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 σ(i)Y_i=X_{\sigma(i)} and (−1) σ(-1)^\sigma is the sign of the permutation σ\sigma.

    Related concepts

    v1, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added:

    Applications

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

    [J,J](X,Y)=2([JX,JY]−[X,Y]−J[X,JY]−J[JX,Y]).[J,J](X,Y)=2([JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]).

    v1, current

    • 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

    diff, v2, current

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

    Added:

    Original definition:

    Refinements for almost complex structures:

    diff, v4, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added an earlier reference:

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

    Further development:

    diff, v5, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 4th 2023

    Added:

    An expository account:

    diff, v6, current

    • 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, v6, current

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