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nLab > Latest Changes: Schouten bracket

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

    Added:

    Definition

    The Schouten bracket on multivector fields is the unuque (up to a multiplication by a constant) natural operation

    Γ(Λ kTM)⊗Γ(Λ lTM)→Γ(Λ k+l−1TM).\Gamma(\Lambda^k TM)\otimes\Gamma(\Lambda^l TM)\to\Gamma(\Lambda^{k+l-1} TM).

    Concretely,

    [X 1∧⋯∧X k,Y 1∧⋯∧Y l]=∑ i,j(−1) i+j[X i,Y j]∧X 1∧⋯X^ i⋯∧X k∧Y 1∧⋯Y^ j⋯∧Y l.[X_1\wedge\cdots\wedge X_k,Y_1\wedge\cdots\wedge Y_l]=\sum_{i,j}(-1)^{i+j}[X_i,Y_j]\wedge X_1\wedge\cdots\hat X_i\cdots\wedge X_k\wedge Y_1\wedge\cdots \hat Y_j\cdots \wedge Y_l.

    For multivector fields regarded as “antifields” in BV-BRST formalism, the Schouten bracket is called the antibracket.

    Related concepts

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added:

    References

    • Jan Schouten, Über Differentialkonkomitanten zweier kontravarianten Grössen, Indagationes Mathematicae 2 (1940), 449–452.

    • Jan Schouten, On the differential operators of the first order in tensor calculus, In: Convegno Int. Geom. Diff. Italia. (1953), 1–7.

    • Albert Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indagationes Mathematicae 17 (1955), 390–403. doi:10.1016/S1385-7258(55)50054-0.

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMay 3rd 2023
    • (edited May 3rd 2023)

    Wjhat do you mean by natural “operation” in 1 ? The link points to natural bundle and here you talk about a morphism of spaces of sections, do you mean induced map on sections by natural transformation among natural bundles viewed as functors ? What are the exact requirements and which reference proves it ?

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Re #3: The requirements are listed in the references at natural bundle. The book by Ivan Kolář, Peter Michor, Jan Slovák has details. Basically, it is a natural transformation of sheaves over the site of smooth families of manifolds: objects are submersions T→U, morphisms are commutative squares that are fiberwise open embeddings, and covering families are given by families inducing an open cover on T.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023

    so let’s add the reference for this fact:

    • Peter W. Michor, Remarks on the Schouten-Nijenhuis bracket, In: Bureš, J. and Souček, V. (eds.): Proceedings of the Winter School “Geometry and Physics” Circolo Matematico di Palermo, Palermo (1987) 207-215 [dml:701423, pdf]

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added:

    Applications

    Suppose P∈Γ(Λ 2TM)P\in\Gamma(\Lambda^2 TM). The the bracket {f,g}=⟨df∧dg,P⟩\{f,g\}=\langle df\wedge dg,P\rangle satisfies the Jacobi identity (and hence is a Poisson bracket) if and only if [P,P]=0[P,P]=0.

    diff, v10, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023
    • (edited May 3rd 2023)

    by the way, typing “$df$” (or “$TM$”) here does not have the desired rendering (this may be lamentable, but we have to live with it). I am fixing these expressions to “$d f$” (or “$T M$”, etc.)

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 3rd 2023

    Added:

    A coordinate-free treatment is given in

    • W. M. Tulczyjew, The Graded Lie Algebra of Multivector Fields and the Generalized Lie Derivative of Forms. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 22:9 (1974), 937–942. PDF.

    diff, v12, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 4th 2023

    Added:

    Textbook account: Chapter 33.2 of

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

    diff, v13, current

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeMay 4th 2023

    5 thanks for the paper

1 to 10 of 10

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