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For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one using the flow $\phi^X_t$, $${\cal L}_X \alpha = \frac{d}{dt}\biggl|_{t=0} \left(\bigl(\Phi^t_X\bigr)^* T\right)$$

Given a multivector field $X$ one can define its Lie derivative by means of Cartan formula, i., $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$. See, for example [1].

My broad question is if the Lie derivative by a multivector means something dynamical. These are some fuzzy questions for which any help or reference would be welcomed.

  • In the simplest case, if $X = X_1\wedge \ldots \wedge X_n$ and the e_i span an integrable distribution, is $\alpha$ constant in some sense on the leaves of this distribution. What happens if the multivector is not integrable?

  • Is there a generalization of the concept of flow for a multivector that applies to this situation?

  • My current geometric understanding of general multivectors is that they are linear combinations of hyperplanes modulo the Plücker relations (whose geometric interpretation feels somewhat obscure to me). I would like to have a better interpretation. On question [2] on this site it is stated that they are global sections of a line bundle over the Grasmannian, some reference of this fact would be useful.

[1] W. M. Tulczyjew, «The Graded Lie Algebra of Multivector Fields and the Generalized Lie Derivative of Forms».

[2] Denis Serre (https://mathoverflow.net/users/8799/denis-serre), Grassmannian as a submanifold of $\Lambda^m(E)$., URL (version: 2011-11-14): Grassmannian as a submanifold of $\Lambda^m(E)$.

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    $\begingroup$ This MO question might be of interest: mathoverflow.net/q/155989/11211 - In that regard, it can be shown that a 2-vector field $X=X_1\wedge X_2$ defines an integrable 2-dimensional distribution iff its Schouten-Nijenhuis bracket with itself vanishes everywhere. This also provides a characterization of when a 2-vector field is a Poisson tensor, check e.g. P. W. Michor, Remarks on the Schouten-Nijenhuis bracket, Rend. Circ. Mat. Palermo Suppl. 16 (1987), pp. 207-215, dml.cz/dmlcz/701423 . I don't know of a similar geometric interpretation of $p$-vector fields for $p>2$. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Dec 9 at 19:07
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    $\begingroup$ This does not generalize to $p$-vector fields with $p>2$. One can compute that if $X = X_1 \wedge X_2 \wedge X_3$ then $[X,X] = 0$ always. $\endgroup$
    – mlainz
    Commented 2 days ago
  • $\begingroup$ This follows from $[X\wedge Y,Z] = X\wedge [Y,Z] \pm [X,Z]\wedge Y $ (see [1]) $\endgroup$
    – mlainz
    Commented 2 days ago
  • $\begingroup$ What about $2p$-vector fields with $p>1$ then? $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented 2 days ago

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