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».
](https://dmitripavlov.org/scans/tulczyjew-the-graded-lie-algebra-of-multivector-fields-and-the-generalized-lie-derivative-of-forms.pdf)

[2] Denis Serre (https://mathoverflow.net/users/8799/denis-serre), Grassmannian as a submanifold of $\Lambda^m(E)$., URL (version: 2011-11-14): https://mathoverflow.net/q/80888