On symmetries of singular foliations

Abstract

This paper shows that a weak symmetry action of a Lie algebra $\mathfrak{g}$ on a singular foliation $\mathcal{F}$ induces a unique up to homotopy Lie$\infty$-morphism from $\mathfrak{g}$ to the DGLA of vector fields on a universal Lie $\infty$-algebroid of $\mathcal{F}$. Such a Lie $\infty$-morphismwas studied by R. Mehta and M. Zambon as $L_{\infty }$-algebra action. We deduce from this general result several geometrical consequences. For instance, we give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space. Last, we introduce the notion of bi-submersion towers over a singular foliation and lift symmetries to those.