On symmetries of singular foliations

TL;DR

This paper explores how weak Lie algebra symmetries induce unique Lie infinity morphisms on singular foliations, leading to new geometric insights and concepts like bi-submersion towers.

Contribution

It establishes the existence of a unique Lie infinity morphism induced by weak symmetry actions on singular foliations, extending previous $L_$-algebra action studies.

Findings

A weak symmetry action induces a unique Lie infinity morphism.

An example shows some Lie algebra actions cannot extend to ambient spaces.

Introduces bi-submersion towers and lifts symmetries to them.

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.