On symmetries of singular foliations

Abstract

This paper shows that a weak symmetry action of a Lie algebra g on a singular foliation F induces a unique up to homotopy Lie ∞-morphism from g to the DGLA of vector fields on a universal Lie ∞-algebroid of F. Such a morphism is known as L∞-algebra action in [1]. 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.