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 N Q-manifold over F. We deduce from this general result several geometric consequences. For instance, we give an example of a weak action of Lie algebra, i.e., an action on the leaf space which cannot be turned into an action on the ambient space. Last, we introduce the notion of tower of bi-submersions over a singular foliations and lift symmetries to those.