Symmetries of singular foliations and Lie infinity actions

Abstract: We investigate singular foliations that admit a universal Lie infinity-algebroid, as defined by Laurent-Gengoux, Lavau, and Strobl.

 We prove that any weak symmetry action of a Lie algebra g on a singular foliation (M,F)—Such an action can be understood as an action of g on the leaf space M/F—induces a unique (up to homotopy) Lie infinity-morphism from g to the differential graded Lie algebra (DGLA) of vector fields on the universal Lie infinity-algebroid of F. This type of Lie infinity-morphism, previously explored by R. Mehta and M. Zambon as a Lie infinity-algebra action, yields several geometric insights. As a specific instance, we give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space.