Universal higher Lie algebras of singular spaces and their symmetries

TL;DR

This paper establishes a categorical equivalence between Lie-Rinehart algebras and homotopy classes of negatively graded acyclic Lie -algebroids, extending universal constructions and analyzing symmetries of singular foliations.

Contribution

It introduces a universal Lie -algebroid framework for singular foliations and explores how symmetries induce -morphisms, with applications to geometric structures.

Findings

Equivalence between Lie-Rinehart algebras and homotopy classes of Lie -algebroids.

Existence of a universal -algebroid for every singular foliation.

Weak symmetry actions induce -morphisms into the DGLA of vector fields.

Abstract

The results of this manuscript is the collection of my articles that I published during my PhD thesis. We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra $\mathcal O$ and homotopy equivalence classes of negatively graded acyclic Lie $\infty$-algebroids. Therefore, this result makes sense of the universal Lie $\infty$-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. This extends to a purely algebraic setting the construction of the universal $Q$-manifold of a locally real analytic singular foliation of Lavau-C.L.-Strobl. Then we apply these results to study symmetries of singular foliations through universal Lie $\infty$-algebroids. More precisely, we prove that a weak symmetry action of a Lie algebra $\mathfrak{g}$ on a singular foliation $\mathfrak F$ (which is morally an action of $\mathfrak g$ on the leaf space $M/\mathfrak F$) induces a unique up to homotopy Lie $\infty$-morphism from $\mathfrak{g}$ to the Differential Graded Lie Algebra (DGLA) of vector fields on a universal Lie $\infty$-algebroid of $\mathfrak F$ (such morphim is known under the name "$L_\infty$-algebra action" in Mehta-Zambon. 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 present the notion of bi-submersion towers over a singular foliation and lift symmetries to those. \end{enumerate} \end{itemize}