Universal higher Lie algebras of singular spaces and their symmetries

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 $\mathrm{\infty }$-algebroids. Therefore, this result makes sense of the universal Lie $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $M\mathrm{/}\mathfrak{F}$) induces a unique up to homotopy Lie $\mathrm{\infty }$-morphism from $\mathfrak{g}$ to the Differential Graded Lie Algebra (DGLA) of vector fields on a universal Lie $\mathrm{\infty }$-algebroid of $\mathfrak{F}$ (such morphim is known under the name “$L_{\mathrm{\infty }}$-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}