Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid

Abstract

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra O and homotopy equivalence classes of negatively graded Lie ∞algebroids over their resolutions (=acyclic Lie ∞-algebroids). This extends to a purely algebraic setting the construction of the universal Q-manifold of a locally real analytic singular foliation of [25,27]. In particular, it makes sense for the universal Lie ∞-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. Also, to any ideal I ⊂ O preserved by the anchor map of a Lie-Rinehart algebra A, we associate a homotopy equivalence class of negatively graded Lie ∞-algebroids over complexes computing TorO(A, O/I). Several explicit examples are given. Contents 38 3.3. Universal Lie ∞-algebroids associated to an affine variety W 40 References 41

Modular class of Lie $\infty$-algebroids and adjoint representation

“…For instance, an unique up to homotopy "universal Lie ∞-algebroid" has also been associated to singular foliations in [15,34,32] and Lie-Rinehart algebras [33].…”