Lie-Rinehart algebras ≃ Acyclic Lie ∞-algebroids

Lie-Rinehart algebras ≃ Acyclic Lie ∞-algebroids

Camille Laurent-Gengoux (1) , Ruben Louis (1)

1 IECL - Institut Élie Cartan de Lorraine

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra 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 [26], [28]. 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 preserved by the anchor map of a Lie-Rinehart algebra , we associate a homotopy equivalence class of negatively graded Lie ∞-algebroids over complexes computing . Several explicit examples are given.

Mots clés: Lie-Rinehart algebras; Lie algebroids Algebras up to homotopy; Lie infinity algebras; Singular foliations; Algebraic geometry

Domaines: Mathématiques [math]

Camille Laurent-Gengoux, Ruben Louis. Lie-Rinehart algebras ≃ Acyclic Lie ∞-algebroids. Journal of Algebra, 2022, 594, pp.1-53. ⟨10.1016/j.jalgebra.2021.11.023⟩. ⟨hal-03462506⟩