Lie algebroids as $L_\infty$ spaces

TL;DR

This paper establishes a deep connection between Lie algebroids and $L_ty$ spaces within derived geometry, constructing functors that relate their categories and structures, including symplectic forms.

Contribution

It introduces a functorial correspondence from Lie algebroids to $L_ty$ spaces and relates their representations and symplectic structures.

Findings

Lie algebroids correspond to $L_ty$ spaces via a faithful functor.

Representations up to homotopy of Lie algebroids relate to vector bundles over $L_ty$ spaces.

Shifted-symplectic structures on dg Lie algebroids induce similar structures on $L_ty$ spaces.

Abstract

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_\infty$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_\infty$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_\infty$ space. Finally, we show that a shifted-symplectic structure on a dg Lie algebroid $L$ produces a shifted-symplectic structure on the associated $L_\infty$ space.