Lie Algebroids As L[infinity] Spaces

Abstract

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) L infinity space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of L infinity 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 infinity space. Indeed, this functor sends the adjoint complex of L to the tangent bundle of the L infinity space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated L infinity space.