1 results
LIE ALGEBROIDS AS
$L_{\infty }$ SPACES
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 2 / March 2020
- Published online by Cambridge University Press:
- 13 February 2018, pp. 487-535
- Print publication:
- March 2020
-
- Article
- Export citation
-
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)
$L_{\infty }$ 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 produces a shifted symplectic structure on the associated 
$L_{\infty }$ space.
