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.

中文翻译：

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作为空间的李代数

在本文中，我们将李代数与 Costello 的派生几何版本联系起来。例如，我们证明了每个李代数——以及对 dg 李代数的自然推广——提供了一个（本质上是唯一的）$L_{\infty }$空间。更准确地说，我们从李代数的范畴构造了一个忠实的函子$L_{\infty }$空格。然后我们证明对于每个李代数$L$，从表示的范畴到同伦的都有一个完全忠实的函子$L$到相关的向量丛的类别$L_{\infty }$空间。事实上，这个函子发送的伴随复合体$L$的切丛$L_{\infty }$空间。最后，我们证明了 dg 李代数上的移位辛结构在相关联上产生了移位辛结构。$L_{\infty }$空间。