$L_{\infty}$-actions of Lie algebroids

TL;DR

This paper explores homotopy actions of Lie algebroids on graded manifolds via $L_{}$-algebra morphisms, establishing a bijective correspondence with a construction involving homological vector fields, and provides explicit examples including extensions and Courant algebroids.

Contribution

It introduces a bijective construction linking homotopy actions of Lie algebroids to homological vector fields, unifying various geometric structures.

Findings

Constructs a homological vector field on the semi-direct product.

Establishes a bijection between homotopy actions and vector fields.

Provides explicit examples involving Lie algebroids and Courant algebroids.

Abstract

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many explicit examples, involving Lie algebroids (including extensions, representations up to homotopy and their cocycles) as well as transitive Courant algebroids.