\( L_\infty \) algebras are natural generalizations of Lie algebras from a homotopy theoretical point of view. The original definition dates back to 1992 by J. Stasheff and T. Lada and the idea is to weaken the condition imposing that the Lie bracket satisfies the Jacobi identity. Instead one asks it to be satisfied only up to homotopy and we must have a whole family of compatible brackets. From a completely different point of view, a \( L_\infty \) algebra can be seen as a differential graded vector space and all the compatible brackets are encoded at a homological vector field. One can generalize the notion of Lie algebroid by following a similar path. T. Voronov formulated this idea by showing that one could identify \( L_\infty \) algebroids with differential graded manifolds. In this talk we intend to review the link between these \( L_\infty \) structures and their associated graded structures.
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