On the Lie algebroid of a derived self-intersection

Let i:X↪Yi:X↪Y be a closed embedding of smooth algebraic varieties. Denote by N the normal bundle of X in Y  . The present paper contains two constructions of certain Lie structure on the shifted normal bundle N[−1]N[−1] encoding the information of the formal neighborhood of X in Y. We also present a few applications of these Lie theoretic constructions in understanding the algebraic geometry of embeddings.