On the Lie algebroid of a derived self-intersection

Let $i:X\hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. Denote by $N$ the normal bundle of $X$ in $Y$. We describe the construction of two Lie-type structures on the shifted bundle $N[-1]$ which encode the information of the formal neighborhood of $X$ inside $Y$. We also present applications of classical Lie theoretic constructions (universal enveloping algebra, Chevalley-Eilenberg complex) to the understanding of the geometry of embeddings.