A trace formula for the quantization of coadjoint orbits

The main goal of this paper is to compute the characteristic class of the Alekseev-Lachowska *-product on coadjoint orbits. We deduce an analogue of the Weyl dimension formula in the context of deformation quantization

A PBW theorem for inclusions of (sheaves of) Lie algebroids

Inspired by the recent work of Chen-Sti\'enon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincar\'e-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforward generalizations of the ones previously developped by Caldararu, Tu and the author for Lie algebra inclusions.Comment: Final version - several changes - 19 page

Derived symplectic geometry

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks. Derived stacks are the main players in derived geometry, the purpose of which is to deal with singular spaces, while symplectic structures are an essential ingredient of the geometric formalism of classical mechanics and classical field theory. In addition to providing an overview of a relatively young field of research, we provide a case study on Casson's invariant.Comment: Review paper. This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physic

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.Comment: final versio