Carlo A. Rossi

Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

Duflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds. All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details. The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory.

Bimodules and branes in deformation quantization

We prove a version of Kontsevichs formality theorem for two subspaces (branes) of a vector space

Caldararu's conjecture and Tsygan's formality

X

Deformation quantization with generators and relations

. The result implies in particular that the Kontsevich deformation quantizations of

A gerbe for the elliptic gamma function

\mathrm{S}(X^*)

Hochschild (Co)homology for Lie Algebroids

and

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

\wedge(X)

Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants ?

associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhets recent paper on Koszul duality in deformation quantization.

Weighted projective spaces and minimal nilpotent orbits

In this paper we complete the proof of Caldararu�s conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured by Tsygan.) We also rely on the fact � recently proved by the first two authors � that Shoikhet�s quasi-isomorphism is compatible with cap products after twisting with a Maurer-Cartan element.

A Note on the Koszul Complex in Deformation Quantization

Abstract In this paper we prove a conjecture of B. Shoikhet which claims that two quantization procedures arising from Fourier dual constructions actually coincide.