138 research outputs found
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
Lectures on Duflo isomorphisms in Lie algebra and complex geometry
International audienceDuflo 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
Universal KZB equations I: the elliptic case
We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB)
connection in genus 1. This is a flat connection over a principal bundle on the
moduli space of elliptic curves with marked points. It restricts to a flat
connection on configuration spaces of points on elliptic curves, which can be
used for proving the formality of the pure braid groups on genus 1 surfaces. We
study the monodromy of this connection and show that it gives rise to a
relation between the KZ associator and a generating series for iterated
integrals of Eisenstein forms. We show that the universal KZB connection
realizes as the usual KZB connection for simple Lie algebras, and that in the
sl_n case this realization factors through the Cherednik algebras. This leads
us to define a functor from the category of equivariant D-modules on sl_n to
that of modules over the Cherednik algebra, and to compute the character of
irreducible equivariant D-modules over sl_n which are supported on the
nilpotent cone.Comment: Correction of reference of Thm. 9.12 stating an equivalence of
categories between modules over the rational Cherednik algebra and its
spherical subalgebr
Hochschild cohomology and Atiyah classes
In this paper we prove that on a smooth algebraic variety the HKR-morphism
twisted by the square root of the Todd genus gives an isomorphism between the
sheaf of poly-vector fields and the sheaf of poly-differential operators, both
considered as derived Gerstenhaber algebras. In particular we obtain an
isomorphism between Hochschild cohomology and the cohomology of poly-vector
fields which is compatible with the Lie bracket and the cupproduct. The latter
compatibility is an unpublished result by Kontsevich. Our proof is set in the
framework of Lie algebroids and so applies without modification in much more
general settings as well.Comment: Reference to work of Cattaneo, Felder and Willwacher adde
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