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Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry
Damien Calaque, ETH, Zurich, Switzerland, and Carlo A. Rossi, Max Planck Institute for Mathematics, Bonn, Germany
A publication of the European Mathematical Society.
cover
EMS Series of Lectures in Mathematics
2011; 114 pp; softcover
Volume: 14
ISBN-10: 3-03719-096-5
ISBN-13: 978-3-03719-096-8
List Price: US$32
Member Price: US$25.60
Order Code: EMSSERLEC/14
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The 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. Kontsevich later refined Duflo's result in the framework of deformation quantization and also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. This book, which arose from a series of lectures by Damien Calaque at ETH, derives these two isomorphisms from a Duflo-type result for \(Q\)-manifolds.

All notions mentioned above are introduced and explained in this book. The only prerequisites are 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 detail.

This book is well suited for graduate students in mathematics and mathematical physics as well as researchers working in Lie theory, algebraic geometry, and deformation theory.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in algebra.

Table of Contents

  • Lie algebra cohomology and the Duflo isomorphism
  • Hochschild cohomology and spectral sequences
  • Dolbeault cohomology and the Kontsevich isomorphism
  • Superspaces and Hochschild cohomology
  • The Duflo-Kontsevich isomorphism for \(Q\)-spaces
  • Configuration spaces and integral weights
  • The map \(\mathcal{U}_Q\) and its properties
  • The map \(\mathcal{H}_Q\) and the homotopy argument
  • The explicit form of \(\mathcal{U}_Q\)
  • Fedosov resolutions
  • Appendix: Deformation-theoretical interpretation of Hochschild cohomology
  • Bibliography
  • Index
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