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Damien Calaque (ETH Zurich, Switzerland)
Carlo A. Rossi (Max Planck Institute for Mathematics, Bonn, Germany)
Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry
ISBN print 978-3-03719-096-8, ISBN online 978-3-03719-596-3DOI 10.4171/096
June 2011, 114 pages, softcover, 17 x 24 cm.
24.00 Euro
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.
Keywords: Lie algebra, Hochschild cohomology, complex manifolds, deformation theory, Kontsevich’s graphical calculus, Atiyah class, Duflo isomorphism, Todd class
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