# Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

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*Damien Calaque; Carlo A. Rossi*

A publication of the European Mathematical Society

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.