Duflo isomorphism

A form of PBW theorem says that the symmetric algebra and the universal algebra of a Lie algebra $g$ are isomorphic as vector spaces (in fact coalgebras and $g$-modules). However this is not an isomorphism of algebras. One can compose the PBW isomorphism with an additional automorphism to get an isomorphism of vector spaces which restricts to isomorphism of **algebras** when restricted to the subalgebras of $g$-invariant functions.

The original proof by Duflo is rather case by case, using the structure theory of Lie algebras. Kontsevich in 1998 gave a new proof which generalizes to some geometric situations in deformation quantization.

- M. Duflo,
*Opérateurs différentiels bi-invariants sur un groupe de Lie*, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265–288 MR56:3188 numdam - Damien Calaque, Carlo A. Rossi,
*Lectures on Duflo isomorphisms in Lie algebra and complex geometry*, European Math. Soc. 2011 - M. Kontsevich,
*Deformation quantization of Poisson manifolds*, Lett. Math. Phys.**66**(2003), no. 3, 157–216;*Operads and motives in deformation quantization*, Lett. Math. Phys.**48**(1999), no. 1, 35–72.

Revised on December 31, 2012 02:43:21
by Zoran Škoda
(31.45.170.107)