nLab Duflo isomorphism

Idea

A form of PBW theorem says that the symmetric algebra and the universal enveloping algebra of a Lie algebra $g$ are isomorphic as vector spaces (in fact as $\mathfrak{g}$-modules, cf. eg. Dixmier 1974, §2.4.10).

However this is not an isomorphism associative algebras, but rather an isomorphism of filtered coalgebras. 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 in Duflo 1977 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 and which refines to a fact at the derived level (expressed in terms of an isomorphism at the level of Hochschild cohomology). At the level of derived geometry this expresses the property of certain derived exponential map constructed using Hochschild-Kostant-Rosenberg map precomposed by a square root of the Todd class in appropriate setup. This has prompted a series of articles by Markarian, Caldararu, Chen and others to explain the appearance of Todd class (or related Atiyah class). Refinements and analogues include Kashiwara-Vergne conjecture.

Related concepts

• PBW theorem

• Wheels theorem

References

Original article:

• Michel Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265–288 (numdam:ASENS_1977_4_10_2_265_0, MR56:3188)

• 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.

Review:

• Damien Calaque, Carlo A. Rossi, Lectures on Duflo isomorphisms in Lie algebra and complex geometry, European Math. Soc. (2011) [doi:10.4171/096, pdf]

See also:

• Wikipedia, Duflo isomorphism

• Adrien Brochier, A Duflo star-product for Poisson groups(arxiv/1604.08450)