KONTSEVICH'S UNIVERSAL FORMULA FOR DEFORMATION QUANTIZATION AND THE CAMPBELL–BAKER–HAUSDORFF FORMULA
- 1 June 2000
- journal article
- Published by World Scientific Pub Co Pte Ltd in International Journal of Mathematics
- Vol. 11 (4), 523-551
- https://doi.org/10.1142/s0129167x0000026x
Abstract
We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on ℝd proposed by Maxim Kontsevich to the Campbell–Baker–Hausdorff (CBH) formula. We show that Kontsevich's formula can be viewed as exp (P) where P is a bi-differential operator that is a deformation of the given Poisson structure. For linear Poisson structures (duals of Lie algebras) his product takes the form exp (C+L) where exp (C) is the ⋆-product given by the universal enveloping algebra via symmetrization, essentially the CBH formula. This is established by showing that the two products are identical on duals of nilpotent Lie algebras where the operator L vanishes. This completely determines part of Kontsevich's formula and leads to a new scheme for computing the CBH formula. The main tool is a graphical analysis for bi-differential operators and the computation of certain iterated integrals that yield the Bernoulli numbers.Keywords
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This publication has 5 references indexed in Scilit:
- A simple geometrical construction of deformation quantizationJournal of Differential Geometry, 1994
- The life and work of Kuo-Tsai ChenIllinois Journal of Mathematics, 1990
- Deformation theory and quantization. I. Deformations of symplectic structuresAnnals of Physics, 1978
- Integration of Paths, Geometric Invariants and a Generalized Baker- Hausdorff FormulaAnnals of Mathematics, 1957
- The formal power series for logexeyDuke Mathematical Journal, 1956