Local Lie algebra determines base manifold

Abstract

It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of differentiability: smooth, real-analytic, and holomorphic.

13 pages, minor corrections, to appear in "From Geometry to Quantum Mechanics, in Honor of Hideki Omori", Y.Maeda et al., eds., Progress in Math. 252, Birkhaeuser 2007