Géométrie des algèbres de Lie locales de Kirillov. (French) Zbl 0562.53029

The paper studies several geometric notions related to a local Lie algebra N, i.e. a Lie algebra structure on the space \(N=C^{\infty}(W)\) of \(C^{\infty}\) functions on a manifold W, with a Lie bracket which does not increase supports. Such a bracket is given by a ”Jacobi structure” on W, a generalization of Poisson structures. One can associate to every \(u\in N\) a (generalized) Hamiltonian vector field \(X_ u\) on W. The fields \(X_ u\) define a (generalized) foliation on W; each leaf carries an induced local Lie algebra structure, which is transitive. Three cases occur: (1) odd-dimensional leaves are Pfaffian manifolds; even-dimensional leaves are ”conformally symplectic”, either (2) globally (their structure is then conformally equivalent to a classical symplectic structure), or (3) only locally.
Cases (1) and (2) are known; in particular, the structure (2) admits nontrivial deformations, whereas (1) has none. A detailed study of case (3) is given here; roughly speaking, this case is intermediate between (1) and (2).
In a neighbourhood of each point of W there exists a special system of coordinates called ”distinguished chart”; it can be constructed by relating first Jacobi structures on W and Poisson structures on \(W\times {\mathbb{R}}\), then applying a result by Weinstein and Marle on Poisson manifolds. The authors determine also all local derivations of the Lie algebra N, all derivations of the Lie algebra L of all infinitesimal conformal automorphisms of the Jacobi structure, and all derivations of the ideal \(L^*\) of L consisting of Hamiltonian vector fields. Finally most results are extended, with no essential change, to local Lie algebras of \(C^{\infty}\) sections of a line bundle on W.
Reviewer: F.Rouviere


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B65 Infinite-dimensional Lie (super)algebras