Lie algebras of vector fields and generalized foliations (Q1325606)

The author proves a Pursell-Shanks type theorem in the smooth, real-analytic and holomorphic cases. Namely, the following main result is stated: Let \({\mathcal F}_i\) be a finitely generated non-singular foliation on a manifold \(M_i\) and \({\mathcal L}_i \subset{\mathfrak X}(M_i)\) a Lie subalgebra of foliation preserving vector fields, with \(i = 1,2\). If \(\Phi : {\mathcal L}_1 \to {\mathcal L}_2\) is a Lie algebra isomorphism, then \(\Phi = \varphi_*\) for a foliation preserving diffeomorphism \(\varphi : M_1 \to M_2\).