Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.

The main result is a Pursell-Shanks type theorem describing isomorphism of the Lie algebras of vector fields preserving generalized foliations. The result includes as well smooth as real-analytic and holomorphic cases.

For any Lie-Rinehart algebra $(A,L)$, B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$-algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-module structures on $A$; likewise, generators for the Gerstenhaber algebra ${\Lambda }_{A}L$ correspond bijectively to right $(A,L)$-connections on $A$. When $L$ is projective as an $A$-module, given a B-V algebra structure $\partial$ on ${\Lambda }_{A}L$, the homology of the B-V algebra $({\Lambda }_{A}L,\partial )$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A,L)$-module structure determined by $\partial$. When...

A complex hypersurface $D$ in ${ℂ}^n$ is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for $n$ at most $4$.By analogy with Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for $D$ if the complex of global logarithmic differential forms computes the complex cohomology of ${ℂ}^n\setminus D$. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the...