Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular a somewhat unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, whether or not the line bundles themselves are isomorphic....

The structure of filtered algebras of Grothendieck's differential operators on a smooth fat point in a curve and graded Poisson algebras of their principal symbols is explicitly determined. A related infinitesimal-birational duality realized by a Springer type resolution of singularities and the Fourier transformation is presented. This algebro-geometrical duality is quantized in appropriate sense and its quantum origin is explained.

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜=𝕂\left[{𝔸}^n\right]$ is said to be of Kostant type, if its centre $Z\left(𝒜\right)$ is freely generated by homogeneous polynomials $F_1,...,F_r$ such that they give Kostant’s regularity criterion on ${𝔸}^n(d_x{F}_i$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$). If the initial Poisson algebra is of Kostant type and $F_i$ satisfy a certain degree-equality, then the contraction is also of Kostant...