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.
For any Lie-Rinehart algebra , B(atalin)-V(ilkovisky) algebra structures on the exterior -algebra correspond bijectively to right -module structures on ; likewise, generators for the Gerstenhaber algebra correspond bijectively to right -connections on . When is projective as an -module, given a B-V algebra structure on , the homology of the B-V algebra coincides with the homology of with coefficients in with reference to the right -module structure determined by . When...
We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.
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 is said to be of Kostant type, if its centre is freely generated by homogeneous polynomials such that they give Kostant’s regularity criterion on are linear independent if and only if the Poisson tensor has the maximal rank at ). If the initial Poisson algebra is of Kostant type and satisfy a certain degree-equality, then the contraction is also of Kostant...
It is proved that there exists a bijection between the primitive ideals of the algebra of regular functions on quantum m × n-matrices and the symplectic leaves of associated Poisson structure.
Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra . We introduce an affine, reduced, irreducible, normal quiver variety which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian...
The Kähler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kähler structure which reflects the geometry of the group. For the group SL(n,ℂ), we interpret the resulting singular Poisson-Kähler geometry of the quotient in terms of complex discriminant varieties and variants thereof.
The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum matrices,...
We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra is of the form with U...