Classification of polarized deformation quantizations

We give a classification of polarized deformation quantizations on a symplectic manifold with a (complex) polarization. Also, we establish a formula which relates the characteristic class of a polarized deformation quantization to its Fedosov class and the Chern class of the polarization.Comment: 40 pages, AMS Late

Quantum G-manifolds

Let $G$ be a Lie group, \g its Lie algebra, and U_h(\g) the corresponding quantum group. We consider some examples of U_h(\g)-invariant one and two parameter quantizations on $G$-manifolds.Comment: Latex2e, 13 pp. A talk at the workshop "New homological and categorical methods in mathematical physics", Manchester, July 5-12, 200

Double quantization of \cp type orbits by generalized Verma modules

It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical" R-matrix. We realize quantization of this Poisson pencil on \cp type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $CP^n$) by means of q-deformed Verma modules.Comment: 21 pages, LaTeX, no figure

Quantum symmetric spaces

Let $G$ be a semisimple Lie group, ${\frak g}$ its Lie algebra. For any symmetric space $M$ over $G$ we construct a new (deformed) multiplication in the space $A$ of smooth functions on $M$. This multiplication is invariant under the action of the Drinfeld--Jimbo quantum group $U_h{\frak g}$ and is commutative with respect to an involutive operator $\tilde{S}: A\otimes A \to A\otimes A$. Such a multiplication is unique. Let $M$ be a k\"{a}hlerian symmetric space with the canonical Poisson structure. Then we construct a $U_h{\frak g}$-invariant multiplication in $A$ which depends on two parameters and is a quantization of that structure.Comment: 16 pp, LaTe