# Kontsevich, and Geometric, Quantization and the Podles sphere

There exist a large family of noncommutative spaces that arise from the quantum matrices. These algebraic objects $q$-deform the coordinate rings of certain varieties. For example, take quantum $SU(2)$, this is the algebra $< a,b,c,d >$ quotiented by the ideal generated by $$ab−qba, ~~ ac−qca, ~~ bc−cb, ~~ bd−qdb, ~~ cd−qdc, ~~ ad−da−(q−q^{−1})bc,$$ and the "q-det" relation $$ad−qbc−1$$ where $q$ is some complex number. Clearly, when $q=1$ we get back the coordinate ring of $SU(2)$. In the classical case $S^2 = SU(2)/U(1)$ (the famous Hopf fibration). This generalises to the q-case: the $U(1)$-action generalises to a $U(1)$-coaction with an invariant subalgebra that q-deforms the coordinate algebra of $S^2$ - the famous Podles sphere. There exist such q-matrix deformations of all flag manifolds.

Since all such manifolds are Kahler, we can also apply Kontsevich deformation to them to obtain a q-defomation. My question is: What is the relationship between these two approaches?

Alternatively, we can apply Kostant-Souriau geometric quantization to a flag manifold. How does alegbra relate to its q-matrix deformation?

As far as I understand, the flag manifolds with Kahler structures mentioned in the question are simply coadjoint orbits of compact Lie groups with the Kirillov-Kostant-Souriau bracket, so their quantizations will yield quotients of the usual enveloping algebra $U(g)$ and will not have to do with quantum groups. I suppose that the q-spaces discussed in the question are meant to be q-deformations of these.
Let $q$ be a variable instead of a complex number, write $q=1+t$ with $t$ also a variable, and complete with respect to the $t$-adic topology. This gives you a formal deformation of the coordinate algebra of the sphere in the sense of Gerstenhaber---this is not tautological: it follows from the fact that the deformation is flat.