Deformation quantization of projective schemes and differential operators

The paper is devoted to noncommutative projective schemes within Kapranov’s framework of noncommutative algebraic geometry. We classify all noncommutative projective schemes obtained from the differential chains in the universal enveloping algebra \(\mathcal {U}\left( \mathfrak {g}_{q}\left( \textbf{x} \right) \right) \) $\mathcal{U}\left({\mathfrak{g}}_q,\left(\mathbf{x}\right)\right)$ of the free nilpotent Lie algebra \(\mathfrak {g}_{q}\left( \textbf{x}\right) \) ${\mathfrak{g}}_q\left(\mathbf{x}\right)$ of index q generated by \(\mathbf {x=}\left( x_{0},\ldots ,x_{n}\right) \) $\mathbf{x}=\left(x_0,,,\dots ,,,x_n\right)$ . The construction proposed allows us to provide a new method of deformation quantization of the commutative projective schemes within the considered framework.