Deformation quantization of projective schemes and differential operators

TL;DR

This paper develops a new method for deformation quantization of commutative projective schemes using noncommutative algebraic geometry, classifying schemes derived from differential chains in universal enveloping algebras.

Contribution

It introduces a classification of noncommutative projective schemes from differential chains and proposes a novel deformation quantization technique within Kapranov's framework.

Findings

Classified all noncommutative projective schemes from differential chains.

Developed a new deformation quantization method for projective schemes.

Provided a framework connecting noncommutative geometry with deformation theory.

Abstract

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 of the free nilpotent Lie algebra of index q generated by x_{0},...,x_{n}. The construction proposed allows us to provide a new method of deformation quantization of the commutative projective schemes within the considered framework.