Noncommutative Complex Structures on Quantum Homogeneous Spaces

TL;DR

This paper introduces a new framework for noncommutative complex geometry on quantum homogeneous spaces, establishing foundational results and conditions for their existence, with quantum projective spaces as key examples.

Contribution

It develops a novel approach combining covariant differential calculi and categorical equivalence to characterize noncommutative complex structures on quantum spaces.

Findings

Established necessary and sufficient conditions for noncommutative complex structures.

Applied framework to quantum projective spaces with Heckenberger--Kolb calculus.

Produced foundational results for noncommutative complex geometry.

Abstract

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. Throughout, the family of quantum projective spaces, endowed with the Heckenberger--Kolb calculus, is taken as the motivating set of examples.