Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces

TL;DR

This paper introduces noncommutative K"ahler structures on quantum homogeneous spaces, extending classical geometric concepts to the quantum setting and analyzing their properties and implications.

Contribution

It defines noncommutative K"ahler structures and demonstrates their fundamental properties, including the development of associated operators and cohomology analysis.

Findings

Quantum projective space with Heckenberger-Kolb calculus has cohomology groups of at least classical dimension.

Many classical K"ahler geometry results extend to the noncommutative setting under these structures.

The theory provides a framework for noncommutative Lefschetz, Hodge, K"ahler-Dirac, and Laplace operators.

Abstract

Building on the theory of noncommutative complex structures, the notion of a noncommutative K\"ahler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical K\"ahler geometry are shown to follow from the existence of such a structure, allowing for the definition of noncommutative Lefschetz, Hodge, K\"ahler-Dirac, and Laplace operators. Quantum projective space, endowed with its Heckenberger-Kolb calculus, is taken as the motivating example. The general theory is then used to show that the calculus has cohomology groups of at least classical dimension.