A Borel-Weil Theorem for the Quantum Grassmannians

TL;DR

This paper extends the classical Borel-Weil theorem to quantum Grassmannians using noncommutative geometry, providing new insights into their structure and applications in quantum Kähler geometry.

Contribution

It develops a noncommutative Borel-Weil theorem for quantum Grassmannians within quantum principal bundles and complex structures, generalizing prior work on quantum projective spaces.

Findings

Noncommutative Borel-Weil theorem established for quantum Grassmannians

Provides a new differential geometric presentation of the quantum Grassmannian coordinate ring

Applications to noncommutative Kähler geometry of quantum Grassmannians

Abstract

We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the quantum Grassmannians. The result is formulated in the framework of quantum principal bundles and noncommutative complex structures, and generalises previous work of a number of authors on quantum projective space. As a direct consequence we get a novel noncommutative differential geometric presentation of the twisted Grassmannian coordinate ring studied in noncommutative projective geometry. A number of applications to the noncommutative K\"ahler geometry of the quantum Grassmannians are also given.