A Borel-Weil theorem for the irreducible quantum flag manifolds

TL;DR

This paper generalizes the classical Borel-Weil theorem to irreducible quantum flag manifolds, providing a new noncommutative geometric framework for their coordinate rings using quantum principal bundles.

Contribution

It extends the Borel-Weil theorem to quantum flag manifolds and introduces a differential geometric approach via quantum principal bundles and principal pairs.

Findings

Established a noncommutative Borel-Weil theorem for quantum flag manifolds.

Provided a new presentation of quantum coordinate rings using differential geometry.

Utilized quantum principal bundles and the Heckenberger-Kolb calculus in the proof.

Abstract

We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$, generalising previous work of a number of authors (including the first and third authors of this paper) on the quantum Grassmannians $\mathcal{O}_q(\mathrm{Gr}_{n,m})$. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings $S_q[G/L_S]$ of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces $\mathcal{O}_q(G/L^{\mathrm{s}}_S)$.