A Borel-Weil theorem for the irreducible quantum flag manifolds

• Published in: arXiv.org
• Main Authors: Carotenuto, Alessandro, Fredy Díaz García, Réamonn Ó Buachalla
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 06.12.2021
• Subjects: Differential calculus; Differential geometry; Flags; Theorems

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 \(\mathcal{O}_q(G/L^{\mathrm{s}}_S)\).