Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

TL;DR

This paper develops covariant q-deformed holomorphic structures for finitely-generated relative Hopf modules over irreducible quantum flag manifolds, generalizing previous work on quantum projective spaces and Podleś spheres.

Contribution

It constructs and proves the uniqueness of covariant holomorphic structures for all finitely-generated relative Hopf modules over these quantum manifolds.

Findings

Holomorphic structures reduce to classical sections in the limit

Unique covariant holomorphic structures for simple modules

Generalizes earlier results on quantum projective spaces

Abstract

We construct covariant $q$-deformed holomorphic structures for all finitely-generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger--Kolb calculi. In the classical limit these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podle\'s sphere, and subsequent work of Khalkhali and Moatadelro for general quantum projective space.