Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

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 Podleś sphere, and subsequent work of Khalkhali and Moatadelro for general quantum projective space.

arXiv comment22 pages, no figures. This article draws heavily from arXiv:1912.08802

References in corpus (5)

• Categories of holomorphic vector bundles on noncommutative two-tori
• Dirac Operators on Quantum Projective Spaces
• Holomorphic structures on the quantum projective line
• Noncommutative complex geometry of the quantum projective space
• Kähler structures on quantum irreducible flag manifolds

Cited by in corpus (5)

• Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
• Nichols Algebras and Quantum Principal Bundles
• Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds
• Quantum Riemannian geometry of quantum projective spaces
• Fubini-Study metrics and Levi-Civita connections on quantum projective spaces