Geometry of Quantum Homogeneous Vector Bundles and Representation Theory of Quantum Groups I

TL;DR

This paper explores the structure of quantum homogeneous vector bundles, their sections, and applications to quantum group representation theory, including quantum Frobenius reciprocity and a generalized Borel-Weil theorem.

Contribution

It introduces a direct description of sections of quantum homogeneous vector bundles and establishes new results like quantum Frobenius reciprocity and a generalized Borel-Weil theorem.

Findings

Sections form projective modules over quantum homogeneous space algebras

Established quantum Frobenius reciprocity

Proved a generalized Borel-Weil theorem

Abstract

Quantum homogeneous vector bundles are introduced by a direct description of their sections in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies inherited from the quantum groups, and their sections furnish projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are investigated, and their applications to the representation theory of quantum groups are explored. In particular, quantum Frobenius reciprocity and a generalized Borel-Weil theorem are established.