Equivariant vector bundles over quantum spheres

TL;DR

This paper develops a quantum analog of vector bundles over even complex spheres, representing them as modules over quantized coordinate rings and analyzing their representations within quantum groups.

Contribution

It introduces two novel methods for quantizing vector bundles over quantum spheres and studies the reducibility of related quantum symmetric pair representations.

Findings

Realized vector bundles as modules over quantized coordinate rings

Established two different constructions for these bundles

Proved complete reducibility of certain quantum symmetric pair representations

Abstract

We quantize homogeneous vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$ and prove their complete reducibility.