Equivariant vector bundles over quantum spheres

Abstract

We quantize $SO(2n+1)$-equivariant 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 linear maps between pseudo-parabolic modules and as induced modules of the orthogonal quantum group. Based on this alternative, we study representations of a quantum symmetric pair related to ${\mathbb{S}}_q^{2n}$ and prove their complete reducibility.