Pulling back noncommutative associated vector bundles and constructing quantum quaternionic projective spaces

Abstract

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. On the level of \(K_{0}\)-groups of vector bundles, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinitely dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, we construct quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.