Pullbacks and nontriviality of associated noncommutative vector bundles

TL;DR

This paper proves that pullbacks of associated noncommutative vector bundles remain noncommutative vector bundles, extends the theory to infinite-dimensional quantum group representations, and constructs noncommutative quaternionic projective spaces with non-trivial bundles.

Contribution

It establishes the nontriviality of associated noncommutative vector bundles under pullback, including infinite-dimensional cases, and introduces noncommutative quaternionic projective spaces with tautological bundles.

Findings

Pullback of associated noncommutative vector bundles preserves nontriviality.

Explicit matrix idempotents realize induced maps on K0-groups.

Noncommutative quaternionic projective spaces have stably non-trivial bundles.

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, 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 infinite dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, using noncommutative Milnor's join construction, we define 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.