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arXiv · 1601.00021

Pullbacks and nontriviality of associated noncommutative vector bundles

Piotr M. Hajac·Tomasz Maszczyk
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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.

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Piotr M. Hajac, Tomasz Maszczyk. 2018-01-02. Pullbacks and nontriviality of associated noncommutative vector bundles. https://arxiv.org/abs/1601.00021

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