Pullbacks and nontriviality of associated noncommutative vector bundles

  • Piotr M. Hajac

    Polish Academy of Sciences, Warsaw, Poland
  • Tomasz Maszczyk

    University of Warsaw, Poland


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. At the level of K0-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.

Cite this article

Piotr M. Hajac, Tomasz Maszczyk, Pullbacks and nontriviality of associated noncommutative vector bundles. J. Noncommut. Geom. 12 (2018), no. 4, pp. 1341–1358

DOI 10.4171/JNCG/307