Pullbacks and nontriviality of associated noncommutative vector bundles
| Autor: | Tomasz Maszczyk, Piotr M. Hajac |
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| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Quantum group Mathematics::Operator Algebras Vector bundle K-Theory and Homology (math.KT) Join (topology) Pullback (differential geometry) Noncommutative geometry Matrix (mathematics) Mathematics::Algebraic Geometry Mathematics::K-Theory and Homology Mathematics - K-Theory and Homology FOS: Mathematics Equivariant map Geometry and Topology Compact quantum group Mathematics::Symplectic Geometry Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1601.00021 |
| Popis: | 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. Comment: 18 pages, to appear in Journal of Noncommutative Geometry |
| Databáze: | OpenAIRE |
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