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Published in:
2020 | OriginalPaper | Chapter
Vector Bundles and Connections
Authors : Edwin J. Beggs, Shahn Majid
Published in: Quantum Riemannian Geometry
Publisher: Springer International Publishing
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This provides the notion of a connection on vector bundle on a differential algebra, where vector bundles are understood as finitely-generated projective modules. The q-sphere, fuzzy sphere and q-fuzzy-sphere provide natural examples with nontrivial line bundles. The chapter also connects up via the associated projector with Connes’ approach to noncommutative geometry based on cyclic cohomology and the Chern-Connes pairing between K-theory and cyclic cohomology. We then turn to the notion of a bimodule connection applicable when the vector bundle is a bimodule, which entails a generalised braiding for the implementation of the 2-sided Leibniz rule.