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arXiv · 1012.1475

Line bundles and the Thom construction in noncommutative geometry

E. J. Beggs·T. Brzezinski
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Abstract

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we can construct Z and N graded algebras, the Z graded algebra being a Hopf-Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the N graded algebra. In this case, we can carry out the associated circle bundle and Thom constructions. Starting with a C* algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C* algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.

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E. J. Beggs, T. Brzezinski. 2010-12-07. Line bundles and the Thom construction in noncommutative geometry. https://arxiv.org/abs/1012.1475

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