Line bundles and the Thom construction in noncommutative geometry

TL;DR

This paper extends classical line bundle and Thom construction concepts to noncommutative geometry using Morita contexts, resulting in new algebraic structures and insights into covariant derivatives and Chern classes.

Contribution

It introduces a noncommutative analogue of line bundles and Thom constructions via Morita contexts and explores associated algebraic and geometric structures.

Findings

Construction of Z and N graded algebras from Morita contexts.

Establishment of star structures and positivity conditions on these algebras.

Analysis of covariant derivatives and Chern classes in the noncommutative setting.

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.