Thomas Weber: Noncommutative differential calculi via Hopf algebraic methods

Abstract

 We give a gentle introduction to noncommutative differential geometry and review how Hopf algebras and Hopf–Galois extensions provide viable tools in this differential graded setting. Our main goal will be the introduction of complete differential calculi, a particular class of differential calculi on quantum principal bundles, which induce basic, horizontal and vertical forms, together with a noncommutative version of the Atiyah sequence. Explicit examples, such as the quantum Hopf fibration and the noncommutative 2-torus, will be discussed in detail. Based on a collaboration with A. Del Donno and E. Latini.