Braiding and exponentiating noncommutative vector fields

Abstract

The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to classical geometers. The definition of vector field used is a specialisation of the Cartan pair definition, and the paper relies on the idea of generalised braidings of 1-forms. The paper considers Kroneker deltas, interior products, Lie derivatives, Lie brackets, exponentiation of vector fields and parallel transport.