We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The
deformation is based on a general twist. This leads to a differential geometry on a …

There are many ways to extend the ideas of classical differential geometry to a
noncommutative world. Our view is that there is no clear answer as to which of these is …

We consider noncommutative geometries obtained from a triangular Drinfeld twist and
review the formulation of noncommutative gravity. A detailed study of the abelian twist …

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-
action, as well as their morphisms and connections. Bimodules naturally arise when …

We construct noncommutative principal fibrations S θ 7→ S θ 4 which are deformations of
the classical SU (2) Hopf fibration over the four sphere. We realize the noncommutative …

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld
twist (2-cocycle). If the twist is associated with the structure group then we have a …

We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a
noncommutative four sphere S_ θ^ 4 and of topological charge equal to 1. These instantons …

We give an introductory survey to the use of Hopf algebras in several problems of
noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded …

With applications in quantum field theory, general relativity and elementary particle physics,
this three-volume work studies the invariance of differential operators under Lie algebras …

We present an introduction to the use of noncommutative geometry for gauge theories by
constructing instantons for a class of four-dimensional toric noncommutative manifolds …