Introduction

In this chapter we present noncommutative geometry (NCG) not as a ‘theory of everything’ but as a bridge between any future, perhaps combinatorial, theory of Quantum Gravity and the classical continuum geometry that has to be obtained in some limit. We consider for the present that NCG is simply a more general notion of geometry that by its noncommutative nature should be the correct setting for the phenomenology and testing of first next-to-classical Quantum Gravity corrections. Beyond that, the mathematical constraints of NCG may give us constraints on the structure of Quantum Gravity itself in so far as this has to emerge in a natural way from the true theory.

Also in this chapter we focus on the role of quantum groups or Hopf algebras as the most accessible tool of NCG, along the lines first introduced for Planck scale physics by the author in the 1980s. We provide a full introduction to our theory of ‘bicrossproduct quantum groups’, which is one of the two main classes of quantum group to come out of physics (the other class, the q-deformation quantum groups, came out of integrable systems rather than Quantum Gravity). The full machinery of noncommutative differential geometry such as gauge theory, bundles, quantum Riemannian manifolds, and spinors (at least in principle) has also been developed over the past two decades; these topics are deferred to a third article.