This chapter is a kind of epilogue, in which we show how the machinery developed in the preceding chapters, some of it quite mathematical, can be used to provide the beginning of a kind of q-deformed geometry. It turns out that the underlying structure here is not so much a quantum group as one of the exotic braided groups which we have encountered in Chapter 9.4.2. Quantum groups still play a role as the quantum symmetry of such q-deformed spaces, but the spaces themselves tend to be braided ones. Thus we will need all the machinery developed so far in this book. Nevertheless, the problem of systematically q-deforming all the geometrical (and other) structures needed in physics is a deep and important one for physics, so we shall try to give here as self-contained and elementary a treatment as possible. It should be possible to come to this chapter directly after Chapter 4, using the intermediate chapters as reference for the mathematical underpinning when required. Also, we cover here only q-deformed or braided versions of ℝn, where the theory is fairly complete. Only when this is thoroughly understood could one reasonably expect to move on to define q-manifolds, etc. The further theory of braided geometry is deferred to a sequel to this book.

We have already covered one standard point of view on q-deforming geometrical structures in Chapter 4, namely as some kind of ‘quantisation’ of an algebra of functions.