The quaternion formalism for Möbius groups in four or fewer dimensions

Abstract

The Möbius group of R N ∪ {∞} defines N-dimensional inversive geometry. This geometry can serve as an alternative to projective geometry in providing a common foundation for spherical Euclidean and hyperbolic geometry. Accordingly the Möbius group plays an important role in geometry and topology. The modern emphasis on low-dimensional topology makes it timely to discuss a useful quaternion formalism for the Möbius groups in four or fewer dimensions. The present account is self-contained. It begins with the representation of quaternions by 2 x 2 matrices of complex numbers. It discusses 2 x 2 matrices of quaternions and how a suitably normalized subgroup of these matrices, extended by a certain involution related to sense reversal, is 2-1 homomorphic to the Möbius group acting on R 4 ∪ {∞}. It provides details of this action and the relation of this action to various models of the classical geometries. In higher dimensions N ⩾ 5, the best description of the Möbius group is probably by means of ( N + 2) × ( N + 2) Lorentz matrices. In the lower dimensions covered by the quaternion formalism, this alternative Lorentz formalism is a source of interesting homomorphisms. A sampling of these homomorphisms is computed explicitly both for intrinsic interest and for an illustration of the ease with which one can handle the quaternion formalism.