Regular Moebius transformations of the space of quaternions

Abstract

Quaternionic Moebius transformations have been investigated for more than 100 years and their properties have been characterized in detail. In recent years G. Gentili and D. C. Struppa introduced a new notion of regular function of a quaternionic variable, which is developing into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the non-commutative setting introduces new phenomena. Unfortunately, the (classical) quaternionic Moebius transformations are not regular. However, in this paper we are able to construct a different class of Moebius-type transformations that are indeed regular. This construction requires several steps: we first find an analog to the Casorati-Weierstrass theorem and use it to prove that the group $${Aut(\mathbb{H})}$$ of biregular functions on $${\mathbb{H}}$$ coincides with the group of regular affine transformations. We then show that each regular injective function from $${\widehat{\mathbb{H}} = \mathbb{H}\cup \{\infty\}}$$ to itself belongs to a special class of transformations, called regular fractional transformations. Among these, we focus on the ones which map the unit ball $${\mathbb{B} = \{q \in \mathbb{H} : |q| < 1 \}}$$ onto itself, called regular Moebius transformations. We study their basic properties and we are able to characterize them as the only regular bijections from $${\mathbb{B}}$$ to itself.