Let H be the real algebra of quaternions. The notion of regular function of a
quaternionic variable recently presented by G. Gentili and D. C. Struppa
developed 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 quaternionic setting introduces new
phenomena. This paper studies regular quaternionic transformations. We first
find a quaternionic analog to the Casorati-Weierstrass theorem and prove that
all regular injective functions from H to itself are affine. In particular, the
group Aut(H) of biregular functions on H coincides with the group of regular
affine transformations. Inspired by the classical quaternionic linear
fractional transformations, we define the regular fractional transformations.
We then show that each regular injective function from the Alexandroff
compactification of H to itself is a regular fractional transformation.
Finally, we study regular Moebius transformations, which map the unit ball B
onto itself. All regular bijections from B to itself prove to be regular
Moebius transformations.Comment: 12 page
