Regular Moebius transformations of the space of quaternions

Abstract

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.

12 pages

References in corpus (4)

• A new functional calculus for non-commuting operators
• Poles of regular quaternionic functions
• The zero sets of slice regular functions and the open mapping theorem
• Möbius transformations and the Poincaré distance in the quaternionic setting

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• Twistor transforms of quaternionic functions and orthogonal complex structures
• The Schwarz-Pick lemma for slice regular functions
• Singularities of slice regular functions
• Regular vs. classical Möbius transformations of the quaternionic unit ball
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• The Bohr Theorem for slice regular functions
• Landau's theorem for slice regular functions on the quaternionic unit ball
• S-regular functions which preserve a complex slice
• Slice starlike functions over quaternions
• Möbius transformation for left-derivative quaternion holomorphic functions
• Some notions of subharmonicity over the quaternions
• Multipoint Schwarz-Pick Lemma for the quaternionic case
• On geometric aspects of quaternionic and octonionic slice regular functions
• Slice regular Malmquist-Takenaka system in the quaternionic Hardy spaces
• Bloch, Besov and Dirichlet spaces of slice hyperholomorphic functions