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.
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Partially supported by GNSAGA of the INdAM, by PRIN “Proprietà geometriche delle varietà reali e complesse” and by PRIN “Geometria Differenziale e Analisi Globale” of the MIUR.
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Stoppato, C. Regular Moebius transformations of the space of quaternions. Ann Glob Anal Geom 39, 387–401 (2011). https://doi.org/10.1007/s10455-010-9238-9
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DOI: https://doi.org/10.1007/s10455-010-9238-9