Regular quaternionic functions and conformal mappings

Abstract

In this paper we study the action of conformal mappings of the quaternionic space on a class of regular functions of one quaternionic variable. We consider functions in the kernel of the Cauchy-Riemann operator

a variant of the Cauchy–Fueter operator. This choice is motivated by the strict relation existing between this type of regularity and holomorphicity w.r.t. the whole class of complex structures on â„. For every imaginary unit p ∈ 𕊲, let J_p be the corresponding complex structure on â„. Let Hol_p (ð›º, â„) be the space of holomorphic maps from (ð›º, J_p) to (â„, L_p), where L_p is defined by left multiplication by p. Every element of Hol_p(ð›º, â„) is regular, but there exist regular functions that are not holomorphic for any p. These properties can be recognized by computing the energy quadric of a function. We show that the energy quadric is invariant w.r.t. three–dimensional rotations of â„. As an application, we obtain that every rotation of the space â„ can be generated by biregular rotations, invertible regular functions with regular inverse. Moreover, we prove that the energy quadric of a regular function can always be diagonalized by means of a three–dimensional rotation.