On the classification of quaternionic Möbius transformations

Abstract

Abstract. It is well known that the dynamics and conjugacy class of a complex Möbius transformation can be determined from a simple rational function of the coefficients of the transformation. We study the group of quaternionic Möbius transformations and identify simple rational functions of the coefficients of the transformations that determine dynamics and conjugacy.

On the Classification of Four-Dimensional Möbius Transformations

“…For g ∈ U(1, 1; H), we have The above observations indicates our classification restricted in U(1, 1; H) is equivalent to the main result in [3].…”

Moebius transformations and the Poincare distance in the quaternionic setting

“…The remaining case in which q 1 , q 2 are R-linearly dependent is straightforward. and it can be written equivalently as (see, e.g., [6])…”

“…In terms of Sp(1, 1), we can rephrase and complete a result of [6] as follows: Sp(1, 1). Moreover, the map…”

“…The structure of the group M of Möbius transformations of ∆ H is studied, for example, in [6], in terms of the (classical) group Sp(1, 1). If H = 1 0 0 −1 , the group Sp(1, 1) is defined (see, e.g., [11] …”

Algebraic Characterization of Isometries of the Hyperbolic 4-Space

“…However, the approaches used in 7, 8 are independent of each other, and hence, the respective characterizations are also of different flavors. Algebraic characterizations of isometries of H 4 are also known due to the work of several authors; most notably among them is the work ofCao et al 6 In all the above works, the authors obtained their characterizations using conjugacy invariants of the isometries. Another approach which has been used recently to characterize the isometries algebraically is in terms of the centralizers, up to conjugacy.…”