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Advances in Applied Clifford Algebras

, Volume 27, Issue 2, pp 1161–1173 | Cite as

Möbius Transformation for Left-Derivative Quaternion Holomorphic Functions

  • Sergio Giardino
Article

Abstract

Holomorphic quaternion functions only admit affine functions; thus, the Möbius transformation for these functions, which we call quaternionic holomorphic transformation (QHT), only comprises similarity transformations. We determine a general group \({\mathsf{X}}\) which has the group \({\mathsf{G}}\) of QHT as a particular case. Furthermore, we observe that the Möbius group and the Heisenberg group may be obtained by making \({\mathsf{X}}\) more symmetric. We provide matrix representations for the group \({\mathsf{X}}\) and for its algebra \({\mathfrak{x}}\). The Lie algebra is neither simple nor semi-simple, and so it is not classified among the classical Lie algebras. We prove that the group \({\mathsf{G}}\) comprises \({\mathsf{SU}(2,\mathbb{C})}\) rotations, dilations and translations. The only fixed point of the QHT is located at infinity, and the QHT does not admit a cross-ratio. Physical applications are addressed at the conclusion.

Keywords

Heisenberg Group Similarity Transformation Hyperbolic Geometry Cross Ratio Heisenberg Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Departamento de Física & Centro de Matemática e AplicaçõesUniversidade da Beira InteriorCovilhãPortugal

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