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The Groups of Two by Two Matrices in Double and Dual Numbers, and Associated Möbius Transformations

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Abstract

Möbius transformations have been studied over the field of complex numbers. In this paper, we investigate Möbius transformations over two rings which are not fields: the ring of double numbers and the ring of dual numbers. We give types of continuous one-parameter subgroups of \(\mathrm {GL}_2({}^2 \mathbb {R}),\) \(\mathrm {SL}_2({}^2 \mathbb {R}),\) \(\mathrm {GL}_2(\mathbb {D}),\) and \(\mathrm {SL}_2(\mathbb {D}).\)

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Acknowledgements

I am grateful to the Iraqi government for its support in a form of a scholarship. I would like to thank Prof. Vladimir Kisil for valuable discussions and important remarks. I am also grateful to the anonymous referees for many useful suggestions which were applied to improve the paper.

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Authors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, UK

    Khawlah A. Mustafa

  2. School of Science, University of Kirkuk, Kirkuk, Iraq

    Khawlah A. Mustafa

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  1. Khawlah A. Mustafa
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Correspondence to Khawlah A. Mustafa.

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Communicated by Rafał Abłamowicz.

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Mustafa, K.A. The Groups of Two by Two Matrices in Double and Dual Numbers, and Associated Möbius Transformations. Adv. Appl. Clifford Algebras 28, 92 (2018). https://doi.org/10.1007/s00006-018-0910-7

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