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Involutions of Complexified Quaternions and Split Quaternions

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Abstract

An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real quaternions were studied by Ell and Sangwine [15]. In this paper we present involutions and antiinvolutions of biquaternions (complexified quaternions) and split quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.

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Authors and Affiliations

  1. Department of Mathematics, Konya Necmettin Erbakan University, 42060, Konya, Turkey

    Murat Bekar

  2. Department of Mathematics, Ankara University, 06100, Ankara, Turkey

    Yusuf Yaylı

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  1. Murat Bekar
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  2. Yusuf Yaylı
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Correspondence to Murat Bekar.

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Bekar, M., Yaylı, Y. Involutions of Complexified Quaternions and Split Quaternions. Adv. Appl. Clifford Algebras 23, 283–299 (2013). https://doi.org/10.1007/s00006-012-0376-y

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  • DOI: https://doi.org/10.1007/s00006-012-0376-y

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