Abstract
A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras \(\mathbb {H}\) (or a subalgebra thereof). An advantage of this formulation is that the hyperquaternion product is defined independently of the choice of the generators. The paper gives an explicit expression of the generators and develops a generalized multivector calculus. Due to the isomorphism \(\mathbb {H}\otimes \mathbb {H}\simeq m(4, \mathbb {R})\), hyperquaternions yield all real, complex and quaternion square matrices. A hyperconjugation is introduced which generalizes the concepts of transposition, adjunction and transpose quaternion conjugate. As applications, simple expressions of the unitary and unitary symplectic groups are obtained. Finally, the hyperquaternions are compared, in the context of physical applications, to another algebraic structure based on octonions which has been proposed recently.
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Acknowledgements
This work was performed in the framework of the LABEX PRIMES (ANR-11-LABX-0063) and LABEX CELYA (ANR-10-LABX-0060) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).
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This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.
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Girard, P.R., Clarysse, P., Pujol, R. et al. Hyperquaternions: A New Tool for Physics. Adv. Appl. Clifford Algebras 28, 68 (2018). https://doi.org/10.1007/s00006-018-0881-8
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DOI: https://doi.org/10.1007/s00006-018-0881-8