Hyperquaternion conformal groups (Q2047499)

The conformal group has many representations in terms of matrices, see for example [\textit{J. Helmstetter}, Adv. Appl. Clifford Algebr. 27, No. 1, 33--44 (2017; Zbl 1381.11027); \textit{J. Lasenby} et al., Adv. Appl. Clifford Algebr. 29, No. 5, Paper No. 102, 9 p. (2019; Zbl 1430.53041); \textit{I. R. Porteous}, Clifford algebras and the classical groups. Cambridge: Cambridge University Press (1995; Zbl 0855.15019); \textit{J. Vince}, Geometric algebra for computer graphics. London: Springer (2008; Zbl 1155.68084)]. Moreover, the conformal groups in \(n\)-dimensional Minkowski space with \(n=p+q\) can be reprented by Clifford algebras \(\mathcal{C}l_{n+2}\left( p+1,q+1\right) \), see for example [\textit{P. Anglès}, Conformal groups in geometry and spin structures. Basel: Birkhäuser (2008; Zbl 1136.53001); \textit{J. Haantjes}, Proc. Akad. Wet. Amsterdam 40, 700--705 (1937; Zbl 0017.42201); \textit{J. Helmstetter}, Adv. Appl. Clifford Algebr. 27, No. 1, 33--44 (2017; Zbl 1381.11027); \textit{H. A. Kastrup}, Ann. Phys. (8) 17, No. 9--10, 631--690 (2008; Zbl 1152.81300); \textit{J. Lasenby} et al., Adv. Appl. Clifford Algebr. 29, No. 5, Paper No. 102, 9 p. (2019; Zbl 1430.53041)]. The authors develop an interesting new representation in term of hyperquaternions. The representation yields simple expressions of the generators, independently of matrices or operators. Ideas based on hyperquaternions were used already by \textit{W. K. Clifford} [Am. J. Math. 1, 350--358 (1878; JFM 10.0297.02)] and [\textit{R. Lipschitz}, C. R. Acad. Sci., Paris 91, 619--621 (1881; JFM 12.0303.01)]. The name hyperquaternions is due to \textit{C. L. E. Moore} [American Acad. Proc. 53, 651--694 (1918; JFM 47.0974.02)]. Hyperquaternions are defined to be a tensor products of quaternion algebras. As an application, the authors present the 4D relativistic conformal group model together with a worked example. They also compare the formalism to the operator representation. Potential applications are in conformal geometry, computer graphics and conformal field theory.