Hyperquaternion Conformal Groups

Patrick Ralph Patrick Girard , Patrick Clarysse , Romaric Pujol , Robert Goutte , Philippe Delachartre (1)

1 Imagerie Ultrasonore

The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as O(p+1,q+1) isomorphic to the nD conformal group with n=p+q. The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory Response to Reviewers: All remarks have been taken into account.

Mots clés: Hyperquaternions; Conformal Groups; Canonical Decomposition; Quaternions

Domaines: Imagerie; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Patrick Ralph Patrick Girard, Patrick Clarysse, Romaric Pujol, Robert Goutte, Philippe Delachartre. Hyperquaternion Conformal Groups. Advances in Applied Clifford Algebras, 2021, 31 (3), ⟨10.1007/s00006-021-01159-y⟩. ⟨hal-03270493⟩