Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 33–44 | Cite as

Conformal Groups and Vahlen Matrices

  • Jacques Helmstetter


This article recalls some facts about the conformal group \({{\rm Conf}(V, Q)}\) of a quadratic space (V, Q); in particular, there is a surjective group morphism \({{\rm O}(V^{\dag}, Q^{\dag}) \rightarrow {\rm Conf}(V, Q)}\), where \({(V^{\dag}, Q^{\dag})}\) is the orthogonal sum of (V, Q) and a hyperbolic plane; its kernel is a group of order two. Then it explains how the elements of \({{\rm O}(V^{\dag}, Q^{\dag})}\) and \({{\rm Conf}(V, Q)}\) can be represented by Vahlen matrices. And finally, it recalls some properties of Vahlen matrices.


Conformal transformations Vahlen matrices 


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  1. 1.
    Anglès, P.: Construction de revêtements du groupe conforme d’un espace vectoriel muni d’une métrique de type (p, q). Ann. Inst. H. Poincaré, Sect. A, XXXIII, 33–51Google Scholar
  2. 2.
    Anglès, P.: Conformal groups in geometry and spin structures. In: Progress in Math. Physics, vol. 50. Birkhäuser, Basel (2008)Google Scholar
  3. 3.
    Cnops, J.: Vahlen matrices for non-definite metrics. In: Abłamowicz, R. et al. (eds.) Clifford Algebras with numeric and symbolic computations. pp. 155–164 Birkhäuser, Basel (1996)Google Scholar
  4. 4.
    Fillmore J.P., Springer A.: Möbius groups over general fields using Clifford algebras associated with spheres. Int. J. Theoret. Phys. 29, 225–246 (1990)CrossRefMATHGoogle Scholar
  5. 5.
    Haantjes J.: Conformal representations of a n-dimensinal Euclidian space with a non-definite fundamental form on itself. Nedel. Akad. Wetensch. Proc. 40, 700–705 (1937)MATHGoogle Scholar
  6. 6.
    Helmstetter, J.: Algèbres de Clifford et algèbres de Weyl. Cahiers mathématiques de Montpellier 25 (1982)Google Scholar
  7. 7.
    Helmstetter J.: Lipschitz monoids and Vahlen matrices. Adv. Appl. Clifford Algebras. 15, 83–122 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Helmstetter J.: Lipschitzian subspaces in Clifford algebras. J. Algebra. 328, 461–483 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Helmstetter J.: A survey of Lipschitz monoids. Adv. Appl. Clifford Algebras. 22, 665–688 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Helmstetter J.: Minimal algorithms for Lipschitz monoids and Vahlen monoids. J. Math. Res. 5(4), 39–51 (2013)CrossRefGoogle Scholar
  11. 11.
    Maks, J.: Clifford algebras and Möbius transformations. In: Micali, A. et al. (eds.) Clifford Algebras and their Applications in Mathematical Physics, pp. 57–63. Kluwer, Dordrecht (1992)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Institut FourierUniversité de Grenoble ISaint-Martin d’HèresFrance

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