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1996 | 37 | 1 | 137-150
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A tutorial on conformal groups

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Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to $ℝ^{p,q}$, the real vector space $ℝ^{p+q}$, furnished with the quadratic form $x^{(2)} = x · x = -x_{1}^{2} - x_{2}^{2} - ... - x_{p}^{2} + x_{p+1}^{2} + ... + x_{p+q}^{2}$, and especially with a description of this group that involves Clifford algebras.
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  • Department of Pure Mathematics, The University of Liverpool, PO Box 147, Liverpool L69 3BX, U.K.
  • [1] L. Ahlfors, Möbius transformations and Clifford numbers, I. Chavel, H.M. Parkas (eds.). Differential Geometry and Complex Analysis. Dedicated to H.E. Rauch, Springer-Verlag, Berlin, (1985), 65-73.
  • [2] É. Cartan, Sur l'espace anallagmatique réel à $n$ dimensions, Ann. Polon. Math. 20 (1947), 266-278.
  • [3] É. Cartan, Deux théorèmes de géométrie anallagmatique réelle à $n$ dimensions, Ann. Mat. Pura Appl. (4)28 (1949), 1-12.
  • [4] W.K. Clifford, (1876) On the Classification of Geometric Algebras, published as Paper XLIII in Mathematical papers. Edited by R. Tucker, Macmillan, London (1882).
  • [5] J. Cnops, Hurwitz Pairs and Applications of Möbius Transformations. Thesis, Universiteit Gent, 1994.
  • [6] J. Fillmore, and A. Springer, Möbius groups over general fields using Clifford algebras associated with spheres, Int. J. Theo. Phys. 29 (1990), 225-246
  • [7] J. Haantjes, Conformal representations of an $n-$dimensional euclidean space with a non-definite fundamental form on itself, Proc. Ned. Akad. Wet. (Math.) 40 (1937), 700-705.
  • [8] R. Hermann, Appendix Kleinian mathematics from an advanced standpoint, A: Conformal and non-Euclidean geometry in $R^3$ from the Kleinian viewpoint, bound with Klein F. Developments of Mathematics in the 19th century. Translated by M. Ackerman, Math. Sci. Press, Brookline, Mass. USA, 1979, 367-376.
  • [9] N.H. Kuiper, On conformally-flat spaces in the large, Ann. Math. 50 (1949), 916-924.
  • [10] J. Liouville, Appendix to Monge, G. Application de l'analyse à la geométrie, 5 éd. par Liouville, 1850.
  • [11] J. Maks, Modulo $(1,1)$ periodicity of Clifford algebras and the generalized (anti-)Möbius transformations. PhD Thesis, Technische Universiteit Delft., 1989.
  • [12] J. Maks, Clifford algebras and Möbius transformations, in A. Micali et al. (eds.) Clifford Algebras and their Applications in Mathematical Physics, Kluwer Acad. Publ., Dordrecht 1992.
  • [13] J.-B.-M.-C. Meusnier, Mémoire sur la courbure des surfaces, Mémoire Div. Sav., 10 (1785), 477-510.
  • [14] I. R. Porteous, Topological Geometry, 2nd Edition, with additional material on Triality, Cambridge University Press, 1981. (The part of this book concerned with Clifford algebras forms part of a new edition entitled Clifford Algebras and the Classical Groups published in 1995 by Cambridge University Press.)
  • [15] I. R. Porteous, Clifford algebra tables in F. Brackx et al (eds.). Clifford Algebras and their applications in Mathematical Physics, Kluwer Academic Publishers, 1993, 13-22.
  • [16] K. Th. Vahlen, Über Bewegungen und complexe Zahlen, Math. Ann. 55 (1902), 585-593.
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