A tutorial on conformal groups

Ian Porteous

Banach Center Publications (1996)

  • Volume: 37, Issue: 1, page 137-150
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Porteous, Ian. "A tutorial on conformal groups." Banach Center Publications 37.1 (1996): 137-150. <http://eudml.org/doc/208590>.

@article{Porteous1996,
abstract = {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.},
author = {Porteous, Ian},
journal = {Banach Center Publications},
keywords = {conformal groups; Liouville theorem; conformal compactification; projective compactification; Möbius transformation; conformal split; para-vectors; Vahlen matrices; Clifford algebra},
language = {eng},
number = {1},
pages = {137-150},
title = {A tutorial on conformal groups},
url = {http://eudml.org/doc/208590},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Porteous, Ian
TI - A tutorial on conformal groups
JO - Banach Center Publications
PY - 1996
VL - 37
IS - 1
SP - 137
EP - 150
AB - 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.
LA - eng
KW - conformal groups; Liouville theorem; conformal compactification; projective compactification; Möbius transformation; conformal split; para-vectors; Vahlen matrices; Clifford algebra
UR - http://eudml.org/doc/208590
ER -

References

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  1. [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. [2] É. Cartan, Sur l’espace anallagmatique réel à n dimensions, Ann. Polon. Math. 20 (1947), 266-278. Zbl0032.11403
  3. [3] É. Cartan, Deux théorèmes de géométrie anallagmatique réelle à n dimensions, Ann. Mat. Pura Appl. (4)28 (1949), 1-12. Zbl0036.37101
  4. [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. [5] J. Cnops, Hurwitz Pairs and Applications of Möbius Transformations. Thesis, Universiteit Gent, 1994. 
  6. [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 Zbl0702.51003
  7. [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. Zbl0017.42201
  8. [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. [9] N.H. Kuiper, On conformally-flat spaces in the large, Ann. Math. 50 (1949), 916-924. Zbl0041.09303
  10. [10] J. Liouville, Appendix to Monge, G. Application de l'analyse à la geométrie, 5 éd. par Liouville, 1850. 
  11. [11] J. Maks, Modulo ( 1 , 1 ) periodicity of Clifford algebras and the generalized (anti-)Möbius transformations. PhD Thesis, Technische Universiteit Delft., 1989. 
  12. [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. Zbl0760.15025
  13. [13] J.-B.-M.-C. Meusnier, Mémoire sur la courbure des surfaces, Mémoire Div. Sav., 10 (1785), 477-510. 
  14. [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. [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. [16] K. Th. Vahlen, Über Bewegungen und complexe Zahlen, Math. Ann. 55 (1902), 585-593. Zbl33.0721.01

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