A tutorial on conformal groups

Porteous, Ian

Porteous, Ian

Banach Center Publications, Tome 37 (1996), p. 137-150 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $x^{(2)} = x \cdot x = - x_{1}^{2} - x_{2}^{2} - . . . - x_{p}^{2} + x_{p + 1}^{2} + . . . + x_{p + q}^{2}$ , and especially with a description of this group that involves Clifford algebras.

Publié le : 1996-01-01

Zbl 0873.15018

EUDML-ID : urn:eudml:doc:208590

@article{bwmeta1.element.bwnjournal-article-bcpv37i1p137bwm,
     author = {Porteous, Ian},
     title = {A tutorial on conformal groups},
     journal = {Banach Center Publications},
     volume = {37},
     year = {1996},
     pages = {137-150},
     zbl = {0873.15018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv37i1p137bwm}
}

Porteous, Ian. A tutorial on conformal groups. Banach Center Publications, Tome 37 (1996) pp. 137-150. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv37i1p137bwm/

Bibliographie

[000] [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.

[001] [2] É. Cartan, Sur l’espace anallagmatique réel à $n$ dimensions, Ann. Polon. Math. 20 (1947), 266-278. | Zbl 0032.11403

[002] [3] É. Cartan, Deux théorèmes de géométrie anallagmatique réelle à $n$ dimensions, Ann. Mat. Pura Appl. (4)28 (1949), 1-12. | Zbl 0036.37101

[003] [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).

[004] [5] J. Cnops, Hurwitz Pairs and Applications of Möbius Transformations. Thesis, Universiteit Gent, 1994.

[005] [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 | Zbl 0702.51003

[006] [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. | Zbl 0017.42201

[007] [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.

[008] [9] N.H. Kuiper, On conformally-flat spaces in the large, Ann. Math. 50 (1949), 916-924. | Zbl 0041.09303

[009] [10] J. Liouville, Appendix to Monge, G. Application de l'analyse à la geométrie, 5 éd. par Liouville, 1850.

[010] [11] J. Maks, Modulo $(1, 1)$ periodicity of Clifford algebras and the generalized (anti-)Möbius transformations. PhD Thesis, Technische Universiteit Delft., 1989.

[011] [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. | Zbl 0760.15025

[012] [13] J.-B.-M.-C. Meusnier, Mémoire sur la courbure des surfaces, Mémoire Div. Sav., 10 (1785), 477-510.

[013] [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.)

[014] [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.

[015] [16] K. Th. Vahlen, Über Bewegungen und complexe Zahlen, Math. Ann. 55 (1902), 585-593. | Zbl 33.0721.01