×

zbMATH — the first resource for mathematics

The holomorphic automorphism group of the complex disk. (English) Zbl 0799.20032
Each holomorphic permutation of the open disk in \(\mathbb{C}\) with positive radius \(c\) and centre 0 can be written as the product of a rotation and a so-called gyration \(z\mapsto(a+z)/(1+\overline{a}z/c^ 2)\). Properties of this product decomposition are axiomatized, giving rise to the notions of a gyrogroup and a gyrosemidirect product.

MSC:
20E22 Extensions, wreath products, and other compositions of groups
22E43 Structure and representation of the Lorentz group
20N05 Loops, quasigroups
30H05 Spaces of bounded analytic functions of one complex variable
32A38 Algebras of holomorphic functions of several complex variables
83A05 Special relativity
30A05 Monogenic and polygenic functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Armstrong, M. A.,Groups and symmetry. Springer, New York, 1988, p. 132. · Zbl 0663.20001
[2] Artzy, R.,Linear geometry. Addison-Wesley, Massachusetts, 1965. · Zbl 0166.16005
[3] Bear, H. S.,Distance-decreasing functions on the hyperbolic plane. Mich. Math. J.39 (1992) 271–279. · Zbl 0758.30035 · doi:10.1307/mmj/1029004522
[4] Bear, H. S.,Part metric and hyperbolic metric. Amer. Math. Monthly98 (1991) 109–123. · Zbl 0742.30039 · doi:10.2307/2323940
[5] Benz, W.,Geometrische Transformationen unter besonderer Berücksichtigung der Lorentztransformationen. Chap. 6. Wissenschaftsverlag, Wien, 1992. · Zbl 0754.51005
[6] Einstein, A.,Zur Elektrodynamik bewegter Körper (On the electrodynamics of moving bodies). Ann. Physik (Leipzig)17 (1905), 891–921. · JFM 36.0920.02 · doi:10.1002/andp.19053221004
[7] Franzoni, T. andVesentini, E.,Holomorphic maps and invariant distances (L. Nachbin, ed.). [Math. Studies, vol. 40]. North Holland, New York, 1980. · Zbl 0447.46040
[8] Friedman, Y. andUngar, A. A.,Gyrosemidirect product structure of bounded symmetric domains. preprint.
[9] Isidro, J. M. andStachó, L. L.,Holomorphic automorphism groups in Banach spaces: An elementary introduction (L. Nachbin, ed.). [Math. Studies, vol. 97]. North Holland, New York, 1984, pp. 29–32.
[10] Jaroszewicz, T. andKurzepa, P. S.,Geometry and spacetime propagation of spinning particles. Ann. Phys.216 (1992), 226–267. · Zbl 0875.53019 · doi:10.1016/0003-4916(92)90176-M
[11] Karzel, H.,Inzidenzgruppen I. Lecture notes by I. Pieper and K. Sörensen, Univ. Hamburg, 1965, pp. 123–135.
[12] Kerby, W. andWefelscheid, H.,The maximal subnear-field of a neardomain. J. Algebra28 (1974), 319–325. · Zbl 0276.16029 · doi:10.1016/0021-8693(74)90043-X
[13] Kikkawa, M.,Geometry of homogeneous Lie loops. Hiroshima Math. J.5 (1975), 141–179. · Zbl 0304.53037
[14] Lang, S.,Complex analysis. 2nd ed., Springer, New York, 1985. · Zbl 0562.30001
[15] Lang, S.,Introduction to complex hyperbolic spaces. Springer, New York, 1987, p. 12. · Zbl 0628.32001
[16] Pflugfelder, H. O.,Quasigroups and loops. Introduction, Heldermann Verlag, Berlin, 1990. · Zbl 0715.20043
[17] Chein, O., Pflugfelder, H. O. andSmith, J. D. H. (eds.),Quasigroups and loops theory and applications. Sigma Series in Pure Mathematics, Vol. 8, Heldermann Verlag, Berlin, 1990. · Zbl 0719.20036
[18] Sabinin, L. V.,On the equivalence of the category of loops and the category of homogeneous spaces (Russian). Dokl. Akad. Nauk. SSR205, no. 3, (1972), 533–536; English trans. Soviet Math. Dokl. 13, no. 4 (1972) 970–974.
[19] Sabinin, L. V.,The geometry of loops (Russian). Mat. Zamethi12, no. 5 (1972), 799–805.
[20] Sexl R. andUrbantke, H. K.,Relativität, Gruppen, Teilchen. Springer, New York, 1992, pp. 40, 138.
[21] Tits, J.,Généralisation des groupes projectifs. Acad. Roy. Belg. Cl. Sci. Mém. Coll. 5e Ser. 35 (1949) 197–208, 224–233, 568–589, 756–773. · Zbl 0034.30504
[22] Ungar, A. A.,Thomas precession and the parametrization of the Lorentz transformation group. Found. Phys. Lett.1 (1988) 57–89. · doi:10.1007/BF00661317
[23] Ungar, A. A.,Weakly associative groups. Resultate Math.17 (1990), 149–168. · Zbl 0699.20055
[24] Ungar, A. A.,Quasidirect product groups and the Lorentz transformation group. In T. M. Rassias (ed.),Constantin Caratheodory: An international tribute, World Sci. Pub., NJ, 1990, pp. 1378–1392. · Zbl 0746.22009
[25] Ungar, A. A.,Solving the velocity composition equation of special relativity. Problem 6659, Amer. Math. Monthly98 (1991) 445–446; and its solution in Amer. Math. Monthly100 (1993) 500–502. · doi:10.2307/2323871
[26] Ungar, A. A.,Thomas precession and its associated grouplike structure. Amer. J. Phys.59 (1991) 824–834. · doi:10.1119/1.16730
[27] Ungar, A. A.,Gyrogroups. 92T-53-49. Abst. Amer. Math. Soc.13 (1992), 293–294.
[28] Ungar, A. A.,The abstract Lorentz transformation group. Amer. J. Phys.60 (1992), 815–828. · Zbl 1219.83029 · doi:10.1119/1.17063
[29] Ungar, A. A.,Gyrogroups. J. Geometry44 (1992), 21–22.
[30] Vigoureux, J. M.,The use of Einstein’s addition law in studies of reflection by stratified planar structures. J. Opt. Soc. Amer. A9 (1992), 1313–1319. · doi:10.1364/JOSAA.9.001313
[31] You, Y. andUngar, A. A.,Equivalence of two gyrogroup structures on unit balls, preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.