Möbius transformations and the poincaré distance in the quaternionic setting

Indiana University Mathematics Journal

Volumn 58, Issue 6, 2009, Pages 2729-2764

  Bisi, Cinzia ^a   Gentili, Graziano ^b  

^a UNIVERSITY OF CALABRIA   (Italy)

^b UNIVERSITY OF FLORENCE   (Italy)

Author keywords

Distance and metric; Functions of hypercomplex variables; Quatemionic poincar ; Quaterionic m bius transformations

Indexed keywords

EID: 77949804703     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2009.58.3706     Document Type: Article

References (21)

1. L.V. AHLFORS, Möbius transformations and Clifford numbers, Differential Geometry and Complex Analysis, Springer, Berlin, 1985, pp. 65-73. MR 780036 (86g:20065)
2. n, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15-27. MR 802464 (87c:20086)
3. n, Complex Analysis (J.. Hersch and A. . Huber, eds.), Birkhäuser, Basel, 1988, pp. 1-15. MR 981397 (90a:30055)
4. H. ASLAKSEN, Quatemionic determinants, Math. Intelligencer 18 (1996), 57-65, http://dx.doi.org/10.1007/BF03024312. MR 1412993 (97j:16028)
5. C. CAO and P.L. WATERMAN, Conjugacy invariants of Möbius groups, Quasiconformal Mappings and Analysis (Ann Abor, MI), Springer, New York, 1998, pp. 109-139. MR 1488448 (98k:30058)
6. W. CAO, J.R. PARKER, and X. WANG, On the classification of quatemionic Möbius transformations, Math. Proc Cambridge Philos. Soc 137 (2004), 349-361, http://dx.doi.org/10.1017/S0305004104007868. MR 2092064 (2005f:30082)
7. H. CARTAN, Cours de Maitrise des Sciences Mathématiques, 1969 (unpublished).
8. N. COHEN and S. DE LEO, The quatemionic determinant, Electron. J. Linear Algebra 7 (2000), 100-111 (electronic). MR 1779435 (2001g:15006)
9. I. GELFAND, V RETAKH, and R.L. WILSON, Quatemionic quasideterminantsanddeterminants, Lie Groups and Symmetric Spaces, Amer. Math. Soc. Transl. Ser. 2, vol.210, Amer. Math. Soc., Providence, RI, 2003, pp. 111-123. MR 2018356 (2004j:15010)
10. G. GENTILI and D.C. STRUPPA, A new theory of regular functions of a quatemionic variable, Adv. Math. 216 (2007), 279-301, http://dx.doi.Org/10.1016/ j.aim.2007.05.010. MR 2353257 (2008h:30052)
11. V.V. GORBATSEVICH, A.L. ONISHCHIK, and É.B. VINBERG, Lie Groups and Lie Algebras, III. Structure of Lie Groups and Lie Algebras (A.L. Onishchik and É.B. Vinberg, editors), Encyclopaedia of Mathematical Sciences, vol.41, Springer-Verlag, Berlin, 1994, ISBN 3-540-54683-9,
12. A translation of Current problems in mathematics. Fundamental directions. Volume 41 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [MR1056485 (91b:22001)]. MR 1349140 (96d:22001)
13. I.N. HERSTEIN, Topics in Algebra, 2nd ed., Xerox College Publishing, Lexington, Mass., 1975. MR 0356988 (50 #9456)
14. R. HEIDRICH and G. JANK, On the iteration of quatemionic Moebius transformations, Complex Variables Theory Appl. 29 ( 1996), 313-318. MR 1390616 (97e:30079)
15. R. KELLERHALS, Collars in PSL(2, ℍ), Ann. Acad. Sci. Fenn. Math. 26 (2001), 51-72. MR 1816562 (2002f:57036)
16. R.M. PORTER, Quatemionic Möbius transformations and loxodromes. Complex Variables Theory Appl. 36 (1998), 285-300. MR 1671478 (99m:20121)
17. V. RETAKH and R.L. WILSON, Advanced Course on Quasideterminants and Universal Localization, Centre de Recerca Matematica (CRM), Bellaterra (Barcelona), Spain, 2007, notes of the course.
18. C.L. SIEGEL, Topics in Complex Function Theory. Vol II. Automorphic Functions andAbelian Integrals, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, ISBN 0-471-60843-2, Translated from the German by A Shenitzer and M. Tretkoff; With a preface by Wilhelm Magnus; Reprint of the 1971 edition; A Wiley-Interscience Publication. MR 1008931 (90h:30003)
19. E. VESENTINI, Capitoli Scelti della Teoria delle Funzioni Olomorfe, Unione Matematica Italiana, 1984.
20. X. WANG, Quatemionic Möbius transformations and subgroups. Complex Var. Theory Appl. 48 (2003), 599-606. MR 1988686 (2004e:30072)
21. J.B. WILKER, The quaternion formalism for Möbius groups in four or fewer dimensions, Linear Algebra Appl. 190 (1993), 99-136, http://dx.doi.org/10. 1016/0024-3795(93)90222-A. MR 1230354 (94c:20086)