Quaternionic modular groups

Linear Algebra and Its Applications

Volumn 295, Issue 1-3, 1999, Pages 159-189

  Johnson, Norman W ^a   Weiss, Asia Ivić ^b  

^a WHEATON COLLEGE   (United States)

^b YORK UNIVERSITY   (Canada)

Author keywords

Clifford algebras; Coxeter groups; Quaternionic integers; Quaternionic matrices; Special linear group

Indexed keywords

EID: 0033423376     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(99)00107-X     Document Type: Article

References (32)

1. L.V. Ahlfors, Möbius transformations and Clifford numbers, in: I. Chavel, H.M. Farkas (Eds.), Differential Geometry and Complex Analysis, H.E. Rauch memorial volume, Springer, Berlin, 1985, pp. 65-73.
2. E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, No. 3, Wiley-Interscience, New York, 1957; Wiley, New York, 1988.
3. H. Aslaksen, Quarternionic determinants, Math. Intelligencer 18 (3) (1996) 57-65.
4. W.K. Clifford, On the classification of geometric algebras, unpublished manuscript (1876) included in: Mathematical Papers, Macmillan, London, 1882, pp. 397-401.
5. J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Grundlehren der mathematischen Wissenschaften, No. 290, Springer, New York, 1988; 2nd ed., 1993.
6. H.S.M. Coxeter, Regular Complex Polytopes, Cambridge University Press, New York, 1974; 2nd ed., 1991.
7. H.S.M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Bd. 14. Springer, Berlin 1957; 4th ed., Springer, Berlin, 1980.
8. L.E. Dickson, Arithmetic of quaternions, Proc. London Math. Soc. (2) 20 (1922) 225-232 (reprinted in: Collected Mathematical Papers, vol. III, Chelsea, New York, 1975, pp. 397-404).
9. L.E. Dickson, Arithmetic of quaternions, Proc. London Math. Soc. (2) 20 (1922) 225-232 (reprinted in: Collected Mathematical Papers, vol. III, Chelsea, New York, 1975, pp. 397-404).
10. L.E. Dickson, Algebras and Their Arithmetics, University of Chicago Press, Chicago, 1923; G.E. Stechert, New York, 1938. Translated by J.J. Burckhardt, E. Schubarth as Algebren und ihre Zahlentheorie, Orell Füssli , Zurich-Leipzig, 1927.
11. L.E. Dickson, Algebras and Their Arithmetics, University of Chicago Press, Chicago, 1923; G.E. Stechert, New York, 1938. Translated by J.J. Burckhardt, E. Schubarth as Algebren und ihre Zahlentheorie, Orell Füssli , Zurich-Leipzig, 1927.
12. J. Dieudonné, Les déterminants sur un corps non-commutatif, Bull. Soc. Math. France 71 (1943) 27-45.
13. P. Du Val, Homographies, Quaternions, and Rotations, Clarendon Press/Oxford University Press, Oxford, 1964.
14. R. Fricke, F. Klein, Vorlesungen über die Theorie der automorphen Funktionen, Bd. I. Teubner, Leipzig, 1897; 2nd ed., 1926.
15. W.R. Hamilton, Quaternion integers, unpublished manuscript (1856); printed as appendix to Mathematical Papers, vol. III, Cambridge University Press, London, 1967, pp. 657-665.
16. W.R. Hamilton, Quaternion integers, unpublished manuscript (1856); printed as appendix to Mathematical Papers, vol. III, Cambridge University Press, London, 1967, pp. 657-665.
17. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, Clarendon Press/Oxford University Press, Oxford, 1938.
18. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 2nd ed., Clarendon Press/Oxford University Press, Oxford, 1945; 5th ed., 1979.
19. A. Hurwitz, Über die Zahlentheorie der Quaternionen, Nach. Königl. Ges. Wiss. Göttingen Math.-Phys. Kl. 1896, pp. 313-340 (reprinted in: Mathematische Werke, Bd. II, Birkhäuser, Basel, 1933, pp. 303-330).
20. A. Hurwitz, Über die Zahlentheorie der Quaternionen, Nach. Königl. Ges. Wiss. Göttingen Math.-Phys. Kl. 1896, pp. 313-340 (reprinted in: Mathematische Werke, Bd. II, Birkhäuser, Basel, 1933, pp. 303-330).
21. N.W. Johnson, A.I. Weiss, Quadratic integers and Coxeter groups, Canad. J. Math. 51 (1999), to appear.
22. F. Klein, Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades, Math. Ann. 14 (1879) 111-172 (reprinted in: Gesammelte mathematische Abhandlungen, Bd. III, Springer, Berlin, 1923, pp. 13-75.
23. F. Klein, Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades, Math. Ann. 14 (1879) 111-172 (reprinted in: Gesammelte mathematische Abhandlungen, Bd. III, Springer, Berlin, 1923, pp. 13-75.
24. R. Lipschitz, Untersuchungen über die Summen von Quadraten, M. Cohen, Bonn, 1886.
25. C. Maclachlan, P.L. Waterman, N.J. Wielenberg, Higher dimensional analogues of the modular and Picard groups, Trans. Amer. Math. Soc. 312 (1989) 739-753.
26. P.M. Neumann, G.A. Stoy, E.C. Thompson, Groups and Geometry, Oxford University Press, Oxford, 1994.
27. I.R. Porteous, Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, No. 50, Cambridge University Press, Cambridge, 1995.
28. E. Schulte, A.I. Weiss, Chirality and projective linear groups, Discrete Math. 131 (1994) 221-261.
29. K.Th. Vahlen, Ueber Bewegungen und complexe Zahlen, Math. Ann. 55 (1902) 585-593.
30. O. Veblen, J. W. Young, Projective Geometry, vol. II, Ginn, Boston, 1918; Blaisdell, New York, 1965.
31. P.L. Waterman, Möbius transformations in several dimensions, Adv. Math. 101 (1993) 87-113.
32. J.B. Wilker, The quaternion formalism for Möbius groups in four or fewer dimensions, Linear Algebra Appl. 190 (1993) 99-136.