Skip to main content Accessibility help
×
×
Home

Reversible maps in the group of quaternionic Möbius transformations

  • ROMAN LÁVIČKA (a1), ANTHONY G. O'FARRELL (a2) and IAN SHORT (a3)
Abstract

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Möbius transformations.

Copyright
References
Hide All
[1]Ahlfors, L. V.. Möbius Transformations in Several Dimensions. University of Minnesota Lecture Notes (Minnesota, 1981).
[2]Ahlfors, L. V.. Möbius transformations and Clifford numbers. Differential Geometry and Complex Analysis (1985), 65–73.
[3]Ahlfors, L. V.. On the fixed points of Möbius transformations in . Ann. Acad. Sci. Fenn. 10 (1985), 1527.
[4]Ahlfors, L. V.. Clifford numbers and Möbius transformations in , Clifford algebras and their applications in physics. NATO Sci. Ser. C. Math. Phy. Sci. 183 (1986), 167175.
[5]Ahlfors, L. V.. Möbius transformations in expressed through 2 × 2 matrices of Clifford numbers. Complex Var. Theory Appl. 5 (1986), 215224.
[6]Cao, W., Parker, J. R. and Wang, X.. On the classification of Möbius transformations. Math. Proc. Camb. Phil. Soc. 137 (2004), 349361.
[7]Chen, S. S. and Greenberg, L.. Hyperbolic Spaces: Contributions to Analysis (Academic Press, 1974).
[8]Coxeter, H. S. M.. Quaternions and reflections. Amer. Math. Monthly 53 (1946), 136146.
[9]Curtis, C. W.. Linear Algebra: An Introductory Approach (Undergraduate Texts in Mathematics, Springer, 1984).
[10]Ellers, E. W.. Conjugacy classes of involutions in the Lorentz group Ω(V) and in SO(V) Linear Algebra Appl. 383 (2004).
[11]Gormley, P. G.. Stereographic projection and the linear fractional transformations of quaternions. Proc. Roy. Irish Acad. 51 (1947), 6785.
[12]Gustafon, W. H., Halmos, P. R. and Radjavi, H.. Products of involutions. Linear Algebra Appl. 13 (1976), 157162.
[13] F. Knüppel and Nielsen, K.. On products of two involutions in the orthogonal group of a vector space. Linear Algebra Appl. 94 (1987), 209216.
[14] F. Knüppel and Nielsen, K.. Products of involutions in O+(V). Linear Algebra Appl. 94 (1987), 217222.
[15]Lamb, J. S. W. and Roberts, J. A. G.. Time-reversal symmetry in dynamical systems: a survey. Phys. D. 112 (1998), 149.
[16] A. G. O'Farrell. Conjugacy, involutions and reversibility for real homeomorphisms. Irish Math. Soc. Bulletin 54 (2004), 4152.
[17]Ratcliffe, J. G.. Foundations of Hyperbolic Manifolds. Number 149 in Graduate Texts (Springer–Verlag, 1994).
[18]Short, I.. Reversible maps in isometry groups of spherical, Euclidean and hyperbolic space. Submitted.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed