Abstract
Let \({{\mathbb G}L(2, \mathbb{H})}\) be the group of invertible 2 × 2 matrices over the division algebra \({\mathbb{H}}\) of quaternions. \({{\mathbb G}L(2, \mathbb{H})}\) acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in \({{\mathbb G}L(2, \mathbb{H})}\) .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors L.V.: Möbius Transformations and Clifford Numbers. Differential Geometry and Complex Analysis, pp. 65–73. Springer, New York (1985)
Aslaksen H.: Quaternionic determinants. Math. Intell. 18, 57–65 (1996)
Beardon A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics 91. Springer- Verlag, Berlin (1983)
Brenner J.L.: Matrices of quaternions. Pacific J. Math. 1, 329–335 (1951)
Cao W., Parker J.R., Wang X.: On the classification of quaternionic Möbius transformations. Math. Proc. Cambridge Philos. Soc. 137(2), 349–361 (2004)
Cao, C. Waterman, P.L.: Conjugacy Invariants of Möbius Groups: Quasiconformal Mappings and Analysis pp. 109–139. Springer (1998)
Cao W.: On the classification of four-dimensional Möbius transformations. Proc. Edinb. Math. Soc. 50(1): 49–62 (2007)
Foreman B.: Conjugacy invariants of \({\mathbb{S}L(2, \mathbb{H})}\) . Linear Algebra Appl. 381, 25–35 (2004)
Gongopadhyay K., Kulkarni R.S.: z-Classes of isometries of the hyperbolic space. Conform. Geom. Dyn. 13, 91–109 (2009)
Gongopadhyay, K.: z-Classes of isometries of pseudo-riemannian geometries of constant curvature. Ph.D. Thesis, IIT Bombay (2008)
Gongopadhyay, K.: Dynamical types of isometries of hyperbolic space of dimension 5. arXiv:math/0511444v1
Kellerhals R.: Collars in \({PSL(2, \mathbb{H})}\) . Ann. Acad. Sci. Fenn. Math. 26, 51–72 (2001)
Kellerhals R.: Quaternions and some global properties of hyperbolic 5-manifolds. Canad. J. Math. 55, 1080–1099 (2003)
Kulkarni R.S.: Dynamical types and conjugacy classes of centralizers in groups. J. Ramanujan Math. Soc. 22(1), 35–56 (2007)
Kulkarni R.S.: Dynamics of linear and affine maps. Asian J. Math. 12(3), 321–344 (2008)
Lee H.C.: Eigenvalues and canonical forms of matrices with quaternion coefficients. Proc. Roy. Irish Acad. Sect. A. 52, 253–260 (1949)
Parker J.R., Short I.: Conjugacy classification of quaternionic Möbius transformations. Comput. Methods Funct. Theory 9, 13–25 (2009)
Ratcliffe J.G.: Foundation of Hyperbolic Manifolds. Graduate Texts in Mathematics 149. Springer, New York (1994)
Waterman P.L.: Möbius groups in several dimensions. Adv. Math. 101, 87–113 (1993)
Wilker J.B.: The quaternion formalism for Möbius groups in four or fewer dimensions. Linear Algebra Appl. 190, 99–136 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gongopadhyay, K. Algebraic characterization of the isometries of the hyperbolic 5-space. Geom Dedicata 144, 157–170 (2010). https://doi.org/10.1007/s10711-009-9394-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-009-9394-x