Osaka Journal of Mathematics

On the classifications of unitary matrices

Krishnendu Gongopadhyay, John R. Parker, and Shiv Parsad

Full-text: Open access

Abstract

We classify the dynamical action of matrices in $\mathrm{SU}(p, q)$ using the coefficients of their characteristic polynomial. This generalises earlier work of Goldman for $\mathrm{SU}(2, 1)$ and the classical result for $\mathrm{SU}(1, 1)$, which is conjugate to $\mathrm{SL}(2, \mathbb{R})$. As geometrical applications, we show how this enables us to classify automorphisms of real and complex hyperbolic space and anti de Sitter space.

Article information

Source
Osaka J. Math., Volume 52, Number 4 (2015), 959-993.

Dates
First available in Project Euclid: 18 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1447856028

Mathematical Reviews number (MathSciNet)
MR3426624

Zentralblatt MATH identifier
1336.51006

Subjects
Primary: 51F25: Orthogonal and unitary groups [See also 20H05]
Secondary: 20H20: Other matrix groups over fields

Citation

Gongopadhyay, Krishnendu; Parker, John R.; Parsad, Shiv. On the classifications of unitary matrices. Osaka J. Math. 52 (2015), no. 4, 959--993. https://projecteuclid.org/euclid.ojm/1447856028


Export citation

References

  • P. Anglès: Conformal Groups in Geometry and Spin Structures, Progress in Mathematical Physics 50, Birkhäuser Boston, Boston, MA, 2008.
  • A.F. Beardon: The Geometry of Discrete Groups, Springer, New York, 1983.
  • A. Cano, J.P. Navarrete and J. Seade: Complex Kleinian Groups, Birkhäuser/Springer Basel AG, Basel, 2013.
  • W. Cao, J.R. Parker and X. Wang: On the classification of quaternionic Möbius transformations, Math. Proc. Cambridge Philos. Soc. 137 (2004), 349–361.
  • D. Chillingworth: The ubiquitous astroid; in The Physics of Structure Formation (Tübingen, 1986), Springer, Berlin, 1987, 372–386.
  • M.P. do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.
  • W.M. Goldman: Trace coordinates on Fricke spaces of some simple hyperbolic surfaces; in Handbook of Teichmüller Theory, II, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009, 611–684.
  • W.M. Goldman: Complex Hyperbolic Geometry, Oxford Univ. Press, Oxford, 1999.
  • W.M. Goldman: Crooked surfaces and anti-de Sitter geometry, Geom. Dedicata 175 (2015), 159–187.
  • W.M. Goldman and J.R. Parker: Complex hyperbolic ideal triangle groups, J. Reine Angew. Math. 425 (1992), 71–86.
  • K. Gongopadhyay: Algebraic characterization of the isometries of the hyperbolic 5-space, Geom. Dedicata 144 (2010), 157–170.
  • K. Gongopadhyay: Algebraic characterization of isometries of the complex and the quaternionic hyperbolic $3$-spaces, Proc. Amer. Math. Soc. 141 (2013), 1017–1027.
  • K. Gongopadhyay and J.R. Parker: Reversible complex hyperbolic isometries, Linear Algebra Appl. 438 (2013), 2728–2739.
  • F. Kirwan: Complex Algebraic Curves, London Mathematical Society Student Texts 23, Cambridge Univ. Press, Cambridge, 1992.
  • G. Mess: Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007), 3–45.
  • J.-P. Navarrete: The trace function and complex Kleinian groups in $\mathbb{P}^{2}_{\mathbb{C}}$, Internat. J. Math. 19 (2008), 865–890.
  • J.R. Parker: Hyperbolic Spaces, Jyväskylä Lectures in Mathematics 2, 2008.
  • J.R. Parker: Traces in complex hyperbolic geometry; in Geometry, Topology and Dynamics of Character Varieties, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 23, World Sci. Publ., Hackensack, NJ, 2012, 191–245.
  • \begingroup J.R. Parker and I. Short: Conjugacy classification of quaternionic Möbius transformations, Comput. Methods Funct. Theory 9 (2009), 13–25. \endgroup
  • J.R. Parker and P. Will: Complex hyperbolic free groups with many parabolic elements; in Geometry, Groups and Dynamics, Contemp. Math. 639, Amer. Math. Soc., Providence RI, 2015, 327–348.
  • T. Poston and I. Stewart: The cross-ratio foliation of binary quartic forms, Geom. Dedicata 27 (1988), 263–280.