Clifford Möbius Geometry

  • Craig A. Nolder
Part of the Trends in Mathematics book series (TM)


The Riemann sphere is a compactification of the complex plane on which the complex Möbius group naturally acts. This Möbius group is isomorphic to the conformal orthogonal group SO+(1, 3). Here we give a unified approach to this compactification and the corresponding Möbius groups for the Clifford algebras of dimensions two and four.


Clifford composition algebras Compactification Conformal Mobius groups 

Mathematics Subject Classification (2010)

Primary 22-06; Secondary 16-06 


  1. 1.
    J. Emanuello, C.A. Nolder, Projective compactification of \(\mathbb {R}^{1,1}\) and its Möbius geometry. Complex Anal. Oper. Theory 9, 329–354 (2015)Google Scholar
  2. 2.
    J. Emanuello, C.A. Nolder, Clifford algebras with induced (semi)-Riemannian structures and their compactifications, in Proceedings of the 9th ISAAC International Conference, Krakow, 5–9 August 2013. Current Trends in Analysis and Its Applications (2015), pp. 499–504Google Scholar
  3. 3.
    J. Emanuello, C.A. Nolder, Linear fractional transformations of the split quaternions (2015)Google Scholar
  4. 4.
    P. Garrett, Sporadic isogenies to orthogonal groups, May 7, 2015, online notesGoogle Scholar
  5. 5.
    J.E. Gilbert, M. Murray, Cliford Algebras and Dirac Operators in Harmonic Analysis (Cambridge University Press, Cambridge, 1991)Google Scholar
  6. 6.
    S. Helgason, Differential Geometry. Lie Groups and Symmetric Spaces (Academic, New York, 1978)zbMATHGoogle Scholar
  7. 7.
    M. Schottenloher, A Mathematical Introduction to Conformal Field Theory, vol. 759 (Springer, Berlin, 2008). ISBN 978-3-540-68628-6zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Craig A. Nolder
    • 1
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA

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