Square Roots of –1 in Real Clifford Algebras

  • Eckhard Hitzer
  • Jacques Helmstetter
  • Rafał Abłamowicz
Part of the Trends in Mathematics book series (TM)


It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [33] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cl 3,0 of ℝ3. Further research on general algebras Cl p,q has explicitly derived the geometric roots of –1for p + q≤4 [20]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of –1f ound in the different types of Clifford algebras, depending on the type of associated ring (ℝ,ℍ,ℝ2,ℍ2, or ℂ). At the end of the chapter explicit computer generated tables of representative square roots of –1 are given for all Clifford algebras with n = 5,7, and s = 3 (mod 4) with the associated ring ℂ. This includes, e.g., Cl 0,5 important in Clifford analysis, and Cl 4,1 which in applications is at the foundation of conformal geometric algebra. All these roots of –1 are immediately useful in the construction of new types of geometric Clifford–Fourier transformations.


Algebra automorphism inner automorphism center centralizer Clifford algebra conjugacy class determinant primitive idempotent trace 


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  1. [1]
    R. Abłamowicz. Computations with Clifford and Grassmann algebras. Advances in Applied Clifford Algebras, 19(3–4):499–545, 2009.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    R. Abłamowicz and B. Fauser. CLIFFORD with bigebra – a Maple package for computations with Clifford and Grassmann algebras. Available at http://math.tntech.edu/rafal/, © 1996–2012.
  3. [3]
    R. Abłamowicz, B. Fauser, K. Podlaski, and J. Rembieliński. Idempotents of Clifford algebras. Czechoslovak Journal of Physics, 53(11):949–954, 2003.MathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Armstrong. The center of an algebra. Weblog, available at http://unapologetic.wordpress.com/2010/10/06/the-center-of-an-algebra/, accessed 22 March 2011.
  5. [5]
    M. Bahri, E. Hitzer, R. Ashino, and R. Vaillancourt. Windowed Fourier transform of two-dimensional quaternionic signals. Applied Mathematics and Computation, 216(8):2366–2379, June 2010.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    M. Bahri, E.M.S. Hitzer, and S. Adji. Two-dimensional Clifford windowed Fourier transform. In E.J. Bayro-Corrochano and G. Scheuermann, editors, Geometric Algebra Computing in Engineering and Computer Science, pages 93–106. Springer, London, 2010.Google Scholar
  7. [7]
    F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman, Boston, 1982.Google Scholar
  8. [8]
    T. Bülow. Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD thesis, University of Kiel, Germany, Institut für Informatik und Praktische Mathematik, Aug. 1999.Google Scholar
  9. [9]
    T. Bülow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fourier transforms of multidimensional signals. In G. Sommer, editor, Geometric computing with Clifford Algebras: Theoretical Foundations and Applications in Computer Vision and Robotics, pages 187–207, Berlin, 2001. Springer.Google Scholar
  10. [10]
    C. Chevalley. The Theory of Lie Groups. Princeton University Press, Princeton, 1957.Google Scholar
  11. [11]
    W.K. Clifford. Applications of Grassmann’s extensive algebra. American Journal of Mathematics, 1(4):350–358, 1878.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    L. Dorst and J. Lasenby, editors. Guide to Geometric Algebra in Practice. Springer, Berlin, 2011.Google Scholar
  13. [13]
    J. Ebling and G. Scheuermann. Clifford Fourier transform on vector fields. IEEE Transactions on Visualization and Computer Graphics, 11(4):469–479, July 2005.CrossRefGoogle Scholar
  14. [14]
    M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis, Christian-Albrechts-Universität, Institut für Informatik und Praktische Mathematik, Kiel, 2002.Google Scholar
  15. [15]
    D. Hestenes. Space-Time Algebra. Gordon and Breach, London, 1966.Google Scholar
  16. [16]
    D. Hestenes. New Foundations for Classical Mechanics. Kluwer, Dordrecht, 1999.Google Scholar
  17. [17]
    D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus. D. Reidel Publishing Group, Dordrecht, Netherlands, 1984.Google Scholar
  18. [18]
    E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    E. Hitzer. OPS-QFTs: A new type of quaternion Fourier transform based on the orthogonal planes split with one or two general pure quaternions. In International Conference on Numerical Analysis and Applied Mathematics, volume 1389 of AIP Conference Proceedings, pages 280–283, Halkidiki, Greece, 19–25 September 2011. American Institute of Physics.Google Scholar
  20. [20]
    E. Hitzer and R. Abłamowicz. Geometric roots of 1in Clifford algebras Cℓ p,q with p + q ≤ 4. Advances in Applied Clifford Algebras, 21(1):121–144, 2010. Published online 13 July 2010.Google Scholar
  21. [21]
    E. Hitzer, J. Helmstetter, and R. Abłamowicz. Maple worksheets created with CLIFFORD for a verification of results presented in this chapter. Available at: http://math.tntech.edu/rafal/publications.html, © 2012.
  22. [22]
    E. Hitzer and B. Mawardi. Uncertainty principle for Clifford geometric algebras Cℓ n,0 , n = 3(mod 4) based on Clifford Fourier transform. In T. Qian, M.I. Vai, and Y. Xu, editors, Wavelet Analysis and Applications, Applied and Numerical Harmonic Analysis, pages 47–56. Birkhäuser Basel, 2007.Google Scholar
  23. [23]
    E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n = 2(mod 4) and n = 3(mod4). Advances in Applied Clifford Algebras, 18(3–4):715–736, 2008.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985.Google Scholar
  25. [25]
    W.M. Incorporated. Maple, a general purpose computer algebra system. http://www.maplesoft.com, © 2012.
  26. [26]
    C. Li, A. McIntosh, and T. Qian. Clifford algebras, Fourier transform and singular convolution operators on Lipschitz surfaces. Revista Matematica Iberoamericana, 10(3):665–695, 1994.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    H. Li. Invariant Algebras and Geometric Reasoning. World Scientific, Singapore, 2009.Google Scholar
  28. [28]
    P. Lounesto. Clifford Algebras and Spinors, volume 286 of London Mathematical Society Lecture Notes. Cambridge University Press, 1997.Google Scholar
  29. [29]
    B. Mawardi and E.M.S. Hitzer. Clifford Fourier transformation and uncertainty principle for the Clifford algebra Cℓ 3,0. Advances in Applied Clifford Algebras, 16(1):41– 61, 2006.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    A. McIntosh. Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains. In J. Ryan, editor, Clifford Algebras in Analysis and Related Topics, chapter 1. CRC Press, Boca Raton, 1996.Google Scholar
  31. [31]
    T. Qian. Paley-Wiener theorems and Shannon sampling in the Clifford analysis setting. In R. Abłamowicz, editor, Clifford Algebras – Applications to Mathematics, Physics, and Engineering, pages 115–124. Birkäuser, Basel, 2004.Google Scholar
  32. [32]
    S. Said, N. Le Bihan, and S.J. Sangwine. Fast complexified quaternion Fourier transform. IEEE Transactions on Signal Processing, 56(4):1522–1531, Apr. 2008.MathSciNetCrossRefGoogle Scholar
  33. [33]
    S.J. Sangwine. Biquaternion (complexified quaternion) roots of 1. Advances in Applied Clifford Algebras, 16(1):63–68, June 2006.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Wikipedia article. Conjugacy class. Available at http://en.wikipedia.org/wiki/ Conjugacy_class, accessed 19 March 2011.
  35. [35]
    Wikipedia article. Inner automorphism. Available at http://en.wikipedia.org/ wiki/Inner_automorphism, accessed 19 March 2011.

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Eckhard Hitzer
    • 1
  • Jacques Helmstetter
    • 2
  • Rafał Abłamowicz
    • 3
  1. 1.College of Liberal Arts, Department of Material ScienceInternational Christian UniversityTokyoJapan
  2. 2.Institut Fourier (Mathématiques)Univesité Grenoble ISaint-Martin d’HèresFrance
  3. 3.Department of MathematicsTennessee Technological UniversityCookevilleUSA

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