Advances in Applied Clifford Algebras

, Volume 24, Issue 4, pp 955–980 | Cite as

Fundaments of Quaternionic Clifford Analysis I: Quaternionic Structure

  • F. Brackx
  • H. De Schepper
  • D. Eelbode
  • R. Lávička
  • V. Souček


Introducing a quaternionic structure on Euclidean space, the fundaments for quaternionic and symplectic Clifford analysis are studied in detail from the viewpoint of invariance for the symplectic group action.


Quaternionic Clifford analysis quaternionic structure 


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  1. 1.
    Abreu Blaya R., Bory Reyes J., Brackx F., De Schepper H., Sommen F.: Cauchy Integral Formulae in Hermitian Quaternionic Clifford Analysis. Compl. Anal. Oper. Theory 6(5), 971–985 (2012)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Abreu Blaya R., Bory Reyes J., Brackx F., De Schepper H., Sommen F.: Matrix Cauchy and Hilbert transforms in Hermitean quaternionic Clifford analysis. Comp. Var. Elliptic Equ. 58(8), 1057–1069 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Brackx F., Bureš J., De Schepper H., Eelbode D., Sommen F., Souček V.: Fundaments of Hermitean Clifford analysis – Part I: Complex structure. Complex Anal. Oper. Theory 1(3), 341–365 (2007)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Brackx F., Bureš J., De Schepper H., Eelbode D., Sommen F., Součcek V.: Fundaments of Hermitean Clifford analysis – Part II: Splitting of h-monogenic equations. Comp. Var. Elliptic Equ. 52(10–11), 1063–1079 (2007)Google Scholar
  5. 5.
    Brackx F., De Schepper H., Eelbode D., Souček V.: The Howe Dual Pair in Hermitean Clifford Analysis. Rev. Mat. Iberoamericana 26(2), 449–479 (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Brackx F., De Schepper H., Sommen F.: The Hermitean Clifford analysis toolbox. Adv. Appl. Cliff. Alg. 18(3–4), 451–487 (2008)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis. Research Notes in Mathematics 76, Pitman, Boston, MA, 1982.Google Scholar
  8. 8.
    Colombo F., Sabadini I., Sommen F., Struppa D.C.: Analysis of Dirac Systems and Computational Algebra. Birkhäuser, Boston (2004)CrossRefMATHGoogle Scholar
  9. 9.
    Damiano A., Eelbode D., Sabadini I.: Quaternionic Hermitian spinor systems and compatibility conditions. Adv. Geom. 11, 169–189 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Delanghe R., Sommen F., Souček V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)CrossRefMATHGoogle Scholar
  11. 11.
    Eelbode D.: A Clifford algebraic framework for sp(m)-invariant differential operators. Adv. Appl. Cliff. Algebras 17, 635–649 (2007)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Eelbode D.: Stirling numbers and Spin–Euler polynomials. Exp. Math. 16(1), 55–66 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Eelbode D.: Irreducible \({\mathfrak{s}\mathfrak{l}}\) (m)-modules of Hermitean monogenics. ComplexVar. Elliptic Equ. 53(10), 975–987 (2008)Google Scholar
  14. 14.
    Fulton W., Harris J.: Representation theory: a first course. Springer, New York (1991)MATHGoogle Scholar
  15. 15.
    Gilbert J., Murray M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
  16. 16.
    K. Gürlebeck, K. Habetha, W. Sprössig, Holomorphic functions in the plane and ndimensional space. Translated from the 2006 German original, Birkhäuser Verlag, Basel, 2008.Google Scholar
  17. 17.
    Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1998)Google Scholar
  18. 18.
    Howe R., Tan E.-C., Willenbring J. F.: Stable branching rules for classical symmetric pairs. Trans. Amer. Math. Soc. 357, 1601–1626 (2005)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Peña-Pñena D., Sabadini I., Sommen F.: Quaternionic Clifford analysis: the Hermitian setting. Complex Anal. Oper. Theory 1, 97–113 (2007)CrossRefMathSciNetGoogle Scholar
  20. 21.
    Rocha-Chavez R., Shapiro M., Sommen F.: Integral theorems for functions and differential forms in \({{\mathbb{C}_{m}}}\) . Research Notes in Math. 428, Chapman&Hall / CRC(New York, 2002).Google Scholar
  21. 22.
    Sabadini I., Sommen F.: Hermitian Clifford analysis and resolutions. Math. Meth. Appl. Sci. 25(16–18), 1395–1414 (2002)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • F. Brackx
    • 1
  • H. De Schepper
    • 1
  • D. Eelbode
    • 2
  • R. Lávička
    • 3
  • V. Souček
    • 3
  1. 1.Clifford Research Group, Dept. of Math. Analysis, Faculty of Engineering and ArchitectureGhent UniversityGentBelgium
  2. 2.University of AntwerpAntwerpenBelgium
  3. 3.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic

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