Hermitian Clifford Analysis on Superspace

  • Hennie De Schepper
  • Alí Guzmán AdánEmail author
  • Frank Sommen


In this paper we first recall the proper algebraic framework, i.e. the radial algebra, needed to extend Hermitian Clifford analysis to the superspace setting. The fundamental objects for this extension then are introduced by means of an abstract complex structure on the Hermitian radial algebra. This leads to a natural representation of this Hermitian radial algebra on superspace.


Hemitian Clifford analysis Superspace Radial algebra 


  1. 1.
    Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F., Souček, V.: Fundaments of hermitean clifford analysis part I: complex structure. Complex Analysis and Operator Theory 1(3), 341–365 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brackx, F., De Schepper, H., Sommen, F.: The hermitian clifford analysis toolbox. Advances in Applied Clifford Algebras 18(3), 451–487 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brackx, F., De Schepper, H., Souček, V.: Fischer decompositions in euclidean and hermitean clifford analysis. Archivum Mathematicum 46(5), 301–321 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra. Progress in Mathematical Physics, vol. 39. Birkhäuser Boston Inc, Boston (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    De Bie, H., Sommen, F.: A clifford analysis approach to superspace. Annals of Physics 322(12), 2978–2993 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Bie, H., Sommen, F.: Correct rules for clifford calculus on superspace. Advances in Applied Clifford Algebras 17(3), 357–382 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Bie, H., Sommen, F., Wutzig, M.: Reproducing kernels for polynomial null-solutions of dirac operators. Constructive Approximation 44(3), 339–383 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De Schepper, H., Adán, A.G., Sommen, F.: The radial algebra as an abstract framework for orthogonal and hermitian clifford analysis. Complex Analysis and Operator Theory 11(5), 1139–1172 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    De Schepper, H., Adán, A.G., Sommen, F.: Spin actions in Euclidean and Hermitian Clifford analysis in superspace. Journal of Mathematical Analysis and Applications 457(1), 23–50 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De Schepper, H., Adan, A. G., Sommen, F.: The spin group in superspace. (2017) (Submitted for publication) Google Scholar
  11. 11.
    Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and Spinor-Valued Functions, Volume 53 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1992). A function theory for the Dirac operator, Related REDUCE software by F. Brackx and D. Constales, With 1 IBM-PC floppy disk (3.5 inch)zbMATHGoogle Scholar
  12. 12.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Fundamental Theories of Physics. D. Reidel Publishing Co., Dordrecht (1984). A unified language for mathematics and physicsCrossRefzbMATHGoogle Scholar
  13. 13.
    Sabadini, I., Sommen, F.: Hermitian clifford analysis and resolutions. Mathematical Methods in the Applied Sciences 25(16–18), 1395–1413 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sabadini, Irene, Sommen, Frank, Struppa, Daniele C: The dirac complex on abstract vector variables: megaforms. Experimental mathematics 12(3), 351–364 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sommen, F.: An extension of Clifford analysis towards super-symmetry. In: Ryan, J., Sprößig, W. (eds) Clifford algebras and their applications in mathematical physics, pp 199–224. Progress in Physics, vol 19. Birkhäuser, Boston, MA (2000)Google Scholar
  16. 16.
    Sommen, F.: An algebra of abstract vector variables. Portugaliae Mathematica 54(3), 287–310 (1997)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sommen, Frank: The problem of defining abstract bivectors. Results in Mathematics 31(1–2), 148–160 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sommen, F.: Clifford analysis on super-space. Advances in Applied Clifford Algebras 11(1), 291–304 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sommen, F.: Clifford analysis on super-space. Progress in Analysis 1, 383–405 (2003)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Clifford Research Group Department of Mathematical Analysis Faculty of Engineering and ArchitectureGhent UniversityGentBelgium

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