Clifford Analysis on the Space of Vectors, Bivectors and ℓ-vectors

  • Irene Sabadini
  • Frank Sommen
Part of the Trends in Mathematics book series (TM)


In this paper we propose several extensions of Clifford analysis. First, we recall the most important definitions and properties for the usual Dirac operator, i.e. the operator defined for functions of a vector variable, and we discuss the importance of using differential forms. Next we discuss Clifford analysis in several vector variables and in particular so called “symplicial monogenics”, that are related to the representation theory of Spin(m). Then we study several canonically defined Clifford algebras valued systems of differential equations defined on the space of bivectors, for which we provide a rather complete theory, and more in general on the space of ℓ-vectors.

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W.W. Adams, C.A. Berenstein, P. Loustaunau, I. Sabadini, D.C. StruppaRegular functions of several quaternionic variables and the Cauchy-Fueter complex, J. Geom. Anal. 9 (1999), 1–16.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of Algebraic Curves, Springer Verlag, New-York, 1985.zbMATHGoogle Scholar
  3. [3]
    W.W. Adams, P. Loustaunau, V. Palamodov, D.C. StruppaHartogs’ phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring, Ann. Inst. Fourier, 47 (1997), 623–640.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman Res. Notes in Math. 76, 1982.zbMATHGoogle Scholar
  5. [5]
    C. ChevalleyThe algebraic theory of spinors, Columbia Univ. Press, New York, 1954.zbMATHGoogle Scholar
  6. [6]
    D. Constales, The relative position of L 2 -domains in complex and Clifford analysis, Ph.D. thesis, Univ. Ghent, 1989–1990.Google Scholar
  7. [7]
    R. DelangheClifford analysis: history and perspective, Comp. Meth. Funct. Theor. 1 (2001), 107–153.MathSciNetzbMATHGoogle Scholar
  8. [8]
    R. Delanghe, F. Sommen, V. SoucekClifford algebra and spinor-valued functions: a function theory for the Dirac operator, Math. and Its Appl. 53, Kluwer Acad. Publ., Dordrecht, 1992.CrossRefzbMATHGoogle Scholar
  9. [9]
    A. Dimakis, F. Mueller-HoisenClifform calculations with applications to classical field theories, Class. Quantum Gravity, 8 (1991), 2093–2132.CrossRefzbMATHGoogle Scholar
  10. [10]
    M.G. Eastwood, P.W. MichorSome remarks on the Plücker relations, Rend. Circ. Mat. Palermo Suppl., 63 (2000), 83–85.MathSciNetGoogle Scholar
  11. [11]
    S.L. Eriksson-Bique, H. LeutwilerHypermonogenic Functions, in Clifford algebras and their applications in mathematical physics. Papers of the 5th international conference, Ixtapa-Zihuatanejo, Mexico, 1999. Clifford analysis. Boston, MA: Birkhuser. Prog. Phys. 19, 287–302 (2000).Google Scholar
  12. [12]
    J. Gilbert, M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Univ. Press 26, Cambridge, 1990.Google Scholar
  13. [13]
    D. Hestenes, G. Sobczyk, Clifford algebras to geometric calculus, D. Reidel, Dordrecht, 1985.Google Scholar
  14. [14]
    W. Hodge, The theory and applications of harmonic integrals, Cambridge Univ. Press, Cambridge, 1959.Google Scholar
  15. [15]
    G. Laville, Une famille de solutions de l’equation de Dirac avec champ electromagnetique quelconque, C. R. Acad. Sc. Paris, 296 (1983), 1029–1032.MathSciNetzbMATHGoogle Scholar
  16. [16]
    H. Leutwiler, Modified Clifford Analysis, Compl. Var., 17 (1992), 153–171.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H. Leutwiler, Modified Quaternionic Analysis in R3, Compl. Var., 20 (1992), 19–51MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P. Lounesto, Spinor valued regular functions in hypercomplex analysis, Ph.D. thesis, Helsinki, 1979.zbMATHGoogle Scholar
  19. [19]
    P. Lounesto, Clifford algebras and spinors, Cambridge Univ. Press, 1997.zbMATHGoogle Scholar
  20. [20]
    H. Malonek, Contributions to a geometric function theory in higher dimensions by Clifford analysis methods: monogenic functions and M-conformal mappings, F. Brackx et al. (eds.), NATO Science Series, II Math., Phys., Chem., 25 (2001), 213–222.Google Scholar
  21. [21]
    L. ManivelSymmetric functions,Schubert polynomials and degeneracy loci, Translated from the 1998 French, SMF/AMS Texts and Monographs, 6 Cours Spécialisés American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001.Google Scholar
  22. [22]
    I.R. PorteousTopological Geometry, Cambridge Univ. Press, Cambridge, 1981.CrossRefzbMATHGoogle Scholar
  23. [23]
    G.B. Rizza, Funzioni regolari nelle Algebre di Clifford, Rend. Mat., XV (1956), 1–27.Google Scholar
  24. [24]
    J. Ryan ed., Clifford Algebras in Analysis and Related Topics, Studies in Advanced Mathematics, CRC Press, 1996.zbMATHGoogle Scholar
  25. [25]
    I. Sabadini, M. Shapiro, D.C. StruppaAlgebraic analysis of the Moisil-Theodorescu system, Compl. Var. 40, (2000) 333–357.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    I. Sabadini, F. SommenSpecial first order systems in Clifford analysis and resolutions, Zeit. Anal. Anw., 21, n. 1,(2002) 27–55.MathSciNetzbMATHGoogle Scholar
  27. [27]
    I. Sabadini, F. Sommen, D.C. StruppaThe Dirac complex on abstract vector variables: megaforms, Exp. Math. 12 (2003), 351–364.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    I. Sabadini, F. Sommen, D.C. Struppa, P. Van LanckerComplexes of Dirac operators in Clifford algebras, Math. Z. 239 (2002), 293–320.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    F. SommenMonogenic Differential Calculus, Trans. Amer. Math. Soc., 326 n. 2 (1991), 613–632.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    F. SommenClifford analysis in two and several vector variables, Appl. An., 73 (1999), 225–253.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    F. SommenClifford analysis on the level of abstract vector variables, NATO Science Series, II Math., Phys., Chem., 25 (2001), 303–322.MathSciNetGoogle Scholar
  32. [32]
    F. Sommen, V. Soucek, Monogenic differential forms, Compl. Var.:Theor. Appl., 19 (1992), 81–90.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    F. Sommen, N. Van Acker, SO(m)-invariant differential operators on Clifford algebra-valued functions, Found. Phys. 23, n. 11 (1993), 1491–1519.MathSciNetCrossRefGoogle Scholar
  34. [34]
    F. Sommen, N. Van Acker, Functions of two vector variables, Adv. Appl. Cliff. Alg. 4, n.1 (1994) 65–72.zbMATHGoogle Scholar
  35. [35]
    E.M. Stein, G. WeissGeneralization of the Cauchy—Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163--196.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    P. Van Lancker, F. Sommen, D. Constales, Models for irreducible representations of Spin(m), Adv. Appl. Cliff. Alg. 11 (2001), 271–289.CrossRefzbMATHGoogle Scholar
  37. [37]
    C. von Westenholz, Differential forms in mathematical physics, Stud. Math. Appl. 3, North Holland, Amsterdam, 1978.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Irene Sabadini
    • 1
  • Frank Sommen
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Department of Mathematical AnalysisState University of GhentGentBelgium

Personalised recommendations