Advertisement

Advances in Applied Clifford Algebras

, Volume 27, Issue 2, pp 1103–1132 | Cite as

Basic Aspects of Symplectic Clifford Analysis for the Symplectic Dirac Operator

  • Hendrik De Bie
  • Marie Holíková
  • Petr Somberg
Article
  • 66 Downloads

Abstract

In the present article we study basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of real dimension 2, this involves the analysis of first order symmetry operators, symplectic Clifford-Fourier transform, reproducing kernel for the symplectic Fischer product and the construction of bases of symplectic monogenics for the symplectic Dirac operator.

Keywords

Symplectic Dirac operator Symplectic spinors Symmetry operators Reproducing kernel Fischer product Bases of symplectic monogenics 

Mathematics Subject Classification

Primary 53C27 Secondary 53D05 81R25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Britten D.J., Lemire F.W.: On modules of bounded multiplicities for the symplectic algebras. Trans. Am. Math. Soc. 351, 3413–3431 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls (Springer Monographs in Mathematics). Springer, Berlin. ISBN: 978-1-4614-6659-8 (2013)Google Scholar
  3. 3.
    De Bie H., Ørsted B., Somberg P., Souček V.: Dunkl operators and a family of realizations of \({\mathfrak{osp}}\)(1|2). Trans. Am. Math. Soc. 364, 3875–3902 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    De Bie, H., Ørsted, B., Somberg, P., Souček, V.: The Clifford deformation of the Hermite semigroup. SIGMA 9, 010 (2013)Google Scholar
  5. 5.
    De Bie, H., Xu, Y.: On the Clifford–Fourier transform. Int. Math. Res. Not. IMRN (22), 5123–5163 (2011)Google Scholar
  6. 6.
    De Bie H., Somberg P., Souček V.: The Howe duality and polynomial solutions for the symplectic dirac operator. J. Geom. Phys. 75, 120–128 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    De Bie, H., Sommen, F., Wutzig, M.: Reproducing kernels for polynomial null-solutions of Dirac operators. Constr. Approx. doi: 10.1007/s00365-016-9326-6 (To appear)
  8. 8.
    Dostálová M., Somberg P.: Symplectic twistor operator on R 2n and the Segal-Shale-Weil representation. Complex Anal. Oper. Theory 8(2), 513–528 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dostálová M., Somberg P.: Symplectic twistor operator and its solution space on \({{\mathbb{R}}^2}\). Archivum Math. 49(3), 161–185 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Eastwood M.: Higher symmetries of the Laplacian. Ann. Math. 161, 1645–1665 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Eastwood, M., Somberg P., Souček, V.: Higher symmetries of the Dirac operator (In preparation)Google Scholar
  12. 12.
    Habermann, K., Habermann, L.: Introduction to Symplectic Dirac Operators, Lecture Notes in Math. Springer-Verlag, Berlin-Heidelberg. ISSN 0075-8434 (2006)Google Scholar
  13. 13.
    Kadlčáková, L.: Dirac operator in parabolic contact symplectic geometry. Ph.D. thesis, Charles University Prague, Prague (2001)Google Scholar
  14. 14.
    Kostant, B.: Symplectic Spinors. XIV, Rome Symposia, pp. 139–152 (1974)Google Scholar
  15. 15.
    Szegö G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Hendrik De Bie
    • 1
  • Marie Holíková
    • 2
  • Petr Somberg
    • 3
  1. 1.Department of Mathematical Analysis, Faculty of Engineering and ArchitectureGhent UniversityGentBelgium
  2. 2.Department of Mathematics and Mathematical Education, Faculty of EducationCharles UniversityPraha 1Czech Republic
  3. 3.Mathematical Institute of Charles UniversityPraha 8-KarlínCzech Republic

Personalised recommendations