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The Weyl algebra, spherical harmonics, and Hahn polynomials

Volume 55 / 2002

Ewa Gnatowska, Aleksander Strasburger Banach Center Publications 55 (2002), 309-322 MSC: Primary 81R10, 33C55; Secondary 16W25, 33C45, 33C80. DOI: 10.4064/bc55-0-15

Abstract

In this article we apply the duality technique of R. Howe to study the structure of the Weyl algebra. We introduce a one-parameter family of “ordering maps”, where by an ordering map we understand a vector space isomorphism of the polynomial algebra on $\mathbb R^{2d}$ with the Weyl algebra generated by creation and annihilation operators $a_1,\,\ldots,\,a_d,\,a_1^+,\,\ldots,\,a_d^+$. Corresponding to these orderings, we construct a one-parameter family of $\mathfrak{ sl}_2$ actions on the Weyl algebra, which enables us to define and study certain subspaces of the Weyl algebra — the space of Weyl spherical harmonics and the space of “radial polynomials”. For the latter we generalize results of Louck and Biedenharn, Bender et al., and Koornwinder describing the radial elements in terms of continuous Hahn polynomials of the number operator.

Authors

  • Ewa GnatowskaDepartment of Mathematical Methods of Physics
    Faculty of Physics
    University of Warsaw
    Hoża 74
    00-682 Warszawa, Poland
    e-mail
  • Aleksander StrasburgerInstitute of Mathematics
    University of Białystok
    Akademicka 2
    15-267 Białystok, Poland
    and
    Department of Mathematical Methods of Physics
    Faculty of Physics,
    University of Warsaw
    Hoża 74
    00-682 Warszawa, Poland
    e-mail
    e-mail

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