The Weyl Algebra, Spherical Harmonics, and Hahn Polynomials
The Weyl Algebra, Spherical Harmonics, and Hahn Polynomials
复制标题
外尔代数、球谐函数和哈恩多项式
DOI：
10.4064/bc55-0-15
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
A. Strasburger
中科院分区：
文献类型：
--
作者：
E. Gnatowska;A. Strasburger
本文应用R. Howe的对偶技术来研究Weyl代数的结构。我们引入了一个单参数的“序映射”族，通过序映射我们理解了$\NR^{2d}$上的多项式代数与由创造和湮灭算子$a_1，…， a_d, a_1^+，…, a_d ^ + $。与这些排序相对应，我们构造了Weyl代数上${\fr sl}_2$作用的单参数族，使我们能够定义和研究Weyl代数的某些子空间——Weyl球谐波空间和“径向多项式”空间。对于后者，我们推广了Luck和Biedenharn， Bender等人，以及Koornwinder用数字算子的连续Hahn多项式描述径向元素的结果。
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 $\NR^{2d}$ with the Weyl algebra generated by creation and annihilation operators $a_1, ..., a_d, a_1^+, ..., a_d^+$. Corresponding to these orderings, we construct a one-parameter family of ${\fr sl}_2$ actions on the Weyl algebra, what 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 Luck and Biedenharn, Bender et al., and Koornwinder describing the radial elements in terms of continuous Hahn polynomials of the number operator.