The Weyl algebra, spherical harmonics, and Hahn polynomials

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 R 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 sl 2 actions on the Weyl algebra, which enables us to define and study certain subspaces of the Weyl algebrathe 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.

A connection between operator orderings and representations of the Lie algebra \mathfrak{sl}_2

“…It can be shown [10] that in general the space H k is independent of the ordering O-for μ-orderings it follows immediately from (5.8), and consequently that this decomposition depends only on the ordering O through the operator R.…”

“…Given w ∈ W, we shall denote by L w : W → W, R w : W → W linear maps of W obtained by left or right multiplication with w, respectively. Following the PhD Thesis of Gnatowska [10] we considered in our paper [11] one-parameter families p(μ) , Q(μ) of linear maps of the Weyl algebra W, with μ ∈ [0, 1], defined by…”

“…has the required properties. We can also check that the following formula for E E = −i p(μ) ad( q) + i ad( p) Q(μ) , defines an operator satisfying (5.6), cf [10,11].…”

“…In the remaining part of the paper we shall detail, adapting to the present context methods of [10,11], the structure of the subspace W 0 of radial elements for the case of μ-orderings. It will be shown that it is in fact a commutative subalgebra of W, the algebra of polynomials in the operator N = O(pq).…”

“…Our interest in the problem was stimulated, on one hand, by the viewpoint of Howe, who explores the role that dual pairs play in the structure of the Weyl algebra [12,13], and on the other hand by the discovery of various connections of this problem with the special functions theory, among others done in the papers [2,3,5,15,16]. Verc ¸in in a recent paper [18] systematically investigated properties of a class of polynomials connected with a certain oneparameter family of orderings, while independently although somewhat later [10] that class of polynomials was investigated from the viewpoint exposed in the present paper by Gnatowska in her Doctor's Thesis at the University of Warsaw-cf also [11].…”