A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible Modules

Abstract

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h generated by elements x, y, which satisfy yxWhen h = 0, the algebras A h are subalgebras of the Weyl algebra A 1 and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of A h and determined its automorphism group Aut F (A h ) and the subalgebra of invariants under Aut F (A h ). Here we determine the irreducible A h -modules. In a sequel to this paper, we completely describe the derivations of A h over any field.

Normally ordered forms of powers of differential operators and their combinatorics

“…Hence, the normally ordered form of the operator (h∂ d ) n , which we have been discussing, yields in particular known expressions for the normally ordered form of elements of this type in the Weyl algebra, with ∂ = ∂ x and h ∈ F[x] (see [5], [6] and [15]). In this section we will apply our results to the more general setting of formal differential operator rings, which include in particular the subalgebras A h of the Weyl algebra studied in [1,2,3] and defined below in (7.1).…”

“…. ; t) in noncommuting variables y i , such that for any ring A, derivation ∂ of A and element h ∈ A, (h∂) n = V n (h, h [1] , . .…”

“…The motivation for this work comes from [1], where the powers of the derivation h d dx appear in the formulas for the action of a certain subalgebra A h of the Weyl algebra A 1 on its irreducible modules. In [1,Sec. 8], some properties of the expression U n for the normally ordered form of h d dx were rediscovered.…”

“…Lastly, in Section 7 it is shown how the polynomials U n,d provide the normally ordered form of elements in formal differential operator rings A[z; ∂], including as special cases the Weyl algebra A 1 and its family of subalgebras A h studied in [1,2,3]. Table 1: The coefficients c n ν of the polynomials U n for n up to 5.…”

“…studied in[1,2,3] and determined by the derivation δ = h∂ x , where h ∈ F[x]. Then A h admits the presentationA h = F x,ŷ | [ŷ, x] = h(x) .= 1 we retrieve the Weyl algebra A 1 and, for an arbitrary nonzero h ∈ F[x], we can view A h as the (unital) F-subalgebra of A 1 generated by x and yh, under the identificationŷ = yh.…”