A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms

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 yx − xy = h, where h ∈ F[x]. We investigate the family of algebras A h as h ranges over all the polynomials in F [x]. When h = 0, the algebras A h are subalgebras of the Weyl algebra A 1 and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of A h over arbitrary fields F and describe the invariants in A h under the automorphisms. We determine the center, normal elements, and height one prime ideals of A h , localizations and Ore sets for A h , and the Lie ideal [A h , A h ]. We also show that A h cannot be realized as a generalized Weyl algebra over F [x], except when h ∈ F. In two sequels to this work, we completely describe the irreducible modules and derivations of A h over any field.

Derivations of a parametric family of subalgebras of the Weyl algebra

“…where f ′ is the usual derivative of f with respect to x. Quantum planes and quantum Weyl algebras are examples of generalized Weyl algebras in the sense of [B, 1.1], and as such, have been studied extensively. In [BLO1,BLO2], we determined the center, normal elements, and prime ideals of the algebras A h , as well as the automorphisms and their invariants, isomorphisms between two algebras A g and A h , and the irreducible A h -modules over any field F. Our aim in this paper is to compute the derivations and first cohomology group of the algebras A h over an arbitrary 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).…”

“…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.…”

On the automorphisms of quantum Weyl algebras

“…Tsuchimoto, [31], and Belov-Kanel and Kontsevich, [8], proved independently that the Dixmier conjecture is stably equivalent to the Jacobian conjecture of Keller [17]. It is natural to ask Dixmier's question for related algebras (see [5,25]), and especially generalizations and quantizations of the Weyl algebras (see [2,10]). In [19], every endomorphism of A α,q (and more generally simple quantum generalized Weyl algebras), when q is not a root of unity, was shown to be an automorphism.…”