Derivations of a parametric family of subalgebras of the Weyl algebra

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]. When h = 0, the algebra A h is subalgebra of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of A h and the Lie structure of the first Hochschild cohomology group HH 1 (A h ) = Der F (A h )/Inder F (A h ) of outer derivations over an arbitrary field. In characteristic 0, we show that HH 1 (A h ) has a unique maximal nilpotent ideal modulo which it is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic, we obtain decomposition theorems for Der F (A h ) and HH 1 (A h ) and describe the structure of HH 1 (A h ) as a module over the center of A h .

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

“…In [BLO1], we determined the center, normal elements, prime ideals, and automorphisms of A h and their invariants in A h . In [BLO2], we determine the derivations of an arbitrary algebra A h over any field, derive expressions for the Lie bracket in the quotient HH 1 (A h ) := Der F (A h )/Inder F (A h ) of Der F (A h ) modulo the ideal Inder F (A h ) of inner derivations, and use these formulas to understand the structure of the Lie algebra HH 1 (A h ). In particular, when char(F) = 0, we construct a maximal nilpotent ideal of HH 1 (A h ) and explicitly describe the structure of the corresponding quotient in terms of the Witt algebra (centreless Virasoro algebra) of vector fields on the unit circle.…”

Normally ordered forms of powers of differential operators and their combinatorics

“…Theorem 6.2. For any n ≥ 0 and d ≥ 1, there is a unique polynomial U n,d ∈ i≥0 Rt i ⊆ R t such that, for any ring A, derivation ∂ of A and central element h in A, [1] , h [2] , . .…”

“…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.…”