Noetherian Algebras of Quantum Differential Operators

Abstract

Abstract. We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum n-space for n ≥ 3. In the former case a set of generators for the quantum differential operators was identified in work by the first author and T. C. McCune but it was not known whether the algebra is Noetherian. We answer this question affirmatively, setting it in a more general context involving the behaviour Noetherian condition under localization at the powers of a single element. In the latter case we determine the algebra of quantum differential operators as a skew group algebra of the group Z n over a quantized Weyl algebra. It follows from this description that this algebra is a simple right and left Noetherian domain.

Simple ambiskew polynomial rings

“…Remark 6.6. The algebras A Λ n can be interpreted in terms of quantum differential operators in the sense of Lunts and Rosenberg [27], see [15].…”

Representations of D q (k[x])

“…Then following [5], the algebra of quantum differential operators on k[x], denoted by D q (k[x]), for a specific bicharacter β was constructed in [4]. In [3], the algebra D q (k[x]) was described in terms of generators and relations.…”

“…Then, D q (k[x]) is the subalgebra of the algebra of k-linear homomorphisms, Hom(k[x], k[x]), generated by the maps {x, ∂ 1 , ∂ −1 , ∂ 0 } ( [4]). The defining relations ( [3]) among these maps are ∂ a x − q a x∂ a = 1, (1.1)…”

“…In [3], it is shown that D q (k[x]) is a left and right Noetherian simple domain of GK dimension 3 when q is transcendental over Q.…”