Following the definition given in [6], we compute the ring of quantum differential operators on the polynomial ring in 1 variable. We further study this ring.
Abstract. After recalling the definitions and the properties of the generic base algebra and of the universal comodule algebra attached to a Hopf algebra by Aljadeff and the second-named author, we determine these algebras for the Taft algebras, the Hopf algebras E(n) and certain monomial Hopf algebras.
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.
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