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Noetherian algebras of quantum differential operators. (English) Zbl 1339.16027
Let $$k$$ be a field of characteristic zero and $$\Gamma$$ an abelian group with a bicharacter $$\beta\colon\Gamma\otimes\Gamma\to k^*$$. Under this assumption one can define on a $$\Gamma$$-graded algebra $$R$$ commutators $$[x,y]_a=xr-\beta(a,d_y)x$$ for any $$a\in\Gamma$$ and homogeneous element $$y\in R_{d_y}$$. Following ideas from a paper by V. A. Lunts and A. L. Rosenberg [Sel. Math., New Ser. 3, No. 3, 335-359 (1997; Zbl 0937.16040)], the algebra $$D(R)$$ of quantum differential operators on $$R$$ is introduced.
The first part of the present paper deals with the case when $$\Gamma=\mathbb Z$$ and $$R$$ is a (Laurent) polynomial algebra in one variable. It is shown that the algebra $$D(R)$$ is left and right Noetherian. This property is preserved under localizations by powers of a single element. In the case of several variables it is shown that the algebra of quantum differential operators is a skew group algebra of the group $$\mathbb Z^n$$ over quantized Weyl algebra. It follows that $$D(R)$$ is simple, left and right Noetherian.

##### MSC:
 16S32 Rings of differential operators (associative algebraic aspects) 16S36 Ordinary and skew polynomial rings and semigroup rings 16P50 Localization and associative Noetherian rings
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