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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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