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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.

16S32 Rings of differential operators (associative algebraic aspects)
16S36 Ordinary and skew polynomial rings and semigroup rings
16P50 Localization and associative Noetherian rings
Full Text: DOI arXiv
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