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A simple localization of the quantized Weyl algebra. (English) Zbl 0833.16025
Generalizing earlier work of the author, this paper describes a general construction of a skew polynomial ring $$R$$ in two variables over an affine $$k$$-algebra $$A$$ ($$k$$ is any field). The construction critically depends on the choice of a normal element in $$A$$, and it generates a normal element in $$R$$ which can then be used for iteration. Various classes of algebras of interest are obtained in this fashion, most notably the quantized Weyl algebras $$A^{\overline {q},\Lambda}_n$$ in $$2n$$ variables which form the main topic of the article. Building on earlier work of J. Alev and F. Dumas [J. Algebra 170, No. 1, 229-265 (1994; Zbl 0820.17015)] and related work by J. C. McConnell and J. J. Pettit [J. Lond. Math. Soc., II. Ser. 38, 47-55 (1988; Zbl 0652.16007)], the author constructs a set $$Z$$ of $$n$$ commuting normal elements in $$A^{\overline {q}, \Lambda}_n$$ such that, provided no member of $$\overline{q}\in(k^\bullet)^n$$ is a root of unity, the localization $$B^{\overline {q}, \Lambda}_n=(A^{\overline{q},\Lambda}_n)_Z$$ is simple. Under the same hypothesis on $$\overline {q}$$, it is also shown that $$B^{\overline {q}, \Lambda}_n$$ has Krull and global dimension $$n$$, all of which perfectly mirrors the situation for the classical $$n$$-th Weyl algebra in characteristic 0.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16S20 Centralizing and normalizing extensions 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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