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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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