Jordan, David A. A simple localization of the quantized Weyl algebra. (English) Zbl 0833.16025 J. Algebra 174, No. 1, 267-281 (1995). 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. Reviewer: M.Lorenz (Philadelphia) Cited in 6 ReviewsCited in 45 Documents 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 Keywords:quantum groups; Krull dimension; skew polynomial rings; affine algebras; normal elements; quantized Weyl algebras; commuting normal elements; localizations; global dimension Citations:Zbl 0820.17015; Zbl 0652.16007 PDFBibTeX XMLCite \textit{D. A. Jordan}, J. Algebra 174, No. 1, 267--281 (1995; Zbl 0833.16025) Full Text: DOI