an:00783865
Zbl 0833.16025
Jordan, David A.
A simple localization of the quantized Weyl algebra
EN
J. Algebra 174, No. 1, 267-281 (1995).
00026929
1995
j
16S36 16S90 16P60 17B37 16S20 16D60
quantum groups; Krull dimension; skew polynomial rings; affine algebras; normal elements; quantized Weyl algebras; commuting normal elements; localizations; global dimension
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 \textit{J. Alev} and \textit{F. Dumas} [J. Algebra 170, No. 1, 229-265 (1994; Zbl 0820.17015)] and related work by \textit{J. C. McConnell} and \textit{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.
M.Lorenz (Philadelphia)
Zbl 0820.17015; Zbl 0652.16007