an:00871360
Zbl 0851.16025
Dumas, Fran??ois; Jordan, David A.
The \(2\times 2\) quantum matrix Weyl algebra
EN
Commun. Algebra 24, No. 4, 1409-1434 (1996).
00032856
1996
j
16S36 17B37 16D25 16P40 16P50 16P60
algebras of differentials on quantum affine spaces; \(2\times 2\) quantum matrices; quantum Weyl algebras; simple localizations; Krull and global dimensions; normal elements; height 1 primes; iterated skew polynomial rings
The algebras of differentials on quantum affine spaces introduced by \textit{G. Maltsiniotis} [Calcul diff??rentiel quantique, Groupe de travail, Universit?? de Paris VII (1992)] have been studied from the point of view of noncommutative ring theory in a number of papers -- e.g., \textit{Akhavizadegan} and the second author [Prime ideals of quantized Weyl algebras (Glasg. Math. J., to appear)]; \textit{J. Alev} and the first author [J. Algebra 170, No. 1, 229-265 (1994; Zbl 0820.17015)]; \textit{G. Cauchon} [J. Algebra 180, No. 2, 530-545 (1996; Zbl 0849.16028)]; \textit{T. H. Lenagan} and the reviewer [J. Pure Appl. Algebra 111, 123-142 (1996)]; the second author [J. Algebra 174, No. 1, 267-281 (1995; Zbl 0833.16025)]; and \textit{L. Rigal} [Beitr. Algebra Geom. 37, No. 1, 119-148 (1996)]. Here the authors consider an algebra \(W_{p,q}\) of differentials on two-parameter \(2\times 2\) quantum matrices defined by \textit{G. Maltsiniotis} [in Commun. Math. Phys. 151, No. 2, 275-302 (1993; Zbl 0783.17007)], and investigate its similarities with the quantum Weyl algebras \(A^{\overline{q},\Lambda}_n\) studied earlier. Similarities: \(W_{p,q}\) has a simple localization of Krull and global dimension 4 obtained by inverting a finite set of normal elements, and this localization is isomorphic to a corresponding localization of an \(A^{\overline{q},\Lambda}_4\) for suitable choices of parameters \(\overline{q},\Lambda\). Dissimilarity: \(W_{p,q}\) has 3 height 1 primes, rather than 4 as in any \(A^{\overline{q},\Lambda}_4\). In particular, \(W_{p,q}\) is not isomorphic to any \(A^{\overline{q},\Lambda}_4\). These properties of \(W_{p,q}\) are derived by presenting the algebra as an iterated skew polynomial ring in such a way that a generalization of the techniques developed by the second author [ibid.] can be applied.
K.R.Goodearl (Santa Barbara)
Zbl 0820.17015; Zbl 0833.16025; Zbl 0783.17007; Zbl 0849.16028