Vanishing of derivations and prime spectra of quantum algebras
type: Item
instance of: scholarly article
title: Vanishing of derivations and prime spectra of quantum algebras
zbMATH Open document ID: 1017.16017
DOI: 10.1016/S0021-8693(02)00542-2
author: Gérard Cauchon
published in: Journal of Algebra
publication date: 2003-04-28T00:00:00Z
review text: The object of study in the paper is an iterated Ore extension \[ k=A_0\subset k[X_1]=A_1\subset k[X_1][X_2;\sigma_2,\delta_2]=A_2\subset \cdots\subset k[X_1][X_2;\sigma_2,\delta_2]\cdots[X_N;\sigma_N,\delta_N]=A_N,\quad N\geq 2, \] where \(k\) is a field, \(\sigma_j\) is an automorphism and \(\delta_j\) is a locally nilpotent \(\sigma_j\)-derivation of \(A_{j-1}\). Moreover \(\sigma_j\delta_j=q_j\delta_j\sigma_j\) for some \(q_j\in k^*\), \(j\geq 2\), which is not a root of 1 and \(\sigma_j(X_i)=\lambda_{ji}X_i\), \(i<j\), where \(\lambda_{ji}\in k^*\). The algebra \(R=A_N\) is an Ore domain with a division ring of fractions \(F=\text{Fract }R\). The paper presents an algorithm for search of a new system of variables \(T_1,\ldots,T_N\) in \(F\) generating a quantum affine space \(\overline R\subset F\) with some nice properties. For example there is a natural embedding \(\phi\colon\text{Spec}\;R\to\text{Spec}\;\overline R\) such that \(\text{Fract}(R/P)\simeq\text{Fract}(\overline R/\phi(P))\) for any \(P\in\text{Spec}\;R\). There is given some additional information concerning the map \(\phi\). For an application the author considers the cases when \(R\) is a quantum universal enveloping algebra, coordinate algebra of a quantum group, quantum Weyl algebra, quantum Euclidean space etc.
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1901834
zbMATH Keywords: iterated Ore extensions
zbMATH Keywords: Ore localizations
zbMATH Keywords: prime spectra
zbMATH Keywords: derivations
zbMATH Keywords: Ore domains
zbMATH Keywords: quantum affine spaces
zbMATH Keywords: quantum universal enveloping algebras
zbMATH Keywords: quantum Weyl algebras
zbMATH Keywords: quantum groups
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00542-2
OpenAlex ID: W1992809880
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description: scientific article; zbMATH DE number 1901834