Discrete valuations on Weyl skew fields
type: Item
instance of: scholarly article
title: Discrete valuations on Weyl skew fields
zbMATH Open document ID: 0937.16050
author: Luc Willaert
published in: Journal of Algebra
publication date: 2000-06-04T00:00:00Z
review text: This paper begins a study of the \(k\)-valuations of \(D_1(k)\), the skew-field of fractions of the first Weyl algebra \(A_1(X,Y)\). Attention is restricted to discrete \(k\)-valuations which are compatible with the Bernstein filtration of \(D_1(k)\) defined by the total degree in \(X\) and \(Y\) and such that the graded algebra associated to the filtration induced by the valuation is commutative. A complete description is given of all such valuations. Apart from the valuation induced by the Bernstein filtration itself, the rest are parameterized by a triple \((p,q,\alpha)\in\mathbb{Z}\setminus\mathbb{N}\times\mathbb{N}\setminus\{0\}\times k^*\) such that \(q\leq-p\). This description relies on a proposition to the effect that the valuations under consideration are completely determined by their restricitions to the subfield \(k(XY^{-1})\) of \(D_1(k)\).
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16W70
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 16K40
zbMATH DE Number: 1014023
zbMATH Keywords: skew fields of fractions
zbMATH Keywords: Weyl algebras
zbMATH Keywords: discrete valuations
zbMATH Keywords: Bernstein filtrations
zbMATH Keywords: graded algebras
reviewed by: T. H. Lenagan
MaRDI profile type: Publication
full work available at URL: jabr.1996.6833
OpenAlex ID: W2046885730
cites work: Sur les algèbres de Weyl
cites work: zariskian filtrations
cites work: Discrete Valuations Extend to Certain Algebras of Quantum Type
cites work: Q5796078
cites work: Valuations on extensions of Weyl skew fields
DOI: 10.1006/JABR.1996.6833
description: scientific article; zbMATH DE number 1014023