On the topology of the Dixmier map
type: Item
instance of: scholarly article
title: On the topology of the Dixmier map
zbMATH Open document ID: 0970.17008
DOI: 10.1007/BF02942549
author: Walter Borho
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2000-03-19T00:00:00Z
review text: Let \(\mathfrak g\) be a finite dimensional semisimple complex Lie algebra with coadjoint group \(G\). The Dixmier map associates a primitive ideal of the universal enveloping algebra \(U({\mathfrak g})\) of \(\mathfrak g\) with any pair \((y,{\mathfrak p})\), where \(y\in{\mathfrak g}^{\ast}={\mathfrak g}\) (by the Killing form) and \(\mathfrak p\) is a polarization of \(y\) (a parabolic subalgebra). For \({\mathfrak g}={\mathfrak s}{\mathfrak l}_n\) one can forget about \(\mathfrak p\) and obtains a mapping \(\text{Dix}:{\mathfrak g}^{\ast}/G\to \mathcal X\), where the orbit space \({\mathfrak g}^{\ast}/G\) is equipped with the factor-topology of the Zariski topology and the space \({\mathcal X}\) of all primitive ideals of \(U({\mathfrak g})\) with the Jacobson topology. In this case \(\text{Dix}\) is a bijection (but not a homeomorphism) on the space \({\mathcal X}^1\) of all completely prime primitive ideals. In the paper under review the author uses a result of [\textit{W. Soergel}, Math. Z. 204, 559-581 (1990; Zbl 0685.17006)] and shows that \(\text{Dix}\) is sheet-wise a homeomorphism onto the image, where a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension (and \(\mathfrak g\) decomposes into finitely many sheets). The above result holds in the general case of any semisimple \(\mathfrak g\) when one has to restrict to a sheet of polarizable orbits where the Dixmier map can be defined. Let \(\mathbf I\) be the annihilator of the generic module induced from a fixed polarization \({\mathfrak p}\subset {\mathfrak g}\). The author describes the centre \(Z\) of the ``relative enveloping algebra'' \(U=U({\mathfrak g})/{\mathbf I}\) in terms of a relative Harish-Chandra isomorphism of the normalization \(\widetilde Z\) with a suitable ring of group invariants. He suggests that the very mild central extension \(\widetilde U\) of \(U\) by \(\widetilde Z\) enjoys good properties and is very suitable for the study of the Dixmier map. In pa…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1416663
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: Dixmier map
zbMATH Keywords: primitive ideals
zbMATH Keywords: semisimple Lie algebra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1416663