Sheets and hearts of prime ideals in enveloping algebras of semisimple Lie algebras
type: Item
instance of: scholarly article
title: Sheets and hearts of prime ideals in enveloping algebras of semisimple Lie algebras
zbMATH Open document ID: 1162.17011
author: Walter Borho
author: Rudolf Rentschler
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: The authors consider the enveloping algebra \(U(\mathfrak g)\) of a complex semisimple Lie algebra \(\mathfrak g\). The heart of a prime ideal \(I\) of \(U(\mathfrak g)\) is the center of the total ring of fractions of \(U(\mathfrak g)/I\). This is an extension field of the field of fractions of the center of \(U(\mathfrak g)/I\). Let \(d\) be the degree of this field extension. The an old problem of J. Dixmier asked whether \(d=1\). A recent paper of the second author [\textit{R. Rentschler}, Commun. Algebra 36, No. 3, 1153--1170 (2008; Zbl 1204.17009), see also the preprint, 2004] gave a negative answer by an example in \(\mathfrak{sl}_4\). Now in the present paper the authors give many more examples, involving the so-called sheets of primitive ideals introduced and studied by the first author and \textit{A. Joseph} [J. Algebra 244, No. 1, 76--167 (2001); corrigendum 259, No. 1, 310--311 (2003); Zbl 1078.17005)]. A sheet corresponds to a prime ideal \(I\) which has a heart of degree \(d\). The main result of this paper is that \(d\) equals the covering degree of the sheet as introduced in [W. Borho, A. Joseph, loc.cit., 8.7].
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 5078828
zbMATH Keywords: sheets
zbMATH Keywords: hearts
zbMATH Keywords: problem of Dixmier
zbMATH Keywords: enveloping algebras
Wikidata QID: Q104089754
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.02.006
OpenAlex ID: W1988322958
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DOI: 10.1016/J.JALGEBRA.2006.02.006
description: scientific article; zbMATH DE number 5078828