The enigma of the missing invariants on the nilradical of a Borel
type: Item
instance of: scholarly article
title: The enigma of the missing invariants on the nilradical of a Borel
zbMATH Open document ID: 1077.17014
author: Anthony Joseph
published in: Bulletin des Sciences Mathématiques
publication date: 2005-04-07T00:00:00Z
review text: The paper under review deals with the open problem to determine the semicentre \(Sz({\mathfrak p})\) of the enveloping algebra \(U({\mathfrak p})\) of a parabolic subalgebra \(\mathfrak p\) of a complex semisimple Lie algebra \(\mathfrak g\) and, in particular, to show that it is a polynomial algebra. In [\textit{F. Fauquant-Millet} and \textit{A. Joseph}, Transform. Groups 6, No. 2, 125--142 (2001; Zbl 1018.17008)] it was shown that the quantum analogue \(Sz_q({\mathfrak p})\) in \(U_q({\mathfrak p})\) is a polynomial algebra. Surprisingly, the standard specialization \(q=1\) does not recover the whole \(Sz({\mathfrak p})\) from \(Sz_q({\mathfrak p})\). In the present paper the author uses an elementary construction due to Kostant to recover most of the generators of \(Sz({\mathfrak b})\), where \(\mathfrak b\) is the standard Borel subalgebra. An extension of the construction gives all the generators except for one generator in type \(E_8\).
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 2153819
zbMATH Keywords: universal enveloping algebras
zbMATH Keywords: semi-invariants
zbMATH Keywords: parabolic subalgebras
MaRDI profile type: Publication
full work available at URL: j.bulsci.2004.02.006
OpenAlex ID: W1997179312
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DOI: 10.1016/J.BULSCI.2004.02.006
description: scientific article; zbMATH DE number 2153819