A slice theorem for truncated parabolics of index one and the Bézout equation
type: Item
instance of: scholarly article
title: A slice theorem for truncated parabolics of index one and the Bézout equation
zbMATH Open document ID: 1141.17007
author: Anthony Joseph
published in: Bulletin des Sciences Mathématiques
publication date: 2007-05-04T00:00:00Z
review text: The paper contains a confirmation of a conjecture stated by the author in the paper [``Parabolic actions in type \(A\) and their eigenslices.'' Transform. Groups 12, No. 3, 515--547 (2007; Zbl 1232.17020)]. The conjecture (called slice conjecture) states that for a parabolic subalgebra \({\mathfrak p}\) of a simple Lie algebra \({\mathfrak g}\), the coadjoint orbits in the canonical truncation \({\mathfrak p}_E\) admits a slice. In the same paper, this conjecture was established for parabolics in type \(A\) which are invariant under the Dynkin diagram involution. In the paper under review, the slice conjecture is confirmed for parabolics of index one in \({\mathfrak {sl}}(n)\). The technique developed leads to another algorithm of solving the Bézout equation \(xp-yq=-1\), with \(p,q\) coprime. The method seems to be less efficient than the Euclidean algorithm, but it is a remarkable connection between the combinatorics of algebraic Lie algebras and elementary number theory.
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 5149357
zbMATH Keywords: simple Lie algebra
zbMATH Keywords: coadjoint action
zbMATH Keywords: parabolic subalgebra
reviewed by: Pavel Kolesnikov
MaRDI profile type: Publication
full work available at URL: j.bulsci.2006.07.003
OpenAlex ID: W1978463769
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DOI: 10.1016/J.BULSCI.2006.07.003
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description: scientific article; zbMATH DE number 5149357