An algebraic slice in the coadjoint space of the Borel and the Coxeter element
type: Item
instance of: scholarly article
title: An algebraic slice in the coadjoint space of the Borel and the Coxeter element
zbMATH Open document ID: 1251.17007
published in: Advances in Mathematics
publication date: 2011-04-20T00:00:00Z
review text: Let \(\mathfrak{g}\) be a complex simple Lie algebra and \(\mathfrak{b}\) a Borel subalgebra. There is a canonically determined ideal \(\mathfrak{b}_E\) of \(\mathfrak{b}\) such that the algebra generated by semi-invariants on \(\mathfrak{b}^*\) coincides with the algebra \(Y(\mathfrak{b}_E)\) of invariants on \(\mathfrak{b}^*_E\). If \(\mathfrak{g}\) is of type \(A\) there exists an algebraic slice to coadjoint action defined to be an affine translate \(y+V\subset\mathfrak{b}^*_E\) of a vector subspace \(V\) of \(\mathfrak{b}^*_E\) such that the restriction map induces an isomorphism of \(Y(\mathfrak{b}^*_E)\) onto the algebra \(R[y+V]\) of regular functions on \(y+V\). This result also holds to all biparabolic subalgebras in type \(A\). The author proves that an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of type \(B(2m)\), \(C(n)\), and \(F(4)\). Outside type \(A\), the slice obtained meets an open dense subset of regular orbits, even though the special point \(y\) of the slice is not itself regular. This explains the failure of the previous construction.
reviewed by: Nikolaos Papalexiou
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 5880833
zbMATH Keywords: invariant
zbMATH Keywords: adapted pairs
zbMATH Keywords: coadjoint action of Borel subalgebra
zbMATH Keywords: simple Lie algebras
author: Anthony Joseph
MaRDI profile type: Publication
full work available at URL: j.aim.2011.02.006
OpenAlex ID: W2004449803
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DOI: 10.1016/J.AIM.2011.02.006
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description: scientific article; zbMATH DE number 5880833