Special transverse slices and their enveloping algebras
type: Item
instance of: scholarly article
title: Special transverse slices and their enveloping algebras
zbMATH Open document ID: 1005.17007
author: Alexander Premet
published in: Advances in Mathematics
publication date: 2002-11-28T00:00:00Z
full work available at URL: GGG.pdf
review text: This paper deals with the singularities arising in Lie theory as the intersection of the nilpotent cone of a Lie algebra and the special slice. It is shown that these singularities admit natural deformations. Let \(G\) be a simply connected algebraic group with Lie algebra \(g\) and \(N(g)\) be the nilpotent cone of \(g\). Let \((E,F,G)\) be an \(sl_2\)-triple in \(g\), denote by \(S_0\) the special transverse slice to the adjoint orbit of \(E\) and by \(X\) the linear function on \(g\) corresponding to \(E\). The author constructs a natural algebra \(H\) naturally associated with \(X\), with a natural filtration such that \(\text{grad}(H)\) is isomorphic to the coordinate ring \(C[S_0]\). This filtration is induced by \textit{C. Moeglin}'s filtration of the enveloping algebra \(U(g)\) [C. R. Acad. Sci., Paris, Sér. I 303, 845-848 (1986; Zbl 0628.17007)]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of \(g\). The author gives some applications of the algebra \(H\) for the theory of \(W\)-algebras. In particular, he conjectures that the BRST quantization of the finite \(W\)-algebra associated with any nilpotent element of \(g\) is isomorphic to \(H\).
reviewed by: Angela Gammella-Mathieu
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 1836810
zbMATH Keywords: deformation quantization
zbMATH Keywords: enveloping algebras
MaRDI profile type: Publication
OpenAlex ID: W2027450954
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DOI: 10.1006/AIMA.2001.2063
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description: scientific article; zbMATH DE number 1836810