Surprising properties of centralisers in classical Lie algebras

[Propriétés surprenantes des centralisateurs dans les algèbres de Lie classiques]

Yakimova, Oksana

Annales de l'Institut Fourier, Tome 59 (2009), p. 903-935 / Harvested from Numdam

Résumé

Soit $\mathfrak{g}$ une algèbre de Lie classique, i.e., $\mathfrak{gl}_n$, $\mathfrak{sp}_n$, ou $\mathfrak{so}_n$, et soit $e$ un élément nilpotent de $\mathfrak{g}$. Nous étudions dans cet article diverses propriétés du centralisateur $\mathfrak{g}_e$ de $e$. Les quatre premières sections concernent des problèmes assez élémentaires portant sur le centre de $\mathfrak{g}_e$, la variété commutante de $\mathfrak{g}_e$, ou encore les centralisateurs des paires commutantes. La seconde partie aborde des questions liées aux différentes structures de Poisson sur $\mathfrak{g}_e^*$ et aux invariants symétriques de $\mathfrak{g}_e$.

Abstract

Let $\mathfrak{g}$ be a classical Lie algebra, i.e., either $\mathfrak{gl}_n$, $\mathfrak{sp}_n$, or $\mathfrak{so}_n$ and let $e$ be a nilpotent element of $\mathfrak{g}$. We study various properties of the centralisers $\mathfrak{g}_e$. The first four sections deal with rather elementary questions, like the centre of $\mathfrak{g}_e$, commuting varieties associated with $\mathfrak{g}_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on $\mathfrak{g}_e^*$ and symmetric invariants of $\mathfrak{g}_e$.

Détail

Publié le : 2009-01-01

MR 2543656 | Zbl 1187.17008

DOI : https://doi.org/10.5802/aif.2451
Classification:  17B45
Mots clés: orbite nilpotente, centralisateurs, invariants symétriques

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