Surprising properties of centralisers in classical Lie algebras

Oksana Yakimova

Displaying similar documents to “Surprising properties of centralisers in classical Lie algebras”

Singular components of Springer fibers in the two-column case

Lucas Fresse (2009)

Annales de l’institut Fourier

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We consider the Springer fiber $ℬ_{u}$ corresponding to a nilpotent endomorphism $u$ of nilpotent order $2$ . As a first result, we give a description of the elements of a given component of $ℬ_{u}$ which are fixed by the action of the standard torus relative to some Jordan basis of $u$ . By using this result, we establish a necessary and sufficient condition of singularity for the components of $ℬ_{u}$ .

A remarkable contraction of semisimple Lie algebras

Dmitri I. Panyushev, Oksana S. Yakimova (2012)

Annales de l’institut Fourier

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Recently, E.Feigin introduced a very interesting contraction $𝔮$ of a semisimple Lie algebra $𝔤$ (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of $𝔤$ . For instance, the algebras of invariants of both adjoint and coadjoint representations of $𝔮$ are free, and also the enveloping algebra of $𝔮$ is a free module over its centre.

Equations of some wonderful compactifications

Pascal Hivert (2011)

Annales de l’institut Fourier

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De Concini and Procesi have defined the wonderful compactification $\bar{X}$ of a symmetric space $X = G / G^{σ}$ where $G$ is a complex semisimple adjoint group and $G^{σ}$ the subgroup of fixed points of $G$ by an involution $σ$ . It is a closed subvariety of a Grassmannian of the Lie algebra $𝔤$ of $G$ . In this paper we prove that, when the rank of $X$ is equal to the rank of $G$ , the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form $w$ on $𝔤$ vanishes...

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Patrick Bonckaert, Freek Verstringe (2012)

Annales de l’institut Fourier

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We explore the convergence/divergence of the normal form for a singularity of a vector field on $ℂ^{n}$ with nilpotent linear part. We show that a Gevrey- $α$ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey- $1 + α$ type with the use of a Gevrey- $1 + α$ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

Different methods for the study of obstructions in the schemes of Jacobi

Roger Carles, M. Carmen Márquez (2011)

Annales de l’institut Fourier

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In this paper the problem of obstructions in Lie algebra deformations is studied from four different points of view. First, we illustrate the method of local ring, an alternative to Gerstenhaber’s method for Lie deformations. We draw parallels between both methods showing that an obstruction class corresponds to a nilpotent local parameter of a versal deformation of the law in the scheme of Jacobi. Then, an elimination process in the global ring, which defines the scheme, allows us to...