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Mouchira Zaiter

Publications and source records attributed to Mouchira Zaiter.

5 recordsLinked to original sources
arXiv2016

On the Commuting variety of a reductive Lie algebra and other related varieties

The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the set of Cartan subalgebras. So someproperties of this variety are given. In particular, it is smooth in codimension $1$.We introduce the generalized isospectral commuting varieties and give some properties.Furthermore, desingularizations of these varieties are given by fiber bundles over adesingularization of the closure in the grassmannian of the set of Cartan subalgebrascontained in a given Borel subalgebra.

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arXiv2015

On the generalized commuting varieties of a reductive Lie algebra

The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.

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arXiv2011

On related varieties to the commuting variety of a semisimple Lie algebra

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and irreducible subvariety of g x g, its normalisation has rational singularities and its normalization morphism is bijective.

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arXiv2011

On the nullcone and the variety $χ_\mathfrak{g}$ of a semisimple lie algebra

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x, y) in g\timesg such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that N is a closed and irreducible subvariety of g \times g whose normalization has rational singularities and such that the normalization morphism is bijective.

math.RT↗