Geodesic
Coadjoint Orbits of Lie Algebras and Cartan Class
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $\mathcal{O}(\alpha)$ at the point $\alpha$ corresponds to the characteristic space associated to the left invariant form $\alpha$ and its dimension is the even part of the Cartan class of $\alpha$. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is $2$ or $4$. We determine also the Lie algebras of dimension $2n$ or $2n+1$ having an orbit of dimension $2n$.
Keywords: Lie algebras; coadjoint representation; contact forms; Frobenius Lie algebras; Cartan class. 
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     author = {Michel Goze and Elisabeth Remm},
     title = {Coadjoint {Orbits} of {Lie} {Algebras} and {Cartan} {Class}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a1/}
}
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Michel Goze; Elisabeth Remm. Coadjoint Orbits of Lie Algebras and Cartan Class. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a1/

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