Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 002, 20 pages      arXiv:1806.07553      https://doi.org/10.3842/SIGMA.2019.002

Coadjoint Orbits of Lie Algebras and Cartan Class

Michel Goze a and Elisabeth Remm b
a) Ramm Algebra Center, 4 rue de Cluny, F-68800 Rammersmatt, France
b) Université de Haute-Alsace, IRIMAS EA 7499, Département de Mathématiques, F-68100 Mulhouse, France

Received September 13, 2018, in final form December 31, 2018; Published online January 09, 2019

Abstract
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 $2n$.

Key words: Lie algebras; coadjoint representation; contact forms; Frobenius Lie algebras; Cartan class.

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