Coadjoint orbits of Lie algebras and Cartan class

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 $2n$.

20 pages

References in corpus (1)

• On filiform Lie algebras. Geometric and algebraic studies

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• Contact Seaweeds