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Register a new userCartan class of Invariant forms on Lie groups
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We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n,R) which doesnt admit left invariant contact form, yet admits a contact form which is invariant by a maximal compact subgroup. We determine also Pfaffian forms on the Heisenberg $3$-dimensional group invariant by a subgroup and obtain the Transport Equation.
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The systematic study of CR manifolds originated in two pioneering 1932 papers of Elie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on $mathrm{SU}_2 = S^3$, deforming the standard `spherical structure. In this paper, mostly expository, we illustrate and clarify Cartans results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that $mathrm{SL}_2(mathbb{R})$ admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of $mathrm{SL}_2(mathbb{R})$ provide CR embedding or immersions of these structures. The same methods apply to all other three-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for $mathrm{SU}_2$, the Heisenberg group, and the Euclidean group.
The Linearization Theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization remains somehow open. We address it here, by first giving a counter-example to a previous conjecture, and then proving a sufficient criterion that uses compatible complete metrics and covers the case of proper group actions. We also show a partial converse that fixes and extends previous results in the literature.
In this paper we propose a (non-linear) smoothing algorithm for group-affine observation systems, a recently introduced class of estimation problems on Lie groups that bear a particular structure. As most non-linear smoothing methods, the proposed algorithm is based on a maximum a posteriori estimator, determined by optimization. But owing to the specific properties of the considered class of problems, the involved linearizations are proved to have a form of independence with respect to the current estimates, leveraged to avoid (partially or sometimes totally) the need to relinearize. The method is validated on a robot localization example, both in simulations and on real experimental data.
Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.
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 $2n$.
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