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Register a new userFoliations formed by generic coadjoint orbits of a class of 7-dimensional real solvable Lie groups
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The paper is a continuation of the authors et al.s work in the first half of the year 2021. It has classified a special class of 7-dimensional real solvable Lie algebras such that the nilradical of each from them is well-known 5-dimensional nilpotent Lie algebra in that work. In this paper, we will consider exponential, connected and simply connected Lie groups which are corresponding to these Lie algebras. Namely, we will describe the geometry of generic (i.e. 6-dimensional) orbits in coadjoint representation of considered Lie groups. Next, we will prove that for each considered group, the family of generic coadjoint orbits forms a measurable foliation in the sense of Connes and give the topological classification of these foliations.
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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 $2n$.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category $mathfrak F_0$ of leaf manifolds containing the orbifold category as a full subcategory. Objects of $mathfrak F_0$ may have non-Hausdorff topology unlike the orbifolds. The topology of some objects of $mathfrak F_0$ does not satisfy the separation axiom $T_0$. It is shown that for every ${mathcal N}in Ob(mathfrak F_0)$ a rigid geometry $zeta$ on $mathcal N$ admits a desingularization. Moreover, for every such $mathcal N$ we prove the existence and the uniqueness of a finite dimensional Lie group structure on the automorphism group $Aut(zeta)$ of the rigid geometry $zeta$ on $mathcal{N}$.
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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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