Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Conformal Killing forms on 2-step nilpotent Riemannian Lie groups

We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.

Nilpotent fuzzy lie ideals

In this paper, we introduce a new definition for nilpotent fuzzy Lie ideal, which is a well-defined extension of nilpotent Lie ideal in Lie algebras, and we name it a good nilpotent fuzzy Lie ideal. Then we prove that a Lie algebra is nilpotent if and only if any fuzzy Lie ideal of it, is a good nilpotent fuzzy Lie ideal. In particular, we construct a nilpotent Lie algebra via a good nilpotent fuzzy Lie ideal. Also, we prove that with some conditions, every good nilpotent fuzzy Lie ideal is finite. Finally, we define an Engel fuzzy Lie ideal, and we show that every Engel fuzzy Lie ideal of a finite Lie algebra is a good nilpotent fuzzy Lie ideal. We think that these notions could be useful to solve some problems of Lie algebras with nilpotent fuzzy Lie ideals.

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

On Marginal Automorphisms of Free Nilpotent Lie Algebras

Let L be the free nilpotent Lie algebra of finite rank over a field of characteristic zero. We define the concepts of marginal ideals and marginal automorphisms of L, and we give some results on marginal automorphisms.

The Schur multipliers of Lie algebras of maximal class

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and [Formula: see text] be its Schur multiplier. It was proved by the second author the dimension of the Schur multiplier is equal to [Formula: see text] for some [Formula: see text]. In this paper, we classify all nilpotent Lie algebras of maximal class for [Formula: see text]. The dimension of Schur multiplier of such Lie algebras is also bounded by [Formula: see text]. Here, we give the structure of all nilpotent Lie algebras of maximal class [Formula: see text] when [Formula: see text] and then we show that all of them are capable.

Certain functors of nilpotent Lie algebras with the derived subalgebra of dimension two

By considering the nilpotent Lie algebra with the derived subalgebra of dimension [Formula: see text], we compute some functors including the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.