Nilpotent Lie Algebras Of Dimension Research Articles

Ad-invariant metrics on nonnice nilpotent Lie algebras

We proved in previous work that all real nilpotent Lie algebras of dimension up to [Formula: see text] carrying an ad-invariant metric are nice, i.e. they admit a nice basis in the sense of Lauret et al. In this paper, we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras admitting an ad-invariant metric exist for every dimension greater than [Formula: see text] and every nilpotency step greater than [Formula: see text]. In the way of doing so, we introduce a method to construct Lie algebras with ad-invariant metrics called the single extension, as a parallel to the well-known double extension procedure.

Real non-degenerate two-step nilpotent Lie algebras of dimension eight

We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.

Bounds for the dimension of the Schur multiplier of finite dimensional nilpotent Lie algebras

Let L be an n-dimensional nilpotent Lie algebra of nilpotency class c≥2 over an arbitrary field F with dim⁡γ2(L)=m and dim(Z(L)Z(L)∩γ2(L))=t, where Z(L) and γ2(L) denote the center ideal and the derived subalgebra of L, respectively. The purpose of this work is to obtain a new bound on the dimension of the Schur multiplier for L, which is given in terms of n,m,c, and t. In the virtue of an old bound due to Niroomand and Russo, namely 12(n−m−1)(n+m−2)+1, we find a finite dimensional nilpotent Lie algebra L of nilpotency class 3 over a field F of characteristic 3 that attains this bound. We then present a complete list of all finite dimensional nilpotent Lie algebras of nilpotency class at least 2 that reach Niroomand–Russo's Formula. Finally, we give the structure of all finite dimensional nilpotent Lie algebras that attain some recent generalizations of Niroomand–Russo's Formula.

On Quasi-para-Sasakian Structures on 5-dimensions

In this study, we investigate the existence of quasi-para-Sasakian structures on five dimensional nilpotent
 Lie algebras. There are six non-abelian nilpotent Lie algebras. We show that quasi-para-Sasakian structures exist only on one of these algebras. Quasi-para-Sasakian structures correspond to the class G_5+G_8 in the classification of almost paracontact metric structures. We show that a quasi-para-Sasakian structure on a five dimensional nilpotent Lie algebra is either in G_5 or G_8.

Nilpotent Lie algebras with two centralizer dimensions over a finite field

A result of Barnea and Isaacs states that if L is a finite dimensional nilpotent Lie algebra with exactly two distinct centralizer dimensions, then nilpotency class of L is either 2 or 3. In this article, we classify all such finite dimensional 3-step nilpotent Lie algebras over a finite field.

Nilpotent Lie algebras of breadth type (0, 3)

For a natural number m, a Lie algebra L over a field k is said to be of breadth type if the co-dimension of the centralizer of every non-central element is m. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type over of odd characteristic up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, and . We also discuss 2-step nilpotent Camina Lie algebras.

The center of the universal enveloping algebras of small-dimensional nilpotent Lie algebras in prime characteristic

We describe the centers of the universal enveloping algebras of nilpotent Lie algebras of dimension at most six over fields of prime characteristic. If the characteristic is not smaller than the nilpontency class, then the center is the integral closure of the algebra generated over the p-center by the same generators that also occur in characteristic zero. Except for three examples, two of which are standard filiform, this algebra is already integrally closed and hence it coincides with the center. In the case of these three exceptional algebras, the center has further generators. Then we show that the center of the universal enveloping algebra of the algebras investigated in this paper is isomorphic to the Poisson center (the algebra of invariants under the adjoint representation). This shows that Braun’s conjecture is valid for this class of Lie algebras.

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

We answer in the affirmative the question posed by Conti and Rossi [7,8] on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.

Examples of Almost Para-Contact Metric Structures on 5-dimensions

In this study, the classes of several almost paracontact metric structures on 5 dimensional nilpotent Lie algebras are determined. It is also shown that there are no $\eta-$ Einstein structures on 5 dimensional nilpotent Lie algebras.

Complete sets of polynomials in bi-involution on nilpotent seven-dimensional Lie algebras

In this paper, we construct complete sets of polynomials in bi-involution on nilpotent Lie algebras of dimension 7 in the list due to Gong. Thus we verify the generalized Mishchenko-Fomenko conjecture for all algebras in this list. Bibliography: 14 titles.

Computation of positively graded filiform nilpotent Lie algebras in low dimensions

We describe a procedure for classifying positively graded filiform nilpotent Lie algebras in low dimensions, over R, C, and Q. Using this procedure, we find all filiform nilpotent Lie algebras of dimension 15 and lower that admit a positive grading.

The construction of real Frobenius Lie algebras from non-commutative nilpotent Lie algebras of dimension

In this present paper, we study real Frobenius Lie algebras constructed from non-commutative nilpotent Lie algebras of dimension ≤ 4. The main purpose is to obtain Frobenius Lie algebras of dimension ≤ 6. Particularly, for a given non-commutative nilpotent Lie algebras N of dimension ≤ 4 we show that there exist commutative subalgebras of dimension ≤ 2 such that the semi-direct sums ɡ = N⊕T is Frobenius Lie algebras. Moreover, T is called a split torus which is a commutative subalgebra of derivation of N and it depends on the given N. To obtain this split torus, we apply Ayala’s formulas of a Lie algebra derivation by taking a diagonal matrix of a standard representation matrix of the Lie algebra derivation of N. The discussion of higher dimension of Frobenius Lie algebras obtained from non-commutative nilpotent Lie algebras is still an open problem.

Almost Para-Contact Metric Structures on 5-dimensional Nilpotent Lie Algebras

In this manuscript, almost para-contact metric structures on 5 dimensional nilpotent Lie algebras are studied. Some examples of para-Sasakian and para-contact structures on five-dimensional nilpotent Lie algebras are given.

Almost Contact B-Metric Structure on 5-Dimensional Nilpotent Lie algebras

The classification of almost contact $B-$metric manifolds is considered. It is shown that $\mathcal{D}-$homothetic deformation of these manifolds in the class $\mathcal{F}_{i}$ $\left( i=0,1,4,5\right) $ remains in the same the class $\mathcal{F}_{i}$. We study almost contact $B-$metric structure on $5-$dimensional nilpotent Lie algebras. The class of the left invariant almost contact $B-$metric structures on corresponding Lie groups is investigated. Finally, we determine the class of $5-$dimensional nilpotent Lie algebras with almost contact $B-$metric structure.

On characterizing nilpotent Lie algebras by their multiplier, $$s(L)=4$$

Let L be a non-abelian nilpotent Lie algebra of dimension n and $$s(L)=\frac{1}{2}(n-1)(n-2)+1- \dim {\mathcal {M}}(L)$$ , where $${\mathcal {M}}(L)$$ denotes the Schur multiplier of L. For a non-abelian nilpotent Lie algebra, we know $$ s(L)\ge 0 $$ and the structure of all nilpotent Lie algebras are well known for $$ s(L) \in \lbrace 0,1,2,3 \rbrace $$ in several papers. The current paper is devoted to obtain the structure of all nilpotent Lie algebras L, when $$ s(L)=4 $$ .

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.

C-capability of Lie algebras with the derived subalgebra of dimension two

The current article is devoted to classify the c-capability of finite dimensional nilpotent Lie algebras with the derived subalgebra of dimension two.

On the dimension of the Schur multiplier of nilpotent lie algebras

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.

Construction of nice nilpotent Lie groups

We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension n up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for n≤9. On every nilpotent Lie algebra of dimension ≤7, we determine the number of inequivalent nice bases, which can be 0, 1, or 2.We show that any nilpotent Lie algebra of dimension n has at most countably many inequivalent nice bases.

On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank

We have recently shown that pseudo-bosonic operators realize concrete examples of finite dimensional nilpotent Lie algebras over the complex field. It has been the first time that such operators were analyzed in terms of nilpotent Lie algebras (under prescribed conditions of physical character). On the other hand, the general classification of a finite dimensional nilpotent Lie algebra l may be given via the size of its Schur multiplier involving the so-called corank t(l) of l. We represent l by pseudo-bosonic ladder operators for t(l)≤6 and this allows us to represent l when its dimension is ≤5.