Dimensional Nilpotent Research Articles

On classification of $7$-dimensional odd-nilpotent Leibniz algebras

In this paper we extend the method of canonical form for congruence of bilinear forms to give the classification of some subclasses of $7-$dimensional nilpotent Leibniz algebras. Odd-nilpotent Leibniz algebras are defined as that its even dimensional ideals in lower central series are all zero and the classification of $7-$dimensional complex odd-nilpotent Leibniz algebras with one dimensional Leib ideal is obtained by applying the aforementioned method.

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.

Tensor product of finite dimensional nilpotent evolution algebras

This paper investigates the tensor product of a finite dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal index of nilpotency has been carried out.

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.

О 7-мерных алгебрах голоморфных векторных полей в $ \Bbb C^4 $, имеющих 5-мерный абелев идеал

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in $ \Bbb C^4 $ we study in this article the 7-dimensional orbits of real Lie algebras in this space. By the well-known Morozov theorem, any nilpotent 7-dimensional Lie algebra has at least a 4-dimensional Abelian ideal. The article considers nilpotent indecomposable 7-dimensional Lie algebras containing a 5-dimensional Abelian ideal. It is proved that in the space $ \Bbb C^4 $ all the orbits of such algebras are Levi degenerate. This statement covers 73 algebras from the complete list of 149 indecomposable 7-dimensional nilpotent Lie algebras.

Isometry groups of six-dimensional nilmanifolds

We determine the 6-dimensional nilpotent metric Lie algebras such that the Lie algebra {mathfrak {n}} has a descending series of ideals invariant under all automorphisms of {mathfrak {n}} and the dimension of the consecutive members of the series decreases by one. We call them metric Lie algebras having a framing determined by ideals. We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on 6-dimensional nilpotent Lie groups having a Lie algebra {mathfrak {n}} as their Lie algebra.

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.

Applications of spherical twist functors to Lie algebras associated to root categories of preprojective algebras

Let ΛQ be the preprojective algebra of a finite acyclic quiver Q of non-Dynkin type and Db(repnΛQ) be the bounded derived category of finite dimensional nilpotent ΛQ-modules. We define spherical twist functors over the root category RΛQ of Db(repnΛQ) and then realize the Weyl group associated to Q as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra g(RΛQ) of RΛQ induced by spherical twist functors. We also present a concrete relation between certain Lie subalgebras of g(RΛQ) and g(RQ), where g(RQ) is the Ringel–Hall Lie algebra associated to the root category RQ of Q.

Central extensions of 2-filiform Leibniz algebras

In this paper we describe the central extensions of some nilpotent Leibniz algebras and one-dimensional central extensions (up to isomorphism) of all naturally graded 2-filiform Leibniz algebras over the field of complex numbers. It is well-know that 2-filiform Leibniz algebras which are not naturally graded have not yet studied. There are such kind of algebras in the algebras obtained along this article. Also in this work we describe some subclasses of 6-dimensional nilpotent Leibniz algebras with condition quotient algebra by two-dimensional center of the algebra is isomorphic to the four-dimensional naturally graded 2-filiform nonsplit Leibniz algebra.

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.

Degenerations of nilpotent algebras

We give a complete description of degenerations of $3$-dimensional nilpotent algebras, $4$-dimensional nilpotent commutative algebras and $5$-dimensional nilpotent anticommutative algebras over $ \mathbb C$. In particular, we correct several mistakes from the paper `Contractions of low-dimensional nilpotent Jordan algebras' by Ancochea Berm\'{u}dez, Fres\'{a}n and Margalef Bentabol.

The geometric classification of nilpotent commutative $$\mathfrak {CD}$$-algebras

We give a geometric classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension $24$ and decomposes into $10$ irreducible components determined by the Zariski closures of a two-parameter family of algebras, three one-parameter families of algebras, and $6$ rigid algebras.

Unified computational approach to nilpotent algebra classification problems

Abstract In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.

Characterizing nilpotent Leibniz algebras by their multiplier

In this paper, we characterize all finite dimensional nilpotent Leibniz algebras L for which (dim⁡(L))2−dim⁡(HL2(L))≤12.

Cohomological induction and uniform measure equivalence

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.

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.

One-generated nilpotent assosymmetric algebras

We give the classification of [Formula: see text]- and [Formula: see text]-dimensional complex one-generated nilpotent assosymmetric algebras.

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.

Classification of some subclasses of 6−dimensional nilpotent Leibniz algebras

This article is a contribution to the improvement of classification theory in Leibniz algebras. We extend the method of congruence classes of matrices of bilinear forms that was used to classify complex nilpotent Leibniz algebras with one dimensional derived algebra. In this work we focus on applying this method to the classification of 6-dimensional complex nilpotent Leibniz algebras with two dimensional derived algebra.