Nilpotent Lie Algebras Of Dimension Research Articles

On Rigid 2-Step Nilpotent Lie Algebras

In this note we consider 2-step nilpotent Lie algebras and give a criterion for the rigidity of this class in the variety [Formula: see text] of 2-step nilpotent Lie algebras of dimension n. We apply this criterion to prove that every rigid Lie algebra in [Formula: see text] is indecomposable, except for [Formula: see text]3 ⊕ ℂ and [Formula: see text]3 ⊕ [Formula: see text]3.

Quasi-Sasakian Structures on 5-dimensional Nilpotent Lie Algebras

In this study, we examine the existence of quasi-Sasakian structures on nilpotent Lie algebras of dimension five. In addition, we give some results about left invariant quasi-Sasakian structures on Lie groups of dimension five, whose Lie algebras are nilpotent. Moreover, subclasses of quasi-Sasakian structures are studied for some certain classes.

The Variety of 7-Dimensional 2-Step Nilpotent Lie Algebras

In this note, we consider degenerations between complex 2-step nilpotent Lie algebras of dimension 7 within the variety N 7 2 . This allows us to obtain the rigid algebras in N 7 2 , whose closures give the irreducible components of the variety.

A description of pseudo-bosons in terms of nilpotent Lie algebras

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.

On dimension of Schur multiplier of nilpotent Lie algebras II

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and put [Formula: see text], where [Formula: see text] denotes the Schur multiplier of [Formula: see text]. Niroomand and Russo in 2011 proved that [Formula: see text] and that [Formula: see text] if and only if [Formula: see text], in which [Formula: see text] is the Heisenberg algebra of dimension [Formula: see text] and [Formula: see text] is the abelian [Formula: see text]-dimensional Lie algebra. In the same year, they also classified all nilpotent Lie algebras [Formula: see text] satisfying [Formula: see text] or [Formula: see text]. In this paper, we obtain all nilpotent Lie algebras [Formula: see text] provided that [Formula: see text].

English

A Lie algebra L is called 2-step nilpotent if L is not abelian and [L,L] lies in the center of L. 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.

2-capability and 2-nilpotent multiplier of finite dimensional nilpotent Lie algebras

In the present context, we investigate to obtain some more results about $2$-nilpotent multiplier $\mathcal{M}^{(2)}(L)$ of a finite dimensional nilpotent Lie algebra $L$. For instance, we characterize the structure of $\mathcal{M}^{(2)}(H)$ when $H$ is a Heisenberg Lie algebra. Moreover, we give some inequalities on $ \mathrm{dim}~ \mathcal{M}^{(2)}(L)$ to reduce a well known upper bound on $2$-nilpotent multiplier as much as possible. Finally, we show that $H(m)$ is 2-capable if and only if m=1.

A class of nilpotent Lie algebras admitting a compact subgroup of automorphisms

The realification of the (2n+1)-dimensional complex Heisenberg Lie algebra is a (4n+2)-dimensional real nilpotent Lie algebra with a 2-dimensional commutator ideal coinciding with the centre, and admitting the compact algebra sp(n) of derivations. We investigate, in general, whether a real nilpotent Lie algebra with 2-dimensional commutator ideal coinciding with the centre admits a compact Lie algebra of derivations. This also gives us the occasion to revisit a series of classic results, with the expressed aim of attracting the interest of a broader audience.

ON THE STRUCTURE OF FACTOR LIE ALGEBRAS

The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if L is a Lie algebra such that dimL/Z(L) = n, then <TEX>$dimL_{(2)}={\frac{1}{2}}n(n-1)-s$</TEX> for some non-negative integer s. In the present paper, we determine the structure of central factor (for s = 0) and the factor Lie algebra <TEX>$L/Z_2(L)$</TEX> (for all <TEX>$s{\geq}0$</TEX>) of a finite dimensional nilpotent Lie algebra L, with n-dimensional central factor. Furthermore, by using the concept of n-isoclinism, we discuss an upper bound for the dimension of <TEX>$L/Z_n(L)$</TEX> in terms of <TEX>$dimL_{(n+1)}$</TEX>, when the factor Lie algebra <TEX>$L/Z_n(L)$</TEX> is finitely generated and <TEX>$n{\geq}1$</TEX>.

The Nash–Moser theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

We apply the Nash–Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space Hk-nil2(g,g), and certain subspaces of it, that provide fine information about the deformations of g in the variety of k-step nilpotent Lie algebras.Then we focus on degenerations and rigidity in the variety of k-step nilpotent Lie algebras of dimension n with n≤7 and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.

Nilradicals of parabolic subalgebras admitting symplectic structures

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures.The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the g-hwv's of H2(n) for a finite dimensional real symplectic nilpotent Lie algebra n with a reductive Lie subalgebra of derivations g acting on it.

Some upper bounds on the dimension of the Schur multiplierof a pair of nilpotent Lie algebras

Let $(L,N)$ be a pair of Lie algebras where $N$ is an ideal of the finite dimensional nilpotent Lie algebra $L$. Some upper bounds on the dimension of the Schur multiplier of $(L,N)$ are obtained without considering the existence of a complement for $N$. These results are applied to derive a new bound on the dimension of the Schur multiplier of a nilpotent Lie algebra.

Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the academic community and the industrial engineering community, since nilpotent Lie algebras are applied in traditional mechanical dynamic problems and current scientific disciplines. The conditions of being quasi-filiform and nilpotent are applied carefully and in several stages, and appropriate changes of the basis are achieved in an iterative and interactive process of simplification. This has been implemented by means of the development of more than thirty Maple modules. The process has led from the first family formulation, with 64 parameters and 215 constraints, to a family of 16 parameters and 17 constraints. This structure theorem permits the exhaustive classification of the quasi-filiform nilpotent Lie algebras of dimension nine with current computational methodologies.

On nilpotent Lie algebras of small breadth

A Lie algebra L is said to be of breadth k if the maximal dimension of the images of left multiplication by elements of the algebra is k. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras.

Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras

The purpose of this paper is to obtain some inequalities and certain bounds for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras. Also, we give an inequality for the dimension of the c-nilpotent multiplier of L connected with dimension of the Lie algebras γd (L) and L / Zd−1 (L) . Finally, we compare our results with the previously known result.

Classification of Nilsoliton metrics in dimension seven

The aim of this paper is to classify Ricci soliton metrics on $7$-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in [Transformation Groups, Volume 17, Number 3 (2012), 639--656]. To this end, we use the classification of $7$-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski in [Rocky Mtn. Journal of Math. 42 (2012), 1521--1549] and [M\"{u}nster J. Math. 3 (2010), 67--88], and of Yuri Nikolayevsky in [Trans. Amer. Math. Soc. 363 (2011), 3935--3958]. Of the $9$ one-parameter families and $140$ isolated $7$-dimensional indecomposable real nilpotent Lie algebras, we have $99$ nilsoliton metrics given in an explicit form and $7$ one-parameter families admitting nilsoliton metrics. Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.

Characterization of Nilpotent Lie Algebras Pair by Their Schur Multipliers

Let (N, L) be a pair of finite dimensional nilpotent Lie algebras, in which N is an ideal in L and ℳ(N, L) is the Schur multiplier of the pair (N, L). In 2011, the second author and colleagues proved that if N admits a complement K say, in L with dim N = n and dim K = m, then , where t(N, L) is a non-negative integer. In the present article, we characterize the pairs (N, L) for which t(N, L) = 0, 1, 2, 3, and 4. Also, in a special case when L is a filiform Lie algebra, we determine all pairs (N, L), when t(N, L) = 0, 1,…, 10.

Nilpotent Lie and Leibniz Algebras

We extend results on finite dimensional nilpotent Lie algebras to Leibniz algebras and counterexamples to others are found. One generator algebras are used in these examples and are investigated further.

Classification of the Quasifiliform Nilpotent Lie Algebras of Dimension 9

On the basis of the family of quasifiliform Lie algebra laws of dimension 9 of 16 parameters and 17 constraints, this paper is devoted to identify the invariants that completely classify the algebras over the complex numbers except for isomorphism. It is proved that the nullification of certain parameters or of parameter expressions divides the family into subfamilies such that any couple of them is nonisomorphic and any quasifiliform Lie algebra of dimension 9 is isomorphic to one of them. The iterative and exhaustive computation with Maple provides the classification, which divides the original family into 263 subfamilies, composed of 157 simple algebras, 77 families depending on 1 parameter, 24 families depending on 2 parameters, and 5 families depending on 3 parameters.

The classification of two step nilpotent complex Lie algebras of dimension 8

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.