Finite-dimensional Nilpotent Lie Algebra Research Articles

Investigation of ID-derivations of finite-dimensional nilpotent Lie algebras

Let [Formula: see text] be a Lie algebra and let [Formula: see text] be the set of all derivations of [Formula: see text]. Then the derivation [Formula: see text] of [Formula: see text] is called an ID-derivation if [Formula: see text] for all [Formula: see text]. The set of all ID-derivations is denoted by [Formula: see text]. Let [Formula: see text] be the set of all central derivations and let [Formula: see text] and [Formula: see text] be, respectively, the set of all center derivations and ID-derivations that map [Formula: see text] to zero. In this paper, we verify relations between [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text]. Also, we prove that if [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text], then [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text].

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.

Finite-Dimensional Nilpotent Lie Algebras with Central Derivation Lie Algebras of Minimal Possible

Finite-Dimensional Nilpotent Lie Algebras with Central Derivation Lie Algebras of Minimal Possible

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.

Finite-Dimensional Nilpotent Lie Algebras with Central Derivations of Nonminimal Dimensions

Finite-Dimensional Nilpotent Lie Algebras with Central Derivations of Nonminimal Dimensions

Image of Lie polynomial of degree 2 evaluated on nilpotent lie algebra

We delineate the image of multilinear Lie polynomial of degree 2 evaluated on L where L is a finite-dimensional nilpotent Lie algebra over field k with

Nilpotent Lie algebras of class 3 with 1-dimensional center and 3-dimensional derived subalgebra

In this paper, we obtain the structure of all non-isomorphic finite-dimensional nilpotent Lie algebras L of class 3 with derived subalgebra of dimension 3 when

Characterization of nilpotent Lie algebras of breadth 3 over finite fields of odd characteristic

In this paper, we give a characterization of finite-dimensional nilpotent Lie algebras of breadth 3 over finite fields of odd characteristic. This characterization parallels to the one for finite p-groups of breadth 3 given earlier in [5].

Bounds for the dimension of the center factor in capable nilpotent Lie algebras of class two

Let L be a finite-dimensional nilpotent Lie algebra of class two over a field of characteristic different from two. The aim of the present paper is to give an upper bound for the dimension of the center factor L/Z(L) of a capable Lie algebra L. As a result, we also find a lower bound for the dimension of the derived subalgebra L 2 for a capable Lie algebra L. Furthermore, we also give some criteria for determining the capability of L.

Finite-dimensional nilpotent Lie algebras of class two and alternating bilinear maps

Let F be a field of characteristic different from two. Suppose that N2 is the category of finite-dimensional nilpotent Lie algebras of class two over the field F and that ALT is the category of alternating bilinear maps of F-vector spaces. We establish a relation between the category N2 and the category ALT. Then we show that the problem of determining the capability of these Lie algebras reduces to determining the epicenter of the corresponding objects in ALT. As an application of this technique, we describe the structure of Lie algebras corresponding to alternating bilinear maps of rank one (that is, to alternating bilinear forms). Also, we describe the epicenter of decomposable nondegenerate alternating bilinear maps of rank two.

The orbit method for locally nilpotent infinite-dimensional Lie algebras

Let n be a locally nilpotent infinite-dimensional Lie algebra over C. Let U(n) and S(n) be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of U(n), and the Poisson topology on the primitive Poisson spectrum of S(n). We provide a homeomorphism between the corresponding topological spaces (at the level of points, it gives a bijection between the primitive ideals of U(n) and S(n)). We also show that all primitive ideals of S(n) from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that n is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals I(λ)⊂S(n) and J(λ)⊂U(n), λ∈n⁎, to be nonzero. Most of these results generalize known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals I(λ), J(λ) are nonzero).

Arens–Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms

Our aim is to give an explicit description of the Arens-Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the Arens-Michael envelope belongs to a class of completions introduced by R.~Goodman in 70s. To find a precise form of this algebra we preliminary characterize the set of holomorphic functions of exponential type on a simply connected nilpotent complex Lie group. This approach leads to unexpected connections to Riemannian geometry and the theory of order and type for entire functions. As a corollary, it is shown that the Arens-Michael envelope considered above is a homological epimorphism. So we get a positive answer to a question investigated earlier by Dosi and Pirkovskii.

Nilpotent Lie algebras of class 4 with the derived subalgebra of dimension 3

The paper is devoted to give a full classification of all finite-dimensional nilpotent Lie algebras L of class 4 with Moreover, we specify the capable ones.

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.

Algorithm to compute minimal matrix representation of nilpotent lie algebras

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

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.

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.

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>.