Nilpotent Lie Algebras Research Articles

Abelian factors in 2-step nilpotent Lie algebras constructed from graphs

We consider real 2-step metric nilpotent Lie algebras associated to graphs with possibly repeated edge labels as constructed by Ray in 2016. We determine how the structure of the edge labeling within the graph contributes to the abelian factor in these Lie algebras. Furthermore, we explicitly compute the abelian factor of the 2-step nilpotent Lie algebras associated with some families of graphs such as star graphs, cycles, Schreier graphs, and properly edge-colored graphs. We also study the singularity properties of these Lie algebras in certain cases.

Locally conformal SKT structures

A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion 3-form [Formula: see text] is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called locally conformal SKT (or shortly LCSKT). More precisely, a Hermitian structure [Formula: see text] is said to be LCSKT if there exists a closed nonzero [Formula: see text]-form [Formula: see text] such that [Formula: see text]. In this paper, we consider nontrivial LCSKT structures, i.e. we assume that [Formula: see text] and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify six-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a six-dimensional 3-step nilpotent Lie algebra admitting a nontrivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a nontrivial LCSKT structure, which allows us to construct explicit examples of six-dimensional unimodular almost abelian Lie algebras admitting a nontrivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a six-dimensional nilpotent or a [Formula: see text]-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is Kähler.

The algebraic and geometric classification of nilpotent Lie triple systems up to dimension four

In this paper we generalize the Skjelbred–Sund method, used to classify nilpotent Lie algebras, in order to classify triple systems with non-zero annihilator. We develop this method with the purpose of classifying nilpotent Lie triple systems, obtaining from it the algebraic classification of the nilpotent Lie triple systems up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Lie triple systems up to dimension four.

The algebraic classification and degenerations of nilpotent Poisson algebras

We generalize the Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Poisson algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Poisson algebras, obtaining from it the algebraic classification of the nilpotent Poisson algebras up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Poisson algebras up to dimension four, by adapting some notions and results used for varieties of algebras with a single multiplication.

Equivalent constructions of nilpotent quadratic Lie algebras

The double extension and the T⁎-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

Classification of a family of 4-dimensional anti-commutative algebras and their automorphisms

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field K of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of K by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by K.

On properties of some nilpotent evolution algebras and their enveloping algebras

In this paper, we consider [Formula: see text]-dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra.

Complex structures on nilpotent Lie algebras with one-dimensional center

We classify the nilpotent Lie algebras of real dimension eight and minimal center that admit a complex structure. Furthermore, for every such nilpotent Lie algebra g, we describe the space of complex structures on g up to isomorphism. As an application, the nilpotent Lie algebras having a non-trivial abelian J-invariant ideal are classified up to eight dimensions.

Finitely Based T-Spaces of Free Lie Nilpotent Algebras of Rank 2

Finitely Based T-Spaces of Free Lie Nilpotent Algebras of Rank 2

The classification of nilpotent Lie-Yamaguti algebras

In this paper, we consider a generalization of the classical Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Lie-Yamaguti algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Lie-Yamaguti algebras, and we obtain from it the algebraic classification of the nilpotent Lie-Yamaguti algebras up to dimension four.

Taylor Spectrum for Modules over Lie Algebras

In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in the case of nilpotent and semisimple Lie algebras. We also show that this result does not hold for solvable Lie algebras and obtain a precise description of the spectrum in the case of Borel subalgebras of semisimple Lie algebras.

Poisson catenarity in Poisson nilpotent algebras

We prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson prime ideals between any two given Poisson prime ideals have the same length.

On the derivations of cyclic Leibniz algebras

Let $L$ be an algebra over a field $F$. Then $L$ is called a left Leibniz algebra, if its multiplication operation $[-,-]$ additionally satisfies the so-called left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of a Leibniz algebra $L$ is called a derivation of an algebra $L$, if $f([a,b])=[f(a),b]+[a,f(b)]$ for all elements $a,b\in L$. It is well known that the set of all derivations $\mathrm{Der}(L)$ of a Leibniz algebra $L$ is a subalgebra of the Lie algebra $\mathrm{End}_{F}(L)$ of all linear transformations of an algebra $L$. The algebras of derivations of Leibniz algebras play an important role in the study of structure of Leibniz algebras. Their role is similar to that played by groups of automorphisms in the study of group structure.
 In this paper, a complete description of the algebra of derivations of nilpotent cyclic Leibniz algebra is obtained. In particular, it was proved that this algebra is metabelian and supersoluble Lie algebra, and its dimension is equal to the dimension of an algebra $L$.

Multipliers of Nilpotent Diassociative Algebras

The paper concerns nilpotent diassociative algebras (also known as associative dialgebras) and their corresponding diassociative Schur multipliers. Using Lie (and group) theory as a guide, we first extend a classic five-term cohomological sequence under alternative conditions in the nilpotent setting. This main result is then applied to obtain a new proof for a previous extension of the same sequence. It also yields a different extension of the sequence that involves terms in the upper central series. Furthermore, we use the main result to obtain a collection of dimension bounds on the multiplier of a nilpotent diassociative algebra. These differ notably from the Lie case. Since diassociative algebras generalize associative algebras, we obtain an associative analogue of the results herein. We conclude by computing both the associative and diassociative multipliers of an associative algebra. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras.

Sheaves of Noncommutative Smooth and Holomorphic Functions Associated with the Non-Abelian Two-Dimensional Lie Algebra

It was shown in papers of Dosi and in a recent article of the author that there is a sheaf of Frechet-Arens-Michael algebras (locally solvable in the complex case and of polynomial growth in the real) on the character space of a nilpotent Lie algebra. For the Lie algebra of the group of affine transformations of the line (the simplest non-nilpotent solvable Lie algebra) we construct analogous sheaves of non-commutative smooth and non-commutative holomorphic functions on a special set of representations.

POINTWISE INNER AUTOMORPHISMS OF RELATIVELY FREE LIE ALGEBRAS

Let $K$ be a field of characteristic zero, and $L_{m,c}$ be the free metabelian nilpotent Lie algebra of class $c$ of rank $m$ over $K$. We call an automorphism $\phi$ pointwise inner, if there exists an inner automorphism $\xi_i$ for each generator $x_i$, $i=1,\ldots,m$, such that $\phi(x_i)=\xi_i(x_i)$. In this study, we exemine the group $PI(L_{m,c})$ of pintwise inner automorphisms of the Lie algebra $L_{m,c}$, and we provide a set of generators for this group.

Hermitian structures on a class of almost nilpotent solvmanifolds

In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients Γ﹨G, where G is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and Γ is a lattice of G. We first obtain a characterization of Hermitian almost nilpotent Lie algebras g whose nilradical n has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In particular, we construct new examples of compact SKT manifolds.Finally, we prove some non-existence results for generalized Kähler structures in real dimension six. In higher dimension we construct the first examples of non-split generalized Kähler structures (i.e., such that the associated complex structures do not commute) on almost abelian Lie algebras. This leads to new compact (non-Kähler) manifolds admitting non-split generalized Kähler structures.

New bounds on the dimension of the Schur multiplier of n-Lie algebras

We give a bound on the dimension of the Schur multiplier of a finite-dimensional nilpotent n-Lie algebra, which sharpens the earlier bounds on the dimension of Schur multiplier of such n-Lie algebras. Then we give the structure of all nilpotent n-Lie algebras that attain this bound. Also, we obtain a new bound on the dimension of the Schur multiplier of a d-dimensional c-step nilpotent n-Lie algebra with the derived subalgebra of dimension d – n, which sharpens the earlier known bounds.

Geometric Structures On 3-Dimensional Hom-Lie Algebras

Some classes of almost contact structures on 5-dimensional nilpotent Lie algebras were investigated in the literature. On the other hand, there are several generalizations of Lie algebras and their characterizations. In particular, an effective characterization of all Hom-Lie algebras of sl(2) type and construction of all the 3-dimensional Hom-Lie algebras for a class of twisting homomorphisms has been presented. A list of twenty 3-dimensional Hom-Lie algebras has been given. In this paper, we consider some classes of almost contact metric structure on all those twenty 3-dimensional Hom-Lie algebras separately. We study semi-cosymplectic, almost cosymplectic and cosymlectic structures. Moreover, we study K-contact and Sasakian structures on 3-dimensional Hom-Lie algebras.

On the classification of nilpotent Lie algebras of maximal rank

On the classification of nilpotent Lie algebras of maximal rank