One-generated nilpotent terminal algebras

The Grunewald-O'Halloran conjecture for nilpotent Lie algebras of rank >= 1

Grunewald and O'Halloran conjectured in 1993 that every complex nilpotent Lie algebra is the degeneration of another, non isomorphic, Lie algebra. We prove the conjecture for the class of nilpotent Lie algebras admitting a semisimple derivation, remaining open for the class of characteristically nilpotent Lie algebras. In dimension 7, where the first characteristically nilpotent Lie algebras appear, we prove the conjecture and we also exhibit explicit nontrivial degenerations to every 7-dimensional nilpotent Lie algebra.

Free nilpotent and nilpotent quadratic Lie algebras

In this paper we introduce an equivalence between the category of the t-nilpotent quadratic Lie algebras with d generators and the category of some symmetric invariant bilinear forms on the t-nilpotent free Lie algebra with d generators. Taking into account this equivalence, t-nilpotent quadratic Lie algebras with d generators are classified (up to isometric isomorphism, and over any field of characteristic zero), in the following cases: d=2 and t<6, d=3 and t<4.

Low dimensional nilpotent n-Lie algebras

In this paper, nilpotent n-Lie algebras of dimension n + 3 as well as nilpotent n-Lie algebras of class 2 and dimension n + 4 are classified.

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

An algebraic classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras is given. This classification is based on an algebraic classification of complex $5$-dimensional nilpotent Jordan algebras.

The algebraic classification of nilpotent commutative algebras

This paper is devoted to the complete algebraic classification of complex $5$-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras.

The algebraic and geometric classification of nilpotent bicommutative algebras

We classify the $4$-dimensional nilpotent bicommutative algebras over $\mathbb C$ from both algebraic and geometric approaches.

The algebraic classification of nilpotent Tortkara algebras

We classify all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C.$

The algebraic and geometric classification of nilpotent binary Lie algebras

The paper is devoted to give the complete algebraic classification of nilpotent binary Lie algebras of dimension $\leq 6$ over an arbitrary base field ${\mathbb{F}}$ of characteristic not $2$ and the complete geometric classification of nilpotent binary Lie algebras of dimension $6$ over $\mathbb C.$ As an application, we have the algebraic and geometric classification of nilpotent anticommutative $\mathfrak{CD}$-algebras of dimension $\leq 6.$

Characterising Semi-Simple Lie Algebras by Their Borel Nilpotent Radical

We show that semi-simple lie algebras can be characterized by their maximal nilpotent subalgebra, which is the same as the nilpotent radical of a Borel subalgebra.

Computational experiments with nilpotent Lie algebra

In this note we present some experimental results on the general matrix nilpotent Lie algebras derived by calculations on a computer

The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).

Lie Algebra Prederivations and strongly nilpotent Lie Algebras

We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify filiform Lie algebras admitting a non-singular prederivation but no non-singular derivation. We prove that any 4-step nilpotent Lie algebra admits a non-singular prederivation. The results can be applied to construct affine structures on Lie algebras.

Free Nilpotent Lie Algebras Admitting Ad-Invariant Metrics

In this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent Lie algebra to be endowed with an ad-invariant metric. For such nilpotent Lie algebras admitting an ad-invariant metric the corresponding automorphisms groups are studied.

On nilpotent index and dibaricity of evolution algebras

An evolution algebra corresponds to a quadratic matrix $A$ of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices $A$. We establish a criterion for an $n$-dimensional nilpotent evolution algebra to be with maximal nilpotent index $2^{n-1}+1$. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any $s=1,...,n-1$ we construct a wide class of $n$-dimensional evolution algebras with nilpotent index $2^{n-s}+1$. We sh…

Finite Dimensional Nilpotent Leibniz Algebras with Isomorphic Maximal Subalgebras

Nilpotent Leibniz algebras with isomorphic maximal subalgebras are considered. The algebras are classified for coclass zero, one, and two. The results are field dependent.

幂等二元与一元 Leibniz 代数的代数与几何分类

The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

研究给出复5维幂零二元Leibniz代数和4维幂零一元Leibniz代数的完整代数与几何分类，并由此推出幂零指数3的复4维幂零代数分类。

The classification of non-characteristically nilpotent filiform Leibniz algebras

In this paper we investigate the derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension is decomposed into three non-intersected families. We found sufficient conditions under which filiform Leibniz algebras of the first family are characteristically nilpotent. Moreover, for the first family we classify non-characteristically nilpotent algebras by means of Catalan numbers. In addition, for the rest two families of filiform Leibniz algebras we describe non-characteristically nilpotent algebras, i.e., those filiform Leibniz algebras which li…

The algebraic classification of nilpotent algebras

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

Generating Hypergraphs, Decomposability and Classification of Two-Step Nilpotent Lie Algebras

In 1973, Gauger proposed a generator-relation method and a duality theory for two-step nilpotent Lie algebras. Based upon these, he classified two-step nilpotent Lie algebras of dimension $8$. In 1999, Galitski and Timashev continued this approach to classify two-step nilpotent Lie algebras of dimension $9$. Their results were partially improved by Ren and Zhu in 2011, Yan and Deng in 2013. Some decomposable two-step nilpotent Lie algebras were excluded in the case of dimension $8$. In this paper, we define generating hypergraph for a two-step nilpotent Lie algebra. The two-step nilpotent Lie …

Lie algebras with associative structures. Applications to the study of 2-step nilpotent Lie algebras

We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.

The algebraic and geometric classification of nilpotent anticommutative algebras

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative algebras, complemented by $130$ additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid $6$-dimensional complex nilpotent anticommutative algebras.

Classification of algebras of level two in the variety of nilpotent algebras and Leibniz algebras

This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the variety of Leibniz algebras. It is shown that, up to isomorphism, there exist three Leibniz algebras of level two, one of which is solvable, and two of which are nilpotent. Moreover we describe all algebras of level two in the variety of nilpotent algebras.

幂等双交换代数的代数分类

The algebraic classification of nilpotent bicommutative algebras

该研究给出了复数域上5维幂零双交换代数的完整代数分类。

The algebraic and geometric classification of nilpotent Novikov algebras

This paper is devoted to give the complete algebraic and geometric classification of $4$-dimensional nilpotent Novikov algebras over $\mathbb C.$

The isomorphism problem for universal enveloping algebras of nilpotent Lie algebras

In this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent Lie algebras. We prove that if the characteristic of the underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie algebra of dimension at most~6 is determined by the isomorphism type of its universal enveloping algebra. Examples show that the restriction on the characteristic is necessary.

The moduli space of 4-dimensional nilpotent complex associative algebras

In this paper, we study 4-dimensional nilpotent complex associative algebras. This is a continuation of the study of the moduli space of 4-dimensional algebras. The non-nilpotent algeras were analyzed in an earlier paper. Even though there are only 15 families of nilpotent 4-dimensional algebras, the complexity of their behaviour warrented a separate study, which we give here.

Characteristically nilpotent Lie algebras : a survey

We review the known results about characteristically nilpotent complex Lie algebras, as well as we comment recent developements in the theory.

The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

We give algebraic and geometric classifications of complex $4$-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of $3$ rigid algebras and $2$ one-parametric families of algebras.

幂零辛交错代数

Nilpotent Symplectic Alternating Algebras

本文研究幂零辛交错代数，建立其结构理论，分类任意域上维数至10的此类代数，并由此发现强幂零群与强可解群两类新群。

One-generated nilpotent assosymmetric algebras

We give the classification of $5$- and $6$-dimensional complex one-generated nilpotent assosymmetric algebras.

The geometric classification of nilpotent Tortkara algebras

We give a geometric classification of all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C$

A characterisation of nilpotent Lie algebras by invertible Leibniz-derivations

Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.

Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.

Classification of 4-dimensional nilpotent complex Leibniz algebras

The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras dimensions less than 3 is extended to the dimension four. {\it AMS Subject Classifications}: 16D70, 17A30, 17A60, 17B30 {\it Key words:} Leibniz algebra, associative algebra, nilpotence, nulfiliform Leibniz algebra, filiform Leibniz algebra.

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.

Degenerations of binary Lie and nilpotent Malcev algebras

We describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over \mathbb{C}. In particular, we describe all irreducible components of these varieties.

幂零辛交错代数 I

Nilpotent symplectic alternating algebras I

本文建立了幂零辛交错代数的结构理论，为该类代数提供了系统性研究框架。

One-generated nilpotent bicommutative algebras

We give a classification of $5$- and $6$-dimensional complex one-generated nilpotent bicommutative algebras.

The algebraic and geometric classification of nilpotent right alternative algebras

We present algebraic and geometric classifications of the $4$-dimensional complex nilpotent right alternative algebras. Specifically, we find that, up to isomorphism, there are only $9$ non-isomorphic nontrivial nilpotent right alternative algebras. The corresponding geometric variety has dimension $13$ and it is determined by the Zariski closure of $4$ rigid algebras and one one-parametric family of algebras.

幂零代数的几何分类

The geometric classification of nilpotent algebras

本文对 n 维幂零代数及其交换与反交换子类进行几何分类，证明相应簇不可约并给出维数与显式泛型族，同时应用于反交换代数长度研究。

Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2

First we describe the Skjelbred-Sund method for classifying nilpotent Lie algebras. Then we use it to classify 6-dimensional nilpotent Lie algebras over any field of characteristic not 2. The proof of this classification is essentially constructive: for a given 6-dimensional nilpotent Lie algebra L, following the steps of the proof, it is possible to find a Lie algebra M that occurs in the list, and an isomorphism L --> M.

The classification of $n$-dimensional nilpotent non-Tortkara anticommutative algebras with $(n-4)$-dimensional annihilator

In this paper, we give a complete classification of $n$-dimensional nilpotent non-Tortkara anticommutative algebras with $(n-4)$-dimensional annihilator over $\mathbb{C}$.

Degenerations of 8-dimensional 2-step nilpotent Lie algebras

In this work, we consider degenerations between 8-dimensional 2-step nilpotent Lie algebras over $\mathbb{C}$ and obtain the geometric classification of the variety $\mathcal{N}^2_8$.

幂零Novikov代数的代数分类

The algebraic classification of nilpotent Novikov algebras

本文针对复5维幂零Novikov代数，开展了完整的代数分类研究工作。

One-generated nilpotent Novikov algebras

We give a classification of 5- and 6-dimensional complex one-generated nilpotent Novikov algebras

A computer-based approach to the classification of nilpotent Lie algebras

We adopt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most~9 over $\F_2$ and those of dimension at most~7 over $\F_3$ and $\F_5$.

Computation of the cohomology of a family of Z_2-graded Nilpotent Algebras

In this paper we compute the cohomology of certain special cases of nilpotent algebras in a complex \zt-graded vector space of arbitrary finite dimension. These algebras are generalizations of the only two nontrivial complex 2-dimensional nilpotent algebras, one from the moduli space of $1|1$-dimensional algebras, and the other from the space of nongraded 2-dimensional algebras.

Degenerations of 7-dimensional nilpotent Lie algebras

We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimension.

On characterizing nilpotent Lie algebra by their multiplier, $ s(L)=6, 7

Let $ L $ be an $ n $-dimensional non-abelian nilpotent Lie algebra and $ s(L)=\frac{1}{2}(n-1)(n-2)+1-\dim \mathcal{M}(L) $ where $ \mathcal{M}(L) $ is the Schur multiplier of a Lie algebra $ L. $ The structures of nilpotent Lie algebras $ L $ when $ s(L)\in \lbrace 0,1,2,3,4,5\rbrace $ are determined. In this paper, we classify all non-abelian nilpotent Lie algebras $ L $ when $ s(L)=6,7. $

Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras

In this paper we investigate pre-derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three non-intersected families. We describe the pre-derivation of filiform Leibniz algebras for the first and second families. We found sufficient conditions under which filiform Leibniz algebras are strongly nilpotent. Moreover, for the first and second families, we give the description of characteristically nilpotent algebras which are non-strongly nilpotent.

Nilpotent Sabinin algebras

In this note we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of Ado theorem for loops whose commutator-associator filtration is of finit…

Zinbiel algebras are nilpotent

In this paper we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result by other authors that they are solvable.

A class of nilpotent Lie algebras whose center acts nontrivially in cohomology

We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.

The classification of p-nilpotent restricted Lie algebras of dimension at most 4

In this paper we obtain the classification of $p$-nilpotent restricted Lie algebras of dimension at most four over a perfect field of characteristic p.