Ivan Kaygorodov, Mykola Khrypchenko, Samuel A. Lopes
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.
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We give an algebraic classification of complex $4$-dimensional nilpotent $\mathfrak{CD}$-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.
We classify the $4$-dimensional nilpotent bicommutative algebras over $\mathbb C$ from both algebraic and geometric approaches.
We classify all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C.$
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 $\leq 6.$
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 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。
This paper is devoted to give the complete algebraic and geometric classification of $4$-dimensional nilpotent Novikov algebras over $\mathbb C.$
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。
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 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。
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.
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).
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.
We give algebraic and geometric classifications of $4$-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are $41$ one-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $18$ two-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $2$ three-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by $21$ additional isomorphism classes. The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the…
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.
The algebraic and geometric classification of nilpotent Leibniz algebras
本文对复5维幂零Leibniz代数进行了完整的代数与几何分类，并确定其代数簇维数为24，包含10个不可约分支及一个刚性代数。
This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension $15$ and decomposes into $3$ irreducible components determined by the Zariski closures of two one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid $4$-dimensional complex nilpotent left symmetric algebras.
8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of char $\ne 2, 3$
本文在特征不等于2,3的代数闭域上，利用代数几何方法和基础线性代数，对8维2步 nil 拉格朗日代数进行了自包含的分类。
We give a computional method to construct and classify nilpotent Jordan algebras over any arbitrary fields by the second cohomolgy of nilpotent Jordan algebras of low dimension "analogue of Skjelbred-Sund method", we see that every nilpotent Jordan algebras can be constructed by the second cohomolgy of nilpotent Jordan algebras of low dimension. We use this method to classify nilpotent Jordan algebras up to dimension three over any field and nilpotent Jordan algebras of dimension four over an algebraic closed field of characteristic not 2 and over the real field R. Also commutative nilpotent a…
We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.
We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.
We introduce a novel concept Frattinian nilpotent Lie algebra. Along with some examples, we show that every Frattinian nilpotent Lie algebra has a central decomposition of its ideals.
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.
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.
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.
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.
In this note we present some experimental results on the general matrix nilpotent Lie algebras derived by calculations on a computer
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.
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.
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…
Nilpotent Leibniz algebras with isomorphic maximal subalgebras are considered. The algebras are classified for coclass zero, one, and two. The results are field dependent.
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…
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 …
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.
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.
We give an algebraic classification of complex $5$-dimensional one-generated nilpotent terminal 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.
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.
We review the known results about characteristically nilpotent complex Lie algebras, as well as we comment recent developements in the theory.
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。
We give the classification of $5$- and $6$-dimensional complex one-generated nilpotent assosymmetric algebras.
We give a geometric classification of all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C$
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.
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.
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.
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.
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.
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。
The algebraic classification of five-dimensional nilpotent right alternative algebras
本文通过中心扩张方法，对无零化子分量且非2步幂零的复数五维幂零右交错代数进行同构分类，得到124个代数和代数族，其中29个有二维零化子，95个有一维零化子。
We give a classification of $5$- and $6$-dimensional complex one-generated nilpotent bicommutative algebras.
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。
Degenerations of 3-dimensional nilpotent associative algebras over an algebraically closed field
本研究确定了特征非2的代数闭域上三维幂零结合代数簇的完整退化图景，开发了适用于任意代数闭域的替代论证技术，发现结果随域特征是否为2呈现二分性。
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.
In this paper, we give a complete classification of $n$-dimensional nilpotent non-Tortkara anticommutative algebras with $(n-4)$-dimensional annihilator over $\mathbb{C}$.
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 $\mathcal{N}^2_8$.
We give a classification of 5- and 6-dimensional complex one-generated nilpotent Novikov 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 $\F_5$.
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.
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.
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