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.
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.
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.
In this article we define the $c$-nilpotent multiplier of a finite dimensional Lie suepralgebra. We characterize the structure of $2$-nilpotent multiplier of finite dimensional nilpotent Lie superalgebras whose derived subalgebras have dimension at most one. Then we give an upper bound on the dimension of $2$-nilpotent multiplier of any finite dimensional nilpotent Lie superalgebra. Moreover, we discuses the $2$-capability of special as well as odd Heisenberg Lie superalgebras and abelian Lie superalgebras.
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.
The paper studies nilpotent $n$-Lie superalgebras. More specifically speaking, we first prove Engel's theorem for $n$-Lie superalgebras. Second, we research some properties of nilpotent $n$-Lie superalgebras, Finally, we give several sufficient conditions that an $n$-Lie superalgebra is nilpotent.
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.
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.
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 $\dim L' \leq 4$.
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.
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.
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.
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 …
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.
We introduce the notion of c-nilpotent Schur Lie-multiplier of Leibniz algebras. We obtain exact sequences and formulas of the dimensions of the underlying vector spaces relating the c-nilpotent Schur Lie-multiplier of a Leibniz algebra Q and its quotient by a two-sided ideal. These tools are used to characterize Lie-nilpotency and c-Lie-stem covers of Leibniz algebras. We prove the existence of c-Lie-stem covers for finite dimensional Leibniz algebras and the non existence of c-covering on certain Lie-nilpotent Leibniz algebras with non trivial c-nilpotent Schur Lie-multiplier, and we provide…
We prove that a Lie nilpotent one-sided ideal of an associative ring $R$ is contained in a Lie solvable two-sided ideal of $R$. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of $R.$ One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form $[... [ [r_1, r_{2}], ... ], r_{n-1}], r_{n}]$ are also studied.
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 we study the varieties of nilpotent Lie superalgebras of dimension $\leq 5$. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a by product we construct rigid nilpotent Lie superalgebras of arbitrary dimension.
We show that the finite dimensional nilpotent complex Lie superalgebras g whose injective hulls of simple U(g)-modules are locally Artinian are precisely those whose even part g_0 is isomorphic to a nilpotent Lie algebra with an abelian ideal of codimension 1 or to a direct product of an abelian Lie algebra and a certain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.
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 $\mathcal{N}^2_8$.
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. $
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 $\F_5$.
We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.
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.
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.
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.
We show that for a given nilpotent Lie algebra $\mathfrak{g}$ with $Z(\mathfrak{g})\subseteq [\mathfrak{g},\mathfrak{g}]$ all commutative post-Lie algebra structures, or CPA-structures, on $\mathfrak{g}$ are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras $F_{g,c}$ and discover a strong relationship to solving systems of linear equations of type $[x,u]+[y,v]=0$ for generator pairs $x,y\in F_{g,c}$. We use results of Remeslennikov and Stöhr concerning these equations to prove that…
We study a certain class of non-maximal rank contractions of the nilpotent Lie algebra $\frak{g}_{m}$ and show that these contractions are completable Lie algebras. As a consequence a family of solvable complete Lie algebras of non-maximal rank is given in arbitrary dimension.
In this paper, we first study properties of the lower central chains for Novikov algebras. Then we show that for every Lie nilpotent Novikov algebra~$\mathcal{N}$, the ideal of~$\mathcal{N}$ generated by the set~$\{ab - ba\mid a, b\in \mathcal{N}\}$ is nilpotent. We secondly provide necessary and sufficient conditions on the graph $E$ and the field $K$ for which the Leavitt path algebra $L_K(E)$ is Lie solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras, and show that the Lie solvability of~$L_K(E)$ and the Lie nilpotency of $[L_K(E),L_K(E)]$ are the…
Consider the special linear Lie algebra $\mathfrak{sl}_n(\mathbb {K})$ over an infinite field of characteristic different from $2$. We prove that for any nonzero nilpotent $X$ there exists a nilpotent $Y$ such that the matrices $X$ and $Y$ generate the Lie algebra $\mathfrak{sl}_n(\mathbb {K})$.
Let $L$ be a nilpotent Lie superalgebra of dimension $(m\mid n)$ and $s(L) = \frac{1}{2}[(m + n - 1)(m + n -2)]+ n+ 1 - \dim \mathcal{M}(L)$, where $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Here $s(L)\geq 0$ and the structure of all non-abelian nilpotent Lie superalgebras with $s(L)=0$ is known((\cite{Nayak2019}). This paper is devoted to obtain all nilpotent Lie superalgebras when $s(L) \leq 2$. Further, we apply those results to list all non-abelian nilpotent Lie superalgebras $L$ with $ t(L) \leq 4$.
In this note we describe the structure of finite dimensional Malcev algebras over a field of real numbers $\mathbb{R}$, which are nilpotent module its Lie center. It is proved that the corresponding analytic global Moufang loops are nilpotent module their nucleus.
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 consider associative algebras over a field. An algebra variety is said to be {\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$ is defined inductively by $[x_1, x_2, ..., x_n]=[[x_1, x_2, ..., x_{n-1}],x_n]$. By Zorn's Lemma every non-Lie nilpotent variety contains a minimal such variety, called {\em almost Lie nilpotent}, as a subvariety. A description of almost Lie nilpotent varieties for algebras over a field of characteristic 0 was made up by Yu.Mal'cev. We find a list of non-pri…
We study post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for nilpotent Lie algebras. First we show that if $\mathfrak{g}$ is nilpotent such that $H^0(\mathfrak{g},\mathfrak{n})=0$, then also $\mathfrak{n}$ must be nilpotent, of bounded class. For post-Lie algebra structures $x\cdot y$ on pairs of $2$-step nilpotent Lie algebras $(\mathfrak{g},\mathfrak{n})$ we give necessary and sufficient conditions such that $x\circ y=\frac{1}{2}(x\cdot y+y\cdot x)$ defines a CPA-structure on $\mathfrak{g}$, or on $\mathfrak{n}$. As a corollary we obtain that every LR-structure on a Heisenberg L…
For a finite dimensional Lie algebra $L$, it is known that $s(L)=\f{1}{2}(n-1)(n-2)+1-\mathrm{dim} M(L)$ is non negative. Moreover, the structure of all finite nilpotent Lie algebras is characterized when $s(L)=0,1$ in \cite{ni,ni4}. In this paper, we intend to characterize all nilpotent Lie algebra while $s(L)=2.$
In this paper we introduce the notion of pure non-characteristically nilpotent Lie algebra and under a condition we prove that a complex maximal extension of a finite-dimensional pure non-characteristically nilpotent Lie algebra is isomorphic to a semidirect sum of the nilradical and its maximal torus. We also prove that such solvable Lie algebras are complete and we specify a subclass of the maximal solvable extensions of pure non-characteristically nilpotent Lie algebras that have trivial cohomology group. Some comparisons with the results obtained earlier are given.
By considering the nilpotent Lie algebra with the derived subalgebra of dimension $ 2$, we compute some functors include the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such 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 $\leq 6.$
In this article, we discuss the category $\mathcal{SN}_2$ where the objects are finite-dimensional nilpotent Lie superalgebras of class two and the category $\mathcal{SSKE}$ where the objects are skew-supersymmetric bilinear maps. We establish relation between $\mathcal{SN}_2$ and $\mathcal{SSKE}$. As a result, we discuss the capability of nilpotent Lie superalgebras of class two.
In this paper, we introduce the concept of (super-)multiplier-rank for Lie superalgeras and classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank $\leq 2$ over an algebraically closed field of characteristic zero. In the process, we also determine the multipliers of Heisenberg superalgebras.
Let $\mathfrak{sp}_{2n}(\mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X \in \mathfrak{sp}_{2n}(\mathbb {K})$ there exists a nilpotent element $Y \in \mathfrak{sp}_{2n}(\mathbb {K})$ such that $X$ and $Y$ generate $\mathfrak{sp}_{2n}(\mathbb {K})$.
We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L+rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also co…
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 maximum extensions of finite-dimensional nilpotent Lie algebras are considered. In particular, it is proved that in the general case such an extension is not unique, which refutes one L. Snoble's assumption.
We introduce the concept of weight graph for the weight system $P\frak{g}(T)$ of a finite dimensional nilpotent Lie algebra $\frak{g}$ and analyze the necessary conditions for a $(p,q)$-graph to be a weight graph for some $\frak{g}$.
In this paper, the structure of all finite-dimensional nilpotent Lie algebras of class two with derived subalgebra of dimension two over an arbitrary field $ \mathbb{F} $ is determined. Furthermore, we give the structure of the Schur multiplier of such Lie algebras.
In this paper, we determine the behavior of the $c$-nilpotent multiplier of Lie algebras with respect to the direct sums. Then we give some results on the $c$-capability of the direct sum of finite dimensional Lie algebras.
