It is already known that any filiform Lie algebra which possesses a codimension 2 solvable extension is naturally graded. Here we present an alternative derivation of this result.

Let $K$ be an associative and commutative ring with $1$, $k$ a subring of $K$ such that $1\in k$, $n\ge 2$ an integer. The paper describes subrings of the general linear Lie ring $g{l}_n\left(K\right)$ that contain the Lie ring of all traceless matrices over $k$.

Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$, there exists a smallest ideal $I$ of $L$ such that $L/I\in 𝒩$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$. In this paper, we define the subalgebra $S\left(L\right)={\bigcap }_{H\le L}{I}_L\left(H^{𝒩}\right)$. Set $S_0\left(L\right)=0$. Define $S_{i+1}\left(L\right)/S_i\left(L\right)=S(L/S_i\left(L\right))$ for $i\ge 1$. By $S_{\infty }\left(L\right)$ denote the terminal term of the ascending series. It is proved that $L=S_{\infty }\left(L\right)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra...

Soient $X$ une variété algébrique complexe, lisse, irréductible, $E$ et $F$ deux espaces vectoriels complexes de dimension finie et $\mu$ un morphisme de $X$ dans l’espace Lin$(E,F)$ des applications linéaires de $E$ dans $F$. Pour $x\in X$, on note $E\left(x\right)$ et $x·E$ le noyau et l’image de $\mu \left(x\right)$, ${\overline{\mu }}_x$ le morphisme de $X$ dans Lin$\left(E\right(x),F/(x·E\left)\right)$ qui associe à $y$ l’application linéaire $v\mapsto \mu \left(y\right)\left(v\right)+x·E$. Soit i${}_{\mu }$ la dimension minimale de $E\left(x\right)$. On dit que $\mu$ ala propriété $\left(𝐑\right)$ en $x$si i${}_{{\overline{\mu }}_x}$ est inférieur à i${}_{\mu }$. Soient $F^{*}$ le dual de $F$, S$\left(F\right)$ l’algèbre symétrique de $F$, ${ℐ}_{\mu }$ l’idéal de ${𝒪}_X{\otimes }_{ℂ}\mathrm{S}\left(F\right)$ engendré par...

We introduce the notion of quasi-trace functions on Lie algebras. As applications we study realizations of 3-dimensional and 4-dimensional 3-Lie algebras. Some comparison results on cohomologies of 3-Lie algebras and Leibniz algebras arising from quasi-trace functions are obtained.