A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $\mathrm{SCAP}$-subalgebra if there is a chief series $0=L_0\subset L_1\subset ...\subset L_t=L$ of $L$ such that for every $i=1,2,...,t$, we have $H+L_i=H+L_{i-1}$ or $H\cap L_i=H\cap L_{i-1}$. This is analogous to the concept of $\mathrm{SCAP}$-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $\mathrm{SCAP}$-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

A pair of sequences of nilpotent Lie algebras denoted by $N_{n,11}$ and $N_{n,19}$ are introduced. Here $n$ denotes the dimension of the algebras that are defined for $n\ge 6$; the first term in the sequences are denoted by 6.11 and 6.19, respectively, in the standard list of six-dimensional Lie algebras. For each of $N_{n,11}$ and $N_{n,19}$ all possible solvable extensions are constructed so that $N_{n,11}$ and $N_{n,19}$ serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating...

We study some properties of generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for $\mathbb{Z}$-graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.

On étudie la structure des algèbres de Lie rigides sur un corps algébriquement clos de caractéristique 0. Elles sont algébriques. Quand le radical est non nilpotent leur dimension est la même que celle de l’algèbre des dérivations. Quand le radical est nilpotent elle appartient à l’un des cas suivants : parfaite, produit direct d’une algèbre parfaite par le corps de base ou encore toutes les dérivations semi-simples sont intérieures.

On donne une majoration de l'indice de certaines algèbres de Lie introduites par V. Dergachev, A. Kirillov et D. Panyushev. On en déduit la preuve d'une conjecture de D. Panyushev. Nous formulons aussi une conjecture concernant l'indice de ces algèbres, et la prouvons dans des cas particuliers. Enfin, nous donnons un résultat concernant l'indice des sous-algèbres paraboliques d'une algèbre de Lie semi-simple.