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.

We give a general definition of branched, self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk’s and Gupta–Sidki’s torsion groups are nil as well as self-similar.We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.

Generalizing Petrogradsky’s construction, we give examples of infinite-dimensional nil Lie algebras of finite Gelfand–Kirillov dimension over any field of positive characteristic.

The paper studies nilpotent $n$-Lie superalgebras over a field of characteristic zero. More specifically speaking, we prove Engel’s theorem for $n$-Lie superalgebras which is a generalization of those for $n$-Lie algebras and Lie superalgebras. In addition, as an application of Engel’s theorem, we give some properties of nilpotent $n$-Lie superalgebras and obtain several sufficient conditions for an $n$-Lie superalgebra to be nilpotent by using the notions of the maximal subalgebra, the weak ideal and the...

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.

In this paper, we continue to investigate some properties of the family $\Gamma$ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras $\Gamma$ are intrinsic. Thereby, we classify these Lie...

We construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic $2$. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain spectral sequences which compare Chevalley-Eilenberg-, commutative- and Leibniz cohomology. These methods are illustrated by a few computations.