Sur les alg\`ebres de Lie quasi-filiformes compl\'etables [On completable quasi-filiform Lie algebras]

TL;DR

This paper classifies quasi-filiform Lie algebras that can be extended to complete Lie algebras and demonstrates the existence of complete Lie algebras with arbitrarily large second cohomology groups.

Contribution

It provides a classification of completable quasi-filiform Lie algebras and shows that for any positive integer, there exists a complete Lie algebra with a large second cohomology group.

Findings

Identified all quasi-filiform Lie algebras that are completable.

Proved the existence of complete Lie algebras with arbitrarily large second cohomology groups.

Abstract

The aim of this work is to determine the quasi-filiform Lie algebras that are completable. We further prove that for any positive integer $m$ there exists a complete Lie algebra, the second cohomology group of which has dimension greater or equal than $m$.