Some remarks about the second Leibniz cohomology group for Lie algebras

TL;DR

This paper compares different Leibniz cohomology groups of Lie algebras using elementary methods, explores properties related to Leibniz deformations, and classifies certain nilpotent Lie algebras based on their cohomological properties.

Contribution

It provides a simple comparison of second Leibniz cohomology groups with classical ones and classifies specific nilpotent Lie algebras with zero Koszul 3-form.

Findings

Comparison of adjoint and trivial Leibniz cohomology spaces with classical cohomology.

Identification of Lie algebras with zero Koszul 3-form including quotients of Borel subalgebras.

Classification of indecomposable nilpotent Lie algebras of dimension ≤7 by Kac-Moody types.

Abstract

We compare by a very elementary approach the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones. Examples are given of coupled cocycles. Some properties are deduced as to Leibniz deformations. We also consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional semisimple Lie algebras. Finally, a list of Kac-Moody types for indecomposable nilpotent Lie algebras of dimension $\leqslant 7$ is given.