A converse to the Whitehead Theorem

TL;DR

This paper characterizes finite-dimensional Lie algebras over characteristic zero fields by their cohomological properties, showing they decompose into semisimple and nilpotent parts when high-degree cohomology vanishes.

Contribution

It provides a converse to the Whitehead Theorem, linking cohomological vanishing to the algebra's structural decomposition.

Findings

High-degree cohomology vanishes for certain Lie algebras.

Such Lie algebras are direct sums of semisimple and nilpotent components.

The result generalizes known theorems about Lie algebra cohomology.

Abstract

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras.