Kurosh theorem for certain Koszul Lie algebras

TL;DR

This paper establishes a Kurosh-type structure theorem for a specific class of positively graded Koszul Lie algebras, extending classical group theory results to a new algebraic context.

Contribution

It proves a Kurosh theorem analogue for certain Koszul Lie algebras with specific cohomological properties, which was previously unknown.

Findings

Subalgebras generated in degree 1 are structured similarly to free products.

The class of Lie algebras considered includes those with all degree 1 subalgebras being Koszul.

The theorem does not hold universally but applies under particular local cohomological conditions.

Abstract

The Kurosh theorem for groups provides the structure of any subgroup of a free product of groups and its proof relies on Bass-Serre theory of groups acting on trees. In the case of Lie algebras, such a general theory does not exists and the Kurosh theorem is false in general, as it was first noticed by Shirshov. However, we prove that, for a class of positively graded Lie algebras satisfying certain local properties in cohomology, such a structure theorem holds true for subalgebras generated in degree 1. Such class consists of Lie algebras, which have all the subalgebras generated in degree $1$ that are Koszul.