The theorems on freedom for free sums of Lie algebras with a relations

TL;DR

This paper proves a theorem on the freedom of free sums of Lie algebras with a single relation and generalizes the Freiheitssatz for such sums, extending classical results in Lie algebra theory.

Contribution

It introduces new theorems on the freedom of free sums of Lie algebras with relations, generalizing Shirshov and Kharlampovich results.

Findings

Proved the theorem on freedom for free sums of Lie algebras with a single relation

Established a generalized Freiheitssatz for free sums of Lie algebras

Extended classical results to broader classes of Lie algebra constructions

Abstract

In this paper we prove the theorem on freedom for free sums of Lie algebras with a single relation (analogous with the well-known result of Shirshov) and a generalized Freiheitssatz for free sums of Lie algebras (analogous with the well-known result of Kharlampovich).