Bass-Serre theory for Lie algebras: a homological approach

TL;DR

This paper extends Bass-Serre theory to Lie algebras using homological methods, introducing a fundamental Lie algebra concept and applying Mayer-Vietoris sequences to analyze their structure and coherence.

Contribution

It develops a homological framework for Lie algebras inspired by Bass-Serre theory, including fundamental Lie algebras and Mayer-Vietoris sequences.

Findings

Defined fundamental Lie algebra of a graph of Lie algebras

Extended group theory results to $ $-graded Lie algebras

Applied Mayer-Vietoris sequences to study coherence

Abstract

We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras.