Koszul Lie algebras and their subalgebras

TL;DR

This paper explores the structure and properties of Koszul and Bloch-Kato Lie algebras, introducing new families and demonstrating their cohomological and decomposition properties using HNN-extensions.

Contribution

It introduces new families of Bloch-Kato Lie algebras, extends the Levi decomposition, and analyzes properties of RAAG Lie algebras within the Koszul framework.

Findings

Bloch-Kato Lie algebras satisfy the Levi decomposition.

They satisfy the Toral Rank Conjecture.

New families of Koszul Lie algebras are constructed.

Abstract

This paper examines (restricted) Koszul Lie algebras, a class of positively graded Lie algebras with a quadratic presentation and specific cohomological properties. The study employs HNN-extensions as a key tool for decomposing and analysing these algebras. Building on a previous work on Koszul Lie algebras ("Kurosh theorem for certain Koszul Lie algebras", S. Blumer), this paper also deals with Bloch-Kato Lie algebras, which constitute a distinguished subclass of that of Koszul Lie algebras where all subalgebras generated by elements of degree $1$ have a quadratic presentation. It is shown that Bloch-Kato Lie algebras satisfy a version of the Levi decomposition theorem and that they satisfy the Toral Rank Conjecture. Two new families of such Lie algebras are introduced, including all graded Lie algebras generated in degree $1$ and defined by two quadratic relations. Throughout the paper, we show many properties of right-angled Artin graded (RAAG) Lie algebras, which form a large class of Koszul Lie algebras.