Koszul Lie algebras and their subalgebras

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.