Koszul Lie algebras and their subalgebras
Simone Blumer
TL;DR
The paper studies quadratic and $p$-restricted Koszul Lie algebras, focusing on the Bloch-Kato subclass and their subalgebra structure. It develops and exploits HNN-extensions as a central tool for decomposing and embedding graded Lie algebras, proving that BK algebras admit a Levi-type decomposition and satisfy the Toral Rank Conjecture, while also establishing embedding results for finitely presented graded Lie algebras into quadratic (Koszul) ones. It analyzes cocyclic ideals, Fröberg's formula, and quadratic embeddings to relate subalgebraKoszulity to global Koszulity, and provides extensive treatment of RAAG Lie algebras as a broad Koszul class. Through a suite of examples, including surface Lie algebras and $2$-relator algebras, the work connects cohomology, eigenvalues, and essential decompositions, highlighting how Koszulity interacts with center structure and finiteness properties. The results illuminate structural constraints and constructive methods for building and analyzing Koszul and BK Lie algebras with applications to related group-theoretic contexts.
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.