On properties of the lower central series of associative algebras

TL;DR

This paper provides an accessible overview of the lower central series in associative algebras, offering clearer proofs, exploring their interplay with geometry and representation theory, and presenting new theoretical and computational insights.

Contribution

It introduces simplified proofs and discusses new results and conjectures on the lower central series of free algebras with relations.

Findings

Shorter, clearer proofs of main results

New theoretical results on lower central series

Computational experiments and conjectures

Abstract

We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and to discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation.