New results on the lower central series quotients of a free associative algebra

TL;DR

This paper advances understanding of the algebraic structure of free associative algebras by establishing bounds on their graded components and confirming related conjectures, contributing to the theory of algebraic series.

Contribution

It provides new bounds on the degrees and coefficients of graded components in the lower central series of free associative algebras, confirming key conjectures.

Findings

Linear bound on tensor field modules in Jordan-Hoelder series

Bound on the leading coefficient of Hilbert polynomial

Confirmation of conjectures on the third graded component

Abstract

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated by B. Feigin and B. Shoikhet. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hoelder series of each graded component, which is conjecturally tight. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.