On the filtration of a free algebra by its associative lower central series

TL;DR

This paper investigates the structure of successive quotients of the associative lower central series in free algebras, confirming a conjecture about their decomposition into tensor field modules and providing explicit calculations for small cases.

Contribution

It proves bounds on the tensor field modules in the Jordan-Hölder series of these quotients, confirming a recent conjecture and computing decompositions for specific small cases.

Findings

Bound on the degree of tensor field modules in the series

Confirmation of Arbesfeld and Jordan's conjecture

Explicit decomposition computations for small n and i

Abstract

This paper concerns the associative lower central series ideals $M_i$ of the free algebra $A_n$ on $n$ generators. Namely, we study the successive quotients $N_i=M_i/M_{i+1}$, which admit an action of the Lie algebra $W_n$ of vector fields on $\Bbb C^n$. We bound the degree $|\lambda|$ of tensor field modules $F_\lambda$ appearing in the Jordan-H\"older series of each $N_i$, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small $n$ and $i$.