On the lower central series of an associative algebra

TL;DR

This paper investigates the structure of lower central series quotients of associative algebras, providing explicit bases and structures for specific cases, extending previous work and confirming conjectures.

Contribution

It offers explicit bases for the second quotient of free algebras, determines structures under nilpotency relations, and confirms conjectures for the third and fourth quotients in two-generator cases.

Findings

Basis for second quotient in free algebra case

Structure of quotients with nilpotent generators

Confirmation of conjectured structures for third and fourth quotients

Abstract

This paper continues the study of the lower central series quotients of an associative algebra A, regarded as a Lie algebra, which was started in math/0610410 by Feigin and Shoikhet. Namely, it provides a basis for the second quotient in the case when A is the free algebra in n generators (note that the Hilbert series of this quotient was determined earlier in math/0610410). Further, it uses this basis to determine the structure of the second quotient in the case when A is the free algebra modulo the relations saying that the generators have given nilpotency orders. Finally, it determines the structure of the third and fourth quotient in the case of 2 generators, confirming an answer conjectured in math/0610410. Finally, in the appendix, the results of math/0610410 are generalized to the case when A is an arbitrary associative algebra (under certain conditions on $A$).