The Lower Central Series of the Quotient of a Free Algebra

TL;DR

This paper investigates the structure of the lower central series of quotients of free associative algebras, establishing an isomorphism for the second term in terms of differential forms under certain conditions.

Contribution

It generalizes previous results on the lower central series of free algebras, providing a new isomorphism for the second quotient in a broader setting.

Findings

Established an isomorphism between $B_2$ and differential forms for specific quotients.

Extended previous work by Balagovic, Balasubramanian, Dobrovolska, Kim, and Ma.

Utilized ideas from Feign and Shoikhet to analyze the structure of $B_i(R)$.

Abstract

Let $L_i(R)$ denote the $i^{\text{th}}$ term of the lower central series of an associative algebra $R$, and let $B_i(R)=L_i(R)/L_{i+1}(R)$. We show that $B_2(\mathbb{C}<x, y>/ P)\cong \Omega^2((\mathbb{C}<x, y>/ P)_{ab})$, for all homogeneous or quasihomogeneous $P$ with square-free abelianization. Our approach generalizes that of Balagovic and Balasubramanian in 2010, which in turn developed from that of Dobrovolska, Kim, and Ma in 2007. We also use ideas of Feign and Shoikhet in 2006, who initiated the study of the groups $B_i(R)$.