The Lower Central Series of the Quotient of a Free Algebra

Let $L_i(R)$ denote the $i^{\text{th}}$ term of the lower central series of an associative algebra $R$, and let $B_i(R)=L_i(R)\mathrm{/}{L}_{i+1}(R)$. We show that $B_2(\mathbb{C}<x,y>\mathrm{/}P)\cong {\mathrm{\Omega }}^2((\mathbb{C}<x,y>\mathrm{/}P{)}_{ab})$, 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 $B_i(R)$.