On the Lower Central Series Quotients of a Graded Associative Algebra

TL;DR

This paper investigates the structure of lower central series quotients in noncommutative associative algebras, specifically describing B_2(A) for certain quotients of free algebras and relating it to Kähler differentials.

Contribution

It provides a detailed description of B_2(A) for quotients of free algebras by generic homogeneous elements, linking it to Kähler differentials on associated varieties.

Findings

B_2(A) is isomorphic to a quotient of Kähler differentials.

Explicit description of B_2(A) for quotients of free algebras.

Connection established between algebraic quotients and geometric differentials.

Abstract

We continue the study of the lower central series L_i(A) and its successive quotients B_i(A) of a noncommutative associative algebra A, defined by L_1(A)=A, L_{i+1}(A)=[A,L_i(A)], and B_i(A)=L_i(A)/L_{i+1}(A). We describe B_{2}(A) for A a quotient of the free algebra on two or three generators by the two-sided ideal generated by a generic homogeneous element. We prove that it is isomorphic to a certain quotient of Kaehler differentials on the non-smooth variety associated to the abelianization of A.