On Successive Quotients of Lower Central Series Ideals for Finitely Generated Algebras

This paper examines the behavior of the successive quotients Ni(A) of the lower central series ideals Mi(A) of a nitely generated associative algebra A over Z. We de ne the lower central series Li(A) by L1(A) = A, Li+1(A) = [A,Li(A)], Mi(A) = A · Li(A) · A, and Ni(A) = Mi(A)/Mi+1(A). We decompose the Ni into its free and torsion components using the structure theorem of nitely generated abelian groups, and we examine patterns in the ranks and torsion of Ni for algebras with various homogeneous relations, including x2 in multiple variables, q-polynomial relation yx − qxy, and xm + ym. In order to do this, we create data tables with the ranks and torsion of various Ni, previously uncalculated, based on calculations done in the program Magma. This paper includes a complete description of Ni for the q-polynomial algebra, Z〈x, y〉 / (yx − qxy) and a proof for the ranks of N2 for A〈x, y〉/(xm+ym), which provides insight into how changing the coe cient or degree of a relation a ects rank and torsion, as well as general patterns for which primes appear in torsion.