Lower Central Series Ideal Quotients Over F_p and Z

Abstract

Given a graded associative algebra $A$, its lower central series is defined by $L_1=A$ and $L_{i+1}=[L_i,A]$. We consider successive quotients $N_i(A)=M_i(A)/M_{i+1}(A)$, where $M_i(A)=A{L}_i(A)A$. These quotients are direct sums of graded components. Our purpose is to describe the $\mathbb{Z}$-module structure of the components; i.e., their free and torsion parts. Following computer exploration using {\it MAGMA}, two main cases are studied. The first considers $A=A_n/(f_1,\dots ,f_m)$, with $A_n$ the free algebra on $n$ generators $\{x_1,\dots ,x_n\}$ over a field of characteristic $p$. The relations $f_i$ are noncommutative polynomials in $x_j^{p^{n_j}},$ for some integers $n_j$. For primes $p>2$, we prove that $p^{\sum n_j}\mid \text{dim}(N_i(A))$. Moreover, we determine polynomials dividing the Hilbert series of each $N_i(A)$. The second concerns $A=\mathbb{Z}⟨x_1,x_2,⟩/(x_1^m,x_2^n)$. For $i=2,3$, the bigraded structure of $N_i(A_2)$ is completely described.

References in corpus (1)

• The lower central series of the symplectic quotient of a free associative algebra