On the Lower Central Series Quotients of a Graded Associative Algebra

Let A be a noncommutative associative algebra, viewed as a Lie algebra via definition of the Lie bracket [x, y] = xy − yx. Define the lower central series filtration inductively by L1(A) = A, and Li+1(A) = [A,Li(A)]. Denote the components of its associated graded space by Bi(A) = Li(A)/Li+1(A). The components of the associated graded space are well understood in the cases when A = An = the free algebra over C with n generators, and when A is a quotient of An by several relations with some additional smoothness conditions on the abelianization of A. In both those cases, the Bi(A) have polynomial growth for i > 1, and B2(A) ∼= Ω(Aab)/dΩ(Aab). Here Ω(Aab) denotes differential forms on the abelianization of A, and d is a diferentiation map. We study the case when A = An/〈P 〉, for P a homogeneous polynomial of any degree. This does not satisfy the aforementioned smoothness conditions. In this case, A and all the Bi(A) are graded by degree of polynomial. We find a basis forB2(An/〈x+y〉) for n = 2, 3,m > 1. Using its Hilbert series and the previous results in the smooth case, we can then conclude that for a generic homogeneous polynomial P and A = An/〈P 〉, we have the isomorphism