On the filtration of a free algebra by its associative lower central series

This paper concerns the associative lower central series ideals Mi of the free algebra An on n generators. Namely, we study the successive quotients Ni=Mi/Mi+1, which admit an action of the Lie algebra Wn of vector fields on Cn. We bound the degree |λ| of tensor field modules Fλ appearing in the Jordan–Hölder series of each Ni, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small n and i.