New results on the lower central series quotients of a free associative algebra

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated in Feigin and Shoikhet (2007) [FS]. We establish a linear bound on the degree of tensor field modules appearing in the Jordan–Hölder series of each graded component. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.