  We continue the study of the lower central series of a free associative
algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410). We
generalize via Schur functors the constructions of the lower central series to
any symmetric tensor category; specifically we compute the modified first
quotient \bar{B}_1, and second and third quotients B_2, and B_3 of the series
for a free algebra T(V) in any symmetric tensor category, generalizing the main
results of (arXiv:math/0610410) and (arXiv:0902.4899). In the case
A_{m|n}:=T(\CC^{m|n}), we use these results to compute the explicit Hilbert
series. Finally, we prove a result relating the lower central series to the
corresponding filtration by two-sided associative ideals, confirming a
conjecture from (arXiv:0805.1909), and another one from (arXiv:0902.4899), as
corollaries.

Rings and Algebras;Quantum Algebra;Representation Theory