Lower central series of free algebras in symmetric tensor categories

Details Lower central series of free algebras in symmetric tensor categories Lower central series of free algebras in symmetric tensor categories

2010-01-08

arxiv.org/abs/1001.1375v2

Mathematics

Rings and Algebras

Corrected an error in the formulation of Conjecture 8.2 pointed out by Darij Grinberg

Scientific paper

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.

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