Lower central series of free algebras in symmetric tensor categories

TL;DR

This paper extends the study of the lower central series of free associative algebras to symmetric tensor categories, providing explicit computations and confirming related conjectures.

Contribution

It generalizes the lower central series constructions to symmetric tensor categories and computes key quotients and Hilbert series for free algebras in these categories.

Findings

Computed the first, second, and third quotients of the lower central series in symmetric tensor categories.

Explicitly determined the Hilbert series for free algebras in super vector spaces.

Proved conjectures relating the lower central series to ideals in associative algebras.

Abstract

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.