Lower central series of free algebras in symmetric tensor categories

2010/01/31 by Asilata Bapat, David Jordan
Mathematics · #math.RA #math.QA #math.RT #msc:16W25

paper · pdf · doi:10.1016/j.jalgebra.2012.10.001

published as Journal of Algebra Volume 373, 1 January 2013, Pages 299-311 · Corrected an error in the proof of Lemma 6.1 pointed out by Alexei Krasilnikov

arxiv created 2012/05/31 · arxiv updated 2016/10/03

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 B₁, and second and third quotients B₂, and B₃ 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.

Citations

• On [A,A]/[A,[A,A]] and on a W_n-action on the consecutive commutators of free associative algebra
• New results on the lower central series quotients of a free associative algebra
• On universal Lie nilpotent associative algebras