An upper bound for the lower central series quotients of a free associative algebra

G. Dobrovolska; P. Etingof

paper

An upper bound for the lower central series quotients of a free associative algebra

arXiv:0801.1997

Abstract

Feigin and Shoikhet conjectured in math/0610410 that successive quotients $B_{m} (A_{n})$ of the lower central series filtration of a free associative algebra $A_{n}$ have polynomial growth. In this paper we give a proof of this conjecture, using the structure of $W_{n}$ -representation on $B_{m} (A_{n})$ which was defined in math/0610410 . We also prove that the number of squares in a Young diagram $D$ corresponding to an irreducible $W_{n}$ -module in the Jordan-Holder series of $B_{m} (A_{n})$ is bounded above by the integer $(m - 1)^{2} + 2 [(n - 2) /2] (m - 1)$ . This bound combined with MAGMA computations by Rains in math/0610410 allows us to confirm the $W_{n}$ -module structure of $B_{3} (A_{3})$ conjectured in math/0610410 .

7 pages; introduction expanded

Cited by in corpus (1)

Membranes and higher groupoids