When does the associated graded Lie algebra of an arrangement group decompose?
Combinatorics
2010-10-26 v3 Group Theory
Abstract
Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum possible rank, given the values the M\"obius function \mu: \L_2\to \Z takes on the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by \phi_r(G)=\sum_{X\in \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We illustrate this new Lower Central Series formula with several families of examples.
Keywords
Cite
@article{arxiv.math/0309324,
title = {When does the associated graded Lie algebra of an arrangement group decompose?},
author = {Stefan Papadima and Alexander I. Suciu},
journal= {arXiv preprint arXiv:math/0309324},
year = {2010}
}
Comments
14 pages, accepted for publication by Commentarii Mathematici Helvetici