When does the associated graded Lie algebra of an arrangement group decompose?

Details When does the associated graded Lie algebra of an arrangement group decompose? When does the associated graded Lie algebra of an arrangement group decompose?

2003-09-19

arxiv.org/abs/math/0309324v3

Commentarii Mathematici Helvetici 81 (2006), no. 4, 859-875

Mathematics

Combinatorics

14 pages, accepted for publication by Commentarii Mathematici Helvetici

Scientific paper

10.4171/CMH/77

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.

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