# Commentarii Mathematici Helvetici

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**Volume 81, Issue 4, 2006, pp. 859–875**

**DOI: 10.4171/CMH/77**

Published online: 2006-12-31

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

Stefan Papadima^{[1]}and Alexander I. Suciu

^{[2]}(1) Romanian Academy, Bucharest, Romania

(2) Northeastern University, Boston, USA

Let $\mathcal{A}$ be a complex hyperplane arrangement, with fundamental group $G$ and holonomy Lie algebra $\mathfrak{H}$. Suppose $\mathfrak{H}_3$ is a free abelian group of minimum possible rank, given the values the MÃ¶bius function $\mu\colon \mathcal{L}_2\to \mathbb{Z}$ takes on the rank $2$ flats of $\mathcal{A}$. Then the associated graded Lie algebra of $G$ decomposes (in degrees $\ge 2$) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by $\phi_r(G)=\sum _{X\in \mathcal{L}_2} \phi_r(F_{\mu(X)})$, for $r\ge 2$. We illustrate this new Lower Central Series formula with several families of examples.

*Keywords: *Hyperplane arrangement, lower central series, associated graded Lie algebra, holonomy Lie algebra, Chen Lie algebra

Papadima Stefan, Suciu Alexander: When does the associated graded Lie algebra of an arrangement group decompose?. *Comment. Math. Helv.* 81 (2006), 859-875. doi: 10.4171/CMH/77