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

### Alexander I. Suciu

Northeastern University, Boston, USA### Stefan Papadima

Romanian Academy, Bucharest, Romania

## Abstract

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 Möbius 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.

## Cite this article

Alexander I. Suciu, Stefan Papadima, When does the associated graded Lie algebra of an arrangement group decompose?. Comment. Math. Helv. 81 (2006), no. 4, pp. 859–875

DOI 10.4171/CMH/77