JournalscmhVol. 81, No. 4pp. 859–875

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

Let A\mathcal{A} be a complex hyperplane arrangement, with fundamental group GG and holonomy Lie algebra H\mathfrak{H}. Suppose H3\mathfrak{H}_3 is a free abelian group of minimum possible rank, given the values the Möbius function μ ⁣:L2Z\mu\colon \mathcal{L}_2\to \mathbb{Z} takes on the rank 22 flats of A\mathcal{A}. Then the associated graded Lie algebra of GG decomposes (in degrees 2\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 ϕr(G)=XL2ϕr(Fμ(X))\phi_r(G)=\sum _{X\in \mathcal{L}_2} \phi_r(F_{\mu(X)}), for r2r\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