JournalscmhVol. 81 , No. 4DOI 10.4171/cmh/77

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

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.