When Does the Associated Graded Lie Algebra of an Arrangement Group Decompose?
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چکیده
Let A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra H. Suppose H3 is a free abelian group of minimum possible rank, given the values the Möbius function μ : L2 → Z takes on the rank 2 flats of A. Then the associated graded Lie algebra of G decomposes (in degrees ≥ 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) = ∑ X∈L2 φr(Fμ(X)), for r ≥ 2. We illustrate this new Lower Central Series formula with several families of examples.
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تاریخ انتشار 2006