Fiber-type arrangements and factorization properties
type: Item
instance of: scholarly article
title: Fiber-type arrangements and factorization properties
zbMATH Open document ID: 0697.55014
DOI: 10.1016/0001-8708(90)90012-C
author: Michel Jambu
published in: Advances in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let \({\mathcal A}\) be an arrangement of hyperplanes in \({\mathbb{C}}^ n\), \(X={\mathbb{C}}^ n-\cup \{H|\) \(H\in {\mathcal A}\}\) and \({\mathcal G}_ X\) the holonomy Lie algebra over \({\mathbb{Q}}\) of X. In this paper, the author reviews the definitions and main properties of a fibre type arrangement and examines the broken-circuit algebra of an arrangement explaining its factorization property. The main result of the paper is the following theorem. Let x be a modular element of rank r(L(\({\mathcal A}))-1\), where L(\({\mathcal A})\) is the intersection lattice of \({\mathcal A}\), and Y be the base space of a fibration associated to x. Then the natural linear map \(\phi\) : Lib(A\({}_ 1)\oplus {\mathcal G}_ Y\to {\mathcal G}_ X\), where \(A_ 1=(\alpha_ 1,...,\alpha_{| A_ 1|})\), is an isomorphism of graded vector spaces. Finally, the author extends his results to affine arrangements.
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 4143039
zbMATH Keywords: 1-minimal model
zbMATH Keywords: lower central series of fundamental group
zbMATH Keywords: Mal'cev completion
zbMATH Keywords: arrangement of hyperplanes in \({\mathbb{C}}^ n\)
zbMATH Keywords: holonomy Lie algebra
zbMATH Keywords: fibre type arrangement
zbMATH Keywords: broken-circuit algebra
zbMATH Keywords: intersection lattice
zbMATH Keywords: affine arrangements
reviewed by: Vasile Cruceanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4143039