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# isDecomposable(CentralArrangement,Ring) -- whether a hyperplane arrangement decomposable in the sense of Papadima-Suciu

## Synopsis

• Function: isDecomposable
• Usage:
isDecomposable(A, R)
isDecomposable A
• Inputs:
• R, a ring, an optional coefficient ring used as the coefficient field for the holonomy Lie algebra. If unspecified, R=QQ
• Outputs:
• , that is true if the hyperplane arrangement decomposes in the sense of Papadima and Suciu over the given coefficient field

## Description

Following Definition 2.3 in Stefan Papadima and Alexander I. Suciu's paper "When does the associated graded Lie algebra of an arrangement group decompose?", Commentarii Mathematici Helvetici (2006) 859-875, arXiv:math/0309324, a hyperplane arrangement is decomposable if the derived subalgebra of its holonomy Lie algebra is a direct sum of the derived subalgebras of free Lie algebras, indexed by the rank-2 flats of the arrangement.

As described in the introduction of Papadima-Suciu, the X3 arrangement is decomposable. The hyperplane arrangement defined by a type $A_3$ root system is not decomposable. The authors show that a graphic arrangement is decomposable over ${\mathbb Q}$ if and only if it is decomposable over any other field. In general, it is not known if there exist arrangements for which property of being decomposable depends on the choice of field.

 i1 : X3 = arrangement "X3" o1 = {x , x , x , x + x , x + x , x + x } 1 2 3 1 2 1 3 2 3 o1 : Hyperplane Arrangement  i2 : assert isDecomposable X3 i3 : assert isDecomposable(X3, ZZ/5) i4 : assert not isDecomposable typeA 3