Decomposition theorems for a generalization of the holonomy Lie algebra of an arrangement

TL;DR

This paper extends decomposition theorems for holonomy Lie algebras of arrangements to a broader class of Lie algebras, providing conditions for their direct product decompositions.

Contribution

It generalizes existing decomposition results for holonomy Lie algebras to a wider class of Lie algebras associated with arrangements.

Findings

Decomposition criteria for generalized Lie algebras

Conditions for direct product decompositions

Extension of previous holonomy algebra results

Abstract

In [7, Papadima and Suciu, When does the associated graded Lie algebra of an arrangement group decompose? Comment. Math. Helv. {\bf 81:4} (2006), 859--875] it is proved that the holonomy Lie algebra of an arrangement of hyperplanes through origo decomposes as a direct product of Lie algebras in degree at least two if and only if a certain (computable) condition is fulfilled. We prove similar results for a class of Lie algebras which is a generalization of the holonomy Lie algebras. The proof methods are the same as in [7].