On the homotopy Lie algebra of an arrangement
Abstract
Let A be a gradedcommutative, connected kalgebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semidirect product of the wellunderstood holonomy Lie algebra h_A with a certain h_Amodule. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincaré polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2005
 arXiv:
 arXiv:math/0502417
 Bibcode:
 2005math......2417D
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 16E05;
 52C35;
 16S37;
 55P62
 EPrint:
 20 pages