ON THE HOMOTOPY LIE ALGEBRA OF AN ARRANGEMENT

GRAHAM DENHAM ALEXANDER I. SUCIU

Abstract. Let A be a graded-commutative, connected k-algebra 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 semi-direct product of the well-understood holonomy Lie algebra 𝔥_{A} with a certain 𝔥_{A}-module. 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.