On the Homotopy Lie Algebra of an Arrangement

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra gA is defined to be the Lie algebra of primitives of the Yoneda algebra, ExtA(k, k). Under certain homological assumptions on A and its quadratic closure, we express gA as a semi-direct product of the well-understood holonomy Lie algebra hA with a certain hA-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. 1. Definitions and statements of results 1.1. Holonomy and homotopy Lie algebras. Let A be a graded, graded-commutative algebra over a field k, with graded piece Ak, k ≥ 0. We will assume that A is locally finite, connected, and generated in degree 1. In other words, A = T (V )/I, where V is a finite-dimensional k-vector space, T (V ) = ⊕ k≥0 V ⊗k is the tensor algebra on V , and I is a two-sided ideal, generated in degrees 2 and higher. To such an algebra A, one naturally associates two graded Lie algebras over k (see for instance [12]). Definition 1.1. The holonomy Lie algebra hA is the quotient of the free Lie algebra on the dual of A1, modulo the ideal generated by the image of the transpose of the multiplication map μ : A1 ∧A1 → A2: (1) hA = Lie(A ∗ 1) / ideal (im(μ : A∗2 → A ∗ 1 ∧A ∗ 1)). Note that hA depends only on the quadratic closure of A: if we put A = T (V )/(I2), then hA = hA. Definition 1.2. The homotopy Lie algebra gA is the graded Lie algebra of primitive elements in the Yoneda algebra of A: (2) gA = Prim(ExtA(k,k)). In other words, the universal enveloping algebra of the homotopy Lie algebra is the Yoneda algebra: (3) U(gA) = ExtA(k,k). 2000 Mathematics Subject Classification. Primary 16E05, 52C35; Secondary 16S37, 55P62.