On the homotopy Lie algebra of an arrangement. (English) Zbl 1198.17012
Summary: 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, \(\text{Ext}_A(k,k)\). 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.
MSC:
| 17B55 | Homological methods in Lie (super)algebras |
| 16S37 | Quadratic and Koszul algebras |
| 17B70 | Graded Lie (super)algebras |
| 16E05 | Syzygies, resolutions, complexes in associative algebras |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
| 55P62 | Rational homotopy theory |
Software:
Macaulay2References:
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