Abstract
Abstract We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the little disks operads. We show that twisting with certain Maurer–Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of these mapping spaces.
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