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Pre-Lie deformation theory. (English) Zbl 1386.18054
Summary: In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for preLie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the d-dbar lemma.

18G55 Nonabelian homotopical algebra (MSC2010)
13D10 Deformations and infinitesimal methods in commutative ring theory
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
18D50 Operads (MSC2010)
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