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)
Full Text: arXiv Link