Pre-Lie deformation theory

Abstract

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie 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 $d{d}^c$-lemma.

Final version. Minor corrections. To appear in the Moscow Mathematical Journal

References in corpus (1)

• Higher Topos Theory

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• L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras
• Universal Lie Formulas for Higher Antibrackets
• Generalized cohomological field theories in the higher order formalism
• Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras