# Moscow Mathematical Journal

Volume 16, Issue 3, July–September 2016 pp. 505–543.

Pre-Lie Deformation Theory

**Authors**:
Vladimir Dotsenko (1), Sergey Shadrin (2), and Bruno Vallette (3)

**Author institution:**(1) School of Mathematics, Trinity College, Dublin 2, Ireland

(2) Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P. O. Box 94248, 1090 GE Amsterdam, The Netherlands

(3) Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Sorbonne Paris Cité, CNRS, UMR 7539, 93430 Villetaneuse, France

**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.

2010 Math. Subj. Class. Primary 18G55; Secondary 13D10, 17B60, 18D50.

**Keywords:**Deformation theory, Lie algebra, pre-Lie algebra, homotopical algebra, operad.

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