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This article is cited in 23 scientific papers (total in 23 papers)
Pre-Lie deformation theory
Vladimir Dotsenkoa, Sergey Shadrinb, Bruno Vallettec a School of Mathematics, Trinity College, Dublin 2, Ireland
b Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P. O. Box 94248, 1090 GE Amsterdam, The Netherlands
c Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Sorbonne Paris Cité, CNRS, UMR 7539, 93430 Villetaneuse, France
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 $\mathrm d$-dbar lemma.
Key words and phrases:
deformation theory, Lie algebra, pre-Lie algebra, homotopical algebra, operad.
Received: July 11, 2015; in revised form March 16, 2016
Citation:
Vladimir Dotsenko, Sergey Shadrin, Bruno Vallette, “Pre-Lie deformation theory”, Mosc. Math. J., 16:3 (2016), 505–543
Linking options:
https://www.mathnet.ru/eng/mmj607 https://www.mathnet.ru/eng/mmj/v16/i3/p505
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