Pre-Lie deformation theory

By Vladimir Dotsenko, Sergey Shadrin, Bruno Vallette. 2015

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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 $dd^c$-lemma.

Subfields: Algebra and Number Theory, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/51075f1f3d85f5c5daaf609617c9362ae314c0e6