Pre-Lie deformation theory
Pre-Lie deformation theory
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DOI：
10.17323/1609-4514-2016-16-3-505-543
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发表时间：
2015-02
期刊：
影响因子：
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通讯作者：
V. Dotsenko;S. Shadrin;B. Vallette
中科院分区：
文献类型：
--
作者：
V. Dotsenko;S. Shadrin;B. Vallette
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.
