Pre-Lie deformation theory
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2015
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| _version_ | 1866910500292395008 |
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| author | Dotsenko, Vladimir Shadrin, Sergey Vallette, Bruno |
| author_facet | Dotsenko, Vladimir Shadrin, Sergey Vallette, Bruno |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1502_03280 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Pre-Lie deformation theory Dotsenko, Vladimir Shadrin, Sergey Vallette, Bruno Quantum Algebra Category Theory K-Theory and Homology Rings and Algebras 18G55, 13D10, 17B60, 18D50 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. |
| title | Pre-Lie deformation theory |
| topic | Quantum Algebra Category Theory K-Theory and Homology Rings and Algebras 18G55, 13D10, 17B60, 18D50 |
| url | https://arxiv.org/abs/1502.03280 |