Geodesic
Deformation Theory with Homotopy Algebra Structures on Tensor Products
Documenta mathematica, Tome 23 (2018), pp. 189-240
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In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven to be compatible with the concepts of homotopy theory: ∞-morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer-Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation ∞-groupoid for differential graded Lie algebras.
DOI : 10.4171/dm/617
Classification : 08C05
Mots-clés : operads, homotopy Lie algebras, homotopy associative algebras, infinity-morphisms, Maurer-Cartan elements 
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     author = {Daniel Robert-Nicoud},
     title = {Deformation {Theory} with {Homotopy} {Algebra} {Structures} on {Tensor} {Products}},
     journal = {Documenta mathematica},
     pages = {189--240},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/617},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/617/}
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Daniel Robert-Nicoud. Deformation Theory with Homotopy Algebra Structures on Tensor Products. Documenta mathematica, Tome 23 (2018), pp. 189-240. doi: 10.4171/dm/617

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