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    Deformation complexes of algebraic operads and their applications

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    TETDEDXPaljug-temple-0225E-121 ...
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    Genre
    Thesis/Dissertation
    Date
    2015
    Author
    Paljug, Brian
    Advisor
    Dolgushev, Vasily
    Committee member
    Futer, David
    Lorenz, Martin, 1951-
    Block, Jonathan, 1960-
    Department
    Mathematics
    Subject
    Mathematics
    Deformation Quantization
    Grothendieck-teichmueller Group
    Homological Algebra
    Operads
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/3379
    
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    DOI
    http://dx.doi.org/10.34944/dspace/3361
    Abstract
    Given a reduced cooperad C, we consider the 2-colored operad Cyl(C) which governs diagrams U: V -> W, where V, W are Cobar(C)-algebras, and U is an infinity-morphism. We then investigate the deformation complexes of Cyl(C) and Cobar(C). Our main result is that the restriction maps between between the deformation complexes Der'(Cyl(C)) and Der'(Cobar(C)) are homotopic quasi-isomorphisms of filtered Lie algebras. We show how this result may be applied to modifying diagrams of homotopy algebras by derived automorphism. We then recall that Tamarkin's construction gives us a map from the set of Drinfeld associators to the homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of V. Drinfeld and T. Willwacher, both the source and the target of this map are equipped with natural actions of the Grothendieck-Teichmueller group GRT. We use our earlier results to prove that this map from the set of Drinfeld associators to the set of homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains is GRT-equivariant.
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