Approche fonctorielle et combinatoire de la propérade des algèbres double Poisson

Abstract : We construct and study the generalization of shifted double Poisson algebras to all additive symmetric monoidal categories. We are especially interested in linear and quadratic double Poisson algebras. We then study the koszulity of the properads DLie and DPois = As ⮽c DLie which encode double Lie algebras and double Poisson algebras respectively. We associate to each, a S-module with a monoidal structure for a new monoïdal product call the connected composition product : we call such monoids protoperads. We show, for any S-module, the existence of the associated free protoperad and we make explicit the underlying combinatorics. We define a bar-cobar adjunction, the notion of Koszul duality and PBW bases for protoperads. We present an attempt of prove a PBW theorem à la Hoffbeck for protoperads, and prove the koszulity of the dioperad associated to the properad DLie.
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Johan Leray. Approche fonctorielle et combinatoire de la propérade des algèbres double Poisson. Mathématiques générales [math.GM]. Université d'Angers, 2017. Français. ⟨NNT : 2017ANGE0027⟩. ⟨tel-01719403⟩

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