Protoperads II: Koszul duality

Leray, Johan

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 897-941.

Dans cet article, on construit une adjonction bar-cobar et une dualité de Koszul pour les protopérades, qui encodent fidèlement des catégories de gèbres avec des symétries diagonales, comme les algèbres double Lie ($𝒟ℒie$). On donne un critère pour montrer qu’une protopérade quadratique binaire est de Koszul, critère que l’on applique avec succès à la protopérade $𝒟ℒie$. Comme corollaire, on en déduit que la propérade $𝒟𝒫ois$ qui encode les algèbres double Poisson est de Koszul. Cela nous permet de décrire les propriétés homotopiques des algèbres double Poisson, qui jouent un role clé en géométrie non commutative.

In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of gebras with diagonal symmetries, like double Lie algebras ($𝒟ℒie$). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad $𝒟ℒie$. As a corollary, we deduce that the properad $𝒟𝒫ois$ which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in non commutative geometry.

Classification : 18D50, 18G55, 17B63, 14A22
Keywords: Properad, protoperad, Koszul duality, double Poisson
Mot clés : Propérades, protopérades, dualité de Koszul, double Poisson

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