Abstract
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 bialgebras 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.
Highlights
This paper develops the Koszul duality theory for protoperads, defined in [Ler19], which are an analog of properads with more symmetries
The main application of this theory is the proof of the Koszulness of the properad which encodes double Lie algebras, from which it follows that the properad encoding double Poisson algebras is Koszul
In order to determine the homotopical properties of a family of algebras, we use the classical strategy, which was already used to understand, for example, the homotopical properties of Gerstenhaber algebras
Summary
This paper develops the Koszul duality theory for protoperads, defined in [Ler19], which are an analog of properads (see [Val, Val07]) with more symmetries. Definition/Proposition 1.16 (Induction functor (see [Ler, Def. 4.1])) We define the induction functor Ind : S-modrked → S-bimodrked which is given, for all reduced S-modules V and, for all finite sets S and E, by: Ind V (S, E) ∼= 0 if S ∼= E k[Aut(E)] ⊗ V (S) otherwise. This functor is exact, has a right adjoint which is the functor of restriction Res, and is monoidal. If P is a weight graded protoperad, so is the shuffle protoperad Psh
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