Koszul Duality for Dioperads

We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various algebraic structures using operads is partly due to the theory of Koszul duality for operads; see eg. [K] or [L] for surveys. However, algebraic structures such as bialgebras and Lie bialgebras, which involve both multiplication and comultiplication, or bracket and cobracket, are defined using PROP’s (cf. [Ad]) rather than operads. Inspired by the theory of string topology of Chas-Sullivan ([ChS], [Ch], [Tr]), Victor Ginzburg suggested to the author that there should be a theory of Koszul duality for PROP’s. The present paper results from the observation that when the defining relations between the generators of a PROP are spanned over trees, then the ”tree-part” of the PROP has the structure of a dioperad. We show that one can set up a theory of Koszul duality for dioperads. In §1, we give the definition of a dioperad and other generalities. In §2, we define the notion of a quadratic dioperad, its quadratic dual, and introduce our main example of Lie bialgebra dioperad. In §3, we define the cobar dual of a dioperad. A quadratic dioperad is Koszul if its cobar dual is quasi-isomorphic to its quadratic dual. The formalism in §2 and §3, in the case of operads, is due to Ginzburg-Kapranov [GiK]. In §4, we prove a proposition to be used later in §5. This proposition is a generalization of a result of Shnider-Van Osdol [SVO]. In §5, we prove that Koszulity of a quadratic dioperad is equivalent to exactness of certain Koszul complexes. In the case of operads, this is again due to Ginzburg-Kapranov, with a different proof by Shnider-Van Osdol. The Koszulity of the Lie bialgebra dioperad follows from this and an adaptation of results of Markl [M2]. 1. Dioperads 1.1. Let C be the category of finite dimensional differential Z-graded super vector spaces over a field k of characteristic 0, and let Hom be the internal hom functor of C. Let Sn denote the automorphism group of {1, . . . , n}. If m = m1 + · · · + mn is an ordered partition and σ ∈ Sn, then the block permutation σm1,... ,mn ∈ Sm is the permutation that acts on {1, . . . ,m} by permuting n intervals of lengths m1, . . . ,mn in the same way that σ permutes 1, . . . , n. If σ1 ∈ Sn1, σ2 ∈ Sn2 and i ∈ {1, . . . , n1}, then define σ1 ◦i σ2 ∈ Sn1+n2−1 by σ1 ◦i σ2 := (σ1)1,... ,1,n2,1... ,1 ◦ (Id× · · · × σ2 × · · · × Id), where σ2 is at the i-th place. (See eg. [MSS] Definition 1.2 or [SVO] p.387. ) 1