Koszul duality for operads
Operad is a system of data that formalizes properties of a collection of maps \(X^n \to X\), a certain set for each \(n = 1, 2, \dots\), which are closed under permutations of arguments of the maps and under all possible superpositions. Operads were introduced by \textit{J. P. May} in 1972 for the needs of homotopy theory. Since then it has been gradually realized that this concept has in fact fundamental significance for mathematics in general. The present paper, in particular, establishes a deep relationship between operads, moduli spaces of stable curves, graph cohomologies (\textit{M. Kontsevich}, 1992-93), and Verdier duality on sheaves. The class of quadratic operads and a distinguished subclass of Koszul operads are introduced. A natural duality on quadratic operads, which is analogous to the duality of \textit{S. B. Priddy} [Trans. Am. Math. Soc. 152, 39-60 (1970; Zbl 0261.18016)] for quadratic associative algebras, is defined. A brief outline of the paper is as follows. In the first chapter the definition of operad in the terms of the category of trees is recalled and a few examples are given. The significance of the operad \(\mathcal M\) formed by Grothendieck-Knudsen moduli spaces is explained: any operad can be described as a collection of sheaves on \(\mathcal M\). The second chapter is devoted to quadratic operads \(\mathcal P\), the ones generated by binary operations subject to relations involving three arguments only. Most of the structures that one encounters in algebra, e.g., associative, commutative, Lie, Poisson, etc. algebras, correspond to quadratic operads. The quadratic dual operad \({\mathcal P}^!\) is defined. It is shown that commutative and Lie operads are quadratic dual to each other, and the associative operad is self dual. The duality of Priddy is recovered by the duality of quadratic algebras over quadratic operads. On the category of quadratic operads the internal hom in the spirit of \textit{Yu. I. Manin} is introduced. The rôle of the Lie operad as dualizing object is shown. This allows the authors to give a natural interpretation of \textit{M. Lazard}'s ``Lie theory'' for formal groups (1955) in terms of Koszul duality. A contravariant duality functor \(D\) on the category of differential graded operads is introduced in Chapter 3 as opposed to the quadratic duality functor \({\mathcal P} \mapsto {\mathcal P}^!\). It is shown that from the algebraic point of view, the duality \(D\) is an analog of the cobar construction and a generalization of the tree part of the graph complex, and that from the geometric point of view, the duality is an analog of the Verdier duality for sheaves. Section 4 is devoted to Koszul operads, the quadratic operads \({\mathcal P}\) whose quadratic dual is canonically quasi-isomorphic to the \(D\)-dual. Equivalent definitions in terms of the Koszul complex or in terms of vanishing of higher homologies for free \({\mathcal P}\)-algebras is given. It is shown that commutative, associative and Lie operads are Koszul. In a previous paper of the first author the operads formed by Clebsch-Gordan spaces for representations of quantum groups and affine Lie algebras were investigated. The authors plan to study Koszulness of such operads.
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