Koszul duality for operads
type: Item
instance of: scholarly article
title: Koszul duality for operads
zbMATH Open document ID: 0855.18006
DOI: 10.1215/S0012-7094-94-07608-4
author: Victor Ginzburg
author: Mikhail Kapranov
published in: Duke Mathematical Journal
publication date: 1996-05-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0709.1228
review text: 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…
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 17A42
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 16S99
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 727824
zbMATH Keywords: sheaf cohomology
zbMATH Keywords: graph cohomologies
zbMATH Keywords: quadratic operads
zbMATH Keywords: Koszul operads
zbMATH Keywords: moduli spaces
zbMATH Keywords: dual operad
zbMATH Keywords: Koszul duality
Wikidata QID: Q29032050
reviewed by: Yuri Bespalov
MaRDI profile type: Publication
arXiv ID: 0709.1228
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