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A Poincaré-Birkhoff-Witt criterion for Koszul operads. (English) Zbl 1207.18009
Koszul duality was introduced by S. B. Priddy in his paper “Koszul resolutions” [Trans. Am. Math. Soc. 152, 39–60 (1970; Zbl 0261.18016)], in order to study the homology of the Steenrod algebra. In this paper Priddy introduces the concept of Koszul algebras for associative algebras. When an algebra $$A$$ is Koszul, $$A$$ has a small homological resolution which is given thanks to its Koszul dual, in this case its homology and its cohomology are much more easier to compute. Thus it is important to have effective criteria to determine whether an algebra is Koszul or not, Priddy gave such a criterion for Koszulness in terms of the existence of a Poincaré-Birkhoff-Witt basis.
Twenty years later, V. Ginzburg and M. Kapranov [Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)] gave a great impetus to the algebraic theory of operads, by introducing the concept of Koszul duality in the operadic framework. Koszul duality for operads explains deep algebraic phenomena, for example the operads $$Com$$ and $$Lie$$ are Koszul dual to each other, this duality explains the relations between the Sullivan model and the Quillen model in rational homotopy theory. Koszul duality is also crucial in order to understand and compute the homology of algebras over an operad. No need to say that it is important to have effective criteria for Koszulness of operads, let us cite I. Kriz [MR1301191 (1996)]:
“A major question still open is if there is a criterion for Koszulness of operads as powerful as Priddy’s for algebras, which would expand applications to examples where calculations of homology are not readily available.” In this paper the author answers positively to this important question. He introduces the concept of Poincaré-Birkhoff-Witt basis for an operad and proves that if an operad has such a basis then it is Koszul. This is not an easy affair, because one has to fight with subtle technicalities in order to put a nice order on trees. At the end of the day the author gives us an easy criterion to test the Koszulness of an operad.
This is truly an important paper, that deserves to be read by anyone interested in Koszul duality and operad theory. Moreover let us mention that the results of this paper are necessary to develop Gröbner bases for operads as done by V. Dotsenko and A. Khoroshkin [Duke Math. J. 153, No. 2, 363–396 (2010; Zbl 1208.18007)].

##### MSC:
 18D50 Operads (MSC2010) 55P48 Loop space machines and operads in algebraic topology 16S37 Quadratic and Koszul algebras
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