Abstract
The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to prove that an operad is Koszul. We define the notion of Poincaré–Birkhoff–Witt basis in the context of operads. Then we show that an operad having a Poincaré–Birkhoff–Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincaré–Birkhoff–Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie, Poisson) have a Poincaré–Birkhoff–Witt basis. We also prove by our methods that new operads are Koszul.
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Hoffbeck, E. A Poincaré–Birkhoff–Witt criterion for Koszul operads. manuscripta math. 131, 87–110 (2010). https://doi.org/10.1007/s00229-009-0303-2
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DOI: https://doi.org/10.1007/s00229-009-0303-2