A Poincaré-Birkhoff-Witt criterion for Koszul operads

2007/09/14 by Eric Hoffbeck, Hoffbeck, Eric · 2 citations
Mathematics · #16S37 #18D50 (55D48) #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:16S37 #msc:18D50

paper · pdf · doi:10.48550/arxiv.0709.2286

24 pages, in english. Changes in v3 : Correction concerning the graded case and some other minor corrections

arxiv created 2008/11/12 · arxiv updated 2009/12/01

Abstract

The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to verify that an operad is Koszul. We define the notion of a Poincare-Birkhoff-Witt basis in the context of operads. Then we show that an operad having a Poincare-Birkhoff-Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincare-Birkhoff-Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie) have a Poincare-Birkhoff-Witt basis.

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• An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence
• Lie-operads and operadic modules from poset cohomology

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