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      A Poincar\'e-Birkhoff-Witt criterion for Koszul operads

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          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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          Author and article information

          Journal
          14 September 2007
          2008-11-12
          0709.2286

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          18D50 (55D48); 16S37
          24 pages, in english. Changes in v3 : Correction concerning the graded case and some other minor corrections
          math.AT

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