A Poincaré-Birkhoff-Witt criterion for Koszul operads

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.

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

Cited by in corpus (12)

• Koszul duality of E_n-operads
• Wheeled props in algebra, geometry and quantization
• (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities
• On the NonKoszulity of (2p+1)-ary partially associative Operads
• Implementing Gröbner bases for operads
• Freeness theorems for operads via Gröbner bases
• Notes on Cohomologies of Ternary Algebras of Associative Type
• Operads for n-ary algebras - calculations and conjectures
• Operad profiles of Nijenhuis structures
• Hom-coassociative ternary coalgebras and infinitesimal bialgebras
• (Co)associative $3$-ary (co)algebras and infinitesimal bialgebras: construction and main properties
• The n-ary algebra of tensors and of cubic and hypercubic matrices