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

TL;DR

This paper introduces a Poincaré-Birkhoff-Witt criterion for identifying Koszul operads, extending classical algebraic criteria and providing a new tool for verifying Koszulness in operad theory.

Contribution

It defines a Poincaré-Birkhoff-Witt basis for operads and proves that having such a basis implies the operad is Koszul, generalizing existing algebraic criteria.

Findings

Operads with a Poincaré-Birkhoff-Witt basis are Koszul.

The Koszul dual of such operads also admits a Poincaré-Birkhoff-Witt basis.

Classical Koszul operads like commutative, associative, and Lie have Poincaré-Birkhoff-Witt bases.

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.