Koszul duality for monoids and the operad of enriched rooted trees

Abstract:

We introduce here the notion of Koszul duality for monoids in the monoidal category of species with respect to the ordinary product. To each Koszul monoid we associate a class of Koszul algebras in the sense of Priddy, by taking the corresponding analytic functor. The operad AM of rooted trees enriched with a monoid M was introduced by the author. One special case of that is the operad of ordinary rooted trees, called in the recent literature the permutative non-associative operad. We prove here that AM is Koszul if and only if the corresponding monoid M is Koszul. In this way we obtain a wide family of Koszul operads, extending a recent result of Chapoton and Livernet, and providing an interesting link between Koszul duality for associative algebras and Koszul duality for operads. © 2009 Elsevier Inc. All rights reserved.

Año de publicación:

2010

Keywords:

• Operads
• Enriched trees
• species
• Koszul duality
• c-Monoids

Fuente:

scopus