Lie-operads and operadic modules from poset cohomology

Abstract

As observed by Joyal, the cohomology groups of the partition posets are naturally identified with the components of the operad encoding Lie algebras. This connection was explained in terms of operadic Koszul duality by Fresse, and later generalized by Vallette to the setting of decorated partitions. In this article, we set up and study a general formalism which produces a priori operadic structures (operads and operadic modules) on the cohomology of families of posets equipped with some natural recursive structure, that we call "operadic poset species". This framework goes beyond decorated partitions and operadic Koszul duality, and contains the metabelian Lie operad and Kontsevich's operad of trees as two simple instances. In forthcoming work, we will apply our results to the hypertree posets and their connections to post-Lie and pre-Lie algebras.