Lie-operads and operadic modules from poset cohomology

TL;DR

This paper develops a general formalism to produce operadic structures on the cohomology of posets with recursive structures, extending previous connections between poset cohomology and Lie operads to a broader framework.

Contribution

It introduces the concept of operadic poset species, generalizing previous results and encompassing new examples like the metabelian Lie operad and Kontsevich's operad of trees.

Findings

Framework produces operads and modules from poset cohomology.

Includes examples like metabelian Lie operad and Kontsevich's operad of trees.

Lays groundwork for future applications to hypertree posets.

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.