Operads on graphs: extending the pre-Lie operad and general construction

TL;DR

This paper introduces a new framework for graph operads, generalizing the pre-Lie operad on rooted trees, and explores their algebraic properties and connections to other operads.

Contribution

It defines two new graph operads, studies their structure, and establishes links with known operads like pre-Lie and Kontsevich-Willwacher, including the Koszul property.

Findings

One of the operads is Koszul, enabling duality computations.

Identified finitely generated sub-operads with links to commutative operads.

Developed a new framework using species and operads for multigraphs.

Abstract

The overall aim of this paper is to define a structure of graph operads, thus generalizing the celebrated pre-Lie operad on rooted trees. More precisely, we define two operads on multigraphs, and exhibit a non trivial link between them and the pre-Lie and Kontsevich- Willwacher operads. We study one of these operads in more detail. While its structure is too involved to exhibit a description by generators and relations, we show that it has interesting finitely generated sub-operads, with links with the commutative and the magmatic commutative operads. In particular, one of them is Koszul and this allows us to compute its Koszul dual. Finally, we introduce a new framework on species and operads and a general way to define operads on multigraphs.