Fine structures inside the PreLie operad

TL;DR

This paper provides a detailed analysis of the PreLie operad, revealing new structural relationships and including a conjecture about a free sub-operad generated by the cyclic Lie module within PreLie.

Contribution

It offers a concrete description of the anticyclic structure of PreLie and establishes an inclusion of the cyclic Lie module into PreLie, advancing understanding of its internal structure.

Findings

Established a concrete link between anticyclic structure and generators of PreLie

Constructed an inclusion of the cyclic Lie module into PreLie

Conjectured the existence of a free sub-operad generated by this inclusion

Abstract

This article aims at a detailed analysis of the PreLie operad. We obtain a more concrete description of the relationship between the anticyclic structure of PreLie and the generators of PreLie as a Lie-module, which was known before only at the level of characters. Building on this, we obtain an inclusion of the cyclic Lie module in the PreLie operad. We conjecture that the image of this inclusion generates an interesting free sub-operad.