The pre-Lie operad as a deformation of NAP

TL;DR

This paper introduces a family of multigraded operads that interpolate between the pre-Lie and NAP operads, revealing a deformation parameter linking these algebraic structures.

Contribution

It defines a new family of operads parametrized by a scalar, unifying pre-Lie and NAP operads through a deformation framework.

Findings

The operads depend smoothly on the scalar parameter.

Special cases recover pre-Lie and NAP operads.

Provides a new perspective on the relationship between these algebraic structures.

Abstract

We define a family of multigraded operads $O_\lambda$ depending on a scalar parameter, such that forgetting the multigraduation gives back the pre-Lie operad when the parameter $\lambda$ is equal to one, and the NAP operad governing Non-Associative Permutative algebras when $\lambda$ is equal to zero.