Doubling pre-Lie algebra of rooted trees

Abstract

We study the pre-Lie algebra of rooted trees $(\Cal T, \rightarrow)$ and we define a pre-Lie structure on its doubling space $(V,⇝)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted respectively by $(\Cal H', \star, Γ)$ and $(\Cal D', \bigstar, χ)$. We prove that $(\Cal D', \bigstar, χ)$ is a module-bialgebra on $(\Cal H', \star, Γ)$ and we find some relations between the two pre-Lie structures.

Accepted for publication in the journal of algebra and its applications (2019)