# CONFLUENTES MATHEMATICI

Enveloping algebras of pre-Lie algebras of rooted trees
Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 11-28.

In this article, we study the insertion pre-Lie algebra of rooted trees $\left(𝒯,⊳\right)$ and we construct a pre-Lie structure on its doubling space $\left(\stackrel{˜}{V},▸\right)$. We prove that $\stackrel{˜}{V}$ is a left pre-Lie module on $𝒯$. Moreover, we describe the enveloping algebras of the two pre-Lie algebras denoted respectively by $\left(𝒦,♦,\Upsilon \right)$ and $\left(𝒲,⧫,\Theta \right)$ and we show that $\left(𝒲,⧫,\Theta \right)$ is a module-bialgebra on $\left(𝒦,♦,\Upsilon \right)$. Finally, we find some relations between the enveloping algebras of the insertion and the grafting pre-lie algebras of rooted trees.

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.75
Classification: 16T10,  16T15,  16T30,  05C90,  81T17
Keywords: Insertion pre-Lie algebras, Rooted Forest, Bialgebras, Enveloping algebras of pre-Lie algebras, Module-bialgebras
Mohamed Belhaj Mohamed 1, 2

1 Laboratoire de mathématiques, physique, fonctions spéciales et applications, Université de Sousse, rue Lamine Abassi 4011 H. Sousse, Tunisie
2 Mathematics Departement, Sciences college, Taibah University, Kingdom of Saudi Arabia
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Mohamed Belhaj Mohamed. Enveloping algebras of pre-Lie algebras of rooted trees. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 11-28. doi : 10.5802/cml.75. https://cml.centre-mersenne.org/articles/10.5802/cml.75/

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