Enveloping algebras of pre-Lie algebras of rooted trees

Mohamed Belhaj Mohamed

doi:10.5802/cml.75

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

Confluentes Mathematici, Tome 13 (2021) no. 2, pp. 11-28.

Résumé

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

Reçu le : 2020-06-29
Révisé le : 2021-10-03
Accepté le : 2021-11-03
Publié le : 2022-03-21

DOI : 10.5802/cml.75

Classification : 16T10, 16T15, 16T30, 05C90, 81T17
Mots clés : Insertion pre-Lie algebras, Rooted Forest, Bialgebras, Enveloping algebras of pre-Lie algebras, Module-bialgebras

Affiliations des auteurs :

Mohamed Belhaj Mohamed ^{1, 2}

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CML_2021__13_2_11_0,
     author = {Mohamed Belhaj Mohamed},
     title = {Enveloping algebras of {pre-Lie} algebras of rooted trees},
     journal = {Confluentes Mathematici},
     pages = {11--28},
     publisher = {Institut Camille Jordan},
     volume = {13},
     number = {2},
     year = {2021},
     doi = {10.5802/cml.75},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.75/}
}

TY  - JOUR
AU  - Mohamed Belhaj Mohamed
TI  - Enveloping algebras of pre-Lie algebras of rooted trees
JO  - Confluentes Mathematici
PY  - 2021
SP  - 11
EP  - 28
VL  - 13
IS  - 2
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.75/
DO  - 10.5802/cml.75
LA  - en
ID  - CML_2021__13_2_11_0
ER  -

%0 Journal Article
%A Mohamed Belhaj Mohamed
%T Enveloping algebras of pre-Lie algebras of rooted trees
%J Confluentes Mathematici
%D 2021
%P 11-28
%V 13
%N 2
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.75/
%R 10.5802/cml.75
%G en
%F CML_2021__13_2_11_0

Mohamed Belhaj Mohamed. Enveloping algebras of pre-Lie algebras of rooted trees. Confluentes Mathematici, Tome 13 (2021) no. 2, pp. 11-28. doi : 10.5802/cml.75. https://cml.centre-mersenne.org/articles/10.5802/cml.75/

Bibliographie
Cité par

[1] M. Belhaj Mohamed, D. Manchon, Doubling bialgebras of rooted trees, Lett Math Phys 107-145 (2017).

[2] M. Belhaj Mohamed, Doubling pre-Lie algebra of rooted trees, journal of algebra and its applications, No 12, 205022 (2020).

[3] D. Calaque, K. Ebrahimi-Fard, D. Manchon, Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series, Advances in Applied Mathematics, 47, $n^{\circ} 2$ , 282-308 (2011).

[4] F. Chapoton, Algèbres pré-Lie et algèbres de Hopf liées à la renormalisation, Comptes- Rendus Acad. Sci., 332 Série I (2001), 681-684.

[5] F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001 (2001), 395-408.

[6] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210, $n^{\circ} 1$ , 249-273 (2000).

[7] A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. in Math. Phys. 199,203-242 (1998).

[8] A. Dzhumadl’daev, C. Löfwall, Trees, free right-symmetric algebras, free Novikov algebras and identities, Homotopy, Homology and Applications, 4(2), 165-190 (2002).

[9] L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I + II, thèse, Univ. de Reims (2002), et Bull. Sci. Math. 126, $n^{\circ} 3$ , 193–239 et $n^{\circ} 4$ , 249–288 (2002).

[10] D. Manchon, A short survey on pre-Lie algebras, E. Schrodinger Institut Lectures in Math. Phys., Eur. Math. Soc, A.Carey Ed. (2011).

[11] D. Manchon. A review on comodule-bialgebras. In Computation and Combinatorics in Dynamics, Stochastics and Control, pages 579-597. Springer International Publishing, (2018).

[12] D. Manchon, A. Saïdi, Lois pré-Lie en interaction, Comm. Alg. vol 39, $n^{\circ} 10$ , 3662-3680 (2011).

[13] R. K. Molnar, Semi-direct products of Hopf algebras, J. Algebra 45, 29-51 (1977).

[14] J. M. Oudom and D. Guin, On the Lie envelopping algebra of a pre-Lie algebra, Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, pp. 147-167, (2008).

Cité par Sources :