On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

2010 ◽  
Vol 92 (2) ◽  
pp. 181-196 ◽  
Author(s):  
Abdellatif Saïdi
Keyword(s):  
Lie Algebra ◽  
Rooted Trees

scholarly journals On the Lie enveloping algebra of a pre-Lie algebra

2008 ◽  
Vol 2 (1) ◽  
pp. 147-167 ◽  
Author(s):  
J.-M. Oudom ◽  
D. Guin
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebra ◽  
Associative Product ◽  
Planar Rooted Trees ◽  
Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

scholarly journals On bijections between monotone rooted trees and the comb basis

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Fu Liu
Keyword(s):  
Lie Algebra ◽  
Rooted Trees ◽  
Close Connection ◽  
Free Lie Algebra ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Lie Brackets ◽  
Element Set

International audience Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$. Soit $A$ un ensemble à $n$ éléments. Soit $\mathscr{L} ie_2(A)$ la partie multilinéaire de l'algèbre de Lie libre sur $A$ avec une paire de crochets de Lie compatibles et $\mathscr{L} ie_2(A, i)$ le sous-espace de$\mathscr{L} ie_2(A)$ généré par tous les monômes en $\mathscr{L} ie_2(A)$ avec $i$ supports d'un même type. L'auteur et Dotsenko-Khoroshkin montrent que la dimension de $\mathscr{L} ie_2(A, i)$ est la taille de la $R_{A,i}$, l'ensemble des arbres enracinés sur $A$ avec $i$ arêtes décroissantes. Il y a trois familles de bases connues pour $\mathscr{L} ie_2(A, i)$ : la base de peigne, la base Lyndon, et la base Liu-Lyndon. Récemment, Gonzalez, D' Léon et Wachs, dans leur étude de (co)-homologie de la poset des partitions pondérés, ont demandé si il y a des bijections jolies entre$R_{A,i}$, et la base de peigne ou la base Lyndon. Nous donnons une définition naturelle de "bijection jolie " , et un conjecture qu'il y a une seule bijection jolie entre $R_{A,i}$, et la base de peigne. Nous montrons que la conjecture est vraie pour les cas extrêmes: $i = 0$, et $n − 1$.

scholarly journals A Note on the $\gamma$-Coefficients of the Tree Eulerian Polynomial

10.37236/5361 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Rafael S. González D'León
Keyword(s):  
Lie Algebra ◽  
Product Formula ◽  
Rooted Trees ◽  
Binary Trees ◽  
Eulerian Polynomial ◽  
Generating Polynomial ◽  
Combinatorial Interpretations ◽  
Set Of Permutations ◽  
Positive Coefficients

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D'León-Wachs and Gonzláez D'León related to the free multibracketed Lie algebra and the poset of weighted partitions.

scholarly journals Weighted partitions

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Rafael González S. D'León ◽  
Michelle L. Wachs
Keyword(s):  
Lie Algebra ◽  
Characteristic Polynomial ◽  
Rooted Trees ◽  
Free Lie Algebra ◽  
International Audience

International audience In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted <mathfrak>S</mathfrak>_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n.

Doubling pre-Lie algebra of rooted trees

2019 ◽  
Vol 19 (12) ◽  
pp. 2050228
Author(s):  
Mohamed Belhaj Mohamed
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebras

We study the pre-Lie algebra of rooted trees [Formula: see text] and we define a pre-Lie structure on its doubling space [Formula: see text]. Also, we find the enveloping algebras of the two pre-Lie algebras denoted, respectively, by [Formula: see text] and [Formula: see text]. We prove that [Formula: see text] is a module-bialgebra on [Formula: see text] and we find some relations between the two pre-Lie structures.

Hamiltonian B-series and a Lie algebra of non-rooted trees

2016 ◽  
Vol 135 (1) ◽  
pp. 97-112
Author(s):  
Geir Bogfjellmo ◽  
Charles Curry ◽  
Dominique Manchon
Keyword(s):  
Lie Algebra ◽  
Rooted Trees
Sign in / Sign up

Export Citation Format

Share Document