On a pre-Lie algebra defined by insertion of rooted trees
type: Item
instance of: scholarly article
title: On a pre-Lie algebra defined by insertion of rooted trees
zbMATH Open document ID: 1243.17001
author: Abdellatif Saïdi
published in: Letters in Mathematical Physics
publication date: 2010-06-11T00:00:00Z
review text: A pre-Lie algebra is a vector space \(A\) with a binary operation \(*\) such that \(x*(y*z) - (x*y)*z = y*(x*z) - (y*x)*z\) for all \(x,y,z\) in \(A\). The free pre-Lie algebra with one generator is isomorphic to the vector space \(\mathcal T\) spanned by the rooted trees, endowed with the grafting Lie product [\textit{F. Chapoton} and \textit{M. Livernet}, Int. Math. Res. Not. 2001, No. 8, 395--408 (2001; Zbl 1053.17001)], [\textit{A. Dzhumaldil'daev} and \textit{C. Löfwall}, Homology Homotopy Appl. 4, No. 2(1), 165--190 (2002; Zbl 1029.17001)]. In the present paper, the author considers the vector space \(\mathcal T'\) spanned by the rooted trees with at least one edge, endowed with the pre-Lie operation given by insertion. It is shown that it is generated by two ladders; furthermore, two families of non-trivial relations are exhibited, so that \(\mathcal T'\) is not a free pre-Lie algebra in these two generators.
reviewed by: Nicolás Andruskiewitsch
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 16T30
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 5719897
zbMATH Keywords: pre-Lie algebra
zbMATH Keywords: rooted trees
Wikidata QID: Q115381890
MaRDI profile type: Publication
full work available at URL: s11005-010-0377-5
OpenAlex ID: W2031547256
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DOI: 10.1007/S11005-010-0377-5
description: scientific article; zbMATH DE number 5719897