On a pre-Lie algebra defined by insertion of rooted trees
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.
scientific article; zbMATH DE number 5719897