Monomial bases for free pre-Lie algebras
As shown by \textit{F. Chapoton} and \textit{M. Livernet} [Int. Math. Res. Not. 2001, No. 8, 395--408 (2001; Zbl 1053.17001)], the free pre-Lie algebra generated by a nonempty set admits a basis given by the set of rooted trees which internal vertices are decorated by elements of the generating set. However, this is not a monomial basis. The present text considers monomial bases of free pre-Lie algebras, such that the leading terms, according to the filtration by the tree-grounded degree, give back the basis of rooted trees. It is shown that these bases are in one-to-one correspondence with sections of the canonical projection from planar rooted trees to rooted trees. Moreover, the matrix of such a monomial basis, expressed in the basis of rooted trees, is unipotent; an explicit formula is given for its coefficients.
scientific article; zbMATH DE number 6335255