Geodesic
Monomial Bases for Free Pre-Lie Algebras
Séminaire lotharingien de combinatoire, Tome 71 (2014-2015)

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We study the concept of a free pre-Lie algebra generated by a (non-empty) set. We review the construction by Agrachev and Gamkrelidze [J. Sov. Math. 17 (1981), 1650-1675] of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial basis vectors in terms of the rooted trees basis exhibited by Chapoton and Livernet [Internat. Math. Res. Notices 8 (2001), 395-408]. Also, we show that this matrix is unipotent, and we find an explicit expression for its coefficients, which uses a similar procedure for the free magmatic algebra at the level of planar rooted trees which has been suggested by Ebrahimi-Fard and Manchon. 

@article{SLC_2014-2015_71_a1,
     author = {Mahdi J. Hasan Al-Kaabi},
     title = {Monomial {Bases} for {Free} {Pre-Lie} {Algebras}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {71},
     year = {2014-2015},
     url = {http://geodesic.mathdoc.fr/item/SLC_2014-2015_71_a1/}
}
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Mahdi J. Hasan Al-Kaabi. Monomial Bases for Free Pre-Lie Algebras. Séminaire lotharingien de combinatoire, Tome 71 (2014-2015). http://geodesic.mathdoc.fr/item/SLC_2014-2015_71_a1/