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Séminaire Lotharingien de Combinatoire, B71b (2014), 19 pp.

# Mahdi J. Hasan Al-Kaabi

# Monomial Bases for Free Pre-Lie Algebras

**Abstract.**
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.

Received: October 17, 2013.
Revised: January 22, 2014.
Accepted: January 29, 2014.

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