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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