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Journal of Lie Theory 28 (2018), No. 4, 941--967
Copyright Heldermann Verlag 2018

Monomial Bases and Pre-Lie Structure for Free Lie Algebras

Mahdi J. Hasan Al-Kaabi
Mathematics Department, College of Science, Mustansiriyah University, Palestine Street, Baghdad, Iraq

Dominique Manchon
LMBP, CNRS-UMR6620, Université Clermont-Auvergne, 3 place Vasarély, 63178 Aubière, France

Frédéric Patras
Laboratoire J. A. Dieudonné, UMR CNRS-UNS N7351, Université de Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France

We construct a pre-Lie structure on the free Lie algebra L(E) generated by a set E, giving an explicit presentation of L(E) as the quotient of the free pre-Lie algebra TE, generated by the (non-planar) E-decorated rooted trees, by some ideal I. The main result in this paper is a description of Gröbner bases in terms of trees.

Keywords: Pre-Lie algebras, NAP algebras, free Lie algebras, monomial bases, rooted trees.

MSC: 05C05, 17D25, 17A50, 17B01

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