Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 629-666.

We compute the expansion of the Catalan family of Lie idempotents introduced in [Menous et al., Adv. Applied Math. 51 (2013), 177–22] on the PBW basis of the Lie module. It is found that the coefficient of a tree depends only on its number of left and right internal edges. In particular, the Catalan idempotents belong to a preLie algebra based on naked binary trees, of which we identify several Lie and preLie subalgebras.

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DOI: 10.5802/alco.224
Classification: 16T30,  05E05,  17D25
Keywords: Noncommutative symmetric functions, Lie idempotents, Free Lie algebra, Dendriform algebras, PreLie algebras.
Foissy, Loïc 1; Menous, Frédéric 2; Novelli, Jean-Christophe 3; Thibon, Jean-Yves 3

1 Fédération de Recherche Mathématique du Nord Pas de Calais FR 2956 Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville Université du Littoral Côte d’Opale-Centre Universitaire de la Mi-Voix 50, rue Ferdinand Buisson, CS 80699, 62228 Calais Cedex, France
2 Laboratoire de Mathématiques Bât. 425 Université Paris-Sud 91405 Orsay Cedex France
3 Laboratoire d’Informatique Gaspard-Monge, Université Gustave Eiffel, CNRS, ENPC, ESIEE-Paris, 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée cedex 2 FRANCE
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Foissy, Loïc; Menous, Frédéric; Novelli, Jean-Christophe; Thibon, Jean-Yves. Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 629-666. doi : 10.5802/alco.224. https://alco.centre-mersenne.org/articles/10.5802/alco.224/

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