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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Keywords: Noncommutative symmetric functions, Lie idempotents, Free Lie algebra, Dendriform algebras, PreLie algebras.
CC-BY 4.0
@article{ALCO_2022__5_4_629_0,
author = {Foissy, Lo{\"\i}c and Menous, Fr\'ed\'eric and Novelli, Jean-Christophe and Thibon, Jean-Yves},
title = {Quadri-algebras, {preLie} algebras, {and~the~Catalan} family of {Lie} idempotents},
journal = {Algebraic Combinatorics},
pages = {629--666},
publisher = {The Combinatorics Consortium},
volume = {5},
number = {4},
year = {2022},
doi = {10.5802/alco.224},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.224/}
}
TY - JOUR AU - Foissy, Loïc AU - Menous, Frédéric AU - Novelli, Jean-Christophe AU - Thibon, Jean-Yves TI - Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents JO - Algebraic Combinatorics PY - 2022 SP - 629 EP - 666 VL - 5 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.224/ DO - 10.5802/alco.224 LA - en ID - ALCO_2022__5_4_629_0 ER -
%0 Journal Article %A Foissy, Loïc %A Menous, Frédéric %A Novelli, Jean-Christophe %A Thibon, Jean-Yves %T Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents %J Algebraic Combinatorics %D 2022 %P 629-666 %V 5 %N 4 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.224/ %R 10.5802/alco.224 %G en %F ALCO_2022__5_4_629_0
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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