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/

[1] Aguiar, Marcelo; Loday, Jean-Louis Quadri-algebras, J. Pure Appl. Algebra, Volume 191 (2004) no. 3, pp. 205-221 | DOI | MR | Zbl

[2] Bandiera, Ruggero; Schaetz, Florian Eulerian idempotent, pre-Lie logarithm and combinatorics of trees (2017) (preprint, https://arxiv.org/abs/1702.08907)

[3] Bialynicki-Birula, I.; Mielnik, Bogdan; Plebański, Jerzy Explicit solution of the continuous Baker–Campbell–Hausdorff problem and a new expression for the phase operator, Annals of Physics, Volume 51 (1969) no. 1, pp. 187-200 | DOI | Zbl

[4] Cartier, Pierre Démonstration algébrique de la formule de Hausdorff, Bull. Soc. Math. France, Volume 84 (1956), pp. 241-249 | DOI | Numdam | MR | Zbl

[5] Chapoton, Frédéric A rooted-trees q-series lifting a one-parameter family of Lie idempotents, Algebra Number Theory, Volume 3 (2009) no. 6, pp. 611-636 | DOI | MR | Zbl

[6] Chapoton, Frédéric Flows on rooted trees and the Menous–Novelli–Thibon idempotents, Math. Scand., Volume 115 (2014) no. 1, pp. 20-61 | DOI | MR | Zbl

[7] Chapoton, Frédéric; Livernet, Muriel Pre-Lie algebras and the rooted trees operad, Internat. Math. Res. Notices (2001) no. 8, pp. 395-408 | DOI | MR | Zbl

[8] Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput., Volume 12 (2002) no. 5, pp. 671-717 | DOI | MR | Zbl

[9] Dynkin, Evgeniĭ B. Calculation of the coefficients in the Campbell–Hausdorff formula, Doklady Akad. Nauk SSSR (N.S.), Volume 57 (1947), pp. 323-326 | MR

[10] Foissy, Loïc Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra, Volume 209 (2007) no. 2, pp. 439-459 | DOI | MR | Zbl

[11] Foissy, Loïc Free brace algebras are free pre-Lie algebras, Comm. Algebra, Volume 38 (2010) no. 9, pp. 3358-3369 | DOI | MR | Zbl

[12] Gelʼfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves Noncommutative symmetric functions, Adv. Math., Volume 112 (1995) no. 2, pp. 218-348 | DOI | MR | Zbl

[13] Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves The algebra of binary search trees, Theoret. Comput. Sci., Volume 339 (2005) no. 1, pp. 129-165 | DOI | MR | Zbl

[14] Klyachko, Alexander A. Lie elements in a tensor algebra, Sibirsk. Mat. Ž., Volume 15 (1974), p. 1296-1304, 1430 | DOI | MR | Zbl

[15] Krob, Daniel; Leclerc, Bernard; Thibon, Jean-Yves Noncommutative symmetric functions. II. Transformations of alphabets, Internat. J. Algebra Comput., Volume 7 (1997) no. 2, pp. 181-264 | DOI | MR | Zbl

[16] Loday, Jean-Louis Opérations sur l’homologie cyclique des algèbres commutatives, Invent. Math., Volume 96 (1989) no. 1, pp. 205-230 | DOI | MR | Zbl

[17] Loday, Jean-Louis Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques à la Mémoire de Jean Leray (Sémin. Congr.), Volume 9, Soc. Math. France, Paris, 2004, pp. 155-172 | MR | Zbl

[18] Loday, Jean-Louis; Frabetti, Alessandra; Chapoton, Frédéric; Goichot, François Dialgebras and related operads, Lecture Notes in Mathematics, 1763, Springer-Verlag, Berlin, 2001, iv+133 pages | DOI | MR | Zbl

[19] Loday, Jean-Louis; Ronco, María O. Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309 | DOI | MR | Zbl

[20] Macdonald, Ian G. Symmetric functions and Hall polynomials. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley. Reprint of the 2008 paperback edition, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | MR | Zbl

[21] Malvenuto, Clauda; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | DOI | MR | Zbl

[22] Menous, Frédéric The well-behaved Catalan and Brownian averages and their applications to real resummation, Proceedings of the Symposium on Planar Vector Fields (Lleida, 1996), Volume 41 (1997), pp. 209-222 | DOI | MR | Zbl

[23] Menous, Frédéric; Novelli, Jean-Christophe; Thibon, Jean-Yves Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras, Adv. in Appl. Math., Volume 51 (2013) no. 2, pp. 177-227 | DOI | MR | Zbl

[24] Mielnik, Bogdan; Plebański, Jerzy Combinatorial approach to Baker–Campbell–Hausdorff exponents, Ann. Inst. H. Poincaré Sect. A (N.S.), Volume 12 (1970), pp. 215-254 | Numdam | MR | Zbl

[25] Novelli, Jean-Christophe; Thibon, Jean-Yves Construction de trigèbres dendriformes, C. R. Math. Acad. Sci. Paris, Volume 342 (2006) no. 6, pp. 365-369 | DOI | MR | Zbl

[26] Novelli, Jean-Christophe; Thibon, Jean-Yves Polynomial realizations of some trialgebras, 18th Formal Power Series and Algebraic Combinatorics (FPSAC’06) (2006) no. 1, pp. 243-254

[27] Oudom, Jean-Michel; Guin, Daniel Sur l’algèbre enveloppante d’une algèbre pré-Lie, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 5, pp. 331-336 | DOI | MR | Zbl

[28] Patras, Frédéric; Reutenauer, Christophe Lie representations and an algebra containing Solomon’s, J. Algebraic Combin., Volume 16 (2002) no. 3, p. 301-314 (2003) | DOI | MR | Zbl

[29] Reutenauer, Christophe Theorem of Poincaré–Birkhoff–Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) (Lecture Notes in Math.), Volume 1234, Springer, Berlin, 1986, pp. 267-284 | DOI | MR | Zbl

[30] Reutenauer, Christophe Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993, xviii+269 pages (Oxford Science Publications) | MR | Zbl

[31] Salvatore, Paolo; Tauraso, Roberto The operad Lie is free, J. Pure Appl. Algebra, Volume 213 (2009) no. 2, pp. 224-230 | DOI | MR | Zbl

[32] Solomon, Louis On the Poincaré-Birkhoff-Witt theorem, J. Combinatorial Theory, Volume 4 (1968), pp. 363-375 | DOI | MR | Zbl

[33] Solomon, Louis A Mackey formula in the group ring of a Coxeter group, J. Algebra, Volume 41 (1976) no. 2, pp. 255-264 | DOI | MR | Zbl

[34] Witt, Ernst Treue Darstellung Liescher Ringe, J. Reine Angew. Math., Volume 177 (1937), pp. 152-160 | DOI | MR | Zbl

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