From left modules to algebras over an operad: application to combinatorial Hopf algebras

Livernet, Muriel

Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 47-96.

Nous étudions en détail le foncteur oubli de la catégorie des $𝕊$-modules dans la catégorie des espaces vectoriels gradués. Cela nous permet de généraliser les résultats de Patras et Reutenauer obtenus dans le cadre associatif à toute opérade $𝒫$ : les $𝒫$-algèbres dans la catégorie des $𝕊$-modules deviennent des $𝒫$-algèbres dans la catégorie des espaces vectoriels gradués. Il en est de même pour les $𝒫$-algèbres de Hopf lorsque l’opérade $𝒫$ est une opérade de Hopf. De plus, si l’opérade est régulière, alors on obtient deux structures de $𝒫$-algèbres (de Hopf) dans la catégorie des espaces vectoriels gradués. Comme application, nous montrons qu’un certain nombre d’opérades de Hopf donne lieu à des algèbres de Hopf combinatoires connues. Le fait que ces algèbres de Hopf soient libres ou colibres est une conséquence directe de la théorie des opérades.

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $𝕊$-modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.

Let $𝒪$ denote the forgetful functor from $𝕊$-modules to graded vector spaces. Left modules over an operad $𝒫$ are treated as $𝒫$-algebras in the category of $𝕊$-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad $𝒫$: the functor $𝒪$ sends $𝒫$-algebras to $𝒫$-algebras. If $𝒫$ is a Hopf operad the functor $𝒪$ sends Hopf $𝒫$-algebras to Hopf $𝒫$-algebras. If the operad $𝒫$ is regular one gets two different structures of Hopf $𝒫$-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal $𝒫$-bialgebras and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as it is the case for various Hopf algebras defined on the faces of the permutohedra and associahedra.

[1] Aguiar, Marcelo; Sottile, Frank Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math., Volume 191 (2005) no. 2, pp. 225-275 | DOI | MR | Zbl

[2] Aguiar, Marcelo; Sottile, Frank Structure of the Loday-Ronco Hopf algebra of trees, J. Algebra, Volume 295 (2006) no. 2, pp. 473-511 | MR | Zbl

[3] Barratt, M. G. Twisted Lie algebras, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II (Lecture Notes in Math.), Volume 658, Springer, Berlin, 1978, pp. 9-15 | MR | Zbl

[4] Bergeron, Nantel; Zabrocki, Mike The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free, J. Algebra Appl., Volume 8 (2009) no. 4, pp. 581-600 | DOI | MR

[5] Chapoton, Frédéric Algèbres de Hopf des permutohèdres, associahèdres et hypercubes, Adv. Math., Volume 150 (2000) no. 2, pp. 264-275 | DOI | MR | Zbl

[6] Chapoton, Frédéric Bigèbres différentielles graduées associées aux permutoèdres, associaèdres et hypercubes, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 4, pp. 1127-1153 | DOI | Numdam | MR | Zbl

[7] Chapoton, Frédéric Construction de certaines opérades et bigèbres associées aux polytopes de Stasheff et hypercubes, Trans. Amer. Math. Soc., Volume 354 (2002) no. 1, p. 63-74 (electronic) | DOI | MR | Zbl

[8] Chapoton, Frédéric Opérades différentielles graduées sur les simplexes et les permutoèdres, Bull. Soc. Math. France, Volume 130 (2002) no. 2, pp. 233-251 | Numdam | MR | Zbl

[9] 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

[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] Fresse, Benoit Koszul duality of operads and homology of partition posets, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory (Contemp. Math.), Volume 346, Amer. Math. Soc., Providence, RI, 2004, pp. 115-215 | MR | Zbl

[12] Livernet, Muriel; Patras, Frédéric Lie theory for Hopf operads, J. Algebra, Volume 319 (2008), pp. 4899-4920 | DOI | MR | Zbl

[13] 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

[14] Loday, Jean-Louis On the algebra of quasi-shuffles, Manuscripta Math., Volume 123 (2007) no. 1, pp. 79-93 | DOI | MR | Zbl

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

[16] Loday, Jean-Louis; Ronco, María Trialgebras and families of polytopes, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory (Contemp. Math.), Volume 346, Amer. Math. Soc., Providence, RI, 2004, pp. 369-398 | MR | Zbl

[17] Loday, Jean-Louis; Ronco, María On the structure of cofree Hopf algebras, J. reine angew. Math., Volume 592 (2006), pp. 123-155 | DOI | MR | Zbl

[18] Malvenuto, Claudia; 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

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

[20] Palacios, Patricia; Ronco, María O. Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, Volume 299 (2006) no. 2, pp. 648-678 | DOI | MR | Zbl

[21] Patras, Frédéric; Reutenauer, Christophe On descent algebras and twisted bialgebras, Mosc. Math. J., Volume 4 (2004) no. 1, p. 199-216, 311 | MR | Zbl

[22] Patras, Frédéric; Schocker, Manfred Trees, set compositions and the twisted descent algebra, J. Algebraic Combin., Volume 28 (2008) no. 1, pp. 3-23 | DOI | MR | Zbl

[23] Poirier, Stéphane; Reutenauer, Christophe Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec, Volume 19 (1995) no. 1, pp. 79-90 | MR | Zbl

[24] Stover, Christopher R. The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring, J. Pure Appl. Algebra, Volume 86 (1993) no. 3, pp. 289-326 | DOI | MR | Zbl

[25] Tonks, Andy Relating the associahedron and the permutohedron, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) (Contemp. Math.), Volume 202, Amer. Math. Soc., Providence, RI (1997), pp. 33-36 | MR | Zbl