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

Muriel Livernet[1]

  • [1] Université Paris 13, CNRS, UMR 7539 LAGA, 99, Avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France.

Annales mathématiques Blaise Pascal (2010)

  • Volume: 17, Issue: 1, page 47-96
  • ISSN: 1259-1734

Abstract

top
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.

How to cite

top

Livernet, Muriel. "From left modules to algebras over an operad: application to combinatorial Hopf algebras." Annales mathématiques Blaise Pascal 17.1 (2010): 47-96. <http://eudml.org/doc/116351>.

@article{Livernet2010,
abstract = {The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $\mathbb\{S\}$-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 $\mathcal\{O\}$ denote the forgetful functor from $\mathbb\{S\}$-modules to graded vector spaces. Left modules over an operad $\mathcal\{P\}$ are treated as $\mathcal\{P\}$-algebras in the category of $\mathbb\{S\}$-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad $\mathcal\{P\}$: the functor $\mathcal\{O\}$ sends $\mathcal\{P\}$-algebras to $\mathcal\{P\}$-algebras. If $\mathcal\{P\}$ is a Hopf operad the functor $\mathcal\{O\}$ sends Hopf $\mathcal\{P\}$-algebras to Hopf $\mathcal\{P\}$-algebras. If the operad $\mathcal\{P\}$ is regular one gets two different structures of Hopf $\mathcal\{P\}$-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal $\mathcal\{P\}$-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.},
affiliation = {Université Paris 13, CNRS, UMR 7539 LAGA, 99, Avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France.},
author = {Livernet, Muriel},
journal = {Annales mathématiques Blaise Pascal},
keywords = {$\mathbb\{S\}$-module; operad; twisted bialgebra; free associative algebra; combinatorial Hopf algebra; -module},
language = {eng},
month = {1},
number = {1},
pages = {47-96},
publisher = {Annales mathématiques Blaise Pascal},
title = {From left modules to algebras over an operad: application to combinatorial Hopf algebras},
url = {http://eudml.org/doc/116351},
volume = {17},
year = {2010},
}

TY - JOUR
AU - Livernet, Muriel
TI - From left modules to algebras over an operad: application to combinatorial Hopf algebras
JO - Annales mathématiques Blaise Pascal
DA - 2010/1//
PB - Annales mathématiques Blaise Pascal
VL - 17
IS - 1
SP - 47
EP - 96
AB - The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $\mathbb{S}$-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 $\mathcal{O}$ denote the forgetful functor from $\mathbb{S}$-modules to graded vector spaces. Left modules over an operad $\mathcal{P}$ are treated as $\mathcal{P}$-algebras in the category of $\mathbb{S}$-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad $\mathcal{P}$: the functor $\mathcal{O}$ sends $\mathcal{P}$-algebras to $\mathcal{P}$-algebras. If $\mathcal{P}$ is a Hopf operad the functor $\mathcal{O}$ sends Hopf $\mathcal{P}$-algebras to Hopf $\mathcal{P}$-algebras. If the operad $\mathcal{P}$ is regular one gets two different structures of Hopf $\mathcal{P}$-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal $\mathcal{P}$-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.
LA - eng
KW - $\mathbb{S}$-module; operad; twisted bialgebra; free associative algebra; combinatorial Hopf algebra; -module
UR - http://eudml.org/doc/116351
ER -

References

top
  1. Marcelo Aguiar, Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math. 191 (2005), 225-275 Zbl1056.05139MR2103213
  2. Marcelo Aguiar, Frank Sottile, Structure of the Loday-Ronco Hopf algebra of trees, J. Algebra 295 (2006), 473-511 Zbl1099.16015MR2194965
  3. M. G. Barratt, Twisted Lie algebras, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II 658 (1978), 9-15, Springer, Berlin Zbl0393.55021MR513566
  4. Nantel Bergeron, Mike Zabrocki, The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free, J. Algebra Appl. 8 (2009), 581-600 Zbl1188.16030MR2555523
  5. Frédéric Chapoton, Algèbres de Hopf des permutohèdres, associahèdres et hypercubes, Adv. Math. 150 (2000), 264-275 Zbl0958.16038MR1749253
  6. Frédéric Chapoton, Bigèbres différentielles graduées associées aux permutoèdres, associaèdres et hypercubes, Ann. Inst. Fourier (Grenoble) 50 (2000), 1127-1153 Zbl0963.16032MR1799740
  7. Frédéric Chapoton, Construction de certaines opérades et bigèbres associées aux polytopes de Stasheff et hypercubes, Trans. Amer. Math. Soc. 354 (2002), 63-74 (electronic) Zbl1035.18006MR1859025
  8. Frédéric Chapoton, Opérades différentielles graduées sur les simplexes et les permutoèdres, Bull. Soc. Math. France 130 (2002), 233-251 Zbl1044.18007MR1924542
  9. Gérard Duchamp, Florent Hivert, Jean-Yves Thibon, Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput. 12 (2002), 671-717 Zbl1027.05107MR1935570
  10. Loïc Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra 209 (2007), 439-459 Zbl1123.16030MR2293319
  11. Benoit Fresse, Koszul duality of operads and homology of partition posets, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic -theory 346 (2004), 115-215, Amer. Math. Soc., Providence, RI Zbl1077.18007MR2066499
  12. Muriel Livernet, Frédéric Patras, Lie theory for Hopf operads, J. Algebra 319 (2008), 4899-4920 Zbl1149.18005MR2423811
  13. Jean-Louis Loday, Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques à la Mémoire de Jean Leray 9 (2004), 155-172, Soc. Math. France, Paris MR2145941
  14. Jean-Louis Loday, On the algebra of quasi-shuffles, Manuscripta Math. 123 (2007), 79-93 Zbl1126.16029MR2300061
  15. Jean-Louis Loday, María Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), 293-309 Zbl0926.16032MR1654173
  16. Jean-Louis Loday, María Ronco, Trialgebras and families of polytopes, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic -theory 346 (2004), 369-398, Amer. Math. Soc., Providence, RI Zbl1065.18007MR2066507
  17. Jean-Louis Loday, María Ronco, On the structure of cofree Hopf algebras, J. reine angew. Math. 592 (2006), 123-155 Zbl1096.16019MR2222732
  18. Claudia Malvenuto, Christophe Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967-982 Zbl0838.05100MR1358493
  19. Jean-Christophe Novelli, Jean-Yves Thibon, Construction de trigèbres dendriformes, C. R. Math. Acad. Sci. Paris 342 (2006), 365-369 Zbl1101.17003MR2209212
  20. Patricia Palacios, María O. Ronco, Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra 299 (2006), 648-678 Zbl1110.16046MR2228332
  21. Frédéric Patras, Christophe Reutenauer, On descent algebras and twisted bialgebras, Mosc. Math. J. 4 (2004), 199-216, 311 Zbl1103.16026MR2074989
  22. Frédéric Patras, Manfred Schocker, Trees, set compositions and the twisted descent algebra, J. Algebraic Combin. 28 (2008), 3-23 Zbl1180.05032MR2420777
  23. Stéphane Poirier, Christophe Reutenauer, Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec 19 (1995), 79-90 Zbl0835.16035MR1334836
  24. Christopher R. Stover, The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring, J. Pure Appl. Algebra 86 (1993), 289-326 Zbl0793.16016MR1218107
  25. Andy Tonks, Relating the associahedron and the permutohedron, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) 202 (1997), 33-36, Amer. Math. Soc., Providence, RI Zbl0873.51016MR1436915

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.