Generalized bialgebras and triples of operads

Loday, Jean-Louis

Astérisque, no. 320 (2008) , 126 p.

@book{AST_2008__320__R1_0,
     author = {Loday, Jean-Louis},
     title = {Generalized bialgebras and triples of operads},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {320},
     year = {2008},
     mrnumber = {2504663},
     zbl = {1178.18001},
     language = {en},
     url = {http://www.numdam.org/item/AST_2008__320__R1_0/}
}

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%A Loday, Jean-Louis
%T Generalized bialgebras and triples of operads
%S Astérisque
%D 2008
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Loday, Jean-Louis. Generalized bialgebras and triples of operads. Astérisque, no. 320 (2008), 126 p. http://numdam.org/item/AST_2008__320__R1_0/

Bibliographie
Cité par

[1] Aguiar (M.) - Infinitesimal bialgebras, pre-Lie and dendriform algebras, in Hopf algebras, Lecture Notes in Pure and Appl. Math., vol. 237, Dekker, New York, 2004, pp. 1-33. | MR | Zbl

[2] Aguiar (M.) - Structure of the Loday-Ronco Hopf algebra of trees, J. Algebra, t. 295 (2006), pp. 473-511. | MR | Zbl | DOI

[3] Aguiar (M.) & Loday (J.-L.) - Quadri-algebras, J. Pure Appl. Algebra, t. 191 (2004), pp. 205-221. | MR | Zbl | DOI

[4] Bergman (G. M.) & Hausknecht (A. O.) - Co-groups and co-rings in the categories of associative rings, Math. Surveys Monogr., vol. 45, American Math. Society, 1996. | MR | Zbl | DOI

[5] Berstein (I.) - On co-groups in the category of graded algebras, Trans. Amer. Math. Soc, t. 115 (1965), pp. 257-269. | MR | Zbl | DOI

[6] Borel (A.) - Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math., t. 57 (1953), pp. 115-207. | MR | Zbl | DOI

[7] Brouder (A.) & Frabetti (A.) - QED Hopf algebras on planar binary trees, J. Algebra, t. 267 (2003), pp. 298-322. | MR | Zbl | DOI

[8] Burgunder (E.) - Eulerian idempotent and the Kashiwara-Vergne conjecture, Ann. Inst. Fourier, t. 58 (2008), pp. 1153-1184. | MR | Zbl | EuDML | Numdam | DOI

[9] Burgunder (E.) - Infinite magmatic bialgebras, Adv. Applied Math., t. 40 (2008), pp. 309-329. | MR | Zbl | DOI

[10] Burgunder (E.) - A symmetric version of Kontsevich graph complex and Leibniz homology, ArXiv:mathQA/0804.2052. | MR | Zbl

[11] Burgunder (E.) & Holtkamp (R.) - Partial magmatic bialgebras, Homology Homotopy Appl., t. 10 (2008), pp. 59-81. | MR | Zbl | DOI

[12] Cartier (P.) - Hyperalgèbres et groupes de Lie formels, Séminaire Sophus Lie, 2e année: Faculté des Sciences de Paris (1955-1956). | Zbl

[13] Chapoton (F.) - Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces, J. Pure Appl. Algebra, t. 168 (2002), pp. 1-18. | MR | Zbl | DOI

[14] Chapoton (F.) & Livernet (M.) - Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Notices, t. 8 (2001), pp. 395-408. | MR | Zbl | DOI

[15] Chapoton (F.) & Vallette (B.) - Pointed and multi-pointed partitions of type A and B, J. Algebraic Combin., t. 23 (2006), pp. 295-316. | MR | Zbl | DOI

[16] Connes (A.) & Kreimer (D.) - Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., t. 199 (1998), pp. 203-242. | MR | Zbl | DOI

[17] Ebrahimi-Fard (K.) & Guo (L.) - On products and duality of binary, quadratic, regular operads, J. Pure Appl. Algebra, t. 200 (2005), pp. 293-317. | MR | Zbl | DOI

[18] Foissy (L.) - Bidendriform bialgebras, trees, and free quasi-symmetric functions, J. Pure Appl. Algebra, t. 209 (2007), pp. 439-459. | MR | Zbl | DOI

[19] Fox (T.) & Markl (M.) - Distributive laws, bialgebras, and cohomology, Operads, Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., t. 202 (1997), pp. 167-205. | MR | Zbl | DOI

[20] Frabetti (A.) - Groups of tree-expanded series, J. Algebra, t. 319 (2008), pp. 377-413. | MR | Zbl

[21] Fresse (B.) - Algèbre des descentes et cogroupes dans les algèbres sur une opérade, Bull. Soc. Math. France, t. 126 (1998), pp. 407-433. | MR | Zbl | EuDML | Numdam | DOI

[22] Fresse (B.)- Cogroups in algebras over an operad are free algebras, Comment. Math. Helv., t. 73 (1998), pp. 637-676. | MR | Zbl | DOI

[23] Fresse (B.) - On the homotopy of simplicial algebras over an operad, Trans. Amer. Math. Soc., t. 352 (2000), pp. 4113-4141. | MR | Zbl | DOI

[24] Fresse (B.) - Koszul duality of operads and homology of partition posets, in Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, Contemp. Math., vol. 346, 2004, pp. 115-215. | MR | Zbl | DOI

[25] Fresse (B.) - Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal, t. 13 (2006), pp. 237-312. | MR | Zbl | EuDML | Numdam | DOI

[26] Gerritzen (L.) & Holtkamp (R.) - Hopf co-addition for free magma algebras and the non-associative Hausdorff series, J. Algebra, t. 265 (2003), pp. 264-284. | MR | Zbl | DOI

[27] Gerstenhaber (M.) & Voronov (A.) - Homotopy $G$ -algebras and moduli space operad, Internat. Math. Res. Notices, t. 3 (1995), pp. 141-153. | MR | Zbl | DOI

[28] Ginzburg (V.) & Kapranov (M.) - Koszul duality for operads, Duke Math. J., t. 76 (1994), pp. 203-272. | MR | Zbl | DOI

[29] Gnedbaye (V.) - Opérades des algèbres $(k + l)$ -aires, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., t. 202 (1997), pp. 83-113. | MR | Zbl

[30] Goncharov (A. B.) - Galois symmetries of fundamental groupoids and non-commutative geometry, Duke Math. J., t. 128 (2005), pp. 209-284. | MR | Zbl | DOI

[31] Grivel (P.-P.) - Une histoire du théorème de Poincaré-Birkhoff-Witt, Expo. Math., t. 22 (2004), pp. 145-184. | MR | Zbl | DOI

[32] Guin (D.) & Oudom (J.-M.) - Sur l'algèbre enveloppante d'une algèbre pré-lie, C. R. Math. Acad. Sci. Paris, t. 340 (2005), pp. 331-336. | MR | Zbl | DOI

[33] Hoffman (M.) - Quasi-shuffle products, J. Algebraic Combin., t. 11 (2000), pp. 49-68. | MR | Zbl | DOI

[34] Holtkamp (R.) - Comparison of Hopf algebras on trees, Arch. Math., t. 80 (2003), pp. 368-383. | MR | Zbl | DOI

[35] Holtkamp (R.) - On Hopf algebra structures over free operads, Adv. Math., t. 207 (2006), pp. 544-565. | MR | Zbl | DOI

[36] Holtkamp (R.), Loday (J.-L.) & Ronco (M.) - Coassociative magmatic bialgebras and the Fine numbers, J. Algebraic Combin., t. 28 (2008), pp. 97-114. | MR | Zbl | DOI

[37] Lafont (Y.) - Interaction combinators, Inform. and Comput., t. 137 (1997), pp. 69-101. | MR | Zbl | DOI

[38] Lafont (Y.) - Towards an algebraic theory of Boolean circuits, J. Pure Appl. Algebra, t. 184 (2003), pp. 257-310. | MR | Zbl | DOI

[39] Lazard (M.) - Lois de groupes et analyseurs, Ann. Sci. École Norm. Sup., t. 72 (1955), pp. 299-400. | MR | Zbl | EuDML | Numdam | DOI

[40] Leray (J.) - Sur la forme des espaces topologiques et sur les points fixes des représentations, J. Math. Pures Appl., t. 24, (1945), pp. 95-167. | MR | Zbl | Numdam

[41] Leroux (P.) - Ennea-algebras, J. Algebra, t. 281 (2004), pp. 287-302. | MR | Zbl | DOI

[42] Livernet (M.) - A rigidity theorem for pre-Lie algebras, J. Pure Applied Algebra, t. 207 (2006), pp. 1-18. | MR | Zbl | DOI

[43] Loday (J.-L.) - Opérations sur l'homologie cyclique des algèbres commutatives, Invent. Math., t. 96 (1989), pp. 205-230. | MR | Zbl | EuDML | DOI

[44] Loday (J.-L.) - Série de Hausdorff, idempotents eulériens et algèbres de Hopf, Exposition. Math., t. 12 (1994), pp. 165-178. | MR | Zbl

[45] Loday (J.-L.) - Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand., t. 77 (1995), pp. 189-196. | MR | Zbl | EuDML | DOI

[46] Loday (J.-L.) - Cyclic homology. Second edition, Grund. Math. Wiss., vol. 301, Springer, Berlin, 1998. | MR | Zbl

[47] Loday (J.-L.) - Dialgebras, in 'Dialgebras and related operads', Springer Lecture Notes in Math., t. 1763 (2001), pp. 7-66. | MR | Zbl | DOI

[48] Loday (J.-L.) - Scindement d'associativité et algèbres de Hopf, Proceedings of the Conference in honor of Jean Leray, Nantes 2002, Séminaire et Congrès, Soc. Math. France, t. 9 (2004), pp. 155-172. | MR | Zbl

[49] Loday (J.-L.) - On the algebra of quasi-shuffles, Manuscripta Math., t. 123 (2007), pp. 79-93. | MR | Zbl | DOI

[50] Loday (J.-L.) & Popov (T.) - Parastatistics algebra and super semistandard Young tableaux, pp. 423-430, Lie Theory and Its Applications in Physics VII, ed. V. K. Dobrev and al., Heron Press, Sofia, 2008. | Zbl

[51] Loday (J.-L.) & Ronco (M.) - Hopf algebra of the planar binary trees, Adv. in Maths, t. 139 (1998), pp. 293-309. | MR | Zbl | DOI

[52] Loday (J.-L.) & Ronco (M.) - Algèbres de Hopf colibres, C. R. Math. Acad. Sci. Paris, t. 337 (2003), pp. 153-158. | MR | Zbl | DOI

[53] Loday (J.-L.) & Ronco (M.) - Trialgebras and families of polytopes, in Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$ -theory, Contemp. Math., vol. 346, 2004, pp. 369-398. | MR | Zbl | DOI

[54] Loday (J.-L.) & Ronco (M.) - On the structure of cofree Hopf algebras, J. reine angew. Math., t. 592 (2006), pp. 123-155. | MR | Zbl

[55] Loday (J.-L.) & Ronco (M.) - Combinatorial Hopf algebras, 2008, preprint ArXiv:math : 08100435. | MR

[56] Loday (J.-L.) & Vallette (B.) - Algebraic Operads, book in progress. | MR | Zbl

[57] Maclane (S.) - Categorical algebra, Bull. Amer. Math. Soc., t. 71 (1965), pp. 40-106. | MR | Zbl | DOI

[58] Markl (M.) - Distributive laws and Koszulness, Ann. Inst. Fourier, t. 46 (1996), pp. 307-323. | MR | Zbl | EuDML | Numdam | DOI

[59] Markl (M.) - Lie elements in pre-Lie algebras, trees and cohomology operations, J. Lie Theory, t. 17 (2007), pp. 241-261. | MR | Zbl

[60] Markl (M.) & Remm (E.) - Algebras with one operation including Poisson and other Lie-admissible algebras, J. Algebra, t. 299 (2006), pp. 171-189. | MR | Zbl | DOI

[61] Markl (M.), Shnider (S.) & Stasheff (J.) - Operads in algebra, topology and physics, Math. Surveys and Monographs, vol. 96, American Math. Society, 2002. | MR | Zbl

[62] May (J. P.) - The geometry of iterated loop spaces, Lectures Notes in Math., vol. 271, Springer, Berlin-New York, 1972. | MR | Zbl

[63] Michaelis (W.) - The dual Poincaré-Birkhoff-Witt theorem, Adv. in Math., t. 57 (1985), pp. 93-162. | MR | Zbl | DOI

[64] Milnor (J. W.) & Moore (J. C.) - On the structure of Hopf algebras, Ann. of Math., t. 81 (1965), pp. 211-264. | MR | Zbl | DOI

[65] Moerdijk (I.) - On the Connes-Kreimer construction of Hopf algebras, Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math., t. 271 (2001), pp. 311-321. | MR | Zbl | DOI

[66] Oudom (J.-M.) - Théorème de Leray dans la catégorie des algèbres sur une opérade, C. R. Acad. Sci. Paris Sér. I Math., t. 329 (1999), pp. 101-106. | MR | Zbl | DOI

[67] Palacios (P.) & Ronco (M.) - Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, t. 299 (2006), pp. 648-678. | MR | Zbl | DOI

[68] Patras (F.) - A Leray theorem for the generalization to operads of Hopf algebras with divided powers, J. Algebra, t. 218 (1999), pp. 528-542. | MR | Zbl | DOI

[69] Pérez-Izquierdo (J. M.) - Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math., t. 208 (2007), pp. 834-876. | MR | Zbl | DOI

[70] Pérez-Izquierdo (J. M.) & Shestakov (I. P.) - An envelope for Malcev algebras, J. Algebra, t. 272 (2004), pp. 379-393. | MR | Zbl | DOI

[71] Pirashvili (T.) - On the PROP corresponding to bialgebras, Cah. Topol. Géom. Différ. Catég., t. 43 (2002), pp. 221-239. | MR | Zbl | EuDML | Numdam

[72] Pirashvili (T.) - Sets with two associative operations, Cent. Eur. J. Math., t. 1 (2003), pp. 169-183. | MR | Zbl | EuDML | DOI

[73] Quillen (D.) - Rational homotopy theory, Ann. of Math., t. 90 (1969), pp. 205-295. | MR | Zbl | DOI

[74] Reutenauer (C.) - 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., t. 1234 (1986), pp. 267-284. | MR | Zbl | DOI

[75] Ronco (M.) - Primitive elements in a free dendriform algebra, New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., t. 267 (2000), pp. 245-263. | MR | Zbl | DOI

[76] Ronco (M.) - Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra, t. 254 (2002), pp. 152-172. | MR | Zbl | DOI

[77] Ronco (M.) - Shuffle bialgebras, arXiv : math/0703437. | MR | Zbl | Numdam | DOI

[78] Shestakov (I. P.) & Umirbaev (U. U.) - Free Akivis algebras, primitive elements, and hyperalgebras, J. Algebra, t. 250 (2002), pp. 533-548. | MR | Zbl | DOI

[79] Vallette (B.) - A Koszul duality for PROPs, Trans. Amer. Math. Soc., t. 359 (2007), pp. 4865-4943. | MR | Zbl | DOI

[80] Vallette (B.) - Homology of generalized partition posets, J. Pure Appl. Algebra, t. 208 (2007), pp. 699-725. | MR | Zbl | DOI

[81] Vallette (B.) - Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. reine angew. Math., t. 620 (2008), pp. 105-164. | MR | Zbl

[82] Voronov (A.) - Homotopy Gerstenhaber algebras, in Conf Moshe Flato 1999, vol. 2, Kluwer Acad. Pub., 2000, pp. 307-331. | MR | Zbl | DOI

[83] Wigner (D.) - An identity in the free Lie algebra, Proc. Amer. Math. Soc., t. 106 (1989), pp. 639-640. | MR | Zbl | DOI