Théorie des opérades de Koszul et homologie des algèbres de Poisson
Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 237-312.
DOI : 10.5802/ambp.219
Fresse, Benoit 1

1 Laboratoire Painlevé Université de Lille 1 et CNRS Cité Scientifique – Bâtiment M2 F-59655 Villeneuve d’Ascq Cedex France
@article{AMBP_2006__13_2_237_0,
     author = {Fresse, Benoit},
     title = {Th\'eorie des op\'erades de {Koszul} et homologie des alg\`ebres de {Poisson}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {237--312},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {13},
     number = {2},
     year = {2006},
     doi = {10.5802/ambp.219},
     zbl = {1141.55006},
     mrnumber = {2275449},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/ambp.219/}
}
TY  - JOUR
AU  - Fresse, Benoit
TI  - Théorie des opérades de Koszul et homologie des algèbres de Poisson
JO  - Annales mathématiques Blaise Pascal
PY  - 2006
SP  - 237
EP  - 312
VL  - 13
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.219/
DO  - 10.5802/ambp.219
LA  - fr
ID  - AMBP_2006__13_2_237_0
ER  - 
%0 Journal Article
%A Fresse, Benoit
%T Théorie des opérades de Koszul et homologie des algèbres de Poisson
%J Annales mathématiques Blaise Pascal
%D 2006
%P 237-312
%V 13
%N 2
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.219/
%R 10.5802/ambp.219
%G fr
%F AMBP_2006__13_2_237_0
Fresse, Benoit. Théorie des opérades de Koszul et homologie des algèbres de Poisson. Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 237-312. doi : 10.5802/ambp.219. http://www.numdam.org/articles/10.5802/ambp.219/

[1] Alev, J.; Lambre, T. Comparaison de l’homologie de Hochschild et de l’homologie de Poisson pour une déformation des surfaces de Klein, Algebra and operator theory, Kluwer Acad. Publ. (1998), pp. 25-38 | MR | Zbl

[2] Balavoine, D. Homology and cohomology with coefficicients of an algebra over a quadratic operad, J. Pure Appl. Algebra, Volume 132 (1998), pp. 221-258 | DOI | MR | Zbl

[3] Berger, C.; Moerdijk, I. Axiomatic homotopy theory for operads, Comment. Math. Helv., Volume 78 (2003), pp. 805-831 | DOI | MR | Zbl

[4] Braconnier, J. Algèbres de Poisson, C. R. Acad. Sci. Paris Sér. A Math., Volume 284 (1977), pp. 1345-1348 | MR | Zbl

[5] Brylinski, J.-L. A differential complex for Poisson manifolds, J. Diff. Geometry., Volume 28 (1988), pp. 93-114 | MR | Zbl

[6] Burghelea, D.; Vigué-Poirrier, M. Cyclic homology of commutative algebras, Lecture Notes in Math., 1318, Springer-Verlag (1988), pp. 51-72 | MR | Zbl

[7] Chen, K. T. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math., Volume 65 (1957), pp. 163-178 | DOI | MR | Zbl

[8] Chen, K. T. Iterated path integral, Bull. Amer. Math. Soc., Volume 83 (1977), pp. 831-879 | DOI | MR | Zbl

[9] Drinfeld, V. G. Quantum groups, Proc. Int. Congr. Math. (1987), pp. 798-820 | MR | Zbl

[10] Drinfeld, V. G. On some unsolved problems in quantum group theory, Quantum groups (Lecture Notes in Math.), Volume 1510, Springer-Verlag (1992), pp. 1-8 | MR | Zbl

[11] Fox, T.; Markl, M. Distributive laws, bialgebras, and cohomology, Operads : Proceedings of renaissance conferences (Contemp. Math.), Volume 202 (1997), pp. 167-205 | MR | Zbl

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

[13] Fresse, B. Homologie de Quillen pour les algèbres de Poisson, C. R. Acad. Sci. Paris Sér. I Math., Volume 326 (1998), pp. 1053-1058 | DOI | MR | Zbl

[14] Fresse, B. Structures de Poisson sur une intersection complète à singularitées isolées, C. R. Acad. Sci. Paris Sér. I Math., Volume 335 (2002), pp. 5-10 | MR | Zbl

[15] Fresse, B. Koszul duality of operads and homology of partition posets, Homotopy theory and its applications (Contemp. Math.), Volume 346 (2004), pp. 115-215 | MR | Zbl

[16] Gelfand, I. M.; Dorfman, I. Y. Hamiltonian operators and the classical Yang-Baxter equation, Funkts. Anal. Prilozh., Volume 16 (1982), pp. 1-9 | MR | Zbl

[17] Gerstenhaber, M.; Schack, S. D. A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra, Volume 48 (1987), pp. 229-247 | DOI | MR | Zbl

[18] Gerstenhaber, M.; Schack, S. D. The shuffle bialgebra and the cohomology of commutative algebras, J. Pure Appl. Algebra, Volume 70 (1991), pp. 263-272 | DOI | MR | Zbl

[19] Getzler, E.; Jones, J. Operads, homotopy algebra and iterated integrals for double loop spaces (1994) (prépublication arXiv :hep-th/9403055)

[20] Ginot, G. Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal, Volume 11 (2004), pp. 95-127 | DOI | Numdam | MR | Zbl

[21] Ginzburg, V.; Kaledin, D. Poisson deformations of symplectic quotient singularities, Adv. Math., Volume 186 (2004), pp. 1-57 | DOI | MR | Zbl

[22] Ginzburg, V.; Kapranov, M. Koszul duality for operads, Duke Math. J., Volume 76 (1995), pp. 203-272 | DOI | MR | Zbl

[23] Hain, R. On the indecomposable elements of the bar construction, Proc. Amer. Math. Soc., Volume 98 (1986), pp. 312-316 | DOI | MR | Zbl

[24] Huebschmann, J. Poisson cohomology and quantization, J. Reine Angew. Math., Volume 408 (1990), pp. 57-113 | DOI | MR | Zbl

[25] Kassel, C. L’homologie cyclique des algèbres enveloppantes, Invent. Math., Volume 91 (1988), pp. 221-251 | DOI | MR | Zbl

[26] Kontsevich, M. Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003), pp. 157-216 | DOI | MR | Zbl

[27] Koszul, J.-L. Crochet de Schouten-Nijenhuis et cohomologie, Elie Cartan et les mathématiques d’aujourd’hui (Astérisque hors série) (1985), pp. 257-271 | Numdam | MR | Zbl

[28] Krasilshschik, I. S. Hamiltonian cohomology of canonical algebras, Dokl. Akad. Nauk., Volume 251 (1980), pp. 1306-1309 | MR | Zbl

[29] Lichnerowicz, A. Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geometry, Volume 12 (1977), pp. 253-300 | MR | Zbl

[30] Livernet, M. Homotopie rationnelle des algèbres sur une opérade, Université Louis Pasteur, Strasbourg (1998) (Ph. D. Thesis) | MR

[31] Loday, J.-L. Opérations sur l’homologie cyclique des algèbres commutatives, Invent. Math., Volume 96 (1989), pp. 205-230 | DOI | MR | Zbl

[32] Loday, J.-L. Cyclic homology, Grund. der Math. Wiss., 301, Springer-Verlag, 1992 | MR | Zbl

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

[34] Mac Lane, S. Homology, Grund. der Math. Wiss., 114, Springer-Verlag, 1963 | MR

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

[36] Markl, M.; Shnider, S.; Stasheff, J. Operads in algebra, topology and physics, Mathematical Surveys and Monographs, 96, American Mathematical Society, 2002 | MR | Zbl

[37] Oh, S.-Q. Poisson enveloping algebras, Comm. Algebra, Volume 27 (1999), pp. 2181-2186 | DOI | MR | Zbl

[38] Patras, F. Construction géométrique des idempotents Euleriens. Filtration des groupes de polytopes et des groupes d’homologie de Hochschild, Bull. Soc. Math. Fr., Volume 119 (1991), pp. 173-198 | Numdam | MR | Zbl

[39] Patras, F. L’algèbre des descentes d’une bigèbre graduée, J. Algebra, Volume 170 (1994), pp. 547-566 | DOI | MR | Zbl

[40] Pirashvili, T. Hodge decomposition for higher Hochschild homology, Ann. Sci. École Norm. Sup., Volume 33 (2000), pp. 151-179 | Numdam | MR | Zbl

[41] Quillen, D. Homotopical algebra, Lecture Notes in Math., 43, Springer-Verlag, 1967 | MR | Zbl

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

[43] Quillen, D. On the (co)-homology of commutative rings, Proc. Symp. Pure Math., 17, Amer. Math. Soc. (1970), pp. 65-87 | MR | Zbl

[44] Reutenauer, C. Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, Combinatoire énumérative (Lecture Notes in Math.), Volume 1234, Springer-Verlag (1986), pp. 267-284 | MR | Zbl

[45] Reutenauer, C. Free Lie Algebras, London Math. Soc. Mon., 7, Clarendon Press, 1993 | MR | Zbl

[46] Rinehart, G. Differential forms on general commutative algebras, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 195-222 | DOI | MR | Zbl

[47] Schlessinger, M.; Stasheff, J. The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra, Volume 38 (1985), pp. 313-322 | DOI | MR | Zbl

[48] Tamarkin, D. Another proof of M. Kontsevich formality theorem (1998) (prépublication arXiv :math.QA/9803025)

[49] Vaisman, I. Lectures on the geometry of Poisson manifolds, Progress in Math., 118, Birkhäuser, 1994 | MR | Zbl

[50] Vigué-Poirrier, M. Cyclic homology of algebraic hypersurfaces, J. Pure Appl. Algebra, Volume 72 (1991), pp. 95-108 | DOI | MR | Zbl

[51] Vigué-Poirrier, M. Décompositions de l’homologie cyclique des algèbres différentielles graduées commutatives, K-theory, Volume 4 (1991), pp. 255-267 | DOI | MR

Cité par Sources :