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Benoit Fresse
Théorie des opérades de Koszul et homologie des algèbres de Poisson
Annales mathématiques Blaise Pascal, 13 no. 2 (2006), p. 237-312
Article PDF | Analyses MR 2275449 | Zbl 1141.55006 | 1 citation dans Cedram
[1] J. Alev and T. Lambre. Comparaison de l’homologie de Hochschild et de l’homologie de Poisson pour une déformation des surfaces de Klein. In Algebra and operator theory, pages 25-38. Kluwer Acad. Publ., 1998 MR 1643407 | Zbl 0931.16007
[2] D. Balavoine. Homology and cohomology with coefficicients of an algebra over a quadratic operad. J. Pure Appl. Algebra, 132:221-258, 1998. MR 1642086 | Zbl 0967.18004
[3] C. Berger and I. Moerdijk. Axiomatic homotopy theory for operads. Comment. Math. Helv., 78:805-831, 2003. MR 2016697 | Zbl 1041.18011
[4] J. Braconnier. Algèbres de Poisson. C. R. Acad. Sci. Paris Sér. A Math., 284:1345-1348, 1977. MR 443000 | Zbl 0356.17007
[5] J.-L. Brylinski. A differential complex for Poisson manifolds. J. Diff. Geometry., 28:93-114, 1988. MR 950556 | Zbl 0634.58029
[6] D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras, Lecture Notes in Math., Springer-Verlag, 1988, p. 51-72 MR 952571 | Zbl 0666.13007
[7] K. T. Chen. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math., 65:163-178, 1957. MR 85251 | Zbl 0077.25301
[8] K. T. Chen. Iterated path integral. Bull. Amer. Math. Soc., 83:831-879, 1977.
Article | MR 454968 | Zbl 0389.58001
[9] V. G. Drinfeld. Quantum groups. In Proc. Int. Congr. Math., pages 798-820, 1987 MR 934283 | Zbl 0667.16003
[10] V. G. Drinfeld. On some unsolved problems in quantum group theory. In Quantum groups, Lecture Notes in Math., pages 1-8. Springer-Verlag, 1992 MR 1183474 | Zbl 0765.17014
[11] T. Fox and M. Markl. Distributive laws, bialgebras, and cohomology. In Operads : Proceedings of renaissance conferences, Contemp. Math., pages 167-205, 1997 MR 1436921 | Zbl 0866.18008
[12] B. Fresse. Algèbre des descentes et cogroupes dans les algèbres sur une opérade. Bull. Soc. Math. Fr., 126:407-433, 1998.
Numdam | MR 1682797 | Zbl 0940.18004
[13] B. Fresse. Homologie de Quillen pour les algèbres de Poisson. C. R. Acad. Sci. Paris Sér. I Math., 326:1053-1058, 1998. MR 1647186 | Zbl 0922.17014
[14] B. Fresse. Structures de Poisson sur une intersection complète à singularitées isolées. C. R. Acad. Sci. Paris Sér. I Math., 335:5-10, 2002. MR 1920425 | Zbl 01777355
[15] B. Fresse. Koszul duality of operads and homology of partition posets. In Homotopy theory and its applications, Contemp. Math., pages 115-215, 2004 MR 2066499 | Zbl 1077.18007
[16] I. M. Gelfand and I. Y. Dorfman. Hamiltonian operators and the classical Yang-Baxter equation. Funkts. Anal. Prilozh., 16:1-9, 1982. MR 684122 | Zbl 0527.58018
[17] M. Gerstenhaber and S. D. Schack. A Hodge-type decomposition for commutative algebra cohomology. J. Pure Appl. Algebra, 48:229-247, 1987. MR 917209 | Zbl 0671.13007
[18] M. Gerstenhaber and S. D. Schack. The shuffle bialgebra and the cohomology of commutative algebras. J. Pure Appl. Algebra, 70:263-272, 1991. MR 1104844 | Zbl 0728.13003
[19] E. Getzler and J. Jones. Operads, homotopy algebra and iterated integrals for double loop spaces. prépublication arXiv :hep-th/9403055, 1994
arXiv
[20] G. Ginot. Homologie et modèle minimal des algèbres de Gerstenhaber. Ann. Math. Blaise Pascal, 11:95-127, 2004.
Cedram | MR 2077240 | Zbl 02207860
[21] V. Ginzburg and D. Kaledin. Poisson deformations of symplectic quotient singularities. Adv. Math., 186:1-57, 2004. MR 2065506 | Zbl 1062.53074
[22] V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Math. J., 76:203-272, 1995.
Article | MR 1301191 | Zbl 0855.18006
[23] R. Hain. On the indecomposable elements of the bar construction. Proc. Amer. Math. Soc., 98:312-316, 1986. MR 854039 | Zbl 0613.55007
[24] J. Huebschmann. Poisson cohomology and quantization. J. Reine Angew. Math., 408:57-113, 1990. MR 1058984 | Zbl 0699.53037
[25] C. Kassel. L’homologie cyclique des algèbres enveloppantes. Invent. Math., 91:221-251, 1988.
Article | MR 922799 | Zbl 0653.17007
[26] M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys., 66:157-216, 2003. MR 2062626 | Zbl 1058.53065
[27] J.-L. Koszul. Crochet de Schouten-Nijenhuis et cohomologie. In Elie Cartan et les mathématiques d’aujourd’hui, Astérisque hors série, pages 257-271, 1985 MR 837203 | Zbl 0615.58029
[28] I. S. Krasilshschik. Hamiltonian cohomology of canonical algebras. Dokl. Akad. Nauk., 251:1306-1309, 1980. MR 570152 | Zbl 0454.58020
[29] A. Lichnerowicz. Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff. Geometry, 12:253-300, 1977. MR 501133 | Zbl 0405.53024
[30] M. Livernet, Homotopie rationnelle des algèbres sur une opérade, thesis, Université Louis Pasteur, 1998 MR 1695322
[31] J.-L. Loday. Opérations sur l’homologie cyclique des algèbres commutatives. Invent. Math., 96:205-230, 1989.
Article | MR 981743 | Zbl 0686.18006
[32] J.-L. Loday. Cyclic homology, volume 301 of Grund. der Math. Wiss.. Springer-Verlag, 1992 MR 1217970 | Zbl 0780.18009
[33] J.-L. Loday. Série de Hausdorff, idempotents Euleriens et algèbres de Hopf. Exposition. Math., 12:165-178, 1994. MR 1274784 | Zbl 0807.17003
[34] S. Mac Lane. Homology, volume 114 of Grund. der Math. Wiss.. Springer-Verlag, 1963 MR 1344215
[35] M. Markl. Distributive laws and Koszulness. Ann. Inst. Fourier, 46:307-323, 1996.
Cedram | MR 1393517 | Zbl 0853.18005
[36] M. Markl, S. Shnider and J. Stasheff. Operads in algebra, topology and physics, volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, 2002 MR 1898414 | Zbl 1017.18001
[37] S.-Q. Oh. Poisson enveloping algebras. Comm. Algebra, 27:2181-2186, 1999. MR 1683858 | Zbl 0936.16020
[38] F. Patras. Construction géométrique des idempotents Euleriens. Filtration des groupes de polytopes et des groupes d’homologie de Hochschild. Bull. Soc. Math. Fr., 119:173-198, 1991.
Numdam | MR 1116844 | Zbl 0752.55014
[39] F. Patras. L’algèbre des descentes d’une bigèbre graduée. J. Algebra, 170:547-566, 1994. MR 1302855 | Zbl 0819.16033
[40] T. Pirashvili. Hodge decomposition for higher Hochschild homology. Ann. Sci. École Norm. Sup., 33:151-179, 2000.
Numdam | MR 1755114 | Zbl 0957.18004
[41] D. Quillen. Homotopical algebra, volume 43 of Lecture Notes in Math.. Springer-Verlag, 1967 MR 223432 | Zbl 0168.20903
[42] D. Quillen. Rational homotopy theory. Ann. of Math., 90:205-295, 1969. MR 258031 | Zbl 0191.53702
[43] D. Quillen, On the (co)-homology of commutative rings, Proc. Symp. Pure Math., Amer. Math. Soc., 1970, p. 65-87 MR 257068 | Zbl 0234.18010
[44] C. Reutenauer. Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers. In Combinatoire énumérative, Lecture Notes in Math., pages 267-284. Springer-Verlag, 1986 MR 927769 | Zbl 0621.20004
[45] C. Reutenauer. Free Lie Algebras, volume 7 of London Math. Soc. Mon.. Clarendon Press, 1993 MR 1231799 | Zbl 0798.17001
[46] G. Rinehart. Differential forms on general commutative algebras. Trans. Amer. Math. Soc., 108:195-222, 1963. MR 154906 | Zbl 0113.26204
[47] M. Schlessinger and J. Stasheff. The Lie algebra structure of tangent cohomology and deformation theory. J. Pure Appl. Algebra, 38:313-322, 1985. MR 814187 | Zbl 0576.17008
[48] D. Tamarkin. Another proof of M. Kontsevich formality theorem. prépublication arXiv :math.QA/9803025, 1998
arXiv
[49] I. Vaisman. Lectures on the geometry of Poisson manifolds, volume 118 of Progress in Math.. Birkhäuser, 1994 MR 1269545 | Zbl 0810.53019
[50] M. Vigué-Poirrier. Cyclic homology of algebraic hypersurfaces. J. Pure Appl. Algebra, 72:95-108, 1991. MR 1115570 | Zbl 0732.16008
[51] M. Vigué-Poirrier. Décompositions de l’homologie cyclique des algèbres différentielles graduées commutatives. K-theory, 4:255-267, 1991. MR 1116926
Benoit Fresse
Théorie des opérades de Koszul et homologie des algèbres de Poisson
Annales mathématiques Blaise Pascal, 13 no. 2 (2006), p. 237-312
Article PDF | Analyses MR 2275449 | Zbl 1141.55006 | 1 citation dans Cedram
Bibliographie
[2] D. Balavoine. Homology and cohomology with coefficicients of an algebra over a quadratic operad. J. Pure Appl. Algebra, 132:221-258, 1998. MR 1642086 | Zbl 0967.18004
[3] C. Berger and I. Moerdijk. Axiomatic homotopy theory for operads. Comment. Math. Helv., 78:805-831, 2003. MR 2016697 | Zbl 1041.18011
[4] J. Braconnier. Algèbres de Poisson. C. R. Acad. Sci. Paris Sér. A Math., 284:1345-1348, 1977. MR 443000 | Zbl 0356.17007
[5] J.-L. Brylinski. A differential complex for Poisson manifolds. J. Diff. Geometry., 28:93-114, 1988. MR 950556 | Zbl 0634.58029
[6] D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras, Lecture Notes in Math., Springer-Verlag, 1988, p. 51-72 MR 952571 | Zbl 0666.13007
[7] K. T. Chen. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math., 65:163-178, 1957. MR 85251 | Zbl 0077.25301
[8] K. T. Chen. Iterated path integral. Bull. Amer. Math. Soc., 83:831-879, 1977.
Article | MR 454968 | Zbl 0389.58001
[9] V. G. Drinfeld. Quantum groups. In Proc. Int. Congr. Math., pages 798-820, 1987 MR 934283 | Zbl 0667.16003
[10] V. G. Drinfeld. On some unsolved problems in quantum group theory. In Quantum groups, Lecture Notes in Math., pages 1-8. Springer-Verlag, 1992 MR 1183474 | Zbl 0765.17014
[11] T. Fox and M. Markl. Distributive laws, bialgebras, and cohomology. In Operads : Proceedings of renaissance conferences, Contemp. Math., pages 167-205, 1997 MR 1436921 | Zbl 0866.18008
[12] B. Fresse. Algèbre des descentes et cogroupes dans les algèbres sur une opérade. Bull. Soc. Math. Fr., 126:407-433, 1998.
Numdam | MR 1682797 | Zbl 0940.18004
[13] B. Fresse. Homologie de Quillen pour les algèbres de Poisson. C. R. Acad. Sci. Paris Sér. I Math., 326:1053-1058, 1998. MR 1647186 | Zbl 0922.17014
[14] B. Fresse. Structures de Poisson sur une intersection complète à singularitées isolées. C. R. Acad. Sci. Paris Sér. I Math., 335:5-10, 2002. MR 1920425 | Zbl 01777355
[15] B. Fresse. Koszul duality of operads and homology of partition posets. In Homotopy theory and its applications, Contemp. Math., pages 115-215, 2004 MR 2066499 | Zbl 1077.18007
[16] I. M. Gelfand and I. Y. Dorfman. Hamiltonian operators and the classical Yang-Baxter equation. Funkts. Anal. Prilozh., 16:1-9, 1982. MR 684122 | Zbl 0527.58018
[17] M. Gerstenhaber and S. D. Schack. A Hodge-type decomposition for commutative algebra cohomology. J. Pure Appl. Algebra, 48:229-247, 1987. MR 917209 | Zbl 0671.13007
[18] M. Gerstenhaber and S. D. Schack. The shuffle bialgebra and the cohomology of commutative algebras. J. Pure Appl. Algebra, 70:263-272, 1991. MR 1104844 | Zbl 0728.13003
[19] E. Getzler and J. Jones. Operads, homotopy algebra and iterated integrals for double loop spaces. prépublication arXiv :hep-th/9403055, 1994
arXiv
[20] G. Ginot. Homologie et modèle minimal des algèbres de Gerstenhaber. Ann. Math. Blaise Pascal, 11:95-127, 2004.
Cedram | MR 2077240 | Zbl 02207860
[21] V. Ginzburg and D. Kaledin. Poisson deformations of symplectic quotient singularities. Adv. Math., 186:1-57, 2004. MR 2065506 | Zbl 1062.53074
[22] V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Math. J., 76:203-272, 1995.
Article | MR 1301191 | Zbl 0855.18006
[23] R. Hain. On the indecomposable elements of the bar construction. Proc. Amer. Math. Soc., 98:312-316, 1986. MR 854039 | Zbl 0613.55007
[24] J. Huebschmann. Poisson cohomology and quantization. J. Reine Angew. Math., 408:57-113, 1990. MR 1058984 | Zbl 0699.53037
[25] C. Kassel. L’homologie cyclique des algèbres enveloppantes. Invent. Math., 91:221-251, 1988.
Article | MR 922799 | Zbl 0653.17007
[26] M. Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys., 66:157-216, 2003. MR 2062626 | Zbl 1058.53065
[27] J.-L. Koszul. Crochet de Schouten-Nijenhuis et cohomologie. In Elie Cartan et les mathématiques d’aujourd’hui, Astérisque hors série, pages 257-271, 1985 MR 837203 | Zbl 0615.58029
[28] I. S. Krasilshschik. Hamiltonian cohomology of canonical algebras. Dokl. Akad. Nauk., 251:1306-1309, 1980. MR 570152 | Zbl 0454.58020
[29] A. Lichnerowicz. Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff. Geometry, 12:253-300, 1977. MR 501133 | Zbl 0405.53024
[30] M. Livernet, Homotopie rationnelle des algèbres sur une opérade, thesis, Université Louis Pasteur, 1998 MR 1695322
[31] J.-L. Loday. Opérations sur l’homologie cyclique des algèbres commutatives. Invent. Math., 96:205-230, 1989.
Article | MR 981743 | Zbl 0686.18006
[32] J.-L. Loday. Cyclic homology, volume 301 of Grund. der Math. Wiss.. Springer-Verlag, 1992 MR 1217970 | Zbl 0780.18009
[33] J.-L. Loday. Série de Hausdorff, idempotents Euleriens et algèbres de Hopf. Exposition. Math., 12:165-178, 1994. MR 1274784 | Zbl 0807.17003
[34] S. Mac Lane. Homology, volume 114 of Grund. der Math. Wiss.. Springer-Verlag, 1963 MR 1344215
[35] M. Markl. Distributive laws and Koszulness. Ann. Inst. Fourier, 46:307-323, 1996.
Cedram | MR 1393517 | Zbl 0853.18005
[36] M. Markl, S. Shnider and J. Stasheff. Operads in algebra, topology and physics, volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, 2002 MR 1898414 | Zbl 1017.18001
[37] S.-Q. Oh. Poisson enveloping algebras. Comm. Algebra, 27:2181-2186, 1999. MR 1683858 | Zbl 0936.16020
[38] F. Patras. Construction géométrique des idempotents Euleriens. Filtration des groupes de polytopes et des groupes d’homologie de Hochschild. Bull. Soc. Math. Fr., 119:173-198, 1991.
Numdam | MR 1116844 | Zbl 0752.55014
[39] F. Patras. L’algèbre des descentes d’une bigèbre graduée. J. Algebra, 170:547-566, 1994. MR 1302855 | Zbl 0819.16033
[40] T. Pirashvili. Hodge decomposition for higher Hochschild homology. Ann. Sci. École Norm. Sup., 33:151-179, 2000.
Numdam | MR 1755114 | Zbl 0957.18004
[41] D. Quillen. Homotopical algebra, volume 43 of Lecture Notes in Math.. Springer-Verlag, 1967 MR 223432 | Zbl 0168.20903
[42] D. Quillen. Rational homotopy theory. Ann. of Math., 90:205-295, 1969. MR 258031 | Zbl 0191.53702
[43] D. Quillen, On the (co)-homology of commutative rings, Proc. Symp. Pure Math., Amer. Math. Soc., 1970, p. 65-87 MR 257068 | Zbl 0234.18010
[44] C. Reutenauer. Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers. In Combinatoire énumérative, Lecture Notes in Math., pages 267-284. Springer-Verlag, 1986 MR 927769 | Zbl 0621.20004
[45] C. Reutenauer. Free Lie Algebras, volume 7 of London Math. Soc. Mon.. Clarendon Press, 1993 MR 1231799 | Zbl 0798.17001
[46] G. Rinehart. Differential forms on general commutative algebras. Trans. Amer. Math. Soc., 108:195-222, 1963. MR 154906 | Zbl 0113.26204
[47] M. Schlessinger and J. Stasheff. The Lie algebra structure of tangent cohomology and deformation theory. J. Pure Appl. Algebra, 38:313-322, 1985. MR 814187 | Zbl 0576.17008
[48] D. Tamarkin. Another proof of M. Kontsevich formality theorem. prépublication arXiv :math.QA/9803025, 1998
arXiv
[49] I. Vaisman. Lectures on the geometry of Poisson manifolds, volume 118 of Progress in Math.. Birkhäuser, 1994 MR 1269545 | Zbl 0810.53019
[50] M. Vigué-Poirrier. Cyclic homology of algebraic hypersurfaces. J. Pure Appl. Algebra, 72:95-108, 1991. MR 1115570 | Zbl 0732.16008
[51] M. Vigué-Poirrier. Décompositions de l’homologie cyclique des algèbres différentielles graduées commutatives. K-theory, 4:255-267, 1991. MR 1116926
© 2010
Annales Mathématiques Blaise Pascal
Editées par le
Laboratoire de mathématiques
CNRS - UMR 6620
Université Blaise Pascal de Clermont-Ferrand
Annales Mathématiques Blaise Pascal
Editées par le
Laboratoire de mathématiques
CNRS - UMR 6620
Université Blaise Pascal de Clermont-Ferrand
