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Theory of Koszul operads and homology of Poisson algebras. (Théorie des opérades de Koszul et homologie des algèbres de Poisson.) (French) Zbl 1141.55006

This article is a detailed exposition of results on (co)-homology of Poisson algebras (over a field of characteristic zero) announced in [B. Fresse, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 9, 1053–1058 (1998; Zbl 0922.17014)]. This Quillen cohomology for Poisson algebras may be described via derived functors of Poisson derivations. The author provides an explicit complex calculating the Quillen homology \(H^{\text{Pois}} _{*} (O,M)\) of a Poisson algebra \(O\) with coefficients in a Poisson representation \(M\). This complex is constructed as a kind of composite of the Hochschild complex (for associative algebra homology) and the Chevalley-Eilenberg complex (for Lie algebra homology).
Detailed background material is covered, including definitions of (differential graded) Poisson algebras, their representations and enveloping algebras and summaries of the weight decomposition of Hochschild homology [M. Gerstenhaber and S. D. Schack, J. Pure Appl. Algebra 48, 229–247 (1987; Zbl 0671.13007)] and of Koszul duality for operads [V. Ginzburg and M. Kapranov, Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)].
The article develops the basic properties of Poisson homology. These include a long exact sequence in homology associated to a short exact sequence of coefficient groups; an identification of the second cohomology group with isomorphism classes of extensions; and a spectral sequence converging to the Poisson homology with \(E^2\) term given by \({Tor}\) groups over the universal enveloping algebra.
The complex for calculating the Poisson homology described above is then introduced and it is shown that this agrees with the complex arising from the theory of Koszul operads. It leads to the main result that there is a spectral sequence \(E^1_{s,t} =\overline{HH}^{(s)}_{s+t}(O,M) \Rightarrow H^{\text{ Pois}} _{s+t} (O,M)\), where \(\overline{HH}^{(s)}_{*}\) denotes the weight \(s\) component of the reduced Hochschild homology. It is shown that the spectral sequence collapses at \(E^1\) in the case of a smooth Poisson algebra, and consequently that the Poisson homology then agrees with the canonical homology defined by Koszul and Brylinski [J.-L. Koszul, Elie Cartan et les mathématiques d’aujourd’hui, The mathematical heritage of Elie Cartan, Semin. Lyon 1984, Astérisque, 1985, 257–271 (1985; Zbl 0615.58029); J.-L. Brylinski, J. Differ. Geom. 28, No. 1, 93–114 (1988; Zbl 0634.58029)].

MSC:

55P48 Loop space machines and operads in algebraic topology
18D50 Operads (MSC2010)
17B55 Homological methods in Lie (super)algebras
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[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, 25-38 (1998) · Zbl 0931.16007
[2] Balavoine, D., Homology and cohomology with coefficicients of an algebra over a quadratic operad, J. Pure Appl. Algebra, 132, 221-258 (1998) · Zbl 0967.18004 · doi:10.1016/S0022-4049(97)00131-X
[3] Berger, C.; Moerdijk, I., Axiomatic homotopy theory for operads, Comment. Math. Helv., 78, 805-831 (2003) · Zbl 1041.18011 · doi:10.1007/s00014-003-0772-y
[4] Braconnier, J., Algèbres de Poisson, C. R. Acad. Sci. Paris Sér. A Math., 284, 1345-1348 (1977) · Zbl 0356.17007
[5] Brylinski, J.-L., A differential complex for Poisson manifolds, J. Diff. Geometry., 28, 93-114 (1988) · Zbl 0634.58029
[6] Burghelea, D.; Vigué-Poirrier, M., Cyclic homology of commutative algebras, 1318, 51-72 (1988) · Zbl 0666.13007
[7] Chen, K. T., Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math., 65, 163-178 (1957) · Zbl 0077.25301 · doi:10.2307/1969671
[8] Chen, K. T., Iterated path integral, Bull. Amer. Math. Soc., 83, 831-879 (1977) · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6
[9] Drinfeld, V. G., Quantum groups, Proc. Int. Congr. Math., 798-820 (1987) · Zbl 0667.16003
[10] Drinfeld, V. G., On some unsolved problems in quantum group theory, Quantum groups, 1510, 1-8 (1992) · Zbl 0765.17014
[11] Fox, T.; Markl, M., Distributive laws, bialgebras, and cohomology, Operads : Proceedings of renaissance conferences, 202, 167-205 (1997) · Zbl 0866.18008
[12] Fresse, B., Algèbre des descentes et cogroupes dans les algèbres sur une opérade, Bull. Soc. Math. Fr., 126, 407-433 (1998) · Zbl 0940.18004
[13] Fresse, B., Homologie de Quillen pour les algèbres de Poisson, C. R. Acad. Sci. Paris Sér. I Math., 326, 1053-1058 (1998) · Zbl 0922.17014 · doi:10.1016/S0764-4442(98)80061-X
[14] Fresse, B., Structures de Poisson sur une intersection complète à singularitées isolées, C. R. Acad. Sci. Paris Sér. I Math., 335, 5-10 (2002) · Zbl 1095.13525
[15] Fresse, B., Koszul duality of operads and homology of partition posets, Homotopy theory and its applications, 346, 115-215 (2004) · Zbl 1077.18007
[16] Gelfand, I. M.; Dorfman, I. Y., Hamiltonian operators and the classical Yang-Baxter equation, Funkts. Anal. Prilozh., 16, 1-9 (1982) · Zbl 0527.58018
[17] Gerstenhaber, M.; Schack, S. D., A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra, 48, 229-247 (1987) · Zbl 0671.13007 · doi:10.1016/0022-4049(87)90112-5
[18] Gerstenhaber, M.; Schack, S. D., The shuffle bialgebra and the cohomology of commutative algebras, J. Pure Appl. Algebra, 70, 263-272 (1991) · Zbl 0728.13003 · doi:10.1016/0022-4049(91)90073-B
[19] Getzler, E.; Jones, J., Operads, homotopy algebra and iterated integrals for double loop spaces (1994)
[20] Ginot, G., Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal, 11, 95-127 (2004) · Zbl 1139.16301 · doi:10.5802/ambp.187
[21] Ginzburg, V.; Kaledin, D., Poisson deformations of symplectic quotient singularities, Adv. Math., 186, 1-57 (2004) · Zbl 1062.53074 · doi:10.1016/j.aim.2003.07.006
[22] Ginzburg, V.; Kapranov, M., Koszul duality for operads, Duke Math. J., 76, 203-272 (1995) · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4
[23] Hain, R., On the indecomposable elements of the bar construction, Proc. Amer. Math. Soc., 98, 312-316 (1986) · Zbl 0613.55007 · doi:10.1090/S0002-9939-1986-0854039-5
[24] Huebschmann, J., Poisson cohomology and quantization, J. Reine Angew. Math., 408, 57-113 (1990) · Zbl 0699.53037 · doi:10.1515/crll.1990.408.57
[25] Kassel, C., L’homologie cyclique des algèbres enveloppantes, Invent. Math., 91, 221-251 (1988) · Zbl 0653.17007 · doi:10.1007/BF01389366
[26] Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216 (2003) · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf
[27] Koszul, J.-L., Crochet de Schouten-Nijenhuis et cohomologie, Elie Cartan et les mathématiques d’aujourd’hui, 257-271 (1985) · Zbl 0615.58029
[28] Krasilshschik, I. S., Hamiltonian cohomology of canonical algebras, Dokl. Akad. Nauk., 251, 1306-1309 (1980) · Zbl 0454.58020
[29] Lichnerowicz, A., Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geometry, 12, 253-300 (1977) · Zbl 0405.53024
[30] Livernet, M., Homotopie rationnelle des algèbres sur une opérade (1998)
[31] Loday, J.-L., Opérations sur l’homologie cyclique des algèbres commutatives, Invent. Math., 96, 205-230 (1989) · Zbl 0686.18006 · doi:10.1007/BF01393976
[32] Loday, J.-L., Cyclic homology, 301 (1992) · Zbl 0780.18009
[33] Loday, J.-L., Série de Hausdorff, idempotents Euleriens et algèbres de Hopf, Exposition. Math., 12, 165-178 (1994) · Zbl 0807.17003
[34] Mac Lane, S., Homology, 114 (1963) · Zbl 0133.26502
[35] Markl, M., Distributive laws and Koszulness, Ann. Inst. Fourier, 46, 307-323 (1996) · Zbl 0853.18005 · doi:10.5802/aif.1516
[36] Markl, M.; Shnider, S.; Stasheff, J., Operads in algebra, topology and physics, 96 (2002) · Zbl 1017.18001
[37] Oh, S.-Q., Poisson enveloping algebras, Comm. Algebra, 27, 2181-2186 (1999) · Zbl 0936.16020 · doi:10.1080/00927879908826556
[38] Patras, F., 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) · Zbl 0752.55014
[39] Patras, F., L’algèbre des descentes d’une bigèbre graduée, J. Algebra, 170, 547-566 (1994) · Zbl 0819.16033 · doi:10.1006/jabr.1994.1352
[40] Pirashvili, T., Hodge decomposition for higher Hochschild homology, Ann. Sci. École Norm. Sup., 33, 151-179 (2000) · Zbl 0957.18004
[41] Quillen, D., Homotopical algebra, 43 (1967) · Zbl 0168.20903
[42] Quillen, D., Rational homotopy theory, Ann. of Math., 90, 205-295 (1969) · Zbl 0191.53702 · doi:10.2307/1970725
[43] Quillen, D., On the (co)-homology of commutative rings, 17, 65-87 (1970) · Zbl 0234.18010
[44] Reutenauer, C., Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, Combinatoire énumérative, 1234, 267-284 (1986) · Zbl 0621.20004
[45] Reutenauer, C., Free Lie Algebras, 7 (1993) · Zbl 0798.17001
[46] Rinehart, G., Differential forms on general commutative algebras, Trans. Amer. Math. Soc., 108, 195-222 (1963) · Zbl 0113.26204 · doi:10.1090/S0002-9947-1963-0154906-3
[47] Schlessinger, M.; Stasheff, J., The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra, 38, 313-322 (1985) · Zbl 0576.17008 · doi:10.1016/0022-4049(85)90019-2
[48] Tamarkin, D., Another proof of M. Kontsevich formality theorem (1998)
[49] Vaisman, I., Lectures on the geometry of Poisson manifolds, 118 (1994) · Zbl 0810.53019
[50] Vigué-Poirrier, M., Cyclic homology of algebraic hypersurfaces, J. Pure Appl. Algebra, 72, 95-108 (1991) · Zbl 0732.16008 · doi:10.1016/0022-4049(91)90132-L
[51] Vigué-Poirrier, M., Décompositions de l’homologie cyclique des algèbres différentielles graduées commutatives, K-theory, 4, 255-267 (1991) · Zbl 0731.19004 · doi:10.1007/BF00533212
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