Twisted Poincaré duality for Poisson homology and cohomology of affine Poisson algebras

Zhu, Can

doi:10.1090/S0002-9939-2014-12411-7

Twisted Poincaré duality for Poisson homology and cohomology of affine Poisson algebras
HTML articles powered by AMS MathViewer

by Can Zhu

Proc. Amer. Math. Soc. 143 (2015), 1957-1967

DOI: https://doi.org/10.1090/S0002-9939-2014-12411-7

Published electronically: December 19, 2014

PDF | Request permission

Abstract:

This paper investigates the Poisson (co)homology of affine Poisson algebras. It is shown that there is a twisted Poincaré duality between their Poisson homology and cohomology. The relation between the Poisson (co)homology of an affine Poisson algebra and the Hochschild (co)homology of its deformation quantization is also discussed, which is similar to Kassel’s result (1988) for homology and is a special case of Kontsevich’s theorem (2003) for cohomology.

References

Roland Berger and Anne Pichereau, Calabi-Yau algebras viewed as deformations of Poisson algebras, Algebr. Represent. Theory 17 (2014), no. 3, 735–773. MR 3254768, DOI 10.1007/s10468-013-9417-z
K. A. Brown and J. J. Zhang, Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras, J. Algebra 320 (2008), no. 5, 1814–1850. MR 2437632, DOI 10.1016/j.jalgebra.2007.03.050
Jean-Luc Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. MR 950556
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Vasiliy Dolgushev, The Van den Bergh duality and the modular symmetry of a Poisson variety, Selecta Math. (N.S.) 14 (2009), no. 2, 199–228. MR 2480714, DOI 10.1007/s00029-008-0062-z
Sam Evens, Jiang-Hua Lu, and Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436. MR 1726784, DOI 10.1093/qjmath/50.200.417
V. Ginzburg, Calabi-Yau algebras, arXiv: math. AG/0612139.
Ji-Wei He, Fred Van Oystaeyen, and Yinhuo Zhang, Cocommutative Calabi-Yau Hopf algebras and deformations, J. Algebra 324 (2010), no. 8, 1921–1939. MR 2678828, DOI 10.1016/j.jalgebra.2010.06.010
G. Hochschild, Bertram Kostant, and Alex Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408. MR 142598, DOI 10.1090/S0002-9947-1962-0142598-8
Johannes Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. MR 1058984, DOI 10.1515/crll.1990.408.57
Johannes Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103–159. MR 1696093, DOI 10.1515/crll.1999.043
Christian Kassel, L’homologie cyclique des algèbres enveloppantes, Invent. Math. 91 (1988), no. 2, 221–251 (French, with English summary). MR 922799, DOI 10.1007/BF01389366
Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MR 2062626, DOI 10.1023/B:MATH.0000027508.00421.bf
Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke, Poisson structures, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 347, Springer, Heidelberg, 2013. MR 2906391, DOI 10.1007/978-3-642-31090-4
Stéphane Launois and Lionel Richard, Twisted Poincaré duality for some quadratic Poisson algebras, Lett. Math. Phys. 79 (2007), no. 2, 161–174. MR 2301394, DOI 10.1007/s11005-006-0133-z
André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253–300 (French). MR 501133
Nicolas Marconnet, Homologies of cubic Artin-Schelter regular algebras, J. Algebra 278 (2004), no. 2, 638–665. MR 2071658, DOI 10.1016/j.jalgebra.2003.11.019
T. Maszczyk, Maximal commutative subalgebras, Poisson geometry and Hochschild homology, arXiv: math.KT/0603386.
C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974
Sei-Qwon Oh, Poisson enveloping algebras, Comm. Algebra 27 (1999), no. 5, 2181–2186. MR 1683858, DOI 10.1080/00927879908826556
R. Sridharan, Filtered algebras and representations of Lie algebras, Trans. Amer. Math. Soc. 100 (1961), 530–550. MR 130900, DOI 10.1090/S0002-9947-1961-0130900-1
Michel Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 213–230. MR 1291019, DOI 10.1007/BF00960862
Michel van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348. MR 1443171, DOI 10.1090/S0002-9939-98-04210-5
Ping Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), no. 3, 545–560. MR 1675117, DOI 10.1007/s002200050540
Amnon Yekutieli, The rigid dualizing complex of a universal enveloping algebra, J. Pure Appl. Algebra 150 (2000), no. 1, 85–93. MR 1762922, DOI 10.1016/S0022-4049(99)00032-8