×

Homological unimodularity and Calabi-Yau condition for Poisson algebras. (English) Zbl 1383.16006

Summary: In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi-Yau algebra if the Poisson structure is unimodular.

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B35 Universal enveloping (super)algebras
17B63 Poisson algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Berger, R., Pichereau, A.: Calabi-Yau algebras viewed as deformations of Poisson algebras. Algebras Represent. Theory 17, 735-773 (2014) · Zbl 1339.16030 · doi:10.1007/s10468-013-9417-z
[2] Brylinski, J.L.: A differential complex for Poisson manifolds. J. Differ. Geom. 28, 93-114 (1988) · Zbl 0634.58029 · doi:10.4310/jdg/1214442161
[3] Brylinski, J.L., Zuckerman, G.: The outer derivation of a complex Poisson manifold. J. Reine Angew. Math. 506, 181-189 (1999) · Zbl 0919.58029
[4] Borel, A.: Algebraic \[D\] D-Modules. Academic Press, London (1987) · Zbl 0642.32001
[5] Brown, K.A., Zhang, J.J.: Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras. J. Algebra 320, 1814-1850 (2008) · Zbl 1159.16009 · doi:10.1016/j.jalgebra.2007.03.050
[6] Chemia, S.: Poincaré duality for \[k-A\] k-A-Lie superalgebras. Bull. Soc. Math. Fr. 122, 371-397 (1994) · Zbl 0840.16032 · doi:10.24033/bsmf.2238
[7] Chemia, S.: A duality property for complex Lie algebroids. Math. Z. 232, 367-388 (1999) · Zbl 0933.32015 · doi:10.1007/s002090050520
[8] Chemia, S.: Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators. J. Algebra 276, 80-102 (2004) · Zbl 1127.17012 · doi:10.1016/j.jalgebra.2003.12.001
[9] Dolgushev, V.A.: The Van den Bergh duality and the modular symmetry of a Poisson variety. Sel. Math. 14, 199-228 (2009) · Zbl 1172.53054 · doi:10.1007/s00029-008-0062-z
[10] Etingof, P., Ginzburg, V.: Noncommutative del Pezzo surfaces and Calabi-Yau algebras. J. Eur. Math. Soc. 12, 1371-1416 (2010) · Zbl 1204.14004 · doi:10.4171/JEMS/235
[11] Ginzburg, V.: Calabi-Yau Algebras (2006). Preprint arXiv:math/0612139 · Zbl 0634.58029
[12] Hartshorne, R.: Residues and duality. In: Hartshorne, R. (ed.) Lecture Notes in Math. vol. 20. Springer, Berlin (1966) · Zbl 0212.26101
[13] Huebschmann, J.: Poisson cohomology and quantization. J. Reine Angew. Math. 408, 57-113 (1990) · Zbl 0699.53037
[14] Huebschmann, J.: Duality for Lie-Rinehart algebras and the modular class. J. Reine Angew. Math. 510, 103-159 (1999) · Zbl 1034.53083
[15] Li, H.-S., van Oystaeyen, F.: Zariskian Filtrations, Kluwer Academic Publishers, K-Monographs in Mathematics, vol. 2. Springer Science+Business Media, B.V., Berlin (1996) · Zbl 0862.16027
[16] Launois, S., Richard, L.: Twisted Poincaré duality for some quadratic Poisson algebras. Lett. Math. Phys. 79, 161-174 (2007) · Zbl 1139.17008 · doi:10.1007/s11005-006-0133-z
[17] Lichnerowicz, A.: Les varieties de Poisson et leurs algebres de Lie associees (French). J. Differ. Geom. 12, 253-300 (1977) · Zbl 0405.53024 · doi:10.4310/jdg/1214433987
[18] Luo, J., Wang, S.-Q., Wu, Q.-S.: Twisted Poincaré duality between Poisson homology and Poisson cohomology. J. Algebra 442, 484-505 (2015) · Zbl 1392.17015 · doi:10.1016/j.jalgebra.2014.08.023
[19] Lü, J.-F., Wang, X.-T., Zhuang, G.-B.: Universal enveloping algebras of Poisson Hopf algebras. J. Algebra 426, 92-136 (2015) · Zbl 1393.17039 · doi:10.1016/j.jalgebra.2014.12.010
[20] Lü, J.-F., Wang, X.-T., Zhuang, G.-B.: Universal enveloping algebras of Poisson Ore-extensions. Proc. Am. Math. Soc. 143, 4633-4645 (2015) · Zbl 1378.16034 · doi:10.1090/S0002-9939-2015-12631-7
[21] Lü, J.-F., Wang, X.-T., Zhuang, G.-B.: DG Poisson algebra and its universal enveloping algebra. Sci. China Math. 59, 849-860 (2016) · Zbl 1404.16006 · doi:10.1007/s11425-016-5127-4
[22] Maszczyk, T.: Maximal Commutative Subalgebras, Poisson Geometry and Hochschild Homology (2006). arXiv: math.KT/0603386 · Zbl 0936.16020
[23] Marconnet, N.: Homologies of cubic Artin-Schelter regular algebras. J. Algebra 278, 638-665 (2004) · Zbl 1067.53064 · doi:10.1016/j.jalgebra.2003.11.019
[24] Oh, S.-Q.: Poisson enveloping algebras. Commun. Algebra 27, 2181-2186 (1999) · Zbl 0936.16020 · doi:10.1080/00927879908826556
[25] Rinehart, G.S.: Differential forms on general commutative algebras. Trans. Am. Math. Soc. 108, 195-222 (1963) · Zbl 0113.26204 · doi:10.1090/S0002-9947-1963-0154906-3
[26] Towers, M.: Poisson and Hochschild Cohomology and the Semiclassical Limit. arXiv:1304.6003 (2013) · Zbl 1353.16009
[27] Umirbaev, U.: Universal enveloping algebras and universal derivations of Poisson algebras. J. Algebra 354, 77-94 (2012) · Zbl 1270.17013 · doi:10.1016/j.jalgebra.2012.01.003
[28] Van den Bergh, M.: Noncommutative homology of some three-dimensional quantum spaces. K-Theory 8, 213-230 (1994) · Zbl 0814.16006 · doi:10.1007/BF00960862
[29] Van den Bergh, M.: Existence theorem for dualizing complexes over non-commutative graded and filtered rings. J. Algebra 195, 662-679 (1997) · Zbl 0894.16020 · doi:10.1006/jabr.1997.7052
[30] Van den Bergh, M.: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 126, 1345-1348 (1998) · Zbl 0894.16005 · doi:10.1090/S0002-9939-98-04210-5
[31] Van den Bergh, M.: Erratum to: a relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 130, 2809-2810 (2002) · doi:10.1090/S0002-9939-02-06684-4
[32] Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23, 379-394 (1997) · Zbl 0902.58013 · doi:10.1016/S0393-0440(97)80011-3
[33] Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200, 545-560 (1999) · Zbl 0941.17016 · doi:10.1007/s002200050540
[34] Yekutieli, A.: Dualizing complexes over non commutative graded algebras. J. Algebra 153, 41-84 (1992) · Zbl 0790.18005 · doi:10.1016/0021-8693(92)90148-F
[35] Zhu, C.: Twisted Poincaré duality for Poisson homology and cohomology of affine Poisson algebras. Proc. Am. Math. Soc. 143, 1957-1967 (2015) · Zbl 1392.17017 · doi:10.1090/S0002-9939-2014-12411-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.