Poincaré duality for Hopf algebroids
DOI:
https://doi.org/10.33044/revuma.2832Abstract
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of van den Bergh. We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds by Chen et al.
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M. van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 no. 5 (1998), 1345–1348. DOI MR Zbl
G. Böhm, Hopf algebroids, in Handbook of algebra, vol. 6, Elsevier/North-Holland, Amsterdam, 2009, pp. 173–235. DOI MR Zbl
G. Böhm and K. Szlachányi, Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra 274 no. 2 (2004), 708–750. DOI MR Zbl
A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics 2, Academic Press, Boston, MA, 1987. MR Zbl
J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 no. 1 (1988), 93–114. MR Zbl Available at http://projecteuclid.org/euclid.jdg/1214442161.
S. Chemla, Poincaré duality for $k$-$A$ Lie superalgebras, Bull. Soc. Math. France 122 no. 3 (1994), 371–397. MR Zbl Available at http://www.numdam.org/item?id=BSMF_1994__122_3_371_0.
S. Chemla, A duality property for complex Lie algebroids, Math. Z. 232 no. 2 (1999), 367–388. DOI MR Zbl
S. Chemla, F. Gavarini, and N. Kowalzig, Duality features of left Hopf algebroids, Algebr. Represent. Theory 19 no. 4 (2016), 913–941. DOI MR Zbl
X. Chen, L. Liu, S. Yu, and J. Zeng, Batalin-Vilkovisky algebra structure on Poisson manifolds with diagonalizable modular symmetry, J. Geom. Phys. 189 (2023), Paper No. 104829, 22 pp. DOI MR Zbl
S. Evens, J.-H. Lu, and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 no. 200 (1999), 417–436. DOI MR Zbl
A. Fröhlich, The Picard group of noncommutative rings, in particular of orders, Trans. Amer. Math. Soc. 180 (1973), 1–45. DOI MR Zbl
J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. DOI MR Zbl
N. Jacobson, Basic algebra. II, W. H. Freeman, San Francisco, CA, 1980. MR Zbl
M. Kashiwara, $D$-modules and microlocal calculus, Translations of Mathematical Monographs 217, American Mathematical Society, Providence, RI, 2003. DOI MR Zbl
Y. Kosmann-Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: a survey, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 005. DOI MR Zbl
J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque no. S131 (1985), 257–271. MR Zbl Available at http://www.numdam.org/item/AST_1985__S131__257_0/.
N. Kowalzig, Hopf algebroids and their cyclic theory, Ph.D. thesis, Utrecht University, Utrecht, Netherlands, 2009. Available at https://dspace.library.uu.nl/handle/1874/34171.
N. Kowalzig and U. Krähmer, Duality and products in algebraic (co)homology theories, J. Algebra 323 no. 7 (2010), 2063–2081. DOI MR Zbl
N. Kowalzig and H. Posthuma, The cyclic theory of Hopf algebroids, J. Noncommut. Geom. 5 no. 3 (2011), 423–476. DOI MR Zbl
S. Launois and L. Richard, Twisted Poincaré duality for some quadratic Poisson algebras, Lett. Math. Phys. 79 no. 2 (2007), 161–174. DOI MR Zbl
J. Luo, S.-Q. Wang, and Q.-S. Wu, Twisted Poincaré duality between Poisson homology and Poisson cohomology, J. Algebra 442 (2015), 484–505. DOI MR Zbl
K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 no. 2 (1994), 415–452. DOI MR Zbl
G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195–222. DOI MR Zbl
P. Schauenburg, Duals and doubles of quantum groupoids ($×_R$-Hopf algebras), in New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 273–299. DOI MR Zbl
M. Takeuchi, Groups of algebras over A⊗Ā, J. Math. Soc. Japan 29 no. 3 (1977), 459–492. DOI MR Zbl
P. Xu, Quantum groupoids, Comm. Math. Phys. 216 no. 3 (2001), 539–581. DOI MR Zbl
A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 no. 1 (1992), 41–84. DOI MR Zbl
C. Zhu, Twisted Poincaré duality for Poisson homology and cohomology of affine Poisson algebras, Proc. Amer. Math. Soc. 143 no. 5 (2015), 1957–1967. DOI MR Zbl
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