Revista de la
Unión Matemática Argentina
Poincaré duality for Hopf algebroids
Sophie Chemla

Volume 67, no. 1 (2024), pp. 123–136    

Published online: April 5, 2024

https://doi.org/10.33044/revuma.2832

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Abstract

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.

References

  1. 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
  2. G. Böhm, Hopf algebroids, in Handbook of algebra, vol. 6, Elsevier/North-Holland, Amsterdam, 2009, pp. 173–235.  DOI  MR  Zbl
  3. 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
  4. 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
  5. 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.
  6. 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.
  7. S. Chemla, A duality property for complex Lie algebroids, Math. Z. 232 no. 2 (1999), 367–388.  DOI  MR  Zbl
  8. 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
  9. 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
  10. 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
  11. A. Fröhlich, The Picard group of noncommutative rings, in particular of orders, Trans. Amer. Math. Soc. 180 (1973), 1–45.  DOI  MR  Zbl
  12. J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113.  DOI  MR  Zbl
  13. N. Jacobson, Basic algebra. II, W. H. Freeman, San Francisco, CA, 1980.  MR  Zbl
  14. M. Kashiwara, $D$-modules and microlocal calculus, Translations of Mathematical Monographs 217, American Mathematical Society, Providence, RI, 2003.  DOI  MR  Zbl
  15. Y. Kosmann-Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: a survey, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 005.  DOI  MR  Zbl
  16. 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/.
  17. 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.
  18. N. Kowalzig and U. Krähmer, Duality and products in algebraic (co)homology theories, J. Algebra 323 no. 7 (2010), 2063–2081.  DOI  MR  Zbl
  19. N. Kowalzig and H. Posthuma, The cyclic theory of Hopf algebroids, J. Noncommut. Geom. 5 no. 3 (2011), 423–476.  DOI  MR  Zbl
  20. 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
  21. 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
  22. K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 no. 2 (1994), 415–452.  DOI  MR  Zbl
  23. G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195–222.  DOI  MR  Zbl
  24. P. Schauenburg, Duals and doubles of quantum groupoids ($\times_R$-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
  25. M. Takeuchi, Groups of algebras over $A\otimes \overline A$, J. Math. Soc. Japan 29 no. 3 (1977), 459–492.  DOI  MR  Zbl
  26. P. Xu, Quantum groupoids, Comm. Math. Phys. 216 no. 3 (2001), 539–581.  DOI  MR  Zbl
  27. A. Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 no. 1 (1992), 41–84.  DOI  MR  Zbl
  28. 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