Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COHOMOLOGY RING OF THE TENSOR PRODUCT OF POISSON ALGEBRAS

  • Zhu, Can (College of Science University of Shanghai for Science and Technology)
  • Received : 2018.11.08
  • Accepted : 2019.07.25
  • Published : 2019.12.30
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Abstract

In this paper, we study the Poisson cohomology ring of the tensor product of Poisson algebras. Explicitly, it is proved that the Poisson cohomology ring of tensor product of two Poisson algebras is isomorphic to the tensor product of the respective Poisson cohomology ring of these two Poisson algebras as Gerstenhaber algebras.

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References

  1. V. P. Cherkashin, Tensor products of Poisson algebras, Math. Notes 54 (1993), no. 3-4, 968-974 (1994); translated from Mat. Zametki 54 (1993), no. 3, 141-151, 160. https://doi.org/10.1007/BF01209563
  2. P. Deligne and J. S. Milne, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, 900, Springer-Verlag, Berlin, 1982.
  3. V. G. Drinfel'd, Quantum groups, J. Soviet Math. 41 (1988), no. 2, 898-915; translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), Differentsial'naya Geometriya, Gruppy Li i Mekh. VIII, 18-49, 193. https://doi.org/ 10.1007/BF01247086
  4. C.-H. Eu and T. Schedler, Calabi-Yau Frobenius algebras, J. Algebra 321 (2009), no. 3, 774-815. https://doi.org/10.1016/j.jalgebra.2008.11.003
  5. M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103. https://doi.org/10.2307/1970484
  6. V. Ginzburg, Calabi-Yau algebras, arXiv: 0612139 [math. AG].
  7. J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57-113. https://doi.org/10.1515/crll.1990.408.57
  8. N. Kowalzig and U. Krahmer, Batalin-Vilkovisky structures on Ext and Tor, J. Reine Angew. Math. 697 (2014), 159-219.
  9. T. Lambre, G. Zhou, and A. Zimmermann, The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, J. Algebra 446 (2016), 103-131. https://doi.org/10.1016/j.jalgebra.2015.09.018
  10. C. Laurent-Gengoux, A. Pichereau, and P. Vanhaecke, Poisson structures, Grundlehren der Mathematischen Wissenschaften, 347, Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-31090-4
  11. J. Le and G. Zhou, On the Hochschild cohomology ring of tensor products of algebras, J. Pure Appl. Algebra 218 (2014), no. 8, 1463-1477. https://doi.org/10.1016/j.jpaa.2013.11.029
  12. A. Lichnerowicz, Les varietes de Poisson et leurs algebres de Lie associees, J. Differential Geometry 12 (1977), no. 2, 253-300. http://projecteuclid.org/euclid.jdg/1214433987 https://doi.org/10.4310/jdg/1214433987
  13. L. Menichi, Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, K-Theory 32 (2004), no. 3, 231-251. https://doi.org/10.1007/s10977-004-0480-4
  14. T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2351-2379. https://doi.org/10.5802/aif.2417
  15. P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), no. 3, 545-560. https://doi.org/10.1007/s002200050540
  16. C. Zhu, F. Van Oystaeyen, and Y. Zhang, On (co)homology of Frobenius Poisson algebras, J. K-Theory 14 (2014), no. 2, 371-386. https://doi.org/10.1017/is014007026jkt276