Skip to main content Accessibility help

On (co)homology of Frobenius Poisson algebras

  • Can Zhu (a1) (a2), Fred Van Oystaeyen (a3) and Yinhuo Zhang (a4)


In this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.



Hide All
Bry88.Brylinski, J. L., A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), 93114.
CE56.Cartan, H., Eilenberg, S., Homological algebra, Princeton university Press, Princeton, 1956.
Do109.Dolgushev, V. A., The Van den Bergh duality and the modular symmetry of a Poisson variety, Selecta Math. 14 (2009), 199228.
Eu10.Eu, C., The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers, J. Pure Applied Algebra 214 (2010), 2846.
ES09.Eu, C., Schedler, T., Calabi-Yau Frobenius algebras, J. Algebra 321 (2009), 774815.
Ger63.Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of Math. 78 (1963), 267288.
Gin06.Ginzburg, V., Calabi-Yau algebras, arXiv: math. AG/0612139.
HKR62.Hochschild, G., Kostant, B., Rosenberg, A., Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383408.
Hue90.Huebschmann, J., Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57113.
Kas88.Kassel, C., L'homologie cyclique des algèbres enveloppantes, (French) [The cyclic homology of enveloping algebras], Invent. Math. 91 (1988), 221251.
Kon03.Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157216.
LR07.Launois, S., Richard, L., Twisted Poincaré duality for some quadratic Poisson algebras, Lett. Math. Phys. 79 (2007), 161174.
LR09.Launois, S., Richard, L., Poisson (co)homology of truncated polynomial algebras in two variables, C. R. Math. Acad. Sci. Paris 347 (2009), 133138.
LPV13.Laurent-Gengoux, C., Pichereau, A. and Vanhaecke, P., Poisson structures, Springer, Heidelberg, 2013.
Lic77.Lichnerowicz, A., Les variétés de Poisson et leurs algèbres de Lie associées, (French), J. Differential Geometry 12 (1977), 253300.
Men04.Menichi, L., Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, K-Theory 32 (2004), 231251.
Oh99.Oh, S. Q., Poisson enveloping algebras, Comm. Algebra 27 (1999), 21812186.
Sm96.Smith, S. P., Some finite dimensional algebras related to elliptic curves, CMS Conf. Proc. 19 (1996), 315348.
Tra08.Tradler, T., The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier 58 (2008), 23512379.
Xu99.Xu, P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Commun. Math. Phys. 200 (1999), 545560.


Related content

Powered by UNSILO

On (co)homology of Frobenius Poisson algebras

  • Can Zhu (a1) (a2), Fred Van Oystaeyen (a3) and Yinhuo Zhang (a4)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.