On the Cohomological Impact of a Quasi-Poisson Structure for Some Poisson Cohomology Spaces

Bruno Iskamlé^1,

¹Department of Mathematics and Computer Science, Faculty of Sciences, University of Maroua, Cameroon

Cite this paper:
Bruno Iskamlé. On the Cohomological Impact of a Quasi-Poisson Structure for Some Poisson Cohomology Spaces. Journal of Mathematical Sciences and Applications. 2022; 9(1):1-6. doi: 10.12691/jmsa-9-1-1

Abstract

In order to construct a Poisson cohomology complex in the quasi-Poisson context, we establish an isomorphism between interesting Poisson cohomology groups for a quasi-Poisson algebra and Poisson cohomology groups for Poisson algebra coming from the Jacobiator of the quasi-Poisson algebra.

Keywords:
(Quasi-) Poisson algebra Jacobiator coboundary Poisson cohomology

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	S. Lie, Math. Ann. 8, 214-303, (1874/75).
[2]	A. M. Vinogradov, I.S. Krasil’shchik. What is Hamiltonian formalism?, (Russian), Uspehi Mat. Nauk, vol.30, no.1, 1975. 173-198.
[3]	J. Braconnier. Algèbres de Poisson, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 21, A1345-A1348.
[4]	A. Lichnerowicz. Les variétés de Poisson et leurs algèbres de Lie associées, (French), J. Di_. Geom, vol.12, (1977); 253-300.
[5]	I. Krasil’shchik. Hamiltonian cohomology of canonical algebras, Dokl. Akad. Nauk SSSR 251 (1980), no.6, 1306-1309.
[6]	A. Weinstein, Poisson geometry, Differential Geometry and its Applications 9 (1998) 213-238.
[7]	J. Block and E. Getzler. Quantization of foliations, Proceedings of XXth International Conference on Differential Geometry Methods in Theoretical Physics, New York, 1991; Vol. 1, 2 (World Scientific, River Edge, NJ, 1992) 471-487.
[8]	D.R. Farkas and G. Letzter. Ring theory from symplectic geometry, J. Pure Appl. Alg. 125 (1998) 155-190.
[9]	F.F. Voronov, On the Poisson envelope of a Lie algebra. ”Noncommutative” moment space, Funct. Anal. Appl. 29 (1995) 196-199.
[10]	P. Xu. Noncommutative Poisson algebras, Amer. J. Math. 116 (1994) 101-125.
[11]	A. Alekseev and Y. Kosmann-Schwarzbach, Manin pairs and moment maps, J. Differential Geometry, 56 (2000) 133-165.
[12]	R. Aminou, Y. Kosmann-Schwarzbach, and E. Meinrenken, Quasi-Poisson manifolds, Canad. J. Math., 54(1):3-29, 2002.
[13]	J. Huebschmann. Poisson structures on certain moduli spaces for bundles on a surface. Ann. Inst. Fourier (Grenoble), 45(1): 65-91, 1995.
[14]	W. Goldman. Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math., 85(2): 263-302, 1986.
[15]	W. Goldman. The symplectic nature of fundamental groups of surfaces. Adv. in Math., 54(2): 200-225, 1984.
[16]	L. Je_rey and J.Weitsman. Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Comm. Math. Phys., 150(3):593-630, 1992.
[17]	I. Vaisman, On the geometric quantization of Poisson manifolds, J. Math. Phys 32(1991), 3339-3345.
[18]	A. Pichereau, Poisson (co)homology and isolated singularities, J. Algebra 299, 2 (2006), 747-777.
[19]	P. Monnier, Poisson cohomology in dimension two, Israel J. Math.,129 (2002), 189-207.
[20]	C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson structures, Grundlerhen der Mathematischen Wissenschaften, 347, Springer, 2013.
[21]	I. Vaismann. Lectures on the Geometry of Poisson manifolds, Birkhauser, Basel, 1994.
[22]	P. Vanhaecke. Integrable systems in the real of algebraic geometry, Vol. 1638 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Second edition, 2001.
[23]	W. Oevel and O. Ragnisco, R-matrices and higher Poisson brackets for integrable systems. Phy. A. 161 (1): 181-220, 1989.
[24]	S. Parmentier. On coproducts of quasi-triangular Hopf algebras, Algebra i Analiz, 6 (4): 204-222, 1994.
[25]	L. C. Li and S. Parmentier. Nonlinear Poisson structures and r-matrices. Comm. Math. Phys., 125 (4): 545-563, 1989.
[26]	W. S. Massey. Algebraic topology: an introduction, Springer-Verlag, New York; 1977. Reprint of the 1967 edition, Graduate Texts in Mathematics, Vol. 56.
[27]	M. Pedroni and P. Vanhaecke. A Lie algebraic generalization of the Mumford system, its symmetries and is multi-Hamiltonian structure. Regul. Chaotic Dyn., 3 (3): 132-160, 1998. J. Moser at 70 (Russian).