Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 687-709, 2018
Remarks on local Lie algebras of pairs of functions
Josef Janyška
Received December 7, 2016. First published December 6, 2017.
Abstract: Starting by the famous paper by Kirillov, local Lie algebras of functions over smooth manifolds were studied very intensively by mathematicians and physicists. In the present paper we study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.
Keywords: almost-cosymplectic-contact structure; almost-coPoisson-Jacobi structure; infinitesimal symmetry; local Lie algebra
References: [1] M. de León, G. M. Tuynman: A universal model for cosymplectic manifolds. J. Geom. Phys. 20 (1996), 77-86. DOI 10.1016/0393-0440(96)00047-2 | MR 1407405 | Zbl 0861.53026
[2] J. Janyška: Special phase functions and phase infinitesimal symmetries in classical general relativity. AIP Conf. Proc. 1460, XX Internat. Fall Workshop on Geometry and Physics (2012), 135-140. DOI 10.1063/1.4733369
[3] J. Janyška: Hidden symmetries of the gravitational contact structure of the classical phase space of general relativistic test particle. Arch. Math., Brno 50 (2014), 297-316. DOI 10.5817/AM2014-5-297 | MR 3303779 | Zbl 1340.70017
[4] J. Janyška: Relations between constants of motion and conserved functions. Arch. Math., Brno 51 (2015), 297-313. DOI 10.5817/AM2015-5-297 | MR 3449110 | Zbl 06537732
[5] J. Janyška: On Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures. Arch. Math., Brno 52 (2016), 325-339. DOI 10.5817/AM2016-5-325 | MR 3610867 | Zbl 06674908
[6] J. Janyška, M. Modugno: Graded Lie algebra of Hermitian tangent valued forms. J. Math. Pures Appl. (9) 85 (2006), 687-697. DOI 10.1016/j.matpur.2005.11.004 | MR 2229665 | Zbl 1113.53020
[7] J. Janyška, M. Modugno: Geometric structures of the classical general relativistic phase space. Int. J. Geom. Methods Mod. Phys. 5 (2008), 699-754. DOI 10.1142/S021988780800303X | MR 2445392 | Zbl 1116.53008
[8] J. Janyška, M. Modugno: Generalized geometrical structures of odd dimensional manifolds. J. Math. Pures Appl. (9) 91 (2009), 211-232. DOI 10.1016/j.matpur.2008.09.007 | MR 2498755 | Zbl 1163.53051
[9] J. Janyška, R. Vitolo: On the characterization of infinitesimal symmetries of the relativistic phase space. J. Phys. A, Math. Theor. 45 (2012), Article ID 485205, 28 pages. DOI 10.1088/1751-8113/45/48/485205 | MR 2998421 | Zbl 1339.70036
[10] A. A. Kirillov: Local Lie algebras. Russ. Math. Surv. 31 (1976), 55-75. DOI 10.1070/RM1976v031n04ABEH001556 | MR 0438390 | Zbl 0357.58003
[11] A. Lichnerowicz: Les variétés de Jacobi et leurs algèbres de Lie associées. J. Math. Pures Appl., IX. Sér. 57 (1978), 453-488. (In French.) MR 0524629 | Zbl 0407.53025
[12] K. C. H. Mackenzie: General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society, Lecture Note Series 213, Cambridge University Press, Cambridge (2005). DOI 10.1017/CBO9781107325883 | MR 2157566 | Zbl 1078.58011
[13] P. J. Olver: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics 107, Springer, New York (1986). DOI 10.1007/978-1-4684-0274-2 | MR 0836734 | Zbl 0588.22001
[14] K. Shiga: Cohomology of Lie algebras over a manifold I. J. Math. Soc. Japan 26 (1974), 324-361. DOI 10.2969/jmsj/02620324 | MR 0368025 | Zbl 0273.58002
[15] P. Sommers: On Killing tensors and constants of motion. J. Mathematical Phys. 14 (1973), 787-790. DOI 10.1063/1.1666395 | MR 0329558
[16] I. Vaisman: Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics 118, Birkhäuser, Basel (1994). DOI 10.1007/978-3-0348-8495-2 | MR 1269545 | Zbl 0810.53019
Affiliations: Josef Janyška Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic, e-mail: janyska@math.muni.cz
