JP Journal of Geometry and Topology

The JP Journal of Geometry and Topology publishes articles in all branches of geometry and topology, with applications to physics. It covers areas such as differential geometry, algebraic topology, and geometric aspects of mathematical physics. Survey articles are also welcome.

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JACOBI STRUCTURES ON A LIE-RINEHART ALGEBRA

Authors

  • N. Mahoungou Moukala
  • M. M. Milounguidi
  • B. G. R. Bossoto

Keywords:

first order differential operator, Jacobi manifold, Schouten-Nijenhuis bracket, Lie-Rinehart algebra.

DOI:

https://doi.org/10.17654/0972415X22009

Abstract

We define the Schouten-Nijenhuis bracket on the algebra of module of Kähler differentials and provide the main features of Jacobi manifolds by using the universal property of first order differential operators. Further, we establish the equivalence between a Lie-Rinehart algebra structure and a Jacobi structure and recover Lichnerowicz’s notion of Jacobi pair on a smooth manifold.

Received: May 2, 2022 
Accepted: June 21, 2022

References

N. Bourbaki, Algèbre, Chapitre 10, Algèbre homologique, Masson, Paris, New York, Barcelona, Milan, 1980.

L. Camille, A. P. Gengoux and P. Vanhaecke, Poisson structures, Grundlehren der Mathematischen Wissenschaften, Springer, Vol. 347, 2013.

A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976), 55-75.

J.-L. Koszul, Chochet de Schouten-Nijenhuis et cohomologie, Astérisques, Numéro Hors Série, 1985, pp. 257-271.

P. Libermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Kluwer, Dordrecht, 1987.

A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9) 57 (1978), 453-488.

C.-M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys. 23(3-4) (1997), 350-359.

E. Okassa, Algèbres de Jacobi et algèbres de Lie-Rinehart-Jacobi, J. Pure Appl. Algebra 208(3) (2007), 1071-1089.

G. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195-222.

I. Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Math. 118, Birkhäuser, Basel, 1994.

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Published

2022-09-30

Issue

Vol. 28 (2022)

Section

Articles

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How to Cite

JACOBI STRUCTURES ON A LIE-RINEHART ALGEBRA. (2022). JP Journal of Geometry and Topology, 28, 55-79. https://doi.org/10.17654/0972415X22009

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