The BV-algebra of a Jacobi manifold

Izu Vaisman

Annales Polonici Mathematici (2000)

  • Volume: 73, Issue: 3, page 275-290
  • ISSN: 0066-2216

Abstract

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We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give new examples of Lie bialgebroids over Poisson manifolds.

How to cite

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Vaisman, Izu. "The BV-algebra of a Jacobi manifold." Annales Polonici Mathematici 73.3 (2000): 275-290. <http://eudml.org/doc/262598>.

@article{Vaisman2000,
abstract = {We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give new examples of Lie bialgebroids over Poisson manifolds.},
author = {Vaisman, Izu},
journal = {Annales Polonici Mathematici},
keywords = {Jacobi manifolds; BV-algebras; Lie bialgebroids; Gerstenhaber algebras; Gerstenhaber algebra; Lie algebroid; Jacobi manifold; BV-algebra; Jacobi homology; Jacobi cohomology; transitive Jacobi manifolds; Lie bialgebroid},
language = {eng},
number = {3},
pages = {275-290},
title = {The BV-algebra of a Jacobi manifold},
url = {http://eudml.org/doc/262598},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Vaisman, Izu
TI - The BV-algebra of a Jacobi manifold
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 3
SP - 275
EP - 290
AB - We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give new examples of Lie bialgebroids over Poisson manifolds.
LA - eng
KW - Jacobi manifolds; BV-algebras; Lie bialgebroids; Gerstenhaber algebras; Gerstenhaber algebra; Lie algebroid; Jacobi manifold; BV-algebra; Jacobi homology; Jacobi cohomology; transitive Jacobi manifolds; Lie bialgebroid
UR - http://eudml.org/doc/262598
ER -

References

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  1. [1] D. Chinea, M. de León and J. C. Marrero, The canonical double complex for Jacobi manifolds, C. R. Acad. Sci. Paris Sér. I 323 (1996), 637-642. Zbl0862.55016
  2. [2] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, New York, 1993. Zbl0717.58026
  3. [3] S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for the Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), 417-436. Zbl0968.58014
  4. [4] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), 265-285. Zbl0807.17026
  5. [5] F. Guédira et A. Lichnerowicz, Géométrie des algèbres de Lie de Kirillov, J. Math. Pures Appl. 63 (1984), 407-484. Zbl0562.53029
  6. [6] J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vil- kovisky algebras, Ann. Inst. Fourier (Grenoble) 48 (1998), 425-440. Zbl0973.17027
  7. [7] Y. Kerbrat et Z. Souici-Benhammadi, Variétés de Jacobi et groupoï des de contact, C. R. Acad. Sci. Paris Sér. I 317 (1993), 81-86. 
  8. [8] Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153-165. Zbl0837.17014
  9. [9] J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in: É. Cartan et les mathématiques d'aujourd'hui, Astérisque, hors série, 1985, 257-271. 
  10. [10] M. de León, J. C. Marrero and E. Padrón, On the geometric quantization of Jacobi manifolds, J. Math. Phys. 38 (1997), 6185-6213. Zbl0898.58024
  11. [11] M. de León, J. C. Marrero and E. Padrón, Cohomologí a y Homologí a Canónica de Lichnerowicz-Jacobi, preprint, 1998. 
  12. [12] P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mecha- nics, D. Reidel, Dordrecht, 1987. 
  13. [13] A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. 57 (1978), 453-488. Zbl0407.53025
  14. [14] A. Lichnerowicz, La géométrie des transformations canoniques, Bull. Soc. Math. Belg. Sér. A 31 (1979), 105-135. Zbl0462.58018
  15. [15] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge Univ. Press, Cambridge, 1987. Zbl0683.53029
  16. [16] K. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452. Zbl0844.22005
  17. [17] I. Vaisman, Remarks on the use of the stable tangent bundle in differential geometry and in the unified field theory, Ann. Inst. H. Poincaré Phys. Théor. 28 (1978), 317-333. Zbl0377.53031
  18. [18] I. Vaisman, Locally conformal symplectic manifolds, Internat. J. Math. Math. Sci. 8 (1985), 521-536. Zbl0585.53030
  19. [19] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkhäuser, Basel, 1994. 
  20. [20] A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379-394. Zbl0902.58013
  21. [21] P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560. Zbl0941.17016

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