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2000 | 51 | 1 | 109-129
Tytuł artykułu

Modular vector fields and Batalin-Vilkovisky algebras

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We show that a modular class arises from the existence of two generating operators for a Batalin-Vilkovisky algebra. In particular, for every triangular Lie bialgebroid (A,P) such that its top exterior power is a trivial line bundle, there is a section of the vector bundle A whose $d_{P}$-cohomology class is well-defined. We give simple proofs of its properties. The modular class of an orientable Poisson manifold is an example. We analyse the relationships between generating operators of the Gerstenhaber algebra of a Lie algebroid, right actions on the elements of degree 0, and left actions on the elements of top degree. We show that the modular class of a triangular Lie bialgebroid coincides with the characteristic class of a Lie algebroid with representation on a line bundle.
Słowa kluczowe
Rocznik
Tom
51
Numer
1
Strony
109-129
Opis fizyczny
Daty
wydano
2000
Twórcy
  • U.M.R. 7640 du C.N.R.S., Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau, France
Bibliografia
  • [1] K. H. Bhaskara and K. Viswanath, Poisson Algebras and Poisson Manifolds, Pitman Res. Notes in Math. 174, Longman, Harlow, 1988.
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  • [4] P. Cartier, Some fundamental techniques in the theory of integrable systems, in: Lectures on Integrable Systems, O. Babelon, P. Cartier and Y. Kosmann-Schwarzbach, eds., World Scientific, Singapore, 1994, 1-41.
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  • [6] S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quarterly J. Math., to appear.
  • [7] M. Gerstenhaber and S. D. Schack, Algebras, bialgebras, quantum groups, and algebraic deformations, Contemp. Math. 134 (1992), M. Gerstenhaber and J. D. Stasheff, eds., 51-92.
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  • [10] J. Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. reine angew. Math. 510 (1999), 103-159.
  • [11] J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann. Inst. Fourier 48 (1998), 425-440.
  • [12] J. Huebschmann, Differential Batalin-Vilkovisky algebras arrising from twilled Lie-Rinehart algebras, this volume.
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  • [14] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré A53 (1990), 35-81, and Dualization and deformation of Lie brackets on Poisson manifolds, in: Differential Geometry and its Applications, J. Janyška and D. Krupka, eds., World Scientific, Singapore, 1990, 79-84.
  • [15] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in: Astérisque, hors série, Elie Cartan et les mathématiques d'aujourd'hui, Soc. Math. Fr., 1985, 257-271.
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  • [17] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge Univ. Press, Cambridge, 1987.
  • [18] K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452.
  • [19] Yu. I. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, Colloq. Publ. 47, Amer. Math. Soc., Providence, RI, 1999.
  • [20] C.-M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys. 23 (1997), 350-359.
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  • [26] P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560.
Typ dokumentu
Bibliografia
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