Divergence operators and odd Poisson brackets
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 419-456.

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, Δ, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).

On définit les opérateurs de divergence sur les algèbres graduées et l’on montre qu’étant donné un crochet de Poisson impair sur l’algèbre, l’opérateur qui associe à un élément la divergence de la dérivation hamiltonienne qu’il définit est un générateur du crochet. C’est le “laplacien impair”, Δ de la quantification de Batalin-Vilkovisky. On étudie alors les générateurs des crochets de Poisson impairs sur les supervariétés, où l’opérateur de divergence peut être défini soit à l’aide d’un volume bérézinien, soit à l’aide d’une connexion graduée. Comme exemples, on trouve des générateurs du crochet de Schouten des multivecteurs sur une variété (la supervariété étant le fibré cotangent, où les coordonnées sur les fibres sont impaires), et ceux du crochet de Koszul-Schouten des formes différentielles sur une variété de Poisson (la supervariété étant le fibré tangent, avec des coordonnées impaires sur les fibres).

DOI: 10.5802/aif.1892
Classification: 17B70,  17B63,  58A50,  81S10,  53D17
Keywords: graded Lie algebras, Gerstenhaber algebra, Batalin-Vilkovisky algebra, Schouten bracket, supermanifold, berezinian volume, graded connection, Maurer-Cartan equation, quantum master equation
Kosmann-Schwarzbach, Yvette 1; Monterde, Juan 2

1 École Polytechnique, Centre de Mathématiques, Plateau de Palaiseau, 91128 Palaiseau Cedex (France)
2 Universitat de València, Departamento de Geometria y Topologia, 46100 Burjasot (València) (Espagne)
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Kosmann-Schwarzbach, Yvette; Monterde, Juan. Divergence operators and odd Poisson brackets. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 419-456. doi : 10.5802/aif.1892. http://www.numdam.org/articles/10.5802/aif.1892/

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