Divergence operators and odd Poisson brackets

Yvette Kosmann-Schwarzbach[1]; Juan Monterde[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)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 2, page 419-456
  • ISSN: 0373-0956

Abstract

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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).

How to cite

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Kosmann-Schwarzbach, Yvette, and Monterde, Juan. "Divergence operators and odd Poisson brackets." Annales de l’institut Fourier 52.2 (2002): 419-456. <http://eudml.org/doc/115985>.

@article{Kosmann2002,
abstract = {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”, $\Delta $, 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).},
affiliation = {École Polytechnique, Centre de Mathématiques, Plateau de Palaiseau, 91128 Palaiseau Cedex (France); Universitat de València, Departamento de Geometria y Topologia, 46100 Burjasot (València) (Espagne)},
author = {Kosmann-Schwarzbach, Yvette, Monterde, Juan},
journal = {Annales de l’institut Fourier},
keywords = {graded Lie algebras; Gerstenhaber algebra; Batalin-Vilkovisky algebra; Schouten bracket; supermanifold; berezinian volume; graded connection; Maurer-Cartan equation; quantum master equation; supermanifold, Berezinian volume},
language = {eng},
number = {2},
pages = {419-456},
publisher = {Association des Annales de l'Institut Fourier},
title = {Divergence operators and odd Poisson brackets},
url = {http://eudml.org/doc/115985},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Kosmann-Schwarzbach, Yvette
AU - Monterde, Juan
TI - Divergence operators and odd Poisson brackets
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 2
SP - 419
EP - 456
AB - 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”, $\Delta $, 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).
LA - eng
KW - graded Lie algebras; Gerstenhaber algebra; Batalin-Vilkovisky algebra; Schouten bracket; supermanifold; berezinian volume; graded connection; Maurer-Cartan equation; quantum master equation; supermanifold, Berezinian volume
UR - http://eudml.org/doc/115985
ER -

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