## Abstract

We detail the construction of a weak Poisson bracket over a submanifold Σ of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle ΠTM, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold Σ may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.

Original language | English |
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Pages (from-to) | 330-344 |

Number of pages | 15 |

Journal | Journal of Geometry and Physics |

Volume | 116 |

Early online date | 22 Feb 2017 |

DOIs | |

Publication status | Published - 1 Jun 2017 |

## Keywords

- BRST theory
- Gauge system
- Homotopy Poisson algebra
- Master equation
- Weak Poisson and Koszul brackets