Odd and even Poisson brackets 1 Y. Kosmann-Schwarzbach

On a Poisson manifold, the divergence of a hamiltonian vector field is a derivation of the algebra of functions, called the modular vector field. In the case of an odd Poisson bracket on a supermanifold, the divergence of a hamilto-nian vector field is a generator of the odd bracket, and it is nontrivial, whether the Poisson structure is symplectic or not. The quantum master equation is a Maurer-Cartan equation written in terms of an odd Poisson bracket (an-tibracket) and of a generator of square 0 of this bracket. The derived bracket of the antibracket with respect to the quantum BRST differential defines the even graded Lie algebra structure of the space of quantum gauge parameters.