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).Comment: 27 pages; new Section 1, introduction and conclusion re-written,
typos correcte