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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”, , 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.
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