Lie superalgebras of differential operators

We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a semi-direct product of the subgroup induced by the supermanifold diffeomorphisms and another subgroup which consists of automorphisms determined by even superdivergences. These superdivergences are proven to exist on any supermanifold and their local form is explicitly described as well