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 of automorphisms induced by the supermanifold diffeomorphisms and another subgroup which consists of automorphisms determined by even superdivergences. We prove the existence of such superdivergences on any supermanifold and we describe their local form.