Operations for modules on Lie-Rinehart superalgebras

Abstract

Letk be a field of characteristic 0 and letA be a supercommutative associativek-superalgebra. LetL be ak−A-Lie-Rinehart superalgebra. From these data, one can construct a superalgebra of differential operatorsV(A,L) (generalizing the enveloping superalgebra of a Lie superalgebra). We will give a difinition of Lie-Rinehart superalgebra morphisms allowing to generalize the notions of inverse image and direct image. We will prove that a Lie-Rinehart superalgebra morphism decomposes into a closed imbedding and a projection. Furthermore, we will see that, under some technical conditions, a closed imbedding decomposes into two closed imbeddings of different nature. The first one looks like a Lie superalgebra morphism. The second one looks like a supermanifold closed imbedding and satisfies a generalization of the Kashiwara’s theorem. Then, as in theD-module theory, we introduce a duality functor. Finally, we will prove that, in the closed imbedding case, the direct image and the duality functor commute.