On classification and deformations of Lie-Rinehart superalgebras

The purpose of this paper is to study Lie-Rinehart superalgebras over the complex field $\mathbb{C}$. We provide a classification in small dimension and a deformation theory. A Lie-Rinehart superalgebra is a pair $(A,L)$ made of an associative superalgebra $A$ and a Lie superalgebra $L$, which are compatible in a certain way. We review their properties and provide some key examples. We give all possible pairs defining a Lie-Rinehart superalgebra for $\dim(A)\leq 2$ and $\dim(L)\leq 4$. Moreover, we develop a theory of formal deformations based on formal power series and a suitable cohomology.