An alternative approach to deformations of restricted Lie-Rinehart algebras

In this paper, we develop a new approach to the deformation theory of restricted Lie-Rinehart algebras in positive characteristic, based on the deformation theory of restricted morphisms introduced in our earlier work. We provide a full cohomology complex adapted to restricted Lie-Rinehart algebras in characteristic 2, define formal deformations, study obstructions and equivalence classes, and show that these are controlled by the cohomology we introduce. In characteristic $p\geq 3$, we construct 2-cocycles and investigate their relationship with formal deformations. Explicit computations are given in characteristic 2 to illustrate the theory.