Cohomologies, deformations and extensions of ResLieDer pairs

We study the ResLieDer pair in positive characteristic, that is, a restricted Lie algebra with a restricted derivation. We define the restricted representation of a ResLieDer pair and the corresponding first and second restricted cohomology complex. The complete cohomology complex of a ResLieDer pair in characteristic 2 is given. We show that a ResLieDer pair is rigid if the second restricted cohomology group is trivial, and a deformation of order n is extensible if and only if its obstruction element, which is defined to be a 3-cochain, is a restricted 3-coboundary. Moreover, we prove that the central extensions of a ResLieDer pair are classified by the second restricted cohomology group. Finally, the extensions of a pair of restricted derivations in characteristic 2 are studied. We show that a pair of restricted derivations is extensible if and only if its obstruction class is trivial.