  The main object of study of this paper is the notion of a LieDer pair, i.e. a
Lie algebra with a derivation. We introduce the concept of a representation of
a LieDer pair and study the corresponding cohomologies. We show that a LieDer
pair is rigid if the second cohomology group is trivial, and a deformation of
order n is extensible if its obstruction class, which is defined to be an
element is the third cohomology group, is trivial. We classify central
extensions of LieDer pairs using the second cohomology group with the
coefficient in the trivial representation. For a pair of derivations, we define
its obstruction class and show that it is extensible if and only if the
obstruction class is trivial. Finally, we classify Lie2Der pairs using the
third cohomology group of a LieDer pair.
