This thesis is associated to the area of mathematics dealing with representation theory of Lie algebras in prime characteristic. This theory is quite different from the corresponding theory in characteristic 0. For example, in prime characteristic all irreducible modules are finite dimensional. On the other hand, there is in most cases no classification of these irreducible modules.
The thesis is concerned with the representation theory of Lie algebras of Cartan type and, in particular, the next-smallest Witt-Jacobson Lie algebra $W(2)$ of rank 2. Work by Holmes on irreducible modules with small "height" is included (the height is a certain invariant attached to irreducible modules for Lie algebras of this type).
The main part of the thesis deals with the remaining irreducible modules (of height at least 2). An approach to this problem is to induce irreducible modules for smaller subalgebras to $W(2)$ and check whether the induced module is irreducible. General criteria for irreducibility are given and applied to the case of our consideration.
In order to classify the irreducible modules, a thorough understanding is needed of the representation theory of supersolvable Lie algebras. The outcome of this study is used to find a dimension formula and a formula for the number of irreducibles of a certain type. For the remaining irreducibles we give examples that illustrate a quite different and more complex situation.