Generalized Restricted Lie Algebras and Representations of the Zassenhaus Algebra

LetFbe a field of characteristicp0,La generalized restricted Lie algebra overF, andP(L) the primitivep-envelope ofL. A close relation betweenL-representations andP(L)-representations is established. In particular, the irreducible κ-reduced modules ofLfor any κ∈L* coincide with the irreducibleκ̄₀-reduced modules ofP(L), whereκ̄₀∈P(L)* is a trivial extension of κ. From this result, the determination of all irreducible representations of the Zassenhaus algebra is completed, and the dimensions of the corresponding modules are also given.