On Irreducible Representations of the Zassenhaus Superalgebras with p-Characters of Height 0

Let n be a positive integer, and \(\mathfrak {A}(n)=\mathbb {F}x]/(x^{p^{n}})\), the divided power algebra over an algebraically closed field \(\mathbb {F}\) of prime characteristic p >?2. Let π(n) be the tensor product of \(\mathfrak {A}(n)\) and the Grassmann superalgebra \(\bigwedge (1)\) in one variable. The Zassenhaus superalgebra \(\mathcal {Z}(n)\) is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra \(\mathcal {Z}(n)\) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained.