Classification of simple modules of the Zassenhaus superalgebras with p-characters of height one

Abstract

Let n be a positive integer, and $A\left(n\right)=F\left[x\right]/\left(x^{p^n}\right)$, $\bigwedge \left(1\right)=F\left[\xi \right]/\left({\xi }^2\right)$ be the divided power algebra and the Grassmann superalgebra of one variable, respectively over an algebraically closed field $F$ of prime characteristic $p>2$. The Zassenhaus superalgebra $Z\left(n\right)$ is by definition the Lie superalgebra of the special super derivations of the superalgebra $\Pi \left(n\right)=A\left(n\right){\otimes }_F\bigwedge \left(1\right)$. In this paper, we study simple modules of the Zassenhaus superalgebra $Z\left(n\right)$ with p-characters of height one. A complete classification of the isomorphism classes of such simple modules and their dimensions are precisely determined. Moreover, a sufficient and necessary condition for irreducibility of Kac modules is given.