Generalized restricted representations of the Zassenhaus superalgebras

Let F be an algebraically closed field of prime characteristic p>2, and n∈N⁺. Let Z(n) be the Zassenhaus superalgebra defined over F, which, as the simplest non-restricted simple Lie superalgebra, is the superversion of the Zassenhaus algebra. More precisely, Z(n) is the Lie superalgebra of the special super-derivations of the superalgebra Π(n). Here Π(n) is the tensor product of the divided power algebra of one variable and the Grassmann superalgebra of one variable. In this paper we study generalized restricted simple modules over the Zassenhaus superalgebra Z(n). Classification of isomorphism classes of generalized restricted simple modules and their dimensions are precisely determined. A sufficient and necessary condition for irreducibility of generalized restricted Kac modules is provided.