Graded Modules Of The Finite-dimensional Cartan Type Modular Lie Superalgebras

In this paper, we consider the graded modules of the finite-dimensional Cartan type modular Lie superalgebras. Firstly, we introduce the development background and basic knowledges of Lie superalgebras and Verma modules, which will be used in this article. The definitions of eight classes of finite-dimensional Cartan type modular Lie superalge-bras are described. Secondly, the relationship between generalized reduced Verma module and coinduced module are summarized. Thus we can prove that the generalized reduced Verma module is isomorphic to the graded modules for modular Lie superalgebras of Cartan type, which are constructed by mixed products. Finally, the graded module of the finite-dimensional Lie superalgebra SHO is studied. By using the method of mixed product, we show that if V is an L-module, then the mixed product (?) is the (?)-graded (?)-module and SHO-module.