Shen-Larsson’s Modules For Simple Generalized Witt Algebras

The extension of Shen-Larsson’s construction was first introduced by Kaplansky in the context of the classification problem of simple finite-dimensional Lie algebras over fields of prime characteristic.The problem studied in the paper is to extend the ShenLarsson’s construction of classical Witt algebras to simple generalized Witt algebras and construct a class of modules.Let P be an admissible module of the generalized Witt algebra W,and V be the general linear Lie algebra gl(D)-module,where D is a nonzero space.Then we define F(P,V)as the Shen-Larsson,s module for the thesis.Since the reducibility of the Shen-Larsson’s module F(P,V)largely depends on the structure of gl(D)-module V,we first introduce some special gl(D)-modules V,which behave like irreducible highest weight modules of fundamental weights,and we illustrate the definitions of these special modules by some examples in this paper.Then we study the necessary and sufficient conditions for the irreducibility of W-module F(P,V),where P is irreducible modules of K,where K is a general Weyl algebra,and V is an irreducible module of gl(D).Finally,the isomorphisms are discussed between these irreducible modules under countable conditions.Thus we obtain a large family of irreducible weight modules other than highest weight modules and many irreducible non-weight modules over the simple generalized Witt algebras.