Irreducible Representations Of Generalized Witt Algebra W(3,1) In Characteristic 2

The study of representation theory of Cartan-type Lie Algebra began with Chang's determination of irreducible modules of Witt algebra W(1,1).Up to now,many results with respect to the subjects have been obtained.These results include Shen Guangyu's characterization of graded irreducible modules and filtered irreducible modules of L=X(m,n),X=W,S,H over an algebraically closed field of characteristic p＞3 and Holmes's classification of the simple modules of generalized Witt algebra with character height at most one.However the research of the representation theory of Cartan-type Lie Algebra in small characteristic has just begun and even the construction and the dimension formula of the simple modules are poorly understood.Recently in[14] and[6]irreducible representations of W(2,1) and S(3,1) in characteristic 2 were realized respectively.According to[5],irreducible modules with character height at most one of the generalized Witt algebra are the quotient of the modules induced from the simple representations of its grade 0 sub-algebra.With this knowledge,by constructing the simple modules of general linear algebras and analyzing the structure of corresponding induced modules,we succeed in describing the structures of the simple modules of W(3,1) with character height less than 1 in characteristic 2 in this dissertation.Finally some general results with respect to the dimensions of the simple restricted W(n,1)-modules are derived.