Irreducible Modules Of Generalized Witt Algebra W(2,1) Over A Field Of Characterize 2

Modular Lie algebras and their representations is one of the most importantbranches in mathematics for their application and their theory. Manymathematicians do a lot of researchs, get achieavements, and make Lie theorydevelop rapidly. For instance, in [2], Prof. Cheng determined all the simplemodules of the Witt algebra W(1,1). Prof. Shen, in [8,9,10], by using his mixedproduct modules, determined simple graded modules and simple filtered modulesfor L=X(m, n), X=W,S,H. On the condition that characteristic of the field islarger than 3. Prof. Hu, in [5,6] determined simple'graded modules and simplefiltered modules for K(m,n) on the condition that characteristic of the field islarger than 3. Prof. Holmes and Prof. Zhang, in [3,4,13] determined the simplemodules for L=X(m,1) (X=W,S,H,K) in the case of the height of the characters isat most one and characteristic of the field is larger than 3. However, the study ofLie algebra over a field whose characteristic is small is more complicate and theresults is limitedly known. In this paper, we try to offer a new way to studyCaftan type Lie algebras with the height of charactistic＜1. First,using thedegree of Cartan type Lie algebras and the basis of the reducible envelop algebra,we determin the maximal vector. Then,using the conditions of a maximal vectorin irreducible module, we determine the maximal vectors in a minimal left idealof the reducible envelop algebra and minimal left ideal of reducible envelopalgebra. Applying this method, we succeed in determining all minimal left idealsof reducible algebra of W(2,1) over a field of characteristic=2 and theirreducible module with a height of characteristic＜1. Finally, we give acomplete representation of the the set contained by the isomorphisic class ofirreducible modules and their dimension.