Lie-Cartan Modules In The Category O For Lie Algebras Of Vector Fields

In the literature [8],the writers introduce the BGG representation category of Lie algebras of vector field W(n)、S(n)、H(n)and conduct depth research.Base on [8],this article gets the conclusion that the BGG category of Lie algebra of polynomial vector field is not an Artin category.The main body of article studies the Lie-Cartan modules.This concept is proposed by Professor Bin Shu by analogy with affine connections in differential geometry.In differential geometry,affine connections combine algebraic structure(structural sheaf)with geometric structure(tangent vector field)organically.Similarly,Lie-Cartan mod-ules will play an important role in representation theory of Lie algebra of vector field.In this article,we introduce the concept of Lie-Cartan modules in the BGG category of Lie algebra of polynomial vector field and reach a conclusion that the actions of polynomi-als on the lowest degree of Lie-Cartan modules are free.The main result of this article is that the mixed product modules are all irreducible Lie-Cartan modules.Furthermore,this article gives a conjecture that all Lie-Cartan modules are mixed product modules.