Quantum Enveloping Algebra Representation In The Areas O

We can obtain many profound properties about some infinite dimensional modules of a semisimple Lie algebra by studying its representation in category O. For example the simplicity criterion of a Verma module has been obtained right now.The definition of the category for a quantum enveloping algebra is given in this paper and some problems about the quantum enveloping algebra Uq (sl2) are solved.This article is divided into four parts.Chapter one is devoted to review some basic properties about finite dimensional semisimple Lie algebra.And the definition of category O for a given semisimple Lie algebra is introduced in chapter two.Some basic properties about quantum group is also reviewed in chapter three. At the chapter four, all the modules for Uq(sl2) are classified by using their central characters.The category O for the common quantum group Uq(g) is introduced in chapter five.Overall, there are many topics to study in the representation theory of Uq(sl2) in category O.