  Let U^>=0 denote the half quantum group for a fixed simple Lie algebra. We
examine some properties and representation of U^>=0. We prove that the Hopf
algebra U^>=0 is not quasi-cocommutative, and hence the category of left
U^>=0-module is not a braided monoidal category. In the weight module category
W, we describe all the simple objects and the projective objects. We also
describe all simple Yetter-Drinfel'd U^>=0-weight modules.
